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The Best Math Teacher I Ever Had Was a Cardboard Box Discover the power of the "Final Answer?" box in teaching math. Read how it transformed one teacher's classroom by revealing misconceptions, and guiding genuinely effective instruction. Gain insights into student thinking, fostering self-reflection, and unlocking the potential of every learner. If I could have been granted one wish as a 5th grade math teacher years ago, I would have pushed for two: I needed to get inside the heads of my students to see why they were getting stuck with what seemed to me to be very simple problems, and I needed them to check the reasonableness of their answers before committing to them. Not a day passed without students submitting answers indicating that I had failed to get across the material, failed to get them to see if their answers made sense, or both. One day, after metaphorically banging my head on the chalkboard for the umpteenth time, a solution popped out. “Who Wants to Be a Millionaire?” was the most popular show in the US at the time, and its climactic catch phrase “Final answer?” was everywhere. I suddenly envisioned a cardboard dropbox marked “Final Answer?” where students could anonymously submit answers to specific math problems - along with their work - on folded slips of paper. Such a box would add drama to the practice portion of each lesson, especially when I revealed the correct answers after a simulated drumroll. More importantly, the collected slips of paper would give me a glimpse into what students actually had in their heads, and the question on the front of the box would remind them to see if their answers made sense before submitting them. I suspected the “Final Answer?” box would give me detailed insight into students’ thinking and help them develop habits of self-reflection, and I was right. (Things like formal assessments and “exit tickets” weren’t enough to give me the kind of anonymous moment-by-moment feedback I needed.) I had no idea, however, that the “Final Answer?” box would change my life. It may sound crazy - and I would have thought it crazy earlier in my career too - but I now know it to be fact: almost everyone can master math given the right materials/instruction. Prior to the “Final Answer?” box, if you had pried open my head, you would have found that I didn’t r-e-a-l-l-y believe that all of my students could learn math. I had seen too many bizarre and seemingly random “solutions” to what should have been easy problems over the years to be convinced that math could ever be understood by all. The “Final Answer?” box showed me in an instant that I was wrong. It happened on a typical day after a typical question. As I started going through the answers the students had dropped in the slot, I sighed - yet again - at how many of them were wrong. But as I unfolded the last one, I got a shock I’ll never forget: half of the class may have gotten the wrong answer - but it was the same wrong answer. The ones who had gotten stuck had all run into the exact same roadblock! “Remove the roadblock,” I thought, “and they should all get the next question right!” I did …and they did! I had chills as the students cheered their own perfect performance. I now knew it wasn’t the math that didn’t make sense to the kids; it was me. This would have depressed me to no end in the past, but at the same moment this problem revealed itself, a means of finding the solution did too: the “Final Answer?” box! The kids never tired of it, it was teaching them to think deeply about their answers, and it could teach me where they were getting stuck and how to “unstick” them! Over the next few years, the “Final Answer?” box taught me far more about teaching basic math than my math training ever had. It taught me that if I didn’t have students total up more than two addends at a time when practicing multi-digit addition, many of them would mistakenly conclude that the regrouping digits were always 1’s, and would develop no understanding of the place value involved. It taught me that a series of multiplication problems like 526 x 3, 526 x 33, and 526 x 333 was far more effective for introducing the concept of place-holding zeros than if I had them solve, say, 753 x 3, 892 x 34, and 615 x 348. It taught me that if I didn’t include zeros in the answers (quotients) of the long division problems I asked students to solve, many of them would get stuck when zeros did appear and settle on answers like 57 remainder 8 when the real answer was 507 remainder 8 or 5,007 remainder 8 - even though they knew these answers were unreasonable! The “Final Answer?” box taught me all of these things and hundreds more besides. I’m now systematically incorporating everything I’ve learned over the years into the example-based You Teach You book series. But far more importantly than any of these things, the “Final Answer?” box taught me that almost anyone can master basic math - for real. Math is logical. Kids understand logic (because it’s logical!). When math is presented logically - in sequence, in stages, in detail, and with an eye toward sticking points and special cases - there’s literally nothing to prevent kids from learning it. It may sound crazy - and I would have thought it crazy earlier in my career too - but I now know it to be fact: almost everyone can master math given the right materials/instruction. And yes, that’s my final answer.	{"url":"https://youteachyou.org/blogs/articles/the-best-math-teacher-i-ever-had-was-a-cardboard-box","timestamp":"2024-11-14T23:45:06Z","content_type":"text/html","content_length":"124691","record_id":"<urn:uuid:f51cfa0d-c053-46a7-9e24-74f8e7851dea>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00830.warc.gz"}
stoimen's web log Each natural number that is divisible only by 1 and itself is prime. Prime numbers appear to be more interesting to humans than other numbers. Why is that and why prime numbers are more important than the numbers that are divisible by 2, for instance? Perhaps the answer is that prime numbers are largely used in cryptography, although they were interesting for the ancient Egyptians and Greeks (Euclid has proved that the prime numbers are infinite circa 300 BC). The problem is that there is not a formula that can tell us which is the next prime number, although there are algorithms that check whether a given natural number is prime. It’s very important these algorithms to be very effective, especially for big numbers. As I said each natural number that is divisible only by 1 and itself is prime. That means that 2 is the first prime number and 1 is not considered prime. It’s easy to say that 2, 3, 5 and 7 are prime numbers, but what about 983? Well, yes 983 is prime, but how do we check that? If we want to know whether n is prime the very basic approach is to check every single number between 2 and n. It’s kind of a brute force. The basic implementation in PHP for the very basic (brute force) approach is as follows. Unfortunately this is one very ineffective algorithm. We don’t have to check every single number between 1 and n, it’s enough to check only the numbers between 1 and n/2-1. If we find such a divisor that will be enough to say that n isn’t prime. Although that code above optimizes a lot our first prime checker, it’s clear that for large numbers it won’t be very effective. Indeed checking against the interval [2, n/2 -1] isn’t the optimal solution. A better approach is to check against [2, sqrt(n)]. This is correct, because if n isn’t prime it can be represented as p*q = n. Of course if p > sqrt(n), which we assume can’t be true, that will mean that q < sqrt(n). Beside that these implementations shows how we can find prime number, they are a very good example of how an algorithm can be optimized a lot with some small changes. Sieve of Eratosthenes Although the sieve of Eratosthenes isn’t the exact same approach (to check whether a number is prime) it can give us a list of prime numbers quite easily. To remove numbers that aren’t prime, we start with 2 and we remove every single item from the list that is divisible by two. Then we check for the rest items of the list, as shown on the picture below. The PHP implementation of the Eratosthenes sieve isn’t difficult. As I said prime numbers are widely used in cryptography, so they are always of a greater interest in computer science. In fact every number can be represented by the product of two prime numbers and that fact is used in cryptography as well. That’s because if we know that number, which is usually very very big, it is still very difficult to find out what are its prime multipliers. Unfortunately the algorithms in this article are very basic and can be handy only if we work with small numbers or if our machines are tremendously powerful. Fortunately in practice there are more complex algorithms for finding prime numbers. Such are the sieves of Euler, Atkin and Sundaram.	{"url":"http://stoimen.com/tag/politics/","timestamp":"2024-11-04T13:50:27Z","content_type":"text/html","content_length":"19485","record_id":"<urn:uuid:3c8572ce-2612-4a64-8a7d-896c43cd0f31>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00899.warc.gz"}
We looked at plots of RTT vs Distance for pairs of Landmarks within 25ms and 1000km radii for Europe and N. America. Typically a formula of the form distance = alpha * RTT is used to derive the distance from the measured RTT. Typical values of alpha are 40 km/ms. However the value of alpha can vary dramatically based on the route the cable takes versus teh great circle distance. Fior example from the US to Japan the route is pretty direct leading to high values of alpha. However from Europe to Japan the route today goes West around the bay of Biscay, through the Mediterranean and the Red Sea, South of India, then passes via Singapore and North up the East coast of Asia to Japan. This result in smaller values of alpha. There are plans afoot to provide more direct routes via the Arctic Ocean. When we measured the RTTs between landmarks of known location and derived the values of alpha, Europe seemed better behaved. We looked at the network topology for the major Academic and Research (A& R) networks in N. America, i.e. Internet2, ESnet and Canarie. They are shown below. Internet2 ESnet Canarie Landmarks We also looked at the European Academic and Research network GEANT, together with the Landmark locations in Europe. See below: The worldwide cable routes can be see: Our initial results depict better (less variability in alpha, density of landmarks (i.e. number of landmarks with the designated radius of each other), and correlation between RTT and distance) intra regional connectivity in Europe than in North America. Both North America and Europe have many nGigabits links connecting one state to another in N.America or one country to another in Europe. We wish to see the state of connectivity of landmarks within a few hundred miles radius around each target landmark since these are the landmarks that are most likely to be used in the eventual In our analysis first we calculated the correlation coefficient for all landmarks within 1000 km (620 miles) and 25 msec. radii (assuming an alpha of 40 km/ms) around them. Below are the two Plot within 1000 KM Distance Radius Plot within 25 MS Delay Radius There are two series in each graph, white balls represent the Correlation coefficient and red balls represent the number of target landmark available within the chosen radius of each landmark (1000km or 25ms). The size of the ball represent the standard deviation of delay associated with each landmark and X-axis represents the ID of the landmark or simply the row number of the spread sheet of raw data. You can find spread sheets of 1000KM_Radius and 25MS_Radius correlation_coefficient_1000km_radius.xlsx and Correlation_Coefficient_25MS_Radius.xlsx(25MS file has the column names). The left-hand (Y-axis) is the Correlation coefficient value and right-hand Y-axis is the number of landmarks within the radius. As you can clearly see the trend of number of adjacent landmarks and correlation coefficient are both high for the whole of Europe but they are more dispersed thoughout North America. These results contradict the perception of very good connectivity inside North America. Thus, we decided to do further testing of this finding. We then took the raw data of some landmark from Europe and North America again within a radii of 1000 kmand 25 ms and plotted delay to distance graphs, we see a similar trend. Below are some sample European Landmarks North American Landmarks All these graphs have delay on X-axis and distance on X-axis and in all graphs with filtered caption we have applied the median filter on the values based upon delay. This was just to see the impact of removing the sensitivity to outliers. In this case outliers are those delay values which are not following the trend. E.g. if there are three distance values 400,500,600,700 and their corresponding delay values 10,12,50,17. Here 50 is clearly an outlier. As you can observe a linear increasing trend in European landmarks which is almose absent in North American nodes. This again is depicting a similar observation that Europe has more direct links in contrast to North America. Raw data is available here, Nashville_NA.rar,Atlanta_NA.rar,Paris_France.rar,Darmstadt_Germany.rar. Optimum Alpha Values for Europe and N. America If we define the optimum value of alpha = alpha(opt) such that the known distance between a landmark and target = alpha(opt) min_RTT *100 km. then for all landmarks within say 500km of each other we can measure the min_RTT each landmark to each other landmark (target) and find the average and standard deviations for each landmark. We did this for Europe and N. America and plots of alpha(opt) vs landmark for landmarks within 500km of each other are shown below in plots from the eu-na-opt-aplha.xlsx spreadsheet. For Europe it is seen that landmarks on the periphery such as the UK and Poland have larger values of Alpha. Possibly this is since there is less meandering on the longer distance links. We are still investigating the above observations against the Intra-State connectivity in North America and through other statistical analysis. New Graph correlation_coefficient_1000_old.xls	{"url":"https://confluence.slac.stanford.edu/pages/diffpagesbyversion.action?pageId=85098983&selectedPageVersions=20&selectedPageVersions=21","timestamp":"2024-11-04T18:43:50Z","content_type":"text/html","content_length":"71307","record_id":"<urn:uuid:f99090e6-2e44-41f0-8050-d2bbf5e6a5ab>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00233.warc.gz"}
Ignite Your Love for Math: The Summer Math Bridge Workbook for Grades 6-7 Mathematics is a subject that can be challenging, exciting, and even fun! However, many students struggle with math and may not have the confidence they need to excel in this subject. To help students prepare for the upcoming school year and ignite their love for math, the Summer Math Bridge Workbook for Grades 6-7 is an excellent resource. In this article, we'll explore how this workbook can help students get ahead this summer, build confidence and mastery in math, and prepare for the upcoming school year. Introducing the Summer Math Bridge Workbook The Summer Math Bridge Workbook is a comprehensive workbook designed to help students improve their math skills during the summer. This workbook is perfect for students who want to get ahead in their math studies or for those who need extra support in math. The workbook is filled with engaging and fun activities that will help students build confidence and mastery in math while preparing for the upcoming school year. Get Ahead this Summer with Math Summer is the perfect time for students to get ahead in their math studies. With the Summer Math Bridge Workbook, students can work through math problems at their own pace, without the pressures of school. This way, they can focus on their weaknesses, improve their skills, and be prepared for the upcoming school year. Build Confidence and Mastery in Math Many students struggle with math because they lack confidence in their abilities. The Summer Math Bridge Workbook is designed to help students build confidence in math. Through engaging activities and clear explanations, students can improve their math skills and feel more confident in their abilities. As they progress through the workbook, they'll gain mastery in math, which will help them succeed in the upcoming school year. Designed Specifically for Grades 6-7 The Summer Math Bridge Workbook is designed specifically for students in grades 6-7. The activities and problems in the workbook are aligned with the standards for these grades, ensuring that students are learning what they need to know to succeed in math. The workbook covers topics such as fractions, decimals, and algebra, which are essential for success in middle school and beyond. Engaging and Fun Activities to Ignite Your Love for Math The Summer Math Bridge Workbook is not your typical boring workbook. It's filled with engaging and fun activities that will ignite your love for math. The activities are designed to be interactive, hands-on, and enjoyable, so students will be motivated to keep learning. From puzzles to games, there's something for everyone in this workbook. Prepare for the Upcoming School Year The Summer Math Bridge Workbook is an excellent resource to prepare for the upcoming school year. By working through the material in the workbook, students will be better prepared for the math concepts they'll encounter in the upcoming school year. This will help them feel more confident and perform better in their math classes. Accessible and Easy to Use The Summer Math Bridge Workbook is accessible and easy to use. The workbook is available in both print and digital formats, making it convenient for students to work on it wherever they are. The instructions are clear and easy to follow, and the problems are designed to be challenging but not overwhelming. Start Your Math Adventure Today! In conclusion, the Summer Math Bridge Workbook is an excellent resource for students who want to get ahead in math, build confidence and mastery in the subject, and prepare for the upcoming school year. With engaging and fun activities, clear explanations, and convenient accessibility, this workbook is a must-have for any student looking to ignite their love for math. So why wait? Start your math adventure today! The Summer Math Bridge: A workbook for Grades 6 to 7: Fractions, Exponents, Roots, Equations, Expressions, Ratio, Proportion, Area, Perimeter, Volume, and more with Step by Step guide and Answers • Chapter. 01: Fractions • Chapter. 02: Pre-Algebra • Chapter. 03: Ratio and Proportion • Chapter. 04: Geometry • Chapter. 05: Metric Weights and Unit Conversion • Chapter. 06: Statistics • Answer Key Summer Math Success: Summer Math Workbook 6-7: 180 Worksheets of Percent, Beginning Statistics, Geometry, Pre Algebra and More • Multiplication with Whole Numbers • Division with Whole Numbers • Mixed Fractions – Multiplication • Mixed Fractions- Division • Fractions: Multiple Operations • Simplifying Fractions • Percent • Percent and Decimals • Percent – Advanced • Ratio Conversions • Cartesian Coordinates • Cartesian Coordinates With Four Quadrants • Plot Lines • Exponents • Scientific Notation • Expressions – Single Step • Number Problems • Pre-Algebra Equations (One Step) Addition and Subtraction • Pre-Algebra Equations (One Step) Multiplication and Division • Pre-Algebra Equations (Two Sides) • Simplifying Expressions • Inequalities – Addition and Subtraction • Find the Area and Perimeter • Find the Volume and Surface Area • Calculate the area of each circle • Calculate the circumference of each circle • Measure of Center – Mean • Measure of Center – Median • Measure of Center – Mode • Measure of Variability – Range • Answers Summer Math Success: Pre Algebra Workbook Grade 6-7: 6th and 7th Grade Pre Algebra Workbook: Solving Single and Double Step Equations and More • Pre-Algebra: Order of Operations • Find Numbers Using Algebraic Thinking • Evaluating Expressions • Solving Equations • Equations (One Variable) • Solving Equations (Two Steps) • Answers Summer Math Success: Math Word Problems Workbook Grades 6-8: Fractions, Numbers, Multi Step, and Percent Word Problems with Answers • Fractions Addition Word Problems • Fractions Subtraction Word Problems • Fractions Multiplication Word Problems • Fractions Division Word Problems • Numbers Word Problems • Mixed Word Problems • Percent Word Problems • Answers Leave a Comment	{"url":"https://thegreateducator.com/ignite-your-love-for-math-the-summer-math-bridge-workbook-for-grades-6-7/","timestamp":"2024-11-02T14:10:35Z","content_type":"text/html","content_length":"151770","record_id":"<urn:uuid:b26df7d9-6862-439f-b496-8abdb73e2b51>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00624.warc.gz"}
Algebra 1 – Everything You Need Original price was: $214.93.Current price is: $199.95. Algebra 1 Online Course - Lifetime Access Elementary Algebra by Harold Jacobs Jacobs Elementary Algebra (Teacher Guide) Jacobs Teacher Guide for use with Algebra 1 - Everything You Need. Includes chapter tests, mid-term, final exams, alternate tests, and answer key to those tests. As of the 2023 printing it also contains solutions to homework problem sets. (The Solutions Manual is discontinued.) Math Designed for Understanding Jacobs’ Elementary Algebra (Algebra 1) utilizes a clear, conversational, engaging approach to teach your student Algebra through practical, real-life application! Jacobs’ guides your student through Algebra, enabling them to discover concepts for themselves and develop a deep understanding of their practical application. 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Required to take this course and included in this package Jacobs Textbook Elementary Algebra Student Textbook: This student text is part of our best-selling Algebra curriculum pack, Jacobs’ Elementary Algebra. • Full-Color Illustrations • 17 sections, covering functions and graphs, integers, rational numbers, exponents, polynomials, factoring, fractions, and more. • Set II exercise solutions provided at the back of the text • Flexible based on the focus & intensity of the course • Homework problems to exercise the concepts for each lesson Jacobs Teacher's Guide Elementary Algebra Teacher’s Guide Includes: □ Tests (chapter, mid-term, final exam, & alternate test versions) □ Test Solutions □ Solutions to exercises in the textbook (formerly the Solutions Manual) Callahan Videos Video Instruction Contains: Guide your student through Jacobs’ Elementary Algebra with Dr. Callahan’s instructional videos! The video instruction leads your student through the Elementary Algebra text with video lessons for each topic and step-by-step examples. Free Homework Support! Dr. Callahan’s Instructional Videos come with FREE Homework Help email support from AskDrCallahan, making this course an ideal solution for parents and students alike looking to learn higher math. ☆ Approximately 6 hours of content ☆ Dr. Callahan selected homework problem ☆ Free Homework Help email support from AskDrCallahan ☆ Links to our pdf teacher's guide (see details below) with syllabus & when to watch videos as well as our own multi-chapter tests, test solutions, and test grading guide. Scientific Calculator: We typically use a TI-30, but we sometimes substitute with an equivalent such as a Casio. This calculator is allowed on the ACT. It can be used throughout your student's academic career and at college. Callahan Teacher's Guide Teacher's Guide by AskDrCallahan. The AskDrCallahan Algebra Teacher's Guide is a PDF download, included FREE inside your Online Video Purchase. ☆ test grading guide ☆ syllabus ☆ Dr. Callahan selected homework problems ☆ Instructions for watching the video instruction ☆ AskDrCallahan Multi-Chapter Tests ☆ AskDrCallahan Multi-Chapter Tests Answer Key Common Questions Other Course Features: • Usable for both self-directed students and teachers & students • Recommended for: 9th Grade / 1 Credit Sample videos from the course Additional information Weight 9 lbs Dimensions 12 × 10 × 8 in	{"url":"https://askdrcallahan.com/product/algebra-1-everything-you-need/","timestamp":"2024-11-02T14:54:17Z","content_type":"text/html","content_length":"368395","record_id":"<urn:uuid:e09c1a04-bf0d-42df-bdb5-7e6d052bf3a6>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00173.warc.gz"}
The Handbook of Mass Measurement - Basculasbalanzas.com The kilogram is one of the most widely used units for mass measurement, and is defined by the International Standards Organization. In the past, kilograms were defined in terms of the Planck constant, which has dimensions of energy times mass x length2 / time. However, the modern definition of the kilogram is based on a simpler model. Despite its shortcomings, the kilogram remains the standard for mass measurement in most countries. Its metric counterpart, the gram, has become obsolete. The Handbook of Mass Measurement is a comprehensive reference for anyone wishing to know the basics of mass measurement. The author has combined fundamentals, history, and technical details to provide a comprehensive overview of the method. The book examines all aspects of mass measurement, including the factors that introduce error. The final chapter describes the methods of mass measurement. In addition, it provides information about the weighing and comparing of different materials. Using mass measurement accurately is important for ensuring product safety. A nuclear measurement requires nuclear technology, such as TOFI or SPEG. In each setting, a large number of nuclei are transmitted. Nuclei with known masses provide calibration and unknown masses. These are essential for precision mass measurements. The final uncertainties of the measurements range from 100 keV for nuclei that are close to stability to lMeV for those near the ends of isotopic chains. The measurement process itself can be lengthy and complicated, so it is advisable to consult the manual before undertaking a nuclear measurement. Nuclear mass measurements are a fundamental probe of the structure of the nuclei. Exotic nuclei are particularly important as they lie at the frontier between known and unknown masses. They serve as reference masses in other mass measurement methods. In addition to their importance for the nuclear industry, they also serve as important benchmarks for atomic structure. And, as they provide the standards for nuclear mass measurements, they are crucial for future studies. And in many ways, nuclear mass measurements have the potential to improve the way we look at the universe. The metric system has a unique history. The kilogram is the basis unit of mass in the SI. This unit is used throughout science, engineering, commerce, and other fields. In addition to its metric equivalent, the kilogram is also referred to as a kilo. This prefix is used to distinguish kilograms from other mass measurements, such as grams. The kilogram has two decimal parts, which is important to understand the meaning of the metric system. The most basic way to measure mass is to weigh the object to determine its weight. In this case, it is easier to weigh the object in question, which means it has a mass. Adding up the weight of an object will help you find its mass. A mass measurement is vital for many fields of science, including physics and engineering. When it comes to weight measurement, it’s important to know how much weight it has to be lifted in order to determine its mass. The definition of mass is different in different scientific disciplines. In classical mechanics, mass is the resistance of an object to acceleration. A higher mass causes a smaller change in force. For the International System of Units, the unit for mass is called a kilogram, which is defined by Planck’s constant, which is 6.62607015 x 10-34 joules per second. For example, one kilogram equals one metre squared per second. Mass measurement is important for scientific research, and there are several methods for determining an object’s mass. One method is to use a balance. A balance uses two scales to determine the weight of an object. This method can be inaccurate, however, because weight changes can affect the weight of the object. In addition to weight, other factors can affect instrumentation, such as metal corrosion or temperature changes. However, in this case, it’s worth knowing that a balance is the most accurate and dependable way to measure mass. Another method of mass measurement is to use a weighing pan. The weighing pan on a balance should be free of dust or other substances that can cause chemical reactions. In addition, the weighing pan should be clean and level before mass measurement. The sample should never be placed directly on the balance. Instead, it should be placed on a weighing sheet, weighted boat, or other container. While using a balance, remember to keep in mind that some chemicals may react with the sample you’re measuring.	{"url":"https://www.basculasbalanzas.com/the-handbook-of-mass-measurement/","timestamp":"2024-11-10T15:04:28Z","content_type":"text/html","content_length":"52917","record_id":"<urn:uuid:b8d28d64-38c0-41d0-a6c5-b1dbc659a4e5>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00096.warc.gz"}
An object of class Nonnegative_quadratic_program_from_iterators describes a convex quadratic program of the form. \begin{eqnarray} \mbox{(QP)}& \mbox{minimize} & \qpx^{T}D\qpx+\qpc^{T}\qpx+c_0 \\ &\mbox{subject to} & A\qpx\qprel \qpb, \\ & & \qpx \geq 0 \end{eqnarray} in $ n$ real variables $ \qpx=(x_0,\ldots,x_{n-1})$. • $ A$ is an $ m\times n$ matrix (the constraint matrix), • $ \qpb$ is an $ m$-dimensional vector (the right-hand side), • $ \qprel$ is an $ m$-dimensional vector of relations from $ \{\leq, =, \geq\}$, • $ D$ is a symmetric positive-semidefinite $ n\times n$ matrix (the quadratic objective function), • $ \qpc$ is an $ n$-dimensional vector (the linear objective function), and • $ c_0$ is a constant. This class is simply a wrapper for existing iterators, and it does not copy the program data. It frequently happens that all values in one of the vectors from above are the same, for example if the system $ Ax\qprel b$ is actually a system of equations $ Ax=b$. To get an iterator over such a vector, it is not necessary to store multiple copies of the value in some container; an instance of the class Const_oneset_iterator<T>, constructed from the value in question, does the job more efficiently. The following example for the simpler model Nonnegative_linear_program_from_iterators<A_it, B_it, R_it, C_it> should give you a flavor of the use of this model in practice. See also	{"url":"https://doc.cgal.org/5.5.2/QP_solver/classCGAL_1_1Nonnegative__quadratic__program__from__iterators.html","timestamp":"2024-11-04T20:08:28Z","content_type":"application/xhtml+xml","content_length":"16991","record_id":"<urn:uuid:4a9407ca-a371-4715-857d-26c99548e436>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00596.warc.gz"}
The fxt demos: graph search The fxt demos: graph search Directory graph: Searching directed graphs, mostly to find combinatorial objects. Find a list of all files in this directory here. An index of all topics is here You may want to look at the outputs first. graph-acgray-out.txt is the output of graph-acgray-demo.cc. Paths through a directed graph: adjacent changes (AC) Gray paths. The demo uses the functions from digraph.h (fxt/src/graph/digraph.h) digraph-paths.h (fxt/src/graph/digraph-paths.h) search-digraph-cond.cc (fxt/src/graph/search-digraph-cond.cc) mk-special-digraphs.h (fxt/src/graph/mk-special-digraphs.h) mk-gray-digraph.cc (fxt/src/graph/mk-gray-digraph.cc) graph-complementshift-out.txt is the output of graph-complementshift-demo.cc. Complement-shift sequence via paths in a directed graph. The demo uses the functions from digraph.h (fxt/src/graph/digraph.h) digraph-paths.h (fxt/src/graph/digraph-paths.h) search-digraph.cc (fxt/src/graph/search-digraph.cc) mk-special-digraphs.h (fxt/src /graph/mk-special-digraphs.h) mk-debruijn-digraph.cc (fxt/src/graph/mk-debruijn-digraph.cc) graph-debruijn-out.txt is the output of graph-debruijn-demo.cc. Find all paths through the binary De Bruijn graph. The demo uses the functions from digraph.h (fxt/src/graph/digraph.h) digraph-paths.h (fxt/src/graph/digraph-paths.h) search-digraph.cc (fxt/src/graph/search-digraph.cc) mk-special-digraphs.h (fxt/src /graph/mk-special-digraphs.h) mk-debruijn-digraph.cc (fxt/src/graph/mk-debruijn-digraph.cc) graph-debruijn-m-out.txt is the output of graph-debruijn-m-demo.cc. Find all paths through the m-ary De Bruijn graph. The demo uses the functions from digraph.h (fxt/src/graph/digraph.h) digraph-paths.h (fxt/src/graph/digraph-paths.h) search-digraph.cc (fxt/src/graph/search-digraph.cc) mk-special-digraphs.h (fxt/src /graph/mk-special-digraphs.h) mk-debruijn-digraph.cc (fxt/src/graph/mk-debruijn-digraph.cc) graph-fibrepgray-out.txt is the output of graph-fibrepgray-demo.cc. Gray codes through Fibonacci representations. The demo uses the functions from fibrep.h (fxt/src/bits/fibrep.h) fibonacci.h (fxt/src/aux0/fibonacci.h) digraph.h (fxt/src/graph/digraph.h) digraph-paths.h (fxt/src/graph/digraph-paths.h) search-digraph-cond.cc (fxt/src/graph/search-digraph-cond.cc) mk-special-digraphs.h (fxt/src/graph/mk-special-digraphs.h) mk-fibrep-gray-digraph.cc (fxt/src/graph/mk-fibrep-gray-digraph.cc) graph-gray-out.txt is the output of graph-gray-demo.cc. Paths through a directed graph: Gray paths. The demo uses the functions from digraph.h (fxt/src/graph/digraph.h) digraph-paths.h (fxt/src/graph/digraph-paths.h) mk-special-digraphs.h (fxt/src/graph/mk-special-digraphs.h) search-digraph.cc (fxt /src/graph/search-digraph.cc) mk-gray-digraph.cc (fxt/src/graph/mk-gray-digraph.cc) graph-lyndon-gray-out.txt is the output of graph-lyndon-gray-demo.cc. Paths through a directed graph: Gray paths through Lyndon words. The demo uses the functions from digraph.h (fxt/src/graph/digraph.h) digraph-paths.h (fxt/src/graph/digraph-paths.h) search-digraph.cc (fxt/src/graph/search-digraph.cc) search-digraph-cond.cc (fxt/ src/graph/search-digraph-cond.cc) mk-special-digraphs.h (fxt/src/graph/mk-special-digraphs.h) mk-lyndon-gray-digraph.cc (fxt/src/graph/mk-lyndon-gray-digraph.cc) lyndon-cmp.cc (fxt/src/graph/ graph-macgray-out.txt is the output of graph-macgray-demo.cc. Paths through a directed graph: modular adjacent changes (MAC) Gray paths. The demo uses the functions from digraph.h (fxt/src/graph/digraph.h) digraph-paths.h (fxt/src/graph/digraph-paths.h) search-digraph-cond.cc (fxt/src/graph/search-digraph-cond.cc) mk-special-digraphs.h (fxt/src/graph/mk-special-digraphs.h) mk-gray-digraph.cc (fxt/src/graph/mk-gray-digraph.cc) graph-monotonicgray-out.txt is the output of graph-monotonicgray-demo.cc. Paths through a directed graph: all canonical monotonic Gray paths starting with zero. The demo uses the functions from digraph.h (fxt/src/graph/digraph.h) digraph-paths.h (fxt/src/graph/digraph-paths.h) mk-special-digraphs.h (fxt/src/graph/mk-special-digraphs.h) mk-gray-digraph.cc (fxt/src/graph/mk-gray-digraph.cc) search-digraph.cc (fxt/src/graph/search-digraph.cc) search-digraph-cond.cc (fxt/src/graph/search-digraph-cond.cc) graph-mtl-out.txt is the output of graph-mtl-demo.cc. Cycles through a directed graph: middle two levels. The demo uses the functions from digraph.h (fxt/src/graph/digraph.h) digraph-paths.h (fxt/src/graph/digraph-paths.h) search-digraph.cc (fxt/src/graph/search-digraph.cc) graph-parengray-out.txt is the output of graph-parengray-demo.cc. Gray codes through valid parentheses strings The demo uses the functions from parenwords.h (fxt/src/bits/parenwords.h) digraph.h (fxt/src/graph/digraph.h) digraph-paths.h (fxt/src/graph/digraph-paths.h) search-digraph.cc (fxt/src/graph/ search-digraph.cc) mk-special-digraphs.h (fxt/src/graph/mk-special-digraphs.h) mk-paren-gray-digraph.cc (fxt/src/graph/mk-paren-gray-digraph.cc) graph-perm-out.txt is the output of graph-perm-demo.cc. Paths through the complete graph: Permutations. The demo uses the functions from digraph.h (fxt/src/graph/digraph.h) digraph-paths.h (fxt/src/graph/digraph-paths.h) search-digraph.cc (fxt/src/graph/search-digraph.cc) mk-special-digraphs.h (fxt/src /graph/mk-special-digraphs.h) mk-complete-digraph.cc (fxt/src/graph/mk-complete-digraph.cc) graph-perm-doubly-adjacent-gray-out.txt is the output of graph-perm-doubly-adjacent-gray-demo.cc. Gray codes through permutations with only adjacent interchanges and successive transpositions overlapping (doubly-adjacent Gray codes). The demo uses the functions from digraph.h (fxt/src/graph/digraph.h) digraph-paths.h (fxt/src/graph/digraph-paths.h) search-digraph.cc (fxt/src/graph/search-digraph.cc) mk-special-digraphs.h (fxt/src /graph/mk-special-digraphs.h) mk-perm-gray-digraph.cc (fxt/src/graph/mk-perm-gray-digraph.cc) graph-perm-pref-rev-out.txt is the output of graph-perm-pref-rev-demo.cc. Permutations by prefix reversals. The demo uses the functions from digraph.h (fxt/src/graph/digraph.h) digraph-paths.h (fxt/src/graph/digraph-paths.h) search-digraph.cc (fxt/src/graph/search-digraph.cc) mk-special-digraphs.h (fxt/src /graph/mk-special-digraphs.h) mk-perm-pref-rev-digraph.cc (fxt/src/graph/mk-perm-pref-rev-digraph.cc) graph-perm-pref-rot-out.txt is the output of graph-perm-pref-rot-demo.cc. Permutations by prefix rotations. The demo uses the functions from digraph.h (fxt/src/graph/digraph.h) digraph-paths.h (fxt/src/graph/digraph-paths.h) search-digraph.cc (fxt/src/graph/search-digraph.cc) mk-special-digraphs.h (fxt/src /graph/mk-special-digraphs.h) mk-perm-pref-rot-digraph.cc (fxt/src/graph/mk-perm-pref-rot-digraph.cc) graph-perm-star-transpositions-out.txt is the output of graph-perm-star-transpositions-demo.cc. Gray codes through permutations with star transpositions. The demo uses the functions from digraph.h (fxt/src/graph/digraph.h) digraph-paths.h (fxt/src/graph/digraph-paths.h) search-digraph.cc (fxt/src/graph/search-digraph.cc) mk-special-digraphs.h (fxt/src /graph/mk-special-digraphs.h) mk-perm-gray-digraph.cc (fxt/src/graph/mk-perm-gray-digraph.cc) lyndon-gray-out.txt is the output of lyndon-gray-demo.cc. Gray cycle through n-bit Lyndon words. Must have n odd, and n < BITS_PER_LONG. By default print (length-7710, base-36) delta sequence for n=17. The demo uses the functions from lyndon-gray.h (fxt/src/graph/lyndon-gray.h) crc64.h (fxt/src/bits/crc64.h) sta-graph-acgray-out.txt is the output of sta-graph-acgray-demo.cc. Paths through a directed graph: adjacent changes (AC) Gray paths. Streamlined standalone routine (for backtracking). sta-graph-macgray-out.txt is the output of sta-graph-macgray-demo.cc. Paths through a directed graph: modular adjacent changes (MAC) Gray paths. Streamlined standalone routine for backtracking.	{"url":"https://jjj.de/fxt/demo/graph/index.html","timestamp":"2024-11-04T23:26:12Z","content_type":"text/html","content_length":"15061","record_id":"<urn:uuid:6368f9f8-b855-4157-928c-9a6da64b0501>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00377.warc.gz"}
wu :: forums - 0.999. wu :: forums medium (Moderators: Grimbal, william wu, Icarus, SMQ, towr, Eigenray, ThudnBlunder) « Previous topic \| Next topic » 4 5 6 7 ... 13 Notify of replies Send Topic Print Author Topic: 0.999. (Read 125744 times) Jeremy Re: 0.999. Newbie « Reply #100 on: Nov 7^th, 2002, 5:56am » Ok, i really suggest we drop this thread... i think if we haven't saved anyone by this time, no one is going to change their opinions. Hopefully we can wait till they get to a pre-calculus class, and then they can take it up with their math teacher. And as for Guess Who's comment about 1/3 not being equal to .333... because if you do long division you will ALWAYS have a remainder umm.... well duh. the 3's just keep going... the ellipse (...) means it's a repeating series. The three's never stop, so you keep having these remainder's of 3, and you have to keep dividing and getting another digit. Really the series wills stop as soon as you divide into a nice even number with no remainder (and that will never happen) towr Re: 0.999. wu::riddles Moderator « Reply #101 on: Nov 7^th, 2002, 5:56am » do we really want to accept that every integer (except 0) has 2 representations? -2 -1.999... -1 -0.999... 1 0.999... Some people are average, some are just mean. 2 1.999... I would argue 0.999... isn't a valid number, but rather bad syntax Posts: 13730 Wikipedia, Google, Mathworld, Integer sequence DB Jeremy Re: 0.999. Newbie « Reply #102 on: Nov 7^th, 2002, 5:59am » and towr, there's already an infinite number of ways to write every number: Posts: 25 towr Re: 0.999. wu::riddles Moderator « Reply #103 on: Nov 7^th, 2002, 6:07am » 5/5 is an expression, not a number.. You need to calculate it's value to get one.. A number is an atom, an equation is a set of operations on atoms.. Some people are average, some are just mean. Posts: 13730 Wikipedia, Google, Mathworld, Integer sequence DB S. Owen Re: 0.999. Full « Reply #104 on: Nov 7^th, 2002, 7:10am » on Nov 7^th, 2002, 5:56am, towr wrote: I would argue 0.999... isn't a valid number, but rather bad syntax Yeah, it would be funny to think that there is more than one base-10 decimal representation for 1. Agreed, the "trick" is that "0.999..." is an abuse of notation. A decimal representation needs to have a finite number of digits. We can still establish a meaning for "0.999...", and its value, 1, but it's not another decimal representation of 1. Posts: As for dropping this thread... somehow I still enjoy arguing about this one. Turn off reply notification if they're bothersome. TimMann Re: 0.999. Senior « Reply #105 on: Nov 7^th, 2002, 10:17am » on Nov 7^th, 2002, 7:10am, S. Owen wrote: Agreed, the "trick" is that "0.999..." is an abuse of notation. A decimal representation needs to have a finite number of digits. Oops, that's a mistake. If you mean that, you'd be saying that irrational numbers and rational numbers that aren't multiples of negative powers of 10 have no decimal representation at all. The only decimal representation of 1/3 is 0.333..., and the only decimal representation of pi is 3.14159... It's more common to do the opposite and say that all decimal representations have an infinite number of digits; for example, 1 = 1.000... This is needed in Cantor's diagonalization proof that the cardinality of the set of reals is greater than that of the set of integers. Posts: There's nothing wrong with either 1.000... or 0.999... as a decimal representation for 1. It's too bad that some numbers turn out to have two infinite decimal representations, but we're 330 stuck with it. The best we can do is choose one of the two and declare it to be canonical. S. Owen Re: 0.999. Full « Reply #106 on: Nov 7^th, 2002, 11:41am » Yeah, let me retract that... I guess I was taking towr to mean 'are there really two finite decimal representations of 1', which "0.999..." isn't, but that's not terribly interesting. Well maybe... does "0.333..." count as an exact decimal representation of 1/3? Certainly if it equals anything, it's 1/3. But I've held the line that "0.333..." isn't really a decimal value as much as a tricky expression of a limit. The answer is probably a matter of context... as far as (finite) computers are concerned, there is no way to exactly represent 1/3 as a base-2 decimal, so the answer's no in that context. But a mathematician might usefully define "0.333..." as the base-10 decimal representation of 1/3 and have no problems. Icarus Re: 0.999. wu::riddles Moderator « Reply #107 on: Nov 7^th, 2002, 4:15pm » on Nov 7^th, 2002, 11:41am, S. Owen wrote: Well maybe... does "0.333..." count as an exact decimal representation of 1/3? Certainly if it equals anything, it's 1/3. But I've held the line that "0.333..." isn't really a decimal value as much as a tricky expression of a limit. Boldly going where even angels fear to tread. 0.333... is a tricky expression of a limit, but that does not stop it from being a "decimal value". All decimals are tricky expressions of mathematical operations. After all, what does 0.3333 (finite) mean but Gender: 3/10 + 3/100 + 3/1000 + 3/1000 ? Posts: 4863 How is saying that any different from saying 0.333... means [sum][i=1]^[supinfty] 3/10^i ? The only difference I see is the sophistication of the concepts, and that 0.333... has a second level of notational convention applied (the ellipsis). However the ellipsis and the summation are both well-defined concepts whose meanings allow us to say 0.333... = 1/3, exactly, and not by "abuse of notation". One other thing. The only real numbers with more than one decimal expression are the non-zero rationals whose denominators are expressable as 2^n5^m. They have exactly two: the terminating one which ends in repeating 0s, and the one that ends in repeating 9s (with the last digit not 9 being 1 less that the equivalent digit in the terminating And, it's not really a matter of whether we want it or not. The two notations are a byproduct of how decimals work. We could decide to throw the repeating 9s notations out, but then we would have to make exceptions every time we discuss decimals: a * (0.bcdef...) is calculated by multiplying a by each digit in turn, summing and performing the carries except when this leads to repeating 9s, then you have to ... I personally think it is much easier to live with double notations, where you only have to state the equivalence once, and then you're covered for life! « Last Edit: Aug 19^th, 2003, 7:07pm by Icarus » "Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? " - Anonymous towr Re: 0.999. wu::riddles Moderator « Reply #108 on: Nov 7^th, 2002, 11:43pm » simplicity makes perfect.. If I answered 0.999... on a math-exam when the answer would be 1, it would be marked as wrong, or at least not right.. The simplest answer is the right one.. Some people are average, some are just mean. Would have been nice to try when I was in highschool though.. I could have driven my math-teacher mad.. Posts: 13730 Wikipedia, Google, Mathworld, Integer sequence DB Icarus Re: 0.999. wu::riddles Moderator « Reply #109 on: Nov 8^th, 2002, 3:45pm » When I was teaching, I would have taken a point off for failure to simplify, but I certainly would not have called it wrong, or even "not right", just not the best form for the answer. Boldly going where even angels fear to « Last Edit: Nov 8^th, 2002, 3:45pm by Icarus » "Pi goes on and on and on ... Gender: And e is just as cursed. Posts: 4863 I wonder: Which is larger When their digits are reversed? " - Anonymous towr Re: 0.999. wu::riddles Moderator « Reply #110 on: Nov 10^th, 2002, 7:49am » well.. a failure to simplify seems to qualify as not right imo.. In any case it's what I meant.. hmm.. Maybe I should teach my niece to try this later when she's in highschool, and how to argue the point its the same.. Some people are average, some are just mean. Posts: 13730 Wikipedia, Google, Mathworld, Integer sequence DB Guest Re: 0.999. Guest « Reply #111 on: Nov 21^st, 2002, 8:22am » To all those claiming 0.999... <1 : It never ceases to amaze me how many stupid idiots on message boards have no idea about this problem. Go off and study how mathematics and analysis works before jumping in with your pathetic uneducated opinions. towr Re: 0.999. wu::riddles Moderator « Reply #112 on: Nov 21^st, 2002, 9:16am » It never ceases to amaze me what spectacular significant contributions some people can bring to a discussion.. You, however, certainly aren't one of them.. Some people are average, some are just mean. Posts: 13730 Wikipedia, Google, Mathworld, Integer sequence DB Guest Re: 0.999. Guest « Reply #113 on: Nov 21^st, 2002, 6:31pm » towr : your posts really are laughable. "well.. a failure to simplify seems to qualify as not right imo.. In any case it's what I meant.. " So if the answer to a calculation is (x+4)(x+5) and I write x^2 + 9x + 20 then I'm wrong am I? Come on. I actually have a degree in mathematics and I know both answers are equally valid, no matter how you're marked in your "math exam". If the question asked to put something in its simplest terms, eg 4/8, then 1/2 or 0.5 would be correct answers and 2/4 would not. "5/5 is an expression, not a number.. You need to calculate it's value to get one.. " Of course its a number, you imbecile. 5/5 and 0.999... are equal real numbers more commonly known as 1. They are the same real number, we just choose to call it 1. 0.999... may be an abuse of notation but its perfectly well 0.999... = lim n-> infinity of (9/10 + 9/100 + 9/1000 + ... + 9/(10^n)) Just look it up in any mathematics textbook. For your information x/2, (x+3)(x+5) and exp((x^2)/2) are expressions. "A number is an atom, an equation is a set of operations on atoms.. " Nice defintion. No really. And your point? "It never ceases to amaze me what spectacular significant contributions some people can bring to a discussion.. You, however, certainly aren't one of them.. " I refer the right honourable cretin to the reply I gave a few moments ago. Jeremiah Smith Re: 0.999. Full Member « Reply #114 on: Nov 21^st, 2002, 7:13pm » It's about time this board got its very own pretentious asshole! I was getting sick and tired of how everyone was getting along so well! Posts: 172 Icarus Re: 0.999. wu::riddles « Reply #115 on: Nov 21^st, 2002, 8:20pm » Uberpuzzler So, "Guest", You have a degree in mathematics! Very good! Now get in line, like the rest. I have a PhD in it myself, and there are several other mathematics degrees on this board, as well as degrees in related fields. The regulars on this board, including towr, are both intelligent and well-educated. You would be wise to consider their words carefully, even when you disagree. Unlike you, they know the value of differing points of view. Even those who are mistaken can bring new depth to your understanding, and suggest things you may not have considered. For instance, Kozo earlier in this thread talks about 0.99...(infinitely many 9s)...9. Of course, YOU know that this isn't a real number. But I ask, do you really know why it isn't? Have you ever considered the possibility of defining numbers that can be expressed in this fashion? Can it be done consistently? If so, how do they behave? How do Boldly going they relate to the real numbers? where even These are useful considerations, but you are so full of yourself I doubt you ever saw them. Just as you failed to understand towr's remarks because you would rather belittle them angels fear to than think! Now, if you have something useful to add this or any other thread, I would love to read it. But if you want to keep up with this self-aggrandizement, stop wasting our time. "Pi goes on and on and on ... And e is just as cursed. Gender: I wonder: Which is larger Posts: 4863 When their digits are reversed? " - Anonymous Guest Re: 0.999. Guest « Reply #116 on: Nov 22^nd, 2002, 6:14am » So, "Icarus", You have a PhD in mathematics! Very good! Then you will agree that everything I have said was mathematically correct if not the tone in which it was written. "For instance, Kozo earlier in this thread talks about 0.99...(infinitely many 9s)...9. Of course, YOU know that this isn't a real number. But I ask, do you really know why it isn't?" Take a never ending sequence of 9s and put a 9 on the end, except you can never reach the end. So that doesn't make sense, it is not well defined, hence the number he is trying to define is not a member of the reals. However we can talk about the limit as n tends to infinity of 0.99... (n 9s) ...9 which is just 0.999... . Similarly the limit of 0.999...(n 9s)...A as n tends to infinity where A is a member of the set {0,1,2,3,4,5,6,7,8} is also equal to 1 and hence 0.999... by standard epsilon proofs. Have you ever considered the possibility of defining numbers that can be expressed in this fashion? Yes before, and when I read Kozo's post. See the above analysis. Because the last digit is on the end of a very long sequence whose length is tending to infinity, it has zero effect on the value of the number. So I ask of the usefulness of this idea. It doesn't work very well in the decimal system that we use but if you were to give definitions that would work better then I would be interested to know about them. Pietro Re: 0.999. K.C. « Reply #117 on: Nov 22^nd, 2002, 8:38am » Member Hauehauehauheauehauh!!! Who would ever have thought that the "0.999... = 1?" question could bring such strong emotions to the fore? I don't believe this "Guest" is for real. I think it's just Wu messing with us, or maybe one of the forum regulars, just for the hell of it. Anyways, as for the "mathematics degree" which you have, why don't you shove it in the Putnam section? If you got a degree just to know facts (which you are so anxious to show), you should have studied botany, or something like that. I would love to see a non-Googled solution of yours to "3 points in a circle", or "bright and dark stars". Thanks! Gender: « Last Edit: Nov 22^nd, 2002, 8:41am by Pietro K.C. » 213 "I always wondered about the meaning of life. So I looked it up in the dictionary under 'L' and there it was --- the meaning of life. It was not what I expected." (Dogbert) FenderStratFatMan Re: 0.999. Newbie « Reply #118 on: Nov 22^nd, 2002, 8:50am » on Jul 29^th, 2002, 6:40am, Kozo Morimoto wrote: OK, nobody seems to be getting the '#' besides me so I'll drop it... How about this. 0.9 is pretty close to 1, an approximation of 1. You add 0.09 to make it 0.99 and its even closer to 1 than 0.9 but still not 1. Repeat to infinity. You get ever closer and closer to 1 from the low side, but you never get Posts: 2 Using the ideas given above, does it mean that 1.000 ... 001 with infinite zeroes between the two 1s make it equal to 1? So Does it mean that 0.999... = 1.000...001 ? How about 0.999...998 = 0.999... = 1.000...001? Wow Kozo your using the same argument as my little brother. Anyway, the reason that would be wrong is because there is no way to write .0000000000000 infinite with one in the last place. Secondly 1 - .9 infinite is 0 try on a mathematical calculator. 1/.9999999999 = 1 and 1 + .99999999999 = 2. Explain any of those. In reality a complex equation is not needed to prove that .999999999 = 1. Pietro Re: 0.999. K.C. « Reply #119 on: Nov 22^nd, 2002, 9:24am » Member Quote: Wow Kozo your using the same argument as my little brother Come on, FenderMan, the point of the discussion is to try and clarify people's thoughts. Kozo's mistake is as valid as any other, and, as I see it, is pretty much the ONLY mistake you can plausibly make regarding this puzzle. This kind of judgment is really uncalled for. Kozo is being polite, the least we can do is return the favor. Second, a calculator will tell you all sorts of things, including absurdities like: Gender: 2/3 = 0.666666667 Posts: pi = 3.141562654 213 1 - 0.9999999999 (terminating) = 0. Don't trust them too much! "I always wondered about the meaning of life. So I looked it up in the dictionary under 'L' and there it was --- the meaning of life. It was not what I expected." (Dogbert) Guest Re: 0.999. Guest « Reply #120 on: Nov 22^nd, 2002, 1:32pm » "Pietro" : To be fair, I can't fault any of your analysis on this thread so far. "I don't believe this "Guest" is for real." Really? "Anyways, as for the "mathematics degree" which you have, why don't you shove it in the Putnam section?" I might do that. I mentioned the "mathematics degree" to add to the argument proving towr and his assumption wrong. What are your qualifications? "If you got a degree just to know facts (which you are so anxious to show), you should have studied botany, or something like that." I imagine botanists would be quite annoyed about that comment. What are you trying to say? I would love to see a non-Googled solution of yours to "3 points in a circle", or "bright and dark stars". Thanks! I will tackle those problems I choose to, not anything laid down as a challenge. As anyone who has done a degree will tell you, it allows one to specialise to a certain extent. I reserve that right here. Analysis is one of my preferred subjects, real and complex. A particularly nice result attributed to Picard is that supposing a complex function f has an isolated essential singularity at a, then in any punctured disc centred at a, f actually assumes every complex value in the plane (except possibly one). [Source : Introduction to Complex Analysis, H. A. Now this is not "anxiousness" to show facts for "self-aggrandizement", it's actually a beautiful and somewhat surprising result. What is pathetic is this : people with zero concept of mathematical definitions of very simple ideas (e.g. numbers) completely disregarding the complexity and beauty of mathematics by a) arguing strongly only with their intuition b) being pretty sure they are right, and c) not listening to reason or being willing to learn. James Re: 0.999. Fingas « Reply #121 on: Nov 22^nd, 2002, 2:02pm » Fellow Forum-goers: I would really hate this thread, which has had a lot of good (and also some not-so-good) discussion on it, to devolve into a flame-war. Guest, I found your post on Nov. 21^st to be rude. If people do not know about mathematics, then there is no better place for them to raise their questions and learn from the answers than in a forum like this. The question is simple enough that people can make up their own ideas, but complex enough to allow some serious mathematical formalization. Although I do not have a degree in math, I appreciate some of the insights that I've seen in this particular thread, shared with all of us by the people who know what they're talking about. If I never ask a stupid question, how Gender: can I come to a deeper understanding of how math works? Posts: 949 Doc, I'm addicted to advice! What should I do? Icarus Re: 0.999. wu::riddles « Reply #122 on: Nov 22^nd, 2002, 4:00pm » Uberpuzzler Yes, you can define numbers with infinitely many nines, and then some more. They are not Real numbers, so your argument against them, which relies on concepts of the Reals, is fallacious. Consider sequences from the countable ordinals into the 10 digits for the definition. They also do not form a field, or even a group, but they do have some interesting behaviors of their own. (Topologically, I believe this is still the long line, though the construction is different.) Yes, the actual math you presented was correct. However your belief that you have any idea what towr was talking about is not. This is evident, since nothing mathematical you said Boldly going where disagrees with what towr's points. If you will read towr's comments again with a less haughty disposition, you will discover that he never argued that 0.999... != 1. His only even angels fear point was that 0.999... should not be considered an actual number at all. I argue that such a definition would create far more problems than the minor one it solves (two to tread. representations for terminating decimal numbers). This does not mean that his point is wrong. There is no "right" or "wrong" on this one, it's a matter of convention. Now, if you have read James' post and understand the truth of what he is saying, and will behave yourself better in the future, I would be more than happy to see you contribute to these forums. But insulting everyone who disagrees with you and being condescending is childish. Gender: "Pi goes on and on and on ... Posts: 4863 And e is just as cursed. I wonder: Which is larger When their digits are reversed? " - Anonymous jon g Re: 0.999. Guest « Reply #123 on: Nov 23^rd, 2002, 1:52am » just drop it, you all have gone too far. Why don't you answer a real question like what is the meaning of life or is there a god? Jeremiah Smith Re: 0.999. Full Member « Reply #124 on: Nov 23^rd, 2002, 11:53am » on Nov 23^rd, 2002, 1:52am, jon g wrote: just drop it, you all have gone too far. Why don't you answer a real question like what is the meaning of life or is there a god? Simple. Those questions haven't been added to William's puzzle pages yet. Posts: 172 4 5 6 7 ... 13 Notify of replies Send Topic Print « Previous topic \| Next topic »	{"url":"https://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?action=display;board=riddles_medium;num=1027804564;start=100","timestamp":"2024-11-11T21:29:30Z","content_type":"text/html","content_length":"113129","record_id":"<urn:uuid:62446380-c38b-416c-852d-deefbf613cda>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00700.warc.gz"}
An Etymological Dictionary of Astronomy and Astrophysics rotational period دورهی ِچرخش dowre-ye carxeš Fr.: période de rotation → rotation period. → rotational; → period. rotational transition گذرش ِچرخشی gozareš-e carxeši Fr.: transition rotationnelle A slight change in the energy level of a molecule due to the rotation of its constituent atoms about their center of mass. → rotational; → transition. rotational velocity تندای ِچرخشی tondâ-ye catxeši Fr.: vitesse de rotation The velocity of a → rotational motion; same as → angular velocity. → rotational; → velocity. shellular rotation چرخش ِپوستهای carxeš-e puste-yi Fr.: rotation coquillaire A rotation mode in which internal rotation of a star depends essentially on depth and little on latitude: Ω(r,θ) = Ω(r), where r is the mean distance to the stellar center of the considered level surface (or → isobar). This particular mode was introduced by J.-P. Zahn (1992, A&A 265, 115) to simplify the treatment of rotational → mixing, but also on more physical grounds. Indeed differential rotation tends to be smoothed out in latitude through → shear turbulence. See also → von Zeipel theorem; → meridional circulation . Shellular, the structure of this term is not clear; it may be a combination of → shell (referring to star's assumed division in differentially rotating concentric shells) + (circ)ular, → circular. The first bibliographic occurrence of shellular is seemingly in Ghosal & Spiegel (1991, On the Thermonuclear Convection: I. Shellular Instability, Geophys. Astrophys. Fluid Dyn. 61, 161). However, surprisingly the term appears only in the title, and nowhere in the body of the article; → rotation. Carxeš, → rotation; puste-yi, adj. of pusté, → shell. sidereal rotation period دورهی ِچرخش ِاختری dowre-ye carxeš-e axtari Fr.: période de rotation sidérale The rotation period of a celestial body with respect to fixed stars. For Earth, same as → sidereal day. → sidereal; → rotation; → period. solar rotation چرخش ِخورشید carxeš-e xoršid (#) Fr.: rotation du Soleil The motion of the Sun around an axis which is roughly perpendicular to the plane of the → ecliptic; the Sun's rotational axis is tilted by 7.25° from perpendicular to the ecliptic. It rotates in the → counterclockwise direction (when viewed from the north), the same direction that the planets rotate (and orbit around the Sun). The Sun's rotation is differential, i.e. the period varies with latitude on the Sun (→ differential rotation). Equatorial regions rotate in about 25.6 days. The regions at 60 degrees latitude rotate more slowly, in about 30.9 days. → solar; → rotation. stellar rotation چرخش ِستاره، ~ ستارهای carxeš-e setâré, é setêre-yi Fr.: rotation stellaire The spinning of a star about its axis, due to its angular momentum. Stars do not necessarily rotate as solid bodies, and their angular momentum may be distributed non-uniformly, depending on radius or latitude.Thus the equator of the star can rotate at a different angular velocity than the higher latitudes. These differences in the rate of rotation within a star may have a significant role in the generation of a stellar magnetic field. → stellar; → rotation. synchronous rotation چرخش ِهمگام carxeš-e hamgâm (#) Fr.: rotation synchrone Of a body orbiting another, where the orbiting body takes as long to rotate on its axis as it does to make one orbit. Therefore it always keeps the same hemisphere pointed at the body it is orbiting. Both bodies are tidally locked (→ tidal locking). This phenomenon is a natural consequence of → tidal braking. Synchronous rotation is common throughout the → solar system. It is found among the satellites of → Mars (→ Phobos and → Deimos), → Jupiter (most of Jupiter satellites, including the → Galilean Moons) and → Saturn (e.g. → Iapetus). Similarly, → Pluto and its moon → Charon are locked in mutual synchronous rotation, with both of them keeping the same faces towards each other. → synchronous; → rotation. Venus rotation چرخش ِناهید carxeš-e nâhid Fr.: rotation de Vénus The → sidereal rotation period of Venus, or its → sidereal day, is 243.025 Earth days (retrograde). The length of a → solar day on Venus (that is one entire day-night period) is 116.75 Earth days, that is significantly shorter than the sidereal day because of the retrograde rotation. One Venusian year is about 1.92 Venusian solar days. → Venus; → rotation. vibrational-rotational transition گذرش ِچرخشی-شیوشی gozareš-e carxeši-šiveši Fr.: transition vibrationnelle-rotationnelle A slight change in the → energy level of a → molecule due to → vibrational transition and/or → rotational transition. → vibrational; → rotational; → transition.	{"url":"https://dictionary.obspm.fr/index.php/?showAll=1&&search=&&formSearchTextfield=ROtation&&page=2","timestamp":"2024-11-11T16:19:29Z","content_type":"text/html","content_length":"23342","record_id":"<urn:uuid:f2bd742b-5a30-4f2f-8d32-a9a334008f68>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00157.warc.gz"}
Convert Katha to Kilometer Square (katha to km2) Katha to Kilometer Square Converter = 0 Kilometer Square Katha To Kilometer Square Conversion Table Unit Conversion Value 1 Katha 0.00 Kilometer Square 2 Katha 0.00 Kilometer Square 5 Katha 0.00 Kilometer Square 10 Katha 0.00 Kilometer Square 20 Katha 0.00 Kilometer Square 50 Katha 0.01 Kilometer Square 100 Katha 0.01 Kilometer Square 200 Katha 0.03 Kilometer Square 500 Katha 0.06 Kilometer Square 1000 Katha 0.13 Kilometer Square 1. What is Katha? Katha is a traditional unit of area measurement commonly used in rural India and Bangladesh that varies in definition but typically ranges from 2,000 to 4,000 square feet. 2. How many Katha are in one square kilometer? There are approximately 14,925 to 15,000 Katha in one square kilometer, depending on the specific regional definition. 3. How do you convert Katha to kilometers squared? To convert Katha to kilometers squared, multiply the number of Katha by 0.000067 km²/Katha. 4. Why is it important to know the conversion of Katha to km²? Knowing the conversion is important for land-use planning, agricultural practices, and property valuations in rural areas. 5. Is the Katha definition universal? No, the definition of Katha can vary between regions, which is why it is best to verify specific local measurements before making conversions. 6. What is the approximate area of one Katha in square meters? One Katha is approximately equal to 670 square meters, but this can vary based on regional definitions. 7. How can farmers benefit from converting Katha to km²? Farmers can use the conversion to effectively communicate and manage their land area for planning, crop production, and when working with government agencies. 8. What challenges might arise from the Katha to km² conversion? Challenges can include discrepancies in definitions, variations in regional practices, and the necessity for precise local knowledge to ensure an accurate conversion. 9. Can Katha be converted into other area units? Yes, Katha can also be converted into acres, square feet, and square meters, but the specific conversion factors will depend on local definitions. 10. Are there tools available for converting Katha to km²? Yes, online calculators and conversion tables can facilitate the conversion process for users who need quick references. About Katha Katha: The Art of Storytelling in Indian Tradition Katha, derived from the Sanskrit word "Katha" which means 'story', encompasses a rich tradition of storytelling that has been an integral part of Indian culture for centuries. This art form combines elements of narrative, performance, and spirituality, offering more than just entertainment — it serves as a medium for moral education, cultural preservation, and community bonding. In this detailed exploration, we delve into the origins, forms, significance, and contemporary expression of Katha. Origins and Historical Context The roots of Katha can be traced back to ancient Indian texts and oral traditions. The Vedas and the Puranas, two of the oldest scriptures in Hinduism, contain narratives that were often recited or performed orally. These texts not only provided spiritual and philosophical guidance but also conveyed stories that taught values and ethics. As societies evolved, so did the methods of storytelling. The tradition of Katha was further enriched during the medieval period by various saints and poets, such as Tulsidas, who narrated the Ramayana through dramatic recitations, and Mirabai, who sung the tales of Lord Krishna. Each of these contributions helped solidify Katha as an important vehicle for religious and moral teachings. The Forms of Katha Katha is not a monolithic form; rather, it varies significantly across regions, languages, and communities in India. Some of the prominent forms include: 1. Ram Katha: Focused on the life and adventures of Lord Rama, this form draws heavily from the "Ramayana". It emphasizes moral and ethical dilemmas faced by Rama and other characters, imparting lessons on duty (dharma), righteousness, and devotion. 2. Krishna Katha: This form centers around the stories of Lord Krishna from his childhood exploits in Vrindavan to his role in the Mahabharata. The playful and divine nature of Krishna's tales engage audiences and often highlight themes of love, devotion, and the complexity of human relationships. 3. Gurbani Katha: Influenced by the teachings of Sikhism, Gurbani Katha involves narrating the messages contained in the Guru Granth Sahib, the holy scripture of Sikhs. It emphasizes spiritual wisdom, ethical living, and social justice. 4. Buddhist Katha: Associated with Buddhist teachings, this form narrates the life of Buddha and his followers, focusing on concepts like compassion, mindfulness, and enlightenment. 5. Sufi Katha: Intertwining with Islamic mysticism, Sufi Katha tells stories that convey the essence of love for God and the pursuit of truth, often through poetic expressions of longing and Each of these forms reflects the region's unique cultural tapestry while sharing universal themes of love, morality, and spirituality. The Art of Katha Performance Katha is traditionally performed by a storyteller known as a 'Katha vachak' or 'Katha kar'. The performance can take many forms — from simple storytelling to elaborate theatrical presentations with music, dance, and dramatic expressions. Elements of Katha Performance • Narrative Style: The Katha vachak employs a captivating narrative style, often using rhetorical questions, dialogues, and vivid descriptions to draw the audience into the story. The storyteller's ability to animate characters and evoke emotions is crucial for engaging the listeners. • Music and Instruments: Music plays an essential role in Katha performances. Instruments like the tabla, harmonium, and dholak are commonly used to accompany the narration. Melodic tunes enhance the emotional weight of the stories, making pivotal moments more impactful. • Audience Interaction: One significant aspect of Katha is the interaction between the storyteller and the audience. Listeners may be invited to respond, ask questions, or reflect on moral lessons, creating a participatory atmosphere that fosters communal learning. • Spiritual and Moral Themes: Most Katha performances aim to impart spiritual wisdom and ethical lessons. These stories often end with a moral or a reflection that encourages self-examination and personal growth among the audience members. Significance of Katha in Indian Culture Katha holds immense significance in Indian society for various reasons: Cultural Preservation Through Katha, historical events, myths, and legends are preserved. As oral history, it keeps alive the cultural identity of different communities, ensuring that traditional values and practices are passed down through generations. Community Bonding Katha sessions often bring people together, fostering a sense of community. Whether held in temples, homes, or public festivals, these gatherings strengthen social bonds and promote collective Spiritual Engagement Katha acts as a medium for spiritual engagement, allowing listeners to connect with their faith and heritage. Many consider attending Katha sessions as a form of worship, where the act of listening becomes a sacred experience. Moral Education The narratives serve as moral compasses, teaching audiences about virtues like honesty, compassion, and forgiveness. The stories often depict characters grappling with dilemmas, providing relatable insights that inspire ethical living. Contemporary Expressions of Katha In recent years, Katha has undergone transformations influenced by modernity and globalization. While traditional forms continue to thrive, contemporary adaptations are gaining popularity. Fusion with Technology Modern storytellers are incorporating technology into Katha presentations, utilizing multimedia, animations, and social media platforms to reach wider audiences. Online streaming of Katha sessions has enabled people from diverse backgrounds to access this art form, breaking geographical barriers. Incorporation of Modern Themes Contemporary Katha performers sometimes weave in modern themes and issues, such as gender equality, environmental concerns, and social justice. This adaptation ensures that Katha remains relevant and resonates with younger audiences. Global Influence The universal appeal of Katha has led to its recognition beyond Indian borders. Global storytellers and artists are exploring Indian narrative traditions, fusing them with their own cultural contexts. This cross-cultural exchange enriches the global storytelling landscape, promoting understanding and appreciation of diverse narratives. Katha, in its essence, is far more than mere storytelling; it embodies a cultural heritage that nurtures the human spirit, connects communities, and imparts moral wisdom. As society continues to evolve, the art of Katha retains its significance, adapting to contemporary challenges while honoring its age-old traditions. By celebrating Katha, we preserve not only the stories of our ancestors but also the values that guide us through the complexities of modern life. About Kilometer Square Understanding Square Kilometers: A Comprehensive Guide In the realm of measurement, particularly in geography and land measurement, the square kilometer (km²) stands out as a fundamental unit. Understanding this unit is essential for various applications—be it environmental studies, urban planning, or even real estate calculations. This article will delve into what a square kilometer represents, how it’s used, its conversions to other units, and its implications in real-world scenarios. What is a Square Kilometer? A square kilometer is a measure of area that is equal to the area of a square with each side measuring one kilometer in length. In mathematical terms, if you take a square where each side is 1 km long, the area can be calculated using the formula: [ \text{Area} = \text{side} \times \text{side} = 1 , \text{km} \times 1 , \text{km} = 1 , \text{km}^2 ] This unit is part of the metric system, a decimal-based system of measurement that is widely used around the world. The square kilometer helps provide a standardized way to quantify areas, making it easier to communicate sizes of regions, properties, and geographical features. Common Applications of Square Kilometers 1. Geographical Measurements: One of the primary uses of square kilometers is in the field of geography. Countries are often measured in terms of their total area in square kilometers, providing an easy way to compare the sizes of different nations. For instance, Russia is the largest country in the world, covering approximately 17.1 million km², while Vatican City is the smallest at about 0.44 km². 2. Urban Planning: Urban planners utilize square kilometers to assess land use and zoning. City size, population density, and the allocation of resources can all be expressed in square kilometers. For example, a city like New York occupies an area of approximately 789 km², which influences its infrastructure development and public service provisions. 3. Environmental Studies: In environmental science, square kilometers are used to quantify ecosystems, wildlife habitats, and conservation areas. When studying deforestation, researchers might report the area of forest lost in square kilometers, emphasizing the scale of environmental impact. 4. Real Estate: In real estate, land area is often expressed in square kilometers for larger plots. While residential properties may be listed in square meters, larger developments or subdivisions can be conveniently represented in square kilometers, especially in rural or sprawling areas. Conversion to Other Units Understanding square kilometers also involves knowing how to convert to other area measurements. Here are some common conversions: • Square Meters (m²): Since one square kilometer is equal to (1,000,000) square meters, converting from square kilometers to square meters can be done by multiplying by this factor. [ 1 , \text{km}² = 1,000,000 , \text{m}² ] • Hectares (ha): One hectare equals 0.01 square kilometers. Therefore, to convert square kilometers to hectares, you multiply by (100). [ 1 , \text{km}² = 100 , \text{ha} ] • Acres: One square kilometer is approximately (247.105) acres. To convert square kilometers to acres, you multiply by (247.105). [ 1 , \text{km}² \approx 247.105 , \text{acres} ] These conversions are vital in fields such as agriculture, where land is often measured in hectares or acres, while large-scale analyses might require square kilometers. Visualization and Context To better comprehend the size of a square kilometer, visualizing the area can be helpful. Picture a square that is 1 km on each side: • Maps and Geography: On many maps, especially topographic or thematic maps, areas are delineated using square kilometers. Each grid square representing 1 km² can provide useful context for understanding the geographic layout and scale. • Urban Areas: Consider a city block. In many cities, a typical block might be around 100 to 200 meters on each side. It takes about 50 to 100 city blocks to cover 1 km², illustrating just how vast this measurement can be when applied to urban settings. Global Context On a global scale, the distribution of land area measured in square kilometers can highlight significant ecological and geopolitical concerns. Here are a few examples that illustrate the importance of square kilometers worldwide: 1. Forests and Natural Reserves: The Amazon Rainforest stretches over approximately 5.5 million km², emphasizing its critical role in biodiversity and climate regulation. Conserving such vast areas is crucial for environmental sustainability. 2. Nation Comparisons: The total area of the world's oceans is about 361 million km². In contrast, land coverage is only about 149 million km². Such comparisons provide perspective on the Earth's surface and the importance of land management policies. 3. Population Density: Understanding square kilometers aids in studying population density. For instance, countries like Bangladesh have a very high population density, with over 1,200 people per square kilometer, while larger nations like Canada have vast areas with sparse populations, resulting in much lower density figures. The square kilometer is more than just a unit of measurement; it serves as a fundamental building block in our understanding of geography, urban planning, environmental science, and much more. Whether comparing the sizes of countries, planning urban spaces, or assessing environmental impacts, the significance of square kilometers cannot be overstated. As we continue to face challenges related to land usage, conservation, and urbanization, grasping the implications of area measurements like square kilometers will be increasingly important in fostering informed decision-making. kathakm2KathaKilometer Squarekatha to km2katha to Kilometer SquareKatha to Kilometer SquareKatha to km2km2 in kathakm2 in KathaKilometer Square in KathaKilometer Square in kathaone katha is equal to how many km2one Katha is equal to how many Kilometer Squareone Katha is equal to how many km2one katha is equal to how many Kilometer Squareone katha equals how many km2one Katha equals how many km2 one Katha equals how many Kilometer Squareone katha equals how many Kilometer Squareconvert katha to km2convert Katha to Kilometer Squareconvert Katha to km2convert katha to Kilometer Squarehow to convert katha to km2how to convert Katha to Kilometer Squarehow to convert Katha to km2how to convert katha to Kilometer Squarehow many km2 are in a kathahow many Kilometer Square are in a Kathahow many Kilometer Square are in a kathahow many km2 are in a Kathahow many km2 to a kathahow many Kilometer Square to a Kathahow many Kilometer Square to a kathahow many km2 to a Kathakatha to km2 calculatorkatha to Kilometer Square calculatorKatha to Kilometer Square calculatorKatha to km2 calculatorkatha to km2 converterkatha to Kilometer Square converterKatha to Kilometer Square converter Katha to km2 converterConvert katha to km2Convert katha to Kilometer SquareConvert Katha to Kilometer SquareConvert Katha to km2	{"url":"https://www.internettoolwizard.com/convert-units/area/katha/km2","timestamp":"2024-11-02T21:28:49Z","content_type":"text/html","content_length":"400126","record_id":"<urn:uuid:bf831538-4eea-4da2-bc5b-004f14c9f975>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00885.warc.gz"}
Pendulum ODE Auto-suggest helps you quickly narrow down your search results by suggesting possible matches as you type. Community Tip - Visit the PTCooler (the community lounge) to get to know your fellow community members and check out some of Dale's Friday Humor posts! X Prime 5 - a constant not function as an answer! 1) wrong syntax of Odesolve (you don't have to provide the function name theta in MC15) 2) the expression on the right hand side of your ODE takes non-real values if theta goes above 45 degreeI coped boldly with the last situation by applying the Re() function 😉 But why I get a constants, not a function - angle of the pendulum as function of time? Total energy is constant throughout, so the value under the square root sign of the numerator of the RHS of your ODE is always zero; hence theta goes nowhere (as you can see from Werner's graph)!! If you wish to get the equations of motion from conservation of energy Val, you need to do the following:	{"url":"https://community.ptc.com/t5/Mathcad/Pendulum-ODE/td-p/665158","timestamp":"2024-11-07T07:38:44Z","content_type":"text/html","content_length":"281101","record_id":"<urn:uuid:af4b9219-d946-4be1-ac14-5de2391443c7>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00086.warc.gz"}
The Speed of Light is Constant (in a Perfect Vacuum) - Fact or Myth? In theory, the speed of light, in a perfect vacuum, measured from an inertial frame, is constant. This isn’t to say that studies haven’t cast doubt on this longstanding theory, and this isn’t to say that light speed has ever been directly observed from an internal frame in a perfect vacuum… it is only to say, the best evidence we have today suggests that the speed of light is constant. The speed of light and the force of gravity,are not independent entities,because,they have a source.Besides,the four? known forces of nature ,is a farce It’s great that you take the time to talk about real science rather than dumbed down soundbites, refreshing to read 🙂 You’ve let a contradiction slip through though: “It is a fact that scientists have constantly found the speed of light (in a perfect vacuum) to be 299,792,458 m/s using mathematics and tests” “In real life there is no empty space, so light is never actually traveling through a perfect vacuum. A perfect vacuum is a theoretical concept.” If scientists can’t make a perfect vacuum, they can’t have measured the speed of light in a perfect vacuum. BIt nitpicky I know, but as you’d taken so much care to try to do it right I thought you wouldn’t mind having your attention drawn to this.	{"url":"http://factmyth.com/factoids/the-speed-of-light-is-constant-in-a-perfect-vacuum/","timestamp":"2024-11-06T14:35:52Z","content_type":"text/html","content_length":"50398","record_id":"<urn:uuid:5bbaf7f3-280a-4b19-bf08-748ea14b93ae>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00528.warc.gz"}
Can someone explain this theorem from von Staudt on denominators of Bernoulli numbers. Can someone explain this theorem from von Staudt on denominators of Bernoulli numbers. This is an extract from Paulo Ribenboim: 13 lectures on Fermats last theorem on page 105. 'In 1845, von Staudt determined some factors of the numerator N_2k. Let 2k = k1k2 with gcd(k1,k2)= 1 such that p\|k2 if and only if p\|D_2k'. Where N_2k and D_2k are the numerators and denominators of Bernoulli number B_2k. So I've actually used the result of this theorem for some other proof, but looking back at it I find it is not true. For example when 2k=74, then 2k=2x37. If we take p=37, we see that 37\|k2=37 and so 37 must divide the denominator D_74 but D_74=6. Im not sure what I'm missing here. Maybe, I have misinterpreted the theorem. Could someone clear this up for me. 1 Answer Sort by ยป oldest newest most voted There are several ways to write 74 as a product: $74 = 1 \times 74 = 2 \times 37 = 37 \times 2 = 74 \times 1$. edit flag offensive delete link more	{"url":"https://ask.sagemath.org/question/47312/can-someone-explain-this-theorem-from-von-staudt-on-denominators-of-bernoulli-numbers/","timestamp":"2024-11-14T17:10:29Z","content_type":"application/xhtml+xml","content_length":"50299","record_id":"<urn:uuid:b4e945c3-bfb4-49d3-8d8a-5f379b5bbd59>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00130.warc.gz"}
Entropy and Self Information This post contains short notes on entropy and self information and why machine learning adopted them from information theory. Information theory is a science concerned with the study of quantification of information. Self information In order to deal with information from a quantitive and not qualitative perspective, Claude Shannon invented a formula to quantify how much information is contained inside an event. The breakthrough idea of Shannon follows from the intuition that the more unlikely an event is, the higher its information - and vice versa. Assuming a set of possible events $X = {x_0, x_1, ... x_n}$ with a corresponding probability distribution such that each $x_i$ has a probability of $p(x_i)$, we want to derive a quantity, which we shall define information content of $x_i$ or $h(x_i)$ such that the higher $p(x_i)$ the lower $h(x_i)$. We define \[h(x_i) = log(\frac{1}{p(x_i)}) = -log(p(x_i))\] to be the self information of event $x_i$. As we know, the number of bits to represent a number $N$ is $log_2(N)$. Therefore, we can look at the self information as the number of bits necessary to represent (store, transmit) the probability of an event. Entropy in information theory (as it also exists in other fields, such as thermodynamics) is the natural extension of the concept of self-information to random variables. Entropy tells us how much information we find on average in a random variable $X$. Assuming we have a random variable $X$, we define the Entropy of $X, H(X)$ as: \[\sum_i p(x_i) h(x_i) = - \sum_i p(x_i) log(p(x_i))\] Entropy is the average number of bits needed to represent the probability of an event from the random variable $X$. The more a probability distribution is balanced, that is, all events are similarly likely, the higher is the entropy. The more unbalanced the distribution, the lower the entropy. Look at the chart from [1] that plots entropy for a probability distribution with only two events. Cross entropy Cross entropy is a popular metric in machine learning, used to measure the difference between two probability distributions. The intuition for this definition comes if we consider a target or underlying probability distribution $P$ and an approximation of the target distribution Q, then the cross-entropy of $Q$ from \ (P\) is the number of additional bits to represent an event using $Q$ instead of $P$. More precisely, in classification problems, the predicted probabilities $Q$ (output of a model) can be compared with the true probabilities $Q$ (one-hot encoded ground truth labels). 1. C. Bishop, Pattern Recognition and Machine Learning	{"url":"https://alessiodevoto.github.io/Entropy-and-Self-Information/","timestamp":"2024-11-10T12:42:54Z","content_type":"text/html","content_length":"16208","record_id":"<urn:uuid:72496a95-6f43-4c27-a49a-f64f117d7ea2>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00335.warc.gz"}
Teaching Math Concepts Using Real-World Problems and Visual Explanations - Concepts for Kids Teaching math concepts can be a challenge, especially when you are dealing with younger students. This is where manipulatives and visual explanations come in handy. To help your students understand math concepts, you can use real-world problems to illustrate your ideas. Lay out the steps for each concept and allow students to ask questions as they struggle to understand it. Remember that math is a sequential subject, so it is best to make the learning process as interesting as possible. Many teachers skip the step that involves the abstraction of numbers. This results in students seeing only symbols and numbers. For example, they wouldn’t understand why 5 x 4 = 20 instead of “5 + 4 = 20”. Counting activities should be varied, using appropriate vocabulary. Students should practice early addition and subtraction by counting objects from two sets and finding one or two more or two fewer than the fixed number. You can also practice early multiplication and division by dividing the number of objects by two. Identifying a pattern from the data can also be useful. Lastly, try linking math concepts with science, economics, civics, and geography. Students will be more engaged when they are given pieces to a puzzle. They can work on three or four problems at once. They may have to break the problem into as many pieces as they need to solve it. This approach works especially well with large classes, since each student can work on three or four problems at once. In addition, students can use symbols to distinguish the parts of the puzzle. Using a game can help students learn concepts, improve their understanding, and foster a home-school connection. A good teacher uses knowledge of students as learners to help them improve their instruction. Teachers can incorporate challenges into their lessons by using open tasks, generative questions, and turn and talk questioning strategies. Teachers can also use argumentative questions to require students to reflect and justify their positions. Students should be given varying levels of support when facing challenges. As the challenges increase, the teacher should release more responsibility. For example, students can start out with the simplest challenges and progress to harder ones, and then move on to the next level. Engaging routines are essential to teach children math. Engaging routines help children learn math concepts by developing their problem-solving, reasoning, and procedural fluency skills. Children can learn math concepts more easily if the routines are fun and engaging. Adding real-world objects will also help kids understand how math works in everyday life. Incorporating activities in your math lessons will help the children retain math concepts. You can even add some history and math trivia to the lesson. Students can get motivated to learn math concepts anywhere in the curriculum. Action learning brings mathematical concepts to life and creates interdisciplinary connections. In this method, mathematics instructors can provide problems whose applications are real-world and which the students must use to find out whether they are useful. They can also conduct research connected with their project experience. Research shows that students are more likely to remember concepts they learn through action than from a lecture method. In fact, action learning improves the retention of math concepts in a way that pure lectures cannot. Hands-on activities are a great way to reinforce math concepts. While they require a lot of time and effort, they can help students retain the concepts much better. For example, a building project that requires students to use angles and geometric shapes, as well as weight capacity, can be a good hands-on experience. When combined with hands-on activities, students learn more in a single experience. The same goes for implementing manipulatives in the classroom. Children develop their own understandings of numbers, patterns, and shapes through problem-solving activities. When they start learning math, they build confidence and capacity in everyday life. They apply their knowledge in everyday situations, such as counting toys, weighing objects, and more. This helps them understand the meaning of numbers and make better decisions and solve problems. In addition to a strong foundation, children will also be able to apply math concepts in different areas of their lives.	{"url":"https://conceptsforkids.us/2022/08/05/teaching-math-concepts-using-real-world-problems-and-visual-explanations/","timestamp":"2024-11-07T06:20:53Z","content_type":"text/html","content_length":"142164","record_id":"<urn:uuid:eb76a6d9-3858-4ca3-b3fc-2a3f70d40b97>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00349.warc.gz"}
Return on Investment (ROI) \| Binance Academy Return on Investment (ROI) Return on Investment, or ROI for short, is a ratio or percentage value that reflects the profitability or efficiency of a certain trade or investment. It is a simple-to-use tool that can generate an absolute ratio (e.g., 0.35) or a value in percentage (e.g., 35%). As such, ROI can also be used when comparing different types of investments or multiple trading operations. Specifically, ROI evaluates the return on an investment in relation to its purchasing cost. This means that the calculation of ROI is simply the return (net profit) divided by the total acquisition costs (net cost). The result may then be multiplied by 100 to get the percentage value. Naturally, a high ROI value indicates that the investment was profitable, while a negative ROI means the return was lower than the costs. The calculation of ROI is based on the following equation: ROI = (Current Value - Total Cost) / Total Cost Alternatively, it may also be written as: ROI = Net Profit / Net Cost As an example, imagine that Alice bought 100 for 1,000 US dollars - paying 10 dollars each. If the current price of is 19 dollars, Alice would have an ROI of 0.90 or 90%. ROI is widely used in both traditional and markets. However, it has some limitations. For instance, Alice may use the ROI formula when comparing two different trades. However, the equation does not take the time into account. This means that in some situations, one investment may seem more profitable than the other when, in reality, its efficiency was lower because it required a much longer period. So, if Alice’s first trade had a 90% ROI but took 12 months to happen, it would be less efficient than a second trade that had, for example, a 70% ROI in 6 months.	{"url":"https://academy.binance.com/lv/glossary/return-on-investment","timestamp":"2024-11-09T16:43:01Z","content_type":"text/html","content_length":"180411","record_id":"<urn:uuid:4fdc2e7c-60bc-485a-be3d-4a030aef946c>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00840.warc.gz"}
Polynomial Multiplication Worksheet Mathematics, particularly multiplication, develops the keystone of many academic techniques and real-world applications. Yet, for several learners, mastering multiplication can position a challenge. To resolve this obstacle, educators and moms and dads have actually accepted an effective tool: Polynomial Multiplication Worksheet. 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Why does MySQL use B+ trees as indexes? - Moment For Technology Sit right into the flower really angry, go out a smile across the river Why does MySQL use B+ trees as indexes? If it is pure guess why MySQL database index uses B+ tree? So the answer to this question is always going to be around what are B+ trees and what are their advantages. (This is not the way I started to think, I read a lot of articles from this dimension of the questions, the answers let me disappointed. Until the other day I asked the 5 year old programmer sitting next to me who was always looking for fish; His languid reply: Why do you think it’s a tree structure? Hey, hearing this answer, suddenly opened my mind, a little interesting! Let’s get rid of all the preconceived answers about what a B+ tree is, what it’s good for. (I didn’t want the hard answer shoved down my throat.) What I want is why? Why choose tree structure when there are so many data structures available for MySQL index? Why so many tree structures? Why B+ trees? That’s what I have to figure out! What I want is not just the answer but the context behind it! I want not only to know what it is, but also to know why. Of all the data structures, why a tree? At the logical level, numerous data structures can be divided into linear structure and nonlinear structure. Linear structure: array, linked list, based on their derived hash table (hash table also known as hash table), stack, queue and so on. The nonlinear structure includes tree and graph. There are other data structures such as: skip tables, bitmaps are also evolved from the basic data structure, different data structures exist to solve some scenarios. If we want to know what data structure index is suitable for, then we need to solve what problems (pain points) and need to play what role, and then look at what data structure to use; The latter is the effect; the former is the cause. MySQL stores data on disk. Even if the device is powered down, data stored on disk is not affected and data loss is guaranteed. This means that MySQL data on disk is persistent. But data storage on disk comes at a cost, in the form of millisecond processing speed, which is nothing compared to the nanosecond speed of memory. You may have no idea what units of time are, but a millisecond is tens of thousands of times slower than a nanosecond. Although the index is stored on the disk, when you use the index to search for data, you can read the index from the disk and save it to the memory, and then find the data from the disk. Then the data read from disk is also put into memory. Indexing allows disk and memory to join forces, taking advantage of memory’s ride and experiencing nanosecond speed. However, no matter how optimized the query process is, as long as the root is still on the disk, it is inevitable that multiple disk I/ OS will occur, and the more disk I/ OS occur, the more time will be consumed. (For those of you who are smart, this is actually a pain point that needs to be addressed.) • Do as little I/O on the disk as possible. But there are so many data structures to choose from. Since the purpose of the index already determines which data structures are available, you can narrow down the selection of other data structures. Start with the primary purpose of why the index itself is built. • Be able to quickly and efficiently range searches by interval. Of course, dynamic data structures that support efficient range queries in addition to their primary purpose can also have operations such as insert updates. So what other data structures can you think of besides tree structures that satisfy these two main conditions? Hash table, hop table Hash table Let’s take a look at a hash table, which we’re all familiar with, because the physical storage of a hash table is an array, and an array is a contiguous address space in memory. Data is stored as keys and values. Hash tables have precise queries, so the time complexity is O(1). And what makes a hash table so fast is that it calculates the index of the array using the Key to find the Value quickly. The simplest method of calculation is the remainder method, which first calculates the HashCode of the key, and then complements the HashCode based on the array length of the hash table. The remainder obtained from the remainder is the array subscript, and finally accesses the key and Value stored in the hash table from the subscript. But the subscript computed by Key might be the same, for example, HashCode 1010 mod 6 is 2, but HashCode 1112 mod 6 is also 2. The hash algorithm randomly calculates the length of the remainder of HashCode and may have the same array subscripts, which is called hash conflict. Hash conflicts are usually solved by linked lists. When a hash conflict occurs, data elements with the same subscript are replaced with storage Pointers, and data elements with different keys are added to the linked list. The search is done by traversing the list with Pointers and matching the correct Key. As you can see above, Key is a “rough seed” string and Value is “don’t forget to follow, like, comment”. We do this by calculating the integer int whose key is HashCode (1010). Then use HashCode (1010) to mod a hash array of length 6 to produce 2, whose Value element is (Key = “a wild seed “,Value =” don’t forget to follow, like, comment “). Although the hash table can be efficient equivalence query. Such as SQL: select * from weixin where username ="A fierce seed."Copy the code However, interval query is not supported. Such as SQL: select * from weixin where age < 18 Copy the code So if the hash table is used as an index, when doing a range query it means a full scan. However, NoSQL databases like Redis, whose storage form is KV, are suitable for equivalent query scenarios, but that is another topic. Jump table Now let’s look at the skip watch. The jumper seems to be a relatively unfamiliar data structure for us, but it is one of the more commonly used data structures in Redis. The underlying essence of a hopelist is an ordered linked list that can be searched in binary order. And add an index layer to the base of the list. It supports dynamic operation such as insert and delete as well as efficient query by interval. The time complexity of search, insert and delete is O(logn). To understand a skip list, let’s look at linked lists. Assuming that linked lists store ordered data, if we want to query one piece of data, in the worst case, we have to go through the entire list from scratch in order n time. As shown in the figure below, a skip list is an index layer on top of a linked list. Can play the role of support interval query. As shown in the figure above, if we want to query a node of 26, the hop table can traverse the index layer first. When traversing the 21 nodes in the index layer, it will find that the next node of the index layer is 36 nodes. Obviously, the node of 26 to be searched is in this interval. At this point, we only need to move down to the original chain layer through the index layer pointer to the original list, and we only need to traverse 2 nodes to find 26. If you had to traverse 10 nodes using the old list, now you only traverse 8 nodes. In the picture below, a picture is worth a thousand words. When the amount of data is large, a linked list containing multiple nodes can be highlighted to see the advantages of the index layer after the establishment of a five-level index. At the same time, note the rule that “with a layer of index, the number of nodes that need to be traversed is reduced and the query efficiency is improved.” (From the user’s point of view, the skip list guy is basically telling the linked list where to start looking faster.) Seeing this, it seems that a jumper is also a good data structure to use as an index. But don’t forget the first condition, “do as little I/O on the disk as possible.” However, the hop table obviously fails to meet this condition. As the amount of data in the hop table increases, the index layer will also increase, which will increase the number of I/O reads, thus affecting performance. So let’s go back to the question of “Of all the data structures, why a tree?” We found that hash tables and skip tables do not well meet the two main conditions for solving disk pain points and indexing purposes. So let’s see why we chose a tree. Tree structure Let’s take a look at the components of a tree in real life, starting with roots, branches, and leaves. The tree structure is also abstracted into the same, the top of the tree structure is the root node (root), the left node is called the left subtree, the right subtree corresponds to the node on the right, the end of the tree without nodes is called the leaf node. The hierarchy of the tree can be divided into upper and lower levels and same level nodes. As shown in the figure below, D and E are the child nodes of NODE B, so node B is their parent node, and node C, which is on the same level with node B, is the brother node of node B. At the same time, the maximum number of levels of the tree is called the height (depth) of the tree. The height of the tree in the figure is 3. In terms of the hierarchy of the tree structure, in fact, the tree structure is not a bit similar to the previous hop list. The reason why jumpers are so fast is because they have an index layer that can be queried efficiently by interval. The tree structure determines the nature of the traversal of the data itself to support the interval query. Plus, trees have the advantage of being non-linear compared to linear arrays, which don’t have to be contiguous. So when a tree inserts new data, it doesn’t have the overhead of moving all the data nodes behind it, like when an array inserts new data. Therefore, the tree structure is more suitable for inserting dynamic data structures such as updates. Tree structure in order to meet the index purpose and other conditions, as to reduce the pain point of disk query operation in fact, we can choose the data structure based on tree structure. That many tree structures? Why B+ trees? Out of all the tree structures, besides B+ trees, what other tree structures can you think of? Binary tree search tree, self-balanced binary tree, B tree. Binary tree Before understanding binary search tree or self-balanced binary tree, we need to simply know what is binary search tree, what is binary search tree. Because if you look at a binary search tree it’s just a combination of these two trees. Let’s take a look at a binary tree. The tree structure of a binary tree defines that each node can have 0 or 1 child node, but no more than 2 child nodes. There are two forms of binary tree: full binary tree and complete binary tree Full binary tree A full binary tree is defined as a tree in which all non-leaf nodes have left and right child nodes, and all child nodes are at the same level. Complete binary tree A complete binary tree is defined as a complete binary tree if all the nodes of the tree are the same as the nodes of a full binary tree of the same depth. The diagram below. Binary search tree Next we will simply look at the binary search tree, the binary search tree of the “two” is not the two, because the “two” that can be said to represent the binary tree, can also represent binary search, because the binary search tree is binary also fusion binary search. First let’s take a look at binary search, binary search can avoid ordered array traversal query, because we know that this case if you want to find a number of the worst case is O(n) complex, the overall query efficiency is not high. If the array is ordered, the query scope can be halved by binary lookup, which is O(logn). As shown in the figure below. Therefore, the binary search tree is different from the ordinary binary search tree, which puts the elements smaller than the root node in the left subtree, while the right subtree is the opposite of the element magnified to the root node. (In plain English, the root node is the median of the left and right subtrees, the left is less than the median, and the right is larger than the median, which is not a binary search algorithm.) As shown in figure, the binary search tree in finding the data, the only need to find elements, comparing with the tree node element, when the element is greater than the root node is the right subtree search, element is less than the root node is the left subtree search, element if happens to be the median is exactly the root node, the binary search trees have efficient query. However, binary trees also have obvious disadvantages. In extreme cases, if the smallest or largest element is inserted each time, the tree structure will degenerate into a linked list. The time complexity of querying the data is O(n), as shown below. When binary lookup trees degenerate into linked lists, we all know that linked lists are not only inefficient queries but also add disk IO operations, so we have to cross the next tree data Self-balanced binary tree Self-balanced binary tree is to solve the problem that binary search tree degenerates into a linked list under the extreme. Self-balanced binary search tree is also called balanced binary search tree (AVL tree). (You can see the evolution from simple binary trees to binary lookup trees to self-balanced binary trees. A simple thing gradually becomes more complex. If we only know the answer, we won’t know the whole story. In fact, balanced binary search tree is to add a constraint on the basis of binary search tree: make the height difference between the left and right subtrees of each node not more than 1. In this way, the left and right subtrees are balanced, and the time complexity of the query data operation is O(logn). As shown in the figure below, a balanced binary lookup tree maintains self-balancing element data for each insertion. A comparison of ordinary non-binary trees and balanced binary trees is shown in the figure below. Of course, there is the red-black tree common in Java collection classes, which is also a self-balanced binary tree. However, no matter whether a self-balanced tree is a balanced binary search tree or a red-black tree, the height of the tree will increase as the number of inserted elements increases, which also means that the number of disk I/O operations will increase, affecting the overall query efficiency. B tree We usually see B+ trees and B- trees, we can’t help but read B- tree as “B minus tree “, but the B- tree is just a hyphen, so B- tree is called B tree. Self-balancing binary tree while to find the time complexity of O (logn), the front also said it is itself a binary tree, each node can only have two child nodes, then increases as the amount of data, the number of nodes, the tree height also increases (that is, the deeper the depth of the tree), increase the number of disk I/O, affect the query efficiency. So if you look at the progression of binary trees in the tree structure, binary trees create new bugs (or pain points) every time they solve a new problem. Just as Lu Xun once said, “when a problem is solved, new problems will appear.” Seeing this, it’s not hard to guess that the emergence of B trees solves the problem of tree height. The b-tree, rather than the “XXX binary tree” in its name, is no longer limited to two children in a parent node, but allows M children (M > 2). Moreover, a node of a B-tree can store multiple elements, which reduces the height of the tree compared to the previous binary tree data structures. The nodes of a B-tree can contain multiple byte points, so a B-tree is a multi-fork tree, and the maximum number of child nodes contained in each node is called the order of the b-tree. The figure below is a b-tree of order 3. Each node in the figure above is called a page. The basic unit of data read in mysql is a page, and a page is the disk block we mentioned above. The P node in the disk block is a pointer to the child node. Pointers are present in the tree structure, as they were in the previous binary tree. So let’s take a look at the figure above. What happens when a b-tree of order 3 looks for elements of 90? Start from the root node, that is, disk block 1. Determine that 90 is between 17 and 35. Locate disk block 4 using pointer P3 in disk block 1. Again, compare between 65 and 87 in disk block 4, and finally find disk block 11 in pointer P3 of disk 4. I’m going to find a key that matches 90. It can be found that when a third-order B-tree searches for leaf nodes, the tree height is only 3, so the search process only needs 3 disk I/O operations at most. It may not be true when the data is small. However, when the amount of data is large, the number of nodes will also increase. In the case of the self-balanced binary tree, the tree can only have two leaves at most and can only expand child nodes vertically. Therefore, the height of the tree will be very high, which means more disk I/O operations are required. B trees, on the other hand, can reduce the height of the tree by extending nodes horizontally, so the efficiency is naturally higher than that of binary trees. (To put it bluntly: to get fat.) Seeing this, I’m sure you also know that if B trees are so good, there’s nothing going on with B+ trees. And then, why don’t we use B plus trees instead of B trees? As you can see, the B-tree already satisfies all of our requirements, reducing disk I/O operations and supporting interval lookups. Note, however, that while b-trees support interval lookups, they are not efficient. For example, in the above example, b-tree can efficiently query the value of 90 through equivalent value, but it is not convenient to query the results of all numbers within the range of 3 to 10 within a period. Since the middle order traversal is required for b-tree range query, the parent node and child node need to constantly switch back and forth, involving multiple nodes, which brings a lot of burden to disk I/O. B + tree B+ tree can be seen from the symbol of + is the upgraded version of B tree, the index underlying data structure of innoDB engine in MySQL uses B+ tree. A B+ tree performs an upgrade over a B tree: none of the non-leaf nodes in a B+ tree store data, but serve only as indexes. The leaf node holds all the data for the entire tree. Leaf nodes form an orderly linked list from small to large, which points to adjacent leaf nodes, that is, an ordered bidirectional linked list is formed between leaf nodes. The structure of B+ tree is shown below. (B+ tree is not a bit like the front of the hop table, data is data, the upper layer is formed by the bottom interval index layer, but it is not like the hop table is longitudinal expansion, but horizontal expansion of the “hop table”. This has the benefit of reducing disk I/O and improving range lookup efficiency. Next, let’s look at the insertion and deletion of THE B+ tree. The B+ tree has a large number of redundant nodes. It can be found from the above that all the elements of the parent node will appear in the child node, so when deleting a node, it can be directly deleted from the leaf node, which is faster. Compared with B+ trees, B trees have no redundant nodes and complex tree deformation will occur when nodes are deleted, while B+ trees have redundant nodes and will not involve complex tree deformation. The same is true for B+ tree insertions, which involve at most one branch path of the tree. B+ trees also do not need more complex algorithms, like black mangrove rotation to automatically balance. It is a question from the title of the article. We did not directly answer why MySQL uses B+ tree as index. Instead, we asked two questions to me, but I don’t know if I have any questions to you. Another question is why B+ trees, of all the tree structures? To get the result, we need to know the cause, we start from two aspects, because MySQL data is placed on disk, and disk processing speed is millisecond level, if the disk IO do too many queries, will bring the query burden, so to minimize the disk I/O operation to do the query. The other is from the primary purpose of the index itself, to be able to range lookup efficiently by interval. With the cause, we can start to explore the effect, so we can answer the first question, “Of all the data structures, why a tree?” Among other data structures linear by logical structure are hash tables and hop tables. Hash table base on array, although can be efficient query, but only equivalent query, and cannot support range query. The bottom layer of the hop table is a linked list. Efficient interval query can be realized through the index layer, but with the increase of data volume, the index layer also increases with the increase of data volume. So the use of tree data structure, tree structure of its characteristics determine the traversal of data itself is purely natural support by the interval query. The tree structure does not have the overhead of a linear array for insertion and other operations, so it is more suitable for inserting and updating data structures for dynamic operations. Then we asked, “Why B+ trees, of all trees?” From the binary tree structure, the parent node can only have two child nodes at most, and then from the binary tree plus binary search tree, the binary tree shows efficient query ability. But binary search trees degenerate into linked lists in extreme cases. So step up to a self-balanced binary tree, where the difference between the left and right subtrees of each node cannot be greater than 1. However, a binary tree can contain a maximum of two child nodes. Therefore, a high tree height may cause excessive I/O query operations on disks. So we finally get to B tree, which is a multi-fork tree, but it can only be efficient single query, not efficient interval query. B+ tree is an upgrade of B tree. All non-leaf nodes are used for indexing. Only leaf nodes store data and are ordered bidirectional linked lists. [DEL:Well, that’s all for today.:DEL] Okay, actually, it’s been here for a couple of weeks. I am a fierce and agile seed, afraid of what infinite truth, into an inch, have into an inch of joy. Welcome to: follow, like, favorites and comment, and we’ll see you next time! The article continues to update, you can search wechat “a fierce and agile seed” to read the first time.	{"url":"https://dev.mo4tech.com/why-does-mysql-use-b-trees-as-indexes.html","timestamp":"2024-11-14T15:07:05Z","content_type":"text/html","content_length":"93980","record_id":"<urn:uuid:e2463c8a-7e75-49f4-a4b1-d5be89e0835a>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00698.warc.gz"}
Listed below are the budgets (in millions of dollars) and the gross receipts (in millions of... Listed below are the budgets (in millions of dollars) and the gross receipts (in millions of... Listed below are the budgets (in millions of dollars) and the gross receipts (in millions of dollars) for randomly selected movies. Construct a scatterplot, find the value of the linear correlation coefficient r, and find the P-value using Is there sufficient evidence to conclude that there is a linear correlation between budgets and gross receipts? Do the results change if the actual budgets listed are $65 comma 000 comma 00065,000,000, $86 comma 000 comma 00086,000,000, $53 comma 000 comma 00053,000,000, and so on? Budget (x) 6565 8686 5353 3232 203203 103103 9393 Gross (y) 6969 6262 4848 5656 513513 143143 4747 What are the null and alternative hypotheses? Construct a scatterplot The linear correlation coefficient r is The test statistic t is The P-value is Because the P-value is ____ (greater or less) than the significance level 0.05 there ____ (is or is not) sufficient evidence to support the claim that there is a linear correlation between between budgets and gross receipts for a significance level of α=0.05 Do the results change if the actual budgets listed are $65,000,000 $86,000,000, $53,000,000, and so on? (options listed below) A. Yes, the results would change because it would result in a different linear correlation coefficient. B. No, the results do not change because it would result in a different linear correlation coefficient. C. Yes, the results would need to be multiplied by 1,000,000. D. No, the results do not change because it would result in the same linear correlation coefficient. Please check the given data is the same or not as your data values are repeated twice. Please do the comment I will edit my answer if there is any change. Budget(x) Gross(y) What are the null and alternative hypotheses? H1:p not equal to 0 Construct a scatterplot The linear correlation coefficient r is 0.926 The test statistic t is 30.058 The P-value is 0.003 Because the P-value is ( less) than the significance level 0.05 there ____ (is) sufficient evidence to support the claim that there is a linear correlation between between budgets and gross receipts for a significance level of α=0.05 Do the results change if the actual budgets listed are $65,000,000 $86,000,000, $53,000,000, and so on? (options listed below) A. Yes, the results would change because it would result in a different linear correlation coefficient Regression Analysis r² 0.857 r 0.926 Std. Error 70.486 n 7 k 1 Dep. Var. Gross(y) ANOVA table Source SS df MS F p-value Regression 1,49,338.6036 1 1,49,338.6036 30.058 .0028 Residual 24,841.3964 5 4,968.2793 Total 1,74,180.0000 6 Regression output confidence interval variables coefficients std. error t (df=5) p-value 95% lower 95% upper Intercept -125.0177 Budget(x) 2.8553 0.5208 5.483 .0028 1.5166 4.1941	{"url":"https://justaaa.com/statistics-and-probability/172276-listed-below-are-the-budgets-in-millions-of","timestamp":"2024-11-06T02:00:10Z","content_type":"text/html","content_length":"61463","record_id":"<urn:uuid:cffb329c-a97d-4980-8a2b-1c40b72d669e>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00320.warc.gz"}
"Noctua 2 Supercomputer" Carsten Bauer, Tobias Kenter, Michael Lass, Lukas Mazur, Marius Meyer, Holger Nitsche, Heinrich Riebler, Robert Schade, Michael Schwarz, Nils Winnwa, Alex Wiens, Xin Wu, Christian Plessl, and Jens Simon Journal of Large-Scale Research Facilities 9 PC2 / NHR "Bridging HPC Communities through the Julia Programming Language" Valentin Churavy, William F Godoy, Carsten Bauer, Hendrik Ranocha, Michael Schlottke-Lakemper, Ludovic Räss, Johannes Blaschke, Mosè Giordano, Erik Schnetter, Samuel Omlin, Jeffrey S. Vetter, and Alan Edelman PC2 - MIT - ORNL - NERSC - HLRS - CSCS - and more "Parallel Quantum Chemistry on Noisy Intermediate-Scale Quantum Computers" Robert Schade, Carsten Bauer, Konstantin Tamoev, Lukas Mazur, Christian Plessl, and Thomas D. Kühne Phys. Rev. Research 4, 033160 PC2 / NHR "Identification of Non-Fermi Liquid Physics in a Quantum Critical Metal via Quantum Loop Topography" George Driskell, Samuel Lederer, Carsten Bauer, Simon Trebst, and Eun-Ah Kim Phys. Rev. Lett. 127, 046601 Cologne - Cornell PhD thesis: "Simulating and machine learning quantum criticality in a nearly antiferromagnetic metal" Advisor: Prof. Dr. Simon Trebst Thesis PDF, Defense Talk "Fast and stable determinant Quantum Monte Carlo" Carsten Bauer SciPost Phys. Core 2, 2 (source code @ GitHub) "Hierarchy of energy scales in an O(3) symmetric antiferromagnetic quantum critical metal: a Monte Carlo study" Carsten Bauer, Yoni Schattner, Simon Trebst, and Erez Berg Phys. Rev. Research 2, 023008 (source code @ GitHub) Cologne - Stanford - Weizmann "Machine Learning Transport Properties in Quantum Many-Fermion Simulations" (record entry) Carsten Bauer, Simon Trebst In NIC Symposium 2020, Vol. 50, pp. 85–92, Forschungszentrum Jülich GmbH Zentralbibliothek, Verlag "Probing transport in quantum many-fermion simulations via quantum loop topography" Yi Zhang, Carsten Bauer, Peter Broecker, Simon Trebst, and Eun-Ah Kim Phys. Rev. B 99, 161120(R), Editors' Suggestion Cologne - Cornell "Nonperturbative renormalization group calculation of quasiparticle velocity and dielectric function of graphene" Carsten Bauer, Andreas Rückriegel, Anand Sharma, and Peter Kopietz Phys. Rev. B 92, 121409(R) Master's thesis: "Quasi-particle velocity renormalization in graphene" Invited talk @ University of Cologne: "Quasi-particle velocity renormalization in graphene" Advisor: Prof. Dr. Peter Kopietz "Microwave-based tumor localization in moderate heterogeneous breast tissue" Jochen Moll, Carsten Bauer, and Viktor Krozer International Radar Symposium (Dresden, Germany), pp.877-884	{"url":"https://carstenbauer.eu/","timestamp":"2024-11-02T23:42:36Z","content_type":"text/html","content_length":"39233","record_id":"<urn:uuid:237b6766-64fd-4424-8ccf-3a00534ffd94>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00626.warc.gz"}
1000 - A+B Calculate a + b The input will consist of a series of pairs of integers a and b,separated by a space, one pair of integers per line. For each pair of input integers a and b you should output the sum of a and b in one line,and with one line of output for each line in input. sample input sample output	{"url":"http://hustoj.org/problem/1000","timestamp":"2024-11-03T20:19:33Z","content_type":"text/html","content_length":"7414","record_id":"<urn:uuid:0c60b248-0772-44d1-ae6f-721027daef8e>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00322.warc.gz"}
Static transition probability analysis under uncertainty Deterministic gate delay models have been widely used to find the transition probabilities at the nodes of a circuit for calculating the power dissipation. However, with progressive scaling down of feature sizes, the variations in process parameters increase, thereby increasing the uncertainty in gate delay. In this work, we propose a novel non-simulative scheme to compute the transition probability waveforms (TPWs) in a single pass of the circuit for continuous gate delay distributions. These TPWs are continuous functions of time as opposed to the deterministic delay case where transitions are constrained to occur at discrete time points. The TPWs are then used to calculate the dynamic power dissipation in a circuit. We show that the corresponding power estimates obtained from deterministic delay models can be off by as much as 75%. Our method has an average error of only 6% and a speed up of 232× when compared to logic simulations. Another important application of our TPWs is in the area of crosstalk noise where the likelihood of signals switching within a certain timing window is required. ASJC Scopus subject areas • Hardware and Architecture • Electrical and Electronic Engineering Dive into the research topics of 'Static transition probability analysis under uncertainty'. Together they form a unique fingerprint.	{"url":"https://nyuscholars.nyu.edu/en/publications/static-transition-probability-analysis-under-uncertainty","timestamp":"2024-11-06T10:24:27Z","content_type":"text/html","content_length":"48907","record_id":"<urn:uuid:35426c15-33bd-4f12-b25c-e144611be9de>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00255.warc.gz"}
benderscut_nogood.h File Reference Detailed Description Generates a no-good cut for solutions that are integer infeasible. Stephen J. Maher The no-good cut is generated for the Benders' decomposition master problem if an integer solution is identified as infeasible in at least one CIP subproblems. The no-good cut is required, because the classical Benders' decomposition feasibility cuts (see benderscut_feas.c) will only cut off the solution $\bar{x}$ if the LP relaxation of the CIP is infeasible. Consider a Benders' decomposition subproblem that is a CIP and it infeasible. Let $S_{r}$ be the set of indices for master problem variables that are 1 in $\bar{x}$. The no-good cut is given by \[ 1 \leq \sum_{i \in S_{r}}(1 - x_{i}) + \sum_{i \notin S_{r}}x_{i} \] Definition in file benderscut_nogood.h. Go to the source code of this file.	{"url":"https://scipopt.org/doc-9.0.0/html/benderscut__nogood_8h.php","timestamp":"2024-11-02T00:08:49Z","content_type":"text/html","content_length":"11753","record_id":"<urn:uuid:b10a65c8-643a-41d7-90dd-8cc82676b351>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00594.warc.gz"}
Indicative uncertainty intervals for the admin-based population estimates: July 2020 1. Measuring statistical uncertainty for the admin-based population estimates We have been working in collaboration with Professor Peter Smith of the University of Southampton to develop measures of statistical uncertainty for Office for National Statistics (ONS) population estimates. "Uncertainty" refers to the quantification of doubt about the estimates. Uncertainty measures for the mid-year local authority population estimates were published in 2017. The methods are described in Methodology for measuring uncertainty in ONS local authority mid-year population estimates: 2012 to 2016. The methodology uses a range of statistical bootstrapping techniques to create plausible simulated distributions for the population estimates. Bootstrapping is a method for assigning measures of accuracy to sample estimates (Efron and Tibshirani, 1993). Uncertainty intervals are taken from the simulated estimates. We have built upon and further developed these methods to derive indicative uncertainty intervals for the experimental admin-based population estimates (ABPEs V3.0). We use a benchmark approach, comparing the ABPEs against the 2011 Census. Ultimately, the intention is that statistical uncertainty for the ABPEs will be derived through a Population Coverage Survey. The ABPEs consist of estimates of the population size produced through linkage of administrative data, including Pay As You Earn (PAYE) and tax credits data, national benefits and Housing Benefit data, English and Welsh school census, Higher Education Statistics Agency (HESA), and data from the Patient Register (PR) and the Personal Demographics Service (PDS). Methods for the production of the ABPEs are described in the Principles of ABPE V3.0 methodology. Back to table of contents We use the following model: P[i,j,k] is the true, unknown population size for local authority (LA), i=1, ... , 348, sex, j=1,2, and age, k=0, ... , 90 ABPE[i,j,k] is the ABPE estimate LSF[i,j,k] is a log-scaling factor (LSF) – a measure of bias in the ABPE estimate is the error. By taking the exponential of both sides of this equation, the model can be written as: We make the simplifying assumption that the error associated with the census is an order of magnitude smaller than the error associated with the ABPE, and therefore set it to zero. Thus: To estimate the log-scaling factor LSF[i,j,k], we compare the 2011 ABPE estimate, abpe[i,j,k], with the 2011 Census, denoted census[i,j,k], as an external benchmark representing the “true” population Thus, we obtain the observed LSF: When interpreting the LSFs, a value greater than zero indicates that fewer people have been captured by the ABPE than in the census data (ABPE undercount). Conversely, a value less than zero indicates that more people have been captured by the ABPE data than in the census data (ABPE overcount). To create plausible simulated values for the LSFs, we need to estimate the error distribution, F. We make the assumption that the variability between LAs within a group of similar LAs is a good proxy for within-LA variability for the LAs within that cluster. We identify groups of similar local authorities through a cluster analysis of the LSF profiles by age and sex. The cluster analysis is run separately for males and females. For each group of local authorities, a Generalised Additive Model (GAM) is fitted to the LSFs, generating predicted and residual values. The residuals are resampled to simulate an error distribution for each LA, by age and sex. This distribution of errors is added to the observed LSFs, and then back-transformed to generate a distribution of plausible population estimates. A step-by-step summary of the entire methodology is provided in Annex A. Back to table of contents Bear in mind the following caveats when considering these indicative uncertainty intervals: • These are interim measures of uncertainty, in place until the ABPE methodology is finalised. • This method does not allow for uncertainty in the 2011 Census estimates. • Nor does it allow for bias in the ABPEs to change as the intercensal decade progresses; instead, it assumes that bias observed with reference to the 2011 Census is propagated through the decade (further refinement may be possible once the ABPE methodology is finalised). • There is some circularity in this interim approach, since ABPEs have been tuned to 2011 Census estimates. Back to table of contents 5. Fitting the Generalised Additive Model Generalised Additive Models (GAMs) are fitted to the log-scaling factors (LSFs) by cluster. This semi-parametric approach was chosen because Rogers-Castro models that were used in mid-year estimate (MYE) uncertainty research were found to be a poor fit for LSF distributions, particularly for females. The GAMs generate fitted values denoted and corresponding residuals, denoted r[i,j,k]. The clusters are denoted c=1, ... , 11, where clusters 1 to 5 correspond to females (four normal clusters and one for the outlier local authorities (LAs)) and 6 to 11 correspond to males (five normal, one outlier). An example of the data set following all the above processes is shown in Table 2. For fitting the GAM, 22 knots are used for the age range of 0 to 90 years and over (one extra knot for ages 86 to 90 years, and an additional knot for age 90 years and over to account for the sharp difference in LSFs between ages 89 to 90 years and over). The knots are placed strategically to allow for more flexibility in the curve at minima and maxima. The knot locations were determined by: • Starting with equally spaced knots (current best) • Always keeping knots at age 0 to 90 years • Randomly selecting 20 ages between 1 year and 89 years • Fitting GAM • Calculating a goodness of fit statistic (AIC) • If the new AIC is better than the current best, then making this knot selection the current best • Repeating 10,000 times (or as much as possible, given time constraints) Table 2: An overview of the data set following the clustering and GAM fitting processes. Cluster LA Sex Age Census ABPE lsf Fitted value r Amber Valley F 0 642 673 -0.047 -0.026 -0.021 3 Amber Valley F 1 631 650 -0.030 0.024 -0.054 …. …. …. …. …. …. …. …. Amber Valley F 90 (+) 816 790 0.032 -0.014 0.046 ... …. …. …. …. …. …. …. …. Wycombe F 0 1150 1161 -0.010 0.013 -0.022 4 Wycombe F 1 1056 1046 0.010 0.025 -0.015 …. …. …. …. …. …. …. …. Wycombe F 90 (+) 878 872 0.007 0.003 0.004 …. …. …. …. …. …. …. …. Download this table Table 2: An overview of the data set following the clustering and GAM fitting processes. .xls .csv Back to table of contents We begin by defining a group as a unique combination of cluster, sex and age. For four normal clusters for females and five for males, plus one outlier cluster for each sex, there is a total of 11 clusters. Since there are 91 ages, there is a total of n = 11 x 91 = 1001 groups, each containing between 5 and 121 residuals (the number of local authorities (LAs) in the corresponding cluster). We standardise the residuals so that the variance for each group is one, sample 1,000 residuals (with replacement) then reverse the standardisation. To standardise the residuals, denoted s[i,j,k], we divide each residual, r[i,j,k], by its within-group standard deviation, σ[c,j,k], that is, We calculate standardised residuals for each group in the “normal” clusters (1 to 4 and 6 to 10), creating 819 groups each with a variance of one. We assume that each group of standardised residuals follows the same underlying distribution^1, and put them together in one pot (“Pot 1”). The outlier clusters (5 and 11) are treated differently because of their unusual log-scaling factor (LSF) profiles. Instead of standardising their residuals, we assume the standard deviation is constant across all groups (within a cluster). Each group contains only five residuals, which is insufficient to obtain an accurate estimate of the standard deviation. Therefore, we allocate the raw residuals from cluster 5 to “Pot 2”, and from cluster 11 to “Pot 3”. The distributions of the residuals in each pot are shown in Figures 5a, 5b and 5c for LAs in the “normal” clusters, in the female outlier cluster and in the male outlier cluster, respectively. Figure 5a: Distribution of the residuals in “Pot 1” (four “normal” clusters for females and five for males) England and Wales, 2011 Source: Office for National Statistics Download this chart Figure 5a: Distribution of the residuals in “Pot 1” (four “normal” clusters for females and five for males) Image .csv .xls Figure 5b: Distribution of the residuals for the female outlier cluster England and Wales, 2011 Source: Office for National Statistics Download this chart Figure 5b: Distribution of the residuals for the female outlier cluster Image .csv .xls Figure 5c: Distribution of the residuals for the male outlier cluster England and Wales, 2011 Source: Office for National Statistics Download this chart Figure 5c: Distribution of the residuals for the male outlier cluster Image .csv .xls For each group in the normal clusters (1 to 4 and 6 to 10) 1,000 standardised residuals are resampled (with replacement) from Pot 1. The selected residuals are un-standardised by multiplying the selected residuals by the group standard deviation, σ[c,j,k]. For each group in clusters 5 and 11, 1,000 residuals are sampled (with replacement) from Pot 2 and 3, respectively. Since these residuals were not standardised, they do not need to be Each set of 1,000 residuals is a simulated error distribution for all the LAs in the group, that is, it is an estimate of in the first equation in Section 2. They are then added to the observed LSFs^2, lsf[i,j,k], to generate 1,000 simulated LSFs for each LA. From the second equation in Section 2, the 1,000 simulated LSFs are back-transformed by exponentiating them to obtain 1,000 plausible scaling factors, and then multiplied by the published ABPE for LA i. Doing so, we obtain a simulated distribution of population estimates for each combination of sex, j, and age, k, within each LA, i. The simulated distributions of population estimates are used to select an empirical 95% uncertainty for the true population size. The uncertainty interval is defined by the 2.5th and 97.5th percentile of the 1,000 simulated estimates. Width of uncertainty intervals The relative width of an uncertainty interval (UI) is calculated as 100*(upper bound – lower bound)/ABPE. The larger uncertainty interval widths are a direct result of the larger group standard Figures 6a and 6b show the relative width of uncertainty intervals by age for local authorities in the normal clusters, for females and males, respectively. Figures 6c and 6d show the equivalent results for local authorities in the outlier clusters. They show that the intervals are larger and more variable in the outlier clusters than in the normal clusters. This is because the outlier clusters contain local authorities whose scaling factor patterns vary greatly within cluster, and therefore the observed standard deviations are large. Uncertainty is typically higher for males than for females. Uncertainty varies by age, typically highest for student and young adult ages. Figure 6a: Relative widths of the uncertainty intervals by age, for females - normal clusters (1 to 4) England and Wales, 2011 Source: Office for National Statistics Download this image Figure 6a: Relative widths of the uncertainty intervals by age, for females - normal clusters (1 to 4) .png (35.9 kB) Figure 6b: Relative widths of the uncertainty intervals by age, for males - normal clusters (6 to 10) England and Wales, 2011 Source: Office for National Statistics Download this image Figure 6b: Relative widths of the uncertainty intervals by age, for males - normal clusters (6 to 10) .png (34.1 kB) Figure 6c: Relative widths of the uncertainty intervals by age, for females - outlier cluster (5) England and Wales, 2011 Source: Office for National Statistics Download this image Figure 6c: Relative widths of the uncertainty intervals by age, for females - outlier cluster (5) .png (15.4 kB) Figure 6d: Relative widths of the uncertainty intervals by age, for males - outlier cluster (11) England and Wales, 2011 Source: Office for National Statistics Download this image Figure 6d: Relative widths of the uncertainty intervals by age, for males - outlier cluster (11) .png (16.4 kB) Location of the ABPEs in their uncertainty intervals We find that 15,689 (25%) ABPEs in the normal clusters and 69 (7.5%) in the outlier clusters lie outside their uncertainty interval (UI). This occurs when the middle 95% of the simulated scaling factors do not include 1. For them to contain 1, the corresponding simulated LSFs must contain 0 (log(1) = 0), and this can only happen if σ is large enough to extend the distribution sufficiently far out from the observed LSF. The census estimate always lies inside the uncertainty interval. Figure 7a shows the distribution of the positions of the published ABPEs in their uncertainty intervals. The position is calculated as where L is the lower bound of the uncertainty interval, and U is the upper bound. Therefore, a position of negative 1 means that the ABPE is the lower bound of the uncertainty interval, 0 means that the ABPE lies in the centre, and 1 means that the ABPE is the upper bound. Figures 7b and 7c show how often the ABPEs fall outside of their uncertainty intervals for each age, by sex. The ABPEs fall outside most frequently for males aged 20 years, of working age, and from age 30 years to retirement, for both sexes. Figure 7a: Distributions of the positions of the ABPEs within their uncertainty intervals, for all local authorities, ages and sexes England and Wales, 2011 Source: Office for National Statistics 1. Four extreme values from clusters 5 and 11 are not shown in the histogram (5.29, 15.44, 16.10, 19.86). Download this chart Figure 7a: Distributions of the positions of the ABPEs within their uncertainty intervals, for all local authorities, ages and sexes Image .csv .xls Figure 7b: The number of times the ABPE falls outside its uncertainty interval, by age for males England and Wales, 2011 Source: Office for National Statistics Download this chart Figure 7b: The number of times the ABPE falls outside its uncertainty interval, by age for males Image .csv .xls Figure 7c: The number of times the ABPE falls outside its uncertainty interval, by age for females England and Wales, 2011 Source: Office for National Statistics Download this chart Figure 7c: The number of times the ABPE falls outside its uncertainty interval, by age for females Image .csv .xls Figure 8 shows that for males aged 25 to 31 years, and 43 to 64 years in Richmondshire, the ABPE falls outside the 95% uncertainty interval. Figure 8: ABPE, census and empirical 95% uncertainty intervals for males Richmondshire, 2011 Source: Office for National Statistics Download this chart Figure 8: ABPE, census and empirical 95% uncertainty intervals for males Image .csv .xls Notes for Uncertainty intervals 1. This assumption is supported by statistical testing of distributional differences using the two-sample Kolmogorov-Smirnof test, with Bonferroni correction for multiple comparisons. 2. The 1,000 resampled residuals are added to the observed LSFs as opposed to the fitted LSFs. This is because observed LSF values reflect relatively stable structural characteristics of LAs that are likely to persist over time. Back to table of contents The ABPE uncertainty intervals currently assume that bias in the ABPEs is unchanged as the decade following the census progresses. The inclusion of contemporaneous covariates would allow us to update the model, to account for change in the ABPE/Census relationship over time. Explore the effects of clustering on the uncertainty intervals, including varying K (the pre-determined number of clusters), using hierarchical clustering, and choosing a subset of ages on which to base the clustering. We also propose that further work could look at local authority-level analysis of the statistical uncertainty (as shown in Figure 8) and how it varies over time, place, and demographic group. Back to table of contents 8. Annex A. Workflow summary for the calculation of the ABPE uncertainty intervals 1. Import Census and ABPE estimates for each local authority (LA), i = 1, ... , 348 by sex, j = 1,2,and single year of age, k = 0, ... , 90(+). 2. Calculate the observed log-scaling factors (LSFs), for all i,j,k 3. Identify unusual LAs separately for females and males based on the distance of their LSFs from the median LSFs and place them in two "outlier" clusters. 4. For each sex, cluster the remaining LAs based on similar patterns of LSFs across ages. 5. Fit a GAM to each cluster to obtain residuals, r[i,j,k]. 6. For each combination of “normal” cluster, sex and age (group), calculate the standard deviation of the residuals, σ[c,j,k]. 7. For normal clusters, calculate the standardised residuals, s[i,j,k], by dividing each of the residuals, r[i,j,k], by their group standard deviation, σ[c,j,k], i.e. 8. Put the standardised residuals into ”Pot 1”. 9. Resample 1,000 standardised residuals (with replacement) from Pot 1. 10. For each group in the normal clusters, un-standardise the selected residuals by multiplying the selected residuals by the group standard deviation, σ[c,j,k]. 11. Put the raw residuals from the female outlier cluster into “Pot 2”, and those from the male outlier cluster into “Pot 3”. 12. For each group in the female (male) outlier cluster resample 1,000 residuals (with replacement) from Pot 2 (3). 13. For each group in all clusters, add the 1,000 residuals to the observed LSFs, lsf[i,j,k], to generate 1,000 simulated LSFs for each LA. 14. To obtain a simulated distribution of population estimates for each LA by age and sex, exponentiate the 1,000 simulated LSFs (generating 1,000 plausible scaling factors), and then multiply each of them by the published ABPE. 15. For each LA by age and sex, use the simulated distribution of population estimates to calculate the uncertainty interval as follows: the lower and upper limits of the uncertainty intervals are the 2.5th and 97.5th percentile of the 1,000 simulated estimates. 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The minimum value of f(x)=\|3-x\|+\|2+x\|+\|5-x\| is -Turito Are you sure you want to logout? The minimum value of A. 0 B. 7 C. 8 D. 10 To find the minimum value of the given function. At x=3, f(x) = 10-x which is minimum. f(3) = 10-3 = 7 Hence, the minimum value of given function is 7. Get an Expert Advice From Turito.	{"url":"https://www.turito.com/ask-a-doubt/maths-the-minimum-value-of-f-x-3-x-2-x-5-x-is-10-8-7-0-qbb791062","timestamp":"2024-11-07T15:46:17Z","content_type":"application/xhtml+xml","content_length":"330823","record_id":"<urn:uuid:bb0191df-cfb9-4d99-8f01-d0f072e9768d>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00803.warc.gz"}
Center of Mass Gravity Most recent answer: 03/30/2018 Textbooks always state that the pull of earth's gravity on an object above its surface can be calculated by Newton's law assuming the mass of the earth is concentrated to a point at its centre. As this is not obvious, I decided to prove it to myself by calculus, summing the contribution from each point within the earth, but got a different and more complex answer. Can you comment? - Peter Fry (age 81) So long as you assume that the Earth is spherically symmetric, you should get Newton's answer. He obtained it the same way you tried, by integral calculus, with the extra step that he had to invent integral calculus to solve this problem. You can see immediately by symmetry that the direction must point toward the center. Calculus is used to show that the strength, for all points outside the Earth's mass, is the same as if all the mass were at the center. Mike W. (published on 03/30/2018)	{"url":"https://van.physics.illinois.edu/ask/listing/206510","timestamp":"2024-11-07T00:49:34Z","content_type":"text/html","content_length":"28052","record_id":"<urn:uuid:ec1b748e-ea3a-4668-813c-14e2e09a5e32>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00347.warc.gz"}
Course - USC Schedule of Classes Stochastic Calculus and Mathematical Finance (3.0 units) Stochastic processes revisited, Brownian motion, Martingale theory, stochastic differential equations, Feynman-Kac formula, binomial models, basic concepts in arbitrage pricing theory, equivalent Martingale measure. Recommended preparation: Math-225, Math-407. Duplicates credit in the former MATH-503. Section Session Type Time Days Registered Instructor Location Syllabus Info 39737R 001 Lecture 2:00-3:15pm Wed, Fri 23 of 40 Jin Ma GFS116	{"url":"https://classes.usc.edu/term-20233/course/math-530a/","timestamp":"2024-11-13T05:58:27Z","content_type":"text/html","content_length":"96210","record_id":"<urn:uuid:2df3e5be-5014-48e5-968b-bd2a324b2646>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00449.warc.gz"}
How to randomize direction in a specific area? Lets say I have a direction: (the line represents the direction our origin is facing) Now, how could I change this direction to a random one within a specific area? I don’t have much knowledge about directions etc. and would really appreciate help! :’) 2 Likes I’m not the smartest with transforming look vectors directly but I do know how to do it with CFrames! local spreadAng = 20 -- later converted to radians local r = math.rad(spreadAng) local look = Vector3.new(0,0,1) local origin = Vector3.new(0,10,0) local cf = CFrame.new(origin,origin + look) local randomUp = (math.random()2 - 1)r local randomRight = (math.random()2 - 1)r local transformedCF = cf * CFrame.Angles(randomUp,randomRight,0) The transformedCF is going to have your new look vector via CF.LookVector. If this doesn’t work you may need to specify the order of the CF like so: local transformedCF = cf * CFrame.Angles(randomUp,0,0) * CFrame.Angles(0,randomRight,0) Fun fact: (math.random()2 - 1) returns a number from -1 to 1, which is helpful in this case. An alternative way to get it (and I think may be better in your case) may be to rotate the CFrame along the Z and then shift it up or down, like so: local randomRot = math.random()math.pi2 local randomUp = (math.random()2 - 1)r local transformedCF = cf CFrame.Angles(0,0,randomRot) * CFrame.Angles(randomUp,0,0) 2 Likes For fun, I came up with a solution that uses trigonometry and does transformations on the directional vectors. I don’t know how much this is functionally different from @Locard’s solution, but here it is anyway: The Code --Returns a random CFrame in the area local function GenerateInArea(originCf, distance, areaRadius) local angle = math.random() * math.pi * 2 local radius = math.random() * areaRadius local lookOffset = originCf.LookVector * radius * math.cos(angle) local rightOffset = originCf.RightVector * radius * math.sin(angle) return originCf + originCf.LookVector * distance + lookOffset + rightOffset local part = Instance.new("Part") part.Name = "ResultPart" part.Anchored = true part.BrickColor = BrickColor.new("Really red") part.Size = Vector3.new(2, 2, 2) part.Parent = workspace --This will place the part in a random place within a circular area of 10 studs that is --20 studs away from the direction of the script.Parent (another part) part.CFrame = GenerateInArea(script.Parent.CFrame, 20, 10) The Math 5 Likes Hey a question how did you get into this deep of math? Did you learn this things on khan academy? 1 Like	{"url":"https://devforum.roblox.com/t/how-to-randomize-direction-in-a-specific-area/1207722","timestamp":"2024-11-05T09:58:30Z","content_type":"text/html","content_length":"31589","record_id":"<urn:uuid:570d1941-041a-4e45-aab9-647b54b4f507>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00358.warc.gz"}
Get started with Geostatistical Analyst in ArcGIS Pro Available with Geostatistical Analyst license. The Geostatistical Analyst extension contains a suite of tools that help you prepare your data for interpolation, fit an interpolation model, and validate the results of the interpolation. The two main components of Geostatistical Analyst are the Geostatistical Wizard and the Geostatistical Analyst toolbox. Geostatistical Wizard The Geostatistical Wizard is accessed through the Analysis Ribbon tab in ArcGIS Pro. The Geostatistical Wizard is a dynamic set of pages that is designed to guide you through the process of constructing and evaluating the performance of an interpolation model. Choices made on one page determine which options will be available on the following pages and how you interact with the data to develop a suitable model. The wizard guides you from the point when you choose an interpolation method all the way to viewing summary measures of the model's expected performance. A simple version of this workflow (for inverse distance weighted interpolation) is represented graphically below: A simple workflow in the Geostatistical Wizard Process of building an interpolation model Geostatistical Analyst includes many tools for analyzing data and producing a variety of output surfaces. While the reasons for your investigations may vary, you're encouraged to adopt the approach described in Geostatistical workflow when analyzing and mapping spatial processes: • Represent data—Create layers and display them in ArcGIS Pro. • Explore data—Examine the statistical and spatial properties of your datasets using, for example, the histogram chart. • Choose an appropriate interpolation method—The choice should be driven by the objectives of the study, your understanding of the phenomenon, and the available output surfaces. • Fit the model—Perform the interpolation, possibly configuring parameters to fit the statistical properties of your data. • Perform diagnostics—Check that the results are reasonable, and evaluate the output surface using cross-validation and validation. This helps you understand how well the model predicts the values at unsampled locations. The Geostatistical Analyst toolbox offers many interpolation methods. You should always have a clear understanding of the objectives of your study and how the predicted values (and other associated information) help you make more informed decisions when choosing a method. To provide some guidance, see Classification trees for a set of classification trees of the diverse methods. Interpolation methods available in Geostatistical Analyst The interpolation functions in Geostatistical Analyst can be divided into two categories: deterministic and geostatistical. Deterministic methods Deterministic techniques have parameters that control either the extent of similarity (for example, inverse distance weighted) of the values or the degree of smoothing (for example, radial basis functions) in the surface. These techniques are not based on a random spatial process model, and there is no explicit measurement or modeling of spatial autocorrelation in the data. Deterministic methods include the following: • Interpolation with barriers (using impermeable or semipermeable barriers in the interpolation process): Geostatistical methods Geostatistical techniques assume at least some of the spatial variation observed in natural phenomena can be modeled by random processes with spatial autocorrelation and require that the spatial autocorrelation be explicitly modeled. Geostatistical techniques can be used to describe and model spatial patterns (variography), predict values at unmeasured locations (kriging), and assess the uncertainty associated with a predicted value at the unmeasured locations (kriging). Learn more about kriging in Geostatistical Analyst Empirical Bayesian kriging is available as a geoprocessing tool. The tool can be used to produce the following surfaces: • Maps of kriging predicted values • Maps of kriging standard errors associated with predicted values • Maps of probability, indicating whether a predefined critical level was exceeded • Maps of quantiles for a predetermined probability level EBK Regression Prediction can be used to create empirical Bayesian kriging models that use explanatory variable rasters to improve the accuracy of the interpolation. GA Layer To Rasters can be used to export these models to the four output types described above. Empirical Bayesian Kriging 3D can be used to interpolate 3D points that have an x-, y-, and z-coordinate along with a measured value to interpolate. Geostatistical Analyst toolbox The Geostatistical Analyst toolbox includes tools for analyzing data, producing a variety of interpolation surfaces, examining and transforming geostatistical layers to other formats, performing geostatistical simulation and sensitivity analysis, and aiding in designing sampling networks. The tools have been grouped into five toolsets:	{"url":"https://pro.arcgis.com/en/pro-app/2.8/help/analysis/geostatistical-analyst/get-started-with-geostatistical-analyst-in-arcgis-pro.htm","timestamp":"2024-11-02T05:51:12Z","content_type":"text/html","content_length":"25460","record_id":"<urn:uuid:703bfd4a-37ac-4d3d-89b9-df0d06b0f21c>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00328.warc.gz"}
Potential future exposure formula parameters in the calculation (current exposure and potential future exposure) was seen to be calculating EAD because the SM used internal models. Potential future exposure (PFE) is a measure of risk in relation to default by a counter-party to a financial transaction. It begins from the assumption that the cost (RC) and the potential future exposure (PFE). Thus, EAD using the PFE terms, Equation (1) differs from EAD using CEM in two important respects:. Potential future exposure (PFE) is a measure of risk in relation to default by a counter-party to a financial transaction. It begins from the assumption that the cost (RC) and the potential future exposure (PFE). Thus, EAD using the PFE terms, Equation (1) differs from EAD using CEM in two important respects:. estimate of potential future exposure (PFE), minus the adjusted value of collateral . approach for calculating capital requirements for counterparty credit risk, For the modelling and calculation of EAD standard formulas; the so-called. CEM/SM or The potential future exposure (PFE) corresponding to the potential. under a single netting set within which partial or full offsetting may be recognised for the purposes of calculating the potential future exposure add-on;. Projecting the replacement cost into the future and is basically retrieving an expected exposure which is just the future's MtM (conditional on being positive). But that's a conditional mean. The PFE, like VaR, is a worst expected exposure with some confidence (probability) so it presupposes a distribution (analytical or simulated). Gregory's 7.1. makes the simplest possible assumption that that the exposure is normally distributed, so it's just solving for a normal quantile. So you are Potential future exposure is an estimate of the risk that subsequent changes in market prices could increase credit exposure. In measuring potential exposure, institutions attempt to determine how much a contract can move in to the money for the institution and out of the money for the counterparty over time. The Current Exposure Methodology is a key part of Leverage Ratio calculations. It dates back to the late 1980s and the first Basel accords on banking capital. CEM calculates the Potential Future Exposure of a derivative trade using a look-up table based on Asset Class and Maturity. CEM is a very simple, notional-based measure of derivatives risk. I have come across a risk measure called "Potential Future Exposure" and I have not really understood the meaning of it. Knowing that this has to do with counterparty credit risk, I read different 3 •Potential Future Exposure (PFE) is defined as the maximum expected credit exposure over a specified period of time calculated at some level of confidence. PFE is a measure of counterparty credit risk. •Expected Exposure (EE) is defined as the average exposure on a future date •Credit Valuation Adjustment (CVA) is an adjustment to the price of a derivative to take maximum potential exposure ($104M) for the position being considered (a 15-year power purchase agreement (PPA)) will occur in 2018. This date is the result of two opposing forces – increased price uncertainty the further out we look and roll off. The effects of roll off are easy enough to figure out, buthow is price uncertainty calculated? The Current Exposure Methodology is a key part of Leverage Ratio calculations. It dates back to the late 1980s and the first Basel accords on banking capital. CEM calculates the Potential Future Exposure of a derivative trade using a look-up table based on Asset Class and Maturity. CEM is a very simple, notional-based measure of derivatives risk. 14 Jan 2020 The structure of the calculation of potential future exposure is flexible and allows to add or delete elements where necessary. Third, the By calculating expected exposure over a range of future times, an exposure The counterparty's credit quality, on the other hand, is likely to be dependent on (Risk-neutral Pricing Formula) In an arbitrage-free complete market M, Potential future exposure (PFE) is the maximum exposure estimated to occur on a are considered in the calculation of the credit exposures). B. Current non-internal replacement cost and the potential future exposure add-on appropriate? IT systems are essential to the calculation of potential future exposure. (PFE). A good system can also automate predeal limit checking, which will significantly For the calculation of the potential future credit exposure according to the above [. ..]. 26 Nov 2018 Calculating the Exposure Amount of Derivative Contracts netting set and the “ potential future exposure” (PFE) of the contract or netting set. Projecting the replacement cost into the future and is basically retrieving an expected exposure which is just the future's MtM (conditional on being positive). But that's a conditional mean. The PFE, like VaR, is a worst expected exposure with some confidence (probability) so it presupposes a distribution (analytical or simulated). Gregory's 7.1. makes the simplest possible assumption that that the exposure is normally distributed, so it's just solving for a normal quantile. So you are Potential future exposure is an estimate of the risk that subsequent changes in market prices could increase credit exposure. In measuring potential exposure, institutions attempt to determine how much a contract can move in to the money for the institution and out of the money for the counterparty over time. The Current Exposure Methodology is a key part of Leverage Ratio calculations. It dates back to the late 1980s and the first Basel accords on banking capital. CEM calculates the Potential Future Exposure of a derivative trade using a look-up table based on Asset Class and Maturity. CEM is a very simple, notional-based measure of derivatives risk. with no analytical solution, such as calculation of potential future exposure (PFE), expected exposure (EE), and credit value adjustment (CVA), an efficient The expected exposure is the mean of all possible probability-weighted replacement costs estimated over the specified time horizon. This calculation may reflect a Hi Guys, Does anyone know what's the formula and advice available to calculate derivatives "PFE" exposures? E.g. Monte Carlo Thanks! Ordinarily when interest rates rise, the discount rate used in calculating the net Counterparty credit risk = (Current net exposure + Potential future exposure) - 19 Sep 2017 means potential future credit exposure can be significantly larger VaR calculations include a calculation of future price uncertainty. Can we For the calculation of the potential future credit exposure according to the formula in □ BIPRU 13.4.17 R perfectly matching contracts included in the netting Modeling Potential Future Exposure. 2 Calculating and Hedging Exposure, Credit Value Adjustment and Economic Capital for Counterparty Credit risk, Evan A part of the regulatory Capital and RWA (Risk weighted asset) calculation PFE (Potential Future Exposure), Average PFE, Effective PFE, indicating any ability 26 Nov 2018 Calculating the Exposure Amount of Derivative Contracts netting set and the “ potential future exposure” (PFE) of the contract or netting set. an amount for potential future changes in credit exposure calculated on the basis of the total notional principal amount of the contract multiplied by the following parameters in the calculation (current exposure and potential future exposure) was seen to be calculating EAD because the SM used internal models. Potential future exposure (PFE) is a measure of risk in relation to default by a counter-party to a financial transaction. It begins from the assumption that the cost (RC) and the potential future exposure (PFE). Thus, EAD using the PFE terms, Equation (1) differs from EAD using CEM in two important respects:. estimate of potential future exposure (PFE), minus the adjusted value of collateral . approach for calculating capital requirements for counterparty credit risk,	{"url":"https://digoptionefpvhdmf.netlify.app/osegueda85154nup/potential-future-exposure-formula-296","timestamp":"2024-11-14T15:29:10Z","content_type":"text/html","content_length":"33620","record_id":"<urn:uuid:51b5499e-1eac-46e5-9acd-2c913e66cba7>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00077.warc.gz"}
American Mathematical Society The problem is to approximate, with local second-order accuracy, the smooth boundary separating a black and a white region in the plane, given discretely located gray values associated with a blurring of that border. "Second-order", here, is with respect to the size h of the scale of the prescribed blurring. The locally determined approximations are line segments. The algorithms discussed here can result in second-order accuracy, but they may not in certain geometric circumstances. Typical local curvature estimates based on adjacent line segments do not converge, but an atypical one does. Consideration of a class of scaled blurrings leads to a type of blurring of borders which is particularly easy for a computer to undo locally, yielding a line which is locally second-order accurate. Some extensions to three (and more) dimensions are appended. References W. A. Beyer & B. Swartz, Halfway Points (a report in preparation), Los Alamos National Laboratory, Los Alamos, NM, 1989. C. de Boor, "Splines as linear combinations of B-splines," in Approximation Theory II (G. G. Lorentz, C. K. Chui &. L. L. Schumaker, eds.), Academic Press, New York, 1976, pp. 1-47. D. S. Carter, G. Pimbley &. G. M. Wing, On the Unique Solution for the Density Function in Phermex, (Declassified) Memo T-5-2023, Los Alamos Scientific Laboratory, Los Alamos, NM, 1957, 7 pp. R. B. DeBar, Fundamentals of the KRAKEN Code, Report UCID-17366, Lawrence Livermore Laboratory, Livermore, CA, 1974, 17pp. G. de Cecco, "Il teorema del sandwich al prosciutto," Archimede, v. 37, 1985, pp. 98-106. (Italian) V. Faber & G. M. Wing, The Abel Integral Equation, Report LA-11016-MS, Los Alamos National Laboratory, Los Alamos, NM, 1987, 48pp. C. Fenske, Math. Rev., 87h:54078, 1987. K. Höllig, "Multivariate splines," in Approximation Theory (C. de Boor, ed.), AMS Short Course Lecture Notes #36, Amer. Math. Soc., Providence, R.I., 1986, pp. 103-127. B. R. Hunt (editor), "Special image processing issue," Proc. IEEE, v. 69, 1981, pp. 499-655. A. Huxley, An Illustrated History of Gardening, Paddington Press (Grosset and Dunlap), New York, 1978. J. M. Hyman, "Numerical methods for tracking interfaces," in Fronts, Interfaces and Patterns (A. R. Bishop, L. J. Campbell & P. J. Channell, eds.), Elsevier, New York, 1984, pp. 396-407. B. D. Nichols & C. W. Hirt, Methods for Calculating Multi-Dimensional, Transient Free Surface Flows Past Bodies, Proc. First Internat. Conf. Numer. Ship Hydrodynamics, Gaithersburg, MD, 1975. W. F. Noh & P. Woodward, "SLIC (Simple Line Interface Calculation)," in Proc. Fifth Internat. Conf. on Numer. Methods in Fluid Dynamics, Lecture Notes in Physics (A. I. van der Vooren & P. J. Zandbergen, eds.), Springer-Verlag, New York, 1976, pp. 330-340. B. Swartz, The Second-Order Sharpening of Blurred Smooth Borders, Report LA-UR-87-2933, Los Alamos National Laboratory, 1987, 35pp. G. M. Wing, A Primer on Integral Equations of the First Kind, Report LA-UR-84-1234, 1984, 98pp. D. L. Youngs, private communication, 1978. D. L. Youngs, "Time-dependent multi-material flow with large fluid distortion," Numerical Methods for Fluid Dynamics (K. W. Morton & M. J. Baines, eds.), Academic Press, New York, 1982, pp. 274-285. D. L. Youngs, An Interface Tracking Method for a 3D Eulerian Hydrodynamics Code, Atomic Weapons Research Establishment Report AWRE-44-92-35, Aldermaston, Berks., 1987, 47pp. C. Zemach, T-Division, Los Alamos National Laboratory; private communication, 1988. Similar Articles • Retrieve articles in Mathematics of Computation with MSC: 65D99, 65P05 • Retrieve articles in all journals with MSC: 65D99, 65P05 Additional Information • © Copyright 1989 American Mathematical Society • Journal: Math. Comp. 52 (1989), 675-714 • MSC: Primary 65D99; Secondary 65P05 • DOI: https://doi.org/10.1090/S0025-5718-1989-0983313-8 • MathSciNet review: 983313	{"url":"https://www.ams.org/journals/mcom/1989-52-186/S0025-5718-1989-0983313-8/?active=current","timestamp":"2024-11-03T03:33:48Z","content_type":"text/html","content_length":"66241","record_id":"<urn:uuid:db0eab28-2f44-4ac6-be34-929773388022>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00074.warc.gz"}
[Discussion] How to estimate conditional probability (cdf) of multivariate dataset? [Discussion] How to estimate conditional probability (cdf) of multivariate dataset? I am sharing the problem I face in Matlab but if you have a solution for this problem even in Python then I would very very happy. I was able to estimate conditional probability (CDF) for a dataset that has two features (X_1 and Y) i.e., P(X_1\|Y) using a Matlab function called “quantilePredict”. It works great. However, when I consider three features X_1, X_2 and Y. Then how can I find the P(X_1,X_2\|Y) without the assumption of conditional independence? How to capture the covariance as well as the CDF while considering quantiles but not mean of the data? Worst case I am fine with how to capture the covariance as well as CDF with mean of the data? TreeBagger is trained (f) by giving “Y” as input and X_1 as output i.e., X_1 = f(Y). We then use the Treebagger model to predict responses for “quantilePredict” but in multivariate case, the Treebagger cannot fit the data where the input is “Y” and output has “X_1, X_2” i.e., Y = f(X_1,X_2) (this idea/pov is probably wrong and naive) ?. submitted by /u/askquestion001 [link] [comments]	{"url":"https://torontoai.org/2019/12/18/discussion-how-to-estimate-conditional-probability-cdf-of-multivariate-dataset/","timestamp":"2024-11-05T21:50:17Z","content_type":"text/html","content_length":"40261","record_id":"<urn:uuid:d2a3f1f3-5bf4-47b3-80ec-0e1920d74fca>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00332.warc.gz"}
the number of ways for distributing Magical Mathematics[Interesting Approach]> please please answer this... the number of ways for distributing "n" different things in to "r" different groups so that no group is left blank is given by?and what is the proof.please please say me this iitians 1 Answers jitender lakhanpal Last Activity: 12 Years ago Dear Nikhil, there can be three cases when n<r then the number of ways are 0 as there will be atleast one group that will be vacant when n = r then the number of ways are n factorial when n>r then the number of ways would be P(n,r) as there would be atleast one object in one group and then remaining n-r objects have to be placed in r things now there would be further 2 cases as case:1 n-r>r then the total number of cases would be P(n,r)P((n-r),r) case:2 n-r<r then the total number of cases would be P(n,r)[(C(n-r,1)r + C(n-r,2)(r-1)2fact + C(n-r,3)(r-2)3fact------- we got this expression by first arranging n objects in r places so we got P(n,r) AND then from remaining n-r objects we select 1 object and arrange in r places we got C(n-r,1)r OR select 2 objects then arrange them in r places OR select 3 objects then arrange them in r places this will go upto n-r objects. and we know that AND means multiplication OR means ADDITION by this we will get the arrangement when no group will be vacant. We are all IITians and here to help you in your IIT JEE preparation. Now you can win by answering the questions on Discussion Forum. So help discuss any query on askiitians forum and become an Elite Expert League askiitian. Now you score 5+15 POINTS by uploading your Pic and Downloading the Askiitians Toolbar : Click here to download the toolbar.. Provide a better Answer & Earn Cool Goodies Enter text here... Ask a Doubt Get your questions answered by the expert for free Enter text here...	{"url":"https://www.askiitians.com/forums/Magical-Mathematics%5BInteresting-Approach%5D/29/41854/please-please-answer-this.htm","timestamp":"2024-11-02T04:50:01Z","content_type":"text/html","content_length":"186397","record_id":"<urn:uuid:60708a27-0edd-47cc-9449-159f02d8a855>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00272.warc.gz"}
The 5-Minute Rule for Physics Vector It’s possible to change their names and they’re sorted by use. There are many ways of measuring where https://www.domyhomeworkfor.me/ you’re. Various apps allow camera to do in vivid and advanced How to Find Physics Vector Velocity could be in any direction, so a particular direction needs to be assigned to it to be able to give complete info. Its direction can be determined by means of a protractor. In the instance of rotational systems, the direction of the vector is the exact same as the direction the most suitable thumb points once the fingers are curled. The New Fuss About Physics Vector The more torque put on the wheel the better, because it will change the rotational motion more, and permit the wheelchair to accelerate more. Now we should think about the wheels. Make certain to have a peek at the projectile motion calculator, which describes a specific case of completely free fall along with horizontal motion. But we aren’t concerned with this kind of detailed workings of the photon. You should also think of whether it’s possible to select the initial speed for the ball and just figure out the angle at which it is thrown. If you own a sense of deja vu, don’t be alarmed. Well, it’s quite easy to grasp once we break down things from the start. Position is a location where someone or something is situated or has been And we should discover the best one, which is frequently referred as the optimal hyperplane. Draw the vectors so the tip of a single vector is joined to the tail of the subsequent 3. https:// admissions.umich.edu/explore-visit/blog/writing-college-essay-little-advice Obviously the unit vectors ensure it is simple to symbolize a vector together with its components. Drag another vector from the bin. The Demise of Physics Vector Within this blog, I’ll illustrate how to create a model that predicts the next word based on a couple of the previous. In fact, the functions within every one of these nodes is executing several times one time for each bit of information in the multi-data stream. The velocity accounts for only the last position and the complete time it got to get there. Whatever They Told You About Physics Vector Is Dead Wrong…And Here’s Why A future that is just beginning. There is an easy and standard method to get this done. With that, we arrive at the conclusion of the series on the Special Relativity. If you own a sense of deja vu, don’t be alarmed. I must bring the info that I’m just a beginner, so maybe I didn’t explain everything right and… yeah. Don’t neglect to answer the question. The direction of a resultant vector can frequently be determined by usage of trigonometric functions. There are more than 1 approach to be a symbol of a vector mathematically. Vectors that aren’t at nice angles want to be handled. When the angle is selected, any of the 3 functions can be utilised to locate the measure of the angle. When you get prequalified you get a specialist estimate of the type of homes for sale in Westminster MD that you are able to be in a position to afford. The cool thing about vectors is that you could pretty much move them around on whatever plane and the values will nonetheless remain the exact same, so long as the directions continue to be the exact same. Text Generation is a sort of Language Modelling issue. Now let’s add the true heart of the algorithm. Discussion it illustrates the addition of vectors using perpendicular components. Vital Pieces of Physics Vector Given all of the positive advantages, it’s imperative that all students have the chance to formally learn physics in their secondary school settings. The access to physics for a course for high school students isn’t equitably distributed throughout the usa. The TensorNetwork library was constructed to promote precisely these types of tasks. I will provide you with some example-code. These 22 deteminants are available quickly. One not superior than the other. The Physics Vector Cover Up This assumption makes it possible for us to avoid using calculus to discover instantaneous acceleration. In calculus terms, velocity is the very first derivative of position related to time. But there’s no reason at all that divisibility should be our only criterion. The issue of locating the values of w and b is known as biology homework help an optimization issue. Lies You’ve Been Told About Physics Vector Below is a diagram in the event you’re not acquainted with it. It behaves like the unspecialized model of vector with the next changes Force is everywhere and it comes in an assortment of sizes directions and sorts. This vector addition diagram is a good example of this kind of situation. Whenever your position changes with time, you can demonstrate this using a position-time graph. Try out playing with the subsequent active diagram. The Ideal Strategy for Physics Vector Combining AAV with gene editing machinery demands a more sensitive way of safe and productive applications. You will realize that some nodes in Grasshopper output data in the kind of trees, and you often must work with data in tree format to achieve certain tasks. This example illustrates the accession of vectors utilizing perpendicular components.	{"url":"https://charityschakras.com/2019/12/24/the-5-minute-rule-for-physics-vector/","timestamp":"2024-11-04T23:38:24Z","content_type":"text/html","content_length":"39450","record_id":"<urn:uuid:1444ba80-ac30-43cd-9062-eaafa72fc3d6>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00192.warc.gz"}
Algebra question type The algebra question type allows algebraic expressions for student answers which are evaluated by instructor provided answers using the basic rules of algebra. It was created by Roger Moore and first offered as contributed code in July 2009. It is currently maintained by Jean-Michel Védrine This is a new question type for Moodle. It implements an algebraic question type where student responses are treated as an algebraic expression and compared to instructor provided answers using the basic rules of algebra. For example if the instructor provided response is ${\displaystyle \sin 2\theta }$ then a student entering ${\displaystyle 2\sin \theta \cos \theta }$ will have their answer treated as matching. The code has been tested in production use with ~230 first year physics students without problems (using the evaluate comparison method only) - mainly thanks to some excellent feedback and testing from Moodle users on the quiz forum before deployment! For the display to work you need to have some way of displaying TeX expressions activated on your Moodle website: either the TeX filter or MathJax enabled. Question Formulation First of, the instructor shall write the description of the question, in the "question text" field. For math expressions she can either use the question editor provided or use TeX syntaxis. In the latter case he'll need to use double dollar sign to start and begin an equation. The sections of the question formulation that are specific for algebra question type are: • Options: Here you define which comparison method is used, three are provided: 1. Evaluation: this is the default and best tested method. It "cheats" by evaluating both the answer and the student response for various random combinations of the variables and ensures that both match to within tolerance at all points. 2. Sage: This method requires the external, open source Sage symbollic algebra system to be installed. A simple XMLRPC server script in Python is included in the package, sage_server.py, and this provides a method to symbollically compare two expressions. (NOT RECOMMENDED FOR PRODUCTION USE) 3. Equivalence: this uses the internal PHP algebra parser to compare the two expressions. It was intended for use in "expand..." and "factor..." type questions where a full symbollic comparison would allow the student to enter the unexpanded (or unfactorized) expression. (NOT RECOMMENDED FOR PRODUCTION USE) • Variable 1: The instructor defines the first independent variable of the question. This is entered in linear syntax notation, not in latex. • Answer 1: The instructor defines the answer. Again, you shall use linear syntax notation. She may choose to give different answers with different values of the grade, but it is required that an answer with the highest grade is included. The instructor fills the text box with the following question: "Find the derivative of ${\displaystyle \\\sin 2x\\}$". In the Variable 1 she fills x, and in the Answer 1 2 cos (2x). The student answer does not have to be exactly this, only shall be evaluated to (very similar) values. For instance, the equivalent answer cos (2x) + cos (2x) would evaluate correctly. The instructor may also include as a second answer cos (2x), but this one gives just 20%, since the student forgot to use the chain rule. Current Features • The students enter responses in a text box, like calculated and numeric questions • The formula they entered is rendered in the box below by clicking on a button. • Option of three different comparison methods: SAGE, Evaluation or Equivalence. • Support for multiple answers so partial credit or assistance can be provided to students. • Core functionality works entirely within Moodle - external programs are optional • Works with Moodle 1.9, Moodle 2.3 and 2.4 New Features Coming Soon • Full, secure Moodle network server implemented in Python to allow connection to Sage. • Separate the evaluated Algebra and symbollic Sage question types. • Enable disallowed expressions for Sage to enable factorize, expand etc. type questions. • Enable use of datasets in Sage question type so that algebraic expressions can vary e.g. "Differentiate {a}x^{n} wrt x" where {a} and {n} will be allowed to vary as they do in calculated • Add Sage filter to allow embedding of 2D and 3D plots in Moodle text e.g. <sage>plot_1d_function(x^2,-1,1)</sage> will insert a plot of the function ${\displaystyle x^{2}}$ between x=-1 and x=1. This version is almost complete - all the basic functionality is present and the new question type is currently being coded.... • Unzip the zip file into the <moodle root>/question/type directory. You should end up with a directory named <moodle root>/question/type/algebra. • Login as an administrator and visit the admin page (http://<my moodle>/admin). This will perform the required database updates to support the question type. • Go to a course, click on question and select the new Algebra question type to add. NB: If you wish to test the current Sage functionality you will also need to install Sage as well as run the python serer code with the command "sage - python sage_server.py". Additionally the questiontype.php file wil need to be edited by hand to point to the server machine and port. This is not for the faint of heart and should not be used in a production system. The full functionality is available without Sage though: just use the evaluate method. • Failed to load question options from the table question_algebra for questionid <n>: this usually occurs because you have entered an algebra question into Moodle without having visiting the admin page (http://<my moodle>/admin) to setup the database first. To fix it go to the admin page while logged in as an administrator to setup the database. Any existing algebra questions that were added (via editing or import) before this will need to be deleted since they are missing important information. See also • Question type: Algebra is a link to the old Modules and plugins database page to download the algebra question type for Moodle 1.9. • Question type: Algebra is a link to the Moodle plugins Directory to download the algebra question type for Moodle 2.3/2.4/2.5. • Discussions:	{"url":"https://docs.moodle.org/25/en/index.php?title=Algebra_question_type&oldid=108020","timestamp":"2024-11-08T18:29:03Z","content_type":"text/html","content_length":"40199","record_id":"<urn:uuid:b4290124-6818-444e-9ac7-41a89e0b2a94>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00762.warc.gz"}
Data Science and Big Data Analytics (WS2014) Teaching Staff: Steffen Herbold, Fabian Korte Dates, Modules, etc. • Lecture: Tuesdays, 14:15-15:45 o'clock, Room 2.101 (first session Oct. 28th) • Exercise: Thursdays, 13:15-14:45 o'clock, Room -1.111 (first session mid of November / to be determined) • Modules: M.Inf.1151 The lecture cannot be recorded or streamed due to the technical restrictions in the lecture room. This lecture requires registration during the first session. The maximum number of participants is 30. Registration and active participation in the exercise is mandatory in order to be allowed to participate in the final exam. The main topic of this lecture is data science, i.e., methods to extract information from data with a scientific approach. We approach this topic from a practical side in this lecture. This means, that we concern ourselves directly with what algorithms do, and where they should be applied. The details of the algorithms and the theory behind them are not part of this lecture. Methods considered in this lecture include: • k-means clustering • Linear regression • Logistic regression • Naive bayes • Decision trees • Text analysis Additionally, we will consider the analysis of Big Data. In this context, we will consider the following topics: • MapReduce • Hadoop • Languages for Hadoop • Mahout The materials for this course are distributed via Stud.IP. 2024 © Software Engineering For Distributed Systems Group Main menu 2	{"url":"https://www.swe.informatik.uni-goettingen.de/lectures/data-science-and-big-data-analytics-ws2014","timestamp":"2024-11-13T02:30:48Z","content_type":"application/xhtml+xml","content_length":"17195","record_id":"<urn:uuid:5f8381fd-b345-432d-a15d-a217e7195651>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00519.warc.gz"}
American Mathematical Society Optimal lower bound for the gap between the first two eigenvalues of one-dimensional Schrödinger operators with symmetric single-well potentials HTML articles powered by AMS MathViewer by Mark S. Ashbaugh and Rafael Benguria Proc. Amer. Math. Soc. 105 (1989), 419-424 DOI: https://doi.org/10.1090/S0002-9939-1989-0942630-X PDF \| Request permission We prove the optimal lower bound ${\lambda _2} - {\lambda _1} \geq 3{\pi ^2}/{d^2}$ for the difference of the first two eigenvalues of a one-dimensional Schrödinger operator $- {d^2}/d{x^2} + V(x)$ with a symmetric single-well potential on an interval of length $d$ and with Dirichlet boundary conditions. Equality holds if and only if the potential is constant. More generally, we prove the inequality ${\lambda _2}[{V_1}] - {\lambda _1}[{V_1}] \geq {\lambda _2}[{V_0}] - {\lambda _1}[{V_0}]$ in the case where ${V_1}$ and ${V_0}$ are symmetric and ${V_1} - {V_0}$ is a single-well potential. Similar Articles • Retrieve articles in Proceedings of the American Mathematical Society with MSC: 81C05, 34B25 • Retrieve articles in all journals with MSC: 81C05, 34B25 Bibliographic Information • © Copyright 1989 American Mathematical Society • Journal: Proc. Amer. Math. Soc. 105 (1989), 419-424 • MSC: Primary 81C05; Secondary 34B25 • DOI: https://doi.org/10.1090/S0002-9939-1989-0942630-X • MathSciNet review: 942630	{"url":"https://www.ams.org/journals/proc/1989-105-02/S0002-9939-1989-0942630-X/?active=current","timestamp":"2024-11-09T16:18:37Z","content_type":"text/html","content_length":"63222","record_id":"<urn:uuid:3c1f6712-e457-4696-b60e-d044c9ee92b3>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00559.warc.gz"}
Acute Angle \| Definition, Formula, Degrees, Images Geometry is a fascinating realm of mathematics, filled with intriguing concepts and angles that shape our understanding of space and form. In this article, we embark on an in-depth exploration of acute angles, covering their definition, degree measurement, visual representations, mathematical formulas, practical examples, properties within triangles, real-world applications, and the crucial distinctions between acute and obtuse angles. Acute Angle Definition: An acute angle is an angle that measures less than 90 degrees. In other words, it is an angle whose magnitude is smaller than a right angle. For example, an angle measuring 20 degrees, 75 degrees, or 89 degrees would all be acute angles. Acute Angle Degree: The measure of an acute angle is always less than 90 degrees. The degree measure of an acute angle could be any value between 0 and 90 degrees, exclusive. Acute Angle Images: Acute Angle Formula: In the context of an acute triangle, the relationships among the lengths of its sides are governed by the following conditions: 1. The sum of the squares of the two shorter sides, denoted as a and b, is greater than the square of the longest side c: a^2 + b^2 > c^2 2. The sum of the squares of the second shortest side, b, and the longest side, c, is greater than the square of the shortest side, a: b^2 + c^2 > a^2 3. The sum of the squares of the shortest side, a, and the second longest side, c, is greater than the square of the remaining side, b: c^2 + a^2 > b^2 Acute Angle Examples: Examples of acute angles could include: • A 30-degree angle. • A 60-degree angle. • Any angle with a measure between 0 and 90 degrees. Acute Angle Triangle: An acute-angled triangle is a triangle in which all three angles are acute angles, meaning each angle is less than 90 degrees. Properties of Acute Triangle: □ Properties of an acute triangle include: ☆ All three angles are acute. ☆ The sum of the interior angles is less than 180 degrees. ☆ No angle is a right angle or obtuse angle. Acute Angle vs Obtuse Angle Characteristic Acute Angle Obtuse Angle Definition An angle that measures greater than 0 degrees An angle that measures greater than 90 degrees Measure Range Between 0 degrees and less than 90 degrees Between 90 degrees and less than 180 degrees Symbolic Representation ∠ABC, where the measure of ∠ABC < 90 degrees ∠PQR, where the measure of ∠PQR > 90 degrees Examples 30 degrees, 45 degrees, 60 degrees 100 degrees, 120 degrees, 150 degrees Triangle Classification All angles in an acute-angled triangle are acute One angle in an obtuse-angled triangle is obtuse Trigonometric Functions Trigonometric functions are defined for acute angles Trigonometric functions are defined for obtuse angles	{"url":"https://www.easonacademy.com/acute-angle/","timestamp":"2024-11-12T20:00:48Z","content_type":"text/html","content_length":"130089","record_id":"<urn:uuid:69b0ec93-78c5-470f-b1d8-200cb6e0a7ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00188.warc.gz"}
An Introduction to Measure and Integration: Second Editionsearch Item Successfully Added to Cart An error was encountered while trying to add the item to the cart. Please try again. Please make all selections above before adding to cart An Introduction to Measure and Integration: Second Edition Hardcover ISBN: 978-0-8218-2974-5 Product Code: GSM/45 List Price: $99.00 MAA Member Price: $89.10 AMS Member Price: $79.20 eBook ISBN: 978-1-4704-2096-3 Product Code: GSM/45.E List Price: $85.00 MAA Member Price: $76.50 AMS Member Price: $68.00 Hardcover ISBN: 978-0-8218-2974-5 eBook: ISBN: 978-1-4704-2096-3 Product Code: GSM/45.B List Price: $184.00 $141.50 MAA Member Price: $165.60 $127.35 AMS Member Price: $147.20 $113.20 Click above image for expanded view An Introduction to Measure and Integration: Second Edition Hardcover ISBN: 978-0-8218-2974-5 Product Code: GSM/45 List Price: $99.00 MAA Member Price: $89.10 AMS Member Price: $79.20 eBook ISBN: 978-1-4704-2096-3 Product Code: GSM/45.E List Price: $85.00 MAA Member Price: $76.50 AMS Member Price: $68.00 Hardcover ISBN: 978-0-8218-2974-5 eBook ISBN: 978-1-4704-2096-3 Product Code: GSM/45.B List Price: $184.00 $141.50 MAA Member Price: $165.60 $127.35 AMS Member Price: $147.20 $113.20 • Graduate Studies in Mathematics Volume: 45; 2002; 424 pp MSC: Primary 28; Secondary 26 Integration is one of the two cornerstones of analysis. Since the fundamental work of Lebesgue, integration has been interpreted in terms of measure theory. This introductory text starts with the historical development of the notion of the integral and a review of the Riemann integral. From here, the reader is naturally led to the consideration of the Lebesgue integral, where abstract integration is developed via measure theory. The important basic topics are all covered: the Fundamental Theorem of Calculus, Fubini's Theorem, $L_p$ spaces, the Radon-Nikodym Theorem, change of variables formulas, and so on. The book is written in an informal style to make the subject matter easily accessible. Concepts are developed with the help of motivating examples, probing questions, and many exercises. It would be suitable as a textbook for an introductory course on the topic or for self-study. For this edition, more exercises and four appendices have been added. The AMS maintains exclusive distribution rights for this edition in North America and nonexclusive distribution rights worldwide, excluding India, Pakistan, Bangladesh, Nepal, Bhutan, Sikkim, and Sri Lanka. Graduate students and research mathematicians interested in mathematical analysis. □ Chapters □ Prologue. The length function □ Chapter 1. Riemann integration □ Chapter 2. Recipes for extending the Riemann integral □ Chapter 3. General extension theory □ Chapter 4. The Lebesgue measure on $\mathbb {R}$ and its properties □ Chapter 5. Integration □ Chapter 6. Fundamental theorem of calculus for the Lebesgue integral □ Chapter 7. Measure and integration on product spaces □ Chapter 8. Modes of convergence and $L_p$-spaces □ Chapter 9. The Radon-Nikodym theorem and its applications □ Chapter 10. Signed measures and complex measures □ Appendix A. Extended real numbers □ Appendix B. Axiom of choice □ Appendix C. Continuum hypotheses □ Appendix D. Urysohn’s lemma □ Appendix E. Singular value decomposition of a matrix □ Appendix F. Functions of bounded variation □ Appendix G. Differentiable transformations □ From reviews for the first edition: Distinctive features include: 1) An unusually extensive treatment of the historical developments leading up to the Lebesgue integral ... 2) Presentation of the standard extension of an abstract measure on an algebra to a sigma algebra prior to the final stage of development of Lebesgue measure. 3) Extensive treatment of change of variables theorems for functions of one and several variables ... the conversational tone and helpful insights make this a useful introduction to the topic ... The material is presented with generous details and helpful examples at a level suitable for an introductory course or for self-study. Zentralblatt MATH □ A special feature [of the book] is the extensive historical and motivational discussion ... At every step, whenever a new concept is introduced, the author takes pains to explain how the concept can be seen to arise naturally ... The book attempts to be comprehensive and largely succeeds ... The text can be used for either a one-semester or a one-year course at M.Sc. level ... The book is clearly a labor of love. The exuberance of detail, the wealth of examples and the evident delight in discussing variations and counter examples, all attest to that ... All in all, the book is highly recommended to serious and demanding students. Resonance — journal of science education • Permission – for use of book, eBook, or Journal content • Book Details • Table of Contents • Reviews • Requests Volume: 45; 2002; 424 pp MSC: Primary 28; Secondary 26 Integration is one of the two cornerstones of analysis. Since the fundamental work of Lebesgue, integration has been interpreted in terms of measure theory. This introductory text starts with the historical development of the notion of the integral and a review of the Riemann integral. From here, the reader is naturally led to the consideration of the Lebesgue integral, where abstract integration is developed via measure theory. The important basic topics are all covered: the Fundamental Theorem of Calculus, Fubini's Theorem, $L_p$ spaces, the Radon-Nikodym Theorem, change of variables formulas, and so on. The book is written in an informal style to make the subject matter easily accessible. Concepts are developed with the help of motivating examples, probing questions, and many exercises. It would be suitable as a textbook for an introductory course on the topic or for self-study. For this edition, more exercises and four appendices have been added. The AMS maintains exclusive distribution rights for this edition in North America and nonexclusive distribution rights worldwide, excluding India, Pakistan, Bangladesh, Nepal, Bhutan, Sikkim, and Sri Graduate students and research mathematicians interested in mathematical analysis. • Chapters • Prologue. The length function • Chapter 1. Riemann integration • Chapter 2. Recipes for extending the Riemann integral • Chapter 3. General extension theory • Chapter 4. The Lebesgue measure on $\mathbb {R}$ and its properties • Chapter 5. Integration • Chapter 6. Fundamental theorem of calculus for the Lebesgue integral • Chapter 7. Measure and integration on product spaces • Chapter 8. Modes of convergence and $L_p$-spaces • Chapter 9. The Radon-Nikodym theorem and its applications • Chapter 10. Signed measures and complex measures • Appendix A. Extended real numbers • Appendix B. Axiom of choice • Appendix C. Continuum hypotheses • Appendix D. Urysohn’s lemma • Appendix E. Singular value decomposition of a matrix • Appendix F. Functions of bounded variation • Appendix G. Differentiable transformations • From reviews for the first edition: Distinctive features include: 1) An unusually extensive treatment of the historical developments leading up to the Lebesgue integral ... 2) Presentation of the standard extension of an abstract measure on an algebra to a sigma algebra prior to the final stage of development of Lebesgue measure. 3) Extensive treatment of change of variables theorems for functions of one and several variables ... the conversational tone and helpful insights make this a useful introduction to the topic ... The material is presented with generous details and helpful examples at a level suitable for an introductory course or for self-study. Zentralblatt MATH • A special feature [of the book] is the extensive historical and motivational discussion ... At every step, whenever a new concept is introduced, the author takes pains to explain how the concept can be seen to arise naturally ... The book attempts to be comprehensive and largely succeeds ... The text can be used for either a one-semester or a one-year course at M.Sc. level ... The book is clearly a labor of love. The exuberance of detail, the wealth of examples and the evident delight in discussing variations and counter examples, all attest to that ... All in all, the book is highly recommended to serious and demanding students. Resonance — journal of science education Permission – for use of book, eBook, or Journal content You may be interested in... Please select which format for which you are requesting permissions.	{"url":"https://bookstore.ams.org/view?ProductCode=GSM/45","timestamp":"2024-11-05T16:18:01Z","content_type":"text/html","content_length":"113837","record_id":"<urn:uuid:8739d319-f46c-4fa6-96e6-fffc9e0e4b72>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00678.warc.gz"}
Robert J Budzyński Though teaching is currently not the #1 item in my job description, I continue to do some teaching as part of my work in the Physics Faculty. In recent times this has been: • Information Technology: actually a code name for an introduction to programming for beginners in Python • Introduction to Databases: basics of processing of (mainly non-numeric) data, and an intro to relational databases and SQL, addressed to students of Neuroinformatics In the more distant past I have taught practice classes in courses including undergraduate Math, Statistical Physics I, and Group Theory.	{"url":"https://budzynski.xyz:443/","timestamp":"2024-11-10T01:47:35Z","content_type":"text/html","content_length":"9165","record_id":"<urn:uuid:a61a0ff4-84f4-4534-93ce-bf6ce484bb05>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00636.warc.gz"}
But if any payments be made before one year's interest hath accrued, then compute the interest on the principal sum due on the obligation, for one year, add- it to the principal, and compute the interest on the sum paid, from the time it was paid up to... The Normal Higher Arithmetic: Designed for Common Schools, High Schools ... - Page 500by Edward Brooks - 1877 - 542 pagesFull view About this book Zephaniah Swift - Law - 1795 - 990 pages ...made before one year's intereft has accrued, then compute the interett on the principal fum due on the obligation, for one year, add it to the principal, and compute the intereft on the fum paid from the time it was paid up to the end of the year, add it to the fum paid,... Nathan Daboll - Arithmetic - 1813 - 244 pages ...madn before one year's interest hath accrued, then compute tl o interest on the principal sum due on the obligation for one year, add it to the principal, and compute the interest or the sum paid, from the time it was paid, up to the end 01' the year : add it to the sum paid, and... Roswell Chamberlain Smith - 1814 - 300 pages ...payments be made before one interest hath accrued, then compute the interest on the principal sum due on the obligation, for one year, add it to the principal,...year ; add it to the sum paid, and deduct that sum from tiio principal and interest, added as above.* " If any payments be made of a less sum than the... Nathan Daboll - Arithmetic - 1815 - 250 pages ...made before one year's interest hath accrued, then compute the interest on the principal sum due on the obligation for one year, add it to the principal, and compute the interest on the eum paid, from the time it was paid, up to the end of the year ; add it to the sum paid, and deduct... Nathan Daboll - Arithmetic - 1817 - 252 pages ...made before one year's interest hath accrued, then compute the interest on t!<e principal sura due on the obligation for one year, add it to the principal,...year ; add it to the sum paid, and deduct that sum from the principal and interest added as above.* " If any payments be made of a less sum than the iuterest... Nathan Daboll - Arithmetic - 1820 - 254 pages ...interest hath accrued, then .compute the interest on the principal sum due on the obligation forone year, add it to the principal, and compute the interest on the sum paki, from the time it was paid, up to the end of the year; add it to the sum paid, and deduct that... Nicolas Pike - Arithmetic - 1822 - 536 pages ...made before one year's interest hath accrued, then compute the interest on the principal sum due on the obligation for one year, add it to the principal...interest on the sum paid, from the time it was paid, lip to the end of the year, add it to the sum paid, and de«hict that sum from the principal and interest,... Nicolas Pike - Arithmetic - 1832 - 538 pages ...made beI'orc one year's inlert- st hath accrued, then compute the interest on the principal sum due on the obligation for one year, add it to the principal...compute the interest on the sum paid, from the time :l was paid, up to the end of the year, add it to the sum paid, anddeifacl that sum from (he principal... Nicolas Pike - Arithmetic - 1822 - 562 pages ...made before one year's interest hath accrued, then compute the interest on the principal sum due on the obligation for one year, add it to the principal...and compute the interest on the sum paid, from the timt' it was paid, up to the end of the year, add it to the sum paid, and deilocl that sum from the... Nathan Daboll - Arithmetic - 1823 - 260 pages ...principal sum due OB the obligation, up to the time of settlement, and likewise find the amount of the sum paid, from the time it was paid, up to the time of final settlement, and deduct ihis amount from the amount of the principal. But if there be...	{"url":"https://books.google.com.jm/books?id=15sXAAAAIAAJ&qtid=fd7e7a5c&lr=&source=gbs_quotes_r&cad=5","timestamp":"2024-11-03T15:27:51Z","content_type":"text/html","content_length":"30059","record_id":"<urn:uuid:8c7b284f-b9ae-4707-9ff9-1c145464840c>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00200.warc.gz"}
C++ Not-Zero Equality Trap What would you expect this C++ code to do? uint8_t val = ~0; If (val == ~0) { } else { The result: NO! How about this? #define DO_TEST( TEST_TYPE ) \ { \ TEST_TYPE val = ~0; \ if (val == ~0 ) { \ printf(#TEST_TYPE ": equal to ~0\n"); \ } else { \ printf(#TEST_TYPE ": NOT equal to ~0\n"); \ } \ } \ The result: uint8_t: NOT equal to ~0 uint16_t: NOT equal to ~0 uint32_t: equal to ~0 uint64_t: equal to ~0 int8_t: equal to ~0 int16_t: equal to ~0 int32_t: equal to ~0 int64_t: equal to ~0 What is so different about uint8_t and uint16_t? … And why do all the signed types seem unaffected? Let’s update the test code to use a literal size-suffix: This will ensure that the literal is a 64-bit value. The new result: uint8_t: NOT equal to ~0 uint16_t: NOT equal to ~0 uint32_t: NOT equal to ~0 uint64_t: equal to ~0 int8_t: equal to ~0 int16_t: equal to ~0 int32_t: equal to ~0 int64_t: equal to ~0 The uint32_t test now fails also. What is going on here? The bitwise negation operator, ~, applied to 0results in a number with all bits set, regardless of the signed-ness. Let’s write the uint8_t expression another way by making Val a literal as well: This evaluates to false also. Let’s also expand out the ~0 to the 32bit default int size. We can see more clearly now that the values are indeed not equal. But why does it succeed when val has more bits than the 32 of default int? … Automatic Integral Promotion! Integral promotion is effective in ensuring that intermediate calculations involving smaller types do not overflow. In the case of the equality operator, both operands are expanded to the size of the larger of the two. Let’s expand the expression for the uint_64 case: 0xffffffffffffffff == ~0x0000000000000000 After applying the binary negation, it is clearly true: 0xffffffffffffffff == 0xffffffffffffffff But what about the signed cases, which always work regardless of integral promotion? Let’s look at the binary representation of -1 with varying type sizes: 8-bits: 11111111 16-bits: 1111111111111111 32-bits: 11111111111111111111111111111111 The first bit indicates that this is a negative number. These negative numbers simply are interpretations of unsigned numbers that wrap around when counting below zero. This representation is known as Two’s Complement. When integral promotion is applied, the numerical value of the negative number is maintained. On a binary level, this is effectively done by repeating the first, “sign bit”, to fill in the newly added bytes. Although ~0 stored in a uint8_t and a int8_t both have the same binary representation, the method of integral promotion applied by the compiler differs. Integral promotion in the expression comparing the two values occurs before the ~ operator. Both the ~0 and val are promoted to the largest type between the two. If val is the larger type, the condition can succeed; otherwise, it will always fail. In the case of negative numbers, integral promotion extends the sign bit to maintain the same numerical value in two’s complement encoding. This results in the values continuing to be equal after	{"url":"https://kearwood.com/content/cpp-trap-not-zero-equality","timestamp":"2024-11-09T07:33:40Z","content_type":"text/html","content_length":"20115","record_id":"<urn:uuid:4df1982f-2075-431c-882c-2b97cca9029c>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00777.warc.gz"}
The number of ways, in which 10 identical objects of one kind, ... \| Filo Question asked by Filo student The number of ways, in which 10 identical objects of one kind, 10 of another kind and 10 of third kind can be divided between two persons so that each person has 15 objects, is a. 136 b. 91 c. 45 d. 36 Not the question you're searching for? + Ask your question Video solutions (1) Learn from their 1-to-1 discussion with Filo tutors. 11 mins Uploaded on: 5/6/2023 Was this solution helpful? Found 3 tutors discussing this question Discuss this question LIVE 12 mins ago One destination to cover all your homework and assignment needs Learn Practice Revision Succeed Instant 1:1 help, 24x7 60, 000+ Expert tutors Textbook solutions Big idea maths, McGraw-Hill Education etc Essay review Get expert feedback on your essay Schedule classes High dosage tutoring from Dedicated 3 experts Students who ask this question also asked View more Stuck on the question or explanation? Connect with our Mathematics tutors online and get step by step solution of this question. 231 students are taking LIVE classes Question Text The number of ways, in which 10 identical objects of one kind, 10 of another kind and 10 of third kind can be divided between two persons so that each person has 15 objects, is Updated On May 6, 2023 Topic Algebra Subject Mathematics Class Class 11 Answer Type Video solution: 1 Upvotes 97 Avg. Video Duration 11 min	{"url":"https://askfilo.com/user-question-answers-mathematics/the-number-of-ways-in-which-10-identical-objects-of-one-kind-33313136323631","timestamp":"2024-11-12T03:41:09Z","content_type":"text/html","content_length":"179703","record_id":"<urn:uuid:977c72d0-2106-4b80-aebd-f6946372e334>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00806.warc.gz"}
Emir Ali B. Arak Scholar of mathematics and astronomy (B. 996 – D. 1036, Harzem). His full name is Ebu Nasr Emir Ali b. Mansur b. Ali b. Arak. Emir Arak studied Islamic and elementary sciences in Harzem Madrasahs which had the quality of an international university at that time and from virtuous scholars who taught in these places. He became a sovereign scholar with a scientific character. He lived in Harzem during 10th and 11th centuries when elementary sciences such as mathematics, astronomy, algebra and medicine were in demand. He joined “Talebe-i Ulum’a Hayır” which mainly consisted of Turks. He gave lessons to all people who were interested in this science and who sought for his help. He raised numerous famous scholars who would remember him with good memories. One of the famous scientists he raised is the famous mathematician and algebra master Birûnî. Emir Arak wrote around 1035 a work titled Risale el-Fihrist, which informed that this valuable Turkish scholar authored twelve books on mathematics, trigonometry, astronomy and other fields and gave their names. Without doubt, the influence of his valuable works continued for long centuries after him. In the 4th century of Hegira (9 A.D.), Ebu Nasr el-Fârabî represented the Islamic culture in philosophy; Emir Nasr b. Ârâk in mathematics and astronomy.^ Ali b. Arak was one of the leading mathematicians according to Cem Saraç. Besides he realized important researches with Ebû'1-Vefa el-Buzcanî about theorems concerning the proportionality of sinus in spherical triangles. He interpreted el-Macestî and Menelaos' work titled Küreviyat. Nasîru'd-Din Tusî in his book titled Şeklû'l-Kutla regarded Ali b. Arak's interpretation as a very successful work. Birûni in his Risâletü'l-Fihrist named twelve books of his master Ali b. Arak on astronomy. The influence of these books lasted for long years. Emir Ali who had a rough life missed his homeland during his last years in Harzem. He passed away here in 1036. Mehmet Şemsettin Günaltay evaluates Emir Ali b. Arak as following: "In 4^th Century of Hegira (10. A.D.), the Islamic culture was mainly represented by three important personalities. All of them were Turks. Among them Ebu Nasr Fârâbi proved the amazing strength of Turkish genie in philosophy, wisdom, sociology and music, Emir Nasr b. Arak in mathematics and astronomy, Ebû Bekr es-Sûli in literature and history.” REFERENCE: Mehmet Şemsettin Günaltay / İslâm Tarihinin Kaynakları (yay. haz. Yüksel Kanar (1991, s. 61), Zeki Velidi Togan / Umumi Türk Tarihine Giriş I (s. 90, 1946), Şaban Döğen / İslâm ve Matematik (s. 97, 2000), Cem Saraç / Bilim Tarihi (1983, s. 57), İhsan Işık / Ünlü Bilim Adamları (Türkiye Ünlüleri Ansiklopedisi, C. 2, 2013) - Encyclopedia of Turkey’s Famous People (2013).	{"url":"https://www.biyografya.com/biyografi/15246","timestamp":"2024-11-02T21:20:56Z","content_type":"text/html","content_length":"67868","record_id":"<urn:uuid:59be81cd-0529-4b2f-90e1-24cd30dfb461>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00837.warc.gz"}
How To Simulate Rolling Dice In Python - Techinima.com How To Simulate Rolling Dice In Python In this tutorial, we will learn how to simulate rolling dice in Python. Simulating dice rolls can be helpful in various applications, such as for games, statistical analyses, or testing the fairness of dice. We will be using the built-in Python library random to create our dice rolls. Step 1: Import the Random Library First, we need to import the random library in our Python code. This library provides various functions for generating random numbers. Step 2: Define the Number of Sides on the Dice and the Number of Rolls Next, we will define variables for the number of sides on the dice and the number of rolls we want to simulate. For the standard six-sided dice, we can use the following code: 1 num_sides = 6 2 num_rolls = 10 You can change the values here to simulate dice with a different number of sides or a different number of rolls. Step 3: Roll the Dice We can now use the randint function from the random library to generate random numbers representing the dice rolls. The randint function accepts two arguments, the lowest and highest number you want to generate (inclusive). In our case, we want to generate numbers from 1 to the number of sides on the dice. We can use a for loop to generate the specified number of rolls, like this: 1 for i in range(num_rolls): 2 roll = random.randint(1, num_sides) 3 print(f"Roll {i+1}: {roll}") This code will print the result of each roll in the specified range. Full Code Here is the full code for simulating the dice rolls using the random library: 2 import random 3 num_sides = 6 4 num_rolls = 10 5 for i in range(num_rolls): 6 roll = random.randint(1, num_sides) 7 print(f"Roll {i+1}: {roll}") Sample Output Roll 1: 3 Roll 2: 6 Roll 3: 5 Roll 4: 1 Roll 5: 4 Roll 6: 2 Roll 7: 2 Roll 8: 6 Roll 9: 1 Roll 10: 5 The output shows the result of each simulated dice roll. Keep in mind that since this is a random process, your output might be different. In this tutorial, we covered how to simulate rolling dice using Python and the random library. With just a few lines of code, we were able to generate random dice rolls, which can be useful in various applications such as games, statistical analyses, or testing the fairness of dice.	{"url":"https://techinima.com/python/simulate-rolling-dice-in-python/","timestamp":"2024-11-07T05:35:53Z","content_type":"text/html","content_length":"46928","record_id":"<urn:uuid:3fa57911-a84c-4318-b240-49a047f76f63>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00757.warc.gz"}
Bit Bashing Aligning the stack Aligning the stack is sure to be a familiar topic for compiler writers and assembly programmers. In x86-64 assembly, functions that contain AVX objects usually align the stack to 32-bytes in the function prologue… andq -32, %rsp If you’re not familiar with AVX, it’s a SIMD extension to x86-64. These instructions operate on vectors of 32-bytes. We like to keep those 32-byte vectors aligned to 32-byte offsets in memory for performance reasons. On x86-64, a penalty is paid when a chunk of memory is loaded from an unaligned memory address. So, how do we align the stack? First, we have to ask ourselves, “Self, is the stack growing up or down?”. The answer will affect the formula we use to calculate the next offset. Stack grows down If the stack grows down, we want to find the next multiple of n smaller than x. x & !(n-1) We will take a real example and work from there. Let’s say that our stack pointer is an 8-bit integer and the address is 25. Let’s also say that we want to align that pointer to the next 8 byte boundary. First, we start with the expression in the parentheses. The binary representation of 8-1 is: - 00000001 And then we NEGATE the result… ! 00000111 The last thing left to do is AND our stack pointer with the result… & 11111000 And that’s it! We can see our result is 24, which is the first multiple of 8 less than 25 (Remember from above? If the stack grows down, we want to find the next multiple of n smaller than x). This works by turning off the least significant bits of our pointer, x, to the right of the 3rd bit position. We chose the 3rd bit position since 8 = 2^3. Let’s look at our example’s source value and the result to convince ourselves that this is correct. Src: 00011001 Dst: 00011000 See how that works? We zeroed out the least significant bits less than 2^3. Stack grows up If the stack grows up, we want to find the next multiple of n larger than x. (x+n) & !(n-1) A careful reader will notice that this equation is not very different from our equation for when the stack grows down. To round up, we merely have to add the value of n to our stack pointer. This will bump up our stack pointer past the next multiple of x by n%x. Then, our problem is as simple as performing the same steps that we covered above when the stack grows down. Align anything in memory Although we focused solely on stack frames above, this concept can also be extended to align any piece of memory. In fact, this technique is often used to optimize the memory returned by a call to alloca. To align a block of memory allocated on the stack to a specific boundary, we simply request the amount of memory required for our data + the desired alignment amount. Then, we can use our Stack grows up formula above to find the properly aligned memory address to place our data. The Real World Another use of this method is used in digital signal processing. It’s called quantization. In the signal processing domain, we round the signal’s value up to the next level or truncate the value down to the next level. Just like a stack! Quantization is preformed on a wide variety of digital signals. One instance, that may be familiar to Computer Scientists, is MP3 Quantization. Let’s say we wanted to convert a vinyl record into an MP3. A record, of course, produces an analogue signal and MP3 is a digital format. Since our MP3 conversion tools will only sample the signal at discrete intervals, there will be some amount of information loss. This information loss comes in the form of quantization, where we round/truncate our signal to the nearest level. This planned rounding error allows for smaller data encodings, but at the cost of lesser signal quality. 345...»Last »	{"url":"http://bitbashing.com/","timestamp":"2024-11-15T00:29:08Z","content_type":"application/xhtml+xml","content_length":"14158","record_id":"<urn:uuid:e1ebef9d-fb26-4d61-b844-d9371e70a700>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00414.warc.gz"}
Corsi di studio e offerta formativa - Università degli Studi di Parma Learning objectives Allow the student to learn and understand the basic principles of chemistry, the properties of the main chemical molecules of biological interest, the functional aspects and inter-relationships of biomolecules and, finally, the etiologic agents present in the environment that are responsible for disease manifestations and onset of tumors. These skills, combined with the acquisition of knowledge of medical terminology will be functional to understand the topics that will be taught in more specialized courses and for future employment.	{"url":"https://corsi.unipr.it/en/ugov/degreecourse/173955","timestamp":"2024-11-05T13:16:58Z","content_type":"text/html","content_length":"62148","record_id":"<urn:uuid:96e8266d-8269-4779-91de-8744b8765234>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00142.warc.gz"}
The Vanishing Gradient Problem Contributed by: Dinesh Introduction to Vanishing Gradient Problem In Machine Learning, the Vanishing Gradient Problem is encountered while training Neural Networks with gradient-based methods (example, Back Propagation). This problem makes it hard to learn and tune the parameters of the earlier layers in the network. The vanishing gradients problem is one example of unstable behaviour that you may encounter when training a deep neural network. It describes the situation where a deep multilayer feed-forward network or a recurrent neural network is unable to propagate useful gradient information from the output end of the model back to the layers near the input end of the model. The result is the general inability of models with many layers to learn on a given dataset, or for models with many layers to prematurely converge to a poor solution. Also Read: Top Deep Learning Interview Questions for 2020 Methods proposed to overcome vanishing gradient problem 1. Multi-level hierarchy 2. Long short – term memory 3. Faster hardware 4. Residual neural networks (ResNets) 5. ReLU Residual neural networks (ResNets) One of the newest and most effective ways to resolve the vanishing gradient problem is with residual neural networks, or ResNets (not to be confused with recurrent neural networks). It was noted before ResNets that a deeper network would have higher training error than the shallow network. The weights of a neural network are updated using the backpropagation algorithm. The backpropagation algorithm makes a small change to each weight in such a way that the loss of the model decreases. How does this happen? It updates each weight such that it takes a step in the direction along which the loss decreases. This direction is nothing but the gradient of this weight (concerning the Using the chain rule, we can find this gradient for each weight. It is equal to (local gradient) x (gradient flowing from ahead), Here comes the problem. As this gradient keeps flowing backwards to the initial layers, this value keeps getting multiplied by each local gradient. Hence, the gradient becomes smaller and smaller, making the updates to the initial layers very small, increasing the training time considerably. We can solve our problem if the local gradient somehow became 1. How can the local gradient be 1, i.e, the derivative of which function would always be 1? The Identity function! As this gradient is back propagated, it does not decrease in value because the local gradient is 1. The ResNet architecture, shown below, should now make perfect sense as to how it would not allow the vanishing gradient problem to occur. ResNet stands for Residual Network. These skip connections act as gradient superhighways, allowing the gradient to flow unhindered. And now you can understand why ResNet comes in flavours like ResNet50, ResNet101 and ResNet152. Also Read: What is Rectified Linear Unit (ReLU)? Rectified Linear Unit Activation Function (ReLU’s) Demonstrating the vanishing gradient problem in TensorFlow Creating the model In the TensorFlow code that I am about to show you, we’ll be creating a seven-layer densely connected network (including the input and output layers) and using the TensorFlow summary operations and TensorBoard visualization to see what is going on with the gradients. The code uses the TensorFlow layers (tf.layers) framework, which allows quick and easy building of networks. The data we will be training the network on is the MNIST hand-written digit recognition dataset that comes packaged up with the TensorFlow installation. To create the dataset, we can run the following: mnist = input_data.read_data_sets("MNIST_data/", one_hot=True) The MNIST data can be extracted from this mnist data set by calling mnist.train.next_batch(batch_size). In this case, we’ll just be looking at the training data, but you can also extract a test dataset from the same data. In this example, I’ll be using the feed_dict methodology and placeholder variables to feed in the training data, which isn’t the optimal method but it will do for these Setup the data placeholders: self.input_images = tf.placeholder(tf.float32, shape=[None, self._input_size]) self.labels = tf.placeholder(tf.float32, shape=[None, self._label_size]) The MNIST data input size (self._input_size) is equal to the 28 x 28 image pixels i.e. 784 pixels. The number of associated labels, self._label_size is equal to the 10 possible hand-written digit classes in the MNIST dataset. We’ll be creating a slightly deep fully connected network – a network with seven total layers including input and output layers. To create these densely connected layers easily, we’ll be using TensorFlow’s handy tf.layers API and a simple Python loop like follows: # create self._num_layers dense layers as the model input = self.input_images for i in range(self._num_layers - 1): input = tf.layers.dense(input, self._hidden_size, activation=self._activation, First, the generic input variable is initialized to be equal to the input images (fed via the placeholder) Next, the code runs through a loop where multiple dense layers are created, each named ‘layerX’ where X is the layer number. The number of nodes in the layer is set equal to the class property self._hidden_size and the activation function is also supplied via the property self._activation. Next we create the final output layer (you’ll note that the loop above terminates before it gets to creating the final layer), and we don’t supply an activation to this layer. In the tf.layers API, a linear activation (i.e. f(x) = x) is applied by default if no activation argument is supplied. # don't supply an activation for the final layer - the loss definition will # supply softmax activation. This defaults to a linear activation i.e. f(x) = x logits = tf.layers.dense(input, 10, name='layer{}'.format(self._num_layers)) Next, the loss operation is setup and logged: # use softmax cross entropy with logits - no need to apply softmax activation to # logits self.loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits_v2(logits=logits, # add the loss to the summary tf.summary.scalar('loss', self.loss) The loss used in this instance is the handy TensorFlow softmax_cross_entropy_with_logits_v2 (the original version is soon to be deprecated). This loss function will apply the softmax operation to the un-activated output of the network, then apply the cross entropy loss to this outcome. After this loss operation is created, it’s output value is added to the tf.summary framework. This framework allows scalar values to be logged and subsequently visualized in the TensorBoard web-based visualization page. It can also log histogram information, along with audio and images – all of these can be observed through the aforementioned TensorBoard visualization. Next, the program calls a method to log the gradients, which we will visualize to examine the vanishing gradient problem: This method looks like the following: def _log_gradients(self, num_layers): gr = tf.get_default_graph() for i in range(num_layers): weight = gr.get_tensor_by_name('layer{}/kernel:0'.format(i + 1)) grad = tf.gradients(self.loss, weight)[0] mean = tf.reduce_mean(tf.abs(grad)) tf.summary.scalar('mean_{}'.format(i + 1), mean) tf.summary.histogram('histogram_{}'.format(i + 1), grad) tf.summary.histogram('hist_weights_{}'.format(i + 1), grad) In this method, first the TensorFlow computational graph is extracted so that weight variables can be called out of it. Then a loop is entered into, to cycle through all the layers. For each layer, first the weight tensor for the given layer is grabbed by the handy function get_tensor_by_name. You will recall that each layer was named “layerX” where X is the layer number. This is supplied to the function, along with “/kernel:0” – this tells the function that we are trying to access the weight variable (also called a kernel) as opposed to the bias value, which would be “/bias:0”. On the next line, the tf.gradients() function is used. This will calculate gradients of the form ∂y/∂x where the first argument supplied to the function is y and the second is x. In the gradient descent step, the weight update is made in proportion to ∂loss/∂W, so in this case the first argument supplied to tf.gradients() is the loss, and the second is the weight tensor. Next, the mean absolute value of the gradient is calculated, and then this is logged as a scalar in the summary. Next, histograms of the gradients and the weight values are also logged in the summary. The flow now returns back to the main method in the class. self.optimizer = tf.train.AdamOptimizer().minimize(self.loss) self.accuracy = self._compute_accuracy(logits, self.labels) tf.summary.scalar('acc', self.accuracy) The code above is fairly standard TensorFlow usage – defining an optimizer, in this case the flexible and powerful AdamOptimizer(), and also a generic accuracy operation, the outcome of which is also added to the summary (see the Github code for the accuracy method called). Finally a summary merge operation is created, which will gather up all the summary data ready for export to the TensorBoard file whenever it is executed: self.merged = tf.summary.merge_all() An initialization operation is also created. Now all that is left is to run the main training loop. Training the model The main training loop of this experimental model is shown in the code below: def run_training(model, mnist, sub_folder, iterations=2500, batch_size=30): with tf.Session() as sess: train_writer = tf.summary.FileWriter(base_path + sub_folder, for i in range(iterations): image_batch, label_batch = mnist.train.next_batch(batch_size) l, _, acc = sess.run([model.loss, model.optimizer, model.accuracy], feed_dict={model.input_images: image_batch, model.labels: label_batch}) if i % 200 == 0: summary = sess.run(model.merged, feed_dict={model.input_images: image_batch, model.labels: label_batch}) train_writer.add_summary(summary, i) print("Iteration {} of {}, loss: {:.3f}, train accuracy: " "{:.2f}%".format(i, iterations, l, acc * 100)) This is a pretty standard TensorFlow training loop (if you’re unfamiliar with this, see my TensorFlow Tutorial) – however, one non-standard addition is the tf.summary.FileWriter() operation and its associated uses. This operation generally takes two arguments – the location to store the files and the session graph. Note that it is a good idea to set up a different sub folder for each of your TensorFlow runs when using summaries, as this allows for better visualization and comparison of the various runs within TensorBoard. Every 200 iterations, we run the merged operation, which is defined in the class instance model – as mentioned previously, this gathers up all the logged summary data ready for writing. The train_writer.add_summary() operation is then run on this output, which writes the data into the chosen location (optionally along with the iteration/epoch number). The summary data can then be visualized using TensorBoard. To run TensorBoard, using command prompt, navigate to the base directory where all the sub folders are stored, and run the following tensorboard –log_dir=whatever_your_folder_path_is Upon running this command, you will see startup information in the prompt which will tell you the address to type into your browser which will bring up the TensorBoard interface. Note that the TensorBoard page will update itself dynamically during training, so you can visually monitor the progress. Now, to run this whole experiment, we can run the following code which cycles through each of the activation functions: scenarios = ["sigmoid", "relu", "leaky_relu"] act_funcs = [tf.sigmoid, tf.nn.relu, tf.nn.leaky_relu] assert len(scenarios) == len(act_funcs) # collect the training data mnist = input_data.read_data_sets("MNIST_data/", one_hot=True) for i in range(len(scenarios)): print("Running scenario: {}".format(scenarios[i])) model = Model(784, 10, act_funcs[i], 6, 10) run_training(model, mnist, scenarios[i]) This should be pretty self-explanatory. Three scenarios are investigated – a scenario for each type of activation reviewed: sigmoid, ReLU and Leaky ReLU. Note that, in this experiment, I’ve set up a densely connected model with 6 layers (including the output layer but excluding the input layer), with each having a layer size of 10 nodes. Analyzing the results The first figure below shows the training accuracy of the network, for each of the activations: Accuracy of the three activation scenarios – sigmoid (blue), ReLU (red), Leaky ReLU (green) As can be observed, the sigmoid (blue) significantly under performs the ReLU and Leaky ReLU activation functions. Is this due to the vanishing gradient problem? The plots below show the mean absolute gradient logs during training, again for the three scenarios: Three scenario mean absolute gradients – output layer (6th layer) – sigmoid (blue), ReLU (red), Leaky ReLU (green) Three scenarios mean absolute gradients – 1st layer – sigmoid (blue), ReLU (red), Leaky ReLU (green) The first graph shows the mean absolute gradients of the loss concerning the weights for the output layer, and the second graph shows the same gradients for the first layer, for all three activation scenarios. First, the overall magnitudes of the gradients for the ReLU activated networks are significantly greater than those in the sigmoid activated network. It can also be observed that there is a significant reduction in the gradient magnitudes between the output layer (layer 6) and the first layer (layer 1). This is the vanishing gradient problem. You may be wondering why the ReLU activated networks still experience a significant reduction in the gradient values from the output layer to the first layer – weren’t these activation functions, with their gradients of 1 for activated regions, supposed to stop vanishing gradients? Yes and no. The gradient of the ReLU functions where x > 0 is 1, so there is no degradation in multiplying 1’s together. However, the “chaining” expression I showed previously describing the vanishing gradient problem, i.e.: isn’t quite the full picture. Rather, the back-propagation product is also in some sense proportional to the values of the weights in each layer, so more completely, it looks something like this: If the weight values are consistently < 0, then we will also see a vanishing of gradients, as the chained expression will reduce through the layers as the weight values < 0 are multiplied together. We can confirm that the weight values in this case are < 0 by checking the histogram that was logged for the weight values in each layer: The diagram above shows the histogram of layer 4 weights in the leaky ReLU scenario as they evolve through the epochs (y-axis) – this is a handy visualization available in the TensorBoard panel. Note that the weights are consistently < 0, and therefore we should expect the gradients to reduce even under the ReLU scenarios. In saying all this, we can observe that the degradation of the gradients is significantly worse in the sigmoid scenario than the ReLU scenarios. The mean absolute weight reduces by a factor of 30 between layer 6 and layer 1 for the sigmoid scenario, compared to a factor of 6 for the leaky ReLU scenario (the standard ReLU scenario is pretty much the same). Therefore, while there is still a vanishing gradient problem in the network presented, it is greatly reduced by using the ReLU activation functions. This benefit can be observed in the significantly better performance of the ReLU activation scenarios compared to the sigmoid scenario. Note that, at least in this example, there is not an observable benefit of the leaky ReLU activation function over the standard ReLU activation function. This post has shown you how the vanishing gradient problem comes about when using the old canonical sigmoid activation function. However, the problem can be reduced using the ReLU family of activation functions. You will also have seen how to log summary information in TensorFlow and plot it in TensorBoard to understand more about your networks. Hope the article helps. If you found this helpful and wish to learn more such concepts, join Great Learning Academy’s free online courses today.	{"url":"https://www.mygreatlearning.com/blog/the-vanishing-gradient-problem/","timestamp":"2024-11-04T09:09:21Z","content_type":"text/html","content_length":"395422","record_id":"<urn:uuid:9210173f-efa9-4307-b0ee-d5a43bb18c8a>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00414.warc.gz"}
Bivariate factorizations via Galois theory, with application to exceptional polynomials Michael E. Zieve: Bivariate factorizations via Galois theory, with application to exceptional polynomials, J. Algebra 210 (1998), 670–689. MR 99m:12001 (The published version is available online.) We give a general method for factoring R[g] := g(X) - g(Y), where g is a polynomial over a field K. Our approach often works when g varies over an infinite family of polynomials. Factorizations of R[g] are especially important when g is an exceptional polynomial, which by definition means that the scalar multiples of X - Y are the only absolutely irreducible factors of R[g] in K[X, Y]. Exceptional polynomials arise in various investigations in case K is finite, since in that case a polynomial g is exceptional if and only if the map a → g(a) induces a bijection on infinitely many finite extensions of K. We apply our method to factor R[g] for each g in the infinite family of exceptional polynomials discovered recently by Lenstra and Zieve. We also apply our method to factor R[g] in case g is one of the Müller–Cohen–Matthews exceptional polynomials; in this case these factorizations had been discovered previously by more complicated methods. Additional comment from September 2008: The results of this paper were used in the subsequent paper by Guralnick and Zieve which, together with a paper by Guralnick, Rosenberg and Zieve, yields a complete classification of indecomposable exceptional polynomials of non-prime power degree. Michael Zieve: home page publication list	{"url":"https://dept.math.lsa.umich.edu/~zieve/papers/biv.html","timestamp":"2024-11-04T12:14:40Z","content_type":"text/html","content_length":"3327","record_id":"<urn:uuid:3c71118a-5062-4419-981c-b7f540a547e7>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00020.warc.gz"}
ist such that all A linked list is a linear data structure where elements are not stored contiguously but randomly. They are instrumental in implementing stacks and queues, graphs, performing arithmetic operations on large integers, and representing sparse matrices. It is a vital topic from the placement point of view, and there are several problems related to it. This blog will discuss one such significant problem, rearranging a linked list such that all even and oddly positioned nodes are together. Also see Data Structures Problem Statement You have been provided with the head pointer of a singly linked list. Your task is to rearrange the elements of the linked list such that all the nodes at the even positions occur together and all the nodes at the odd positions occur together. Let us look at a few examples to get a clear understanding. Output: 1→3→5→2→4 Output: 2→3→6→7→1→5→4 Before solving the problem, one thing that should be kept in mind is the different cases possible in the linked list: • No nodes • One node • Two nodes • Odd number of nodes • Even the number of nodes So, keeping in mind all the above cases, we can design the following algorithm: • Maintain 2 pointers ‘ODD’ and ‘EVEN’ for the node at the odd and even positions, respectively. • Traverse the original linked list and put the odd nodes in the odd linked list and the even nodes in the even linked list. • Also, store the first node of the even linked list so that the even linked list can be attached at the end of the odd list after all the odd and even nodes have been connected in different lists. STEP 1: STEP 2: STEP 3: // C++ program to rearrange a linked list such that all even and odd positioned nodes are together. #include <iostream> using namespace std; // 'NODE' class to store linked list nodes. class Node int data; Node next; // Constructor to create a node with given data. Node(int data) this->data = data; next = NULL; // 'LINKED_LIST' class to create linked list objects. class LinkedList // 'HEAD' pointer to store the address of the first node. Node head; head = NULL; // Function to print the linked list elements. void printLinkedList() 'CURRENT' node pointer traverses through the linked list. Set 'CURRENT' node equal to the first node of the linked list. Node current = head; // Loop to traverse the linked list. while (current) cout << current->data << " "; current = current->next; cout << endl; // Function to insert nodes in the linked list. void insert(int data) / If linked list is empty, create a new node and direct the 'HEAD' node to the newly created node. if (head == NULL) head = new Node(data); // Else traverse the linked list until the end of the list is encountered. Node current = head; while (current->next) current = current->next; // Create and insert a new node at the end of the linked list. current->next = new Node(data); // Function to rearrange a linked list such that all even and odd positioned nodes are together. void rearrangeEvenOdd() if (head == NULL) // Initialising the first nodes of even and odd lists. Node odd = head; Node even = head->next; // Storing the first node of the linked list so that it can be connected at the end of the odd list. Node evenFirst = even; while (1) // If the odd list and the even lists have ended, the first node of the even linked list is connected to the last node of the odd linked list. if (!odd \|\| !even \|\| !(even->next)) odd->next = evenFirst; // Connecting the odd nodes. odd->next = even->next; odd = even->next; // If there are no more even nodes after the current odd node. if (odd->next == NULL) even->next = NULL; odd->next = evenFirst; // Connecting the even nodes. even->next = odd->next; even = odd->next; int main() int n; // Taking user input. cout << "Enter the size of the linked list: "; cin >> n; cout << "Enter the values of the nodes: "; LinkedList *givenList = new LinkedList(); for (int i = 0; i < n; i++) int data; cin >> data; // Calling the function to rearrange the linked list such that all even and odd positioned nodes are together. // Printing this modified linked list. cout << "The modified linked list is: "; return 0; You can also try this code with Online C++ Compiler Run Code Enter the size of the linked list: 5 Enter the values of the nodes: The modified linked list is: Complexity Analysis Time Complexity O(N), where N is the size of the linked list. Since we are traversing the entire linked list with N nodes only once, the time complexity is given by O(N). Space Complexity The space complexity is O(1). The space complexity is constant as we are not using extra space except for variables and pointers. Recommended Topic, Floyds Algorithm	{"url":"https://www.naukri.com/code360/library/rearrange-a-linked-list-such-that-all-even-and-odd-positioned-nodes-are-together","timestamp":"2024-11-09T20:18:11Z","content_type":"text/html","content_length":"409838","record_id":"<urn:uuid:66d847b3-c142-45ed-a9d8-0b3a62d0755d>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00848.warc.gz"}
InBetsment Academy - Statistical concepts for your sports bets Statistical concepts that will help you analyze the betting tips of your punters Fundamental concepts In this section we present the statistical concepts used in InBetsment in order to assess the quality of tipsters and forecasts: Yield measures, in percentage, the profit in relation to the total amount bet for a determined number of bets. For example, if a tipster places 15 bets during one month (let´s consider he applies 2 units each) and the net profit is 15 units, the yield is 15/30 = 50%. It is of course one of the most important tipster statistical values to check, but perhaps it is overrated, since that yield is calculated taking into account the odds at time of publication. However, there are different factors that make the users sometimes bet below that odds value. This makes the real customer yield to decrease below the official one. InBetsment has developed a term called ‘score’ that assesses all the important factors to give our tipsters a rating reflecting the quality of their service. It is the profit or loss obtained after placing a bet. It is measured in units, like the stake. It is possible to estimate the probability that a bookie is giving to an event to occur, through the odds. If you want to know how odds are translated into the real probability a bookie has assigned to an event, we need first to know the pay-out percentage indicator, which indicates the profits bookies are making. The best way to explain this is with an example: Let’s use the match Real Betis - Valencia CF, with the following odds: Real Betis = 1.75 Draw = 3.30 Valencia = 4.20 The profit of the bookie is calculated as follows: Profit = (1 / 1.75) + (1 / 3.30) + (1 / 4.20) = 0.57 + 0.30 + 0.24 = 1.11 Therefore, the pay-out, or payment percentage, is: Pay-out = (1 / E) * 100 = (1 / 1.11) * 100 = 90% The probability assigned by the bookmaker is the inverse of the odds, multiplied by the payment percentage, namely: Real Betis win probability = (1 / 1.75) * 90% = 51% Draw probability = (1 / 3.30) * 90% = 27% Valencia win probability = (1 / 4.20) * 90% = 21% Value refers to the odds that are overpaid, regarding to the probability of occurrence. They are bets that you can individually win or lose but in a long term, they are very profitable. Our specialists are experts in finding value: they dominate a particular market, or sport, and are able to estimate or limit the probability of that occurrence and, therefore, estimate which odds are the best. If they find a bookie whose odds for an event are overpaid, they will publish that pick. It is the minimum odds value to which the specialist recommends to bet. Below this value, it is recommended not to bet, considering that this forecast is worthless. For example: Betsgrowth, premium tipster, sends a pick with odds at 2.10 and stake 2/5 The tipster determines that the threshold value is 1.90. If the user, for whatever reason, sees the pick below this value, he should dismiss the bet and ignore the forecast. We could define this as the capacity of a forecast to make a profit, without the market closing, or odds value going down. Within the vast range of bets that bookies offer, there are huge disparities in market liquidity. The more important is a competition, the more liquidity, and vice versa. This is one of the details that tipsters need to know, in order to offer quality picks to their customers, since when a certain amount of money enters into a market, the bookie will automatically decrease all odds related to that market. Therefore, this circumstance does not allow the rest of bettors to bet within the values advised by the tipster. This is directly related to the duration of the odds, which is a very relevant factor when it is assumed that the official stats are the real ones that customers can get. Therefore, it is a factor that we have included in the tipster ‘Score’. Betvalue is a statistic number that measures the expected return in the long term if you often repeat bets on similar picks. In statistical jargon it would be called “the expectation of profit”. It is calculated as follows: Betvalue = Odds actual event probability Scenario: Wimbledon Final: Nadal vs Djokovic. A tipster sees no favourite, so the probability to win of either is 50%, which would mean that we could expect odds of around 1.9 for each player. If the bookie releases odds of 1.53 vs 2.37, the tipster will probably send the forecast for Djokovic to win. The betvalue of that pick would be 50% 2.37 = 1.185, which means that for each euro bet you make, on average, €1.185. This serves as a rough estimate of yield. The yield, if we bet repeatedly to these forecasts, would Betvalue -1. In our example: 18.5% ROI (Return Of Investment) The ROI, or return on investment, in betting, is a financial term that refers to the profitability of the bank over a period of time. It is normally calculated yearly, ie: the profitability of your bank after a year of betting. For example, if we start on the 1^st of January with a 1,000 € bank and at the end of the year, on 31^th of December, we have 3,000 €, the ROI is 300%. The ROI of your investment in betting is directly comparable, for example, to the APR of savings accounts (usually 1% - 3%) or the profitability of an investment fund (rarely achieving values above It is common for customers who are not familiar with betting argot to mix up ROI and yield. These are completely different terms, even though both measure performance. The yield, as we explained above, gives the average winning percentage of a tipster per 100 units or euros bet. The ROI measures the total profit gained for every 100€ of the bank, over a period (usually a year). With examples: 1^st January: bank of 1000 € and 1000 bets/year: 500 equal bets: I bet 10 € with a net profit of 15 € 500 equal bets: I bet 10 € with a net profit of -10 € The yield will be Y = (500 bets * profit 15 € - 500 bets profit 10 €) / (10 bets 10 €) = 25% ROI will be R = 2500/1000 = 250% The turnover is not more than the volume of money wagered. That is, we refer to the turnover of the month as the volume bet in that month. It is a little used concept but it refers to how many times the bank has bet in a period of time. For example, if you have a bank of 1000 € and in one year you have bet 4000 €, you have rotated the bank 4 times in a year.	{"url":"https://inbetsment.com/en/2/estadisticos-conceptos-avanzados/73","timestamp":"2024-11-10T11:17:23Z","content_type":"text/html","content_length":"33075","record_id":"<urn:uuid:e67d94ed-ca96-4784-a772-5b6281457886>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00375.warc.gz"}
Table of Contents A brief introduction to uncertainty analysis In practice one can never measure something exactly. The goal of uncertainty analysis is to determine an estimate $\overline{x}$ from a finite amount of measurements and to give an uncertainty $\ Delta x$. Have a look at John R. Taylor's “An introduction to error analysis” to get a very approachable introduction to uncertainties and their propagation. Systematic uncertainties • E.g. a ruler/tape measure does not have the stated length, or its divisions are wrong. • Voltage of a power supply, or value of a resistor, … is too large or too small. A systematic error is the misrepresentation of a measured quantity in one direction. Statistical (unsystematic) uncertainties • When repeating a measurement results in slight variations, this is usually a random uncertainty. This can be due to fluctuations in the experimental setup or environment, friction in mechanical measurement devices (needle of an analog voltmeter)… In every measurement you are interested in the true value of a physical quantity. Systematic uncertainties will (systematically) shift it one one direction. Random uncertainties give equal probabilities for the measurement of a certain smaller or larger value. We are interested in both, and in practice statistical and systematic uncertainties are quoted separately $x\pm \Delta x_\text {syst.}\pm \Delta x_\text{stat.}$. Even though historically not much distinction has been made, we have to distinguish between uncertainties and errors. Errors are definite deviations or hard bounds on a measurement, while uncertainties follow a certain statistical distribution, for example they can be random. Uncertainty propagation The best estimate for a quantity is given by using averages in the functional equation. For example a density might depend on volume $V$ and mass $m$, then the density $\rho$ depends on those, $\ overline\rho=f(\overline V,\overline m)$, where $\overline x$ are averages and $f(V,m)=m/V$. The standard deviation of quantity is calculated as the square sum of weighted standard deviations of the individually measured quantities. For example for the density this would be: $$\Delta \rho = \sqrt{ \left( \frac{\partial f}{\partial V} \Delta V \right)^2 + \left(\frac{\partial f}{\partial m} \Delta m \right)^2 }\,. $$ Note that this formula generalizes straight-forwardly to arbitrary functions $f(x_1,x_2,x_3,\ldots)$ depending on many quantities $x_i$. Also note that this formula is only valid when assuming independent and random uncertainties! ^1) An error propagation that always works (assuming a first-order Taylor expansion is applicable) and gives upper error bounds is given by: $$ \Delta \rho = \left\| \frac{\partial f}{\partial V} \right\| \Delta V + \left\|\frac{\partial f}{\partial m}\right\| \Delta m\,.$$ You can use this formula if you are not sure if your uncertainties are independent and random. Estimates of individually measured quantities To combine multiple measurements with random uncertainties into an improved estimate, e.g. measurement values $l_i$ for $i=1,\ldots,N$, then you can take the arithmetic average: $$ \overline l = \ frac{1}{N} \sum_{j=1}^N{l_j}\,. $$ The standard deviation $\Delta l$ can be obtained from the individual measurements as^2) $$ \Delta l = \sqrt{\frac{1}{N} \sum_{j=1}^N ( l_j - \overline l)^2}\,. $$ The standard deviation indicates how accurately the mean represents the sample data. More advanced and goes beyond this lab: In many practical measurements one has to distinguish between a data sample and the full data population. In some cases only a sample of the data can be used. Then, the standard error of the mean $\Delta l / \sqrt{N}$ is the standard deviation of the theoretical distribution of the sample mean. It indicates the likely discrepancy compared to that of a larger population of data. Read about the Gaussian or Normal distribution. Also note that this is based on the first-order derivatives of $f$, and is therefore a good estimate of the standard deviation of $\rho$ or $f$ as long as the input standard deviations ($\Delta V$ and $\Delta m$) are small enough. These quantities are estimates of the parameters of a Gaussian normal distribution.	{"url":"https://smu-labs.tobias-neumann.eu/doku.php?id=intro-error-analysis","timestamp":"2024-11-07T19:41:49Z","content_type":"text/html","content_length":"13750","record_id":"<urn:uuid:d33c3e07-b7bd-4efa-8097-9b1468fcae69>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00428.warc.gz"}
CURSO COMPLETO RHINOCERS 3D #10 - COORDENADAS TLDR: Explore coordinate systems in Rhino3D for precision and creativity. 📍 Article Source CURSO COMPLETO RHINOCERS 3D #10 - COORDENADAShttps://www.youtube.com/watch?v=aa08v6jxn68 Creating Curvatures in the Command Region In this tutorial, I will be discussing the fascinating process of creating curvatures in the command region and entering values using coordinates. This is an essential skill for any Rhino3D user, and it opens up a world of possibilities for creative design and precision engineering. Types of Coordinates I'll be delving into the three types of coordinates that are crucial for working with Rhino3D: absolute, relative, and polar. Each type has its unique advantages and applications, and mastering them will greatly enhance your proficiency in 3D modeling. Absolute Coordinates Let's start by exploring absolute coordinates. I will demonstrate how to use absolute coordinates to define exact x, y, and z coordinates on the construction plan. This precision is fundamental for accurate and detailed design work, and I'll show you some practical examples of how to leverage absolute coordinates effectively. Relative Coordinates Next, I'll discuss relative coordinates and their significance in Rhino3D. You'll learn how relative coordinates are relative to a specific point, and I'll explain the connection to the base point. Understanding and utilizing relative coordinates is a key aspect of optimizing your workflow and efficiency in 3D modeling. Polar Coordinates We will then delve into the concept of polar coordinates. I'll introduce the concept, which indicates both length and angle from a base point. This method offers a unique way of defining locations in 3D space and provides a fresh perspective on precision modeling. Accuracy and Visualization I'll highlight the significance of using coordinates for accuracy in projects. Precision is paramount in the world of 3D design, and I'll emphasize how mastering coordinate systems is essential for achieving accurate and high-quality outcomes. Additionally, I'll touch on the alternative of working with a base image to visualize and draw on top of it, offering valuable insights into different approaches for achieving your design goals. Timestamped Summary Renounce has three types of coordinates • Renounce's command region triggers commands and allows us to input values • Perspective is important to visualize projects • Working with coordinates is essential in Renounce Understanding coordinate system in construction plans • Coordinates are used to indicate the exact position of axes on a plan • The X, Y and Z axes are located at the left corner of the plan with 10 units on X, 5 units on Y and 5 units on Z axis Coordinates of a point in 3D space • Coordinates consist of x, y, and z axes • Units can be measured in absolute values Explanation of absolute and relative coordinates. • Absolute coordinate is defined by a fixed point. Relative coordinate is defined by a point of reference. • Relative coordinates are relative to a point of desire. Understanding absolute and relative coordinates • Absolute coordinates are fixed and referenced to a construction plan • Relative coordinates are based on a previous point and can change Introduction to polar coordinates • Polar coordinates are a way to represent position using an angle and a length • To create a polar coordinate, a specific length value must first be chosen Explaining polar coordinates and angles • The polar coordinate is symbolized using a smaller symbol • The angle is kept at 50 degrees or its opposite direction Accurate values of curvatures is necessary for the project • We can work with coordinates to guarantee accuracy • If project requires, can use base image to draw on top of Related Questions What is the absolute coordinate in Rhino 3D? The absolute coordinate in Rhino 3D refers to a specific point on the construction plan, indicated by its exact x, y, and z coordinates. In the video, Soft3D - Eduardo Loja demonstrates how to use the absolute coordinate feature when working with the 'poline' command to define precise points in the project. What is the relative coordinate and how is it used in Rhino 3D? The relative coordinate in Rhino 3D is a coordinate that is relative to a specific point of reference. Soft3D - Eduardo Loja explains that the 'r' command is used to indicate that a coordinate will be relative to the last activated point. By using the relative coordinate, users can create lines and shapes based on a reference point, providing flexibility in design and construction. How are polar coordinates used in Rhino 3D? Polar coordinates in Rhino 3D are coordinates that define a length and an angle from a base point. In the video, Soft3D - Eduardo Loja demonstrates the use of polar coordinates to specify the length and angle of a polyline, allowing precise positioning of design elements. The use of polar coordinates is valuable for creating accurate and detailed projects in Rhino 3D. What are the three types of coordinates in Rhino 3D? The three types of coordinates in Rhino 3D are absolute coordinates, relative coordinates, and polar coordinates. Soft3D - Eduardo Loja explains each type in the video, providing a comprehensive overview of how designers and engineers can utilize these coordinates to work with precision and accuracy in Rhino 3D projects. How can designers utilize coordinates in Rhino 3D projects? Designers can utilize coordinates in Rhino 3D projects to specify precise locations, lengths, and angles for design elements. By understanding and using absolute, relative, and polar coordinates, as demonstrated by Soft3D - Eduardo Loja in the video, designers can achieve accuracy and control in their 3D designs, ensuring that projects meet specific requirements and standards.	{"url":"https://www.youtubesummaries.com/people-and-blogs/aa08v6jxn68","timestamp":"2024-11-10T15:00:12Z","content_type":"text/html","content_length":"95275","record_id":"<urn:uuid:fce5c52c-10a9-4472-8514-dec9fc9d3389>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00676.warc.gz"}
Semantic Analytics Group @ Uniroma2 KeLP supports natively a multiple representation formalism. It is useful, for example, when the same data can be represented by different observable properties. For example, in NLP one can decide to derive features of a sentence for different syntactic levels (e.g. part-of-speech, chunk, dependency) and treat them in a learning algorithms with different kernel functions. As an example, consider the following representation: 1 service \|BV\| _.:1.0 _and:1.0 _good:1.0 _is:1.0 _look:1.0 _sharp:1.0 _staff:1.0 _the:1.0 _they:1.0 _too:1.0 _very:1.0 \|EV\|\|BDV\| 0.37651452,0.32109955,0.07726285,0.053550426,-0.06682896,-0.20111458,- 0.14017934,... \|EDV\| \|BS\| The staff is very sharp and they look good too . \|ES\| \|BS\| 35820984#608922#3 \|ES\| It is composed by 1. label (i.e. the class to be learned, service). 2. Sparse vector, whose boundaries are delimited by the special tokens \|BV\| \|EV\|; in this example, a bag of word is used. Note that features can be string! 3. Dense vector, whose boundaries are delimited by the special tokens \|BDV\| \|EDV\|. 4. Two String representation, delimited by \|BS\| \|ES\|; they can be used both for comments, or for advanced kernel functions (e.g. sequence kernels). On this representation a multiple kernel learning algorithm can be applied. Let’s look at an example of code: The first part load a dataset, print some statistics and define the basic objects for our learning procedure. 3 // Read a dataset into a trainingSet variable 4 SimpleDataset trainingSet = new SimpleDataset(); 5 trainingSet.populate("src/main/resources/multiplerepresentation/train.dat"); 6 // Read a dataset into a test variable 7 SimpleDataset testSet = new SimpleDataset(); 8 testSet.populate("src/main/resources/multiplerepresentation/test.dat"); 9 // define the positive class 10 StringLabel positiveClass = new StringLabel("food"); 11 // print some statistics 12 System.out.println("Training set statistics"); 13 System.out.print("Examples number "); 14 System.out.println(trainingSet.getNumberOfExamples()); 15 System.out.print("Positive examples "); 16 System.out.println(trainingSet .getNumberOfPositiveExamples(positiveClass)); 17 System.out.print("Negative examples "); 18 System.out.println(trainingSet.getNumberOfNegativeExamples(positiveClass)); 19 System.out.println("Test set statistics"); 20 System.out.print("Examples number "); 21 System.out.println(testSet.getNumberOfExamples()); 22 System.out.print("Positive examples "); 23 System.out.println(testSet.getNumberOfPositiveExamples(positiveClass)); 24 System.out.print("Negative examples "); 25 System.out.println(testSet.getNumberOfNegativeExamples(positiveClass)); 26 // instantiate a passive aggressive algorithm 27 KernelizedPassiveAggressiveClassification kPA = new KernelizedPassiveAggressiveClassification(); 28 // indicate to the learner what is the positive class 29 kPA.setLabel(positiveClass); 30 // set an aggressiveness parameter 31 kPA.setAggressiveness(2f); The kernel function is the only that has knowledge about the representation on which it will operate. To use multiple representations, each with a specific kernel function, we must specify for each kernel what representation to use. Note that to have comparable scores with different kernels, we normalize each kernel, by applying a NormalizationKernel. // Kernel for the first representation (0-index) Kernel linear = new LinearKernel("0"); // Normalize the linear kernel NormalizationKernel normalizedKernel = new NormalizationKernel(linear); // Apply a Polynomial kernel on the score (normalized) computed by // the linear kernel Kernel polyKernel = new PolynomialKernel(2f, normalizedKernel); // Kernel for the second representation (1-index) Kernel linear1 = new LinearKernel("1"); // Normalize the linear kernel NormalizationKernel normalizedKernel1 = new NormalizationKernel(linear1); // Apply a Polynomial kernel on the score (normalized) computed by // the linear kernel Kernel rbfKernel = new RbfKernel(1f, normalizedKernel1); // tell the algorithm that the kernel we want to use in learning is // the polynomial kernel A weighted linear combination of kernel contribution is simply obtained by instantiating a LinearKernelCombination, and by using the addÂ method on it. Finally we set the kernel on the passive aggressive algorithm. LinearKernelCombination linearCombination = new LinearKernelCombination(); linearCombination.addKernel(1f, polyKernel); linearCombination.addKernel(1f, rbfKernel); // normalize the weights such that their sum is 1 // set the kernel for the PA algorithm Then, we learn a prediction function, and we apply it on the test data. 2 // learn and get the prediction function 3 kPA.learn(trainingSet); 4 Classifier f = kPA.getPredictionFunction(); 5 // classify examples and compute some statistics 6 int correct = 0; 7 for (Example e : testSet.getExamples()) { 8 ClassificationOutput p = f.predict(testSet.getNextExample()); 9 if (p.getScore(positiveClass) > 0 && e.isExampleOf(positiveClass)) 10 correct++; 11 else if (p.getScore(positiveClass) < 0 && !e.isExampleOf(positiveClass)) 12 correct++; 13 } 14 System.out.println("Accuracy: " + ((float) correct / (float) testSet.getNumberOfExamples()));	{"url":"http://sag.art.uniroma2.it/demo-software/kelp/kelp-multiple-representation-formalism/","timestamp":"2024-11-12T15:24:59Z","content_type":"text/html","content_length":"94985","record_id":"<urn:uuid:e728d5c9-872e-4832-a277-376acab90d0e>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00222.warc.gz"}
I’m Francisco Adams Over the last ten odd years I’ve had the pleasure of working with some great companies, working side by side to design and develop new apps and improve upon existing products. See for yourself! #Photoshop, #Illustrator, #CSS, #Python, #Ruby, #Photography The term “portfolio” refers to any combination of financial assets such as stocks, bonds and cash. Portfolios may be held by individual investors and/or managed by financial professionals, hedge funds, banks and other financial institutions. It is a generally accepted principle that a portfolio is designed according to the investor’s risk tolerance, time frame and investment objectives. The monetary value of each asset may influence the risk/reward ratio of the portfolio. When determining a proper asset allocation one aims at maximizing the expected return and minimizing the risk. This is an example of a multi-objective optimization problem: many efficient solutions are available and the preferred solution must be selected by considering a tradeoff between risk and return. In particular, a portfolio A is dominated by another portfolio A‘ if A‘ has a greater expected gain and a lesser risk than A. If no portfolio dominates A, A is a Pareto-optimal portfolio. The set of Pareto-optimal returns and risks is called the Pareto efficient frontier for the Markowitz portfolio selection problem.	{"url":"http://domeni-grundel.de/im-francisco-adams/","timestamp":"2024-11-13T17:30:26Z","content_type":"text/html","content_length":"32052","record_id":"<urn:uuid:fbc66ed4-c96a-4dc1-9587-71c9de26d2d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00180.warc.gz"}
Simpson's Paradox \| Brilliant Math & Science Wiki Simpson's paradox occurs when groups of data show one particular trend, but this trend is reversed when the groups are combined together. Understanding and identifying this paradox is important for correctly interpreting data. For example, you and a friend each do problems on Brilliant, and your friend answers a higher proportion correctly than you on each of two days. Does that mean your friend has answered a higher proportion correctly than you when the two days are combined? Not necessarily! This seemingly unintuitive possibility is referred to as Simpson's paradox. Let's go back to our example on problem accuracy competition to see how this can occur. • On Saturday, you solved $7$ out of $8$ attempted problems, but your friend solved $2$ out of $2.$ You had solved more problems, but your friend pointed out that he was more accurate, since $\dfrac{7}{8} < \dfrac{2}{2}$. Fair enough. • On Sunday, you only attempted $2$ problems and got $1$ correct. Your friend got $5$ out of $8$ problems correct. Your friend gloated once again, since $\dfrac{1}{2} < \dfrac{5}{8}$. However, the competition is about the one who solved more accurately over the weekend, not on individual days. Overall, you have solved $8$ out of $10$ problems whereas your friend has solved $7 $ out of $10$ problems. Thus, despite your friend solving a higher proportion of problems on each day, you actually won the challenge by solving the higher proportion for the entire weekend! While your friend got furious, you calmly pointed him to this page: you had just shown an instance of Simpson's paradox. On this page, we'll give a formal definition of the paradox, show some interesting real-world examples, and provide an opportunity for you to add your own encounters with Simpson's paradox. In layman's term, Simpson's paradox occurs when some groups of data show a certain relationship in each group, but when the data is combined, that relationship is reversed: In the example above, we saw that when the problems were grouped into Saturday and Sunday, your friend solved a higher proportion correctly each day, but when the problems were combined into both days, you actually solved a higher proportion correctly. This common form of Simpson's paradox can be defined as follows: Consider $n$ groups of data such that group $i$ has $A_i$ trials and $0 \leq a_i \leq A_i$ "successes". Similarly, consider an analogous $n$ groups of data such that group $i$ has \ (B_i\) trials and $0 \leq b_i \leq B_i$ "successes". Then, Simpson's paradox occurs if \[\frac{a_i}{A_i} \leq \frac{b_i}{B_i} \text{ for all }i=1,2,\ldots,n \ \text{ but } \ \frac{\sum_{i=1}^{n} a_i}{\sum_{i=1}^{n} A_i} \geq \frac{\sum_{i=1}^{n} b_i}{\sum_{i=1}^{n} B_i},\] and at least one of the inequalities is strict (meaning that it is not in the equality case). Of course, we could also flip the inequalities and still have the paradox, since $A$ and $B$ are chosen arbitrarily. $_\square$ To gain an intuition, let's see how this definition applies to our example above. There, the $A_i/B_i$ are the number of problems you/your friend attempt on each day, and the $a_i/b_i$ are the number you/your friend solve correctly on each day. As seen before, \[ \frac{7}{8}= \frac{a_1}{A_1} < \frac{b_1}{B_1}=\frac{2}{2} \text{ and } \frac{1}{2}=\frac{a_2}{A_2} < \frac{b_2}{B_2}=\frac{5}{8},\text{ yet }\frac{7+1}{8+2} = \frac{a_1+a_2}{A_1+A_2} > \frac However, this is not the only way in which Simpson's paradox can occur. In general, Simpson's paradox is exhibited whenever there is a trend projected by individual categories of data, but the trend reverses when all categories are combined. While this template only considers binary "successes", where each individual datum contributes a single "yes" or a single "no", this can be easily generalized into numbers where the measure used for the trend is its average. We can even use some other measure (such as median). This is discussed in the section "Other Applications" below. Now, let's try an example to test your ability to identify if Simpson's paradox is occurring! It's a draw. Your friend You You and your friend decide to spend the whole weekend doing a Brilliant marathon. The winner is whoever has answered the most number of questions right out of a massive set of 1500 problems at the end of the 2 day weekend. On the first day, you answer 1200 questions, out of which about 62.2% were right. Your friend answers 700 questions, out of which about 63.6% were correct. Your friend teases you over the phone about how he has got a higher percentage of questions correct than you at the end of the first day. On the second day, you answer 300 questions, out of which about 58.3% were correct. Your friend answers 800 questions, out of which he answers about 58.8% were correct. Once again, your friend teases you because the percentage of the number of questions he has got correct is greater than yours. Who won this marathon? You've seen the results, but why does it occur? Normally, one would expect that winning all groups means winning overall. However, this is only guaranteed to be the case if the group sizes are equal. When group sizes differ, the totals for each side might be dominated by particular groups, but these groups belong to different categories. In the introductory example above, the totals are dominated by the days where each player solved 8 problems, and in this case, you actually won ($7/8 > 5/8$), which explains why you could win overall (when both days are combined). As an exaggerated example, consider a variant: Day You Your friend Saturday $\frac{98}{99} = 98.99\%$ $\frac{1}{1} = \color{red}{\mathbf{100\%}}$ Sunday $\frac{0}{1} = 0\%$ $\frac{1}{99} = \color{red}{\mathbf{1.01\%}}$ Total $\frac{98}{100} = \color{red}{\mathbf{98\%}}$ $\frac{2}{100}= 2\%$ The dominating groups are clearly those with $99$ problems attempted. Those with $1$ only affects the winner of the corresponding day; they barely do anything to the total. When we line up the groups with equal sizes, we can see this paradox vanishing: Size You Your friend Big $\frac{98}{99} = \color{red}{\mathbf{98.99\%}}$ $\frac{1}{99} = 1.01\%$ Small $\frac{0}{1} = 0\%$ $\frac{1}{1} = \color{red}{\mathbf{100\%}}$ Total $\frac{98}{100} = \color{red}{\mathbf{98\%}}$ $\frac{2}{100} = 2\%$ Let's try another example to illustrate these ideas: Two new drugs AntiCynicismia and AntiMisantropia are currently in a clinical trial phase, where the pharmacists determine whether the drugs are safe to use. The trial was divided into five distinct groups and the results of the experiment are as follows: Drug name AntiCynicismia AntiMisantropia Group A 436 out of 545 people were cured, or 80% success rate 9 out of 10 people were cured, or 90% success rate Group B 245 out of 350 people were cured, or 70% success rate 16 out of 20 people were cured, or 80% success rate Group C 48 out of 80 people were cured, or 60% success rate 21 out of 30 people were cured, or 70% success rate Group D 10 out of 20 people were cured, or 50% success rate 180 out of 300 people were cured, or 60% success rate Group E 2 out of 5 people were cured, or 40% success rate 320 out of 640 people were cured, or 50% success rate It may seem like AntiMisantropia is a more effective drug based on the success rates on different groups. However, this is not true! Given that $x\%$ is the difference in success rates between these two drugs, find the value of $x$. The above definition provides one common form of Simpson's paradox. However, it can occur in other ways. The data, instead of "yes" and "no" (binary successes), may be arbitrary real numbers, and we can still have Simpson's paradox with the averages: Category Faction 1 Faction 2 Category 1 $6, 7, 8, 9 \to 7.5$ $10 \to \color{red}{\mathbf{10}}$ Category 2 $0 \to 0$ $1, 2, 3, 4 \to \color{red}{\mathbf{2.5}}$ Total $0, 6, 7, 8, 9 \to \color{red}{\mathbf{6}}$ $1, 2, 3, 4, 10 \to 4$ One real-world example using the average of real numbers occurs with income tax. Between 1974 and 1978, the U.S. tax rate decreased for every category of earning (under $5000, $5000-$10000, etc). When aggregated across all of the people, however, the average tax rate increased! The trend can also be the median instead of the average: Category Faction 1 Faction 2 Category 1 $6, 7, 8, 9 \to 7.5$ $10 \to \color{red}{\mathbf{10}}$ Category 2 $0 \to 0$ $1, 2, 3, 4 \to \color{red}{\mathbf{2.5}}$ Total $0, 6, 7, 8, 9 \to \color{red}{\mathbf{7}}$ $1, 2, 3, 4, 10 \to 3$ In fact, one real-world example of Simpson's paradox involves median wages. Median US wage between 2000 and 2012 has risen (about 1%). However, median US wage across the same period has fallen for every subgroup: high school dropouts, high school graduates but no college education, college education, and Bachelor/higher degree. While technically the trend can be exhibited by many functions, the best trends are those that people wouldn't expect to be able to be reversed when the groups are combined. Average (mean) and median make good trends; these are likely counter-intuitive, which exactly explains the name paradox. University of California Admission Rates A study showed that, overall, men were accepted more than women (44% vs 35%). However, looking at each department, women were usually accepted at a rate equal to or higher than the rate at which men were accepted. What was happening? In fact, women tended to apply to departments which were harder to be admitted into. Kidney Stone Treatment / Ambulances vs. Helicopters Advanced surgical procedures should perform better than traditional treatment on kidney stones. When the data is grouped into treating small kidney stones and large kidney stones, the advanced surgical procedures outperform traditional treatment in each group. However, when all of the cases are combined, the traditional treatment outperforms! How could this be? Well, the advanced surgical procedures were used more frequently when the kidney stones were large. Accordingly, these cases had high failure rates relative to smaller stones. Thus, because the advanced surgical procedure was used most in "tough" surgeries, it performed "worse" overall than traditional treatment. In fact, there is an analogous result with medical evacuation helicopters and traditional ambulances. In the overall data, the helicopters actually do worse at saving lives than ambulances, but this is because they are sent to the higher-risk situations. Low Birth Weight Paradox Babies born to smokers have a higher mortality rate than babies born to non-smokers. Babies can be born under-weight. It turns out that normal-weight babies born to smokers have an equal mortality rate to normal-weight babies born to non-smokers. However, under-weight babies born to smokers have a lower mortality rate than under-weight babies born to non-smokers. Can you guess why this might be the case? Batting Averages A baseball player can have higher batting average than another on each of two years, but lower than the other when the two are combined. In one case, David Justice had a higher batting average than Derek Jeter in 1995 and 1996, but across the two years, Jeter's average was higher. It is possible to win a higher percentage of votes in multiple areas, yet lose the overall vote. This is a real-world phenomenon that can be partially seen in the U.S. electoral college model. Try your hand at this "Simpson's" example: In a recent election, Sideshow Bob and Joe Quimby decided to run for election for Mayor of Springfield. The decision depends on the results in two districts: City and Countryside. Whichever candidate wins both districts wins the election. The results shows that In the City: $ 15000 $ out of $ 25000 $ voted for Sideshow Bob, while $ 4000 $ out of $ 5000 $ voted for Joe Quimby. In the Countryside: $ 1000 $ out of $ 5000 $ voted for Sideshow Bob, while $ 7500 $ out of $ 25000 $ voted for Joe Quimby. Tabulation of data: \[ \def \arraystretch{2.5} \begin{array} { \| l \| l \|l \|} \hline \text{Candidate} & \text{City} & \text{Countryside} \\ \hline \text{Sideshow Bob} & \dfrac{15000}{25000} = 60\% & \dfrac{1000}{5000} = 20\% \\ \hline \text{Joe Quimby} & \dfrac{4000}{5000} = \color{red}{\mathbf{80\%}} & \dfrac{7500}{25000} = \color{red}{\mathbf{30\%}} \\ \hline \end{array} \] Because there's a higher percentage of people who have voted for Joe Quimby in both the City and Countryside, Joe Quimby was re-elected as the Mayor of Springfield. If the decision does not depend on winning individual districts, but instead on the winning more votes in the entire population, then show that Sideshow Bob would have won the election. Also, if Sideshow Bob beat Joe Quimby by $x \% $ of the total voters, what is the value of $x?$ Image Credit: Simpsons Wikia. Feel free to add your own examples below! If you can polish it up, your example might even be featured above. Andrew: I saw a real life example of Simpson's paradox when I did data analysis for my old company. The passing rates on the algebra end of course exam went up in each grade from 2012 to 2013, but the overall passing rate went down! Write yours below! Here's a few interesting examples to try your hand at. This first one is pretty straightforward: More people prefer cars with automatic transmission. More people prefer cars with manual transmission. There is no difference. A survey was conducted on different age groups to determine whether more people prefer cars with automatic transmission or manual transmission. Age 16 to 21 Age 22 to 30 Age 31 to 50 Age 51 and higher Automatic 90% of 100 people 60% of 200 people 50% of 300 people 50% of 400 people Manual 80% of 400 people 40% of 300 people 40% of 200 people 40% of 100 people According to the survey result above, what is the conclusion? Image Credit: Flickr Sophie. Here's more of a challenge: I have four boxes, each containing a number of red marbles and blue marbles. Box A Box B Box C Box D \[ \text{Red marbles}\] \[70\] \[y\] \[2\] \[7\] \[\text{Blue marbles}\] \[30\] \[3\] \[98\] \[53\] If the probability of randomly selecting a red marble from Box A is $a$, and the probability of randomly selecting a red marble from Box B is $b$, then $a < b$. Suppose we group all the marbles in Box A and Box C into another Box AC; likewise we group all the the marbles in Box B and Box D into another Box BD. Now, there is a higher probability of randomly selecting a red marble from Box AC than from Box BD. What is the sum of the smallest and the largest possible values of $y$ for which the above criteria is satisfied? Image Credit: Flickr Lyle. If you're looking for something really tough, check this one out: Find the smallest example of (strict) Simpson's paradox; that is, construct such table where the number of cases is minimum. Formally, suppose that $a,b,x,y$ are nonnegative integers and $A,B,X,Y $ are positive integers such that $a \le A, b \le B, x \le X, y \le Y$, and also $\dfrac{a}{A} > \dfrac{x}{X}$, $\dfrac{b}{B} > \dfrac{y}{Y}$, but $\dfrac{a+b}{A+B} < \dfrac{x+y}{X+Y}$. Determine the minimum value of $A+B+X+Y$. Example: There are two kinds of kidney stone problems, those with small stones and those with large stones. There are also two kinds of treatments, a simple treatment and a complex treatment. The number of success cases, divided by the number of cases for each stone/treatment combination, is displayed in the table below. Small stone Large stone Both Complex treatment 81/87 (93%) 192/263 (73%) 273/350 (78%) Simple treatment 234/270 (87%) 55/80 (69%) 289/350 (83%) As one can see, the complex treatment performs better with small stone cases, and so as with large stone cases, but when the data is combined, the simple treatment performs better. In the sample above, there are a total of 700 cases considered, with 350 complex treatments and 350 simple treatments (or alternatively 357 small stone cases and 343 large stone cases). This problem asks for the minimum possible total number of cases considered. Clarification: In usual Simpson's paradox, it's allowed to have several weak inequalities (some of the inequalities above may actually be equalities). This problem thus has a stronger form of Simpson's paradox, where none of the inequalities may be an equality.	{"url":"https://brilliant.org/wiki/simpsons-paradox/?subtopic=paradoxes&chapter=paradoxes-in-probability","timestamp":"2024-11-09T07:32:35Z","content_type":"text/html","content_length":"79664","record_id":"<urn:uuid:893311bc-8a15-4a44-ada3-0f9d7ee4c696>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00341.warc.gz"}
Items where Year is 2022 Number of items: 50. MoL-2022-19: Almeida, Rodrigo Nicolau (2022) Polyatomic Logics and Generalised Blok-Esakia Theory with Applications to Orthologic and KTB. [Report] PP-2022-05: Aloni, Maria and van Ormondt, Peter (2022) Modified numerals and split disjunction: the first-order case. [Pre-print] (Submitted) MoL-2022-11: Arteche Echeverría, Noel (2022) Parameterized Compilability. [Report] PP-2022-06: Baltag, Alexandru and Bezhanishvili, Nick and Fernandez Duque, David (2022) The Topology of Surprise. [Pre-print] DS-2022-01: Bellomo, Anna (2022) Sums, Numbers and Infinity: Collections in Bolzano’s Mathematics and Philosophy. Doctoral thesis, University of Amsterdam. PP-2022-07: Bezhanishvili, Nick and Cleani, Antonio M. (2022) Translational embeddings via stable canonical rules. [Pre-print] PP-2022-04: Bezhanishvili, Nick and Martins, Miguel and Moraschini, Tommaso (2022) Bi-intermediate logics of trees and co-trees. [Pre-print] DS-2022-04: Brokkelkamp, Ruben (2022) How Close Does It Get? From Near-Optimal Network Algorithms to Suboptimal Equilibrium Outcomes. Doctoral thesis, ILLC. DS-2022-05: Bussière-Carae, Lwenn (2022) No means No! Speech Acts in Conflict. Doctoral thesis, University of Amsterdam. MoL-2022-02: Carr, James (2022) Hereditary Structural Completeness over K4: Rybakov’s Theorem Revisited. [Report] MoL-2022-17: Chernev, Anton (2022) Degrees of FMP in extensions of bi-intuitionistic logic. [Report] MoL-2022-10: Cruchten, Mike (2022) Topics in Ω-Automata: A Journey through Lassos, Algebra, Coalgebra and Expressions. [Report] DS-2022-02: Czajkowski, Jan (2022) Post-Quantum Security of Hash Functions. Doctoral thesis, University of Amsterdam. X-2022-06: Dekker, Paul (2022) Truth and Agreement. [Report] MoL-2022-05: Felderhoff, Lukas (2022) Single-Peaked Electorates in Liquid Democracy. [Report] MoL-2022-15: Filipetto, Valentino (2022) Constructing queries from data examples. [Report] MoL-2022-22: Grotenhuis, Lide (2022) Natural Axiomatic Theories and Consistency Strength: A Lakatosian Approach to the Linearity Conjecture. [Report] MoL-2022-04: Janssens, Nicolien S. (2022) Communicate and Vote: Collective Truth-tracking in Networks. [Report] PP-2022-08: Khomskii, Yurii and Oddsson, Hrafn Valtýr (2022) Paraconsistent and Paracomplete Zermelo-Fraenkel Set Theory. [Pre-print] (Submitted) MoL-2022-14: Klochowicz, Tomasz (2022) Investigating semantic and selectional properties of clause-embedding predicates in Polish. [Report] HDS-33: Klop, Jan Willem (2022) Combinatory Reduction Systems. Doctoral thesis, Rijksuniversiteit Utrecht. MoL-2022-24: Knudstorp, Søren Brinck (2022) Modal Information Logics. [Report] MoL-2022-07: Koudijs, Raoul (2022) Learning Modal Formulas via Dualities. [Report] MoL-2022-06: klein Goldewijk, Thomas (2022) Fairness in Perpetual Participatory Budgeting. [Report] MoL-2022-16: Leijnse, Koen (2022) On the Quantum Hardness of Matching Colored Triangles. [Report] MoL-2022-01: Loustalot Knapp, Daniela (2022) Justification of Matching Outcomes. [Report] MoL-2022-20: McCloskey, Erin (2022) Relative Weak Factorization Systems. [Report] DS-2022-06: Mojet, Emma (2022) Observing Disciplines: Data Practices In and Between Disciplines in the 19th and Early 20th Centuries. Doctoral thesis, Universiteit van Amsterdam. MoL-2022-21: Osso, Gian Marco (2022) Some results on the Generalized Weihrauch Hierarchy. [Report] MoL-2022-27: Otten, Daniël D. (2022) De Jongh’s Theorem for Type Theory. [Report] MoL-2022-18: Oudshoorn, Anouk Michelle (2022) Cost Fixed Point Logic. [Report] MoL-2022-03: Peng, Zichen (2022) Simultaneous Substitution Algebras. [Pre-print] DS-2022-03: Ramotowska, Sonia (2022) Quantifying quantifier representations: Experimental studies, computational modeling, and individual differences. Doctoral thesis, University of Amsterdam. MoL-2022-12: Romanovskiy, Vasily (2022) "A-ha, I hadn’t thought of that": the Bayesian Problem of Awareness Growth. [Report] MoL-2022-26: Samwel, Rover Junior (2022) Explorations in Coalgebraic Predicate Logic (With a Focus on Interpolation). [Report] MoL-2022-08: Schmidtlein, Marie Christin (2022) Voting by Axioms. [Report] MoL-2022-13: Soeteman, Arie W. (2022) Artificial Understanding. [Report] X-2022-01: ten Cate, Balder (2022) Lyndon Interpolation for Modal Logic via Type Elimination Sequences. [Report] MoL-2022-25: Vrijbergen, Pepijn (2022) Validity, Logic, and Models. [Report] PP-2022-03: van Benthem, Johan and Bezhanishvili, Nick (2022) Modal structures in groups and vector spaces. [Pre-print] PP-2022-02: van Benthem, Johan and Li, Lei and Shi, Chenwei and Yin, Haoxuan (2022) Hybrid sabotage Modal Logic. [Pre-print] (In Press) MoL-2022-09: van Harskamp, Jasmijn (2022) The National Contests Behind International Success: A Musical Comparison of the Eurovision Song Contest, the Festival di Sanremo and the Melodifestivalen. X-2022-05: van Ulsen, Paul (2022) Bibliografie E.W. Beth. [Report] X-2022-02: van Ulsen, Paul (2022) Onderwijs en onderzoek in logica en wetenschapsfilosofie. [Report] X-2022-04: van Ulsen, Paul (2022) Organisaties en genootschappen. [Report] X-2022-03: van Ulsen, Paul (2022) Wetenschapsfilosofie. [Report] MoL-2022-29: Weststeijn, Nikki (2022) The Relation Between Shannon Information and Semantic Information. [Report] DS-2022-07: Witteveen, Freek Gerrit (2022) Quantum information theory and many-body physics. Doctoral thesis, Universiteit van Amsterdam. MoL-2022-23: Zhang, Tianwei (2022) Bisimulations over Parity Formulas. [Report] PP-2022-01: Özgün, Aybüke and Schoonen, Tom (2022) The Logical Development of Pretense Imagination. 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Harlan Brothers, founder of Brothers Technology, had his first success as an inventor in the early nineties with his sale of the Bathtub Buddy to a major manufacturer of small appliances, Salton Inc.&nbsp Salton incorporated this unique water alarm into its popular Wet Tunes line of products. Since then Harlan has obtained five patents and worked as a design consultant. His research into the problem of creating and authenticating tamper-proof digital recordings led to a patent for The Event Verification System (EVS). EVS offers a broad solution that fulfills the ever-growing need for irrefutable authentication of digital information. The patent was sold to a well-established intellectual property firm. Current projects range from novel consumer devices to commercial encryption techniques and educational tools. In the area of pure research, Harlan has a long-standing interest in number theory and its applications.&nbsp He has discovered formulas and relationships relating to the constants e, pi, and Euler's gamma.&nbsp His paper entitled "Improving the Convergence of Newton's Series Approximation for e" includes the fastest known methods for computing this fundamental constant of nature.&nbsp The article appears in the January 2004 issue of The College Mathematics Journal.&nbsp Here is a presentation on the subject from the Third Annual Citizen Science Conference.&nbsp For six years he worked with Michael Frame and Benoit Mandelbrot at Yale University to explore the use of fractals in mathematics education.&nbsp Projects at Yale included a lecture and workshop on the subject of fractal music composition and analysis. Here is a brief introduction to fractals in PDF format [1.7MB]. Working with Benoit inspired Harlan to explore new places to apply fractal geometry, both in the natural world and in number theory. Harlan has published and lectured on fractal geometry in music, has used it to explore an original 3D extension of Pascal's triangle, and used Iterated Function Systems to uncover biases in the distribution of prime numbers. The following links reference early research on one of the fundamental constants of Nature, the base of the natural logarithm, e: NASA (Serendipit-e, John Knox) Mathematical Association of America (Science News Online, Ivars Peterson) Science Magazine (Random Samples, Dana Mackenzie) Wolfram Research (Eric Weisstein) UAB Magazine (Dan Willson) Here are to more information on e. H. J. Brothers, "Using Iterated Function Systems to Reveal Biases in the Distribution of Prime Numbers." arXiv e-prints, arXiv:1701.00698, December 2016. H. J. Brothers, "The Nature of Fractal Music," in Benoit Mandelbrot - A Life in Many Dimensions, edited by Michael Frame, World Scientific Publishing (May, 2015). (Supplementary material) N. Neger and H. J. Brothers, "Benoit Mandelbrot: Educator," in Benoit Mandelbrot - A Life in Many Dimensions, edited by Michael Frame, World Scientific Publishing (May, 2015). M. F. Barnsley, M. Berry, M. Frame, I. Stewart, D. Mumford, K. Falconer, R. Eglash, H. J. Brothers, N. Lesmoir-Gordon, J. Barrallo, Glimpses of Benoit Mandelbrot (1924-2010). Notices of the American Mathematical Society, Vol. 8, No. 59, 2012; pages 1056-1063. H. J. Brothers, "Pascal's prism." The Mathematical Gazette, Vol. 96, No. 536, 2012; pages 213-220. (Supplementary material) H. J. Brothers, "Pascal's triangle: The hidden stor-e ." The Mathematical Gazette, Vol. 96, No. 535, 2012; pages 145-148. H. J. Brothers, "Finding e in Pascal's triangle." Mathematics Magazine, Vol. 85, No. 1, 2012; page 51. H. J. Brothers, "Mandel-Bach Journey: A marriage of musical and visual fractals." Proceedings of Bridges Pecs, 2010; pages 475-478. H. J. Brothers, "Intervallic scaling in the Bach cello suites." Fractals, Vol. 17, No. 4, 2009; pages 537-545. (Supplementary material can be found here.) H. J. Brothers, "How to design your own pi to e converter." The AMATYC Review, Vol. 30, No. 1, 2008; pages 29 35. H. J. Brothers, "Structural scaling in Bach's cello suite no. 3." Fractals, Vol. 15, No. 1, 2007; pages 89 95. (Supplementary material can be found here.) The following files are in Adobe PDF format. H. J. Brothers, Improving the convergence of Newton's series approximation for e. College Mathematics Journal, Vol. 35, No. 1, 2004; pages 34-39. [723KB] (The above article appears with permission of CMJ. Supplementary material can be found here.) J. A. Knox and H. J. Brothers, Novel series-based approximations to e. College Mathematics Journal, Vol. 30, No. 4, 1999; pages 269-275. [126KB] (NOTE: The above paper was selected by mathematicians Ron Larson, Robert P. Hostetler, and Bruce H. Edwards as one of the fifty best articles on calculus from MAA periodicals. It is now a supplement to their textbook, Calculus with Analytic Geometry, Seventh Edition.) H. J. Brothers and J. A. Knox, New closed-form approximations to the Logarithmic Constant e. The Mathematical Intelligencer, Vol. 20, No. 4, 1998; pages 25-29. [1,143KB]	{"url":"http://www.brotherstechnology.com/math/index.html","timestamp":"2024-11-10T06:23:46Z","content_type":"application/xhtml+xml","content_length":"15532","record_id":"<urn:uuid:9fb2499b-fcd4-4bdd-a293-f1a1f9e749eb>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00049.warc.gz"}
How Many Pennies Are In $100? How many pennies can be realized from 100 dollars? This question could be difficult to answer, that’s if you do not know how to make the Penny to Dollar calculation, or Dollar to Penny. In this article, I will be walking you through the process of knowing how much a penny is in a dollar. How Many Pennies Are In $100? Before understanding how many pennies are in 100 dollars, is it important to know how many pennies will go for a dollar? However, for every dollar, there are 100 pennies, hence for every 100 dollars, you will get 10000 pennies. How Much Is 1000 Pennies Worth? Alright, let’s do the reverse calculation in knowing how much dollars can be gotten from pennies. Since a dollar equals 100 pennies, therefore, 1000 pennies will equal $10. Here is how you can do the calculation; you divide the available pennies, with the number of pennies for a dollar, which is 1000 % 100, which will be equal to $10. Therefore, for every 1000 pennies, you will get a sum of $10. How Much Is a Penny? A Penny since to be the lowest amount of money in Dollars when converted. A Penny doesn’t have any Worth currently, because for every penny you get, you will have as much as $0.01. What’s 100000 Pennies in Dollars? For every 100000 pennies, you will get a sum of one thousand dollars in total. How Many Dollars Is 5 Million Pennies? If you convert the sum of 5 million pennies into dollars, you will have the sum of $50, 000 dollars. How Much Is 80000 Pennies in Cash? Since a penny is $0.01, therefore 80000 pennies will be equal $800. I Need 25 Dollars Right Now (How To Make 25 dollars Fast) Tips on One Dollar A Day Savings Plan What To Do With 6000 Dollars Explained In this article, I have provided you with the calculation on knowing how many pennies are for 100 dollars.The method of calculation done here, also shows how much dollars will go for a certain amount of Penny.	{"url":"https://www.banktransfercodes.com/how-many-pennies-are-in-100/","timestamp":"2024-11-06T04:04:03Z","content_type":"text/html","content_length":"52068","record_id":"<urn:uuid:30ecff88-e2b9-490a-8bb2-8d9e6168d301>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00833.warc.gz"}
French paper explaining renormalizability "To conclude, we see that although the renormalization procedure has not evolved much these last thirty years, our interpretation of renormalization has drasti-cally changed10: the renormalized theory was assumed to be fundamental, while it is now believed to be only an effective one; Λ was interpreted as an artificial parameter that was only useful in intermediate calculations, while we now believe that it corresponds to a fundamental scale where new physics occurs; non-renormalizable couplings were thought to be forbidden, while they are now interpreted as the remnants of interaction terms in a more fundamental theory. Renormalization group is now seen as an efficient tool to build effective low energy theories when large fluctuations occur between two very different scales that change qualitatively and quantitatively the physics. We know now that the invisible hand that creates divergences in some theories is actually the existence in these theories of a no man’s land in the energy (or length) scales for which cooperative phenomena can take place, more precisely, for which fluctuations can add up coherently.10 In some cases, they can destabilize the physical picture we were relying on and this manifests itself as divergences. Renormalization, and even more renormalization group, is the right way to deal with these fluctuations. ... Let us draw our first conclusion. Infinities occur in the perturbation expansion of the theory because we have assumed that it was not regularized. Actually, these divergences have forced us to regularize the expansion and thus to introduce a new scale Λ. Once regularization has been performed, renormalization can be achieved by eliminating g0. The limit Λ → ∞ can then be taken. The process is recursive and can be performed only if the divergences possess, order by order, a very precise structure. This structure ultimately expresses that there is only one coupling constant to be renormalized. This means that imposing only one prescription at x = μ is enough to subtract the divergences for all x. In general, a theory is said to be renormalizable if all divergences can be recursively subtracted by imposing as many prescriptions as there are independent parameters in the theory. In QFT, these are masses, coupling constants, and the normalization of the fields. An important and non-trivial topic is thus to know which parameters are independent, because symmetries of the theory (like gauge symmetries) can relate different parameters (and Green functions). Let us once again recall that renormalization is nothing but a reparametrization in terms of the physical quantity gR. The price to pay for renormalizing F is that g0 becomes infinite in the limit Λ → ∞, see Eq. (12). We again emphasize that if g0 is a non-measurable parameter, useful only in intermediate calculations, it is indeed of no consequence that this quantity is infinite in the limit Λ → ∞. That g0 was a divergent non-physical quantity has been common belief for decades in QFT. The physical results given by the renormalized quantities were thought to be calculable only in terms of unphysical quantities like g0 (called bare quantities) that the renormalization algorithm could only eliminate afterward. It was as if we had to make two mistakes that compensated each other: first introduce bare quantities in terms of which everything was infinite, and then eliminate them by adding other divergent quantities. Undoubtedly, the procedure worked, but, to say the least, the interpretation seemed rather obscure. ... A very important class of field theories corresponds to the situation where g0 is dimensionless, and x, which in QFT represents coordinates or momenta, has dimensions (or more generally when g0 and x have independent dimensions). In four-dimensional space-time, quantum electrodynamics is in this class, because the fine structure constant is dimensionless; quantum chromodynamics and the Weinberg-Salam model of electro-weak interactions are also in this class. In four space dimensions the φ^4 model relevant for the Ginzburg-Landau-Wilson approach to critical phenomena is in this class too. This particular class of renormalizable theories is the cornerstone of renormalization in field theories. ... Note that we have obtained logarithmic divergences because we have studied the renormalization of a dimensionless coupling constant. If g0 was dimensional, we would have obtained power law divergences. This is for instance what happens in QFT for the mass terms ... there should exist an equivalence class of parametrizations of the same theory and that it should not matter in practice which element in the class is chosen. This independence of the physical quantity with respect to the choice of prescription point also means that the changes of parametrizations should be a (renormalization) group law ... if we were performing exact calculations: we would gain no new physical information by implementing the renormalization group law. This is because this group law does not reflect a symmetry of the physics, but only of the parametrization of our solution. This situation is completely analogous to what happens for the solution of a differential equation: we can parametrize it at time t in terms of the initial conditions at time t0 for instance, or we can use the equation itself to calculate the solution at an intermediate time τ and then use this solution as a new initial condition to parametrize the solution at time t. The changes of initial conditions that preserve the final solution can be composed thanks to a group law. ... Our main goal in this section is to show that, independently of the underlying physical model, dimensional analysis together with the renormalizability con- straint, determine almost entirely the structure of the divergences. This underlying simplicity of the nature of the divergences explains that there is no combinatorial miracle of Feynman diagrams in QFT as it might seem at first glance. 8) The cut-off Λ, first introduced as a mathematical trick to regularize integrals, has actually a deep physical meaning: it is the scale beyond which new physics occur and below which the model we study is a good effective description of the physics. In general, it involves only the renormalizable couplings and thus cannot pretend to be an exact description of the physics at all scales. However, if Λ is very large compared with the energy scale in which we are interested, all non-renormalizable couplings are highly suppressed and the effective model, retaining only renormalizable couplings, is valid and accurate (the Wilson RG formalism is well suited to this study, see Refs. 25 and 26). In some models — the asymptotically free ones — it is possible to formally take the limit Λ → ∞ both perturbatively and non-perturbatively, and there is therefore no reason to invoke a more fundamental theory taking over at a finite (but large) Λ. ... (1) The long way of renormalization starts with a theory depending on only one parameter g0, which is the small parameter in which perturbation series are expanded. In particle physics, this parameter is in general a coupling constant like an electric charge involved in a Hamiltonian (more precisely the fine structure constant for electrodynamics). This parameter is also the first order contribution of a physical quantity F. In particle/ statistical physics, F is a Green/correlation function. The first order of perturbation theory neglects fluctuations — quantum or statistical — and thus corresponds to the classical/mean field approximation. The parameter g0 also is to this order a measurable quantity because it is given by a Green function. Thus, it is natural to interpret it as the unique and physical coupling constant of the problem. If, as we suppose in the following, g0 is dimensionless, so is F. Moreover, if x is dimensional — it represents momenta in QFT — it is natural that F does not depend on it as is found in the classical theory, that is, at first order of the perturbation expansion. (2) If F does depend on x, as we suppose it does at second order of perturbation theory, it must depend on another dimensional parameter, through the ratio of x and /\ . If we have not included this parameter from the beginning in the model, the x-dependent terms are either vanishing, which is what happens at first order, or infinite as they are at second and higher orders. This is the very origin of divergences (from the technical point of view). (3) These divergences require that we regularize F. This requirement, in turn, requires the introduction of the scale that was missing. In the context of field theory, the divergences occur in Feynman diagrams for high momenta, that is, at short distances. The cut-off suppresses the fluctuations at short distances compared with /\^−1. " Note with quantum gravity tiny black holes form at short distances giving a natural cut off at Lp at least where delta(x) ~ h/delta(p) + Lp^2delta(p)/h the second term on RHS is added to the Heisenberg uncertainty principle. "In statistical physics, this scale, although introduced for formal reasons, has a natural interpretation because the theories are always effective theories built at a given microscopic scale. It corresponds in general to the range of interaction of the constituents of the model, for example, a lattice spacing for spins, the average intermolecular distance for fluids. In particle physics, things are less simple. At least psychologically. It was indeed natural in the early days of quantum electrodynamics to think that this theory was fundamental, that is, not derived from a more fundamental theory. More precisely, it was believed that QED had to be mathematically internally consistent, even if in the real world new physics had to occur at higher energies. Thus, the regulator scale was introduced only as a trick to perform intermediate calculations. The limit /\ → ∞ was supposed to be the right way to eliminate this unwanted scale, which anyway seemed to have no interpretation. We shall see in the following that the community now interprets the renormalization process differently (4) Once the theory is regularized, F can be a nontrivial function of x. The price is that different values of x now correspond to different values of the coupling constant (defined as the values of F for these x). Actually, it does no longer make sense to speak of a coupling constant in itself. The only meaningful concept is the pair (μ, gR(μ)) of coupling constants at a given scale. The relevant question now is, “What are the physical reasons in particle/statistical physics that make the coupling constants depend on the scale while they are constants in the classical/mean field approximation?” As mentioned, for particle physics, the answer is the existence of new quantum fluctuations corresponding to the possibility of creating (and annihilating) particles at energies higher than mc^2. What was scale independent in the classical theory becomes scale dependent in the quantum theory because, as the available energy increases, more and more particles can be created. The pairs of (virtual) particles surrounding an electron are polarized by its presence and thus screen its charge. As a consequence, the charge of an electron depends on the distance (or equivalently the energy) at which it is probed, at least for distances smaller than the Compton wavelength. Note that the energy scale mc^2 should not be confused with the cut-off scale . mc^2 is the energy scale above which quantum fluctuations start to play a significant role while /\ is the scale where they are cut-off. Thus, although the Compton wave length is a short distance scale for the classical theory, it is a long distance scale for QFT, the short one being /\^−1. There are thus three domains of length scales in QFT: above the Compton wave length where the theory behaves classically (up to small quantum corrections coming from high energy virtual processes), between the Compton wave length and the cut-off scale /\^−1 where the relativistic and quantum fluctuations play a great role, and below /\^−1 where a new, more fundamental theory has to be In statistical physics, the analogue of the Compton wave length is the correlation length which is a measure of the distance at which two microscopic constituents of the system are able to influence each other through thermal fluctuations.38 For the Ising model for instance, the correlation length away from the critical point is the order of the lattice spacing and the corrections to the mean-field approximation due to fluctuations are small. Unlike particle physics where the masses and therefore the Compton wavelengths are fixed, the correlation lengths in statistical mechanics can be tuned by varying the temperature. Near the critical temperature where the phase transition takes place, the correlation length becomes extremely large and fluctuations on all length scales between the microscopic scale of order /\^−1, a lattice spacing, and the correlation length add up to modify the mean-field behavior (see Refs. 21, 22 and also Ref. 23 for a bibliography in this subject). We see here a key to the relevance of renormalization: two very different scales must exist between which a non-trivial dynamics (quantum or statistical in our examples) can develop. This situation is a priori rather unnatural as can be seen for phase transitions, where a fine tuning of temperature must be implemented to obtain correlation lengths much larger than the microscopic scale. Most of the time, physical systems have an intrinsic scale (of time, energy, length, etc) and all the other relevant scales of the problem are of the same order. All phenomena occurring at very different scales are thus almost completely suppressed. The existence of a unique relevant scale is one of the reasons why renormalization is not necessary in most physical theories. In QFT it is mandatory because the masses of the known particles are much smaller than a hypothetical cut-off scale , still to be discovered, where new physics should take place. This is a rather unnatural situation, because, contrary to phase transitions, there is no analogue of a temperature that could be fine-tuned to create a large splitting of energy, that is, mass, scales. The question of naturalness of the models we have at present in particle physics is still largely open, although there has been much effort in this direction using supersymmetry. (5) The classical theory is valid down to the Compton/correlation length, but cannot be continued naively beyond this scale; otherwise, when mixed with the quantum formalism, it produces divergences. Actually, it is known in QFT that the fields should be considered as distributions and not as ordinary functions. The need for considering distributions comes from the non-trivial structure of the theory at very short length scale where fluctuations are very important. At short distances, functions are not sufficient to describe the field state, which is not smooth but rough, and distributions are necessary. Renormalizing the theory consists actually in building, order by order, the correct “distributional continuation” of the classical theory. The fluctuations are then correctly taken into account and depend on the scale at which the theory is probed: this non-trivial scale dependence can only be taken into account theoretically through the dependence of the (analogue of the) function F with x and thus of the coupling with the scale μ. (6) If the theory is perturbatively renormalizable, the pairs (μ, g(μ)) form an equivalence class of parametrizations of the theory. The change of parametrization from (μ, g(μ)) to (μ′, g(μ′)), called a renormalization group transformation, is then performed by a law which is self-similar, that is, such that it can be iterated several times while being form-invariant.19,20 This law is obtained by the integration of ... In particle physics, the β-function gives the evolution of the strength of the interaction as the energy at which it is probed varies and the integration of the β-function resums partially the perturbation expansion. First, as the energy increases, the coupling constant can decrease and eventually vanish. This is what happens when α > 0 in Eqs. (65) and (66). In this case, the particles almost cease to interact at very high energies or equivalently when they are very close to each other. The theory is then said to be asymptotically free in the ultraviolet domain.3,5 Reciprocally, at low energies the coupling increases and perturbation theory can no longer be trusted. A possible scenario is that bound states are created at a sufficiently low energy scale so that the perturbation approach has to be reconsidered in this domain to take into account these new elementary excitations. Non-abelian gauge theories are the only known theories in four spacetime dimensions that are ultraviolet free, and it is widely believed that quantum chromodynamics — which is such a theory — explains quark confinement. The other important behavior of the scale dependence of the coupling constant is obtained for α < 0 in which case it increases at high energies. This corresponds for instance to quantum electrodynamics. For this kind of theory, the dramatic increase of the coupling at high energies is supposed to be a signal that the theory ceases to be valid beyond a certain energy range and that new physics, governed by an asymptotically free theory (like the standard model of electro-weak interactions) has to take place at short distances. (7) Renormalizability, or its non-perturbative equivalent, self-similarity, ensures that although the theory is initially formulated at the scale μ, this scale together with g0 can be entirely eliminated for another scale better adapted to the physics we study. If the theory was solved exactly, it would make no difference which parametrization we used. However, in perturbation theory, this renormalization lets us avoid calculating small numbers as differences of very large ones. It would indeed be very unpleasant, and actually meaningless, to calculate energies of order 100GeV, for instance — the scale μ of our analysis — in terms of energies of order of the Planck scale ? 10^19 GeV, the analogue of the scale /\ . In a renormalizable theory, the possibility to perturbatively eliminate the large scale has a very deep meaning: it is the signature that the physics is short distance insensitive or equivalently that there is a decoupling of the physics at different scales. The only memory of the short distance scale lies in the initial conditions of the renormalization group flow, not in the flow itself: the β-function does not depend on / \. We again emphasize that, usually, the decoupling of the physics at very different scales is trivially related to the existence of a typical scale such that the influence of all phenomena occurring at different scales is almost completely suppressed. Here, the decoupling is much more subtle because there is no typical length in the whole domain of length scales that are very small compared with the Compton wave length and very large compared with /\^−1. Because interactions among particles correspond to non-linearities in the theories, we could naively believe that all scales interact with each others — which is true — so that calculating, for instance, the low energy behavior of the theory would require the detailed calculation of all interactions occurring at higher energies. Needless to say that in a field theory, involving infinitely many degrees of freedom — the value of the field at each point — such a calculation would be hopeless, apart from exactly solvable models. Fortunately, such a calculation is not necessary for physical quantities that can be calculated from renormalizable couplings only. Starting at very high energies, typically , where all coupling constants are naturally of order 1, the renormalization group flow drives almost all of them to zero, leaving only, at low energies, the renormalizable couplings. This is the interpretation of non-renormalizable couplings. They are not terrible monsters that should be forgotten as was believed in the early days of QFT. They are simply couplings that the RG flow eliminates at low energies. If we are lucky, the renormalizable couplings become rather small after their RG evolution between /\ and the scale μ at which we work, and perturbation theory is valid at this scale. We see here the phenomenon of universality: among the infinitely many coupling constants that are a priori necessary to encode the dynamics of the infinitely many degrees of freedom of the theory, only a few ones are finally relevant.25 All the others are washed out at large distances. This is the reason why, perturbatively, it is not possible to keep these couplings finite at large distance, and it is necessary to set them to zero.39 The simplest non-trivial example of universality is given by the law of large numbers (the central limit theorem) which is crucial in statistical mechanics.21 In systems where it can be applied, all the details of the underlying probability distribution of the constituents of the system are irrelevant for the cooperative phenomena which are governed by a gaussian probability distribution.24 This drastic reduction of complexity is precisely what is necessary for physics because it lets us build effective theories in which only a few couplings are kept.10 Renormalizability in statistical field theory is one of the non-trivial generalizations of the central limit theorem. (8) The cut-off , first introduced as a mathematical trick to regularize integrals, has actually a deep physical meaning: it is the scale beyond which new physics occur and below which the model we study is a good effective description of the physics. In general, it involves only the renormalizable couplings and thus cannot pretend to be an exact description of the physics at all scales. However, if is very large compared with the energy scale in which we are interested, all non-renormalizable couplings are highly suppressed and the effective model, retaining only renormalizable couplings, is valid and accurate (the Wilson RG formalism is well suited to this study, see Refs. 25 and 26). In some models — the asymptotically free ones — it is possible to formally take the limit /\ → ∞ both perturbatively and non-perturbatively, and there is therefore no reason to invoke a more fundamental theory taking over at a finite (but large) . Let us emphasize here several interesting points. (i) For a theory corresponding to the pair (μ, gR(μ)), the limit /\ → ∞ must be taken within the equivalence class of parametrizations to which (μ, gR(μ)) belongs.40 A divergent non-regularized perturbation expansion consists in taking = ∞ while keeping g0 finite. From this viewpoint, the origin of the divergences is that the pair ( u = ∞, g0) does not belong to any equivalence class of a sensible theory. Perturbative renormalization consists in computing g0 as a formal powers series in gR (at finite ), so that (μ0, g0) corresponds to a mathematically consistent theory; we then take the limit /\ → ∞. (ii) Because of universality, it is physically impossible to know from low energy data if /\ is very large or truly infinite. (iii) Although mathematically consistent, it seems unnatural to reverse the RG process while keeping only the renormalizable couplings and thus to imagine that even at asymptotically high energies, Nature has used only the couplings that we are able to detect at low energies. It seems more natural that a fundamental theory does not suffer from renormalization problems. String theory is a possible candidate. 27 To conclude, we see that although the renormalization procedure has not evolved much these last thirty years, our interpretation of renormalization has drastically changed10: the renormalized theory was assumed to be fundamental, while it is now believed to be only an effective one; was interpreted as an artificial parameter that was only useful in intermediate calculations, while we now believe that it corresponds to a fundamental scale where new physics occurs; non-renormalizable couplings were thought to be forbidden, while they are now interpreted as the remnants of interaction terms in a more fundamental theory. Renormalization group is now seen as an efficient tool to build effective low energy theories when large fluctuations occur between two very different scales that change qualitatively and quantitatively the physics." complete paper e.g. the spin 1 tetrad vector e^I theory of gravity is to Einstein's spin 2 metric tensor guv theory of gravity as the SU2 gauge theory of the weak force is to the 4spinor model of Fermi that was not	{"url":"https://stardrive.org/index.php/all-blog-articles/2660-french-paper-explaining-renormalizability","timestamp":"2024-11-11T23:35:02Z","content_type":"text/html","content_length":"40790","record_id":"<urn:uuid:f4fa3717-6013-4516-b54a-4375940ebcc3>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00137.warc.gz"}
SAG Mill Feed Particle Size Analyser - 911Metallurgist We describe here one of the very first pioneering SAG Mill Feed Particle Size Analysers: A set of preliminary experiments was carried out in order to assess the hardware-software’s behaviour. Static ore was used to obtain the right sequence of image processing algorithms. The camera was installed 3 mts away from the commercial sizer over the feed conveyor of the SAG Mill. This analyzer was calibrated following the manufacturer’s procedure. The process characteristics are those of any normal working day: Conveyor belt width………………………………………………………………..1 mt Conveyor belt speed……………………………………………………………..2 mt” sec Capacity…………………………………………………………………………….650 MTPH 100%, of then ore under…………………………………………………………….8″ The T.V. analyzer ranges were adjusted to be the same as the ARMCO sizer: 1″, 2″, 3″, 4″. The data output from the commercial sizer was recorded every 1 minute over 5 hours operation. Data from the T.V. analyzer was recorded 10 times per second and then compressed off-line. The graphs in figures 2 and 4 show the close agreement between both analyzers. From the table in figure 3 it can be inferred that both analyzers give the same output. The statistics of the differences of averages µO, µI are not significant for t. 995 = 2.75.0 for all the sizes. One of the tests consisted in the analysis of the behavior of both analyzer under a “step” of increment of the capacity. There is a statistically significant difference between both analyzer’s output for the 500 TPH capacity. See Figure 4. “t” numbers and statistics procedures form Dillon (1969). As mentioned in previous paragraphs, the size distribution percentages are calculated as a fraction of “chord lengths”. The study of the correlation of the T.V. analyzer output and a reference weight distribution from a calibrated stock of ore was carried out in the pilot plant. Original data from weight distribution was converted to a linear distribution using the formula where: F1(x) = Linear frequency distribution os size x chord length based F3(x) = Weight frequency distribution of size x. Calculation were made using a discrete formula, taking into account size range frequencies and the mean value of the limiting sizes of each range. Experimental data shows an adequate correspondence between the outputs of both analyzers for a steady capacity of 600 TMPH. Under dynamic conditions, both analyzers responded with the some speed, although there is a significant output difference for the small size ranges at a 500 TPH capacity. The “t” statistics are showing a significant difference for the outputs of the ARMCO and T.V. based analyzer. The ARMCO analyzer measure lower percentages of ore for the 3″, 2″ and 1″ ranges. This is due to the segregation of the fine particle induced by the lower capacity. This effect tends to group the coarse particles at the centre of the conveyor belt. At a lower TPH, the ARMCO sizer detect less percentage of fines particles due to its measurement principle. The T.V. based analyzer uses exploration lines which are less sensitive to the segregation effects due to their orientation. There is also a good correlation between the output of the T.V. analyzer and the transformed weight distribution of the reference stock. See Figure 5. The Image Analyzer has demonstrated to be an efficient option to ether commercially available products. Since the T.V. analyzer is computer based, more power and sophistication on data minipulation and analysis are possible. Customized analysis is also attractive depending on the particular requirements of the application. The field instrument, the T.V. camera is already a proven apparatus which has demonstrated to withstand a good number of industrial environments. Modular implementation introduces important saving costs since a multipoint analysis uses one computer and several camera linked by a suitable switcher to the digitizer-computer unit. The flexibility of this software based machine makes possible to configure a very powerful operation tool such as automatic “chute” and feeder monitoring. – Material segregation on the conveyor implies a “bias” on the measurement because it is made ever the surface of the material being transported – Measurement of length along exploration lines do not correspond to standard ore size distribution analysis, such as weight distribution or sieve size.	{"url":"https://www.911metallurgist.com/blog/sag-mill-feed-particle-size-analyzer/","timestamp":"2024-11-02T05:01:57Z","content_type":"text/html","content_length":"128427","record_id":"<urn:uuid:7b0bee3f-18ee-4406-b8b9-28c094ecd212>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00586.warc.gz"}
Photonics: The fast lane towards useful Quantum Machine Learning? Machine Learning Over the last decade, machine learning (ML) and Artificial Intelligence (AI) have dominated the information-technology landscape, to the degree where most enterprises use ML-powered solutions. Here are a few examples you may be familiar with: • Virtual assistants like Siri or Alexa use ML to analyse speech patterns. • The proposed self-driving cars of Google and Tesla are the essence of ML, studying and reproducing the behaviour of a human driver. • Translation apps are built from a marriage of machine learning and linguistic rule creation. • Online recommendations from companies like Amazon and Netflix are based on a (worryingly accurate) online profile of your behaviour, built by ML-algorithms Machine Learning focuses on the use of data and algorithms to imitate the way that humans learn, gradually improving its accuracy. These algorithms build a model based on sample data (known as training sets), in order to make predictions or decisions without being explicitly programmed to do so. We can imagine a ML algorithm like a human baby, who using the sample data of the words they hear makes a model of language which can be used to make new sentences. With the emergence and growing power of quantum computers, technology is bracing for its next big upheaval. But the question remains: How can we best use our new quantum computers to improve current machine learning algorithms, or even develop entirely new ones. If you need a rundown on how Quantum Computers work and their advantages, you can check out this article where we give you the basics. Quantum Machine Learning There has been a huge influx of research into this question. In October 2022 alone, there were over 50 articles on the physics archive which mention ‘quantum machine learning’ in the abstract. This enthusiasm is well-placed. Improved ML through faster structured prediction is listed as (one of) the first quantum computing enabled breakthroughs by major management consulting firms such as McKinsey & Company. There are two primary reasons why ML and Quantum Computers are a good fit: 1. Lower requirements on fidelities and error rates compared to other quantum computer applications. Quantum computing devices are inherently vulnerable to noise and information loss, particularly in the NISQ* era. When it comes to ML algorithms, we run small circuits many times, reducing the effects of errors on our results. Like classical ML, we have variable elements that can alter our circuit many times throughout the runtime of the algorithm. This allows the circuit to adapt and overcome errors as they emerge. 2. Machine learning represents data in high-dimensional parameter spaces. Quantum circuits have an exponential advantage for dimensionality.** One of the major advantages of quantum computers is their ability to efficiently encode information. This can greatly decrease the dimension of the required data for an ML algorithm. When we use ML-techniques with Quantum Processing Units (QPUs), we enter a new field of study called Quantum Machine Learning (QML). Why Photonics? Photonic quantum computers are quickly establishing themselves as the dominant architectures not just for near-term quantum applications, but also full-scale fault-tolerant quantum computers. Aside from their advantages for quantum communication, scalability and security, photonic QPUs also offer some unique benefits for data-embedding which have tangible benefits for QML applications. You can find a description of how computations work on an optical quantum computer here! The data-embedding process is one of the bottlenecks of quantum ML. Indeed, classical data cannot be loaded as it is into a quantum chip — it should be encoded somehow into a quantum state. If the encoding process is too costly, it could potentially slow down the overall computation enough to negate any quantum speedups. Considering this, more effective data-encoding strategies are necessary. Photonic quantum computers have the unique advantage of being able to store information in a more efficient space than that of other hardwares. A space is a very general mathematical concept, which in this case simply refers to the structure of our data. You can imagine our quantum states as living in a certain space. With photonic devices we can store our information in the Fock space instead of the Hilbert space used for other quantum processing units. The diagram below outlines the difference between storing information in the Hilbert space compared to the Fock space. In the Hilbert space, each qubit is represented as its own mode, you can imagine the modes as the houses the data lives in. This means that every qubit lives in its own matrix in Hilbert space. Whereas in the Fock space, it’s a much more communal setup, where the photons can live together, and we can count the number of indistinguishable particles in each mode. In this diagram, we can see the Hilbert and Fock space representation of the same state. The Hilbert space has a mode for each particle, but the Fock space counts the particles in each mode, it is in the state \|2,2> which means we have two photons in the first mode and two in the second. In the Hilbert space, each additional photon doubles our possible combinations, leading to a scaling of 2𝑛 where n is the number of photons. This follows from the fact that in our Hilbert space description of qubits, we take each qubit as a 2-level system, meaning each qubit adds 2 new degrees of freedom. The limiting aspect of the Hilbert space qubit representation of quantum data is that it is only ever useful to model our qubits as a 2-level system, despite the number of available modes. In the Fock space, our available combinations scale by a binomial factor of: $$ \Huge \binom{n+m-1}{n}; \text{ where } \binom{n}{k} = \frac{n!}{k!(n-k)!} $$ n is the number of photons and m is the number of modes. It may not be immediately obvious why this is the case, so let’s examine the situation when we have 4 photons and 3 modes to distribute them between in Fock Space: http://xxSo in this case (n=4, m=3), We have 4 photons to be distributed among 3 modes. This problem now has the form of the famous stars and bars theorem. Note : these are 3 of the 15 possible combinations We can imagine that we have 4 indistinguishable balls that must be placed in 3 bins, and we must count all possible combinations. If we count the possibilities, we end up with $$ \Huge \binom{3+4-1}{4} = 15 \text{ possible combinations.} $$ The stars and bars theorem states that this relationship always scales as a binomial coefficient. However, in experimental reality, it is not always possible to access every state in the Fock space. This is because we cannot always measure multiple photons in a single mode with perfect accuracy. This is an active area of research. This scaling advantage for encoding our data into the photonic Fock space is shown for when the number of modes is twice the number of photons (m = 2n), in the following graph: In summary, photonic devices allow us to access the Fock space which has a very useful scaling of the computational space. We should take advantage of this when designing algorithms to run on photonic quantum computers. A Universal Function Approximator (Fourier Series) A tremendously useful application of a photonic quantum circuit arranged in this way is that it can be used to simulate Fourier Series. A Fourier series is an expansion of a mathematical function in terms of an infinite sum of sine and cosine functions. To the non-mathematicians in the audience, this may not sound like a particularly exciting application, but those who have sweated through a multivariable analysis module know that these Fourier Series can be used to approximate ANY function defined over a given interval to an arbitrary degree of accuracy. This means; if we can show that any Fourier Series can be generated by a quantum circuit then we have made a universal function generator. A Fourier Series takes the form: $$ \Huge f(x) = \sum_{\omega \in \Omega} c_\omega e^{i\omega x} $$ where x are our data points, cω are variable coefficients and ω are integer-valued multiples of a base frequency ω0. Ω is our range of frequencies. In this paper, they discuss in detail the conditions necessary for a quantum circuit to universally express a Fourier series. To briefly summarise: the more photons we send in, the more expressive the data encoding operation becomes, and the larger the available spectrum of the output function becomes. Circuit description Our quantum circuits look a bit like a classical Neural Network. In a traditional neural network, you have nodes or neurons connected by weights where you try to optimise these weights in order to minimise a cost function. In ML, cost functions are used to estimate how badly models are performing. Put simply, a cost function is a measure of how wrong the model is in terms of its ability to estimate the relationship between the input of a function and its output. So the lower the value of the cost function the better our model is. In traditional Machine learning, we would input data points into trainable circuit blocks which have a tunable parameter which we can adjust through multiple runs of our circuit to minimise our defined cost function. So far, so classical. What’s unique about our QML circuits is that our data is fed into a quantum circuit which encodes our data in to a quantum state (such as our Fock state) which is then manipulated by adjustable quantum gates. The parameters θ for these quantum gates are optimised using a classical algorithm and fed back into the quantum circuit to give us our ideal quantum output. This is illustrated in the diagram below: We can see that the parameters of our quantum gates are altered to give us our desired output. We refer to this as a hybrid algorithm as it makes use of both classical and quantum methods. Classifier Algorithm Everything we have discussed so far can be neatly implemented in to a QML — classifier algorithm. A classifier algorithm is used to train a circuit to recognize and distinguish different categories of data. We will take a look at a two-dimensional binary classifier algorithm, which is the simplest example as it only distinguishes between two categories of data. For the binary classifier algorithm, the objective is simple. We have a set of data, with elements belonging to one of two classes, Class A or Class B. We must train an algorithm to distinguish the Our algorithm is based on the following circuit: Data input and output Like a typical ML algorithm, we first input training data. This is encoded by the S(x) blocks in to our Fock states: $$ \Huge x_i = \left\| n_i \ldots n_m \right\rangle $$ where n is the number of photons sent into mode i. This data is manipulated by our training blocks W(θ) and outputs in the form of a Fourier Series f(xi), as discussed earlier. Cost function We then need to define and minimize a cost function to train our circuit with this training data. Our aim is that when we give the trained circuit some new test data, it will be able to categorize and divide it. The test data has a predefined classification Yxi which defines the class it belongs to. So, our cost function then takes the form: $$ \Huge \lVert f_{x_i} – y_{x_i} \rVert $$ This is the difference between the values f (the classification to be predicted by our circuit) and y (the actual classification of the data). By tuning the parameters θ of the W(θ) blocks, our circuit can be trained to perfectly identify the training data by minimising this cost function as much as possible By adapting the circuit to bring this cost function to a minimum, we have prepared a circuit that can output the correct classification of a given input. Our circuit is then ready to classify some new test data. In the picture below, we can see an illustration of how the data may be classified and separated. This is the simplest version of this type of algorithm, but it gives you an idea of what is possible. These algorithms are already being applied to applications such as the classification of different molecules. This could be very powerful for the discovery of new chemical products such as medicines. QML Skepticism For the sake of balance, it is worth noting that alongside the great enthusiasm for QML- techniques, there is some (valid) scepticism about the current hype surrounding the topic. Maria Schuld and Nathan Killoran outline a number of criticisms of our current attitudes towards this field. They remind researchers to avoid the ‘tunnel vision of quantum advantage’. It’s an astute point that many people are rushing to implement quantum versions of classical algorithms even if the quantum version might not be particularly useful. They can then try to get a better performance with the quantum algorithm and claim quantum advantage. Sometimes this increase in performance is obtained only on a specific dataset that is carefully chosen, or the classical counterpart could perform better if it was optimised more. In the same way that quantum computers are not expected to be better than classical computers for all problems, but only for some specific ones (such as factoring with Shor’s algorithm), we expect that QML is going to beat ML only in specific scenarios. It is now up to the researchers to find the scenarios where QML will truly excel.	{"url":"https://www.quandela.com/resources/blog/photonics-the-fast-lane-towards-useful-quantum-machine-learning/","timestamp":"2024-11-06T21:40:22Z","content_type":"text/html","content_length":"167806","record_id":"<urn:uuid:2dd29796-cb36-451f-9a1c-d12114a00221>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00520.warc.gz"}
mathspp.com feedSolving diophantine equations with random walks Here's how I like to solve my equations: just walk around randomly until I trip over a solution! Random walks (check my older post here) and Diophantine equations are two simple mathematical beasts. After a college seminar, I tried putting them together to make something neat, and came up with this: just pick a Diophantine equation, simulate a random walk, and try to see if the random walk went over any solutions! In this report I briefly go over how I came up with this, and show the code I wrote/some results. The Matlab code is all in here (I also have some Python code about random walks here, check my post!). In the image above, the blue line represents a path taken by the random walk and in orange/red the nodes tell us how far we are from a solution (the smaller the node, the closer we are). Note that this notion of proximity isn't given by actually computing the distance to a known solution, but just by looking at how similar the two sides of the equation are. The image below shows the evolution of three independent random walks trying to find an example solution of $\frac{4}n = \frac1x + \frac1y + \frac1z$ with $n = 25$ (that expression is the expression of the Erdös-Straus conjecture).	{"url":"https://mathspp.com/blog/tags/diophantine-equations.atom","timestamp":"2024-11-02T21:50:49Z","content_type":"application/atom+xml","content_length":"4641","record_id":"<urn:uuid:2e1770dd-768d-4b74-bad7-3504e2205566>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00044.warc.gz"}
Learning mathematics creates opportunities for and enriches the lives of all Australians. The Australian Curriculum: Mathematics provides students with essential mathematical skills and knowledge in Number and Algebra, Measurement and Geometry, and Statistics and Probability. It develops the numeracy capabilities that all students need in their personal, work and civic life, and provides the fundamentals on which mathematical specialties and professional applications of mathematics are built. Last reviewed 26 August 2021 Last updated 26 August 2021	{"url":"https://dalbyss.eq.edu.au/curriculum/subjects-and-programs/maths","timestamp":"2024-11-10T12:23:32Z","content_type":"text/html","content_length":"119063","record_id":"<urn:uuid:f7b5b9bd-47cf-44bc-8524-3ed16333e2bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00264.warc.gz"}
Category: Learning Communities 0 Comments This week I visited a Grades 3 and Grades 3/4 class at Betty Huff Elementary. The two teachers and I met during the first week of school and worked collaboratively to discuss how to begin the year. I am working with these teachers this term to support their professional inquires which focus on developing their students curricular competencies in relation to content, as well exploring what inquiry-based practices could look like in mathematics. We began by creating a mini-unit exploring the following questions: What is math? What does it mean to do math? Where does math live in the world? Additionally, we wanted students to begin to explore their own identity as a math learner. This is an important aspect of BC's redesigned curriculum. The Positive Personal and Cultural Identity is one of our core competencies and includes the awareness, understanding, and appreciation of all the facets that contribute to a healthy sense of oneself . We wondered how our students saw themselves as Mathematicians. We began with an activity I had seen on Graeme Anshaw's blog where he asked students to share their personal histories with mathematics. The teachers gave each student a sticky and ask them to write or draw how they felt about math and why. Then students were invited to share their thoughts with the class and place their sticky on the line spectrum. The teachers commented how through this activity they were able to learn about the mindsets students held with regard to mathematics. As you can see the Grade 3/4 class held more positive views, while the Grade 3 class was more mixed. The teachers remarked how curious they were to dig more deeply into these stories and work with these students to shift some of their negative feelings. Grades 3 class Grade 3/4 class I visited the class on the second day of the mini-unit. We began with the question "What is Mathematics?", and surprisingly, we learned that not all children had an understanding of the word "Mathematics". They didn't realize it was just a longer version of the word "Math"... an important reminder that all children are "mathematics language learners" and we must pay attention to the language we use. Grade 3 class Grades 3/4 class When reflecting upon the charts, we were able to see similarities to the work that Tracy Zager and Deborah Nichols reported hearing from a Grades 1/2 class (pg. 11 - 13 in Becoming The Math Teacher You Wish You'd Had). It seemed that our students felt mathematics was mostly about computation. Their experiences had been many worksheets, which they felt was really hard. Inspired by Tracy Zager's suggestions to use books to help expand children's view of mathematics, we read aloud On A Beam of Light: A Story of Albert Einstein by Jennifer Berne. Before beginning the book, we passed out "Notice/Wonder" sticks to the students. We asked them to raise their "Notice" glasses if they noticed anything in the book that they thought represented mathematics, which we would add to the chart. Additionally, student were asked to raise their "Wonder" question marks if they thought they might have seen or heard something that could be added to our chart, but were unsure. Credits go out to Beth Kobett who shared the idea of Notice and Wonder sticks on Twitter. The students liked using their notice/wonder sticks to share their thoughts. We added to the chart using a blue marker to show our new information. The teachers and I were impressed with how the students beliefs about math had begun to broaden to include some important verbs, such as noticing, thinking, questioning, understanding, discovering, and dreaming. Following this, we split into a two groups and went on a math hunt throughout the school. We asked ourselves "Where does Math live in the world?" We took pictures of what we saw using iPads and shared our images with the class. The students noticed many mathematical concepts that had been missed on our initial brainstorm including patterns, geometry, and measurement. Below is the mini-unit we created including using ideas from Tracy Zager's book Jo Boaler's inspirational week of maths, and Graeme Anshaw's blog. Feel free to leave any feedback you have on our mini-unit. We welcome your ideas and suggestions! I am curious how did you start your year with your Mathematicians? 0 Comments I love returning to school in September. I get giddy with excitement purchasing school supplies and can't wait to meet my new students. This year my two kids have started Grades 6 and 8. They also share my love of picking out colourful pencils, pens, binders, and such, but since we didn't know what was required for high school I told my son we would purchase his supplies at the end of the first week. Last weekend, I took the necessary time to review my son's course outlines. Generally they were as I expected but it was a comment from my son that caused me pause. He told me that he needed more pencils and a bigger eraser for Math. He said that all his Math work had to be done in pencil and the erasers at the end of pencil weren't going to be big enough to erase mistakes. Herein, is where my mind started buzzing with questions. This long-standing tradition of using only pencil in Mathematics so students can erase their errors doesn't make any sense to me! We don't insist on using only pencil in any other subjects, so why Math? In writing, when students review and revise their work, they simply write on top or under what they have written, cross one word out and add another and add indentation marks to show where they have added. This makes it possible for teachers to see the trajectory of learning. Likewise, when children read to us and they make a mistake, we don't halt them and say "stop, that's incorrect"; instead we give them time to see if they catch their mistake and self-correct and then we discuss the error and strategies to help the student. The practice of writing in pencil so that we can erase mistakes in Math runs contradictory to all the current research we know about developing growth mindsets, honouring process over product, and capitalizing on opportunities to move our students' learning forward. Jo Boaler writes Studies of successful and unsuccessful business people show something surprising: what separates the more successful people from the less successful people is not the number or successes but the number of mistakes they make, with the more successful people making more mistakes (Mathematical Mindsets, 2016, p.g. 15) If mistakes are a necessary part of our learning process, then why are Math teachers asking children to erase them? I believe mistakes should be seen positively, as they show what the child currently understands and any misconceptions are opportunities for learning to occur. Furthermore, we know from both research and experience that when we build classroom communities that are safe to take risks and use errors for instruction, we have the potential to increase our students' understanding (Bray, 2013). Why would we give up this opportunity by encouraging erasing? I have wrestled with this idea before. Two years ago I sat frustrated trying to figure out my son's understanding about adding fractions with different denominators.I knew he had some misconceptions but I couldn't make sense of his thought process because the work he brought home from school had been fully erased. He had shown his teacher his work and was told to erase it and re-do it. I then asked his teacher to please refrain from having my son erase and explained my rationale. Here we are two years later with more knowledge about the importance of mistakes as part of learning and I am unsure if we are further ahead. I would like to be able to tell you that both my children are using only pen in their Math classes but that wouldn't be the truth. They don't want to stand out against their peers or make their teachers unhappy so they continue to use pencil... but they no longer erase (at least that I can see). As I move forward supporting both teachers and students in Math classes this year, I will be bringing with me a beautiful tin of colourful pens, with all sorts of interesting designs, including some that are smelly! We will use these to record our Mathematical thinking and understanding and there will be NO ERASING! All thoughts will be valued and mistakes will be seen positively, as part of our learning and growing process! I am interested to hear others thoughts on this topic and the role pencils and erasers play in your Mathematics classes. Please feel free to add a comment. Boaler, J. (2016). Mathematical mindsets: Unleashing students' potential through creative math, inspiring messages, and innovative teaching. Bray, Wendy S. 2013. "How to Leverage the Potential of Mathematical Errors." Teaching Children Mathematics 19 (7): 424-431. 1 Comment As a parent of two kids, aged 10 and 12 I am learning many things about tweens! I am learning new acronyms for texting, "bad" words I wasn't aware of, and that bathing everyday does not necessarily occur without adult reminders. My new understanding of tweens could fill an entire blog post.. but my children might not appreciate that. This weekend in a discussion with my two tweens, I learned something I wasn't necessary surprised by, but was deeply saddened about. We were talking about the subject of Mathematics and I was asked what they were doing at school. In unison both my children groaned about their dislike for the subject. As a passionate Math teacher, I commented that they just needed to try seeing the beauty of the subject. Both started laughing and said "Mom, don't you know what MATH stands for?" Puzzled I said "no". They then they proceeded to inform me that in their minds, and apparently the minds of "all of their peers" Math stood for "Mental Abuse To Humans". Perhaps I have been living under some rock of Math utopia, but I was surprised that my kid's generation thought this way. I prodded both kids a little further and asked if they really believed this saying about Math? How could my kids not value Math? I mean they have had some experiences at home with me and I know they have had some excellent teachers who have taught Math in engaging ways... But something I have noticed, is that as they have moved up into the intermediate grades much of their experiences with Math have been through interactions with text books. I am not saying that using a text book is all bad or that teachers who use text books are wrong to do so, but I think texts tend to focus on paper and pencil exercises where discrete skills and concepts with little context are the focus. Similarly facts and algorithms are told to students, as opposed to having the learners discover different strategies that work and make sense to them. Texts also do not tend to emphasize collaboration; therefore, much of the work is done in isolation. Looking at Math through these types of experiences, I could see how my kids viewed Math as boring and tortuous (yes, that was a word they used!). Teachers have tough jobs, especially at the Elementary level where they are required to be knowledgeable of many curricular areas. Teachers want to engage students in Mathematics but often either don't have the time to or knowledge of what resources needed to shift their practice. Teacher preparation schools (e.g., Universities) aren't much help either. Often, teacher candidates get less than 24 hours of Math instruction to prepare them to teach Math from Kindergarten to Grade Seven. My point in all this discussion is that teachers like myself, who are lucky enough to focus solely on Math education education, need to create places where we share key resources, ideas, and stories from our classes. Similar to our students, we need to connect, collaborate, and learn from each other. As disappointed as was with my discussion with my kids and their negative disposition to Mathematics, it reminded me of the urgency. We must unite in our plight to develop children who love Mathematics! Hearing my children's voices about Math was just the nudge I needed to push me to finally curate this site... it had been percolating in my mind for awhile. Our Mathematical moments/stories have value and they have the power to inspire others. I am ready to begin sharing my stories. Are you? If you are a Mathematics educator who blogs about your practice, I encourage you to share your stories! We can make positive change! 0 Comments	{"url":"http://www.meaningfulmathmoments.com/musings/category/learning-communities","timestamp":"2024-11-11T16:36:36Z","content_type":"text/html","content_length":"76085","record_id":"<urn:uuid:4e69c853-c29e-405f-929c-417a66a441e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00025.warc.gz"}
Data Parallelism Matrix multiplication is a compute intensive operation that can leverage data parallelism. Figure Data Parallelism shows a G program with 8 sequential frames to demonstrate the performance improvement via data parallelism. Figure 10.2 Data Parallelism The Create Matrix function generates a square matrix based of size indicated by Size containing random numbers between 0 and 1. The Create Matrix function is shown in Figure Creating a Square Matrix. Figure 10.3 Creating a Square Matrix The Split Matrix function determines the number of rows in the matrix and shifts right the resulting number of rows by one (integer divide by 2). This value is used to split the input matrix into the top half and bottom half matrices. The Split Matrix function is shown in Figure Split Matrix into Top & Bottom. Figure 10.4 Split Matrix into Top & Bottom Sequence Frame Operation Description First Frame Generates two square matrices initialized with random numbers Second Frame Records start time for single core matrix multiply Thrid Frame Performs single core matrix multiply Fourth Frame Records stop time of single core matrix multiply Fifth Frame Splits the matrix into top and bottom matrices Sixth Frame Records start time for multicore matrix multiply Seventh Frame Performs multicore matrix multiply Eighth Frame Records stop time of multicore matrix multiply The rest of the calculations determine the execution time in milliseconds of the single core and multi-core matrix multiply operations and the performance improvement of using data parallelism in a multicore computer. The program was executed in a dual core 1.83 GHz laptop. The results are shown in Figure Data Parallelism Performance Improvement. By leveraging data parallelism, the same operation has nearly a 2x performance improvement. Similar performance benefts can be obtained with higher multicore processors Figure 10.5 Data Parallelism Performance Improvement	{"url":"http://www.opentextbooks.org.hk/ditatopic/8152","timestamp":"2024-11-11T10:47:48Z","content_type":"text/html","content_length":"95895","record_id":"<urn:uuid:da247814-0c8b-447a-be35-c6411bb24497>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00535.warc.gz"}
Is dark energy vacuum energy at scale of universe?(10/12/21) Is dark energy vacuum energy at scale of universe?(10/12/21) Planck length, Planck time, and Planck mass [1] Planck length, Planck time and Planck time will be defined from the fundamental constants of physics which are the velocity of light noted c, the gravitational constant noted G, and Planck’s constant noted h, [2] by using dimensional arguments. In the MKS system, their values are: c = 299 792 458 m s^– 1, G ≈ 6.674 30 × 10^−11 m^3 kg^− 1 s^− 2, h ≈ 6.626 070 040 × 10^−34 kg m^2s^− 1 [S:h:S] ≈ 1.054 571 800 × 10^−34 kg m^2 s^− 1 For c, this is an exact value (by definition), the others are measured values so are approximated values, the value of all these constants is not predicted by the theory, they are called free parameters.[3] The dimensions of these constants are listed. The Planck length l[P] will therefore be defined by the simplest product of these constants which has the dimension of a length, for the time t[P] and the Planck mass m[P], it is the same principle but for a time and a mass. This gives: m[P] = (hc /G)^1/2 ≈ 2,177.10 ^-8 kg, t[P] = (hG /c^5)^1/2 ≈5,391.10^-44 s l[P] = c.t[P]= (hG /c^3)^1/2 ≈1,616.10 ^-35m, We can check that these values have the correct dimension and with the values of the constants c, G, h, in the MKS system, that their values are correct. Planck force By using the law f = m γ, where f is the force applied to mass m and γ the resulting acceleration, by using Planck’s values for the operands, we get: f[P] = m[P] γ[P], with γ[P] = l[P] / (t[P] ²) →f[P] = c^4/G ≈1.21. 10^44 Newtons (m.kg.s^– 2) This huge value is independant of the value of the Planck constant! In other words, whatever the value of h is, we get this result! Planck force is an invariant of physics, which will be also found in other phenomena, such as: Planck Force invariance applied to the universe. Rough estimated mass of the universe This c^4/G factor seems to play a general role. Since this “Planck” force is independent of the value of Planck’s constant, let us consider the value of a modified Planck constant, denoted h[u], (h universe) where “the modified Planck mass” would be equal to that of the universe. Current data assigns a mass of roughly one hundred billion (10^11) solar masses to our galaxy and the number of galaxies is estimated to be around 1000 billion, 10^12 (these figures are recent, earlier figures have been revised upwards). With 10^12 galaxies of 10^11 solar masses and a Solar mass of ≈ 2.10^30kg, we get: Mass of the universe = M[u] ≈2.10^53 kg. According to the Planck mass, (m[p] ≈ 2 .10^-8 kg) it is necessary to multiply by a factor K ≈ 10^61, for getting the mass of the universe. As in the definition of Planck’s mass, it is its square root that is involved, the h[u] constant to be used, instead of Planck’s constant, is such that: h[u] ≈ 10^122 h. Vacuum energy and cosmological constant For explaining the acceleration of the expansion of the universe, one introduced the concept of dark energy. One mathematical solution for dark energy is the cosmological constant. The vacuum energy was proposed as a physical solution of this cosmological constant. This hypothesis was discarded because of a huge discrepancy (around 10^122) between the calculated value of the cosmological constant resulting from vacuum energy and its current value, measured by cosmologists. There is an annex in my book explaining that, in more details. Due to this huge discrepancy, there are issues about the physical representation of vacuum energy in cosmology, this being considered as a major problem! But, per our dimensional analysis, the vacuum energy of the whole universe should be calculated not with the physical Planck constant h but with h[u], which introduces a 10^122 factor, between the calculated vacuum energies. Estimate of the vacuum energy by this dimensional analysis The dimension of the cosmological constant λ being [L]^-2 (reverse of a squared length). This can be obtained by calculating (T.c)^-2, where T is the age of the universe and c the celerity of light. The calculation with an age of 13.7 billion years gives a value of λ =0.6 x 10^-52 m^-2, compared with the measured value which is 1.088 x 10^-52. This undervalued value is explained by the fact that, in our dimensional analysis, we do not take into account matter (baryonic and black) whose effect, contrary to that of the cosmological constant, is to slow the expansion. The value obtained by dimensional analysis corresponds to a phenomenology that would be only due to the cosmological constant. It is therefore undervalued, because it does not take into account the opposite effect of matter. We find that the energy of the vacuum, such defined, reflects, in order of magnitude, the observed black energy and the value of the associated cosmological constant. Calculation of the cosmological constant by using the Newton second law By using the law: f[planck ]= M.γ where M is the mass of the universe, and γ the acceleration associated with the force of Planck we obtain: γ = 0.6 x 10^-9 m / sec² By converting, by analysis dimensional, λ, the cosmological constant [L] ^-2 to obtain an acceleration [L] [T] ^-2, on the scale of the universe, it must be multiplied by c^3 and by t = age of the universe ( in seconds). γ = λ. c^3 .t (age of the universe). –> λ =. c^-3 .t^-1 γ^-1 This calculation will show that we obtain the same value for λ . This result is obtained by two different methods, which is interesting. Let us add, that if we are able to measure these values precisely that we will give a more realistic value of the mass of the universe, since they depend on it. Does cosmological constant depend on time? This last equation shows, on the other hand, that the value of the cosmological constant depends on the age of the universe, which seems contradictory with its definition. But perhaps the physical phenomenon at work in the acceleration of the universe, which appears to us as a cosmological constant, is not one, especially as if it is a question of an energy of vaccuum, it is not inconceivable that it depends on time. This issue is under investigation curently. An article listed on the main page « Hubble constant problem » reviews how the concept ofPlanck’s force and its related cosmological constant may solve this issue. But above all, and fundamentally, let us remember that the age of the universe is linked to one specific description of cosmology and only reflects the value of a past in relation to our present, in a specific coordinate system (Robetson-Walker) where, moreover, time (called cosmological time) is not ours. Remember overall that the universe, in relativity, is a space-time, where the notion of age of this space-time has no meaning … From this, we should consider the dimensional analysis not only as as brain storming but a way aimed to stimulate our mind on this topic…. Justification of the physical relevance of the change of scale. From a physical point of view this proposition can be supported by statistical considerations. You can divide the universe into microscopic cells. At the level of a microscopic cell, the vacuum state which undergoes fluctuations linked to the process of creation / annihilation of particle-antiparticle pairs can be represented by a random variable governed by a statistical law, binomial law or Poisson law when the probability of events is low. Anywhere in the vacuum throughout the universe the random variables associated with these micro-cells are independent. Statistics tell us that the distribution law of the state of the universe, which is the combination of the states of the huge number of microscopic cells, at the scale of the universe, resulting from all these independent random variables at microscopic scale, is a random variable which tends, whatever the statistical law is at the microscopic level, towards a normal distribution law (Gaussian) whose parameters, likelihood and variance, are calculated from microscopic laws and the configuration of the universe in microscopic terms. Under these conditions, a vacuum energy at the scale of the universe is a physical hypothesis. Planck length and Planck time at the universe’s scale. To calculate the length L and the time T associated with the scale of the universe, with the same “Planck force”, the values of the Planck scale must be multiplied by: K ≈ 10^61. By applying this, we get [4] : L ≈ lp x 10^61 ≈ 1.6 10^-35 x10^61 m ≈1, 6 x 10^26 m ≈1.7 10^10 al. That is 17 billion light years. T = tp x 10^61 ≈ 5.4 10^-44 x10^61s ≈5.4 x 10^17 s ≈17 x10^10 years. That’s 17 billion years. Given the inaccuracies in the estimates of the mass of the universe, we see that we get figures that are of the order of magnitude of what is adopted today. Conversely, to obtain the correct figures, we just need to correct the mass of the universe where, for example, the number of galaxies should be estimated at 800 billion, instead of 1000 billion. [1] This topic follows an amazing proposal from Edouard Bassinot, to define a “force of Planck”, defined by the second law of Newton with the Planck parameters. I was intrigued by the proposal and tried to explore its possible consequences. One of them would be that the gravitational constant G and the celerity of light c would be not free parameters in the theory. They would be enforced, by the physical parameters (age, size, and mass) of the universe. [2] [S:h :S]= h / 2π, in place of h, is often used, because angular pulsation θ/s is more convenient than frequency in physics. The factor 2π results from the fact that 1 Herz is equal to 2π radians/ [3] Let us stress that the value of the constants G and c are not predicted by the theory. They are what we call “free parameters”. The exact value of c results from a convention allowing to define the units of length and time! For the constant h, also unpredictable (therefore free), the relation E = h.f, where f is the frequency of a photon and E its energy allows to measure its value. The constant h, first introduced in physics for the radiation of the black body (see another item on the website), is ubiquitous in quantum mechanics. [4] In a year there are 3600x 24x 365 ≈ 3.15 10^7 seconds and in a light year 9.45 10^15 meters.	{"url":"https://vous-avez-dit-bigbang.fr/?page_id=343","timestamp":"2024-11-05T23:31:29Z","content_type":"text/html","content_length":"44986","record_id":"<urn:uuid:40991323-cb6c-48c0-ad11-af0245188011>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00571.warc.gz"}
perplexus.info :: Numbers : N^2 and N^3 I know one answer. In base 10, 69^2=4761 and 69^3=328509. This was in my puzzle Pandigital powers. I also know that there cannot be any solutions in a base b of the form 5x+1. If N has x digits, then N^2 has at most 2x digits and N^3 has at most 3x digits. Then, N^2 and N^3 have at most 5x digits together. If N has x+1 digits, then N^2 has at least 2x+1 digits and N^3 has at least 3x+1 digits. Then, N^2 and N^3 have at least 5x+2 digits together. Therefore, N^2 and N^3 cannot have 5x+1 digits, so there are no solutions with b=5x+1. Posted by Math Man on 2023-11-08 13:50:55	{"url":"http://perplexus.info/show.php?pid=13825&cid=70352","timestamp":"2024-11-03T22:42:06Z","content_type":"text/html","content_length":"12372","record_id":"<urn:uuid:c045de93-3f75-4d2a-993c-9463055ca694>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00077.warc.gz"}
Additional Function Extensions \| Details \| Hackaday.io So there exists the IN-15A and IN-15B nixie tubes, which are a pair of tubes designed for meters. The IN15A outputs unit scalars like +, -, u, m, k, M and the like. The IN-15B displays unit symbols W, F, Hz, H, V, S, A and an omega. So now I'm challenged to design entirely analog meters for all of these units. If I stick to my as-designed voltmeter as a base, all I need to do is conjure up entirely analog means of translating every unit into voltage, in a completely linear manner. Most of these were pretty straightforward but also pretty entertaining. I will leave out voltage measure because it's already described in the comparator-ladder section so that's taken care of. The rest we'll see what I got. First, resistance. A constant current is driven through the unknown resistor. Since V=IR, if I is constant, V ~ R. Done. Second, capacitance. This one gave me some fits. Use a constant current source to charge the unknown capacitor up to a threshold value. Since the voltage across a capacitor increases linearly with a constant current, the time required to charge the unknown capacitor to the preset value is directly proportional to the value of the capacitor. it=VC -> (i/V)t = C and if i and V are constant, t ~ C. At the same time - and for the same duration - we charge a second, known capacitor with a second constant-current source. The capacitance and current are constant in this case. it=VC -> V=(i/C)t so we have a voltage proportional to the time. Since the charge time is the same for both capacitors, and the output voltage is proportional to the time is proportional to the unknown capacitance, V ~ C. Done. Third, current. Run the current through a small fixed resistance and measure the voltage drop across it. Since V=IR, if R is constant, V ~ I. Done. Fourth, inductance. This one was also fun, and I'm not sure how well it'll work. For a nonsaturated inductor, V=Ldi/dt. If we push current through the unknown inductor in series with a known inductor such that we maintain a constant voltage across the known inductor, we have a constant di/dt. di/di = V1/L1 A constant di/dt should induce a constant voltage across the known inductor. V2 = L2di/dt so V2 = L2(V1/L1) so V ~ L. This will only work for short current pulses because in DC the inductors will saturate, so some timing logic will be required but in general, Done. Fifth, conductance (siemens). This is basically the inverse of resistance, so one way to do it is calculate resistance and then invert it. But that's a division step which, in purely analog circuits, usually requires a log transform, subtraction, and antilog transform. Which is difficult and hard to keep precise. So instead we'll do something different. Starting with ohm's law V=IR -> V/R=I, and knowing S=1/R, I=SV. We run a current source through a known resistor R1 and unknown resistor R2, where the current source is keyed to maintain the voltage across R2 at a constant known value. We then measure the voltage across the known resistor R1. S2Vr2=i, and Vr1=iR1 -> i=Vr1/R1 so S2Vr2=Vr1/R1 -> S2=Vr1(1/R1Vr2) and since R1 and Vr2 are known constants, S ~ V. Done. Sixth, frequency. This one was pretty tricky. The obvious solution is to use a binary counter with a fixed duration during which to sample pulses, and an analog DAC in the form of a weighted summing amplifier. The problem with this is philosophical - it's heavily logic-based and not really an analog measure at all. Additionally, many ICs are required and the DAC might have precision issues? So we thought of something else, in the form of a state machine. The initial state is really what matters most, the rest is just there to make sure the input and output values are latched properly. So, here goes. We use a precision 32.768KHz RTC oscillator divided down into a 1-second timebase. This will have exactly 50% duty cycle, so a high pulse duration of exactly one half second. We AND this half-second pulse with the incoming measured pulsetrain and use that to trigger a one-shot with a fixed pulse duration of, say, 800uS. This is based on the maximum frequency directly measurable being 1KHz (other higher ranges achieved by decade dividers). Every incoming pulse will generate a single constant-duration 800uS pulse which will enable a constant-current source to charge a fixed capacitor. This results in fixed-duration fixed-current pulses charging the capacitor, which means bursts of constant charge resulting in a piecewise-linear increase in capacitor voltage. When the half-second gating pulse is over, we now have a capacitor C which has received nQ charge, where Q is i*800uS and n is the number of incoming pulses. Now we're seeing nQ=VC -> V=nQ/C and since Q/C is constant, and since n is pulses per half-second (f/2), V ~ f. Done. Seventh, power. I really wasn't sure how to do this without log multiplication, which as noted above, sucks. However, I might have figured it out. First, a modification to the diagram needs to be mentioned - instead of the EN line feeding back to a pulse generator, pretend it's going straight into both current sources. Now we're ready to break it down. First we require three probes, to measure voltage V1 and current I1 simultaneously. We charge a known capacitor C2 with a constant current source I3 until its voltage reaches a threshold proportional to V1. Q=VC and since I = dQ/dt, if I is constant then Q=It so I3t~V1C. Since I3 and C are constant, V1~(I3/C)t -> t~V1 At the same time, and for the same length of time, we charge a second known capacitor C1 with a current source proportional to the measured current value I1. Q=VC so (kI1)t=Vc1C1 and since t~V1, I1V1~Vc1C1 and since C1 is constant, I1V1~Vc1 so V ~ I1V1 so V ~ Power. Done. Well, that was all quite fun and will require quite a bit of algebra and calibration to make any of it actually useful. We'll start with making the voltage measurement and display work first, and thet move up in complexity probably starting with current measurement. Hopefully I can get this thing done sometime this year.	{"url":"https://hackaday.io/project/1458-state-based-nixie-digital-voltmeter/log/3771-additional-function-extensions","timestamp":"2024-11-08T20:49:39Z","content_type":"text/html","content_length":"32057","record_id":"<urn:uuid:d32809c9-fd86-403e-8f29-d773799cee0d>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00443.warc.gz"}
How To Calculate Centroid Of An Image - the meta pictures How To Calculate Centroid Of An Image How To Calculate The Centroid Of A Two Dimensional Shape 16 how to calculate centroid of an image how to calculate centroid of an image is important information with HD images sourced from all websites in the world. Download this image for free by clicking "download button" below. If want a higher resolution you can find it on Google Images. Note: Copyright of all images in how to calculate centroid of an image content depends on the source site. We hope you do not use it for commercial purposes. How To Find The Centroid Of A Triangle Video Lesson Transcript Calculating The Centroid Of Compound Shapes Using The Method Of Troubles Figuring Out How To Calculate The Centroid For A How To Find The Centroid Of Simple Composite Shapes Youtube How To Calculate Centroid Of Polygon In C Stack Overflow Calculate The Centroid Of A Beam Section Skyciv Cloud Structural Finding Centroid Of An Area Mathematics Stack Exchange Steps To Calculate Centroids In Cluster Using K Means Clustering How To Calculate The Centroid Of A Two Dimensional Shape 16 Example C4 3 Centroid Of Composite Bodies Statics No Comment Add Comment	{"url":"https://www.themetapictures.com/2020/01/how-to-calculate-centroid-of-image.html","timestamp":"2024-11-13T16:14:13Z","content_type":"text/html","content_length":"65623","record_id":"<urn:uuid:facfac43-462b-487d-ab87-8c3dab141e69>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00004.warc.gz"}
Copyright (C) 2017 Tim McGilchrist License BSD-style (see the file LICENSE) Maintainer timmcgil@gmail.com Stability experimental Portability portable Safe Haskell Safe-Inferred Language Haskell98 exceptT :: Monad m => (x -> m b) -> (a -> m b) -> ExceptT x m a -> m b Source # Map over both arguments at the same time. Specialised version of bimap for ExceptT. mapExceptT :: (m (Either e a) -> n (Either e' b)) -> ExceptT e m a -> ExceptT e' n b # Map the unwrapped computation using the given function. bimapExceptT :: Functor m => (x -> y) -> (a -> b) -> ExceptT x m a -> ExceptT y m b Source # Map the unwrapped computation using the given function. handleIOExceptT :: MonadIO m => (IOException -> x) -> IO a -> ExceptT x m a Source # Try an IO action inside an ExceptT. If the IO action throws an IOException, catch it and wrap it with the provided handler to convert it to the error type of the ExceptT transformer. Exceptions other than IOException will escape the ExceptT transformer. Note: IOError is a type synonym for IOException. handleExceptT :: (MonadCatch m, Exception e) => (e -> x) -> m a -> ExceptT x m a Source # Try any monad action and catch the specified exception, wrapping it to convert it to the error type of the ExceptT transformer. Exceptions other that the specified exception type will escape the ExceptT transformer. • Warning*: This function should be used with caution! In particular, it is bad practice to catch SomeException because that includes asynchronous exceptions like stack/heap overflow, thread killed and user interrupt. Trying to handle StackOverflow, HeapOverflow and ThreadKilled exceptions could cause your program to crash or behave in unexpected ways. handlesExceptT :: (Foldable f, MonadCatch m) => f (Handler m x) -> m a -> ExceptT x m a Source # Try a monad action and catch any of the exceptions caught by the provided handlers. The handler for each exception type needs to wrap it to convert it to the error type of the ExceptT transformer. Exceptions not explicitly handled by the provided handlers will escape the ExceptT transformer. handleLeftT :: Monad m => (e -> ExceptT e m a) -> ExceptT e m a -> ExceptT e m a Source # Handle an error. Equivalent to handleError in mtl package. bracketExceptT :: Monad m => ExceptT e m a -> (a -> ExceptT e m b) -> (a -> ExceptT e m c) -> ExceptT e m c Source # Acquire a resource in ExceptT and then perform an action with it, cleaning up afterwards regardless of left. This function does not clean up in the event of an exception. Prefer bracketExceptionT in any impure setting. bracketExceptionT :: MonadMask m => ExceptT e m a -> (a -> ExceptT e m c) -> (a -> ExceptT e m b) -> ExceptT e m b Source # Acquire a resource in ExceptT and then perform an action with it, cleaning up afterwards regardless of left or exception. Like bracketExceptT, but the cleanup is called even when the bracketed function throws an exception. Exceptions in the bracketed function are caught to allow the cleanup to run and then rethrown. hushM :: Monad m => Either e a -> (e -> m ()) -> m (Maybe a) Source # Convert an Either to a Maybe and execute the supplied handler in the Left case.	{"url":"http://hackage.haskell.org/package/transformers-except-0.1.2/docs/Control-Monad-Trans-Except-Extra.html","timestamp":"2024-11-10T01:44:03Z","content_type":"application/xhtml+xml","content_length":"35621","record_id":"<urn:uuid:b819b816-365a-4b6a-bef2-cbe69c71c83a>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00510.warc.gz"}
How does the PV and NPV function work in Excel? How does the PV and NPV function work in Excel? Difference between PV and NPV in Excel 1. The NPV function can calculate uneven (variable) cash flows. The PV function requires cash flows to be constant over the entire life of an investment. 2. With NPV, cash flows must occur at the end of each period. How do you calculate PVA in Excel? The basic annuity formula in Excel for present value is =PV(RATE,NPER,PMT). PMT is the amount of each payment. Example: if you were trying to figure out the present value of a future annuity that has an interest rate of 5 percent for 12 years with an annual payment of $1000, you would enter the following formula: =PV(. What is the PV function? PV, one of the financial functions, calculates the present value of a loan or an investment, based on a constant interest rate. Use the Excel Formula Coach to find the present value (loan amount) you can afford, based on a set monthly payment. At the same time, you’ll learn how to use the PV function in a formula. What is the difference between NPV and PV? Present value (PV) is the current value of a future sum of money or stream of cash flow given a specified rate of return. Meanwhile, net present value (NPV) is the difference between the present value of cash inflows and the present value of cash outflows over a period of time. What is NPV function in Excel? The Excel NPV function is a financial function that calculates the net present value (NPV) of an investment using a discount rate and a series of future cash flows. What is type in Excel PV? The Excel PV function is a financial function that returns the present value of an investment. You can use the PV function to get the value in today’s dollars of a series of future payments, assuming periodic, constant payments and a constant interest rate. type – [optional] When payments are due. What is PV Nper formula? Nper Required. The total number of payment periods in an annuity. For example, if you get a four-year car loan and make monthly payments, your loan has 412 (or 48) periods. You would enter 48 into the formula for nper. How do you use PV function? The built-in function PV can easily calculate the present value with the given information. Enter “Present Value” into cell A4, and then enter the PV formula in B4, =PV(rate, nper, pmt, [fv], [type], which, in our example, is “=PV(B2,B1,0,B3).” Since there are no intervening payments, 0 is used for the “PMT” argument. What is an example of PV in Excel? PV in excel is based on the concept of time value of money. For example, receiving Rs. 5,000 now is worth more than Rs. 5,000 earned next year because the money received now could be invested to get an additional return till next year. How do you calculate PV annuity? The PV calculation uses the number of payment periods to apply a discount to future payments. You can use the following formula to calculate an annuity’s present value: PV of annuity = P [1 – ((1 + r) ^(-n)) / r] How to use the Excel NPV function? Example of how to use the NPV function: Set a discount rate in a cell. Establish a series of cash flows (must be in consecutive cells). Type “=NPV (” and select the discount rate “,” then select the cash flow cells and “)”. What is the use of FV function in Excel? The Excel FV function is a financial function that returns the future value of an investment. You can use the FV function to get the future value of an investment assuming periodic, constant payments with a constant interest rate.	{"url":"https://www.theburningofrome.com/users-questions/how-does-the-pv-and-npv-function-work-in-excel/","timestamp":"2024-11-11T08:39:52Z","content_type":"text/html","content_length":"124755","record_id":"<urn:uuid:2de7b84d-872f-4e51-81b6-3601495729b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00776.warc.gz"}
In statistics, sampling errors are incurred when the statistical characteristics of a population are estimated from a subset, or sample, of that population. Since the sample does not include all members of the population, statistics of the sample (often known as estimators), such as means and quartiles, generally differ from the statistics of the entire population (known as parameters). The difference between the sample statistic and population parameter is considered the sampling error.^[1] For example, if one measures the height of a thousand individuals from a population of one million, the average height of the thousand is typically not the same as the average height of all one million people in the country. Since sampling is almost always done to estimate population parameters that are unknown, by definition exact measurement of the sampling errors will not be possible; however they can often be estimated, either by general methods such as bootstrapping, or by specific methods incorporating some assumptions (or guesses) regarding the true population distribution and parameters thereof. Sampling Error The sampling error is the error caused by observing a sample instead of the whole population.^[1] The sampling error is the difference between a sample statistic used to estimate a population parameter and the actual but unknown value of the parameter.^[2] Effective Sampling In statistics, a truly random sample means selecting individuals from a population with an equivalent probability; in other words, picking individuals from a group without bias. Failing to do this correctly will result in a sampling bias, which can dramatically increase the sample error in a systematic way. For example, attempting to measure the average height of the entire human population of the Earth, but measuring a sample only from one country, could result in a large over- or under-estimation. In reality, obtaining an unbiased sample can be difficult as many parameters (in this example, country, age, gender, and so on) may strongly bias the estimator and it must be ensured that none of these factors play a part in the selection process. Even in a perfect non-biased sample, the sample error will still exist due to the remaining statistical component; consider that measuring only two or three individuals and taking the average would produce a wildly varying result each time. The likely size of the sampling error can generally be reduced by taking a larger sample.^[3] Sample Size Determination The cost of increasing a sample size may be prohibitive in reality. Since the sample error can often be estimated beforehand as a function of the sample size, various methods of sample size determination are used to weigh the predicted accuracy of an estimator against the predicted cost of taking a larger sample. Bootstrapping and Standard Error As discussed, a sample statistic, such as an average or percentage, will generally be subject to sample-to-sample variation.^[1] By comparing many samples, or splitting a larger sample up into smaller ones (potentially with overlap), the spread of the resulting sample statistics can be used to estimate the standard error on the sample. In Genetics The term "sampling error" has also been used in a related but fundamentally different sense in the field of genetics; for example in the bottleneck effect or founder effect, when natural disasters or migrations dramatically reduce the size of a population, resulting in a smaller population that may or may not fairly represent the original one. This is a source of genetic drift, as certain alleles become more or less common), and has been referred to as "sampling error",^[4] despite not being an "error" in the statistical sense. See also Wikimedia Commons has media related to Sampling error.	{"url":"https://www.knowpia.com/knowpedia/Sampling_error","timestamp":"2024-11-14T04:08:46Z","content_type":"text/html","content_length":"78522","record_id":"<urn:uuid:331b0b81-896d-4bb8-b8c6-dbf60923df10>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00701.warc.gz"}
Mathematics Professor Martin Weissman wins 2020 Guggenheim Fellowship Martin Weissman, professor of mathematics at UC Santa Cruz, has been awarded a Guggenheim Fellowship for 2020. Weissman will use the $50,000 award to support his work on the visualization of prime Weissman is one of 175 writers, scholars, artists, and scientists chosen this year from among nearly 3,000 applicants for recognition by the John Simon Guggenheim Memorial Foundation. Guggenheim Fellows are appointed on the basis of prior achievement and exceptional promise. Weissman's research involves number theory, representation theory, and geometry, and he has a particular interest in design and information visualization. His 2018 book An Illustrated Theory of Numbers won acclaim for its novel approach to visualizing concepts in mathematics. The Guggenheim Fellowship will support a new project, which he plans to begin in 2021, to convey the nature of prime numbers through a series of 60 images. According to Weissman, visualization can allow both experts and non-experts to explore data in new ways, and prime numbers can be thought of as the most important data set in mathematics. The primes are often called the “building blocks” of “My hope is to visualize the nature of prime numbers, from ancient history to the current research,” he said. “I want to create images that convey not just a mathematical form—a shape or pattern native to mathematicians—but which convey a mathematical narrative. I hope to create didactic art that provides a visual mathematical experience. Weissman said he envisions publishing this work as a book about prime numbers, driven by images and written for a scientifically literate public. Weissman earned his Ph.D. in mathematics at Harvard University in 2003 and joined the UCSC faculty in 2006.	{"url":"https://news.ucsc.edu/2020/04/weissman-guggenheim.html","timestamp":"2024-11-14T17:07:51Z","content_type":"application/xhtml+xml","content_length":"19380","record_id":"<urn:uuid:885627e0-d38d-4b8c-a520-8913515d5836>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00041.warc.gz"}
How to plot the output of A.eigenvectors_right()? How to plot the output of A.eigenvectors_right()? asked 2014-09-08 06:06:34 +0100 This post is a wiki. Anyone with karma >750 is welcome to improve it. Trying to calculate and plot eigenvectors of a matrix A. The call A.eigenvectors_right() works, but the output is a mixed list. How to extract the eigenvectors from it to plot them? The attempt: s= Lambda.eigenvectors_right();v=s[1];v[1];v1=v[1];plot(v1) gives an error message... Thank you. 2 Answers Sort by » oldest newest most voted Something like that maybe ? sage: A = matrix([[2, 3], [3, 5]]) sage: l = flatten([u[1] for u in A.eigenvectors_right()]) sage: add(plot(v) for v in l) edit flag offensive delete link more It works!... DVD ( 2014-09-08 21:46:20 +0100 )edit As you can see, v1 is a list containing a single tuple: sage: v1 [(1, 1.618033988749895?)] You want to plot the vector described by the inner tuple, not the list. The following should work: sage: plot(v1[0]) edit flag offensive delete link more Looks like "flatten()" is needed (see below)... DVD ( 2014-09-08 21:44:47 +0100 )edit	{"url":"https://ask.sagemath.org/question/24042/how-to-plot-the-output-of-aeigenvectors_right/","timestamp":"2024-11-05T09:16:44Z","content_type":"application/xhtml+xml","content_length":"60283","record_id":"<urn:uuid:b1848e4d-bfb3-4801-97c3-ee6e4875bf82>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00162.warc.gz"}
Stand Counts – Perennials In use from 2009-01-01 Stand counts can be used to evaluate seedling emergence, density, and distribution. We use the Vogel-Masters stand count method (Vogel and Masters, 2001) in herbaceous perennial treatments to measure frequency of occurrence or stand percentages of a single target species (i.e., switchgrass, miscanthus) or of a given growth form (i.e., native grasses). Dividing the frequency of occurrence by the total area examined provides a conservative estimate of plant density (plants/m2). This protocol should be completed annually during the establishment phase. Make or obtain a frequency grid. The grid is comprised of a metal or PVC frame, 75 cm x 75 cm, that has been divided into 25 cells, each 15 cm x 15 cm. Place the grid on the soil surface at a random location in the plot. Avoid areas that are not representative of the plot (e.g., adjacent to sampling equipment, on plot edges). Count the number of cells that contain at least one plant of the target species. Flip the grid over, end-to-end, and count again. Repeat two more times, for a total of 4 counts (100 cells). Add the 4 counts together. To calculate the frequency of occurrence, divide the total number of cells that contain the target plant by 100 (the total number of cells counted). To estimate the number of plants per m2, divide the total number of cells that contain the target plant by 2.25 (area of each cell= 0.0225 m2, multiplied by 100 cells). Vogel, K.P. and R.A. Masters. 2001. Frequency grid: A simple tool for measuring grassland establishment. Journal of Range Management. 54: 653-655. Date modified: Thursday, May 09 2024 5/9/2024: JS updated to clarify calculations and include mixed communities Edit \| Back	{"url":"https://data.sustainability.glbrc.org/protocols/175","timestamp":"2024-11-01T19:44:46Z","content_type":"text/html","content_length":"7790","record_id":"<urn:uuid:9bddad15-95d8-466b-9c04-ce9785eb5b79>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00353.warc.gz"}
The Use of a Computer Display Exaggerates the Connection Between Education and Approximate Number Ability in Remote Populations Piazza et al. reported a strong correlation between education and approximate number sense (ANS) acuity in a remote Amazonian population, suggesting that symbolic and nonsymbolic numerical thinking mutually enhance one another over in mathematics instruction. But Piazza et al. ran their task using a computer display, which may have exaggerated the connection between the two tasks, because participants with greater education (and hence better exact numerical abilities) may have been more comfortable with the task. To explore this possibility, we ran an ANS task in a remote population using two presentation methods: (a) a computer interface and (b) physical cards, within participants. If we only analyze the effect of education on ANS as measured by the computer version of the task, we replicate Piazza et al.’s finding. But importantly, the effect of education on the card version of the task is not significant, suggesting that the use of a computer display exaggerates effects. These results highlight the importance of task considerations when working with nonindustrialized cultures, especially those with low education. Furthermore, these results raise doubts about the proposal advanced by Piazza et al. that education enhances the acuity of the approximate number sense. In order to understand the universal properties of human thought, there has been a burgeoning interest in cross-cultural research focused on remote, nonindustrialized cultures (Henrich, Heine, & Norenzayan, 2010; Norenzayan & Heine, 2005). However, differences in behavior must always be interpreted with care, as culture often unexpectedly influences performance in ways that complicate interpretation (e.g., Berry, 2002; Cole & Scribner, 1974; Medin, Bennis, & Chandler, 2010). Recently, computer interfaces have gained popularity for collecting behavioral data from remote cultures. A danger in interpreting such data is that the participants may be unfamiliar with the testing devices, leading them to perform less well than they might otherwise. A recent example of a potential overinterpretation of results obtained from an indigenous culture using a computer interface comes in the domain of number cognition. Piazza, Pica, Izard, Spelke, and Dehaene (2013) used a computerized display to evaluate the ability to estimate approximate quantities (e.g., Dehaene, 2011) in the Munduruku, an indigenous population in the Brazilian Amazon. Piazza et al. reported a strong correlation between education and approximate number sense (ANS) acuity over a small sample of adults (N = 38). This result is potentially important because it could mean—as Piazza et al. speculate—that “symbolic and nonsymbolic numerical thinking mutually enhance one another over the course of mathematics instruction” (p. 1037). For example, practice with arithmetic might afford a learner the opportunity to calibrate and sharpen their approximate number judgments. One possible confound, however, is that participants with less education were simply less comfortable with the computer displays, potentially leading to worse performance based solely on their comfort with the testing situation. Piazza et al. attempt to control for this confound by showing that participants were matched on their ability to perform a separate task on a computer display—choosing the larger of two discs—but participants were near ceiling on this task (mean accuracy = 95%), suggesting that this task was too simple to reliably differentiate among individuals. Thus, it is still possible that education may simply predict comfort with a computer display in this indigenous population, rather than participants’ ability in an approximate number task. To investigate the role of computer displays in number cognition in a population that has little familiarity with computers, we worked with the Tsimane’, a native Amazonian group living in the lowlands of Bolivia (Huanca, 2008). The Tsimane’ live in small groups, hunt, and farm (to a limited extent) for subsistence. Unlike people from industrialized cultures, many Tsimane’ adults have never attended school, and those that have attended often begin school at a later age than individuals in industrialized countries, and they often leave school earlier. Hence their education level is highly variable across the population. We constructed the present experiment to test whether possible discomfort with the computer presentation would manifest itself in the measurement of ANS acuity, much like effects of task comfort on success that have been observed in U.S. children (Odic, Hock, & Halberda, 2014). For our purposes, such a baseline shift in performance would prevent “fair” comparison of acuity levels across cultures; more generally, a variable influence of task might preclude comparison across any two populations, including adults and children. Even more problematically, interactions between the task effect and education would lead to spurious (or exaggerated) education effects in correlations (as in Piazza et al., 2013): When only computerized displays were used, it would appear as though education improved ANS acuity, when in fact increased education might just allow participants to be comfortable in the testing paradigm.^^1 One hundred and forty-five adults (mean age: 36.8 years; SD: 16.3 years; range: 17–77 years) were recruited from six Tsimane’ communities near the town of San Borja in the Bolivian Amazon, in collaboration with the Centro Boliviano de Investigación y de Desarrollo Socio Integral (CBIDSI), which provided interpreters, logistical coordination, and expertise in Tsimane’ culture. Participants first completed a short demographic survey, including reporting the highest number of years of education they had achieved (a whole number between 0 and 16), their age, gender, Spanish proficiency, and household size. Tsimane’ education consists of classroom work in the village, with the local teacher (usually the most educated person in the village). Children learn the basics of arithmetic, reading, writing, Spanish language, and training in needed skills for village living, such as how to build houses. Our main task consisted of two parts, performed in a random order for each participant. Area-controlled, intermixed dot stimuli were presented in two different ways to each participant: (a) via a touchscreen laptop computer and (b) via laminated cards that were presented by the experimenters. The stimuli consisted of black and red dots of varying sizes, intermixed inside a disc (see Figure 1 ). For each version of the task, participants were asked to report whether there were more black or red dots in the display. The sets of black and red dots were matched for the size of the biggest and smallest dots in each set. The sets varied in ratios among the following ratios, going from least to most complex to discriminate: 1:3, 1:2, 2:3, 3:4, 4:5, 5:6, 6:7, 7:8, 8:9, 9:10, 10:11, and 11:12. In order to minimize spurious differences among the perceivable ratios, we kept the total number of dots as close to a total of 20 as possible, given these ratios (i.e., 5:15, 7:14, 8:12, 9:12, 8:10, 10:12, 12:14, 7:8, 8:9, 9:10, 10:11, and 11:12). There were eight versions of each of the ratios, each with four trials where “red” was the correct answer, and four where “black” was Participants were instructed to touch a red square below the presentation disc if there were more red dots or a black square if there were more black dots. These squares were on the screen of the touchscreen computer version, or on a laminated card in the card version. Stimuli remained in front of the participant until they touched one of the squares. In the cards version of the task, the correct answer was printed on the card in a coded form. The experimenter would put his thumb over this code when placing the card in front of the participant, so that neither he nor the participant could see it until the trial was complete. This way, the experimenter could not provide cues to the participants about correct answers. Participants were trained on eight practice trials consisting of dots in a 1:3 ratio. If they made any mistakes, the experimenter would explain the task again, and then they would repeat the set of eight practice trials, in a different random order. Participants were given at most three attempts to complete the practice trials correctly. Four participants failed this criterion and no further data were collected for them. Once a participant finished one version of the task (computer or cards), they would start the other version (cards or computer). We report data from the 141 participants who succeeded in the practice trials (78 who did the cards version first; 63 who did the computer version first). In the test part of each task, participants performed 30 total trials, starting at the 1:2 ratio, and using a two-up, one-down staircase design, such that if they got two answers in a row correct at one level, they moved up a level of difficulty. If, however, they got an answer wrong, then they were moved down a level. Participants who got trials incorrect at the 1:2 ratio continued at the 1:2 ratio. The eight trials per level were randomized/shuffled before each participant. All subjects’ behavior on each task was characterized by fitting a Weber fraction (W) to their entire set of responses. The Weber fraction W indexes the amount of variance in participants’ ANS number representations, such that a smaller Weber fraction indicates better performance and sharper Gaussian curves. We use Piantadosi’s (2016) method for fitting W, which is closely related to the maximum likelihood fitting used widely in the field (e.g., Halberda, Mazzocco, & Feigenson, 2008), but introduces a weak prior bias (for small W) in order to combat the problem that high W are difficult to distinguish statistically. The small bias decreases the variance of each subjects’ estimated W, while introducing a negligible influence on the mean of the estimate, leading to quantifiably better estimates.^^2 We treat all subjects’ fit Ws as point estimates of their acuity in each ANS task. We predict Log W from our dependent features. Data and analyses are available at http://osf.io/ctaj4 (Gibson, 2017). A statistical evaluation of the relationship between Education and Log W for the sum-coded card and computer versions of the task is provided in Tables 1 and 2. Figure 2 shows the relationship between years of education and Log W for the two versions of the task. As is visually apparent in the figure, there is a reliable interaction between Tsimane’ education and task, such that there is a strong correlation between education and Log W for the computer version of the task on the right, but much less so in the card version on the left. These correlations are presented in Table 2. Table 1. AIC . BIC . logLik . deviance . df.resid . 401.8 423.7 −194.9 389.8 276 Scaled residuals: Min 1Q Median 3Q Max −2.1824 −0.6418 −0.0226 0.4943 4.9975 Random effects: Groups Name Variance SD Subject (Intercept) 0.02481 0.1575 Residual 0.20978 0.4580 Number of obs: 282, groups: subject, 141 Fixed effects: Estimate SE t value (Intercept) −1.252289 0.041399 −30.249 Education −0.042551 0.008400 −5.066 task1 −0.165655 0.037229 −4.450 Education:task1 0.031667 0.007554 4.192 Correlation of fixed effects: (Intr) Eductn task1 Education −0.681 task1 0.000 0.000 −0.681 AIC . BIC . logLik . deviance . df.resid . 401.8 423.7 −194.9 389.8 276 Scaled residuals: Min 1Q Median 3Q Max −2.1824 −0.6418 −0.0226 0.4943 4.9975 Random effects: Groups Name Variance SD Subject (Intercept) 0.02481 0.1575 Residual 0.20978 0.4580 Number of obs: 282, groups: subject, 141 Fixed effects: Estimate SE t value (Intercept) −1.252289 0.041399 −30.249 Education −0.042551 0.008400 −5.066 task1 −0.165655 0.037229 −4.450 Education:task1 0.031667 0.007554 4.192 Correlation of fixed effects: (Intr) Eductn task1 Education −0.681 task1 0.000 0.000 −0.681 Note: summary(lmer(W_value_lg ∼ Education * task + (1 \| subject), REML=F, data=gathered_d)) Linear mixed model fit by maximum likelihood [’lmerMod’] Formula: W_value_lg ∼ Education * task + (1 \| Table 2. Residuals: . Min . 1Q . Median . 3Q . Max . −1.1209 −0.4343 −0.1016 0.3188 2.5396 Estimate SE t value Pr(>\| t \|) (Intercept) −1.08663 0.07125 −15.251 <2e–16 *** Education −0.07422 0.01446 −5.134 9.39e–07 * Min 1Q Median 3Q Max −0.75909 −0.20395 0.02276 0.23466 0.63624 Estimate SE t value Pr(>\| t \|) (Intercept) −1.417943 0.034819 −40.723 <2e–16 * Education −0.010884 0.007065 −1.541 0.126 Residuals: . Min . 1Q . Median . 3Q . Max . −1.1209 −0.4343 −0.1016 0.3188 2.5396 Estimate SE t value Pr(>\| t \|) (Intercept) −1.08663 0.07125 −15.251 <2e–16 * Education −0.07422 0.01446 −5.134 9.39e–07 * Min 1Q Median 3Q Max −0.75909 −0.20395 0.02276 0.23466 0.63624 Estimate SE t value Pr(>\| t \|) (Intercept) −1.417943 0.034819 −40.723 <2e–16 *** Education −0.010884 0.007065 −1.541 0.126 Note: lm(formula = W_value_lg ∼ Education, data = just_comp) ^†p < .1. p < .05. p < .01. p < .001. Figure 3 shows another visualization of these data, giving the difference score (Cards minus Computer) as a function of education. A smoothed nonparametric fit (loess) is shown for each of males and females, with 95% confidence bands. This figure demonstrates that for low education, Log Card W is less than Log Computer W, meaning that participants perform better on the card task. However, the positive trend of the average line indicates that the effect disappears with high education. In addition, this figure reveals no obvious trends with respect to age and the difference score, but one can see that high education participants tend to be younger and male, reflecting current Tsimane’ demographics. To assess statistical significance, we computed a linear regression predicting the difference in Log W (Cards) minus Log W (Computers) from demographic and task factors (see Table 3). The resulting fit suggests that adults with no education perform significantly worse when the task is administered on a computer interface, as the intercept is significantly less than zero. The regression reveals a significant effect of education such that the difference between the tasks vanishes with increasing education. The magnitude of the coefficients accords with Figure 3: The tasks do not differ after approximately 5 or 6 years of education (.287 / .051 = 5.6 years). The regression also included a (sum-coded) predictor for which task was run first within participants: when the computer version was run first, the difference between the computer and cards version was larger. In addition, the regression reveals no effect of (standardized) age, but a marginal effect of gender (sum-coded), such that the difference between the computer and cards version was slightly larger for women than men.^^3 When we omit participants with more than 10 years of education, we find very similar statistical trends as in the analyses reported in the table: All reliable effects are also reliable here. Thus it is not the 13 Tsimane’ participants with 12+ years of education that are driving the observed effects. Finally, the regression in Table 4 shows that Tsimane’ education is largely predicted by age and gender: More educated Tsimane’ participants tend to be young and male. Table 3. Residuals: . Min . 1Q . Median . 3Q . Max . −2.4145 −0.3365 0.1191 0.4094 1.3290 Estimate SE t value Pr( >\|t\|) (Intercept) −0.28733 0.08092 −3.551 0.000528 * Education 0.05106 0.01663 3.071 0.002579 Comp.First.sum −0.38043 0.10809 −3.519 0.000589 * scale(Age) 0.01967 0.05797 0.339 0.734911 Gender1 −0.09847 0.05915 −1.665 0.098286 ^† Residuals: . Min . 1Q . Median . 3Q . Max . −2.4145 −0.3365 0.1191 0.4094 1.3290 Estimate SE t value Pr( >\|t\|) (Intercept) −0.28733 0.08092 −3.551 0.000528 * Education 0.05106 0.01663 3.071 0.002579 Comp.First.sum −0.38043 0.10809 −3.519 0.000589 *** scale(Age) 0.01967 0.05797 0.339 0.734911 Gender1 −0.09847 0.05915 −1.665 0.098286 ^† Note: lm(formula = CardsMinusComputers.lg ∼ Education + Comp.First.sum + scale(Age) + Gender, data = d). ^†p < .1. p < .05. p < .01. p < .001. Table 4. Residuals: . Min . 1Q . Median . 3Q . Max . −5.6181 −2.0344 −0.3949 1.2314 11.9871 Estimate SE t value Pr( >\|t\|) (Intercept) 3.5706 0.2849 12.532 < 2e–16 * scale(Age) −1.2454 0.2818 −4.420 1.99e–05 * Gender1 −1.2057 0.2849 −4.232 4.22e–05 * scale(Age):Gender1 −0.4277 0.2818 −1.518 0.131 Residuals: . Min . 1Q . Median . 3Q . Max . −5.6181 −2.0344 −0.3949 1.2314 11.9871 Estimate SE t value Pr( >\|t\|) (Intercept) 3.5706 0.2849 12.532 < 2e–16 * scale(Age) −1.2454 0.2818 −4.420 1.99e–05 * Gender1 −1.2057 0.2849 −4.232 4.22e–05 *** scale(Age):Gender1 −0.4277 0.2818 −1.518 0.131 Note: lm(formula = Education ∼ scale(Age) * Gender, data = d). ^†p < .1. p < .05. p < .01. **p < .001. Our results demonstrate that participants with lower education levels performed worse on the task with the computer display than with the card display, whereas participants with higher education levels did just as well on the card or computer versions of the task. These results emphasize the importance of task comfort and understanding, particularly when working with populations that are unfamiliar with experimental psychology and behavioral paradigms. The fact that task performance is influenced by education level suggests that, if we had not noticed the potential confound with task, we might have found a spurious education effect. Indeed, as seen in Table 2, if we only analyze the effect of education on W, as measured by the computer version of the task, the effect is statistically significant, even though the effect of education on the card version of the task is not. The variable effect of education within the Tsimane’ population on these task factors might therefore have led us to conclude that education strongly influenced ANS if we had run only the computer version. Of course, the influence of task does not show that there is no education effect, only that if task is not controlled, we cannot be sure. This is a plausible alternative explanation for the findings of Piazza et al. (2013), discussed above. Though it is plausible that education influences ANS in these populations (we find a small nonsignificant tendency in this direction), detailed controls for task are required to rule out alternative explanations. A comparison of the slope of the effect in our computer task—about .034 W / year (note that the regression in Table 2 is computed over Log W, not W)—to the slope of the effect in Piazza et al. (2013) —about .25 W / year—suggests that the Piazza et al. effect is much larger, even on the matched computer task. But this comparison assumes that the education years are matched. Alternatively, it is possible that the education in the Munduruku is more organized than in the Tsimane’, leading to a larger effect each year. Note that the effect is miniscule for the Tsimane’ cards task—.003 W / year—and this is not significant in the regression. There are several plausible explanations for the strong education effect on task that we observed, due to factors that correlate with educational level. First, it is not the case that experience with computers could explain the observed differences, because almost none of the participants had ever seen a computer or computer tablet before, independent of their educational level, according to their own self-reports. One possible source for the education effect on task is more experience with technology more generally, such as radios, TV screens, and phones. People with more education are more likely to travel to the local Bolivian (Spanish) towns, where there is access to technology, such as a television in the town square. Another possibility is that people with greater education might have better developed cognitive control (Brod, Bunge, & Shing, 2017; Burrage et al., 2008; Morrison, Smith, & Dow-Ehrensberger, 1995; Roebers, Röthlisberger, Cimeli, Michel, & Neuenschwander, 2011) so that they can better ignore irrelevant aspects of the testing situation, such as the novel computer presentation. People with lower education might have a harder time focusing on the relevant aspects of the task in the novel situation. Thus, removing computer interfaces from cross-cultural studies would not address the current concerns. These results also have important ramifications beyond cross-cultural research: anywhere where familiarity with technology may covary with another dimension, such as age, socio economic class, or gender. In such cases, participants might be unfamiliar with computers, and researchers should therefore be careful to show that their participants don’t behave differently depending on how the task is administered. Our results show that one should be careful when designing tasks with participants who are not used to cognitive research, and careful about interpreting results from studies on remote populations, especially if a study purports to show performance that is different from a population with industrialized education. If the remote group performs similarly to industrialized nation participants, then the remote group understood the task as well as the industrialized participants (e.g., Dehaene, Izard, Pica, & Spelke, 2006, 2008; Izard, Pica, Dehaene, Hinchey, & Spelke, 2011; Izard, Pica, Spelke, & Dehaene, 2011; McCrink, Spelke, Dehaene, & Pica, 2012; Pica, Jackson, Blake, & Troje, 2011; Pica, Lemer, Izard, & Dehaene, 2004). But when they perform relatively poorly, this may not reflect a genuine cognitive difference between groups. Perhaps most importantly, our results suggest that the effect of education on ANS that Piazza et al. (2013) had observed in the Munduruku may be much weaker—if it exists at all—than suggested by Piazza et al.’s study. Future work will be needed to see if there is a reliable correlation between education and ANS that is unconfounded by task. EG, JJE, and STP designed the study. EG, JJE, and RL carried out the experiment. STP did the analyses. EG and STP drafted the manuscript. JJE and RL provided critical feedback on the manuscript, and all authors contributed to the final draft of the manuscript. We thank Ricardo Godoy and Tomas Huanca for logistical help. Dino Nate Añez, Robertina Nate Añez, and Salomon Hiza Nate helped with translating and running the task. We thank Evelina Fedorenko and Rachel Ryskin for comments on earlier drafts of this paper. Research reported in this publication was supported by National Science Foundation Grant 1022684 from the Research and Evaluation on Education in Science and Engineering (REESE) program to EG. The project was also supported by the Eunice Kennedy Shriver National Institute of Child Health & Human Development of the National Institutes of Health under Award Number F32HD070544 to STP. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Although there is a large literature on computer vs. paper tasks in the education literature, this literature predominantly involves reading, such as the TOEFL (Test of English as a Foreign Language) or SAT tasks (for a review, see Noyes & Garland, 2008). The results from this literature suggest that there is either a slight benefit for doing tasks on paper rather than computer, across all levels of participants, or no benefit either way (which seems to be the tendency in the more recent literature, possibly due to [a] better computer screens for task presentation and [b] people being more used to working on computer screens in recent years). Whereas this literature is potentially related to our research question, our ANS task involves no reading whatsoever. Furthermore, we were most interested in whether there are differences according to education level. But we are not aware of any literature on computer versus paper tasks reporting interactions with education. The statistical patterns are very similar when we used standard maximum likelihood fits: Any effect that we report as significant using Piantadosi’s methods is also significant using maximum likelihood fits. In another analysis, we investigated a potential interaction between education and task order, but this effect was nonsignificant (p = .12), so we left this interaction term out of the presented analysis. Although not significant, the direction of this interaction was such that lower education participants had a larger difference score. The regressions in Table 2 are similar if we predict W instead of Log W. In particular, for the computer task the relationship is significant (beta = −.034, t = −3.13, p = .002), but for the card task the relationship is not (beta = −.0029, t = −1.67, p = .098). J. W. Cross-cultural psychology: Research and applications Cambridge, England Cambridge University Press S. A. , & Y. L. Does one year of schooling improve children’s cognitive control and alter associated brain activation? Psychological Science . doi: M. S. C. C. E. A. B. C. A. M. , & F. J. Age-and schooling-related effects on executive functions in young children: A natural experiment Child Neuropsychology , & Culture & thought: A psychological introduction New York, NY John Wiley The number sense: How the mind creates mathematics New York, NY Oxford University Press , & E. S. Core knowledge of geometry in an Amazonian indigene group E. S. , & Log or linear? Distinct intuitions of the number scale in Western and Amazonian cultures 2017, August 10 M. M. , & Individual differences in non-verbal number acuity correlate with maths achievement S. J. , & The weirdest people in the world? Behavioral and Brain Sciences Tsimane’ oral tradition, landscape, and identity in tropical forest La Paz, Bolivia South-South Exchange Programme for Research on the History of Development (SEPHIS) , & E. S. Geometry as a universal mental construction . In Attention and performance Vol. 24 Space, time and number in the brain: Searching for the foundations of mathematical thought Oxford, England Oxford University Press E. S. , & Flexible intuitions of Euclidean geometry in an Amazonian indigene group Proceedings of the National Academy of Sciences E. S. , & Non-symbolic halving in an Amazonian indigene group Developmental Science , & Culture and the home-field disadvantage Perspectives on Psychological Science . doi: F. J. , & Education and cognitive development: A natural experiment Developmental Psychology , & S. J. Psychological universals: What are they and how can we know? Psychological Bulletin J. M. , & K. J. Computer-vs. paper-based tasks: Are they equivalent? , & Hysteresis affects approximate number discrimination in young children Journal of Experimental Psychology: General S. T. Efficient estimation of Weber’s W Behavior Research Methods , & Education enhances the acuity of the non-verbal approximate number system Psychological Science , & N. F. Comparing biological motion perception in two distinct human societies , & Exact and approximate arithmetic in an Amazonian indigene group C. M. , & School enrolment and executive functioning: A longitudinal perspective on developmental changes, the influence of learning context, and the prediction of pre-academic skills European Journal of Developmental Psychology Author notes Competing Interests: The authors have no competing interests to declare. © 2017 Massachusetts Institute of Technology Published under a Creative Commons Attribution 4.0 International (CC BY 4.0) license Massachusetts Institute of Technology Published under a Creative Commons Attribution 4.0 International (CC BY 4.0) license This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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Critical colored-RVB states in the frus SciPost Submission Page Critical colored-RVB states in the frustrated quantum Heisenberg model on the square lattice by Didier Poilblanc, Matthieu Mambrini, Sylvain Capponi This is not the latest submitted version. This Submission thread is now published as Submission summary Authors (as registered SciPost users): Sylvain Capponi · Matthieu Mambrini · Didier Poilblanc Submission information Preprint Link: https://arxiv.org/abs/1907.03678v1 (pdf) Date submitted: 2019-07-11 02:00 Submitted by: Poilblanc, Didier Submitted to: SciPost Physics Ontological classification Academic field: Physics • Condensed Matter Physics - Theory Specialties: • Quantum Physics Approach: Theoretical We consider a family of SU(2)-symmetric Projected Entangled Paired States (PEPS) on the square lattice, defining colored-Resonating Valence Bond (RVB) states, to describe the quantum disordered phase of the $J_1-J_2$ frustrated Heisenberg model.For $J_2/J_1\sim 0.55$ we show the emergence of critical (algebraic) dimer-dimer correlations -- typical of Rokhsar-Kivelson (RK) points of quantum dimer models on bipartite lattices -- while, simultaneously, the spin-spin correlation length remains short. Our findings are consistent with a spin liquid or a weak Valence Bond Crystal in the neighborhood of an RK point. Current status: Has been resubmitted Reports on this Submission Report #2 by Anonymous (Referee 2) on 2019-8-28 (Invited Report) • Cite as: Anonymous, Report on arXiv:1907.03678v1, delivered 2019-08-28, doi: 10.21468/SciPost.Report.1136 -study of an important model in the field -state of the art results -fresh insights for a new critical spin liquid phase - too technical for general audience The Authors report a numerical study of the spin-1/2 J1-J2 Heisenberg model on the square lattice, based on a family of variational, SU(2)-symmetric PEPS wavefunctions. This model is one of the first and simplest frustrated models studied, with potential spin liquid states in the region J2~J1/2. Despite many studies, the nature of the ground state(s) in this region is still unsettled. The Authors focus at the parameter point J2=0.55J1 and offer fresh insights for a possible critical phase, different from the one at J2=0.5J1. While these results do not settle the issue fully (the Authors point out possible connections to VBC phases reported previously), this study should be interesting for people working in this field and will motivate further investigations in this very old model. I would therefore recommend the article for publication. The paper is well written, although a bit too technical for non-specialists. Report #1 by Anonymous (Referee 1) on 2019-8-26 (Invited Report) • Cite as: Anonymous, Report on arXiv:1907.03678v1, delivered 2019-08-26, doi: 10.21468/SciPost.Report.1124 1. Paper explores an interesting and long-standing problem regarding the nature of the quantum disordered region for the spin-$1/2$ $J_1$-$J_2$ Heisenberg model on the square lattice. 2. A rather careful variational wave-function analysis is presented using the iPEPS method. 3. Signatures of an interesting state with short-ranged spin-spin correlations but critical dimer-dimer correlations found for $J_2/J_1 \sim 0.55$. 1. The technicalities related to the variational determination of the ground state using iPEPS are a bit difficult to follow. The properties of the ground state of the spin-$1/2$ $J_1$-$J_2$ Heisenberg model on the square lattice in its quantum disordered region (in the vicinity of $J_2/J_1 \sim 0.5$) has attracted considerable attention for several years now due to the interplay of strong quantum fluctuations and frustration. Despite this, the nature of the phase(s) in the quantum disordered region remains Here, the authors consider a family of PEPS with the full space group symmetry of the lattice and the $SU(2)$ spin rotation symmetry incorporated in their calculations. Within this variational approach, they find an interesting RVB-like state with short-ranged spin-spin correlations but (almost) critical dimer-dimer correlations for $J_2/J_1 \sim 0.55$. This state is rather different from a gapless spin liquid (where the spin-spin correlations are also algebraic) obtained at $J_2/J_1=0.5$ using a similar framework. Could the authors comment on the following? Can they detect some signatures of an enlarged symmetry (possibly $U(1)$) from the wave-function at $J_2/J_1 \sim 0.55$, e.g., by looking at columnar and plaquette like dimer correlation functions as a function of $r$? Secondly, when they compute the dimer-dimer correlations using long stripes, how do they take into account of the reduced lattice symmetry in their variational parameters? The results presented here are definitely interesting and the numerics has been carefully done. After getting appropriate response from the authors to my questions above, I will be happy to recommend this manuscript for publication. Requested changes 1. Few typos need to be corrected. E.g. Page 2, second paragraph->"which are specially designed to "describe" SU(2)-invariant", Page 6, first paragraph->"Hence, we have "improved" the CTMRG", Page 11, last paragraph->"(at least) two "scenarios""	{"url":"https://www.scipost.org/submissions/1907.03678v1/","timestamp":"2024-11-06T13:42:22Z","content_type":"text/html","content_length":"38174","record_id":"<urn:uuid:5a2edc20-d668-4d31-99f9-bb8f3d3dad40>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00696.warc.gz"}
Supervised Learning: Bayesian Learning This is the 10th in a series of class notes as I go through the Georgia Tech/Udacity Machine Learning course. The class textbook is Machine Learning by Tom Mitchell. This chapter is less theoretical than VC Dimensions, but still more focusing on the theoretical basis of what we do in Supervised Learning. The Core Question What we've been trying to do is: learn the best hypothesis that we can, given some data and domain knowledge. We are going to refine what our notion of "best" is to be "most probable" hypothesis, out of the hypothesis space. Bayes Rule I'll assume you know this (more info, wonderful counterintuitive application here): Applied to ML: • A is our hypothesis • B is our Data • our hypothesis given the data is P(A\|B), what we want to maximize • our data sampling is P(B) • our prior domain knowledge is represented by P(A) • our updating (the running of the algorithm) is P(B\|A), also considered accuracy Since P(B) is constant among hypotheses and we just care about maximizing P(A\|B), we can effectively ignore it. We've had many ways of representing domain knowledge so far - SVM kernels, choosing to use kNN (closer is more similar) - P(A) is a more general form of this. Maximum Likelihood: In fact if we further assume we have no priors (aka assume a uniform distribution for P(A)), P(A) can effectively be ignored. This means maximizing P(A\|B) (our most likely hypothesis given data) is the same as maximizing P(B\|A) (the most likely data labels we see given our hypothesis) if we don't have a strong prior. This is the philosophical justification for making projections based on data, but as you can see, even a little bit of domain knowledge will skew/improve our conclusions dramatically. However, it is impractical to iterate through all hypotheses and mechanically update priors like this. What we are aiming for with Bayes is more of an intuition about the levers we can pull rather than an actual applicable formula. The Relationship of Bayes and Version Spaces No Noise: This proof is too mundane to write out, but basically given no priors, it is a truism that the probability of any particular hypothesis given data (P(h\|D)) is exactly equal to 1/\|VS\| (VS for version space). AKA if all hypotheses are equally likely to be true, then the probability of any one hypothesis being true is 1 / the number of hypotheses. Not exactly exciting stuff, but it derives from Bayes. With Noise: If we assume that noise has a normal distribution, we then use this proof to arrive at the sum of squares best fit formula: The Minimum Description Objective By adding a logarithm (a monotonic function, so it doesnt change the end result), we can translate, decompose, and flip the maximization goal we defined above to a minimization with tradeoffs: Here L stands for Length, for example the length of a decision tree, which indicates the complexity of the model/hypothesis. L(h) is the complexity of the model we have selected, while L(D\|h) is the data that doesn't fit the model we have selected, aka the error. We want to minimize this total "description length", aka seek the minimum description length. This is a very nice mathematical expression of Occam's razor with a penalty for oversimplification. There are practical issues with applying this, for example the unit comparability of errors vs model complexity, but you can decide on a rule for trading them off. The Big Misdirection Despite everything we laid out here and in prior chapters, we don't -really- care about finding the single best or most likely hypothesis. We care about finding the single most likely label. A correct hypothesis will help us get there every time, sure, but if every remaining hypothesis in our version spaces in aggregate point towards a particular label, then that is the one we actually Thus, Bayesian Learning is merely a stepping stone towards Bayesian Classification, and building a Bayes Optimal Classifier. We will explore this in the next chapter on Bayesian Inference. Next in our series Further notes on this topic: Hopefully that was a good introduction to Bayesian Learning. I am planning more primers and would love your feedback and questions on: • Supervised Learning • Unsupervised Learning □ Clustering - week of Feb 25 □ Feature Selection - week of Mar 4 □ Feature Transformation - week of Mar 11 • Reinforcement Learning □ Markov Decision Processes - week of Mar 25 □ "True" RL - week of Apr 1 □ Game Theory - week of Apr 15 Top comments (0) For further actions, you may consider blocking this person and/or reporting abuse	{"url":"https://dev.to/swyx/supervised-learning-bayesian-learning-403l","timestamp":"2024-11-11T13:14:15Z","content_type":"text/html","content_length":"70382","record_id":"<urn:uuid:ce40eb17-4002-4a8d-b3fb-bb292c602eec>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00228.warc.gz"}
Types of Matrices and Matrix Addition Professor Dave Explains 24 Oct 201806:45 TLDRThe script introduces matrices and defines terminology related to them like square, diagonal, triangular, vector, and row vector matrices. It explains scalar multiplication and matrix addition, which requires matrices to have identical dimensions. Addition is commutative but subtraction is not. It also shows the vector form of a linear system, where the coefficient vectors are multiplied by scalars and added to the constant vector. These basic matrix operations like scalar multiplication, addition, and subtraction are explained. • π Matrices are arrays of numbers that often represent coefficients and constants from systems of linear equations. • π Square matrices have the same number of rows and columns. The main diagonal goes from the top left to the bottom right. • π Ί Diagonal, upper triangular, and lower triangular are special types of square matrices based on where the zero entries are. • β Vectors are special cases of matrices - a column vector has one column, and a row vector has one row. • β Matrices can be added together if they have the same dimensions. Each entry is the sum of the corresponding entries. • β Matrices can also be subtracted using the same rule by subtracting corresponding entries. • π Matrix addition is commutative, but matrix subtraction is not. • β Scalar multiplication multiplies every entry in a matrix by a scalar value. • π The vector form of a linear system uses separate vectors for the coefficients of each variable. • π These basic matrix operations allow matrices to be manipulated and linear systems to be expressed in multiple ways. Q & A • What is a square matrix? -A square matrix is one where the number of rows and columns is the same. • What is a diagonal matrix? -A diagonal matrix is one where all the entries that are not part of the main diagonal are zero. • What is an identity matrix? -An identity matrix is a diagonal matrix where all the entries on the main diagonal are one and the rest are zero. • What is an upper triangular matrix? -An upper triangular matrix is a square matrix where all the entries below the main diagonal are zero. • What is a lower triangular matrix? -A lower triangular matrix is a square matrix where all the entries above the main diagonal are zero. • What is a vector in matrix terminology? -A vector is a matrix with a single column. It is commonly used to represent the solution to a system of equations. • What is a row vector? -A row vector is a matrix with a single row. • What operation allows you to multiply every entry in a matrix by a scalar? -Scalar multiplication allows you to multiply every entry in a matrix by a scalar value. • What conditions must be met to add two matrices together? -To add two matrices, they must have the same number of rows and columns - identical dimensions. • Is matrix addition commutative? -Yes, matrix addition is commutative, meaning the order does not matter. π Defining Matrices and Performing Basic Matrix Operations Paragraph 1 defines matrices, describes types of matrices like square, diagonal, upper/lower triangular matrices, vectors, and row vectors. It also explains basic matrix operations like scalar multiplication, matrix addition/subtraction, noting that addition is commutative while subtraction is not. β Checking Comprehension on Matrix Operations Paragraph 2 states that matrix addition is commutative while matrix subtraction is not. It then prompts the reader to check their comprehension on the matrix operations just covered. π ‘matrix A matrix is an array of numbers arranged in rows and columns. Matrices are used to represent systems of linear equations. The video discusses different types of matrices and operations that can be performed on them. π ‘square matrix A square matrix has the same number of rows and columns. Square matrices have a main diagonal going from the top left to the bottom right. They are important because matrix operations often require the matrices to have the same dimensions. π ‘diagonal matrix A diagonal matrix is a square matrix where all entries except those on the main diagonal are zero. Getting an augmented matrix into diagonal form is an important step when using matrices to solve systems of linear equations. π ‘identity matrix An identity matrix is a diagonal matrix where all the entries on the main diagonal are 1. Identity matrices represent the multiplicative identity in matrix operations. π ‘scalar multiplication Scalar multiplication involves multiplying every entry in a matrix by a scalar value. This scales the matrix up or down in size. π ‘matrix addition To add matrices, they must have the same dimensions. Each entry in the sum matrix is the sum of the corresponding entries from the matrices being added. π ‘vector A vector is a matrix with only one column or one row. Vectors are used to represent solutions to systems of equations. The linear system can also be expressed in vector form. π ‘commutative An operation like addition is commutative if changing the order of the operands does not change the result. Matrix addition is commutative but matrix subtraction is not. π ‘row vector A row vector is a 1 x n matrix with just a single row. Column vectors are more commonly used than row vectors in linear algebra. π ‘upper triangular matrix An upper triangular matrix has all its entries below the main diagonal equal to zero. Converting a matrix into triangular form is often an intermediate step when solving systems of equations. The transcript discusses using machine learning models to understand protein folding, which could have major impacts on drug discovery and disease research. The AlphaFold system was able to predict protein structures with high accuracy, outperforming previous approaches. Understanding protein folding at the molecular level provides insights into biological mechanisms and can enable targeted drug design. DeepMind's results show that AI can solve complex molecular biology problems, with performance exceeding human experts. Accurate protein structure prediction has been a grand challenge in biology for over 50 years. The deep learning system was trained on hundreds of thousands of known protein sequences and structures. The breakthrough demonstrates the power of AI to make fundamental advances in the life sciences. Understanding protein folding may provide insights into diseases like Alzheimer's and Parkinson's. Predicting protein structures reduces the need for expensive and time-consuming lab techniques like cryo-electron microscopy. The ability to computationally determine protein structures is a major milestone with far-reaching benefits. The AlphaFold system opens up new possibilities for accelerated drug discovery and personalized medicine. Public availability of these protein structure predictions will empower researchers around the world. Overall, this work represents a transformational advance in computational biology and the application of AI to the life sciences. In the future, even more accurate models could be developed as increased biological data becomes available. The ability to computationally predict protein structures may be one of the most significant applications of AI with huge potential. Rate This 5.0 / 5 (0 votes)	{"url":"https://learning.box/video-390-Types-of-Matrices-and-Matrix-Addition","timestamp":"2024-11-06T09:06:02Z","content_type":"text/html","content_length":"106818","record_id":"<urn:uuid:8d3f3387-dfad-47d2-b6ab-21e066ce5b59>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00196.warc.gz"}
What is significance of dispersion number? What is significance of dispersion number? The dispersion number represents the overall extent of axial dispersion in the system under consideration and is the reciprocal of the Peclet number (Pe = unL/Da). In the dispersion number expression, Da is the axial dispersion coefficient, un is the mean velocity and L is the length between the two measurement points. What is dispersion number of CSTR? keff = (kkg)1/2. What is an ideal CSTR? An ideal CSTR will exhibit well-defined flow behavior that can be characterized by the reactor’s residence time distribution, or exit age distribution. Not all fluid particles will spend the same amount of time within the reactor. What is dispersion model in CRE? Dispersion Model. The one parameter to be determined in the dispersion model is the dispersion coefficient, Da. The dispersion model is used most often for non-ideal tubular reactors. The dispersion coefficient can be found by a pulse tracer experiment. What is dispersion number of reactor? Vessel dispersion number is defined as: The variance of a continuous distribution measured at a finite number of equidistant locations is given by: (5) Where mean residence time τ is given by: (6) How is dispersion number calculated? Add together your differences from the mean and divide by the number of data values you have. In the example, 2.66 plus 0.33 plus 3.33 equals 6.32. Then, 6.32 divided by 3 equals an average deviation of 2.106. What is the importance of damkohler number? A Damköhler number (Da) is a useful ratio for determining whether diffusion rates or reaction rates are more ‘important’ for defining a steady-state chemical distribution over the length and time scales of interest. What are CSTRs used for? CSTRs are most commonly used in industrial processing, primarily in homogeneous liquid-phase flow reactions where constant agitation is required. However, they are also used in the pharmaceutical industry and for biological processes, such as cell cultures and fermenters. What is non ideal reactor? NON-IDEAL REACTORS Definition: reactors do not meet ideal conditions of flow and mixing due. to: Dispersion deviates from ideal plug flow conditions. Short-circuiting and dead spaces deviate from ideal mixing and. plug flow conditions. What is space velocity in reactor? In chemical engineering and reactor engineering, space velocity refers to the quotient of the entering volumetric flow rate of the reactants divided by the reactor volume which indicates how many reactor volumes of feed can be treated in a unit time. It is commonly regarded as the reciprocal of the reactor space time. What is a good coefficient of dispersion? The International Association of Assessing Officers (2013) states that the performance standard for the coefficient of dispersion is 15% or less for residential properties and 20% or less for commercial properties.	{"url":"https://www.ammarksman.com/what-is-significance-of-dispersion-number/","timestamp":"2024-11-03T15:44:07Z","content_type":"text/html","content_length":"39199","record_id":"<urn:uuid:39055b28-28c6-4e48-8140-53426020c584>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00296.warc.gz"}
Re: Bug Report - Two numerical values for a same variable • To: mathgroup at smc.vnet.net • Subject: [mg54468] Re: Bug Report - Two numerical values for a same variable • From: "Drago Ganic" <drago.ganic at in2.hr> • Date: Mon, 21 Feb 2005 03:44:37 -0500 (EST) • References: <cv9af3$l1s$1@smc.vnet.net> • Sender: owner-wri-mathgroup at wolfram.com The Mathematica type Real is NOT a subset of the mathematical field Reals. Those are quit different things. On the other hand, the Mathematica domain Reals is intended to mimic all properties of the mathematics Reals field. The same is tru for Complex/Complexes. The Mathematica Real type is CS stuff (for programming as Andrzej said). It represents an approximate (real) number. Approximate real numbers, as defined in Mathematica, are "intervals" [x - d/2, x + d/2] and NOT "points" on a (infinite) number line. That fact is used in the (significance) arithmetic of Real numbers which is different than the arithmetic used for real numbers. The name Float could also been used instead of Real and would maybe be a better alternative. I'm not a Mathematician as you two guys, so I could be wrong :-) Greetings from Croatia, "Richard Fateman" <fateman at cs.berkeley.edu> wrote in message news:cv9af3$l1s$1 at smc.vnet.net... > Andrzej Kozlowski wrote: >> On 19 Feb 2005, at 20:54, Richard Fateman wrote: >>> I suspect that Mathematica's routines for testing for membership in >>> domains like Reals etc. are pretty good. I don't, however, expect it >>> to understand anything. Just symbolic manipulation. That's different. >> Well, if you really know any people who think that computer programs >> "understand" anything than you move in curious circles. But just a >> moment, who was it who wrote: >>> The fact that these are stand-ins for well-known mathematically real >>> quantities is irrelevant to Mathematica, since it only understands Real, >>> and Real is a subset of mathematical real. >> Understands? >> Andrzej Kozlowski > I quote from you, "Mathematica certianly undersands Reals," > by which I believe you intend for us to believe that Mathematica > understands the concept of real numbers in mathematics. > If you mean Mathematica has a heuristic program that is intended to figure > out if an expression is guaranteed to be mathematically real, then > But it is not a decision procedure to determine membership. > If that is what you meant by "understands", I don't disagree. > When I said Mathematica only understands Real, I meant the PROGRAM > named "Real" is part of Mathematica, and it is an executable constructor > for a data type. The possibilities for encoding data as object > with Head Real constitute a subset of the mathematical real numbers. > While anthropomorphizing has its dangers, I think these two > examples are different. > RJF	{"url":"http://forums.wolfram.com/mathgroup/archive/2005/Feb/msg00627.html","timestamp":"2024-11-12T02:29:43Z","content_type":"text/html","content_length":"32988","record_id":"<urn:uuid:8ca83043-5894-482d-834e-82cd1d222ca0>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00831.warc.gz"}
\] is superimposed on an alternating current \ Hint We can obtain the equation of the resulting current through the wire by adding the direct current to the alternating current. The effective value of current is equal to the root mean square value of the current. So we need to take the square, calculate its mean by integrating it over a time period, and take the square root to obtain the RMS value which will be the required effective Complete step-by-step solution: Since the direct current is superimposed over the alternating current, the resulting current will be equal to the sum of the direct and the alternating current. So we have ${I_R} = {I_{dc}} + {I_{ac}}$ According to the question, ${I_{dc}} = 5{\text{A}}$ and ${I_{ac}} = 10\sin \omega t$. Substituting these in the above equation, we get the resultant current as ${I_R} = 5 + 10\sin \omega t$ Now, we know that the effective value of the current is nothing but the RMS, or the root mean square value of the current. By the definition of RMS, we have to take the square, then take the mean, and finally take the square root of the current. So on squaring both sides of the above equation, we get ${I_R}^2 = {\left( {5 + 10\sin \omega t} \right)^2}$ We know that ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$. So we write the above equation as ${I_R}^2 = {5^2} + 2\left( 5 \right)\left( {10\sin \omega t} \right) + {\left( {10\sin \omega t} \right)^2}$ $ \Rightarrow {I_R}^2 = 25 + 100\sin \omega t + 100{\sin ^2}\omega t$ (1) Now, we know that the mean of a function is given by $M = \dfrac{1}{T}\int_0^T {f\left( t \right)dt} $ The period of the function ${I_R}^2$ is clearly equal to $\dfrac{{2\pi }}{\omega }$. So we integrate it from $0$ to $\dfrac{{2\pi }}{\omega }$ to get its mean as \[M = \dfrac{1}{{2\pi /\omega }}\int_0^{\dfrac{{2\pi }}{\omega }} {{I_R}^2dt} \] \[ \Rightarrow M = \dfrac{\omega }{{2\pi }}\int_0^{\dfrac{{2\pi }}{\omega }} {{I_R}^2dt} \] Putting (1) above, we get \[ \Rightarrow M = \dfrac{\omega }{{2\pi }}\int_0^{\dfrac{{2\pi }}{\omega }} {\left( {25 + 100\sin \omega t + 100{{\sin }^2}\omega t} \right)dt} \] \[ \Rightarrow M = \dfrac{\omega }{{2\pi }}\left( {\int_0^{\dfrac{{2\pi }}{\omega }} {25dt} + \int_0^{\dfrac{{2\pi }}{\omega }} {100\sin \omega tdt} + \int_0^{\dfrac{{2\pi }}{\omega }} {100{{\sin }^ 2}\omega tdt} } \right)\] We know that the average value of the sinusoidal functions is equal to zero. So we can put \[\int_0^{\dfrac{{2\pi }}{\omega }} {100\sin \omega tdt} = 0\] in the above equation to get \[M = \dfrac{\omega }{{2\pi }}\left( {\int_0^{\dfrac{{2\pi }}{\omega }} {25dt} + \int_0^{\dfrac{{2\pi }}{\omega }} {100{{\sin }^2}\omega tdt} } \right)\] We know that ${\sin ^2}\theta = \dfrac{{1 - \cos 2\theta }}{2}$. So we write the above equation as\[M = \dfrac{\omega }{{2\pi }}\left( {\int_0^{\dfrac{{2\pi }}{\omega }} {25dt} + \int_0^{\dfrac{{2\pi }}{\omega }} {100\left( {\dfrac{{1 - \cos 2\omega t}}{2}} \right)dt} } \right)\] \[ \Rightarrow M = \dfrac{\omega }{{2\pi }}\left( {25\int_0^{\dfrac{{2\pi }}{\omega }} {dt} + 50\int_0^{\dfrac{{2\pi }}{\omega }} {\left( {1 - \cos 2\omega t} \right)dt} } \right)\] \[ \Rightarrow M = \dfrac{\omega }{{2\pi }}\left( {25\int_0^{\dfrac{{2\pi }}{\omega }} {dt} + 50\int_0^{\dfrac{{2\pi }}{\omega }} {dt} - \int_0^{\dfrac{{2\pi }}{\omega }} {\cos 2\omega tdt} } \right) We know that $\int {dt} = t$. So we get \[ \Rightarrow M = \dfrac{\omega }{{2\pi }}\left( {25\left[ t \right]_0^{\dfrac{{2\pi }}{\omega }} + 5025\left[ t \right]_0^{\dfrac{{2\pi }}{\omega }} - 50\int_0^{\dfrac{{2\pi }}{\omega }} {\cos 2\ omega tdt} } \right)\] \[ \Rightarrow M = \dfrac{\omega }{{2\pi }}\left( {25\left( {\dfrac{{2\pi }}{\omega }} \right) + 50\left( {\dfrac{{2\pi }}{\omega }} \right) - \int_0^{\dfrac{{2\pi }}{\omega }} {\cos 2\omega tdt} } \ Since $\cos 2\omega t$ is also sinusoidal, its average over a cycle will also be zero, that is, \[\int_0^{\dfrac{{2\pi }}{\omega }} {\cos 2\omega tdt} = 0\]. \[ \Rightarrow M = \dfrac{\omega }{{2\pi }}\left( {25\left( {\dfrac{{2\pi }}{\omega }} \right) + 50\left( {\dfrac{{2\pi }}{\omega }} \right)} \right)\] \[ \Rightarrow M = \left( {25 + 50} \right) = 75{{\text{A}}^2}\] Now, we take the square root of this mean value to get the final RMS value of the resulting current as $RMS = \sqrt M $ \[ \Rightarrow RMS = \sqrt {75} {\text{A}} = 5\sqrt 3 {\text{A}}\] Thus the effective value of the resulting current is equal to \[5\sqrt 3 {\text{A}}\]. Hence, the correct answer is option B. Note: We can integrate the square of the current between any time interval which makes a complete time period. But it will be convenient to set the lower limit of the integral equal to zero. So we chose the period from $0$ to $2\pi $.	{"url":"https://www.vedantu.com/question-answer/a-direct-current-of-5a-is-superimposed-on-an-class-12-physics-cbse-600928834ee4305c5790eb24","timestamp":"2024-11-13T02:46:35Z","content_type":"text/html","content_length":"195325","record_id":"<urn:uuid:4a161573-4e42-4f07-9652-c90a6379abc7>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00535.warc.gz"}
3 discount rates Hallo :) I'm trying to use Global Discounting on my Markov model. The problem is the discount rate is cycle dependent, so from cycle 1 to 46 = 3,5%, 47-69 = 2,5% and > 70 = 1,5%. I figured that I need a table to make this happen, but regardless of which format i use it just does'nt seem to work. I have tried to enter a fixted discount rate on 3% and it worked fine. Anyone familiar with using tables to discount? Best regards 3 comments • Official comment Andrew Munzer The Global Discounting function only supports a fixed interest rate over Markov cycles. There is no way to use this function for a non-fixed interest rate. You should be able to modify the custom discounting function in the example model Special Features > Custom Function Discount.trex. The function within that model delays discounting for 5 cycles. You want something different, but the basic structure of the custom function will remain the same. Note that if you use this approach, you will have to place the CustomDiscount function around every cost and utility in the model. It is also critical that you manually test the values you will see in the Markov Cohort trace because the discounted values will not be shown separately. Comment actions Anne Sofie Jensen I had a chat with a colleague and a simple table without cycle 0 solved the problem with different rates and global discounting. Comment actions Andrew Munzer TreeAge Pro's global discounting uses a single interest rate calculated at _stage 0 for every Markov cycle. You can use the Discount function to apply a different rate by cycle. However, I recommend against it. Even with the Discount function, discounting would not be applied in a graduated manner (diff rate for diff years). It would be applied using the discount rate set for that specific _stage value. I believe that the discount function should generally be applied with a fixed rate. Comment actions Please sign in to leave a comment.	{"url":"https://treeage.zendesk.com/hc/en-us/community/posts/4409683779739-3-discount-rates-","timestamp":"2024-11-06T11:12:22Z","content_type":"text/html","content_length":"28044","record_id":"<urn:uuid:eed65d0e-504d-412b-87eb-d2a034c2d9db>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00473.warc.gz"}
The Impact of the Creative Problem-Solving (CPS) Learning Model on the Mathematical Reasoning Skills of 8th-grade Junior High School Students Studying Systems of Linear Equations with Two Variables - International Journal of Research and Innovation in Social ScienceThe Impact of the Creative Problem-Solving (CPS) Learning Model on the Mathematical Reasoning Skills of 8th-grade Junior High School Students Studying Systems of Linear Equations with Two Variables - International Journal of Research and Innovation in Social Science Submission Deadline-15th November 2024 November 2024 Issue : Publication Fee: 30$ USD Submit Now Submission Deadline-05th November 2024 Special Issue on Economics, Management, Sociology, Communication, Psychology: Publication Fee: 30$ USD Submit Now Submission Deadline-20th November 2024 Special Issue on Education, Public Health: Publication Fee: 30$ USD Submit Now The Impact of the Creative Problem-Solving (CPS) Learning Model on the Mathematical Reasoning Skills of 8th-grade Junior High School Students Studying Systems of Linear Equations with Two Variables • Abdul Hamid B • Cynthia Tri Octavianti • Yosep Jaka Nagha • 929-938 • Feb 3, 2024 The Impact of the Creative Problem-Solving (CPS) Learning Model on the Mathematical Reasoning Skills of 8th-grade Junior High School Students Studying Systems of Linear Equations with Two Variables Abdul Hamid B^1, Cynthia Tri Octavianti^2[,] Yosep Jaka Nagha^3 ^1Lecturer, Department of Teacher Professional Education, Wisnuwardhana University of Malang, Indonesia, 65139 ^2Lecturer, Department of Mathematics Education, Wisnuwardhana University of Malang,Â Indonesia, 65139 ^3Student, Department of Mathematics Education, Wisnuwardhana University of Malang, Indonesia, 65139 Corresponding Author DOI: https://dx.doi.org/10.47772/IJRISS.2024.801070 Received: 25 December 2023; Revised: 04 January 2024; Accepted: 08 January 2024; Published: 03 February 2024 The goal of this research is to examine the effect of the Creative Problem Solving (CPS) Learning Model on Students’ Mathematical Reasoning Abilities in the Topic of Systems of Linear equations with Two Variables for 8th-grade students in a Public Junior High School I Jabung of Malang.Â This study used an Experimental Design. The research subjects consist of 60 students from two groups. The sampling technique used is Random Cluster Sampling. The research instrument used is a test. The data analysis techniques include descriptive statistics, inferential statistics, and SPSS. The results of this study indicate a significant effect of the creative problem-solving Learning model on students’ mathematical reasoning abilities. This is evident from the average score in the experimental group, which is 72.600, and the average score in the control group, which is 2.200. Based on the results of the inferential statistical analysis, the post-test value obtained is t-value = 0.694, which is greater than the t-table value of 0.58. Therefore, the null hypothesis (H[0]) is injected, and the alternative hypothesis (H[1]) is accepted. This means that the Creative Problem-Solving learning model significantly affects students’ mathematical reasoning. Keywords: Creative Problem Solving, Mathematical Reasoning Ability, Systems of linear equations with two Variables. Reasoning ability is one of the objectives of mathematics education. Teachers, as educators and instructors, should be able to develop students’ reasoning abilities. Students with strong reasoning skills will easily grasp mathematical concepts, while those with low mathematical reasoning abilities will find it challenging to understand mathematical materials. This is because mathematics content and mathematical reasoning are two inseparable things. Mathematics content is understood through reasoning, and reasoning is understood through learning mathematics content (Shadiq, 2004). Using reasoning, students can build knowledge and skills in working on mathematics problems easily (Napitupulu et.al., 2016). The importance of having mathematical reasoning abilities in students is essentially under the vision of mathematics, especially in meeting future needs (Ria et al., 2021). mathematical reasoning abilities enable students to (a) recognize reasoning and proof as basic aspects of mathematics; (b) make and examine mathematical conjectures; and (c) develop and evaluate mathematical arguments and proofs (Wibowo, 2017). In a preliminary study conducted (Aprilianti & Zhanty, 2019) it was explained that mathematical reasoning has a very important role for students, students should not only understand and practice questions but students should actively participate in solving problems in mathematics learning. Due to the crucial role of mathematics in human life, there is a need for an enhancement of the quality of mathematics education in schools. Various efforts can be made through the development of teaching strategies, teaching models, and the use of teaching media aimed at facilitating the delivery of instructional materials to students in the classroom (Dimyati and Mudijono, 2006). Under the results of the researcher’s observation during the study at State Junior High School 1 Jabung, Malang, it was found that several students do not enjoy mathematics lessons. These students lack enthusiasm for learning mathematics; during math lessons, they display a very passive attitude, meaning that if the teacher does not instruct them to complete assignments, the students will not do the tasks given by the teacher. When faced with new problems, some students are unwilling to work on them because they can only solve problems similar to the examples given previously. Furthermore, some students struggle with shortcuts and question new numbers that suddenly appear, desiring a more systematic approach to learning. There are still students who cannot solve new problems because they do not utilize their reasoning optimally.Â In terms of the application of teaching models, the selection of an appropriate teaching model is crucial so that students can learn effectively and efficiently. Therefore, a teacher must master and choose the right model in line with the teaching material, the environmental conditions, and the students themselves. The chosen teaching model should also be able to influence students’ motivation to learn in the classroom. The appropriateness of using the model can also affect students’ abilities, especially their mathematical reasoning skills. Based on the results of these observations, the researcher selects the Creative Problem-Solving teaching model to address the difficulties experienced by students. With the Creative problem-solving teaching model, students not only learn more systematically but also have the opportunity to verify the correctness of their answers by checking. The Creative Problem-Solving model is an instructional approach aimed at fostering students’ ability to creatively and innovatively solve problems. This model promotes creative thinking, analytical skills, and collaborative abilities, all of which are vital skills for addressing complex real-world issues. Shoimin (2014) states that the Creative Problem model focuses on teaching problem-solving skills, followed by skill reinforcement. When confronted with a question, students can utilize problem-solving skills to select and develop their responses. Instead of memorization without thinking, problem-solving skills enhance the thinking process.Â Using this teaching model is an effort to enhance the process, and achievement, and serve a greater function, namely providing the foundation for the development of classroom teaching activities, fostering student engagement in self-assessment, and raising awareness of their personal development. Therefore, it is expected that with a teaching model that encourages students to think collaboratively and is driven by their intrinsic motivation, the learning objectives can be achieved. The creative Problem-Solving model can guide students to generate ideas, acquire ideas, and provide solutions in problem-solving, especially in mathematics. In this approach, the teacher presents problems for the students to work on, which are related to everyday life, sparks students’ curiosity, and helps them formulate problem-solving strategies, enabling students to find solutions using the best strategies. This is in line with the findings of Syazali (2015), which indicate that the implementation of the Creative Problem-Solving teaching model has an impact on students’ mathematical problem-solving abilities. Similarly, the results of research conducted by researchers such as Nopitasari (2016) and Muin et al. (2018) demonstrate that students’ mathematical problem-solving skills improve more when following the Creative Problem-Solving model compared to conventional teaching models because the creative problem-solving model can foster the development of ideas and critical Research Design This study is an Experimental Research. The experimental research method is one of the scientific approaches used to understand the cause-and-effect relationship between specific variables in a phenomenon. In this method, researchers intentionally manipulate a variable called the independent variable to observe its impact on the dependent variable. The design used in this study is the inequivalent Control Group Design. In this design, both the experimental and control groups were not selected randomly. The design incorporates a pretest to determine the initial condition of the average mathematics learning outcomes of students in both the experimental and control groups. The post-test is used for data analysis for both the experimental and control groups. The research design is as follows: O[1] Â Â Â Â Â Â Â Â xÂ Â Â Â Â Â Â Â Â Â Â Â Â O[2] O[3 Â ] Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â O[4] Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â O1 = pretest for the experimental group Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â O2 = posttest for the experimental group Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â O3 = pretest for the control group Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â O4 = posttest for the control group Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â XÂ = treatment for the form of Creative Problem Solving model implementation The Subject of research The subjects of this research are 60 students from 8th grade D and 8th grade E at Public Junior High School I of Jabung, comprising 30 students in 8th grade D and 30 students in 8th grade E. The sample selection was carried out using the cluster random sampling technique. The sampling method involved selecting 2 groups randomly from all available groups. Similarly, from the 2 selected groups, further random selection was conducted to determine the experimental group and the control group. As a result, 8th grade D was chosen as the experimental group, which received the Creative Problem-Solving learning model, and 8th grade E was selected as the control group, which received the conventional learning model. Research Instruments The research instruments used in this study are tests, namely pretests, and posttests. Pretests are employed to assess students’ mathematical reasoning abilities before any intervention is given. Meanwhile, post-tests are used to determine the level of students’ mathematical reasoning abilities after being exposed to the treatment using the Creative Problem-Solving Model. Data Analysis Techniques There are two types of data analysis used in this research: (1) Descriptive Statistical Analysis, and (2) Inferential Statistical analysis. Data analysis in this study was conducted using the Statistical Package for Social Science (SPSS). The decisions were made to determine the differences and the effects of independent variables on the dependent variable based on a significance level of Î± = 0.05 (5% error rate) or a 95% confidence level. For data analysis and hypothesis testing, the researcher used the t-test for two sets of data used as a comparative statistical test. The t-test produces a t-value that can be used to validate the hypothesis. The statistical analysis method employed in this study is the paired t-test, which compares data points before and after the treatment or intervention, also known as a time-based t-test Descriptive Statistical Analysis From the pretest and posttest results conducted in both groups, descriptive statistical analysis was carried out and is depicted in the following table: Table 1. Mean of pretest and post-test results for the experimental and control groups │ │pretest │posttest │ │Descriptive Analysis ├──────────┬───────┼──────────┬───────┤ │ │Experiment│Control│Experiment│Control│ │Number of Samples │30 │30 │30 │30 │ │Means │71,47 │72,03 │79,17 │81,37 │ Based on the table above, it is evident that the experimental and control groups had similar initial abilities. This can be observed from the relatively small difference in average scores, and there was an improvement in students’ mathematics learning outcomes from the pretest to the post-test. In the pretest, the average mathematics learning outcomes for the experimental group and the control group differed by 1.44. Furthermore, when looking at the post-test data, there was an increase in the average mathematics learning outcomes for the experimental group compared to the control group, with a difference of 2.20. This indicates that after applying the Creative Problem-Solving model to the experimental group, the average mathematics learning outcomes improved, implying an impact on the reasoning abilities and mathematics learning outcomes of the 8th-grade students at SMP Negeri 1 Jabung. To determine more accurately whether there is an influence of the creative problem-solving model on the reasoning abilities and learning outcomes of the experimental and control groups, inferential analysis will be conducted. Inferential Statistical Analysis The data analysis technique used in this research is a two-sample t-test. The t-test is conducted to determine whether there is an influence before and after the treatment. Both the pretest and posttest consist of 3 open-ended questions related to linear systems of two variables story problems. The pretest and posttest scores are statistically analyzed using tests for normality, homogeneity, and the two-sample t-test. Inferential Analysis of Pretest Data Pretest scores are obtained from the mathematics evaluation test of the students before any intervention is applied. After conducting the pretest, the teaching and learning process is carried out using the Creative Problem Solving (CPS) learning model in the experimental group and conventional teaching in the control group. The pretest data analysis is processed as follows: Normality Testi Results for Pretest Scores The data analyzed in this normality test includes the pretest scores of the experimental and the control group. The informality test is conducted to determine whether the data from each group is normally distributed or not. This is because one of the assumptions that must be met before conducting the homogeneity of variance test is that the data from both groups follows a normal distribution. In this study, the normality test is computed using the Kolmogorov-Smirnov test with the assistance of IBM Statistics. The results of the normality test for pretest data can be seen in Table 2 as follows: Table 2. Normality Test for Pretest in the Experimental and Control Group │ │Kolmogorof-smirnov│ │ │Group ├────┬─────────────┤Conclusion │ │ │N │Sig │ │ │Experiment│30 │ │ │ ├──────────┼────┤0,306 │Data is normally distributed │ │Control │30 │ │ │ Using the Kolmogorov-Smirnov test, the results for the pretest in the experimental and the control group showed a significant value of 0.306 > Î± (Î± = 0.05), which leads to the conclusion that H[0] is accepted, indicating that the data follows a normal distribution. Homogeneity of Variance iTest Results for Pretest Scores In addition to the normality test, one of the prerequisites before conducting a comparison of two sample groups is that they should have the same characteristics before receiving different treatments, which is tested through the homogeneity of variance. Since the pretest scores in both groups are normally distributed, the homogeneity of variance test for pretest data is performed. The purpose of the homogeneity test is to determine whether the experimental and the control groups have the same variance or not. To determine whether the values of both groups are homogenous or not homogenous, it is done by comparing the F-[statistic] and F-[table] values. The F-[test] is obtained by comparing the largest variance with the smallest variance. The results of the homogeneity test calculations can be seen in the appendix and summarized as follows: Table 3. Homogeneity Test of Pretest Score Data for experimental and iControl Group │Group │Variance│N │F-[statistic]│F[table]│explanation │Conclusion │ │Experiment │164,49 │30│1,21 │1,84 │F[statistic]< F[table] │Homogeneous│ │Â Â Control│135,92 │30│ │ │ │ │ Based on the Table 3 above, it is found that F-statistic < F-table, which is 1.21 < 1.84. This means that the variances of the experimental group with the creative problem-solving learning model and the Control group with conventional learning are inhomogeneous. In addition to manual calculations, the results of the homogeneity of variance test were assisted by using SPSS. The results of the homogeneity test for pretest data can be seen in the following Table 4. Homogeneity of Variance Test for Pretest in the Experimental and Control Group │ │Levene Statistic │Sig. │explanation │ │Based on Means│0,467 │0,497│Homogeneous Data │ Based on Table 4 above, with statistics based on the mean, a significance value of 0.497 > Î± (Î± = 0.05) is obtained. Therefore, H[o] is accepted, indicating that the data is homogeneous. Two-Sample Mean Difference Test for Pretest Scores Because the variances of the experimental and the control groups are homogeneous, the statistical test for comparing the two mean learning outcomes is the t-test. The results of the t-test calculation for pretest scores in the experimental and the control group can be seen in Table 5 below: │Group │N │ │S[ig]│F-statistic│Â F-table│Â Â Â Â Â Â Â Â Â Â Â Â Â Explanation │ │Experiment│30│36,25│ │ │ │ │ ├──────────┼──┼─────┤12,25│-8. 397 │58 │F[table]<F[statistic]< F[table] │ │Control │30│41,93│ │ │ │ │ Table 5. Results of the t-test for Pretest Data in the experimental and control group Based on Table 5 above, we have obtained t-statistic = -8.398, and t-table = 58. This means that -F-table < F-statistic < F-table, which is -58 < -8.398 < 58. This indicates that H[0] is accepted and H[I] is rejected. In other words, it means that the average reasoning abilities of the students in the experimental group are the same as the average reasoning abilities of the students in the control group before the treatment. Furthermore, the researcher also used SPSS to calculate the t-test. The results of the t-test can be seen in the following table. │ │t-test for Equality of Means│ │ ├─────┬──────┬───────────────┤ │ │T │Df │Sig. (2-tailed)│ │Equal variances assumed │0,157│58 │0,875 │ │equal variances not assumed │0,157│57,509│0,875 │ Table 6. Test of Two Pretest Means for Experimental and Control Group Based on the table above, it is evident that the significance value (2-tailed) is 0.875, which means that the significance value is greater than Î±. Therefore, H0 is accepted. This indicates that the average reasoning abilities of the students in the experimental group are also the same as the average reasoning abilities of the students in the control group before the treatment. Inferential Analysis of Post-Test Because the pretest scores data exhibit the same capabilities, the analysis proceeds to the post-test. Post-test scores are statistically analyzed using tests for normality, homogeneity, and a two-sample mean difference test (t-test). Post-test scores are obtained from the mathematics evaluation test of the students after the treatment has been administered. The analysis of post-test data is as follows. The results of the normality Test for Post-Test Scores The data analysed in this normality test consists of the post-test scores for both the experimental and the control groups. The purpose of this normality test is to determine whether the data from each group follows a normal distribution or not. This is because one of the assumptions that must be met before conducting the homogeneity of variance test is that the data from both groups should be normally distributed. In this research, the normality test is computed using the Kolmogorov-Smirnov test with the assistance of SPSS/IBM Statistics. The results of the normality test for post-test data can be seen in T able 7 below: Table 7. Post-test Normality iTest for Experimental and iControl Group │ │Kolmogorov-Smirnov │ │ │Group ├─────┬─────────────┤Conclusion │ │ │N │Sig │ │ │Experiment│30 │ │ │ ├──────────┼─────┤0,349 │Distributed data │ │Control │30 │ │ │ Using the Kolmogorov-Smirnov test, the results for the post-test in the experimental and the control group show a significance value of 0.349, which is greater than Î± (Î± = 0.05). Therefore, it can be concluded that H[0] is accepted, indicating that the data follows a normal distribution. Post-test Value Variance Homogeneity Test Results Because the post-test scores data in both groups are normally distributed, the homogeneity of variance test for post-test data is conducted. The results of the homogeneity test can be seen in the appendix and are summarized in table 8 below: Table 8. Homogeneity Test Results for Post-Test Data in the experimental and control Group │Group │variance│N │F-[statistic]│F-[table]│Explanation │conclusion │ │Experiment│163,799 │30│ │ │ │ │ ├──────────┼────────┼──┤1,60 │1,84 │F-[statistic ]< F-[table] │Homogeneous│ │Control │137,757 │30│ │ │ │ │ Based on the table above, it is found that F-[statistic] < F-[table], which is 1.60 < 1.84. This indicates that the variances of the experimental group with the Creative Problem-Solving learning model and the control group with conventional learning are homogeneous. In addition to manual calculations, the results of the homogeneity of variance test were aided by using SPSS. The results of the homogeneity test for variance in pretest data can be seen in the following table. Table 9. Homogeneity of Variance Test for Post-Test in the Experimental and Control Group │ │Levene Statistic │Sig. │Explanation │ │Based on means│0.713 │0, .402│Homogenous Data│ Based on the table above, with statistics based on the mean, a significance value of 0.402 is obtained, which is greater than Î± (Î± = 0.05). Therefore, H[o] is accepted, indicating that the data is also homogeneous. Two-Sample Mean Difference Test for Post-Test Scores Because the variances in the experimental and the control groups are homogeneous, the statistical test for comparing the two mean learning outcomes is the t-[test]. The results of the t-[test] calculation for post-test scores in the experimental and the control group can be seen in the appendix in Table 10 below: Table 10. Results of the it-Test for Post-Test Data in the Experimental and Control Group. │Group │N │ │Sig │t[-statistic] │T-[table]│ │Experiment │30│72,600│ │ │ │ ├───────────┼──┼──────┤0,491│0,694 │58 │ │Control │30│2,200 │ │ │ │ Based on the table above, we obtain t-statistic = 0.694, and t-table = 58. Thus, t-statistic > t-table, which is 0.694 > 58. It means that H[o] is rejected, and H[I ]is accepted. In other words, the average reasoning ability of students in the experimental group is better than the average reasoning ability of students in the control group after the treatment. The pretest results for the experimental group range from a highest score of 61 to a lowest score of 20, while the control group obtained scores with a highest of 60 and a lowest of 20. This condition indicates that neither group achieved the Minimum Passing Standard. This is due to the passive learning process, where students did not pay close attention to their teachers, and hesitated to ask questions when encountering difficulties during practice, leading to students attempting problems without knowing the correct solutions. Based on the experience gained by the researcher during the implementation of the Creative Problem-Solving model, there was an increase in learning activity in the classroom. During the introductory phase, the researcher provided motivation and apperception by linking mathematics to everyday life, and most of the students responded positively to what the researcher conveyed. Then, in the core activities, students followed the instructions on the Student Activity Sheet provided. Although at the beginning, some students were confused when working on the Student Activity Sheet in the second meeting, the teacher assisted them and asked them to clarify the issues, and the students responded well. In the subsequent meetings, students were able to work on the Student Activity Sheet, and develop, and solve the problems presented on the sheet. The student’s response was consistently active throughout the learning process, actively seeking creative solutions to problems and engaging in discussions with other students. In addition, the creative problem-solving learning model trains students to design discoveries, think and act creatively, solve problems realistically, and make school education more relevant to everyday life. As a result, students become active, motivated, and eager to seek solutions to every challenge they face. After students have completed the activities on the Student Activity Sheet, they proceed to present the results of their discussions, and other groups respond to the presenting group if there are different answers. In the closing phase of the activity, students are capable of drawing conclusions and working on the exercises provided. In the post-test results, the experimental group obtained the highest score of 100 and the lowest score of 47, while the control group achieved the highest score of 81 and the lowest score of 40. The mathematics learning outcomes in both groups showed an average improvement from the pretest to the post-test. The average learning outcomes of the experimental and the control group before the treatment were similar. The analysis then continued with posttest data. From the analysis of the post-test, it can be observed that the average score for the experimental group is 72.600, while the average score for the control group is 2.200. Based on the results of the inferential statistical analysis, it was found that t-statistic = 0.694 > t-table = 58, which leads to the rejection of the null hypothesis (H[0]) and the acceptance of the alternative hypothesis (H[I]). This means that there is a significant influence of the Creative Problem Solving learning model on students’ mathematical reasoning. This is supported by the research findings of Nopitasari (2016), Senjayawati and Nurfauziah (2018), Ningsih (2021), Tabunan (2021), and Marbun et al. (2022), which demonstrate that the creative problem-solving model can enhance students mathematical reasoning. This is because students become more creative in solving mathematical problems when exposed to the Creative Problem-Solving learning model. In other words, the creative problem-solving learning model not only improves students’ reasoning but also enhances their creativity. Triyono et al. (2017) have stated that the Creative problem-solving learning model can enhance students’ creativity. Based on the observations and data analysis conducted, it is evident that the average score for the experimental group is 72.600, while the average score for the control group is 2.200. The results of the inferential statistical analysis show that t-[statistic] = 0.694 > t-[table] = 58, leading to the rejection of the null hypothesis (H[0]) and the acceptance of the alternative hypothesis (H [I]). This means that there is a significant effect of creative problem-solving and learning models on students’ mathematical reasoning abilities. 1. Aprilianti,Â Y.,Â &Â Zhanty,Â L.Â S.Â 2019.Â AnalisisÂ KemampuanÂ PenalaranÂ MatematikÂ SiswaÂ SMPÂ padaÂ Materi Segiempat dan Segitiga. Journal on Education, Vol. 1(2), 524-532Â 2. Dimyati & Mudjiono. (2006). Belajar dan pembelajaran. Jakarta: Rineka Cipta 3. Marbun, H. D.; Siahaan, T.M,; Sauduran, G.N. 2022. Pengaruh Model Pembelajaran Creative Problem Solving terhadap Kemampuan Penalaran Adaptif Matematis Siswa di kelas VIII SMP Swasta Trisakti Pematangsianta, Jurnal Pendidikan dan Konseling, Vol.4 (5), 8192-8202 https://journal.universitaspahlawan.ac.id/index.php/jpdk/article/view/8009 4. Muin, A;Â Hanifah S.HÂ & Diwidian, F. 2018. The effect of creative problem solving on studentsâ€™ mathematical adaptive reasoning. Â Journal of Physics: Conference Series 948 (1), 012001. 5. Napitupulu,Â E. ; Suryadi,Â D. &Â Kusumah,Â Y.Â S.Â (2016).Â Cultivating upper secondary studentsâ€™ mathematical reasoning-ability and attitude towards mathematics through problem-based learning. Journal on Mathematics Education, 7(2), 117â€“128. https://doi.org/10.22342/jme.7.2.3542.117-128 6. Ningsih, R.E.; Wahidin; Pradipta, T.R. 2021. Pengaruh Model Pembelajaran Creative Problem Solving Berbantu Maple Terhadap Kemampuan Penalaran Matematis Siswa. Euclid, Vol. 8 (1), 62-71. http:// 7. Nopitasari, D. (2016). Pengaruh Model Pembelajaran Creative Problem Solving (CPS) terhadap kemampuan penalaran adaptif matematis siswa. in jurnal matematika dan pendidikan matematika, Vol.1(2), 103â€“112. https://doi.org/10.31943/mathline.v1i2.22 8. Ria,Â Y.; Risalah,Â D.Â &Â Sandie.Â (2021). KemampuanÂ PenalaranÂ MatematisÂ SiswaÂ DalamÂ MenyelesaikanÂ SoalÂ Higher Jurnal Educatio, 8(2), 2022, 505-511511Open Access: https:// ejournal.unma.ac.id/index.php/educatioOrder Thinking Skills (HOTS) Pada Materi Teorema Phytagoras Siswa Kelas VIII SMP Negeri 2 Monterado. 1(5), 767â€“772. 9. Senjayawati, E; Nurfauzia, P. 2018.Â Peningkatan Kemampuan Penalaran Matematik dan Self Efficacy Siswa SMK dengan Menggunakan Pendekatan Creative Problem Solving. Jurnal Ilmiah P2M STKIP Siliwangi, Vol. 5 (2), 117-129. https://doi.org/10.22460/p2m.v5i2p117-129.1085 10. Shadiq (2004) Pemecahan Masalah. Penalaran dan Komunikasi. 11. Shadiq, Fadjar. 2014. Pembelajaran Matematika: Cara Meningkatkan Kemampuan Berpikir Siswa. Yogyakarta: Graha Ilmu. 12. Shoimin, Aris. 2014. 68 Model Pembelajaran Inovatif Dalam Kurikulum 2013. Yogyakarta : AR-Ruzz Media. 13. Syazali, M. 2015. Pengaruh model pembelajaran Creative Problem Solving berbantuan maple II terhadap kemampuan pemecahan masalah matematis. Journal of Chemical Information and Modeling, 6 (1), 91â€“98. http://dx.doi.org/10.24042/ajpm.v6i1.58 14. Tambunan, L.O. 2021. Model Pembelajaran Creative Problem Solving untuk Meningkatkan Kemampuan Penalaran dan Komunikasi Matematis. JNPM (Jurnal Nasional Pendidikan Matematika) Vol. 5, (2),362-373. 15. Triyono; Senam; Jumadi & Wilujeng, I. 2017. The Effects of Creative Problem Solving-based Learning Towards Students’ Creativities. Jurnal Kependidikan Penelitian Inovasi Pembelajaran, Vol 1 (2), 214-226. DOI: 10.21831/jk.v1i2.9429 16. Wibowo,Â A.Â 2017.Â PengaruhÂ pendekatanÂ pembelajaranÂ matematikaÂ realistikÂ danÂ saintifikÂ terhadapÂ prestasi belajar,Â kemampuanÂ penalaranÂ matematisÂ danÂ minatÂ belajar. 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For the K Desktop Environment (KDE, https://www.kde.org/), I have in particular developed the program Kpl. Kpl is used for two- and three-dimensional graphical visualization of data sets and functions. In addition, nonlinear fits of function parameters to data sets can be performed. Also general linear regression and smoothing spline interpolation are possible. Online calculators Here you can find the following online calculators, implemented in JavaScript: SEIR model calculator With the SEIR model calculator calculations on epidemics as the COVID-19 pandemic can be performed on base of the SEIR model (https://en.wikipedia.org /wiki/Compartmental_models_in_ epidemiology# Dew point calculator The dew point temperature (https://en.wikipedia.org/wiki/Dew_point) of humid air is the temperature at which the water starts to condense when it cools down. Using the dew point calculator you can calculate the dew point temperature from the air temperature and the relative air humidity. Relative air humidity φ in dependence of the air temperature ϑ for different constant values of the dew point temperature τ. Air is experienced as muggy when the dew point temperature exceeds 16 °C. This is the case when the relative air humidiy at the actual air temperature is above the black curve for τ = 16 °C. Sunrise and sunset With this calculator the times of sunrise and sunset can be calculated for locations with given geographical coordinates. If geolocation is supported (particularly using GPS) by the device, the actual position can be determined for this. Times (Central European Summer Time CEST: MESZ) of sunrise and sunset for Freiburg im Breisgau, Germany, in the year 2023. UTM converter With the UTM converter geographic coordinates can be converted from longitude and latitude to UTM coordinates (https://en.wikipedia.org/wiki/Universal_Transverse_Mercator_coordinate_system) and vice versa. If geolocation is supported (particularly using GPS) by the device, the actual position can be determined for this.	{"url":"https://werner-stille.de/programming-en.html","timestamp":"2024-11-05T01:04:33Z","content_type":"text/html","content_length":"4697","record_id":"<urn:uuid:b6da3f77-73d6-4975-834c-bc30839a483b>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00677.warc.gz"}
[molpro-user] problems with contraction in RS2c ramon at buzon.uaem.mx ramon at buzon.uaem.mx Thu Jul 19 15:34:36 BST 2007 we are doing RS2 type calculations on a weakly bound complex, when using the RS2c module we observe a jump or discontinuity in the potential at long range(see the table below). When looking at the output of rs2c we notice that there is a corresponding change in the number of contracted internal and N-1 configurations used in the neighborhood of the jump. If we use the RS2 module the problem disappears but we would like to use the more efficient RS2c. A similar problem had been already discussed in the mailing list(31/10/03) for acpf calculations and Prof. Werner suggested a solution. We first tried adding the option thresh,pnorm=0,thrdlp=1.d-8 but this had no effect on the calculated energies. Then we tried fixing the pair list using the option save,5000.2;start,5000.2 and in this case the problem that appears is that when RS2 reads the wavefunction from 5000.2(in the second point of the calculation) it complains that its overlap with the previous CI vector is nearly zero and exits(see below the table) Your help is appreciated Ramon Hernandez R(ang) R ERS2C INT-POT (INT-POT)*27211.4 37.042390 70.0 -300.3143994 -0.00079168 0.0000 10.583540 20.0 -300.3144003 -0.00079257 -0.02440715 8.466832 16.0 -300.3144039 -0.00079514 -0.09422729 7.408478 14.0 -300.3144158 -0.00079960 -0.21562064 6.350124 12.0 -300.3151163 -0.00146832 -18.41250990 5.291770 10.0 -300.3152382 -0.00150677 -19.45864921 Overlap between present and previous reference vectors 1 -.0000017 1 -.0000017 ovmax= 0.170665286226771260E-05 ovref= 0.170665286226771260E-05 ? Error ? Insufficient overlap ? The problem occurs in cihdia CURRENT STACK: CIPRO MAIN More information about the Molpro-user mailing list	{"url":"https://www.molpro.net/pipermail/molpro-user/2007-July/002284.html","timestamp":"2024-11-12T11:12:14Z","content_type":"text/html","content_length":"4726","record_id":"<urn:uuid:03270040-9c89-4c48-bde4-d0547751ea3a>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00826.warc.gz"}
Directly and inversely proportional- equations Slide 1 Directly and inversely proportional Use with student whiteboards, or paper Objective- a mini lesson on writing equations based on definitions that use the words directly and inversely proportional Slide 2 A is directly proportional to B A is inveresly proportional to C Slide 3 A is directly proportional to B A is directly proportional to C Slide 4 In the equation A is inveresly proportional to B A is inveresly proportional to C Slide 5 On your whiteboards- practice writing the equations according to the definitions provided. Slide 6 weight(w) is directly proportional to mass(m) and directly proportional to acceleration due to gravity(g) Slide 7 Pressure(P) is directly proportional to the Force(F)and inversely proportional to the area(A) Slide 8 Frequency(f) is inversely proportional to the time(t) Slide 9 Acceleration(a) is directly proportional to the Force(F) and inversely proportional to the mass(m) Slide 10 Jis directly proportional to the , X directly proportional to the L and inversely proportional to theN Slide 11 Slide 12 weight(w) is directly proportional to mass(m) and directly proportional to acceleration due to gravity(g) Slide 13 Pressure(P) is directly proportional to the Force(F)and inversely proportional to the area(A) Slide 14 Frequency(f) is inversely proportional to the time(t) Slide 15 Acceleration(a) is directly proportional to the Force(F) and inversely proportional to the mass(m)	{"url":"https://www.sliderbase.com/spitem-243-1.html","timestamp":"2024-11-04T07:03:50Z","content_type":"text/html","content_length":"16815","record_id":"<urn:uuid:294151a4-e29e-4630-b4d4-7dcb6730b841>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00018.warc.gz"}
AVERAGEIF Function in Excel What Is AVERAGEIF Function In Excel? The AVERAGEIF function in Excel determines the average/mean of the selected cells in a dataset that satisfies the required criteria. The function is used to evaluate the central tendency, a measure representing the center of a set of data values in a statistical distribution. The AVERAGEIF Excel function is an inbuilt Statistical function, so we can insert the formula from the “Function Library” or enter it directly in the worksheet. For example, the first table below contains a list of fruits and their quantity, and the second table shows the criterion to determine the quantity values provided in the first table to get the We will use the Excel AVERAGEIF function to calculate the average. Select cell F6, enter the formula =AVERAGEIF(B2:B7,F3), and press “Enter.” The output is shown above. The function averages the values in the given data range B2:B7, which are greater than 0. Thus, the result, 450 pounds, is the average of the quantities 500, 450, 550, and Key Takeaways • The AVERAGEIF function in excel evaluates the average of the cell values in a supplied range that fulfill the given conditional criteria. • The AVERAGEIF() accepts two mandatory arguments, range and criteria, and one optional argument, average_range, as input. The range can include numbers, names, arrays, or cell references to numbers. And the criteria can be number, text, expression, or cell reference. • The AVERAGEIF() allows wildcard characters, ‘?’ and ‘’, in the criteria argument to match a single character or sequence of characters, respectively. AVERAGEIF() Excel Formula The syntax of the Excel AVERAGEIF formula is: The arguments of the Excel AVERAGEIF formula are: • range: The cell range of a dataset to test against the specified criteria. It is a mandatory argument. • criteria: The criteria to check the specified cells to average. It is a mandatory argument. • average_range: The actual range of cells that we require to average. It is an optional argument. Excel VBA – All in One Courses Bundle (35+ Hours of Video Tutorials) If you want to learn Excel and VBA professionally, then Excel VBA All in One Courses Bundle (35+ hours) is the perfect solution. Whether you’re a beginner or an experienced user, this bundle covers it all – from Basic Excel to Advanced Excel, Macros, Power Query, and VBA. How To Use AVERAGEIF Excel Function? We can use the AVERAGEIF excel function in 2 methods, namely, 1. Access from the Excel ribbon. 2. Enter in the worksheet manually. Method #1 – Access from the Excel ribbon First, choose an empty cell for the output → select the “Formulas” tab → go to the “Function Library” group → click the “More Functions” option drop-down → click the “Statistical” option right arrow → select the “AVERAGEIF” function, as shown below. The “Function Argument” window appears. Enter the argument values in the “Range, Criteria, and Average_range” fields → click “OK”, as shown below. Method #2 – Enter in the worksheet manually First, ensure the given range is not empty/blank and does not include only text values. 1. Select the target cell for the output. 2. Type =AVERAGEIF( in the cell. [Alternatively, type =A or =AV and double-click the AVERAGEIF function from the Excel suggestions.] 3. Enter the arguments as cell values or cell references in Excel. 4. Close the brackets and press Enter to execute the function. Let us take an example to understand this function. We will find the state-wise average household income for the state specified using Excel AVERAGEIF function. The table below lists the top ten richest US cities, their states, and the respective average household income details. The steps to find the average using AVERAGEIF function in Excel are, 1. Select cell B18, enter the formula =AVERAGEIF(B2:B11,B15,C2:C11), and press Enter. [Also, enter the formula as =AVERAGEIF(B2:B11,”CA“,C2:C11), cell value instead of a cell reference.] 2. Select cell B18 and set the Number Format in the Home tab as Currency to view the required average household income in the state of CA as a currency value. The output is shown above. First, the function checks the cell range B2:B11 for the state name, “CA”, and finds three instances in cells A2, A5, and A6. Then considers the corresponding cell values from the cell range C2:C11 and returns the average value as $403,332.67. [Alternatively, we can apply the AVERAGEIF() as follows: click Formulas → More Functions → Statistical → AVERAGEIF, which will open the Function Arguments window. Then, enter the respective AVERAGEIF() arguments in the Function Arguments window. And finally, click OK in the Function Arguments window to execute the function.] We will understand some advanced scenarios using the Excel AVERAGEIF function examples. Example #1 We can use the wildcard characters, ‘?’ or ‘’, in the criteria argument, when the given criterion involves a partial match and apply the AVERAGEIF Excel function. The first table in the image below contains a list of inventory items and their quantity details. The procedure to find the average number of boxes of all oil varieties using the AVERAGEIF function in Excel is, Select cell E6, enter the formula =AVERAGEIF(A2:A12,E3,B2:B12), and press Enter. The output is shown above. [Output Observation: The wildcard character asterisk, ‘*’, helps match a sequence of characters. So, when we use it in the AVERAGEIF(), the function tries to find a partial match for the phrase “oil” in the given cell range, A2:A12. And it gets the required matches in cells A2, A5, A7, A11, and A12. Then, it calculates the average of the corresponding column B cell values, 200, 250, 100, 100, and 50, in cells B2, B5, B7, B11, and B12, and returns the average number of boxes of all oil varieties, Example #2 Let us see how the Excel AVERAGEIF function works if the range of values to find the average includes blank and non-blank cells. The below table contains a list of students, their scores, and scholarship details. The procedure to determine the average using the Excel AVERAGEIF function is, Select cell B18, enter the formula =AVERAGEIF(B2:B11, “<>”,C2:C11), and press Enter. The output is shown above. [Output Observation: The above AVERAGEIF() checks for non-empty cells in column B as the students with a scholarship have the scholarship status as Yes in column B. Here the criteria argument includes ‘<>’ to check for non-empty cells, B3, B4, B6, B9, and B11. And then, the function averages the scores in the corresponding column C cells (C3, C4, C6, C9, and C11), 495, 478, 453, 486, and 470, to return the average total score as 476.4.] Example #3 We will use the Excel AVERAGEIF function for multiple criteria. The following table shows a list of participants from different states and their respective timing details during a competition. It shows more than one participant belonging to the same state, such as GA and NY. The procedure to use the Excel AVERAGEIF function for multiple criteria is: Select cell G7, enter the formula =AVERAGE(AVERAGEIF(B2:B16,G3,C2:C16),AVERAGEIF(B2:B16,G4,C2:C16)), and press Enter. The output is shown above. [Output Observation: The above formula determines the average time taken by participants from the states, GA and NY, separately. And then, find the average of the two average values the Excel AVERAGEIF function returns. So, the first AVERAGEIF() checks for the participants from GA and then averages the corresponding time values, 77, 63, 70, and 73, to return the value of 70.75 seconds. Next, the second AVERAGEIF() checks for the participants from NY and then averages the corresponding time values, 80, 61, and 50, to return the value of 63.66 seconds. And finally, the AVERAGE() returns the average of the values 70.75 and 63.66, 67.20833 seconds.] Important Things to Note • If no cells in the supplied range satisfy the specified criteria, or an empty range is selected, or a range with only text values, then we get the #DIV/0 error. • If the criteria argument contains an empty cell, the function assumes it as 0. • The AVERAGEIF() ignores cells containing logical values, TRUE or FALSE, in the given range. And the function also ignores empty cells in average_range. • If we ignore the argument average_range, the function uses the given range. Frequently Asked Questions (FAQs) 1. Is there an AVERAGEIF function in Excel? There is an AVERAGEIF function in Excel. It is in the Formulas tab, and we can select Formulas → More Functions → Statistical → AVERAGEIF to apply it in the required cell. 2. How to use AVERAGEIF() with the date criterion in Excel? We can use AVERAGEIF() with the date criterion in Excel if we need the average units dispatched after 4/1/2022. For example, the following table contains a list of items and their dispatch details. The procedure to apply the Excel AVERAGEIF function with the required date criterion is: Select cell G6, enter the formula =AVERAGEIF(B2:B11,G3,C2:C11), and press Enter. The above formula checks for dates after 4/1/2022 in column B and then determines the average of the corresponding units, 2000, 900, 1170, 2300, and 2710, 1816 units. 3. How to use the AVERAGEIF() in Excel with a logical operator in criteria? We can use the AVERAGEIF() in Excel with a logical operator in the criteria and determine the average contribution by volunteers aged 25 and above. For example, the below table contains the details of the contributions made by a set of volunteers. The procedure to apply the AVERAGEIF Excel function by volunteers aged 25 and above is, Select cell G6, enter the formula =AVERAGEIF(B2:B11,”>=”&G3,C2:C11), and press Enter. [Note: However, we may also enter the formula as =AVERAGEIF(B2:B11,”>=25″,C2:C11)] The output is shown above. [Output Observation: In this case, we must provide the required logical operator and the number in double quotations when supplying them as the criteria argument. In the above formulas, the function checks for cells containing ages 25 and above in column B. And then, it averages the corresponding contribution values in column C (200, 120, 250, 300, and 350) to return the value of $244.] Download Template This article must help understand the AVERAGEIF Excel function, with its formula and examples. We can download the template here to use it instantly. Recommended Articles This has been a guide to Excel AVERAGEIF Function. Here we find average of cell values with set criteria, formula, examples & a downloadable excel template. You can learn more from the following articles –	{"url":"https://www.excelmojo.com/averageif-function-in-excel/","timestamp":"2024-11-10T12:21:37Z","content_type":"text/html","content_length":"230672","record_id":"<urn:uuid:292ec1d8-c3b3-4a88-ab30-0bcaed919324>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00627.warc.gz"}
Problem C The use of units is ubiquitous in science. Physics uses units to distinguish distance (e.g., meters, kilometers, etc.), weight (e.g., kilograms, grams), and many other quantities. Computer scientists have specialized units to describe storage capacity (e.g., kilobytes, megabytes, etc.). You are to write a program to display the conversion factors for a set of units. Specifying the relationship between various units can be done in many different, but equivalent, ways. For example, the units for metric distance can be specified as the group of relationships between pairs for units: $1$ km = $1\, 000$ m, $1$ m = $100$ cm, and $1$ cm = $10$ mm. An alternative set of pairs consists of: $1$ km = $100\, 000$ cm, $1$ km = $1\, 000\, 000$ mm, and $1$ m = $1\, 000$ mm. In either presentation, the same relationship can be inferred: $1$ km = $1\, 000$ m = $100\, 000$ cm = $1\, 000\, 000$ mm. For this problem, the units are to be sorted according to their descending size. For example, among the length units cm, km, m, mm, km is considered the biggest unit since $1$ km corresponds to a length greater than $1$ cm, $1$ m, and $1$ mm. The remaining units can be sorted similarly. For this set, the sorted order would be: km, m, cm, mm. This problem is limited to unit-systems whose conversion factors are integer multiples. Thus, factors such as $1$ inch = $2.54$ cm are not considered. Further, the set of units and the provided conversions are given such that all units can be expressed in terms of all other units. The input consists of several problems. Each problem begins with the number of units, $N$. $N$ is an integer in the interval $[2, 10]$. The following line contains $N$ unique case-sensitive units, each of which consists of at most $5$ characters (using only a–z and A–Z). Following the set of units are $N-1$ unique lines, each specifying a relationship between two different units, with the format containing the following four space-separated pieces: name of the unit; an “=”; a positive integer multiplier larger than $1$; and the name of a second unit that is smaller than the one to the left of the equal sign. Each of these lines establishes how many units are equivalent to the larger unit on the left. Each unit appears in the set of $N-1$ lines and is given in such a way to ensure the entire system is defined. The set of multiples yields conversion factors that do not exceed $2^{31} - 1$. The sentinel value of $N=0$ indicates the end of the input. For each set of units, produce one line of output that contains the equivalent conversions. The conversions should be sorted left to right, with the largest unit appearing on the left. The conversion factors should be defined with respect to the leftmost unit (i.e., the largest unit) and should be separated by “ = ”. Sample Input 1 Sample Output 1 km m mm cm km = 1000 m m = 100 cm cm = 10 mm m mm cm km km = 100000 cm km = 1000000 mm m = 1000 mm 6 1km = 1000m = 100000cm = 1000000mm MiB Mib KiB Kib B b 1km = 1000m = 100000cm = 1000000mm B = 8 b 1MiB = 8Mib = 1024KiB = 8192Kib = 1048576B = 8388608b MiB = 1024 KiB 1MiB = 8Mib = 1024KiB = 8192Kib = 1048576B = 8388608b KiB = 1024 B Mib = 1048576 b Mib = 1024 Kib Kib B MiB Mib KiB b B = 8 b MiB = 1048576 B MiB = 1024 KiB MiB = 8192 Kib MiB = 8 Mib	{"url":"https://open.kattis.com/contests/csiahp/problems/units","timestamp":"2024-11-11T04:05:41Z","content_type":"text/html","content_length":"33146","record_id":"<urn:uuid:ad59f2f9-b715-419b-92c5-f841092ada70>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00775.warc.gz"}
Argument Writing in Math: Getting Started Are you an English teacher prepping a training on argument writing for the teachers in your building/district (and you want to be applicable to those STEM folks)? Are you a teacher that just went through an argument writing training that left you wanting more? Are you a math teacher that wants to integrate more logic, reasoning, and writing into your course? ( Let's be friends! ) Whoever you are, I wanted to share a few thoughts and resources to get you started on your argument writing (and thinking) in a math class. If you'd like more of a primer on argument writing in general, check out this post I wrote after I practiced some argument "writing" with my 3 year old daughter last year. ( Teaching Argument Writing for Preschoolers...and anyone else! The most natural application of argument writing is in proofs, which most often come up in Geometry, and should also usually be used in Algebra (but rarely are in my experience). To make a common core connection, standard for mathematical practice #3 requires students to "Construct viable arguments and critique the reasoning of others." Most students experience with proofs (and perhaps you remember your own) is with column proofs like this, for proving things about angles, line segments, or shapes. Have I induced any terror sweats yet? But proofs can also be written in paragraph form , which is where the English training and Hillcock can be applied. Reasons are the warrants, the statements are evidence, and claims will be what must be proved. "Prove that angle A is a supplementary angle," or "prove that the lines defined by y=2x+3 and y=2x-25 are parallel" But...I don't even know how to explain what a proof is! Watch this adorable TED-Ed video introducing and explaining the basis and application of mathematical proof. Want some more? Here's a resource from Berkeley on mathematical logic "First, a proof is an explanation which convinces other mathematicians that a statement is true. A good proof also helps them understand why it is true. The dialogue also illustrates several of the basic techniques for proving that statements are true. Table 1 summarizes just about everything you need to know about logic. It lists the basic ways to prove, use, and negate every type of statement. In boxes with multiple items, the first item listed is the one most commonly used. Don’t worry if some of the entries in the table appear cryptic at first; they will make sense after you have seen some examples. In our first example, we will illustrate how to prove ‘for every’ statements and ‘if. . . then’ statements, and how to use ‘there exists’ statements. These ideas have already been introduced in the dialogue." - from Introduction to Mathematical Arguments A couple more resources for your classroom: 1. "Making arguments with equations, figures, and images" - ( As soon as you have students start writing more in math class, some of them will start trying to write out EVERYTHING. The point of this post is that sometimes mathematical symbols are still most 2. "Developing argument writing in math using crime scene investigations" -( This blog post from a teacher directly integrates an argument writing text by George Hillocks, Jr., Teaching Argument Writing, Grades 6-12 (with a bonus handout!) as a strategy for students to attack math word problems	{"url":"https://blog.chucklearns.com/2014/06/argument-writing-in-math-getting-started.html","timestamp":"2024-11-07T10:46:46Z","content_type":"text/html","content_length":"79915","record_id":"<urn:uuid:317de07c-3672-4621-9d10-e034a60b2546>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00107.warc.gz"}
How do you simplify (n^2-4)/(n^2-2n)n/(n^2+10n+16)? \| HIX Tutor How do you simplify #(n^2-4)/(n^2-2n)n/(n^2+10n+16)#? Answer 1 Sign up to view the whole answer By signing up, you agree to our Terms of Service and Privacy Policy Answer 2 To simplify the expression (n^2-4)/(n^2-2n)*n/(n^2+10n+16), we can follow these steps: 1. Factorize the numerator and denominator separately: □ The numerator (n^2-4) can be factored as (n+2)(n-2). □ The denominator (n^2-2n) can be factored as n(n-2). □ The second numerator n does not require factoring. □ The denominator (n^2+10n+16) cannot be factored further. 2. Simplify the expression by canceling out common factors: □ Cancel out the common factor (n-2) in the numerator and denominator. □ Cancel out the common factor n in the numerator and denominator. 3. After canceling out the common factors, the simplified expression becomes: (n+2)/(n+8) Sign up to view the whole answer By signing up, you agree to our Terms of Service and Privacy Policy Answer from HIX Tutor When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some Not the question you need? HIX Tutor Solve ANY homework problem with a smart AI • 98% accuracy study help • Covers math, physics, chemistry, biology, and more • Step-by-step, in-depth guides • Readily available 24/7	{"url":"https://tutor.hix.ai/question/how-do-you-simplify-n-2-4-n-2-2n-n-n-2-10n-16-8f9af9c081","timestamp":"2024-11-09T23:38:40Z","content_type":"text/html","content_length":"576344","record_id":"<urn:uuid:88018977-af6f-4ddf-b27b-f0a92306b4dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00282.warc.gz"}
American Journal of Computational Mathematics Vol. 1 No. 1 (2011) , Article ID: 4449 , 4 pages DOI:10.4236/ajcm.2011.11003 Conjugate Effects of Radiation and Joule Heating on Magnetohydrodynamic Free Convection Flow along a Sphere with Heat Generation ^1Department of Mathematics, Dhaka Commerce College, Dhaka, Bangladesh ^2Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka, Bangladesh ^3Department of Mathematics, Jahangirnagar University, Dhaka, Bangladesh E-mail: mirajaknd@gmail.com, maalim@math.buet.ac.bd, sazzad67@hotmail.com Received February 11, 2011; revised February 24, 2011; accepted March 1, 2011 Keywords: Natural Convection, Thermal Radiation, Prandtl Number, Joule Heating, Heat Generation, Magnetohydrodynamics, Nusselt Number The conjugate effects of radiation and joule heating on magnetohydrodynamic (MHD) free convection flow along a sphere with heat generation have been investigated in this paper. The governing equations are transformed into dimensionless non-similar equations by using set of suitable transformations and solved numerically by the finite difference method along with Newton’s linearization approximation. Attention has been focused on the evaluation of shear stress in terms of local skin friction and rate of heat transfer in terms of local Nusselt number, velocity as well as temperature profiles. Numerical results have been shown graphically for some selected values of parameters set consisting of heat generation parameter Q, radiation parameter Rd, magnetic parameter M, joule heating parameter J and the Prandtl number Pr. 1. Introduction The conjugate effects of radiation and joule heating on magnetohydrodynamic (MHD) free convection boundary layer on various geometrical shapes such as vertical flat plate, cylinder, sphere etc, have been studied by many investigators and it has been a very popular research topic for many years. Nazar et al. [1], Huang and Chen [2] considered the free convection boundary layer on an isothermal sphere and on an isothermal horizontal circular cylinder both in a micropolar fluid. Molla et al. [3] have studied the problem of natural convection flow along a vertical wavy surface with uniform surface temperature in presence of heat generation or absorption. Miraj et al. [4] studied the effect of radiation on natural convection flow on a sphere in presence of heat generation. Amin [5] also analyzed the influences of both first and second order resistance, due to the solid matrix of non-darcy porous medium, Joule heating and viscous dissipation on forced convection flow from a horizontal circular cylinder under the action of transverse magnetic field. Hossain [6] studied viscous and joule heating effects on magnetohydrodynamic (MHD) free convection flow with variable plate temperature. Alam et al. [7] studied the viscous dissipation effects with MHD natural convection flow on a sphere in presence of heat generation. In the present work, the effects of joule heating with radiation heat loss on natural convection flow around a sphere have been investigated. The governing partial differential equations are reduced to locally non-similar partial differential forms by adopting appropriate transformations. The transformed boundary layer equations are solved numerically using implicit finite difference method with Keller box scheme described by Keller [8] and later by Cebeci and Bradshaw [9]. The results have been shown in terms of the velocity, temperature profiles, the skin friction and surface heat transfer 2. Formulation of the Problem A steady two-dimensional magnetohydrodynamic (MHD) natural convection boundary layer flow from an isothermal sphere of radius a, which is immersed in a viscous and incompressible optically dense fluid with heat generation and radiation heat loss is considered. The physical configuration considered is as shown in Figure 1. Under the above assumptions, the governing equations for steady two-dimensional laminar boundary layer flow problem under consideration can be written as With the boundary conditions The above equations are further non-dimensionalised using the new variables: The radiation heat flux is in the following form Figure 1. Physical model and coordinate system. Substituting (5), (6) and (7) into Equations (1), (2) and (3) leads to the following non-dimensional equations With the boundary conditions (4) become To solve Equations (10) and (11) with the help of following variables where y is the stream function defined by Equation (10) can be written as Equation (11) becomes Along with boundary conditions It can be seen that near the lower stagnation point of the sphere, i.e., ξ » 0, Equations (15) and (16) reduce to the following ordinary differential equations: Subject to the boundary conditions In practical applications, skin-friction coefficient C[f] and Nusselt number Nu can be written in non-dimensional form as Putting the above values in Equation (21), we have 3. Results and Discussion Solutions are obtained for some test values of Prandtl number Pr = 2.00, 5.00, 7.00, 9.00; radiation parameter Rd =1.00, 2.00, 3.00, 4.00, 5.00; heat generation parameter Q = 0.00, 0.05, 0.10, 0.15, 0.20; magnetic parameter M = 0.50, 1.00, 1.50, 2.00, 3.00 and joule heating parameter J = 0.10, 0.50, 1.00, 1.50, 2.00 in terms of velocity and temperature profiles, skin friction coefficient and heat transfer coefficient. The effects for different values of radiation parameter Rd the velocity and temperature profiles in case of Prandtl number Pr = 0.72, heat generation parameter Q = 0.10, magnetic parameter M = 2.00 and joule heating parameter J = 0.50 are shown in Figures 2(a) and 2(b). In Figures 3(a) and 3(b) are shown that when the Prandtl number Pr increases with radiation parameter Rd = 1.00, heat generation parameter Q = 0.10, magnetic parameter M = 2.00 and joule heating parameter J = 0.50, both the velocity and temperature profiles are decrease. For different values of heat generation parameter Q with radiation parameter Rd = 1.00, Prandtl number Pr = 0.72, magnetic parameter M = 2.00 and joule heating parameter J = 0.50 and display results in Figures 4 (a,b) that as the heat generation parameter Q increases, the velocity and the temperature profiles increase. It has been seen from Figure 5(a) that as the magnetic parameter M increases, the velocity profiles decrease up to the position of h = 4.10555 after that position velocity profiles increase with the increase of magnetic parameter. We see that in Figure 5(b) temperature profiles increase for increasing values of magnetic parameter M with radiation parameter Rd = 1.00, Prandtl number Pr = 0.72, heat generation parameter Q = 0.10 and joule heating parameter J = 0.50. It has been seen from Figures 6(a,b) that as the joule heating parameter J increases, the velocity and the temperature profiles increase. In Figures 7(a,b) shown that the radiation parameter Rd increases, both the skin friction coefficient and heat transfer coefficient increase. The variation of the local skin friction coefficient C[f ] and local rate of heat transfer Nu[ ]for different values of Prandtl number Pr while radiation parameter Rd = 1.00, heat generation parameter Q = 0.10, magnetic parameter M = 2.00 and joule heating parameter J = 0.50 are shown in Figures 8(a) and 8(b). From Figure 2. (a) Velocity and (b) Temperature profiles for different values of Rd when Pr = 0.72, Q = 0.1, M = 2.0 and J = 0.5. Figure 3. (a) Velocity and (b) Temperature profiles for different values of Pr when Rd = 1.0, Q = 0.1, M = 2.0 and J = 0.5. Figure 4. (a) Velocity and (b) Temperature profiles for different values of Q when Rd = 1.0, Pr = 0.72, M = 2.0 and J = 0.5. Figure 5. (a) Velocity and (b) Temperature profiles for different values of M when Rd = 1.0, Pr = 0.72, Q = 0.1 and J = 0.5. Figure 6. (a) Velocity and (b) Temperature profiles for different values of J when Rd = 1.0, Pr = 0.72, Q = 0.1 and M = 2.0. Figure 7. (a) Skin friction coefficient and (b) Rate of heat transfer for different values of Rd when Pr = 0.72, Q = 0.1, M = 2.0 and J = 0.5. Figure 8. (a) Skin friction coefficient and (b) Rate of heat transfer for different values of Pr when Rd = 1.0, Q = 0.1, M = 2.0 and J = 0.5. Figure 9(a) we observed that the skin friction coefficient C[f] increases significantly as the heat generation parameter Q increases and Figure 9(b) show that heat transfer coefficient Nu decreases for increasing values of heat generation parameter Q with relevant parameters. It reveals that the rate of heat transfer decreases along the ξ direction from lower stagnation point to the downstream. Figures 10(a,b) shown that skin friction coefficient C[f] and heat transfer coefficient Nu decrease for increasing values of magnetic parameter M while radiation parameter Rd = 1.00, Prandtl number Pr = 0.72,heat generation parameter Q = 0.10, and joule heating parameter J = 0.50. From Figure 11(a) we observed that the skin friction coefficient C[f] increases and Figure 11(b) shown that heat transfer coefficient Nu decreases for increasing values of joule heating parameter J. 4 Conclusions The conjugate effects of radiation and joule heating on magnetohydrodynamic free convection flow along a sphere with heat generation has been investigated for different values of relevant physical parameters including the magnetic parameter M. From the present investigation the following conclusions may be drawn: All the velocity profile, temperature profile, the local skin friction coefficient C[f] and the local rate of heat transfer Nu increase significantly when the values of radiation parameter Rd As joule heating parameter J increases, both the velocity and the temperature profiles increase and also the local skin friction coefficient C[f] and the local rate of heat transfer Nu increase Figure 9. (a) Skin friction coefficient and (b) Rate of heat transfer for different values of Q when Rd = 1.0, Pr = 0.72, M = 2.0 and J = 0.5. Figure 10. (a) Skin friction coefficient and (b) Rate of heat transfer for different values of M when Rd = 1.0, Pr = 0.72, Q = 0.1 and J = 0.5. Figure 11. (a) Skin friction coefficient and (b) Rate of heat transfer for different values of J when Rd = 1.0, Pr = 0.72, Q = 0.1 and M = 2.0. For increasing values of Prandtl number Pr leads to decrease the velocity profile, the temperature profile and the local skin friction coefficient C[f] but the local rate of heat transfer Nu An increase in the values of M leads to decrease the velocity profiles but to increase the temperature profiles and also both the local skin friction coefficient C[f] and the local rate of heat transfer Nu decrease. 5. References 1. R. Nazar, N. Amin, T. Grosan and I. Pop, “Free Convection Boundary Layer on an Isothermal Sphere in a Micropolar Fluid,” International Communications in Heat and Mass Transfer, Vol. 29, No. 3, 2002, pp. 377-386. doi:10.1016/S0735-1933(02)00327-5 2. M. J. Huang and C. K. Chen, “Laminar Free Convection from a Sphere with Blowing and Suction,” Journal of Heat Transfer, Vol. 109, 1987, pp. 529-532. doi:10.1115/1.3248117 3. M. M. Molla, M. A. Taher, M. M. K. Chowdhury and Md. A. Hossain, “Magnetohydrodynamic Natural Convection Flow on a Sphere in Presence of Heat Generation,” Nonlinear Analysis: Modelling and Control, Vol. 10, No. 4, 2005, pp. 349-363. 4. M. Miraj, M. A. Alim and M. A. H. Mamun, “Effect of Radiation on Natural Convection Flow on a Sphere in Presence of Heat Generation,” International Communications in Heat and Mass Transfer, Vol. 37, No. 6, 2010, pp. 660-665. doi:10.1016/j.icheatmasstransfer.2010.01.013 5. M. F. El-Amin, “Combined Effect of Viscous Dissipation and Joule Heating on MHD Forced Convection over a Non-Isothermal Horizontal Cylinder Embedded in a Fluid Saturated Porous Medium,” Journal of Magnetism and Magnetic Materials, Vol. 263, No. 3, 2003, pp. 337-343. doi:10.1016/S0304-8853(03)00109-4 6. M. A. Hossain, “Viscous and Joule Heating Effects on MHD-Free Convection Flow with Variable Plate Temperature,” International Journal of Heat and Mass Transfer, Vol. 35, No. 12, 1992, pp. 3485-3487. doi:10.1016/0017-9310(92)90234-J 7. M. M. Alam, M. A. Alim and M. M. K. Chowdhury, “Viscous Dissipation Effects with MHD Natural Convection Flow on a Sphere in Presence of Heat Generation,” Nonlinear Analysis: Modelling and Control, Vol. 12, No. 4, 2007, pp. 447-459. 8. H. B. Keller, “Numerical Methods in Boundary Layer Theory,” Annual Review of Fluid Mechanics, Vol. 10, 1978, pp. 417-433. doi:10.1146/annurev.fl.10.010178.002221 9. T. Cebeci and P. Bradshaw, “Physical and Computational Aspects of Convective Heat Transfer,” Springer, New York, 1984.	{"url":"https://file.scirp.org/Html/3-1100003_4449.htm","timestamp":"2024-11-05T22:19:50Z","content_type":"application/xhtml+xml","content_length":"44428","record_id":"<urn:uuid:cbf4fea5-1c12-4720-b709-5d0e133cfeb1>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00558.warc.gz"}
Aptitude Test Numerical Aptitude Questions And Answers Aptitude Test Numerical Aptitude Questions And Answers have ended up being essential in evaluating a person's inherent abilities and capabilities. As the employment landscape develops and schools seek much more efficient analysis methods, comprehending Aptitude Test Numerical Aptitude Questions And Answers is important. 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A: Typically, there are no strict age limitations, but the importance of the test may vary based upon the person's career stage.	{"url":"https://mushroomhead.15ru.net/en/aptitude-test-numerical-aptitude-questions-and-answers.html","timestamp":"2024-11-12T20:21:46Z","content_type":"text/html","content_length":"29204","record_id":"<urn:uuid:9f59aea4-7519-4774-8b45-9072478da8a2>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00889.warc.gz"}
Note Frequency Calculator - Calculate the Musical Note Frequency Our Musical Note Frequency Calculator is a tool that helps determine the precise frequency of a given musical note. It is an essential resource for musicians, sound engineers, and anyone working with audio or music production. This tool takes into account the fundamental principles of sound waves and musical theory to provide accurate results. By providing accurate and reliable frequency calculations, the Musical Note Frequency Calculator ensures that musicians, audio professionals, and others working with sound waves can achieve precise and high-quality results in their respective fields. Note Frequency Calculator What is the frequency of the C5 note? To find the frequency of the C5 note, we use the formula: f = f0 × 2^(n/12) • f0 = 440 Hz (reference frequency for A4) • n = number of semitones away from A4 C5 is 9 semitones above A4, so n = 9. Substituting the values: f = 440 × 2^(9/12) = 523.25 Hz The frequency of the C5 note is 523.25 Hz. What is the note with a frequency of 311 Hz? To find the note corresponding to a given frequency, we rearrange the formula: n = 12 × log2(f/f0) Substituting the values: • f = 311 Hz • f0 = 440 Hz (reference frequency for A4) n = 12 × log2(311/440) = -5.87 Since n = -5.87 is approximately -6 semitones away from A4, the note with a frequency of 311 Hz is Eb4 (E-flat 4). What is the frequency of the G#2 note? G#2 is 8 semitones above A2, and A2 is an octave below A4 (440 Hz). First, we find the frequency of A2: f_A2 = 440 × 2^(-1) = 220 Hz Then, we find the frequency of G#2 using the formula: f_G#2 = 220 × 2^(8/12) = 207.65 Hz The frequency of the G#2 note is 207.65 Hz. Note Frequency Calculation Formula The frequency of a musical note is calculated based on a formula that considers the note’s position on the musical scale and the reference frequency (typically A4 = 440 Hz). The formula is as follows: f = f0 × 2^(n/12) In this formula: • f is the frequency of the desired note • f0 is the reference frequency (usually A4 = 440 Hz) • n is the number of semitones away from the reference note For example, to calculate the frequency of the C4 note (four semitones below A4), the formula would be: f = 440 × 2^((-4)/12) = 261.63 Hz Related Tools: What is Musical Note Frequency? Musical Note Frequency refers to the number of vibrations per second produced by a sound wave, which is perceived as a specific pitch or note. Each note on the musical scale corresponds to a specific frequency, and these frequencies are determined by the principles of physics and the properties of sound waves. The human ear is capable of perceiving frequencies ranging from approximately 20 Hz to 20,000 Hz, with the most sensitive range being between 2,000 Hz and 4,000 Hz. Musical notes typically fall within the audible range, with the lowest note on a standard piano (A0) having a frequency of 27.5 Hz and the highest note (C8) having a frequency of 4,186 Hz. Why Use Musical Note Frequency Calculator? There are several reasons why musicians, sound engineers, and audio professionals may need to use a Musical Note Frequency Calculator: 1. Tuning Instruments: Ensuring that instruments are properly tuned is crucial for producing accurate and harmonious sounds. The calculator helps determine the exact frequency for each note, allowing musicians to tune their instruments precisely. 2. Audio Synthesis and Sampling: In the field of audio synthesis and sampling, knowing the precise frequencies of musical notes is essential for creating accurate and realistic sounds. 3. Acoustics and Room Design: Architects and acoustic engineers use musical note frequency calculations to optimize the acoustics of concert halls, recording studios, and other sound-sensitive 4. Music Theory and Composition: Composers and music theorists may use the calculator to explore the relationships between different notes, intervals, and musical scales, aiding in the creative 5. Audio Editing and Processing: Sound engineers and audio editors often need to apply specific filters or effects to certain frequencies, making the Note Frequency Calculator a valuable tool for identifying and isolating desired frequencies. What note is 147 Hz? To find the note corresponding to 147 Hz, we need to rearrange the formula: n = 12 × log2(f/f0) Substituting the values: • f = 147 Hz • f0 = 440 Hz (reference frequency for A4) n = 12 × log2(147/440) = -16.63 Since n = -16.63 is approximately -17 semitones away from A4, the note with a frequency of 147 Hz is D3. What note is 392 Hz? Using the same approach: n = 12 × log2(392/440) = -1.49 Since n = -1.49 is approximately -2 semitones away from A4, the note with a frequency of 392 Hz is G4. What is 432 Hz as A note? To find the note corresponding to 432 Hz, we can use the formula directly: n = 12 × log2(432/440) = 0.32 Since n = 0.32 is slightly higher than 0 semitones away from A4, the note with a frequency of 432 Hz is considered A4+ or a slightly sharp A4. In summary: • The note with a frequency of 147 Hz is D3. • The note with a frequency of 392 Hz is G4. • A frequency of 432 Hz corresponds to A4+ or a slightly sharp A4.	{"url":"https://ctrlcalculator.com/misc/note-frequency-calculator/","timestamp":"2024-11-04T20:02:23Z","content_type":"text/html","content_length":"105084","record_id":"<urn:uuid:9983f8aa-8afe-45d2-9a3b-0e35270a198d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00177.warc.gz"}
SKM 2023 – wissenschaftliches Programm DY 34.2: Vortrag Mittwoch, 29. März 2023, 17:00–17:15, ZEU 250 Time-reversal symmetries and equilibrium-like Langevin equations — •Lokrshi Prawar Dadhichi and Klaus Kroy — ITP, Leipzig University, Brüderstraße 15, 04103, Leipzig Graham has shown in Z. Physik B 26, 397-405 (1977) that a fluctuation-dissipation relation can be imposed on a class of non-equilibrium Markovian Langevin equations that admit a stationary solution of the corresponding Fokker-Planck equation. Here we show that the resulting equilibrium form of the Langevin equation is associated with a nonequilibrium Hamiltonian and ask how precisely the broken equilibrium condition manifests itself therein. We find that this Hamiltonian need not be time reversal invariant and that the "reactive" and "dissipative" fluxes loose their distinct time reversal symmetries. The antisymmetric coupling matrix between forces and fluxes no longer originates from Poisson brackets and the "reactive" fluxes contribute to the ("housekeeping") entropy production, in the steady state. The time-reversal even and odd parts of the nonequilibrium Hamiltonian contribute in qualitatively different but physically instructive ways to the entropy. Finally, this structure gives rise to a new, physically pertinent instance of frenesy.	{"url":"https://www.dpg-verhandlungen.de/year/2023/conference/skm/part/dy/session/34/contribution/2","timestamp":"2024-11-09T14:23:52Z","content_type":"text/html","content_length":"7650","record_id":"<urn:uuid:9c9d2015-c597-4010-8a37-09cbb792213a>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00403.warc.gz"}
Iris feature extraction with 2D Gabor wavelets Iris feature extraction aims to extract the most discriminative features from iris images while reducing the dimensionality of data by removing unrelated or redundant data. Unlike 2D face images that are mostly defined by edges and shapes, iris images present rich and complex texture with repeating (semi-periodic) patterns of local variations in image intensity^[1]. In other words, iris images contain strong signals in both spatial and frequency domains and should be analyzed in both. Examples of iris images can be found on John Daugman's website. Gabor filtering Research has shown that the localized frequency and orientation representation of Gabor filters is very similar to the human visual cortex’s representation and discrimination of texture^[2]. A Gabor filter analyzes a specific frequency content at a specific direction in a local region of an image. It has been widely used in signal and image processing for its optimal joint compactness in spatial and frequency domain. Fig. 1 Constructing a Gabor filter is straightforward. The product of (a) a complex sinusoid signal and (b) a Gaussian filter produces (c) a Gabor filter. As shown above, a Gabor filter can be viewed as a sinusoidal signal of particular frequency and orientation modulated by a Gaussian wave. Mathematically, it can be defined as $G_{\lambda,θ,\phi,\sigma,γ}(x, y)=\exp(-\frac{x'^2+γ^2y'^2}{2\sigma^2})\exp(j(2\pi\frac{x'}{\lambda}+\phi))$ \left [ \begin{aligned} &x'\\ &y' \end{aligned}\right]=\left [ \begin{array}c \cos\theta & \sin\theta\\ -\sin\theta & \cos\theta \end{array}\right]\left [ \begin{aligned} &x\\ &y \end{aligned}\right] Among the parameters, $σ$ and $γ$ represent the standard deviation and the spatial aspect ratio of the Gaussian envelope, respectively, $λ$ and $ϕ$ are the wavelength and phase offset of the sinusoidal factor, respectively, and $θ$ is the orientation of the Gabor function. Depending on its tuning, a Gabor filter can resolve pixel dependencies best described by narrow spectral bands. At the same time, its spatial compactness accommodates spatial irregularities. For more details see^[3]. The following figure shows a series of Gabor filters at a $45^∘$ angle in increasing spectral selectivity. While the leftmost Gabor wavelet resembles a Gaussian, the rightmost Gabor wavelet follows a harmonic function and selects a very narrow band from the spectrum. Best for iris feature extraction are the ones in the middle between the two extremes. Fig. 2 Varying wavelength (a-d) from large to small can change the spectral selectivity of Gabor filters from broad to narrow. Because a Gabor filter is a complex filter, the real and imaginary parts act as two filters in quadrature. More specifically, as shown in the figures below, (a) the real part is even-symmetric and will give a strong response to features such as lines; while (b) the imaginary part is odd-symmetric and will give a strong response to features such as edges. It is important that we maintain a zero DC component in the even-symmetric filter (the odd-symmetric filter already has zero DC). This ensures zero filter response on a constant region of an image regardless of the image intensity. Fig. 3 Giving a closer look at the complex space of a Gabor filter where (a) the real part is even-symmetric and (b) the imaginary part is odd-symmetric. Multi-scale Gabor filtering Like most textures, iris texture lives on multiple scales (controlled by $σ$). It is therefore natural to represent it using filters of multiple sizes. Many such multi-scale filter systems follow the wavelet building principle, that is, the kernels (filters) in each layer are scaled versions of the kernels in the previous layer, and, in turn, scaled versions of a mother wavelet. This eliminates redundancy and leads to a more compact representation. Gabor wavelets can further be tuned by orientations, specified by $θ$. The figure below shows the real part of 28 Gabor wavelets with four scales and 7 orientations. Fig. 4 Constructing Gabor wavelets with multiple scales (vertically) and orientations (horizontally) to extract texture features with various frequencies and directions. In our feature extraction process, we use a small set of filters that concentrate within the range of scales and orientations of the most discriminative iris texture. Phase-quadrant demodulation and encoding After a Gabor filter is applied to an iris image, the filter response at each analyzed region is then demodulated to extract its phase information^[4]. This process is illustrated in the figure below, as it identifies in which quadrant of the complex plane each filter response is projected to. Note that only phase information is recorded because it is more robust than the magnitude, which can be contaminated by extraneous factors such as illumination, imaging contrast, and camera gain. Fig. 5 Demodulating the phase information of filter response into four quadrants of the complex space. The resulting cyclic codes are used to produce the final iris code. Another desirable feature of the phase-quadrant demodulation is that it produces a cyclic code. Unlike a binary code in which two bits may change, making some errors arbitrarily more costly than others, a cyclic code only allows a single bit change in rotating between any adjacent phase quadrants^[4]. Importantly, when a response falls very closely to the boundary between adjacent quadrants, its resulting code is considered a fragile bit. These fragile bits are usually less stable and could flip values due to changes in illumination, blurring or noise. There are many methods to deal with fragile bits, and one such method could be to assign them lower weights during matching. When multi-scale Gabor filtering is applied to a given iris image, multiple iris codes are produced accordingly and concatenated to form the final iris template. Depending on the number of filters and their stride factors, an iris template can be two to three orders of magnitude smaller than the original iris image. Robustness of iris codes Because iris codes are generated based on the phase responses from Gabor filtering, they are considered to be rather robust against illumination, blurring and noise. To measure this quantitatively, we add each effect, namely, illumination (gamma correction), blurring (Gaussian filtering), and Gaussian noise to an iris image, respectively, in slow progression and measure the drift of the iris code. The amount of added effect is measured by the Root Mean Square Error (RMSE) of pixel values between the modified and original image, and the amount of drift is measured by the Hamming distance between the new and original iris code. Mathematically, RMSE is defined as: $\textrm{RMSE} = \sqrt{\frac{1}{N}\sum^{N}_{p=1}(I'_{p}-I_{p})^2}$ where $N$ is the number of pixels in the original image $I$ and the modified image $I^′$. The Hamming distance is defined as: $\textrm{HD} = \frac{1}{K}\sum^{K}_{p=1}\|C'_p - C_p\|$ where $K$ is the number of bits (0/1) in the original iris code $C$ and the new iris code $C^′$. A Hamming distance of 0 means a perfect match, while 1 means the iris codes are completely opposite. The Hamming distance between two randomly generated iris codes is around 0.5. The following figures help us understand both visually and quantitatively the impact of illumination, blurring and noise on the robustness of iris codes. For illustration purposes, these results are not generated with the actual filters we deploy but nevertheless demonstrate the property in general of Gabor filtering. Also, the iris image has been normalized from a donut shape in the cartesian coordinates to a fixed-size rectangular shape in the polar coordinates. This step is necessary to standardize the format, mask-out occlusion and enhance the iris texture. As shown in the figure below, iris codes are very robust against grey-level transformations associated with illumination as the HD barely changes with increasing RMSE. This is because increasing the brightness of pixels reduces the dynamic range of pixel values, but barely affects the frequency or spatial properties of the iris texture. Fig. 6 Demonstrating the impact of illumination on the robustness of iris codes using Root Mean Square Error (RMSE) between images (blue line) vs Hamming Distance (HD) between corresponding iris codes (green line). Blurring, on the other hand, reduces image contrast and could lead to compromised iris texture. However, as shown below, iris codes remain relatively robust even when strong blurring makes iris texture indiscernible to naked eyes. This is because the phase information from Gabor filtering captures the location and presence of texture rather than its strength. As long as the frequency or spatial property of the iris texture is present, though severely weakened, the iris codes remain stable. Note that blurring compromises high frequency iris texture, therefore, impacting high frequency Gabor filters more, which is why we use a bank of multi-scale Gabor filters. Demonstrating the impact of blurring on the robustness of iris codes using Root Mean Square Error (RMSE) between images (blue line) vs Hamming Distance (HD) between corresponding iris codes (green Finally, we observe bigger changes in iris codes when Gaussian noise is added, as both spatial and frequency components of the texture are polluted and more bits become fragile. When the iris texture is overwhelmed with noise and becomes indiscernible, the drift in iris codes is still small with a Hamming distance below 0.2, compared to matching two random iris codes ($≈0.5$). This demonstrates the effectiveness of iris feature extraction using Gabor filters even in the presence of noise. Fig. 8 Demonstrating the impact of noise on the robustness of iris codes using using Root Mean Square Error (RMSE) between images (blue line) vs Hamming Distance (HD) between corresponding iris codes (green In this blog post, we discussed iris feature extraction as a necessary and important step in iris recognition. It reduces the dimensionality of the iris representation from a high resolution image to a string of binary code, while preserving the most discriminative texture features using a bank of Gabor filters. It is worth noting that Gabor filters have their own limitations, for example, one cannot design Gabor filters with arbitrarily wide bandwidth while maintaining a near-zero DC component in the even-symmetric filter. This limitation can be overcome by using the Log Gabor filters^[5 ]. In addition, Gabor filters are not necessarily optimized for iris texture, and machine-learned iris-domain specific filters (e.g. BSIF) have potential to achieve further improvement in feature extraction and recognition performance in general^[6]. Moreover, we are investigating novel approaches to leverage higher quality images and the latest advances in the field of deep metric learning and deep representation learning to push the state of the art in iris recognition. As we showcased the resilience of iris feature extraction amidst external factors, it is crucial to note that even minor fluctuations in iris code variability hold significant importance when dealing with a billion people, as the tail-end of the distribution dictates the error rates, thus influencing the number of false rejections. We will elaborate further on this subject in an upcoming blog Yi Chen, Tanguy Jeanneau and Christian Brendel The information in this article is over 12 months old and may be outdated. Please visit for the most recent information about the project.	{"url":"https://world.org/blog/engineering/iris-feature-extraction","timestamp":"2024-11-10T02:19:46Z","content_type":"text/html","content_length":"226944","record_id":"<urn:uuid:d06d0075-c197-46fd-ab59-4cd2a285aae1>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00218.warc.gz"}
When total product is increasing at a decreasing rate? When total product is increasing at a decreasing rate? Finally, when total product is increasing at an increasing rate the total cost is increasing at a decreasing rate. When total product is increasing at a decreasing rate, the total cost is increasing at an increasing rate. 1. What is the formula for total product? For any degree of an input, the sum of marginal products of every foregoing unit of that input gives the total product. So, the total product is the sum of marginal products. When total product increases at decreasing rate what happened to marginal product? Marginal Product should be decreasing. When TP is increasing at a decreasing rate is known as point? This stage is called the stage of diminishing returns to a factor. It refers to the phase where TP increases at a diminishing rate and reaches its maximum. In this phase, MP is declining but note that it still remains positive. The stage ends where MP = 0. Why does MP decrease with an increase in TP at decreasing rate? Because the MP curve is derived from the TP curve, it reflects the information in the TP curve. For example, when the slope of the TP curve is increasing, MP is increasing because of specialization and teamwork. In the middle range where TP is increasing at a decreasing rate, MP is positive but falling. How is tp AP MP calculated? We calculate it as APL=TPL/L, where APL is the average product of labour, TPL is the total product of labour and L is the amount of labour input used. 3. Marginal product: Marginal product of an input is defined as the change in output per unit of change in the input when all other inputs are held constant. When total product is increasing at an increasing rate marginal product is? As if marginal product ≥ 0 it is profitable to increase production. If marginal product ≤ 0 it is profitable to decrease production. Due to the fixed resources initially being underutilized, the total product increases as a result of adding more workers, this is when the marginal product is rising. When TP increases at an increasing rate what happens to MP? Explanation: MP falls, but remains positive. When TP increases at a decreasing rate what happens to the MP? When MP is decreasing TP increases at which rate? 1 Answer. No. When MP is decreasing, TP increases at a diminishing rate. When TP increases at decreasing rate what happens to MP? When TP increases at a diminishing rate, MP falls. When TP reaches its maximum point MP becomes zero. When TP starts decreasing, MP becomes negative. As long as MP is more than AP, AP rises. When TP increases at a rate MP starts decreasing? Answer. Answer: When MP is decreasing, only an addition to TP is decreasing i.e. TP continues to increase, though at a diminishing rate. TP starts declining only when MP becomes negative. How is tp AP and MP calculated? What is the relationship between TP and MP? Relationship between Total Product and Marginal Product The relationship between TP and MP is explained through the Law of Variable Proportions. As long as the the TP increases at an increasing rate, the MP also increases. This goes on till MP reaches maximum. When TP increases at a diminishing rate, MP declines. When total product is increasing at an increasing rate marginal product is positive and decreasing? If the total product curve rises at an increasing rate, the marginal product of labormarginal product of laborIn economics, the marginal product of labor (MPL) is the change in output that results from employing an added unit of labor. It is a feature of the production function, and depends on the amounts of physical capital and labor already in use.https://en.wikipedia.org › wiki › Marginal_product_of_laborMarginal product of labor – Wikipedia curve is positive and rising. If the total product curve rises at a decreasing rate, the marginal product of labor curve is positive and falling. 8. In which stage total product increase at an increasing rate? First Stage or Stage of Increasing returns: In this stage, the total product increases at an increasing rate. This happens because the efficiency of the fixed factors increases with addition of variable inputs to the product. When TP increases at an increasing rate what happens to MP Class 11? When MP increases TP increases at? Explain the relationship between the marginal products and the total product of an input. Note. Since MPP is addition to TPP, it also implies that if MPP is positive, TPP must be increasing and if MPP is negative (-), TPP must be decreasing. (i) When MPP rises, TPP rises at an increasing rate. When MP is falling TP always decreases? Under diminishing returns to a factor, the marginal productmarginal productMarginal Physical Product is the change in output produced by employing one additional unit of the variable input. It can be calculated as : MPPn=ΔTPPΔUnits of variable input.https://byjus.com › define-marginal-physical-productDefine Marginal Physical Product. – Byju’s tends to fall. Falling marginal product implies that total product increases at a diminishing rate. TP is maximum when MP=0. After this point, MP becomes negative and TP then starts to fall. When TP increases at increasing rate MP * 1 point a Falls B rises C remains constant D all of these? False; when MP is constant, TP increases at a constant rate. How do you calculate TP and MP in AP? When TP is rising at increasing rate MP is? When TP increases at an increasing rate, MP will. Uh-Oh! That’s all you get for now. When MP is positive and falling TP rises at a decreasing rate? Expert-verified answer When the MP rises the TP also rises at an increasing rate. On the other hand, if the MP is falling the TP is also falling. Hence from this, we can say that all the given options are correct. Hence, option (d) is correct. What will happen to MP when TP increases at an increasing rate? TP increases at an increasing rate when MP increases. This pattern provides a Total Product Curve with a shape of convex. How is AP and TP calculated? Average product: Average product is defined as the output per unit of variable input. We calculate it as APL=TPL/L, where APL is the average product of labour, TPL is the total product of labour and L is the amount of labour input used.	{"url":"https://mattstillwell.net/when-total-product-is-increasing-at-a-decreasing-rate/","timestamp":"2024-11-02T21:20:35Z","content_type":"text/html","content_length":"42817","record_id":"<urn:uuid:a465cfb0-edf7-4148-b01b-433b895188ec>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00880.warc.gz"}
1.2: Electrical Fundamentals Last updated Page ID $ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $ $ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ ( \newcommand{\kernel}{\mathrm{null}\,}\) $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\\| #1 \\|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\\| #1 \\|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$ $ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$ $ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$ $ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vectorC}[1]{\textbf{#1}} $ $ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $ $ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $ $ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $ $ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $ $\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\ evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\ newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y} $ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real} {\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec} [3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array} {r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\ wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\ newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var} {\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left\|#1\right\|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\ bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\ widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$ Hopefully by the time you’re considering wiring up motors and their controls, you already have a knowledge of electricity and how it behaves. The following is just a brief refresher, but any technician considering the wiring of three-phase motors should have a solid understanding of electrical fundamentals. If you do not have this understanding, please use your favorite education resource to complete training. Electrons moving around in a never-ended loop is considered a circuit. Circuits that are closed allow for electron flow. Circuits that are open (broken) do not allow for electron flow. Electricity has three main components. Voltage is the amount of potential electrical energy between two points, and it is usually represented by the letter E (for electromotive force) and is measured in volts, abbreviated with V. Current is the flow of electrons from negatively-charge atoms toward atoms with a positive charge. We measure current in Amperes or Amps (abbreviated A), and the symbol for current is I. Current that travels in only one direction we call direct current (DC). Current that changes direction at regular intervals we call alternating current (AC). The final component is resistance. Resistance (abbreviated R) is the opposition to current flow, and it is measured in ohms (abbreviated with the Greek letter Omega, Ω). These three terms are related to one another in the following way: E = I * R. Knowing this, if we ever know any two of these variables in a circuit, we can always calculate the third. Two other ways to express the same equation are: I = E / R and R = E / I. In electricity, we often use scientific notation and prefixes with the units we measure. If you’re unfamiliar with these concepts, you may want to study them a bit more. With electricity, some multiples and prefixes are used more often than others. These are in bold in the table below. When building electrical circuits, components can be connected in two basic ways: either in series with one another or in parallel. When there is only one path for current flow, the components are said to be in series. Look at this example of a series circuit. The circuit has three resistors (labeled R1, R2, and R3), and these resistors are in series with each other. Notice too that the push button (labeled PB) is also in series with the resistors, as is the 9-volt battery. There is only one path for electrons to take, and all electrons must follow the same path through the circuit when the push button is pushed. Because the electrons only have one path, the current flow is the same throughout the entire circuit, and all resistances are cumulative. Let’s say that R1 has a resistance of 10Ω, R2 has a resistance of 20Ω, and R3 has a resistance of 30Ω. Because all resistances are cumulative and current is the same through, voltage at any particular component is determined by that component’s resistance. The formula look like this: R[Total] = R[1] + R[2] + R[3] + … [ITotal] = I[1] = I[2] = I[3] = … [ETotal] = E[R1] + E[R2] + E[R3] + … In our example, total resistance is 60 ohms (10Ω + 20Ω + 30Ω). Since the total resistance is 60Ω in a 12-volt system, the total current must be 0.2A, or 200mA (12V/60Ω). Now that total current is known, voltage at each load can be calculated. For instance, the voltage at R1 (E=IR) is going to be 0.2A * 10Ω, or 2V. Voltage at R2 is 4V and voltage at R3 is 6V. Notice that all of our voltages (2V, 4V, 6V) add up to the source voltage of 12V. This rule applies to all series circuits. In parallel circuits, current flow has more than one path. This changes the behavior of the electrons in the circuit. First, the voltage across parallel components is no longer split among them. Since they are in parallel with one another (electrically common), they each get full voltage. Notice on the diagram below where the black connecting dots are located. Each of those represent two (or more) locations where voltage is going to be available equally in either direction. So, in this example, when the push button is pushed, 12 volts becomes available to every resistor in the circuit, since they are each parallel with each other (the push button, however, is still in series with all resistors). So, if every resistor has the full source voltage available, how much current will be flowing through the circuit? This depends on the resistance of each “branch” or “rung” of our circuit. We simply calculate the current through each resistor, then add them together to get the total current (see below). [ETotal] = E [R1] = E [R2] = E [R3] = … I [Total] = I [1] + I [2] + I [3] + … R[Total] = 1 / (1/R[1] + 1/R[2] + 1/R[3] + …] Total resistance, however, is now a little more difficult, because as resistors are added in parallel to our circuit, we’re providing more paths for electrons to flow. Each new path that’s added decreases the total resistance of the whole circuit. In fact, the total resistance of our parallel circuit MUST be lower than the resistance of the lowest resistor. Let’s use the same resistance values as the series circuit (R1 has a resistance of 10Ω, R2 is 20Ω, and R3 is 30Ω). Knowing that our total resistance is now going to be less than 10Ω, let’s do the R[Total] = 1 / (1/10 + 1/20 + 1/30) R[Total] = 1 / (0.1 + 0.05 + 0.0333) R[Total] = 1 / 0.1833 R[Total] = 5.45Ω Clever technicians always look for ways to verify results. In this case, we could calculate current for our circuit at each resistor. I[R1] = 12/10 I[R2] = 12/20 I[R3] = 12/30 I[R1] = 1.2A I[R2] = 0.6A I[R3] = 0.4A [ITotal] = 1.2A + 0.6A + 0.4A = 2.2A of total current Using Ohm’s Law, our total resistance should be equal to total voltage divided by total current, and indeed, if we divide 12V by 2.2A, we get a total resistance of 5.45Ω. matching our previous result. Nice. Testing Components Switches and contacts are designed to allow and stop the flow of current. When a switch or set of contacts are closed, the resistance should be very low (near zero), meaning that very little voltage is dropped (near zero). When a switch or set of contacts are open, the resistance should be infinite (OL on many meters), meaning that all potential voltage is available across those open contacts (assuming there aren’t other opens in series with the one being measured). Protection Devices Testing fuses, circuit breakers, or overloads is similar to testing switches, except they are designed to always allow current flow. It’s only when a protection device has experienced current higher than its specified amount that the device opens the circuit, stopping current. When testing resistance, these devices should have very low resistance. Since they are placed in series with the circuit they are designed to protect, we don’t want them using up available voltage. A protection device with infinite resistance is “blown” or in its “tripped” state. When measuring voltage across one of these devices, when good there should be very little voltage dropped. If source voltage is present, the device is “blown” or in its “tripped” state. Loads are designed to use applied voltage to do the work of the circuit. This could be running a motor, lighting a bulb, or actuating a relay. Loads need to have some amount of resistance (this amount will vary by load and can be found by checking the manufacturer’s specifications or measuring similar “known good” parts). Loads that measure either no resistance or an infinite amount of resistance are not good. Because loads are designed to use the available voltage, measuring voltage at a load may not always tell you if the load is good or bad. If the load has an internal open, source voltage will be read at the load. If the load is working properly, source voltage will also be found at the load. If the load is shorted, most likely, any protection devices in the circuit (fuses, circuit breakers, overloads) will be tripped, due to the increased current. Because relays operate like an electrically-controlled switch, you have two components to test. First is the coil of the relay, which acts as a load. Next are the contacts within the relay, which act as switches. The contacts of the relay are often in a different circuit than the coil, but all of these components are tested as described above. Lastly are transformers. Transformers are typically used to change AC voltage by either stepping it up or stepping it down. Transformers consist of two coils of wire wrapped around an iron core. The coils of wire behave like a typical load and can be tested similarly, as described above. 1. Test primary coil 2. Test secondary coil 3. Measure Resistance (Rs) from primary to secondary coil (should be no continuity) 4. Measure Rs from Primary coil to transformer housing (should be no continuity) 5. Measure Rs from Secondary coil to transformer housing (should be no continuity)	{"url":"https://workforce.libretexts.org/Bookshelves/Electronics_Technology/Troubleshooting_Motors_and_Controls_(Dickson-Self)/01%3A_Lesson_1/1.02%3A_Electrical_Fundamentals","timestamp":"2024-11-10T14:55:35Z","content_type":"text/html","content_length":"140088","record_id":"<urn:uuid:20bcf1a5-f56a-4587-abd3-32da4e00f21e>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00795.warc.gz"}