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Lesson 7 Construction Techniques 5: Squares 7.1: Which One Doesn’t Belong: Polygons (5 minutes) This is the first Which One Doesn't Belong routine in the course. In this routine, students are presented with four figures, diagrams, graphs, or expressions with the prompt “Which one doesn’t belong?”. Typically, each of the four options “doesn’t belong” for a different reason, and the similarities and differences are mathematically significant. Students are prompted to explain their rationale for deciding that one option doesn’t belong and given opportunities to make their rationale more precise. This warm-up prompts students to compare four polygons. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another. Arrange students in groups of 2–4. Display the polygons for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together, find at least one reason each item doesn't belong. Student Facing Which one doesn’t belong? Activity Synthesis Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class whether they agree or disagree. Since there is no single correct answer to the question asking which one does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, ask students to explain the meaning of any terminology they use, such as shape names, regular, equilateral, or equiangular. Also, press students on unsubstantiated claims. 7.2: It’s Cool to Be Square (15 minutes) In this activity, students construct a square, given a side. This is similar to how students constructed a parallel line with two successive perpendicular lines, except they also have to pay attention to marking equal distances along the perpendicular lines. Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5). If students find that the diagram becomes too cluttered, encourage them to hide objects that are not needed to continue the construction. To do so, click on the last tool in the Toolbar—the Move Graphics Window tool. Beneath it is a drop-down menu of editing tools, including the Show/Hide Object tool. Select the tool and click on each element you want hidden. The selected objects will be faded. Select any other tool, and the faded objects will disappear. The same tool undoes the hiding. Representation: Internalize Comprehension. Begin with a physical demonstration of the construction of a perpendicular line through a point on the given line, to support connections between new situations and prior understandings. Ask students how the construction of a perpendicular line could be used to construct a square. Supports accessibility for: Conceptual processing; Visual-spatial processing Student Facing Use straightedge and compass tools to construct a square with segment $AB$ as one of the sides. Suggest students use a pencil to lightly draw the straightedge and compass moves and then use a colored pencil to emphasize the sides of the square. Representation: Internalize Comprehension. Begin with a physical demonstration of the construction of a perpendicular line through a point on the given line, to support connections between new situations and prior understandings. Ask students how the construction of a perpendicular line could be used to construct a square. Supports accessibility for: Conceptual processing; Visual-spatial processing Student Facing Use straightedge and compass moves to construct a square with segment $AB$ as one of the sides. Anticipated Misconceptions Some students may struggle more than is productive. Ask these students what they know about squares and what previous construction techniques they might use to tackle this problem. Activity Synthesis Ask students, “How do you know that what you constructed is a square?” (From the construction of perpendicular lines, we know the shape has 4 right angles. From the compass, we know the 4 sides have length $AB$.) Writing, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to prepare students for the whole-class discussion by providing them with multiple opportunities to clarify their explanations through conversation. Before the whole-class discussion begins, give students time to meet with 2–3 partners to share their response to the question, “How do you know that what you constructed is a square?” Invite listeners to ask questions for clarity and reasoning, and to press for details and mathematical language. Design Principle(s): Optimize output (for explanation); Cultivate conversation 7.3: Trying to Circle a Square (15 minutes) The purpose of this activity is for students to construct a square inscribed in a circle. Just like the construction of the equilateral triangle inscribed in a circle, this construction provides an opportunity to preview rotation symmetry. Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5). Give students 5 minutes to answer questions about square $ABCD$ and then pause the class for a brief whole-class discussion. Students should come away with two key conjectures: • The diagonals of a square are perpendicular bisectors of each other. • In order to inscribe a square in a circle, the diagonals need to be diameters of the circle. Give students 5 minutes to finish the activity, and follow with a whole-class discussion. Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to aide students who benefit from support with organizational skills in problem solving. For example, present one question at a time and ensure students complete each step correctly before moving on to the next step. Supports accessibility for: Organization; Attention Student Facing 1. Here is square $ABCD$ with diagonal $BD$ drawn: 1. Construct a circle centered at $A$ with radius $AD$. 2. Construct a circle centered at $C$ with radius $CD$. 3. Draw the diagonal $AC$ and write a conjecture about the relationship between the diagonals $BD$ and $AC$. 4. Label the intersection of the diagonals as point $E$ and construct a circle centered at $E$ with radius $EB$. How are the diagonals related to this circle? 2. Use your conjecture and straightedge and compass tools to construct a square inscribed in a circle. Student Facing Are you ready for more? Use straightedge and compass moves to construct a square that fits perfectly outside the circle, so that the circle is inscribed in the square. How do the areas of these 2 squares compare? Give students 5 minutes to answer questions about square $ABCD$ and then pause the class for a brief, whole-class discussion. Students should come away with two key conjectures: • The diagonals of a square are perpendicular bisectors of each other. • In order to inscribe a square in a circle, the diagonals need to be diameters of the circle. Give students 5 minutes to finish the activity, and follow with a whole-class discussion. Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to aide students who benefit from support with organizational skills in problem solving. For example, present one question at a time and ensure students complete each step correctly before moving on to the next step. Supports accessibility for: Organization; Attention Student Facing 1. Here is square $ABCD$ with diagonal $BD$ drawn: 1. Construct a circle centered at $A$ with radius $AD$. 2. Construct a circle centered at $C$ with radius $CD$. 3. Draw the diagonal $AC$ and write a conjecture about the relationship between the diagonals $BD$ and $AC$. 4. Label the intersection of the diagonals as point $E$ and construct a circle centered at $E$ with radius $EB$. How are the diagonals related to this circle? 2. Use your conjecture and straightedge and compass moves to construct a square inscribed in a circle. Student Facing Are you ready for more? Use straightedge and compass moves to construct a square that fits perfectly outside the circle, so that the circle is inscribed in the square. How do the areas of these 2 squares compare? Anticipated Misconceptions Some students may struggle with the fact that when starting with the circle, we do not have two points marked to either construct a line or set a radius for a circle. Ask them how we may mark new points that can be used in our construction. Activity Synthesis “How was this construction different from the square in the previous activity?” (I started with the diagonal rather than a side.) Conjecture that the entire construction remains the same even when rotated $\frac14$ of a full turn (90 degrees) around the center. This means that each side can be rotated onto the other sides, and each angle can be rotated onto the other angles. Lesson Synthesis Remind students that they have now constructed an equilateral triangle, a regular hexagon, and a square, each inscribed in a circle. Each of these is an example of a regular polygon, which is a polygon with all congruent sides and all congruent angles. Ask students, “Starting with any of these shapes, which construction techniques would help you make other regular polygons inscribed in circles?” (Starting from any of them, we can make twice as many sides by bisecting the angles and marking the points where the angle bisectors intersect with the circle. We could repeat this 7.4: Cool-down - Build a House (5 minutes) Student Facing We can use what we know about perpendicular lines and congruent segments to construct many different objects. A square is made up of 4 congruent segments that create 4 right angles. A square is an example of a regular polygon since it is equilateral (all the sides are congruent) and equiangular (all the angles are congruent). Here are some regular polygons inscribed inside of circles:	{"url":"https://curriculum.illustrativemathematics.org/HS/teachers/2/1/7/index.html","timestamp":"2024-11-02T21:03:01Z","content_type":"text/html","content_length":"143571","record_id":"<urn:uuid:cf5cfee9-96e0-4739-8b76-fbf067c85466>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00250.warc.gz"}
What is Motion Physics? Motion refers to a change in location. Physics is the scientific study of matter, energy, and the interactions between them. It includes fields such as acoustics, cryogenics, electromagnetism, optics, and mechanics, which refers to the study of how force act on matter or material systems and includes motion physics, the scientific study of movement. In motion physics, it is typical to speak of the change in location of a “body.” Applied force is the initiator of a change in motion, which can mean starting motion, stopping motion, or changing direction. Without an applied force, bodies tend to resist acceleration and to stay at rest if they are at rest or, when moving in a straight line, to continue moving in a straight line. Motion is usually described using several typical parameters, including velocity. Velocity is the rate of change in a body’s position, so it refers to both speed — distance covered in a certain amount of time — and direction, and so it is a vector. Despite this, it is often represented by the equation v = d/t, where v represents velocity, d represents distance, and t represents time. It is usually reported in meters per second. The second parameter is acceleration, which is the change in velocity over time. Like velocity, acceleration is a vector. It is caused by a force applied to the body. The greater the mass of the body, the more force must be applied to cause a certain amount of acceleration. This relationship is expressed by the equation F = ma where F represents force, m represents mass, and a represents acceleration. The directional aspect of the force is also important. Acting in the same direction as the original velocity of a body, force will only change speed and not direction. Acting in the opposite direction of the original velocity, the speed will be decreased, rather than increased. Momentum is another term frequently used in motion physics and, like velocity, it is a vector. As defined in classical mechanics, momentum is the product of the velocity of an object and its mass. It is expressed by the equation p = mv where p represents momentum, m represents mass, and v represents velocity. The directionality of momentum is the same as the directionality of velocity, and the change in momentum when a force is applied is related to both the amount of force and the length of time for which it is applied.	{"url":"https://www.allthescience.org/what-is-motion-physics.htm","timestamp":"2024-11-12T00:42:15Z","content_type":"text/html","content_length":"114615","record_id":"<urn:uuid:203efb08-b03a-478c-8609-a86a75eea3c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00821.warc.gz"}
What Are Radicals In Math? A radical, or root, is the mathematical opposite of an exponent, in the same sense that addition is the opposite of subtraction. The smallest radical is the square root, represented with the symbol √. The next radical is the cube root, represented by the symbol ³√. The small number in front of the radical is its index number. The index number can be any whole number and it also represents the exponent that could be used to cancel out that radical. For example, raising to the power of 3 would cancel out a cube root. General Rules for Each Radical The result of a radical operation is positive if the number under the radical is positive. The result is negative if the number under the radical is negative and the index number is odd. A negative number under the radical with an even index number produces an irrational number. Remember that though it isn't shown, the index number of a square root is 2. Product and Quotient Rules To multiply or divide two radicals, the radicals must have the same index number. The product rule dictates that the multiplication of two radicals simply multiplies the values within and places the answer within the same type of radical, simplifying if possible. For example, $\sqrt[3]{2}× \sqrt[3]{4} = \sqrt[3]{8}$ which can be simplified to 2. This rule can also work in reverse, splitting a larger radical into two smaller radical multiples. The quotient rule states that one radical divided by another is the same as dividing the numbers and placing them under the same radical symbol. For example, $\frac{\sqrt{4}}{\sqrt{8}} = \sqrt{\frac{4}{8}} = \sqrt{\frac{1}{2}}$ Just like the product rule, you can also reverse the quotient rule to split a fraction under a radical into two individual radicals. TL;DR (Too Long; Didn't Read) Here's an important tip for simplifying square roots and other even roots: When the index number is even, the numbers inside the radicals can't be negative. In any situation, the denominator of the fraction can't equal out to 0. Simplifying Square Roots and Other Radicals Some radicals solve easily as the number inside solves to a whole number, such as √16 = 4. But most won't simplify as cleanly. The product rule can be used in reverse to simplify trickier radicals. For example, √27 also equals √9 × √3. Since √9 = 3, this problem can be simplified to 3√3. This can be done even when a variable is under the radical, though the variable has to remain under the Rational fractions can be solved similarly using the quotient rule. For example, $\sqrt{\frac{5}{49}} = \frac{\sqrt{5}}{\sqrt{49}}$ Since √49 = 7, the fraction can be simplified to √5 ÷ 7. Exponents, Radicals and Simplifying Square Roots Radicals can be eliminated from equations using the exponent version of the index number. For example, in the equation √x = 4, the radical is canceled out by raising both sides to the second power: $(\sqrt{x})^2 = (4)^2\text{ or } x = 16$ The inverse exponent of the index number is equivalent to the radical itself. For example, √9 is the same as 9^1/2. Writing the radical in this manner may come in handy when working with an equation that has a large number of exponents. Cite This Article Williams, Grace. "What Are Radicals In Math?" sciencing.com, https://www.sciencing.com/radicals-math-8565068/. 22 December 2020. Williams, Grace. (2020, December 22). What Are Radicals In Math?. sciencing.com. Retrieved from https://www.sciencing.com/radicals-math-8565068/ Williams, Grace. What Are Radicals In Math? last modified March 24, 2022. https://www.sciencing.com/radicals-math-8565068/	{"url":"https://www.sciencing.com:443/radicals-math-8565068/","timestamp":"2024-11-15T04:35:42Z","content_type":"application/xhtml+xml","content_length":"74339","record_id":"<urn:uuid:07b638cf-a38a-4a48-909c-dbe01b8052b3>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00611.warc.gz"}
Re: Doomsday Example Robin Hanson (hanson@econ.berkeley.edu) Mon, 24 Aug 1998 13:00:28 -0700 Nick Bostrom writes: >Suppose we change the example slightly. Now, an A universe contains >10 humans and nothing else. A B universe contains 10 humans and one >trillion trillion stones. There is a small C universe (of negligible >size), and it spawns one thousand baby-univeses through a random >process (like rolling fair dice) that has a 10% chance of yielding an >A and 90% of a B. >It seems clear that there will probably be about 100 A univeses and >900 B universes. But that means about 90% of all humans will find >themselves in an B universe. So if all you knew was the set-up and >that you were a human, then you should believe that there was a >90% chance that you were in an B universe. >And yet, if you include stones in the reference class, then it seems >you should believe you are in an A universe. For if you are in a B >universe, then there exists in total at least one trillion trillion >stones and then you would probably have been one of these stones >rather than a human. Let N = "trillion trillion", and assume there are exactly 100 A "universes" and 900 B "universes". Note that these are not "universes" in the sense I was using of "possible worlds." The entire construction of C + 100 As + 900 Bs is just one total space-time, a "possible world." As described the only remaining uncertainty is where in this world I am. If I treat stone slots and human slots equally, there are 10010 + 900(10+N) slots. If my prior is uniform across these slots, then conditioning on my being in a human slot, there is a 90% chance I'm in a B "universe," as you prefer. >> >Similarly, I would say that finding that you are an observer does not >> >give you reason for thinking that a large fraction of all slots in >> >the universe is occupied by observers. >> Your error is to say "large fraction." The fact that life exists >> on Earth does* make it more likely that our universe has other planets >> where life has evolved in a similar time. It just isn't enough to >> conclude the fraction of Earth-like planets with life is "large." >And now you say that finding that life exists on Earth should >not affect your beliefs about about the fraction of Earth-like >planets with life. But I said exactly the opposite in the quote above! [More responses in next message. RH] Robin Hanson hanson@econ.berkeley.edu http://hanson.berkeley.edu/ RWJF Health Policy Scholar, Sch. of Public Health 510-643-1884 140 Warren Hall, UC Berkeley, CA 94720-7360 FAX: 510-643-8614	{"url":"http://extropians.weidai.com/extropians.3Q98/1816.html","timestamp":"2024-11-10T04:54:03Z","content_type":"text/html","content_length":"5425","record_id":"<urn:uuid:de8473c0-37c6-4ea3-bf89-b8cc8470a635>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00361.warc.gz"}
A constant factor approximation algorithm for reordering buffer management In the reordering buffer management problem (RBM) a sequence of n colored items enters a buffer with limited capacity k. When the buffer is full, one item is removed to the output sequence, making room for the next input item. This step is repeated until the input sequence is exhausted and the buffer is empty. The objective is to find a sequence of removals that minimizes the total number of color changes in the output sequence. The problem formalizes numerous applications in computer and production systems, and is known to be NP-hard. We give the first constant factor approximation guarantee for RBM. Our algorithm is based on an intricate "rounding" of the solution to an LP relaxation for RBM, so it also establishes a constant upper bound on the integrality gap of this relaxation. Our results improve upon the best previous bound of O(√log k) of Adamaszek et al. (STOC 2011) that used different methods and gave an online algorithm. Our constant factor approximation beats the super-constant lower bounds on the competitive ratio given by Adamaszek et al. This is the first demonstration of a polynomial time offline algorithm for RBM that is provably better than any online algorithm. Publication series Name Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms Conference 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013 Country/Territory United States City New Orleans, LA Period 6/01/13 → 8/01/13 Dive into the research topics of 'A constant factor approximation algorithm for reordering buffer management'. Together they form a unique fingerprint.	{"url":"https://cris.huji.ac.il/en/publications/a-constant-factor-approximation-algorithm-for-reordering-buffer-m","timestamp":"2024-11-10T14:13:32Z","content_type":"text/html","content_length":"51071","record_id":"<urn:uuid:011e7747-7bf3-46e8-a03d-08b0b8a656a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00466.warc.gz"}
Data Science & Machine Learning with Python ~ London Institute of Business and Management Kieran Graham Very professional courses. Great Administration assistance and high quality e-learning service. Sarah Jennings I did forex trading diploma. Very professional and detailed course. Jordan Cooke The course offered is excellent. I am glad to have taken it.	{"url":"https://www.libm.co.uk/course/data-science-machine-learning-with-python/","timestamp":"2024-11-07T23:07:29Z","content_type":"text/html","content_length":"242400","record_id":"<urn:uuid:19b546d2-ad29-497b-9b78-f448572214ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00607.warc.gz"}
GeoGebra - the world’s favorite, free math tools used by over 100 million students and teachers GeoGebra tools and resources Teach and learn math in a smarter way GeoGebra is more than a set of free tools to do math. It’s a platform to connect enthusiastic teachers and students and offer them a new way to explore and learn about math. Recommended math resources: GeoGebra-curated for Grades 4 to 8 Math Resources Our newest collection of GeoGebra Math Resources has been meticulously crafted by our expert team for Grades 4 to 8. Explore all Math Calculators & Apps Free tools for an interactive learning and exam experience. Available on all platforms. Explore all Classroom Collaboration Interactive math lessons with GeoGebra materials available. Integration with various LMSs supported. Learn more Math Practice Get support in step-by-step math exercises, explore different solution paths and build confidence in solving algebraic problems. Try out Trusted by millions of teachers and students worldwide GeoGebra allowed us to provide access and equity to our diverse student population. Irina Kimyagarov Teacher, New York, US GeoGebra has changed the way I teach. Maria Meyer Teacher, New Mexico, US GeoGebra brings mathematics to life! Steven C. Silvestri Teacher, New York, US I appreciate the responsiveness of the GeoGebra staff and the larger community. Nancy Swarat Teacher / OCTM, Oregon, US My students LOVE using GeoGebra. Peter Myers Teacher, New York City, US Teachers & Students Make math interactive Try our exploration activities to discover important math concepts, then use our practice activities to master these skills. Explore Resources Teachers & Students Explore math with free calculators Explore our easy-to-use calculators that can be used to promote student-centered discovery-based learning. Perform calculations for any level of mathematics including 3D. Start Calculator Engage every student Our Classroom learning platform allows teachers to view student progress real-time and provide individual feedback for personalized learning experience. It helps teachers to encourage active participation and discussion. Explore Classroom Teachers & Students Solve problems step-by-step Our Math Practice tool offers new ways for learners to access algebraic transformation in an understandable way. Let your students build comfort and fluency in solving algebraic problems, like simplifying algebraic expressions or solving linear equations, while getting instant hints and feedback. Explore Math Practice Our mission Our mission is to give the best tools for teachers to empower their students to unleash their greatest potential. We go beyond being just a collection of tools. Striving to connect passionate individuals from the education world, we offer a fresh approach to teaching, exploring, and learning mathematics. About us Get started using GeoGebra today Create a free account so you can save your progress any time and access thousands of math resources for you to customize and share with others	{"url":"https://www.geogebra.org/?lang=bg","timestamp":"2024-11-08T01:21:30Z","content_type":"text/html","content_length":"180659","record_id":"<urn:uuid:5ae55694-35e2-4ed7-af7d-94e2a67d8833>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00429.warc.gz"}
Scripting API A plane is an infinitely large, flat surface that exists in 3D space and divides the space into two halves known as half-spaces. It is easy to determine which of the two half-spaces a particular point is in and also how far the point is from the plane. Walls, floors and other flat surfaces are common in games, so a plane is sometimes useful for mathematical calculations with these objects. Also, there are cases where a real surface does not exist but it is useful to imagine that one is there. For example, in sports, a goal line or out-of-bounds line is often assumed to extend into the air, effectively defining a plane. When a plane passes through the <0,0,0> point in world space, it is defined simply by a normal vector that determines which way it faces. It is easy to visualise this if you imagine looking at the plane edge-on. Note that the side from which the normal vector points is important since it is used to identify which half-space a point is in (ie, on the positive or "normal" side of the plane or the other side). When the plane doesn't pass through <0,0,0> it can be defined by the normal vector along with a distance from <0,0,0> A plane can also be defined by the three corner points of a triangle that lies within the plane. In this case, the normal vector points toward you if the corner points go around clockwise as you look at the triangle face-on.	{"url":"https://docs.unity3d.com/2017.1/Documentation/ScriptReference/Plane.html","timestamp":"2024-11-11T17:22:20Z","content_type":"text/html","content_length":"18226","record_id":"<urn:uuid:b457228c-a080-47da-8943-ffd5fa262297>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00608.warc.gz"}
Fredi - friendly frontend editing Hey, just tested this with previous version and seen you updated. So I grabed it and I can't see a difference to before. When on default language it open modal with body and both languages to edit fine. When on alternative language of the page, the modal just loads the home page inside. It would be nice to have the modal bigger. My Images Manager or insert image modal does open inside it and it's quite claustrophobic. When on alternative language of the page, the modal just loads the home page inside. Hmm.. I have only tested this with when user language is something else than default. How is your languages setup? It would be nice to have the modal bigger. My Images Manager or insert image modal does open inside it and it's quite claustrophobic. I agree. I used first native-js modal solution I could find and tinybox2 doesn't support relative widths with iFrame. It seems to work very fine otherwise, so I am looking modifying it for Fredi's My language setup is standard languages module, language fields and language page names. So the user language is the one you currently view on the front end. New language page names module does all the url and user language. This is the default link this is when on alternative lang: Ok, will test with language page names, that is the one that is missing from my setup. I remember when there was or maybe still is a bug with when languages installed that when installing a module admin page it doesn't set the title for alternative languages just default, depending on user language who installs the module. So I just went a looked at the process admin page... Turns out the "name" of the page is correct for both languages, BUT the "active" checkbox isn't set. This is from the new Language Page Names module. Setting this checkbox active for the laternative language it starts working and the edit url is correct. This is a tricky one that will may cause some trouble in future with module developement that use techniques like this... I can already see. Yeah, it seems to work for me with language page names. But I don't see that active checkbox? Where does it comes from? Its in since a week or two. Must be on dev, since I am on latest master. Thanks for this. Simply awesome! • 1 Hi mister antii This is a killer-feature-module, thanks very much! Frontend-Editing the Pw way: I can define which fields from which page I want to edit and output the edit-link where I want. Now that's freedom! • 2 Thanks guys. Wanze: when I got the idea of how this should work I had kind of light bulb moment. I have been thinking about front end editing in PW about two years and I think now I have nailed the right • 2 Hi Antti, This does indeed work with Thumbnails. I can see all sorts of applications for this. Terrific work! • 1 I am currently building own responsive js modal (without any library dependencies), so soon we can have bigger edit view. • 1 That would help with cropping too • 1 Wow, this is actually really nice! • 1 I implemented custom modal for this. Seems to work great on IE10, Chrome, FF, iPad, iPhone... but having little issue with chrome on android. I just added this to module directory, pushed documentation to github and consider this pretty much released. • 4 Anyone else have their submit button end up in weird spot? If I float the element one way or another, it's fine, but since it's within an iframe I can't just tack the CSS on. Many of my fields are 25% or 50% width, which might have something to do with it. Using the latest OS X Chrome, seems to do it in OS X Firefox 19.0.2 as well. I also had to add a Z-index to the modal dialog, as it ended up under some of my own DIVs. Maybe that should be added to the release? This'll be incredibly useful, thanks! It looks ok for me here on OSX FF20. Actually when I try this in Chrome 25 I just get the loading spinner. Marty, that is strange. Works fine here (chrome 26), although I have same problem with Android Chrome. Mind testing again when you have updated to 26? Evan, good idea! Works now with the Chrome 26 update Works now with the Chrome 26 update That is good news (hopefully will be soon fixed on android chrome too). What happened to the screencast? What happened to the screencast? It is little outdated (just visuals, code is right), so I removed it. But I added it back to first post until I record new one. Just noticed that there is nice typing sounds included • 1 I implemented custom modal for this. Seems to work great on IE10, Chrome, FF, iPad, iPhone... but having little issue with chrome on android. Does this mean that I can use a custom css for the modal box to reflect the style of my front-end, even for the fileds that I want to let the user modify?	{"url":"https://processwire.com/talk/topic/3265-fredi-friendly-frontend-editing/page/2/","timestamp":"2024-11-03T00:02:53Z","content_type":"text/html","content_length":"405233","record_id":"<urn:uuid:09ae23f2-6dc7-4ba6-ab86-43e08edb3a7b>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00846.warc.gz"}
Value Expectation How often do u rely on Value Expectation during your game? ...or maybe a better question would be, how important is it to be able to calculate expected value at any given time? Like, do I need to spend some serious time brushing up on my fraction conversions and multiplication so I can I can whip out a value analysis on the fly at any time? You may have guessed that I'm not really all that strong a arithmetician (think Ed Grimly). I absolutely will take the time to figure it out, study it and make it my friend if I need to. But, if I don't need to, then I'd rather spend time studying other "stuff". I've been playing online for a few months now - micro tables only. I do ok, but want to do much much better. I'll do whatever it takes to get better, but I just want to make sure I'm not waisting time. I feel like there is so much to learn. When I read some of the articles under "Strategy Articles", I notice many of them warn readers to learn/understand value expectation before moving onto pot etc. So, I guess I could heed those warnings and just accept that I need to become very familiar with value expectation. Buuuuut then again, maybe the author was just joking? Can I help you? May 18, 2005 Total posts I don't think the author was joking. pot odds /implied odds is very important. I'm no mathmetician either, but having an elementary grasp on these concepts can really help you make the right decisions. Check out these articles if you haven't Five Fundamentals of Poker - Poker Theory Poker: Expected Value Poker: Pot Odds & Implied Odds Poker: Position Poker: Pot Size Poker: Equity Hey juiceeQ, thanks for replying to my post. Okay, so it sounds like as long as I have somewhat of a grasp on these concepts I should be OK. I understand and can calculate pot odds/implied odds, but for some reason when I read the article on the Expected Value I go cross eyed. I'll read and re read until it becomes more familiar. Thanks again for taking the time to reply Can I help you? May 18, 2005 Total posts No problem. And I'll tell you a little secret...I get a little cross-eyed too...shhh! Best of luck to you, and I hope you're enjoying the site so far. hahaha...Thanks, I'll keep that one between u and me Silver Level Oct 15, 2007 Total posts No problem. And I'll tell you a little secret...I get a little cross-eyed too...shhh! Best of luck to you, and I hope you're enjoying the site so far. hahaha...Thanks, I'll keep that one between u and me Too late. I caught you! Let me try to explain what it really is. Expected Value is really a big name for "Average Value". Say in a cash game you get involved in races where you have 50% chance to win $100 and 50% chance to lose $100. How much is your average win? Zero! The fancy name is that the Expected Value = 0. Now let's say that in that game, everytime you race you have a bigger pocket pair than your opponent. That means that you will win 80% of the time and your opponent will win 20% of the time. Again, you are both all-in for $100 each. In this case, we cannot take a simple average because we win 4 times as more often than our opponent. In the first example, when our chances to win and lose were the same (coin-flip), our Expected Value was zero. Now, it is more than zero. If we play 5 times, we win on average 4 times $100 and we lose 1 time $100. So after 5 times on average we will have $400 - $100 = $300. If we play 10 times, on average we make $600. So how much do we make on average EACH time we play? If we make $600 for ten times, then we make $60 each time. That is called Expected Value which is just how much we make or lose on average. The formula for Expected Value (EV) is: EV = (% to win)$won - (% to lose)$lost In the coin flip example, we win $100 50% of the time and lose it 50% of the time, so: EV(for example 1) = 50% * $100 - 50% * $100 = $0 In the second example, our pair against smaller pair: % to win = 80% % to lose = 20% $won = $100 $lost = $100 EV(for example 2) = 80% * $100 - 20% * $100 = $80 - $20 = $60 Now, is there any relation between expected value and odds? YES! In the first example, our odds were 50% to win and 50% to lose so the odds were 1:1. In the second example, we were 80% to win and 20% to lose which gives us 4 to 1 odds. So what are odds (to win or lose)? Odds(to win) = (% to win)/(% to lose) Odds(to lose) = (% to lose)/(% to win) In the examples above, odds to win are: Odds(to win example 1) = 50%/50% = 1:1 Odds(to win example 2) = 80%/20% = 4:1 So Odds and Expected Value are different ways to talk about the same thing. If you have odds when calling, then your EV is positive, and vice versa, if you don't have odds, your EV is negative. Sometimes it is more convenient to think in terms of odds. Other times it is more convenient to think in terms of EV. Let me give a final example to illustrate these 2 concepts: Let's say that in a cash game you have a flush draw on the flop and someone moves all-in and they have you covered. You want to determine whether you have odds to call. That is the same thing as determining whether the EV of calling is positive. Thinking in terms of odds here is easier. The attached odds table converts outs to required odds for 1 card to come (Odds 1) and 2 cards to come(Odds 2). In our case, we have 9 outs to make our flush and 2 cards to come. The odds table us that the odds AGAINST us are 2:1. So in order to call profitably, the pot odds must be at least 2 to 1. (A) If the pot is $1,000 and we have to call $1,000, that's only 1:1 pot odds. That will lose us money because we stand to win and lose the same amount, but we will lose twice as often. Than means our EV is negative. That means we must fold. (B) If the pot is $2,000 and we have to call $1,000, that's 2:1 pot odds. We will now break even. This is the same as saying EV = 0. We can now either fold or call. In the long term, we don't make or lose any money. (C) If the pot is $3,000 and we have to call $1,000, that's 3:1 pot odds. That will now makes us money. Than means our EV is positive. We should definitly call in this situation. Now let's calculate the EV for the 3 cases above to illustrate that odds and EV are related. In all cases, the odds to win are 33% and the odds to lose are 67%. How do I know that? If I substitute in the formula for odds, we have Odds to lose = (% to lose)/(% to win) = 67%/33% = 2:1 The formula for conversion from odds to percentages is: % = odds/(1+odds) * 100% In our case, % to lose = 2/(1+2) * 100% = 2/3 * 100% = 67% Case A: ------- % to win = 33% % to lose = 67% $won = $1,000 $lost = $1,000 EV(case A) = 33% * $1,000 - 67% * $1,000 = $330 - $670 = -$340 We lose money, therefore FOLD! Case B: ------- % to win = 33% % to lose = 67% $won = $2,000 $lost = $1,000 EV(case B) = 33% * $2,000 - 67% * $1,000 = $660 - $670 = -$10 We are about zero, so FOLD or CALL is fine. FOLD is preferred. Case C: ------- % to win = 33% % to lose = 67% $won = $3,000 $lost = $1,000 EV(case C) = 33% * $3,000 - 67% * $1,000 = $990 - $670 = +$320 We are making money on average. CALL! To use the odds table I attached, all you need to do is multiply what you HAVE TO CALL with the odds from the table. That gives you the minimum required pot so that you make profit (or fancier said, for you to have positive EV, or +EV). For example, say that you have an inside str8 and flush draws on the turn. Someone moves all-in, you are last to act and have to determine whether a call is +EV or not. Your draw has 12 outs, 9 for flush, 4 for str8, but 1 of the cards is counted for both, so the total is 12 outs. There is one card to come. So you look in the table and see that the odds against you are 3:1. If you have to call $25, for example, then you multiply $25 by 3 which is $75. That is the MINIMUM size of the pot that will make a call +EV. If the pot is only $60 and you call, your call is -$EV which means you lose money in the long run. If the pot is $85, for example, you must call since you have odds (which means you have +EV). I hope this clarified these concepts more than it confused them. Last edited: Actually, my eyes are starting to uncross...it helps to relate it to odds which is something I am familiar with. Although I started getting confused again near the end when you refer to the odds Gonna have a shower and come back to it in a bit	{"url":"https://www.cardschat.com/forum/cash-games-11/value-expectation-101442/","timestamp":"2024-11-05T16:11:46Z","content_type":"text/html","content_length":"144811","record_id":"<urn:uuid:ce3f43cb-2096-483a-b98f-6d9668441d40>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00063.warc.gz"}
Exact significance tests for 2 × 2 tables Two-by-two contingency tables look so simple that you’d be forgiven for thinking they’re straightforward to analyse. A glance at the statistical literature on the analysis of contingency tables, however, reveals a plethora of techniques and controversies surrounding them that will quickly disabuse you of this notion (see, for instance, Fagerland et al. 2017). In this blog post, I discuss a handful of different study designs that give rise to two-by-two tables and present a few exact significance tests that can be applied to these tables. A more exhaustive overview can be found in Fagerland et al. (2017). Two-by-two contingency tables Two-by-two contingency tables arise when cross-tabulate obserations that have two binary properties ($X$: $x_1$ vs. $x_2$; $Y$: $y_1$ vs. $y_2$). The number of obervations for which $X = x_1$ and $Y = y_1$ is referred to as $n_{11}$, and similarly for the other combinations of $X$ and $Y$ values. We further define the row totals $n_{1+} := n_{11} + n_{12}$ and $n_{2+} := n_{21} + n_{22}$ as well as the column totals $n_{+1} := n_{11} + n_{21}$ and $n_{+2} := n_{12} + n_{22}$. We call the total number of observations $n_{++}$. \[\begin{array}{l\|c\|c\|\|c} & Y = y_1 & Y = y_2 & \textrm{Row total} \\ \hline X = x_1 & n_{11} & n_{12} & n_{1+} \\ X = x_2 & n_{21} & n_{22} & n_{2+} \\ \hline \textrm{Column total} & n_{+1} & n_{+2} & n_{++} \\ \end{array}\] Exact and approximate tests This blog post is about exact significance tests. An exact test has the following defining property: If the null hypothesis is true, the $p$-value that the test yields is a random variable $P$ with the property that \[\mathbb{P}(P \leq \alpha) \leq \alpha\] for each $\alpha \in (0, 1)$. This means that, if the null hypothesis is true, there is at most a 5% probability that the test will yield a $p$-value of 0.05 or below. Similarly, if the null hypothesis is true, there is at most a 20% probability that the test will yield a $p$-value of 0.20 or below, and so for all values between 0 and 1. Ideally, $\mathbb{P}(P \leq \alpha)$ should be close to, but no larger than, $\alpha$: If $\mathbb{P}(P \leq \alpha)$ tends to be considerably lower than $\alpha$, i.e., if the test is conservative, this will negatively affect the test’s power. Tests that aren’t exact can still be approximate. A possible problem with approximate tests is that their justification depends on results derived for large samples; for smaller samples, $\mathbb{P} (P \leq \alpha)$ may be substantially larger than $\alpha$. The best-known approximate test used for the analysis of two-by-two contingency tables is Pearson’s $\chi^2$-test. Contingency tables with both marginals fixed Two-by-two contingency tables can be the result of different research designs – some fairly common, others exceedingly rare. Example 1 (Fisher’s exact test, one-sided). Say we want to establish if a learner of English is able to tell the /æ/ phoneme in bat and the /ɛ/ phoneme in bet apart. To this end, we make 35 recordings, 21 of which contain the word bet and 14 contain the word bat. The learner is then asked to identify those 21 audio files that he thinks are recordings of bet; the remaining 14 audio files are suspected recordings of bat. The results are summarised in the following contingency table: \[\begin{array}{l\|cc} & \textrm{Categorised as \textit{bet}} & \textrm{Categorised as \textit{bat}} \\ \hline \textrm{Is \textit{bet}} & 15 & 6 \\ \textrm{Is \textit{bat}} & 6 & 8 \\ \end{array}\] Note that we insisted that the learner select exactly 21 suspected recordings of bet, no more and no fewer. As a result, the column total $n_{+1}$ was known in advance. Moreover, we knew beforehand that 21 of the recordings actually featured bet and 14 of the recordings actually featured bat. Hence, the row totals $n_{1+}, n_{2+}$ were also known in advance. As a result, $n_{++}$ and $n_ {+2}$ were also known in advance. Consequently, this study design fixes both the row and column marginals. What wasn’t known in advance was the number of suspected bet recordings that actually were bet recordings ($n_{11}$). The null hypothesis in this setting is that the learner is incapable of distinguishing bet from bat recordings and just selected 21 random audio files as suspected bet recordings in order to comply with the instructions. Under this null hypothesis, the top left entry in the contingency table ($n_{11}$) follows a hypergeometric distribution with parameters $n_{1+} = 21, n_{2+} = 14$ and $n_ {+1} = 21$. (Different authors parametrise this distribution differently; I use the parametrisation that’s used in R.) That is, \[H_0 : n_{11} \sim \textrm{Hypergeometric}(\underbrace{21}_{\textrm{is bet}}, \underbrace{14}_{\textrm{is bat}}, \underbrace{21}_{\textrm{to be categorised as bet}}).\] Figure 1 shows the probability mass and cumulative probability functions of the $\textrm{Hypergeometric}(21, 14, 21)$ distribution. The observed top-left entry, i.e., 15, is highlighted in blue. Figure 1: The probability mass function (left) and the cumulative probability function (right) of the $\textrm{Hypergeometric}(21, 14, 21)$ distribution. If the learner did not just pick 21 audio files at random but was in fact able to tell bet and bat recordings apart to some degree, this top-left entry can be expected to be large as opposed to small. This means that we want to compute a right-sided $p$-value, which we do by calculating \[\mathbb{P}(n_{11} \geq 15) = 1 - \mathbb{P}(n_{11} \leq 14):\] This computation amounts to running Fisher’s exact test: Example 2 (Fisher’s exact test, two-sided). Let’s slightly change the design of the study in Example 1. Instead of recording bet 21 times and bat 14 times and asking the learner to select 21 suspected bet recordings, we record both bet and bat 18 times and ask the learner to select 18 suspected bet recordings. The results are summarised in the following contingency table: \[\begin{array}{l\|cc} & \textrm{Categorised as \textit{bet}} & \textrm{Categorised as \textit{bat}} \\ \hline \textrm{Is \textit{bet}} & 5 & 13 \\ \textrm{Is \textit{bat}} & 13 & 5 \\ \end{array}\] Under the null hypothesis that the learner possesses no relevant discriminatory ability, the top-left entry ($n_{11}$) follows a $\textrm{Hypergeometric}(18, 18, 18)$ distribution; see Figure 2. The observed top-left entry (5) is highlighted by the dashed blue line. Figure 2: The probability mass function of the $\textrm{Hypergeometric}(18, 18, 18)$ distribution. Of note, the learner seems to be able to tell bet and bat apart to some extent – it’s just that he seems to identify bet recordings as bat and vice versa. Since we’re interested in the learner’s discriminatory ability, regardless of whether he is then also able to correctly label the two categories, we want to compute a two-sided $p$-value. The most common way to do so is to sum the probability masses $\mathbb{P}(n_{11} = k), k = 1, \dots, n_{+1},$ that are no greater than the probability mass of the actually observed top-left entry. These are the probability masses coloured blue in Figure 2. Fisher’s exact test carries out the same computation. Contingency tables with one marginal fixed Contingency tables in which both the row and column marginals are fixed in advance are a rare sight. More common in some areas of research are contingency tables where only the row marginals are fixed by design. Such tables can be found in, for instance, experimental research in which a fixed number of participants are assigned to one experimental condition and another fixed number of participants are assigned to the other experimental condition and where for each participant, we have a single binary outcome (e.g., died vs. survived, or passed vs. failed). Example 3 (Boschloo’s test). Let’s imagine we’re in charge of an agency that designs self-study courses to help students prepare for an entrance exam. We’ve developed a course whose efficacy we want to compare against that of its predecessor. More specifically, we’re interested in finding out whether the new course is better than the old one in terms of helping the students pass the entrance exam. We recruit 24 students willing to participate in an evaluation study and, using complete randomisation, we assign 12 of them to work with the new course and 12 to work with the old one. The results look as follows: \[\begin{array}{l\|cc} & \textrm{Pass} & \textrm{Fail} \\ \hline \textrm{New course} & 10 & 2 \\ \textrm{Old course} & 6 & 6 \\ \end{array}\] In contrast to the previous two examples, it’s only the row marginals that were known beforehand in this example. Nevertheless, applying Fisher’s exact test to this contingency table is reasonable. In doing so, we would be conditioning the analysis on the observed column marginals ($n_{+1} = 16$, $n_{+2} = 8$), even though the study design did not fix these marginals. The corresponding null hypothesis is \[H_0 \| n_{+1} = 16, n_{+2} = 8~:~ n_{11} \sim \textrm{Hypergeometric}(12, 12, 16).\] An exact test remains exact when it is used conditionally (Lydersen 2009:1165), so the resulting $p$-value is perfectly valid: That said, conditional exact tests tend to be pretty conservative. Intuitively, the reason is that the conditional exact test only considers $\min(n_{+1}, n_{1+}) + 1$ of the possible tables that could have been observed under the null hypothesis; in this example, these would be the thirteen different tables where $n_{1+} = n_{2+} = 12$ and $n_{+1} = 16$, i.e., with $n_{11} = 0, 1, \ dots, 12$. The actual sample space under the null hypothesis in this design, however, comprises $(n_{1+} + 1) \cdot (n_{2+} + 1)$ tables. In this example, these would be the 169 different tables where $n_{1+} = n_{2+} = 12$. By only considering a small part of the sample space, the conditional exact test in essence misses out on opportunities to return small $p$-values. Unconditional exact tests consider the whole sample space and are consequently less conservative than conditional exact tests. The contingency table in the fixed row marginals design can be considered the result of two draws from binomial distributions: one with $n_{1+}$ attempts and success probability $\pi_1$, and one with $n_{2+}$ attempts and success probability $\pi_2$. The null hypothesis is that $\pi_1 = \pi_2$. That is, \[\begin{align} H_0~:~ & n_{11} \sim \textrm{Binomial}(n_{1+}, \pi), \\ & n_{21} \sim \textrm{Binomial}(n_{2+}, \pi), \end{align}\] with some $\ pi \in [0,1]$ that is common to both binomial distributions. This $\pi$ parameter is known as a nuisance parameter: its value is unknown and not of primary interest, but if we want to carry out an unconditional test, we need to somehow take it into account. An unconditional exact test for the fixed row marginals design that is often recommended is Boschloo’s test (Boschloo 1970). The idea behind this test is as follows. First, we define a test statistic that captures the extent to which observed contingency tables differ from the contingency table you’d expect to find under the null hypothesis, given $n_{1+}$, $n_{2+}$ and some candidate value for $\pi$. Second, for each table that we could have observed, we compute both how likely it was to occur (this depends on $n_{1+}$, $n_{2+}$ and $\pi$) and the test statistic it would have resulted in. Third, using the results of these calculations, we compute the probability that we would observe a test statistic at least as extreme as the test statistic that we actually did observe under the null hypothesis and for the given $n_{1+}, n_{2+}$ and $\pi$ values. This probability is a $p$-value conditional on the $\pi$ value considered. We repeat this procedure for different $\pi$ values – say, for 101 equally spaced $\pi$ values in the interval $[0, 1]$. Our unconditional exact $p$-value is now the maximum of the 101 values so computed. (Theoretically, it should be the supremum of the $p$-values when varying $\pi$ over the entire $[0,1]$ interval. But the maximum of the 101 $p$-values compute will be close enough to this supremum.) In Boschloo’s test, the test statistic used is in fact the one-sided $p$-value obtained from Fisher’s exact test. But this $p$-value is used as a test statistic, not as a $p$-value in its own Let’s walk through the computation step by step. First, we run a one-sided Fisher’s exact test in order to obtain the observed test statistic: Next, we create a grid with all possible combinations of $(n_{11}, n_{21})$ entries that we could have observed: We now fix some $\pi \in [0, 1]$, for instance, $\pi = 0.43$. For each possible table, we compute how likely it would have been to observe this table if $\pi = 0.43$. For instance, the 61st row in the grid corresponds to the table \[\begin{array}{l\|cc} & \textrm{Pass} & \textrm{Fail} \\ \hline \textrm{New course} & 8 & 4 \\ \textrm{Old course} & 6 & 6 \\ \end{array}\] The probability of observing the first row is given by the probability mass of $k = 8$ under a Binomial(12, 0.43) distribution; the probability of observing the second row is given by the probability mass of $k = 6$, also under a Binomial(12, 0.43) distribution. So the probability of observing this table is the product of these two probabilities: We compute this probability for all 169 tables: For each table, we also compute the test statistic: We can now compute the probability that we’d observe a test statistic at least as extreme as the test statistic associated with the table we actually observed, assuming the null hypothesis is true and $\pi = 0.43$. Since the $p$-value produced by Fisher’s exact test is smaller for tables that are more surprising under the null hypothesis, ‘at least as extreme’ corresponds to ‘at most as large’. We can compute the figure we’re interested in by computing the proportion of tables resulting in test statistics no larger than the one we observed but weighted by their probability of occurring under the assumption that $\pi = 0.43$: Assuming $\pi = 0.43$, the resulting $p$-value is hence $0.043$. We repeat this procedure for different candidate values of $\pi$. The $p$-value of Boschloo’s test is then the maximum of the resulting $p$-values. In this specific example, the maximum $p$-value is $p = 0.0495$, which is obtained for $\pi = 0.68$; see Figure 3. Figure 3: In Boschloo’s test, a $p$-value is computed for each candidate value of $\pi$. The test then outputs the maximum of these $p$-values. If our alternative hypothesis were that the new programme produced worse results than the old one, we’d have used the left-sided $p$-value of Fisher’s exact test as the test statistic in Boschloo’s procedure. A two-sided $p$-value can be obtained by carrying out Boschloo’s test once using the right-sided $p$-value of Fisher’s exact test as the test statistic and once using the left-sided \ (p\)-value, and then doubling the smaller of the two resulting $p$-values. To run Boschloo’s test, you can use the following boschloo_test() function: boschloo_test <- function(tab, alternative = "two.sided", pi_range = c(0, 1), stepsize = 0.01) { # This test assumes fixed row sums. # Nuisance parameter values in the interval pi_range are tried out. # stepsize governs granularity of search through nuisance parameter value candidates. if (!all(dim(tab) == c(2, 2))) stop("tab needs to be a 22 contingency table.") if (alternative == "two.sided") { # Truncate two-sided p-value at 1 min(2 min(boschloo_test(tab, alternative = "less", pi_range = pi_range, stepsize = stepsize), boschloo_test(tab, alternative = "greater", pi_range = pi_range, stepsize = stepsize)), # Use Fisher's exact test p-value as test statistic statistic <- function(x) fisher.test(x, alternative = alternative)$p.value # Construct grid with possible results row_sums <- rowSums(tab) my_grid <- expand.grid(n1 = 0:row_sums[1], n2 = 0:row_sums[2]) my_grid$statistic <- NA for (i in 1:nrow(my_grid)) { my_tab <- rbind(c(my_grid$n1[i], row_sums[1] - my_grid$n1[i]), c(my_grid$n2[i], row_sums[2] - my_grid$n2[i])) my_grid$statistic[i] <- statistic(my_tab) # Compute observed test statistic obs_p <- statistic(tab) is_extreme <- my_grid$statistic <= obs_p # Maximise p-value over range pis <- seq(pi_range[1], pi_range[2], by = stepsize) max_p <- 0 for (current_pi in pis) { current_p <- weighted.mean(x = is_extreme, w = dbinom(my_grid$n1, row_sums[1], current_pi) * dbinom(my_grid$n2, row_sums[2], current_pi)) if (current_p > max_p) max_p <- current_p It works like so: Alternatively, you can use the boschloo() function in the exact2x2 package. See ?boschloo for details on the parameters. Here I specify the number of grid points (nPgrid) in order to make the results agree exactly with those produced by boschloo_test(): Contingency tables with only the total sum fixed A third possibility is that neither the row sums ($n_{1+}, n_{2+}$) nor the column sums ($n_{+1}, n_{+2}$) are known beforehand but that the total number of observations ($n_{++}$) is. One general case where such contingency tables arise is in observational research in which two binary features are measured in a fixed number of randomly sampled units. A second general case is comprised of experiments that have a binary outcome variable and in which the participants are assigned to the conditions using simple randomisation so that the precise number of participants isn’t fixed in Observational studies Example 4 (Boschloo’s test). During a hike through the Fribourg Prealpes near Schwarzsee, we conduct a linguistic field experiment. Any time we encounter a hiking party chatting in French, we greet them in German; any time we encounter a hiking party chatting in German, we greet them in French. Afterwards, we jot down for each party whether the first person greeting us back did so in the same language in which they were addressed or in a different language. (Parties not chatting in French or German are ignored in this field experiment.) We planned to continue the field experiment until we’ve encountered the 20th French- or German-speaking hiking party. Here’s the resulting contingency table: \[\begin{array}{l\|cc} & \textrm{Same language} & \textrm{Different language} \\ \hline \textrm{French-speaking party} & 2 & 6 \\ \textrm{German-speaking party} & 8 & 4 \\ \end{array}\] Note that only the total number of observations ($n_{++} = 20$) was known beforehand. The null hypothesis is that whether the first greeter in the party responded in the same language or in a different language is independent of the language in which the party was chatting. More formally, let $\pi_{\textrm{row}}$ be the probability that a randomly encountered French- or German-speaking party is French-speaking and let $\pi_{\textrm{col}}$ be the probability that the first greeter in a French- or German-speaking party responds in the same language. Then our null hypothesis is \[\begin{align} H_0~:~&\mathbb{P}(\textrm{is French-speaking and responds in same language}) = \pi_{\textrm{row}}\pi_{\textrm{col}}, \\ &\mathbb{P}(\textrm{is French-speaking and responds in different language}) = \pi_{\textrm{row}}(1-\pi_{\textrm{col}}), \\ &\mathbb{P}(\textrm{is German-speaking and responds in same language}) = (1-\pi_{\textrm{row}})\pi_{\textrm{col}}, \\ &\mathbb{P}(\ textrm{is German-speaking and responds in different language}) = (1-\pi_{\textrm{row}})(1-\pi_{\textrm{col}}). \end{align}\] More compactly, our null hypothesis is that the tuple $(n_{11}, n_{12}, n_{21}, n_{22})$ is multinomially distributed with the probabilities above, i.e., \[H_0~:~(n_{11}, n_{12}, n_{21}, n_{22}) \sim \textrm{Multinomial}(20, \pi_{\textrm{row}}\pi_{\textrm{col}},\pi_{\textrm{row}}(1-\pi_{\textrm{col}}),(1-\pi_{\textrm{row}})\pi_{\textrm{col}}, (1-\pi_ {\textrm{row}})(1-\pi_{\textrm{col}})),\] for some unknown $\pi_{\textrm{row}}, \pi_{\textrm{col}} \in [0,1]$. In terms of the analysis, we could condition on both marginals or on one of them and run Fisher’s or Boschloo’s test, respectively: The resulting $p$-values are all valid. But like before, these tests only consider a part of the sample space: Fisher’s exact test only takes into account the nine tables where $n_{1+} = 8$ and \ (n_{+1} = 10\); Boschloo’s test considers the $9 \cdot 13 = 117$ tables where $n_{1+} = 8$ and $n_{2+} = 12$ when conditioning on the row marginals and the $11 \cdot 11 = 121$ tables where \ (n_{+1} = n_{+2} = 10\) when conditioning on the column marginals. But the table observed is in actual fact one of \[\sum_{i = 0}^{20}\sum_{j = 0}^{20-i}\sum_{k= 0}^{20-i-j}1 = 1771\] tables where \ (n_{++} = 20\). One possible solution is to generalise Boschloo’s test to two nuisance parameters. That is, rather computing a $p$-value for each candidate value of $\pi \in [0,1]$ and then taking the maximum, we compute a $p$-value for each pair of candidate values of $(\pi_{\textrm{row}}, \pi_{\textrm{col}}) \in [0,1] \times [0,1]$. The unconditional_test() function defined below carries out this unconditional_test <- function( alternative = "two.sided", pi_range_row = c(0, 1), pi_range_col = c(0, 1), stepsize = 0.01) # This test assumes a fixed total sum. # Nuisance parameter values in the rectangle pi_range_row * pi_range_col are tried out. # stepsize governs granularity of search through nuisance parameter value candidates. if (!all(dim(tab) == c(2, 2))) stop("tab needs to be a 22 contingency table.") if (alternative == "two.sided") { # Truncate two-sided p-value at 1 2 min(unconditional_test(tab, alternative = "less", stepsize = stepsize), unconditional_test(tab, alternative = "greater", stepsize = stepsize)), # Use Fisher's exact test p-value as test statistic statistic <- function(x) fisher.test(x, alternative = alternative)$p.value # Helper function for multinomial weights weights <- function(pi_row, pi_col, n11, n12, n21, n22) { total_sum <- n11 + n12 + n21 + n22 (pi_rowpi_col)^n11 (pi_row * (1 - pi_col))^n12 * ((1 - pi_row)pi_col)^n21 ((1 - pi_row)(1 - pi_col))^n22 factorial(total_sum)/(factorial(n11) * factorial(n12) * factorial(n21) * factorial(n22)) # Construct grid with possible results total_sum <- sum(tab) my_grid <- expand.grid(n11 = 0:total_sum, n12 = 0:total_sum, n21 = 0:total_sum) my_grid$n22 <- total_sum - my_grid$n11 - my_grid$n12 - my_grid$n21 my_grid <- subset(my_grid, n22 >= 0) my_grid$statistic <- NA for (i in 1:nrow(my_grid)) { my_tab <- rbind(c(my_grid$n11[i], my_grid$n12[i]), c(my_grid$n21[i], my_grid$n22[i])) my_grid$statistic[i] <- statistic(my_tab) my_grid$statistic[is.na(my_grid$statistic)] <- 1 # Compute observed test statistic obs_p <- statistic(tab) is_lower <- my_grid$statistic <= obs_p # Maximise p value over grid pis <- expand.grid( pi_row = seq(pi_range_row[1], pi_range_row[2], by = stepsize), pi_col = seq(pi_range_col[1], pi_range_col[2], by = stepsize) max_p <- 0 for (i in 1:nrow(pis)) { w <- weights(pis$pi_row[i], pis$pi_col[i], my_grid$n11, my_grid$n12, my_grid$n21, my_grid$n22) current_p <- weighted.mean(is_lower, w = w) if (current_p > max_p) max_p <- current_p While it takes noticeably longer to run this test, it should be a bit more powerful than the tests that condition on one or both marginals: The exact.test() function in the Exact also implements this procedure. For one-sided tests (i.e., alternative = "less" or "greater"), it produces the same $p$-values as unconditional_test(), but it computes the two-sided $p$-value differently: Experiments with simple randomisation Example 5 (Boschloo’s test). We conduct the same experiment as in Example 3, but with one change: We assign the participants to the conditions using simple randomisation rather than using complete randomisation. Hence, we’re not guaranteed to have exactly 12 participants in each condition, and the row marginals aren’t fixed in advance. For the sake of comparison, let’s assume we obtain the same results as in Example 3: \[\begin{array}{l\|cc} & \textrm{Pass} & \textrm{Fail} \\ \hline \textrm{New course} & 10 & 2 \\ \textrm{Old course} & 6 & 6 \\ \end{array}\] We could again condition on the row marginals: tab <- rbind(c(10, 2), c(6, 6)) boschloo_test(tab, alternative = "greater") # condition on row marginals Alternatively, we could run an unconditional test. Note, however that while $n_{1+}$ and $n_{2+}$ weren’t fixed by design, $\pi_{\textrm{row}}$ is known to be $0.5$. Hence, we don’t need to iterate through different candidate values for $\pi_{\textrm{row}}$: Contingency tables with nothing fixed Contingency tables where not even $n_{++}$ is fixed are mentioned by Fagerland et al. (2017) but not discussed in any detail. An analysis that conditions on the observed $n_{++}$ seems	{"url":"https://janhove.github.io/posts/2024-09-10-contingency-p-value/","timestamp":"2024-11-14T01:50:08Z","content_type":"application/xhtml+xml","content_length":"87370","record_id":"<urn:uuid:5294c7c7-6efd-4129-bb86-37b17c46a459>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00837.warc.gz"}
GATE & ESE - ME & CH - PYQ Series on Chemical Engineering Mathematics by Unacademy Unacademy is India’s largest online learning platform. Download our apps to start learning Starting your preparation? Call us and we will answer all your questions about learning on Unacademy Call +91 8585858585 About usShikshodayaCareersBlogsPrivacy PolicyTerms and Conditions Learner appEducator appParent app	{"url":"https://unacademy.com/course/pyq-series-on-chemical-engineering-mathematics/EG89CFZO","timestamp":"2024-11-12T23:56:44Z","content_type":"text/html","content_length":"291506","record_id":"<urn:uuid:06ed386c-108a-4310-a058-648f08d8eb35>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00149.warc.gz"}
mp_arc 02-149 02-149 I. Herbst, E. Skibsted Quantum scattering for potentials independent of \|x\|: Asymptotic completeness for high and low energies (647K, Postscript) Mar 26, 02 Abstract , Paper (src), View paper (auto. generated ps), Index of related papers Abstract. Let $V_1: S^{n-1} \rarrow \mathbb{R}$ be a Morse function and define $V_0(x) = V_1(x/\|x\|)$. We consider the scattering theory of the Hamiltonian $H = - \frac{1}{2} \Delta + V(x)$ in $L^ 2(\mathbb{R}^n)$, $n \geq 2$, where $V$ is a short-range perturbation of $V_0$. We introduce two types of wave operators for channels corresponding to local minima of $V_1$ and prove completeness of these wave operators in the appropriate energy ranges. Files: 02-149.src( 02-149.keywords , hom050302.ps )	{"url":"http://kleine.mat.uniroma3.it/mp_arc-bin/mpa?yn=02-149","timestamp":"2024-11-06T17:18:26Z","content_type":"text/html","content_length":"1749","record_id":"<urn:uuid:e948f7ca-7aa0-46db-9029-3cd701f004b8>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00718.warc.gz"}
The Operational Ratios of the Points-based System Chapter 8 The Operational Ratios of the Points-based System April 19, 2023 1 minutes • 115 words Table of contents The following ratios determine the useful productivity of the users of our points-based system just as capitalism has ratios such as Earning per Share, ROI, Return on Assets, etc. Realization Ratio This is the ratio of the actual realized points versus the total points. Realized Points / Total Net Points Average Time to Realize This is the average time that it takes to realize a point from the time when that point was created or given. AVE(Time of Point Debit - Time of Point Credit) Transaction Usefulness Ratio This is the number of transaction instances that were realized, relative to the total number of transactions done. Realized Transaction Instances / Total Number of Transactions Done	{"url":"https://www.superphysics.org/social/economics/principles/part-3/chapter-08/","timestamp":"2024-11-13T17:42:09Z","content_type":"text/html","content_length":"38793","record_id":"<urn:uuid:1d41482d-9d73-4d9d-8d8b-6472f756b10e>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00856.warc.gz"}
Ordinary Differential Equations An ordinary differential equation (also abbreviated as ODE), in Mathematics, is an equation which consists of one or more functions of one independent variable along with their derivatives. A differential equation is an equation that contains a function with one or more derivatives. But in the case ODE, the word ordinary is used for derivative of the functions for the single independent In case of other types of differential equations, it is possible to have derivatives for functions more than one variable. The types of DEs are partial differential equation, linear and non-linear differential equations, homogeneous and non-homogeneous differential equation. In mathematics, the term “Ordinary Differential Equations” also known as ODE is an equation that contains only one independent variable and one or more of its derivatives with respect to the variable. In other words, the ODE is represented as the relation having one independent variable x, the real dependent variable y, with some of its derivatives. y’,y”, ….y^n ,…with respect to x. The order of ordinary differential equations is defined to be the order of the highest derivative that occurs in the equation. The general form of n-th order ODE is given as; F(x, y,y’,….,y^n ) = 0 Note that, y’ can be either dy/dx or dy/dt and y^n can be either d^ny/dx^n or d^ny/dt^n. An n-th order ordinary differential equations is linear if it can be written in the form; a[0](x)y^n + a[1](x)y^n-1 +…..+ a[n](x)y = r(x) The function a[j](x), 0 ≤ j ≤ n are called the coefficients of the linear equation. The equation is said to be homogeneous if r(x) = 0. If r(x)≠0, it is said to be a non- homogeneous equation. Also, learn the first-order differential equation here. The ordinary differential equation is further classified into three types. They are: • Autonomous ODE • Linear ODE • Non-linear ODE Autonomous Ordinary Differential Equations A differential equation which does not depend on the variable, say x is known as an autonomous differential equation. Linear Ordinary Differential Equations If differential equations can be written as the linear combinations of the derivatives of y, then they are called linear ordinary differential equations. These can be further classified into two • Homogeneous linear differential equations • Non-homogeneous linear differential equations Non-linear Ordinary Differential Equations If the differential equations cannot be written in the form of linear combinations of the derivatives of y, then it is known as a non-linear ordinary differential equation. ODEs has remarkable applications and it has the ability to predict the world around us. It is used in a variety of disciplines like biology, economics, physics, chemistry and engineering. It helps to predict the exponential growth and decay, population and species growth. Some of the uses of ODEs are: • Modelling the growth of diseases • Describes the movement of electricity • Describes the motion of the pendulum, waves • Used in Newton’s second law of motion and Law of cooling Examples of ODE Some of the examples of ODEs are as follows; $y’=x^2 – 1\\ \frac{d y}{d x}=(x+y)^{6} \\ x y^{\prime}=\sin x \\ \frac{d^{2} y}{d t^{2}}+2 \frac{d y}{d t}+5 y=0, y(0)=0, y^{\prime}(0)=2 \\ y^{\prime \prime}=y^{\prime}+x e^{x}$ Problems and Solutions The solutions of ordinary differential equations can be found in an easy way with the help of integration. Go through the below example and get the knowledge of how to solve the problem. Question 1: Find the solution to the ordinary differential equation y’=2x+1 Given, y’=2x+1 Now integrate on both sides, ∫ y’dx = ∫ (2x+1)dx y = 2x^2/2 + x + C y =x^2 + x + C Where C is an arbitrary constant. Question 2: Solve y^4y’+ y’+ x^2 + 1 = 0 Take, y’ as common, Now integrating on both sides, we get Where C is an arbitrary constant. For more maths concepts, keep visiting BYJU’S and get various maths related videos to understand the concept in an easy and engaging way. Frequently Asked Questions – FAQs What is Ordinary differential equation? Give an example. An ordinary differential equation is an equation which is defined for one or more functions of one independent variable and its derivatives. It is abbreviated as ODE. y’=x+1 is an example of ODE. What are the types of Ordinary differential equations? There are basically three types of ODEs: Autonomous Ordinary Differential Equations Linear Ordinary Differential Equations Non-linear Ordinary Differential Equations What is the order of ordinary differential equations? The order of ordinary differential equations is defined to be the order of the highest derivative that occurs in the equation. What is an explicit ordinary differential equation? If x is independent variable and y is dependent variable and F is a function of x, y and derivatives of variable y, then explicit ODE of order n is given by the equation: F(x, y, y’, …., y^n-1) = y^n What is an implicit ordinary differential equation? If x is independent variable and y is dependent variable and F is a function of x, y and derivatives if variable y, then implicit ODE of order n is given by the equation: F(x, y, y’, y’’, …., y^n-1) = 0 What is an autonomous differential equation? When the differential equation is not dependent on variable x, then it is called autonomous. What are the uses of ordinary differential equations? The application of ordinary differential equations can be seen in modelling the growth of diseases, to demonstrate the motion of pendulum and movement of electricity.	{"url":"https://mathlake.com/Ordinary-Differential-Equations","timestamp":"2024-11-06T23:55:33Z","content_type":"text/html","content_length":"18047","record_id":"<urn:uuid:96c3cd27-72e9-4f70-ad2d-bf92d7de3ad4>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00403.warc.gz"}
Understanding Mathematical Functions: How To Find A Linear Function Eq Introduction: Understanding the Foundation of Linear Functions Linear functions are essential building blocks in mathematics, providing a fundamental understanding of relationships between variables. In this chapter, we will delve into the importance of linear functions in both mathematical contexts and real-world applications, the definition of a linear function, and the key components that define it. Importance of linear functions in mathematics and real-world applications Linear functions play a crucial role in various mathematical concepts, from algebra to calculus. They provide a simple yet powerful way to model and analyze relationships between variables. In real-world applications, linear functions are used extensively in fields such as economics, physics, and engineering to predict outcomes, make decisions, and solve problems. Definition of a linear function and its general form y = mx + b A linear function is a mathematical function that can be represented by the equation y = mx + b, where y represents the dependent variable, x represents the independent variable, m is the slope of the line, and b is the y-intercept. Overview of the components of a linear function: slope (m) and y-intercept (b) The slope (m) of a linear function determines the rate at which the dependent variable changes with respect to the independent variable. A positive slope indicates an upward trend, while a negative slope indicates a downward trend. The y-intercept (b) is the value of the dependent variable when the independent variable is zero, representing the point where the function intersects the y-axis. Key Takeaways • Identify the slope and y-intercept. • Use the formula y = mx + b. • Substitute values to find the equation. • Check your work by graphing the function. • Practice with different examples for mastery. Identifying Components of a Linear Function When it comes to understanding mathematical functions, linear functions are one of the most basic and fundamental types. In order to find the equation of a linear function, it is important to identify its key components: the slope and the y-intercept. A Detailed explanation of the slope as the rate of change The slope of a linear function represents the rate of change of the function. It indicates how much the function's output (y-value) changes for a given change in the input (x-value). Mathematically, the slope is calculated as the ratio of the change in y-values to the change in x-values between two points on the function. For example, if the slope of a linear function is 2, it means that for every 1 unit increase in the x-value, the y-value increases by 2 units. A positive slope indicates an upward trend, while a negative slope indicates a downward trend. Understanding the y-intercept as the starting point of the function on the y-axis The y-intercept of a linear function is the point where the function intersects the y-axis. It represents the starting point of the function when x=0. The y-intercept is a crucial component of the function as it helps determine the initial value of the function. For example, if the y-intercept of a linear function is 3, it means that the function starts at the point (0,3) on the y-axis. This point is where the function crosses the y-axis and serves as a reference point for graphing the function. Examples of determining slope and y-intercept from a graph One common way to determine the slope and y-intercept of a linear function is by analyzing its graph. By looking at the graph, you can identify two points on the function and calculate the slope as the ratio of the change in y-values to the change in x-values between those points. Similarly, the y-intercept can be determined by identifying the point where the function crosses the y-axis on the graph. This point gives you the initial value of the function and helps in writing the equation of the linear function. By understanding the slope as the rate of change and the y-intercept as the starting point of the function on the y-axis, you can easily find the equation of a linear function and interpret its behavior on a graph. How to Derive a Linear Function from Two Points When given two points on a graph, you can find the equation of a linear function that passes through those points. This process involves calculating the slope of the line and then using one of the points to solve for the y-intercept. Let's break down the steps involved in finding a linear function equation from two points. Explanation of the formula (y2 - y1) / (x2 - x1) for calculating slope The slope of a line is a measure of how steep the line is. It is calculated using the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points given. This formula represents the change in y-values divided by the change in x-values between the two points. Step-by-step guide on using the slope and one point to solve for b (y-intercept) Once you have calculated the slope using the formula mentioned above, you can use one of the points given to solve for the y-intercept, denoted as 'b' in the linear function equation y = mx + b. Substitute the coordinates of one point into the equation, along with the calculated slope, and solve for 'b'. Practical example of calculating a linear function equation from two points Let's consider the points (2, 3) and (4, 7) on a graph. First, calculate the slope using the formula: (7 - 3) / (4 - 2) = 4 / 2 = 2. Now, choose one of the points, say (2, 3), and substitute into the equation y = 2x + b. Solving for 'b', we get 3 = 2(2) + b, which simplifies to b = -1. Therefore, the linear function equation passing through the points is y = 2x - 1. Graphing Linear Functions for Better Visual Understanding Understanding how to graph linear functions is essential in mathematics as it provides a visual representation of the relationship between two variables. By plotting a linear function on a graph, you can easily interpret the slope, y-intercept, and overall behavior of the function. Techniques for plotting a linear function on a graph • Step 1: Identify the slope (m) and y-intercept (b) of the linear function in the form y = mx + b. • Step 2: Plot the y-intercept on the y-axis. This is the point where the line intersects the y-axis. • Step 3: Use the slope to find additional points on the line. The slope represents the rate of change of the function. • Step 4: Connect the points with a straight line to graph the linear function. How to use the slope and y-intercept to sketch the line The slope of a linear function determines the steepness of the line, while the y-intercept indicates where the line crosses the y-axis. By understanding these two components, you can easily sketch the line on a graph. For example, if the slope is 2 and the y-intercept is 3, you would start by plotting the point (0, 3) on the graph. Then, using the slope of 2, you would move up 2 units and over 1 unit to plot another point. Connecting these points will give you a straight line representing the linear function. Tools and software that can aid in graphing linear functions There are various tools and software available that can assist in graphing linear functions, making the process more efficient and accurate. • Graphing calculators: Graphing calculators allow you to input the linear function equation and plot the graph instantly. • Online graphing tools: There are many online tools that provide graphing capabilities, allowing you to input equations and visualize the linear functions. • Math software: Programs like MATLAB, Mathematica, and Desmos offer advanced graphing features for linear functions and other mathematical concepts. Real-World Applications of Linear Functions Linear functions play a crucial role in various real-world applications, providing a simple yet powerful tool for modeling relationships between variables. Let's explore some common applications where linear functions are used: A. Demonstrating the use of linear functions in budgeting and finance In budgeting and finance, linear functions are frequently employed to analyze and predict financial trends. For example, a company may use a linear function to create a budget based on past revenue data. By plotting revenue over time and fitting a straight line to the data points, financial analysts can estimate future revenue and make informed decisions about investments and expenses. Furthermore, linear functions can be used to calculate break-even points, which is the point at which total revenue equals total costs. This information is invaluable for businesses looking to optimize their operations and maximize profits. B. Application in physics for calculating speed or distance over time In physics, linear functions are commonly used to describe the motion of objects. For instance, when an object moves at a constant speed, its position can be modeled by a linear function. By plotting distance over time, physicists can determine the object's speed by calculating the slope of the line. Linear functions are also utilized to predict future positions of objects based on their initial velocity and acceleration. This predictive capability is essential in various fields, such as astronomy, engineering, and robotics. C. Linear functions in economics for cost and revenue modeling In economics, linear functions are employed to analyze the relationship between costs, revenues, and profits. By using linear functions, economists can estimate the cost of producing goods or services, forecast revenue based on sales volume, and determine the optimal pricing strategy. Moreover, linear functions are crucial for understanding the concept of elasticity, which measures how sensitive the quantity demanded is to changes in price. By analyzing the slope of the linear demand curve, economists can assess consumer behavior and make informed decisions about pricing and market strategies. Common Pitfalls and Troubleshooting When working with linear functions, it is important to be aware of common pitfalls that can lead to errors in finding the correct equation. By understanding these potential pitfalls and implementing strategies for troubleshooting, you can ensure accuracy in your mathematical calculations. A. Misinterpreting the slope as a y-intercept and vice versa One common mistake when finding a linear function equation is misinterpreting the slope as the y-intercept, and vice versa. The slope of a linear function represents the rate of change, while the y-intercept is the point where the line intersects the y-axis. To avoid this pitfall: • Remember that the slope is the coefficient of x in the equation y = mx + b, while the y-intercept is the constant term. • Double-check your calculations to ensure that you have correctly identified the slope and y-intercept in the equation. B. Errors in slope calculation due to inaccurate plotting of points Another common error when finding a linear function equation is inaccurately plotting points on a graph, leading to errors in calculating the slope. The slope of a linear function is determined by the change in y divided by the change in x between two points on the line. To troubleshoot this issue: • Ensure that you accurately plot the points on the graph and calculate the correct change in y and change in x. • Double-check your calculations to verify that you have accurately determined the slope of the line. C. Strategies for verifying the accuracy of a derived linear function equation After deriving a linear function equation, it is important to verify its accuracy to ensure that your calculations are correct. By implementing strategies for verification, you can catch any errors and make necessary corrections. Some strategies for verifying the accuracy of a derived linear function equation include: • Substitute known points into the equation to check if they satisfy the equation. • Graph the linear function and compare it to the plotted points to see if they align. • Use mathematical software or calculators to perform calculations and verify the results. Conclusion: Mastery of Linear Functions & Best Practices A Recap of the importance and utility of understanding how to find linear function equations • Understanding the importance: Linear functions are fundamental in mathematics and have wide applications in various fields such as physics, economics, and engineering. Being able to find linear function equations allows us to model and predict real-world phenomena accurately. • Utility of linear functions: Linear functions help us analyze trends, make predictions, and solve problems efficiently. They provide a simple yet powerful tool for understanding relationships between variables. B Emphasizing the value of practice and real-world application for mastery • Practice makes perfect: Like any skill, mastering the art of finding linear function equations requires practice. The more you practice, the more comfortable and confident you will become in dealing with different types of problems. • Real-world application: Applying linear functions to real-world scenarios not only enhances your understanding but also makes the learning process more engaging and practical. It allows you to see the direct impact and relevance of linear functions in everyday life. C Best practices: double-check calculations, use of graphing tools for clarity, and staying curious about real-world linear relationships • Double-check calculations: Accuracy is key when dealing with linear function equations. Always double-check your calculations to avoid errors and ensure the correctness of your results. • Use of graphing tools: Graphing tools such as graphing calculators or software can help visualize linear functions and their relationships. They provide a clear and visual representation that aids in understanding and • Stay curious: Keep exploring real-world linear relationships and their applications. Stay curious and ask questions to deepen your understanding and discover new insights. The more you engage with real-world examples, the better you will grasp the concepts of linear functions.	{"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-find-linear-function-equation","timestamp":"2024-11-09T04:21:21Z","content_type":"text/html","content_length":"222536","record_id":"<urn:uuid:f5777ab5-5ff3-4916-9b37-80465e80ca39>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00506.warc.gz"}
How to Fix TypeError: Can't Multiply Sequence by Non-Int of Type Numpy.float64 How to Fix TypeError: Can’t Multiply Sequence by Non-Int of Type Numpy.float64 Have you ever encountered a TypeError like “can’t multiply sequence by non-int of type numpy.float64”? If so, you’re not alone. This error is a common one, and it can be caused by a number of different things. In this article, we’ll take a look at what this error means, what causes it, and how you can fix it. We’ll start by discussing what a TypeError is. Then, we’ll take a look at the specific error message “can’t multiply sequence by non-int of type numpy.float64”. We’ll discuss what this error means and what causes it. Finally, we’ll show you how to fix this error so that you can get your code running smoothly again. By the end of this article, you’ll have a good understanding of what this error is and how to fix it. So let’s get started! Error Cause Solution TypeError: can’t multiply sequence by non-int of type You are trying to multiply a numpy.float64 array by a Cast the non-integer value to an integer before multiplying it by the numpy.float64 non-integer value. numpy.float64 array. In this tutorial, you will learn what the error `TypeError: can’t multiply sequence by non-int of type numpy.float64` means and how to fix it. What is the error `TypeError: can’t multiply sequence by non-int of type numpy.float64`? This error occurs when you try to multiply a sequence by a value that is not an integer. For example, if you try to multiply the sequence `[1, 2, 3]` by the float ` 2.0`, you will get the error `TypeError: can’t multiply sequence by non-int of type numpy.float64`. This is because NumPy arrays are immutable, which means that you cannot change their values once they have been created. When you try to multiply a NumPy array by a non-integer value, NumPy will try to convert the non-integer value to an integer. However, if the non-integer value cannot be converted to an integer, NumPy will throw an error. How to fix the error `TypeError: can’t multiply sequence by non-int of type numpy.float64` There are two ways to fix the error `TypeError: can’t multiply sequence by non-int of type numpy.float64`: 1. Cast the non-integer value to an integer. For example, you can cast the float `2.0` to an integer by using the `int()` function. >>> a = [1, 2, 3] >>> b = int(2.0) >>> a * b [2, 4, 6] 2. Use the `numpy.multiply()` function. The `numpy.multiply()` function can be used to multiply a NumPy array by a scalar value. The scalar value can be any type, including integers, floats, and strings. >>> a = [1, 2, 3] >>> b = 2.0 >>> numpy.multiply(a, b) [2.0, 4.0, 6.0] In this tutorial, you learned what the error `TypeError: can’t multiply sequence by non-int of type numpy.float64` means and how to fix it. You can fix this error by casting the non-integer value to an integer or by using the `numpy.multiply()` function. 3. What causes the error `TypeError: can’t multiply sequence by non-int of type numpy.float64`? This error occurs because NumPy arrays are immutable, which means that you cannot change their values after they have been created. When you try to multiply a NumPy array by a non-integer value, NumPy tries to convert the non-integer value to an integer. If it cannot do this, it will raise the error `TypeError: can’t multiply sequence by non-int of type numpy.float64`. For example, the following code will raise an error: >>> a = np.array([1, 2, 3]) >>> a * ‘a’ Traceback (most recent call last): File ““, line 1, in TypeError: can’t multiply sequence by non-int of type numpy.float64 This is because the string `’a’` is not an integer, and NumPy cannot convert it to an integer. 4. How can you fix the error `TypeError: can’t multiply sequence by non-int of type numpy.float64`?** There are a few ways to fix this error. One way is to convert the non-integer value to an integer before multiplying it by the NumPy array. For example, you could use the `int()` function to convert the float `1.5` to an integer `1`: >>> a = np.array([1, 2, 3]) >>> a * int(1.5) array([1.5, 3.0, 4.5]) Another way to fix this error is to use the `numpy.multiply()` function to multiply the NumPy array by the non-integer value. The `numpy.multiply()` function will automatically convert the non-integer value to an integer before multiplying it by the NumPy array. For example, the following code will not raise an error: >>> a = np.array([1, 2, 3]) >>> a * np.multiply(1.5, 2) array([1.5, 3.0, 4.5]) Finally, you can also use the `numpy.astype()` function to cast the non-integer value to an integer before multiplying it by the NumPy array. For example, the following code will not raise an error: >>> a = np.array([1, 2, 3]) >>> a * np.astype(1.5, int) array([1.5, 3.0, 4.5]) 5. Examples of code that will raise the error `TypeError: can’t multiply sequence by non-int of type numpy.float64`** The following code will raise the error `TypeError: can’t multiply sequence by non-int of type numpy.float64`: >>> a = np.array([1, 2, 3]) >>> a * ‘a’ Traceback (most recent call last): File ““, line 1, in TypeError: can’t multiply sequence by non-int of type numpy.float64 >>> a = np.array([1, 2, 3]) >>> a * [‘a’, ‘b’, ‘c’] Traceback (most recent call last): File ““, line 1, in TypeError: can’t multiply sequence by non-int of type numpy.float64 6. Examples of code that will not raise the error `TypeError: can’t multiply sequence by non-int of type numpy.float64`** The following code will not raise the error `TypeError: can’t multiply sequence by non-int of type numpy.float64`: >>> a = np.array([1, 2, 3]) >>> a * int(1.5) array([1.5, 3.0, 4.5]) >>> a = np.array([1, 2, 3]) >>> a * np.multiply(1.5, 2) array([1.5, 3.0, 4.5]) >>> a = np.array([1, 2, 3]) >>> a * np.astype(1.5, int) array([1.5, 3.0, 4.5]) Q: What does the error “TypeError: can’t multiply sequence by non-int of type numpy.float64” mean? A: This error occurs when you try to multiply a NumPy array by a non-integer value. For example, the following code will raise an error: >>> a = np.array([1, 2, 3]) >>> a * ‘a’ Traceback (most recent call last): File ““, line 1, in TypeError: can’t multiply sequence by non-int of type ‘str’ The reason for this error is that NumPy arrays are immutable, which means that you cannot change their values after they have been created. When you try to multiply a NumPy array by a non-integer value, NumPy interprets this as an attempt to change the values of the array, and it raises an error. To avoid this error, you can either cast the non-integer value to an integer before multiplying it by the array, or you can use the `numpy.multiply()` function to perform the multiplication. For example, the following code will work without raising an error: >>> a = np.array([1, 2, 3]) >>> a * int(‘a’) array([1, 2, 3]) >>> a = np.array([1, 2, 3]) >>> a = np.multiply(a, int(‘a’)) >>> a array([1, 2, 3]) Q: How can I fix the error “TypeError: can’t multiply sequence by non-int of type numpy.float64”? A: There are two ways to fix this error. You can either cast the non-integer value to an integer before multiplying it by the array, or you can use the `numpy.multiply()` function to perform the To cast the non-integer value to an integer, you can use the `int()` function. For example, the following code will fix the error: >>> a = np.array([1, 2, 3]) >>> a * int(‘a’) array([1, 2, 3]) To use the `numpy.multiply()` function, you can pass the array and the non-integer value as arguments to the function. For example, the following code will also fix the error: >>> a = np.array([1, 2, 3]) >>> a = np.multiply(a, int(‘a’)) >>> a array([1, 2, 3]) Q: What are some common causes of this error? A: There are a few common causes of this error. One common cause is trying to multiply a NumPy array by a string. For example, the following code will raise an error: >>> a = np.array([1, 2, 3]) >>> a * ‘a’ Traceback (most recent call last): File ““, line 1, in TypeError: can’t multiply sequence by non-int of type ‘str’ Another common cause is trying to multiply a NumPy array by a float. For example, the following code will also raise an error: >>> a = np.array([1, 2, 3]) >>> a * 1.0 Traceback (most recent call last): File ““, line 1, in TypeError: can’t multiply sequence by non-int of type ‘float’ To avoid these errors, you should make sure that you are only multiplying NumPy arrays by integers. Q: How can I prevent this error from happening in the future? A: There are a few things you can do to prevent this error from happening in the future. First, you can make sure that you are only multiplying NumPy arrays by integers. Second, you can use the `numpy.multiply()` function to perform the multiplication. Third, you can cast the non-integer value to an integer before multiplying it by the array. By following these tips, you can help to prevent the “TypeError: can’t multiply sequence by non-int of type numpy.float64” error from occurring. In this blog post, we discussed the common error `TypeError: can’t multiply sequence by non-int of type numpy.float64`. We explained the cause of the error and provided several solutions for how to fix it. We hope that this blog post was helpful and that you were able to resolve the error on your own. Here are the key takeaways from this blog post: • The `TypeError: can’t multiply sequence by non-int of type numpy.float64` error occurs when you try to multiply a NumPy array by a non-integer value. • To fix this error, you can either convert the non-integer value to an integer or use the `numpy.multiply()` function. • The `numpy.multiply()` function takes two arrays as input and returns a new array that is the product of the two input arrays. • You can also use the `numpy.dot()` function to multiply two arrays. The `numpy.dot()` function takes two arrays as input and returns a single value that is the dot product of the two input Author Profile Hatch, established in 2011 by Marcus Greenwood, has evolved significantly over the years. Marcus, a seasoned developer, brought a rich background in developing both B2B and consumer software for a diverse range of organizations, including hedge funds and web agencies. Originally, Hatch was designed to seamlessly merge content management with social networking. We observed that social functionalities were often an afterthought in CMS-driven websites and set out to change that. Hatch was built to be inherently social, ensuring a fully integrated experience for users. Now, Hatch embarks on a new chapter. While our past was rooted in bridging technical gaps and fostering open-source collaboration, our present and future are focused on unraveling mysteries and answering a myriad of questions. We have expanded our horizons to cover an extensive array of topics and inquiries, delving into the unknown and the unexplored.	{"url":"https://hatchjs.com/typeerror-cant-multiply-sequence-by-non-int-of-type-numpy-float64/","timestamp":"2024-11-08T11:45:33Z","content_type":"text/html","content_length":"93975","record_id":"<urn:uuid:18f9fb6c-0a04-45f6-9623-c86c6047ea0d>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00786.warc.gz"}
Implementing different groebner_basis() algorithms Implementing different groebner_basis() algorithms I'm not sure if this is the correct forum to ask this type of question, but I'll try anyway. I am curious if there is a way to do the following in Sage. I would like a way to make a Groebner basis calculation to not worry about calculations involving polynomials of too high degree. Macaulay2's gb command has the ability to specify this with an option called HardDegreeLimit, for instance. The Sage documentation on groebner_basis seems to imply that I can use Macaulay2's gb command with this option, but I don't know how to make it work. Singular's std command also has a degBound option, but I again don't know how to use this in my Sage implementation. 1 Answer Sort by ยป oldest newest most voted With the current Groebner basis commands in Sage, there is not a good way to do this. None of the underlying methods like _groebner_basis_macaulay2 support a degree limit. As a workaround, you can do something like the following: sage: P.<a,b,c> = PolynomialRing(QQ,3, order='lex') sage: I = sage.rings.ideal.Katsura(P,3) sage: I.groebner_basis() #compute the normal Groebner basis [a - 60c^3 + 158/7c^2 + 8/7c - 1, b + 30c^3 - 79/7c^2 + 3/7c, c^4 - 10/21c^3 + 1/84c^2 + 1/84c] sage: singular.eval('degBound = 2;') #set the degree bound in Singular 'degBound = 2;' sage: gb = Sequence(map(P, singular(I).std())); gb [10bc - b + 12c^2 - 4c, 4b^2 + 2bc - b, a + 2b + 2c - 1] sage: singular.eval('degBound = 0;') #reset it back to 0 (unlimited) 'degBound = 0;' I've made this ticket #9789. edit flag offensive delete link more I agree with your answer, but I think your example is misleading. The Singular manual states that "degBound should not be used for a global ordering with inhomogeneous input". I'm pretty sure that the output for a non-homogeneous ideal is potentially not the degree truncation of an actual GB. Volker Braun ( 2010-08-23 21:57:44 +0100 )edit	{"url":"https://ask.sagemath.org/question/7619/implementing-different-groebner_basis-algorithms/?answer=11523","timestamp":"2024-11-04T15:08:57Z","content_type":"application/xhtml+xml","content_length":"56398","record_id":"<urn:uuid:4b794c1a-5f18-4c70-826f-b5076da58569>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00280.warc.gz"}
UE4 Transform Calculus - Part 1 \| Unreal Engine Community Wiki UE4 Transform Calculus - Part 1 UE4 Transform Calculus - Part 1 Motivation Slate often has to deal with deep and wide hierarchies of widgets, expressing the sizes and positions of children in terms of their parents. Somet... UE4 Transform Calculus - Part 1 Slate often has to deal with deep and wide hierarchies of widgets, expressing the sizes and positions of children in terms of their parents. Sometimes these are simple relationships like text in a button, but sometimes those relationships are much more complex, like a graph editing panel that might be panning and zooming across a virtual canvas. As hierarchies get deeper, this relationship becomes layered. For instance, if we apply a global DPI (dots per inch) scale to our application, which itself sits in a window at some arbitrary location on the desktop, there are a series of zooms and offsets that combine to form the final position of a widget on the screen. [The complex relationship of slate widgets. A UE4 Editor view displaying blueprint widgets within a virtual panel within a tabwell within a window, all having a global application scale applied. The Blueprint editor must reason about the widgets in virtual canvas space, implement mouse events in desktop space, and render the widgets in window space.] Slate must not only be able to unwind these hierarchies internally, but also provide straightforward ways for widget authors to reason about widgets in multiple spaces (ie, local, virtual canvas, window, desktop). With the addition of Render Transforms to Slate, the math for computing a child's relationship to its parent became much more involved than a simple scale and offset. [SlateViewer can apply an arbitrary render transform to the entire application. Now what's the relationship of one of these widgets to its parent, or the window, or the desktop?] In fact, the historical simplicity of a child widget's transform with respect to its parent in Slate contributed to an overall daunting task of updating hundreds of widgets to support arbitrary render transforms. We chose to focus on the core widgets that most widgets are composed of, but lots of legacy functions still exist in Slate that are used by other widgets that don't support render transforms. This series of posts will focus on a new notation for manipulating transforms, along with a C++ implementation in UE4 that Slate is using to assist with this sometimes mind-bending task. Target Audience For this discussion, I assume you are already aware of basic linear algebra and have at least cursory experience working with rigid body hierarchies in code. You should be comfortable with vectors and matrices, be able to perform basic calculations with them, and be familiar with the idea of using them to represent transformations like translation, rotation, scale, shear, etc. Some familiarity with alternate representations like quaternions etc may also be useful when we start discussing the code implementation. If you've ever tried to use a physics engine or render something in OpenGL or Direct3D, you likely have more than enough exposure to these concepts. I will generally try to keep a conversational tone, but will occasionally stray into math-heavy territory. When I do, I'll try to summarize the salient points and provide links to read up more if you'd like. The reason for getting mathy sometimes is to establish the mathematical foundation for the notation we are developing, not to rigorously prove it. I'm a programmer, not a mathematician. Why Calculus? So, a word about why I chose the term Transform Calculus instead of something less presumptive (like Framework or API). This document in fact describes a logical calculus, or a formalization of a logical theory[1]. These are concepts that provide a mathematical notation for uniformly expressing and manipulating transforms that transcends the representation (ie, a matrix or quaternion). There happens to be an implementation in UE4, but that implementation is secondary to the underlying concepts. Also, the implementation sometimes has to make compromises to work within the C++ language and UE4 Core types, so is not a pure expression of these concepts. To be clear, I'm not inventing anything "novel", I'm just wrapped some well-known math concepts in a higher level abstraction. That said, I do feel all the focus on notation is a critical part of the journey, as one can't use the library effectively without understanding the notation. For that reason, simply referring to this as a Framework or API seemed insufficient. One could perhaps more accurately refer to this an an algebra, as it's closed over the affine vector space. But I don't prove this, and frankly I've been using the term calculus for years, so it seemed too late to turn back now. C'est la vie. :) Rigid Body Hierarchies Slate UI hierarchies are conceptually identical to rigid body hierarchies that physics engines utilize. We'll start with a quick overview of rigid body hierarchies to establish the concepts. I expect anyone reading this to be familiar already, so I won't dwell on details. Hierarchies as Attachments One way to think about the hierarchy is as each part being attached to its parent by a transformation. This transformation places a child part relative to its parent. To place a part in the world we start at the root of the tree (world space) and successively transform the part all the way down to the leaf node where the part is located. If one thinks of these transformations as simple offsets, then it is easy to conceptualize. However, most rigid body systems allow more complex transformations like scale, rotate, and/or shear. Reasoning about parts' relations to each other in such a hierarchy can quickly become very complicated. Hierarchies as Frames of Reference Another way to look at the hierarchy is that each part is a camera, and the transformation is taking the part from it's local the coordinate system into that of the parent. So each part is essentially in it's own coordinate system, or frame of reference relative to its parent. The representation is logically equivalent to the attachment point of view described above, but instead of viewing all parts as placed at different points in the same coordinate system, each part has its own local coordinate system. This makes it easier to conceptualize the parent-child relationship as more than a simple offset. To place a child in the world, we chain, or composite, these transformations in succession just as we did when thinking of them as attachments. Transform Calculus This notion of a hierarchy as a chain of transformations taking us from one frame of reference to another is very powerful. In linear algebra, a frame of reference is like a vector basis[2], and the transformation to another frame of reference is a change of basis. In code, we often represent this transformation as a matrix and composite them using matrix multiplication[3]. For efficiency, we sometimes represent a transform using narrower representations like euler angles, quaternions, translations, or even scalars. Regardless of the representation we use, a transformation is essentially a function over a vector space that maps one frame of reference to another. We can represent this function along with operations to manipulate it using a common notation, or calculus: Transform Calculus Operations Composition ($\oplus \oplus$) Note that composition ($\oplus \oplus$) is a transitive relation -- the output frame of the first must match the input frame of the second, otherwise the composition is invalid: Composition is Transitive: $TA\to B\oplus TB\to CTA \rightarrow B\oplus TB \rightarrow C$ Valid (output Frame B matches input Frame B) $TA\to B\oplus TC\to BTA\rightarrow B\oplus TC\rightarrow B$ Invalid (output Frame B mismatches input Frame C) $TA\to B\oplus TC\to B-1=TA\to B\oplus TB\to CTA\rightarrow B\oplus TC\rightarrow B-1 = TA\rightarrow B\oplus TB \rightarrow C$ Valid (inversion swaps the input and output Frame) I'm using the mysterious symbol for composition because it is a conceptual operation, and the calculation is not important right now. For instance, something like the multiply or addition operator might seem attractive to use instead, but could be misleading. For instance, two matrices are indeed composited using multiplication, but two translation vectors are composited using addition. However, they are both conceptually a composition of two transformations. I want to convey the concept of composition without focusing on the specific math required to achieve it, which is more tied to the representation used for the transform. I'll discuss this more in the next post. Let's go back to our rigid body tank example and express some transformations using this new calculus: $Tgun\to world=Tgun\to turret\oplus Tturret\to chassis\oplus Tchassis\to worldTgun\rightarrow world = Tgun\rightarrow turret\oplus Tturret\rightarrow chassis\oplus Tchassis\rightarrow world$ $Tgun\to wheel1=Tgun\to turret\oplus Tturret\to chassis\oplus Twheel1\to chassis-1Tgun\rightarrow wheel1 = Tgun\rightarrow turret\oplus Tturret\rightarrow chassis\oplus Twheel1\rightarrow chassis-1$ This is makes logical sense: to determine the transformation from the gun's frame of reference to the world's, chain the gun-to-turret transform to the turret-to-chassis transform to the chassis-to-world transform. Each step moves UP the hierarchy to the root. Note in the second example how we use the inverse to go back DOWN the hierarchy to get to the lower wheel1 node, preserving the transitive chain of operations. Transforming Vectors and Points As discussed, a transformation is a mapping function over the vector space. But we use points to describe geometry. So what's the difference between a vector and a point? Well, a point is a distinct location in space, while a vector is a displacement between two points (like a vertex normal). Luckily, both points AND vectors can be represented using homogeneous coordinates, the former with a homogeneous coordinate of 1, and the latter with a 0[4]. By thinking of our transforms as operating on affine spaces using homogeneous coordinates, we can transform points and vectors the same way: $TA\to B\left(P\right)TA\rightarrow B\left(P\right)$ = Transformation of homogeneous point $PP$ from frame $AA$ to frame $BB$ $TA\to B\left(V\right)TA \rightarrow B\left(V\right)$ = Transformation of homogeneous vector $VV$ from frame $AA$ to frame $BB$ We have outlined a formal notation, or calculus, for expressing a rigid body hierarchy as a tree of coordinate transformations taking us from the frame of reference of a child node to its parent. We can composite and invert these transformations using a logical notation that allows us to reason about any node from the perspective of any other node, regardless of how that transform is represented. Finally, we can apply these transformations to a set of vectors or points to reason about specific geometry associated with those nodes. This is something Slate has to do all the time. In the next post I'll discuss how these operations are actually implemented using several transformation representations available in UE4 and demonstrate how the calculus allows us to simplify real-world code by expressing the concept rather than focusing on the math itself.	{"url":"https://unrealcommunity.wiki/ue4-transform-calculus-part-1-4kjryw05","timestamp":"2024-11-06T13:59:26Z","content_type":"text/html","content_length":"76624","record_id":"<urn:uuid:61514f04-cf07-4fb2-afb5-d1b9a2131522>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00830.warc.gz"}
TF by DL.AI — Course 4: Sequences, TS, Prediction • Sequence models: focus on time series (there are others) — stock, weather,... • At the end, we wanna model sunspot actitivity cycles which is important to NASA and other space agencies. • Using RNN on time series data. 📙 Notebook: introduction to time series . + explaining video . → How to create synthetic time series data + plot them. • Time series is everywhere: stock prices, weather focasts, historical trends (Moore's law),... • Univariate TS and Miltivariate TS. • Type of things can we do with ML over TS: □ Any thing has a time factor can be analysed using TS. □ Predicting a forecasting (eg. birth & death in Japan -> predict future for retirement, immigration, impacts...). □ Imputation: project back into the past. □ Fill holes in the data. □ Nomalies detecction (website attacks). □ Spot patterns (eg. speed recognition). • Common patterns in TS: □ Trend: a specific direcion that they're moving in. □ Seasonality: patterns repeat at predictable intervals (eg. active users for a website). □ Combinition of both trend and seasonality. □ Stationary TS □ Autocorrelated TS: a time series is linearly related to a lagged version of itself.. There is no trend, no seasonality. □ Multiple auto correlation. □ May be trend + seasonality + autorrelation + noise. □ Non-stationary TS In this case, we base just on the later data to predict the future (not on the whole data). • Fixed partitioning (this course focuses on) = splitting TS data into training period, validation period and test period. □ If TS is seasonal, we want each period contains the whole number of seasons. • We can split + train + test to get a model and then re-train with the data containing also the test period so that the model is optimized! In that case, the test set comes from the future. • Roll-forward partitioning: we start with a short training period and we gradually increase it (1 day at a time or 1 week at a time). At each iteration, we train the model on training period, use it to focast the following day/week in the validation period. = Fixed partitioning in a number of times! For evaluating models: 1errors = forecasts - actual 3# Mean squared error (square to get rid of negative values) 4# Eg. Used if large errors are potentially dangerous 5mse = np.square(errors).mean() 6# Get back to the same scale to error 7rmse = np.sqrt(mse) 9# Mean absolute error (his favorite) 10# this doesn't penalize large errs as much as mse does, 11# used if loss is proportional to the size of err 12mae = np.abs(errors).mean() 14# Mean abs percentage err 15# idea of the size of err compared to the values 16mape = np.abs(errors / x_valid).mean() 1# MAE with TF 2keras.metrics.mean_absolute_error(x_valid, naive_forecast).numpy() Moving average: a simple forecasting method. Calculate the average of blue lines within a fixed "averaging windows". • This can eliminate noises and doesn't anticipate trend or seasonality. • Depend on the "averaging window", it can give worse result than naive forecast. Take the average on each yellow window. MAE=7.14 (optimal is 4). 1def moving_average_forecast(series, window_size): 2 """Forecasts the mean of the last few values. 3 If window_size=1, then this is equivalent to naive forecast""" 4 forecast = [] 5 for time in range(len(series) - window_size): 6 forecast.append(series[time:time + window_size].mean()) 7 return np.array(forecast) Differencing: remove the trend and seasonality from the TS. We study on the differences between points and their previous neighbor in period. Left image: we find the differencing of original values, then we find the average (orange line). Right image: restore the trend and seasonality. MAE=5.8 (optimal is 4). Above method still get the noises (because we add the differencing to the previous noise). If we remove past noise using moving average on that. Smoothing both past and present values. MAE=4.5 (optimal is 4). Keep in mind before using Deep Learning, sometimes simple approaches just work fine! • We need to split our TS data into features and labels so that we can use them in ML algos. • In this case: features=#values in TS, label=next_value. □ Feature: window size and train to predict next value. □ Ex: 30 days of values as features and next value as label. □ Overtime, train ML to match 30 features to match a single label. 1def windowed_dataset(series, window_size, batch_size, shuffle_buffer): 2 dataset = tf.data.Dataset.from_tensor_slices(series) 3 dataset = dataset.window(window_size + 1, shift=1, drop_remainder=True) 4 dataset = dataset.flat_map(lambda window: window.batch(window_size + 1)) 5 dataset = dataset.shuffle(shuffle_buffer).map(lambda window: (window[:-1], window[-1])) 6 dataset = dataset.batch(batch_size).prefetch(1) 7 return dataset Sequence bias is when the order of things can impact the selection of things. It's ok to shuffle! 1# Simple linear regression (1 layer NN) 2dataset = windowed_dataset(x_train, window_size, batch_size, shuffle_buffer_size) 3l0 = tf.keras.layers.Dense(1, input_shape=[window_size]) 4model = tf.keras.models.Sequential([l0]) 5model.compile(loss="mse", optimizer=tf.keras.optimizers.SGD(lr=1e-6, momentum=0.9)) 7print("Layer weights {}".format(l0.get_weights())) 9forecast = [] 11for time in range(len(series) - window_size): 12 forecast.append(model.predict(series[time:time + window_size][np.newaxis])) 13 # np.newaxis: reshape X to input dimension that used by the model 15forecast = forecast[split_time-window_size:] 16results = np.array(forecast)[:, 0, 0] 1# A way to choose an optimal learning rate 2lr_schedule = tf.keras.callbacks.LearningRateScheduler( 3 lambda epoch: 1e-8 * 10*(epoch / 20)) 4optimizer = tf.keras.optimizers.SGD(lr=1e-8, momentum=0.9) 5model.compile(loss="mse", optimizer=optimizer) 6history = model.fit(dataset, epochs=100, callbacks=[lr_schedule], verbose=0) 1lrs = 1e-8 (10 ** (np.arange(100) / 20)) 2plt.semilogx(lrs, history.history["loss"]) 3plt.axis([1e-8, 1e-3, 0, 300]) Loss w.r.t different learning rates. We choose the lowest one, around 8e-6. • RRN is a NN containing Recurrent layer. • The different from DNN is the input shape is 3 dimensional (batch_size x #time_step x dims_input_at each_timestep). • Re-use 1 cell multiple times in different layers (in this course). Idea of how RNN works with TS data. The current location can be impacted more by the nearby locations. • Suppose: window size of 30 time steps, batch size of 4: Shape will be 4x30x1 and the memory cell input will be 4x1 matrix. • If the memory cell comprises 3 neurons then the output matrix will be 4x3. Therefore, the full output of the layer will be 4x30x3. • Below figure: input and also output a sequence. Dimension of input to RNN. • Sometimes, we want only input a sequence but not output. This called sequence-to-vector RNN. I.E., ignore all of the outputs except the last one!. In tf.keras, it's default setting! Sequence to vector RNN. 1# Check the figure below as an illustration 2model = tf.keras.models.Sequential([ 3tf.keras.layers.SimpleRNN(20, return_sequences=True, input_shape=[None, 1]), 4 # input_shape: 5 # TF assumes that 1st dim is batch size -> any size at all -> no need to define 6 # None -> number of time steps, None means RNN can handle sequence of any length 7 # 1 -> univariate TS 9 # if there is `return_sequences=True` -> sequence-to-sequence RNN Illustration with keras. 1model = tf.keras.models.Sequential([ 2 tf.keras.layers.Lambda(lambda x: tf.expand_dims(x, axis=-1), # expand to 1 dim (from 2) so that we have 3 dims: batch size x #timesteps x series dim 3 input_shape=[None]), # can use any size of sequences 4 tf.keras.layers.SimpleRNN(40, return_sequences=True), 5 tf.keras.layers.SimpleRNN(40), 6 tf.keras.layers.Dense(1), 7 tf.keras.layers.Lambda(lambda x: x * 100.0) 8 # default activation in RNN is tanh -> (-1, 1) -> scale to -100, 100 • Loss function Huber (wiki): less sensitive to outliers. => we use this because our data in this case get a little bit noisy! 1# clear internal variables 3dataset = windowed_dataset(x_train, window_size, batch_size, shuffle_buffer_size) 5model = tf.keras.models.Sequential([ 6 tf.keras.layers.Lambda(lambda x: tf.expand_dims(x, axis=-1), 7 input_shape=[None]), 8 # LSTM here 9 tf.keras.layers.Bidirectional(tf.keras.layers.LSTM(32, return_sequences=True)), 10 tf.keras.layers.Bidirectional(tf.keras.layers.LSTM(32, return_sequences=True)), 11 tf.keras.layers.Bidirectional(tf.keras.layers.LSTM(32)), 12 # 13 tf.keras.layers.Dense(1), 14 tf.keras.layers.Lambda(lambda x: x * 100.0)	{"url":"https://dinhanhthi.com/note/deeplearning-ai-tensorflow-course-4/","timestamp":"2024-11-11T00:25:54Z","content_type":"text/html","content_length":"606935","record_id":"<urn:uuid:123297a2-6394-4a61-a917-eeb1994d25e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00375.warc.gz"}
An Iterative Approach to Solve a System of Equations In this tutorial, we are going to solve the following two algebraic equations iteratively. X₁ = X₂ - 1 X₂ = 2.5 X₁ - 0.5 Using fixed point iterative method and taking initial assumptions according to the question as, X₁ = X₂ = 0, the solutions do not converge with this present arrangement of equations. However, if we rearrange these equations in the following forms, the solutions indeed converge. X2 = X1 + 1 X₁ = X₂ / 2.5 + 0.2 A MATLAB program has been developed to solve the modified equations iteratively by creating a user defined function called "iterative". This function may be used to solve other equations iteratively which is the essence of the user defined function. The previous forms of equations from the question are also fed into the program to see the solutions, but the program indicates divergence. The following MATLAB program written in “m-file” subsequently enlightens essential commands in the program by additional notes. MATLAB Codes: % Definition of a function "iterative" to solve equations iteratively function [x1, x2, ea1, ea2] = iterative(X1, X2, etol1, etol2) % Input X1=0, X2=0, etol1, etol2=Tolerance definition for errors % Output x1, x2=Solutions of equations, ea1, ea2= Calculated errors in loop % Program Initialization solution1 = 0; solution2 = 0; % Iterative Calculations while (1) solutionprevious1 = solution1; solutionprevious2 = solution2; solution2 = X1+1; % Problem equation 1 solution1 = (X2/2.5)+0.2; % Problem equation 2 if solution1~=0 && solution2~=0 % Approximate percent relative errors ea1=abs((solution1 - solutionprevious1)/solution1)100; ea2=abs((solution2 - solutionprevious2)/solution2)100; if ea1<=etol1 && ea2<=etol2, % Conditions to meet specified error tolerances x1 = solution1; x2 = solution2; % Display of output parameters 2 comments: 1. David L.January 5, 2021 at 4:05PM Thank you for sharing the code. 1. My pleasure!	{"url":"https://www.modellingsimulation.com/2019/08/iterative-method.html?m=0","timestamp":"2024-11-04T11:49:38Z","content_type":"application/xhtml+xml","content_length":"125779","record_id":"<urn:uuid:537e4f60-9d63-4b3d-aaef-755ecc71be0b>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00550.warc.gz"}
Tarjan's algorithm is a procedure for finding strongly connected components of a directed graph. A strongly connected component is a maximum set of vertices, in which exists at least one oriented path between every two vertices. Tarjan's algorithm is based on depth first search (DFS). The vertices are indexed as they are traversed by DFS procedure. While returning from the recursion of DFS, every vertex Tarjan's algorithm is only a modified depth first search, hence it has an asymptotic complexity index = 0 * Runs Tarjan's algorithm * @param g graph, in which the SCC search will be performed * @return list of components List executeTarjan(Graph g) Stack s = {} List scc = {} //list of strongly connected components for Node node in g if (v.index is undefined) tarjanAlgorithm(node, scc, s) return scc * Tarjan's algorithm * @param node processed node * @param SCC list of strongly connected components * @param s stack procedure tarjanAlgorithm(Node node, List scc, Stack s) v.index = index v.lowlink = index s.push(node) //add to the stack for each Node n in Adj(node) do //for all descendants if n.index == -1 //if the node was not discovered yet tarjanAlgorithm(n, scc, s, index) //search node.lowlink = min(node.lowlink, n.lowlink) //modify parent's lowlink else if stack.contains(n) //if the component was not closed yet node.lowlink = min(node.lowlink, n.index) //modify parents lowlink if node.lowlink == node.index //if we are in the root of the component Node n = null List component //list of nodes contained in the component n = stack.pop() //pop a node from the stack component.add(n) //and add it to the component while(n != v) //while we are not in the root scc.add(component) //add the compoennt to the SCC list	{"url":"http://www.programming-algorithms.net/article/44220/Tarjan's-algorithm","timestamp":"2024-11-09T12:19:42Z","content_type":"text/html","content_length":"21192","record_id":"<urn:uuid:659a35fb-2cdc-4c2c-92a3-fd9135299bb2>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00752.warc.gz"}
multivariable calculus calculator Free Multivariable Calculus calculator - calculate multivariable limits, integrals, gradients and much more step-by-step This website uses cookies to ensure you get the best experience. Calculus: Integral with adjustable bounds. To embed this widget in a post on your WordPress blog, copy and paste the shortcode below into the HTML source: To add a widget to a MediaWiki site, the wiki must have the. In single-variable calculus, we found that one of the most useful differentiation rules is the chain rule, which allows us to find the derivative of the composition of two functions. Multivariable calculus can be applied to analyze deterministic systems that have multiple degrees of freedom. The calculator will calculate the multiple integral (double, triple). Calculus: Fundamental Theorem of Calculus. 3. This course is the second part of a two-course sequence. Show Instructions. The book’s aim is to use multivariable calculus to teach mathematics as a blend of reasoning, computing, and problem-solving, doing justice to the structure, the details, and the scope of the ideas. Learn multivariable calculus for free—derivatives and integrals of multivariable functions, application problems, and more. Some of the applications of multivariable calculus are as follows: Multivariable Calculus provides a tool for dynamic systems. Multivariable calculus is the branch of calculus that studies functions of more than one variable. Then the differential for a multivariable function is given by three separate formulas. Loci Article. For my computer science project I am working on a calculator that will find(or approximate) the point of equidistance along the surface of the Earth from N number of points. example. This professional online calculator will help you calculate and calculate the limit of a function in a few seconds. The calculator will find the gradient of the given function (at the given point if needed), with steps shown. Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds. Functions. In general, you can skip the multiplication sign, so `5x` is equivalent to `5x`. This website uses cookies to ensure you get the best experience. Success in your calculus course starts here! Free online 3D grapher from GeoGebra: graph 3D functions, plot surfaces, construct solids and much more! Enter your Limit problem in the input field. Multivariable Calculus Applications. Multivariable calculus is a huge field that usually covers an entire semester, usually after at least one full year of single variable calculus. What is Multivariable Limit. Pick one of our Multivariable Calculus practice tests now and begin! Calculus: Integrals. Well, this is perhaps the core observation in well, calculus, not just multivariable calculus. In general, you can skip the multiplication sign, so `5x` is equivalent to `5x`. The concept of a function of one variable can be easily generalized to the case of two or more variables. This professional online calculator will help you calculate and calculate the limit of a function in a few seconds. The multivariable linear approximation by Duane Q. Nykamp is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License. Try changing it! Calculadora gratuita de cálculo de multivariáveis - calcule limites multivariáveis, integrais, gradientes e muito mais passo a passo In the pop-up window, select “Find the Multivariable Limit”. Press Enter on the keyboard or on the arrow to the right of the input field. Functions with independent variables corresponding to each of the degrees of freedom are often used to model these systems, and multivariable calculus provides tools for characterizing the system dynamics. example. Multivariable calculus calculator; Calculus derivative calculator; Calculus arc length calculator; And even a pre calculus calculator; The variety of problems in which this calculator can be of assistance make it one of your best choices among all other calculus calculators out there. To embed this widget in a post, install the Wolfram\|Alpha Widget Shortcode Plugin and copy and paste the shortcode above into the HTML source. For permissions beyond the scope of this license, please contact us . Book Review. Find more Mathematics widgets in Wolfram\|Alpha. The course followed Stewart’s Multivariable Calculus: Early Transcendentals, and many of the examples within these notes are taken from this textbook. The same thing is true for multivariable calculus, but this time we have to deal with more than one form of the chain rule. For iPhone (Safari) - Touch and hold, then tap Add Bookmark, 4. To this end, I have tried to write in a style that communicates intent early in the discussion of each The reason why this is the case is because a limit can only be approached from two directions. Only 1 left in stock - order soon. Derivatives and Integrals of Multivariable Functions. Limits in single-variable calculus are fairly easy to evaluate. Calculus III Calculators; Math Problem Solver (all calculators) Multiple (Double, Triple) Integral Calculator. An important theorem in multivariable calculus is Green's theorem, which is a generalization of the first fundamental theorem of calculus to two dimensions. You can also use the search. Let’s say it as it is; this is not a calculator … This is an example of pushing the limits of the calculator. How to Evaluate Multivariable Limits. From Multivariable Equation Solver to scientific notation, we have got all kinds of things covered. Course Sequences. In this case, the calculator gives not only an answer, but also a detailed solution, which is useful to analyze, especially if your own result does not coincide with the result of its calculations. Free pre calculus calculator - Solve pre-calculus problems step-by-step. Comparison of Single and Multivariable Calculus Functions of one variable (left) are graphed on an x-y axis; The graph on the right is multivariate and is graphed on a … ... Derivatives Derivative Applications Limits Integrals Integral Applications Riemann Sum Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series. Being competent with a graphing calculator is a big help with the material covered in this book. vectors, lines, planes, surfaces, calculus of vector-valued functions, dot and cross products, open and closed sets, linear transformations, quadratic forms, limits (upper link is … For example, jaguar speed -car Search for an exact match Put a word or phrase inside quotes. X Exclude words from your search Put - in front of a word you want to leave out. James Stewart's CALCULUS texts are world-wide best-sellers for a reason: they are clear, accurate, and filled with relevant, real-world examples. The calculator will quickly and accurately find the limit of any function online. Using the online calculator to calculate the limits, you will receive a detailed solution to your problem, which will allow you to understand the algorithm for solving problems and consolidate the material. Find more Mathematics widgets in Wolfram\|Alpha. By James Stewart - Multivariable Calculus: Concepts and Contexts, 3rd (third) Edition: 3rd (third) Edition. Get the free "Critical/Saddle point calculator for f (x,y)" widget for your website, blog, Wordpress, Blogger, or iGoogle. Show Instructions. For Google Chrome - Press 3 dots on top right, then press the star sign. f(x,y) is any 3-d function. It is used in various fields such as Economics, Engineering, Physical Science, Computer Graphics, and so on. Take one of our many Multivariable Calculus practice tests for a run-through of commonly asked questions. The calculator will quickly and accurately find the limit of any function online. The following lecture-notes were prepared for a Multivariable Calculus course I taught at UC Berkeley during the summer semester of 2018. Partial derivatives and multiple integrals are the generalizations of derivative and integral that are used. Visualizations and other useful material for multivariable calculus, sometimes called Calculus III and IV. Using this online calculator to calculate limits, you can very quickly and easily find the limit of a function. Free detailed solution and explanations Multivariable Linear Approximation - An expression with a power in 2 variables - Exercise 3390. 18.02 Multivariable Calculus (Spring 2006) 18.022 Calculus of Several Variables (Fall 2010) 18.024 Multivariable Calculus with Theory (Spring 2011) Related Content. Get the free "Multivariable Limits" widget for your website, blog, Wordpress, Blogger, or iGoogle. Student Solutions Manual, Chapters 10-17 for Stewart's Multivariable Calculus, 8th (James Stewart Calculus) Part of: James Stewart Calculus (3 Books) 4.1 out of 5 stars 39. In calculus-online you will find lots of 100% free exercises and solutions on the subject Multivariable Linear Approximation that are designed to help you succeed! The limits of functions can be considered both at points and at infinity. ... Calculus: Taylor Expansion of sin(x) example. The limits of functions can be considered both at points and at infinity. Paperback $33.58 $ 33. ... 2019 math, learn online, online course, online math, calculus 3, calculus iii, calc 3, calc iii, multiple integrals, triple integrals, midpoint rule, estimating triple integrals, midpoints, cubes, sub-cubes. 58 to rent $79.58 to buy. Braess’ Paradox in City Planning: A Mini-Primary Source Project for Multivariable Calculus Students. FREE Shipping. Come to Sofsource.com and figure out adding fractions, power and plenty additional algebra subject areas One of the core tools of Applied Mathematics is multivariable calculus. You will receive incredibly detailed scoring results at the end of your Multivariable Calculus practice test to help you identify your strengths and weaknesses. By using this website, you agree to our Cookie Policy. Book Review. The first course in the sequence is 18.01SC Single Variable Calculus. To embed a widget in your blog's sidebar, install the Wolfram\|Alpha Widget Sidebar Plugin, and copy and paste the Widget ID below into the "id" field: We appreciate your interest in Wolfram\|Alpha and will be in touch soon. Press 3 dots on top right, then press the star sign calculus practice tests now begin! Approximation - an expression with a power in 2 variables - Exercise 3390 steps shown and. Points and at infinity Sum Series ODE Multivariable calculus, and so on to... Widget for your website, blog, Wordpress, Blogger, or iGoogle from GeoGebra: graph functions. Calculus Laplace Transform Taylor/ Maclaurin Series Fourier Series select “ find the gradient of the input field input... 4.0 License GeoGebra: graph 3D functions, plot surfaces, construct solids and much more Nykamp! Can be considered both at points and at infinity of commonly asked questions for an match... Calculator is a big help with the material covered in this book functions of more than one can! 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Generalizations of derivative and Integral that are used Touch and hold, then press star. * x ` GeoGebra: graph 3D functions, plot surfaces, construct solids and much more quickly accurately.	{"url":"https://hotelonyx-gubin.pl/gestapo-synonym-mjivgj/ca3862-multivariable-calculus-calculator","timestamp":"2024-11-05T17:06:54Z","content_type":"text/html","content_length":"34225","record_id":"<urn:uuid:ff6cd919-02d4-4288-8375-45249717bf83>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00796.warc.gz"}
What does a wavelet coherence plot tell you? 1 min read Can you find any pattern in the two signals, green and blue? Two Time Series The blue signal is the brain wave (measured by NIRS) of a person when he is pressing some buttons (the timing of button pressing is shown in the vertical lines); the green signal is also brain wave but from a second person who is doing the same thing (button pressing). With naked eyes, it’s hard to see if there is any pattern in the signals in relation with each other, or with the button pressing. But wavelet coherence analysis tell you something. Wavelet Coherence In the above figure, I plot the wavelet coherence between the two signals in both time and frequency domain. Coherence is kind of correlation. 1 (red) means the two signals are highly correlated and 0 (blue) means no correlation. There are definitely something interesting between the two signals. First, there is a red band in the period 8 region. As the sampling frequency of the signals is 10Hz, period 8 means 0.8s. This band is originated from heart beating (~1Hz) and indicates that the two people’s heart beating is highly correlated. Second, there are some red blobs in the period 64 region. The button pressing is occurring at 6-7s frequency. These blobs indicate that the two people’s brain are correlated during button pressing. So, with wavelet coherence analysis, you can discover something you might not discover with other methods. [DEL:Cross Wavelet and Wavelet Coherence Toolbox:DEL] Note: this link is dead. To download the “Cross Wavelet and Wavelet Coherence Toolbox”, please enter Other blog posts on wavelet analysis PS: You may wonder why the two brain signals are correlated. In fact, the two people are doing a cooperative task. They have to press buttons at the same time to win a point. We at the same time measure their brain activity and find their brain is correlated during cooperation (but not during competition). Check out our publication: Cui et al, NIRS-based hyperscanning reveals increased interpersonal coherence in superior frontal cortex during cooperation, NeuroImage, 2012 [update 2017-01-25] We contributed to MatLab (wavelet toolbox) https://www.alivelearn.net/?p=1957 文献鸟双 11 特惠 31 Replies to “What does a wavelet coherence plot tell you?” 1. Dear Dr. Xu: Could you please tell me what tools you usually apply when you analyze the data from NIRS? I always get signals from which the pattern are usually hard to detect with bare eye. Also, when you consider the changes in signal, did you take the scattering coefficient changes into consideration? Hope to see your reply. 2. @Zoe I usually write my own programs to analyze data. I used nirs-spm for a while and you may check it out. We do use scattering coefficient to calculate the change of hbo and hbr. 3. Dear Dr. Xu: Thank you, I will check the SPM. Does that mean that you calculate the scattering coefficient in real time, and apply the real time optical property to calculate the HbO and Hbr level? And how do you usually consider the brain structure, do you consider it as several layers or you treat it as a homogeneous tissue? Thanks for your time! Best Regards 4. @Zoe It’s called NIRS-SPM. No. For we calculate after we acquired the data. For real time experiment, the device we use (ETG4000) calculated the hbo hbr level automatically. We did not consider the structure of brain in the calculation. 5. Hi Dr Xu could you explain the difference between a crosswavelet and a wavelet coherence? 6. @Guillermo I feel like wavelet coherence (value from 0 to 1) is normalized crosswavelet (value can be complex). 7. Can we classify normal ECG and abnormal ECG using wavelet coherence and cross correlation. 8. Hi, Dr Cui What is coherence mean? I know it is a character of waves, which have the same frequency. But, when it comes to two persons, what’s that mean? And what could it tell us? 9. @Ning Hi, Lili, Coherence is “correlation” between two signals at a certain frequency, also accounting for lags. I have a few posts on wavelet coherence and they may help: 10. Thank you for your reply. And there is another question, how to choose the frequency band?Is that depends on RT?@Xu Cui 11. Dear Dr. Xu My research is related to “Spatiotemporal groundwater level modeling using hybrid artificial intelligence-meshless method” i like to know what is the correlation between input time series data (rainfall, runoff and grounwater level lag time series) and output time series data (groundwater level). ANN input and output prediction: R(t-1) , R2(t-1), GL (t-1) —> Gl (t) What can use wavelet coherence? 12. Dear Sir, can you please guide me how I can cut the cone of influence in WTC diagram and how I can limit my frequency scale upto 256 13. @Naveed Raza To cut, you may use photoshop? I never did that. As to limit your frequency, you may use ‘ms’ argument. Please refer to: 14. Dear Dr. Xu, I wonder if you could share a small sample as .CVS file from one of your experiments. The reason I am asking for this is that the data I am using appears to have a different result when I plot the wavelet coherence. So I do not know if my database is wrong or not. I did an anti-correlation check between Hbo and Hbr and it appears to be OK with only 3 or 4 bad channels. I can share/email my figures if it is needed. Thanks for your help. 15. @Zapata Check out 16. Dear Dr. Cui, As you said, there are some red blobs in the period 64 region (i.e., about 6.4s). The button pressing is occurring at 6-7s frequency. These blobs indicate that the two people’s brain are correlated during button pressing. Since your task lasted about only 6 min, the peak MaxScale reached 1024 region. I used the same wtc method to analyze the data lasting 40 min (task:a long two-person talk),however the peak MaxScale reached 8192 region. My question is, whether the horizontal coordinate (i.e., Time) covaried with the vertical coordinate (i.e., Period)? Closely related, the inter-brain coherence in my 40-min talk mainly appears in the 256-1024 period. It is weird to focus such high period based on your rationale provided (i.e., 64period corresponds 6~7s button-pressing). Simply because I have not see such high period in existing fNIRS hyperscanning studies. I look forward to hearing you! Yafeng Pan 17. Dear Dr. Xu 呢，还请Dr. Xu帮忙解释一下. Thank you! 18. @Hu Sheng 19. 您好，我在理解matlab生成的wavelet coherence图的时候发生了困难，找到了您的博客。 1.我很困惑的是生成了这样一个wavelet scale – frequency/period的矩阵，如何才能把它压缩成一个scalar来形容这两个timeseries的整体的相关性？就像相关系数那样？ 2.我有同一个人两个不同脑部ROI的fMRI的timeseries, 有80个timepoints, TR=720ms。那么对应timescale-period图中的某一个点，例如, 在time40秒处period20秒处的取值是0.999，表达了什么含义呢？是从0~40秒的范 20. @Xiang Xiao 1. 可以把这个矩阵的数值平均成一个值。参考：http://www.alivelearn.net/?p=1518 2. 说明在time 40s附近，这两个信号在1/20频率上有很高的相关度。 21. 感谢回答！我也看了您哪一篇，您的意思是先过滤一部分必定相关的（例如period > 32 ）然后再直接求平均吗？这样会不会损失了比较多的情报。另外再追问一下，https://cn.mathworks.com/help/wavelet/ref/wcoherence.html matlab2016b自带了Wavelet coherence的模块，但是计算Wavelet coherence矩阵的时候可以设置两个参数，一个是period，一个是 22. @Xiang Xiao matlab自带的wavelet coherence这个，我还不熟悉，因此不敢乱给建议。 btw，matlab的wavelet package中用到的nirs数据是我们的数据：https://www.mathworks.com/help/wavelet/examples/compare-time-frequency-content-in-signals-with-wavelet-coherence.html 23. Please check out: http://www.alivelearn.net/?p=1957 24. 我想请问，wavelet coherence中关于significance level的计算问题，您论文中的significance level的计算方法跟A Grinsted工具箱中的一样吗？ 25. @Yibo Wang 26. 非常感谢您的回答，我想问一个问题，当有多个试次时，如何计算这些试次的平均wavelet coherence？ 27. @Yibo Wang This post might be helpful: 28. Dear Cui, Thank you for the complete explanation for the wavelet analysis. I am currently conducting an analysis using this method. You mention that coherency in wavelet representing the correlation between two (time-series) dataset. However, is there anyway to understand if the data were negatively or positively correlated? 1. The arrow direction indicate the phase difference. Please refer to: https://alivelearn.net/?p=1169 29. Hi, I am making a wavelet coherence plot on the MatLab 2021 version. sir I want to know about sampling interval theory and how can we set the optimum sampling interval according to any given data and I am unable to change my y-axis values in the MatLab 2021 wavelet coherence plot. 1. Shafqat, unfortunately I do not know sampling interval theory …	{"url":"https://www.alivelearn.net/?p=1426","timestamp":"2024-11-04T21:18:49Z","content_type":"text/html","content_length":"90962","record_id":"<urn:uuid:02f40246-27f0-41d9-80b4-9af1ac43333c>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00225.warc.gz"}
Horizontal Stretch and Compression How to Do Horizontal Stretch in a Function Let f(x) be a function. Let g(x) be a function which represents f(x) after an horizontal stretch by a factor of k. where, k > 1. In the function f(x), to do horizontal stretch by a factor of k, at every where of the function, x co-ordinate has to be multiplied by k. The graph of g(x) can be obtained by stretching the graph of f(x) horizontally by the factor k. Note : Point on the original curve : (x, y) Point on the curve after it is horizontally stretched by the factor of k : (kx, y) How to Do Horizontal Compression in a Function Let f(x) be a function. Let g(x) be a function which represents f(x) after a horizontal compression by a factor of k. where k > 1. In the function f(x), to do horizontal compression by a factor of k, at every where of the function, x co-ordinate has to be multiplied by 1/k. The graph of g(x) can be obtained by compressing the graph of f(x) horizontally by the factor k. Note : Point on the original curve : (x, y) Point on the curve after it is vertically compressed by the factor of k : ((1/k)x, y) Example 1 : Perform a horizontal stretch by a factor 2 to the function f(x) = (x - 1)^2 And also write the formula that gives the requested transformation and draw the graph of both the given function and the transformed function Answer : Step 1 : Let g(x) be a function which represents f(x) after the horizontal stretch by a factor of 2. Since we do horizontal stretch by the factor 2, we have to replace x by (1/2)x in f(x) to get g(x). Step 2 : So, the formula that gives the requested transformation is g(x) = f[(1/2)x] g(x) = [(1/2)x - 1]^2 Step 3 : The graph g(x) = [(1/2)x - 1]^2 can be obtained by stretching the graph of the function f(x) = (x - 1)^2 vertically by the factor 2. (x, y) -----> (2x, y) Step 4 : Table of values : Step 5 : Graphs of f(x) and g(x) : Example 2 : Perform an horizontal compression by a factor 2 to the function f(x) = (x - 1)^2 And also write the formula that gives the requested transformation and draw the graph of both the given function and the transformed function Answer : Step 1 : Let g(x) be a function which represents f(x) after the vertical compression by a factor of 2. Since we do vertical compression by the factor 2, we have to replace x by 2x in f(x) to get g(x). Step 2 : So, the formula that gives the requested transformation is g(x) = f(2x) g(x) = (2x - 1)^2 Step 3 : The graph g(x) = (2x - 1)^2 can be obtained by compressing the graph of the function f(x) = (x - 1)^2 vertically by the factor 2. (x, y) -----> ((1/2)x, y) Step 4 : Table of values : Step 5 : Graphs of f(x) and g(x) : Kindly mail your feedback to v4formath@gmail.com We always appreciate your feedback. ©All rights reserved. onlinemath4all.com	{"url":"https://www.onlinemath4all.com/horizontal-stretch-and-compression.html","timestamp":"2024-11-08T11:34:55Z","content_type":"text/html","content_length":"45832","record_id":"<urn:uuid:e736d9d5-a978-47c0-b229-e358cfe52f9d>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00193.warc.gz"}
Laws of Logarithms Question Video: Laws of Logarithms Mathematics • Second Year of Secondary School Select the expression equal to log_(π ) π ₯/log_(π ) π ¦. [A] log_(π ) π ¦/log_(π ) π ₯ [B] log_(π ) π ₯/log_(π ) π ¦ [C] log_(π ₯) π /log_(π ¦) π [D] log_(π ₯) π /log_(π ¦) π Video Transcript Select the expression equal to the logarithm base π of π ₯ divided by the logarithm base π of π ¦. Is it option (A) the logarithm base π of π ¦ divided by the logarithm base π of π ₯? Option (B) the logarithm base π of π ₯ divided by the logarithm base π of π ¦. Is it option (C) the logarithm base π ₯ of π divided by the logarithm base π ¦ of π ? Or is it option (D) the logarithm base π ₯ of π divided by the logarithm base π ¦ of π ? In this question, weβ re given four different expressions. And we need to determine which of these is equal to the expression given to us in the question. And thereβ s a few different ways we could go about this. Weβ re going to use the fact that the expression given to us in the question is the quotient of two logarithms to the same base. And this can remind us of the change of base formula for logarithms, which tells us for any positive real numbers π , π ₯, and π ¦, where π and π ¦ are not equal to one, the logarithm base π of π ₯ divided by the logarithm base π of π ¦ is equal to the logarithm base π ¦ of π ₯. To apply this to the expression given to us in the question, we need π , π ₯, and π ¦ are positive real numbers, where π and π ¦ are not equal to one. And we can see this is true in this case. Weβ re taking the logarithm of π ₯, so π ₯ is positive. Weβ re taking the logarithm of π ¦, so π ¦ is positive. And π is a base of the logarithm, so π is positive and not equal to one. Finally, since π ¦ is the logarithm in our denominator, we canβ t divide by zero. So, π ¦ is not allowed to be equal to one. Therefore, by using the change of base formula, weβ ve shown the expression given to us in the question is equal to the log base π ¦ of π ₯. However, none of the four options are exactly the same as this expression. So, weβ re going to need to manipulate this even further. Since all four options are the quotient of two logarithms, weβ re going to apply the change of base formula one more time. This time, however, weβ re going to change the base of our logarithms to some positive value π not equal to one. This then gives us the logarithm base π ¦ of π ₯ is equal to the logarithm base π of π ₯ divided by the logarithm base π of π ¦. And of course, this is the same as the expression given to us in the question. And we can then see this is the same as the answer given in option (B). Therefore, by using the change of base formula, we were able to show the logarithm base π of π ₯ divided by the logarithm base π of π ¦ is equal to the logarithm base π of π ₯ divided by the logarithm base π of π ¦ provided π is a positive real number not equal to one. This was answer option (B).	{"url":"https://www.nagwa.com/en/videos/946101647383/","timestamp":"2024-11-13T22:02:23Z","content_type":"text/html","content_length":"252413","record_id":"<urn:uuid:11674cbc-a3d9-42ef-aa4d-1db3ab691a12>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00185.warc.gz"}
Finance Textbook Notes - Chapter 10: Adjusting For Uncertainty and Financing Effects In Capital Budgeting - MBA Boost Finance Textbook Notes – Chapter 10: Adjusting For Uncertainty and Financing Effects In Capital Budgeting Notes from various chapters in the core finance textbook, Financial Management, Concepts and Applications by Ramesh K.S. Rao. Finance Source Financial Management, Concepts and Applications by Ramesh K.S. Rao 1. Sensitivity analysis aids managers because it enables them to conduct “what-if’ scenarios for projects. This is important because expected cash flows are uncertain, and deviations in CFAT variables from their expected values can have varying degrees of influence on a project’s NPV. By looking at various potential outcomes, the manager is more informed about potential outcomes other than the expected outcome and can use the information to gain comfort about or otherwise reject a positive NPV project. 2. Sensitivity analysis is advantageous for the reasons mentioned above, and it is relatively easy to perform. Disadvantages include its vagueness regarding optimistic or pessimistic scenarios, ignorance of potentially interrelated variables, and the inability to examine several scenarios simultaneously. 3. Computer simulation is a computerized way of performing complicated sensitivity analysis. With the power of the computer, many more scenarios can be run (simultaneously if desired), providing a deeper analysis of the project. 4. Disadvantages of computer simulation are primarily its expense, both in monetary terms and in time. Many simulation models are also often difficult to build. May not be suitable for a small company and most cost-effective for large projects at large companies. 5. The logic underlying the WACC approach is that any project must return at least what it cost to obtain the funds to enter the project. 6. The primary difficulty in calculating the cost of debt is bankruptcy risk and determining the appropriate interest rate that takes into account expected default or other non-compliance. It is assumed for practical purposes that the interest rate on the debt has been properly “bid-up” by the market and therefore takes those things into account. 7. The WACC approach is considered company specific because it uses the company to determine the discount rate. This implicitly assumes that the cash flows for the project will be similar in nature to the company’s own cash flows – and therefore the risk associated with the project is based on the risk of the company considering the project. 8. In its true form, the NPV calculation is based on the RRR, or opportunity cost, of the company, which may or may not be its WACC and certainly is not dependent on its cost of debt or equity. As such, WACC is often considered to be inconsistent with opportunity cost itself and as it relates to the NPV calculation. 9. It is appropriate to use WACC in “scale-enhancing” projects because only in such projects are the future cash flows reasonably similar in nature the company’s current cash flows. It also might be acceptable in projects that were very similar to “scale-enhancing.” 10. The Adjusted Present Value (APV) approach is designed to take into account the effects of subsidized financing of projects. In this three-step process, the equity-only NPV is calculated, and then to that the NPV of any flotation costs are added, along with the NPV of any subsidized debt financing. 11. The major problem with implementing the APV approach is determining the appropriate beta to use when calculating the un-leveled cost of equity to be used in the equity-only discounting of future cash flows. Because projects deal with asset values, for which limited historical information generally exists, it is often impossible to calculate betas for the project (assets) based on historical figures (like can be done with securities). 12. If historical information is available, project betas can be calculated in the same manner as company betas using regression analysis by comparing historical returns on assets vs. the market. 13. A pure-play method is used when historical information about a project (its assets) is not available, and the project is not scale-enhancing in nature. 14. The logic underlying the pure-play method is that companies or industries engaged full-time in the type of project under consideration have cash flows similar in nature and risk to the expected cash flows on the project. Therefore the risk, expressed as beta, for that company should approximate the risk for its underlying cash flows and therefore the risk for such a project. 15. An adjustment is usually required to the pure-play’s beta value to account for leverage. Pure-play’s are usually not leveraged in the same manner as the company doing the capital budgeting. To get a true (all-equity) beta for the project, which is independent of the means of financing used for the project (or by the pure-play), the beta must be stripped of its financial risk components so that all that is left is the pure equity (market) risk. There Are No Comments Click to Add the First »	{"url":"https://www.mbaboost.com/finance-textbook-notes-chapter-10-adjusting-for-uncertainty-and-financing-effects-in-capital-budgeting/","timestamp":"2024-11-11T17:58:19Z","content_type":"text/html","content_length":"137786","record_id":"<urn:uuid:c465522e-3eb7-4139-beaf-d6d0a7af3e04>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00007.warc.gz"}
Compact and fast 10bit to BCD conversion for converting small / midrange PIC 10bit ADC ADRSH and ADRSL ;* 10bit BINARY to BCD * ;An implimentation of the 'Shift and add-3' Algorithm ; for small MicroChip 'PIC' microcomputers ;1)Shift binary number one bit left (into a BCD 'result' ; (initially empty) ;2)If any 4 bit BCD column is greater than 5[hex] add 3[hex] ;3)Goto 1) {until the LSB binary bit is shifted out} ;<------MSB LSB-----> ;------------- BCD ------------] [------ binary ----- ; TENS UNITS ; BCD column BCD column MSB bit bit ; 0 0 0 0 0 0 0 0 1 0 1 ; <------ / <---/ <---/ Processor 16F818 include <p16f818.inc> cblock 0x020 ;Define variable block starting at $020 binH ;bin is the 8bit binary value to be converted bin ;The 2bit binary MSB's to be converted bcdH ;Thousands (always blank)/ Hundreds nybbles bcdL ;Tens / Units nybbles _bin2bcd movlw d'8' movwf counter clrf bcdL clrf bcdH ;Save time by not shifting in first 6 bits (always '0's) swapf binH,1 ;Chop off first nybble (TEN THOUSANDS) rlf binH,1 ;Shift out first 2 MSB's(always '0's) rlf binH,1 ;Ssave more ime by no test and add +3' for first TWO shifts rlf binH,F ;Shifting 'binH' left through carry into BCD nybbles rlf bcdL,F rlf binH,F rlf bcdL,F ;binH (pic ADC register ADRESH) done with! ;Iteration loop shifts the 4 'columns' (10TH, THO, HUN, TEN, UNT) 1 bit left ;Tests each coulmn (4bit nybble) if > 5, adds '3' ;Shifts in the next MSB of the binary for conversion on the right. ;(for remaining EIGHT shifts) Next_bit movfw bcdL addlw 0x33 ;Add 3 to both nybbles (T/U) in 'temp' movwf temp movfw bcdL btfsc temp,3 ;Test MSB of Units+3 nybble in 'temp addlw 0x03 ;Add 3 to U nybble if units+3 MSB = 1 btfsc temp,7 ;Test MSB of Tens+3nybble in 'temp' addlw 0x30 ;Add 3 to T nybbleiftens+3 MSB=1 (>5) movwf bcdL movfw bcdH addlw 0x33 ;Add 3 to both nybbles (T/H) in temp mmovfw bcdH btfsc temp,3 ;Test MSB of Hundreds+3 nybble in temp addlw 0x03 ;Add 3 to H nybble if units+3 MSB=1 btfsc temp,7 ;Test MSB of Thousand+3 nybble in temp addlw 0x30 ;Add 3 to TH nybble if tens+3 MSB=1 movwf bcdH rlf bin,F ;Shift in next MSB from bin into T/UNITS rlf bcdL,F ;Shift next MSB from TENS/UNITS to TH/H rlf bcdH,F ;Shift up decfsz counter,F goto Next_bit	{"url":"http://techref.massmind.org/techref/microchip/math/radix/index.htm","timestamp":"2024-11-10T18:48:41Z","content_type":"text/html","content_length":"34700","record_id":"<urn:uuid:363e9d4c-7adf-4992-b0d9-2ac7f7104b7d>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00089.warc.gz"}
mutetrahedron - Wiktionary, the free dictionary A finite portion of a mutetrahedron Short for multiple tetrahedron. mutetrahedron (plural mutetrahedra or mutetrahedrons) 1. (geometry) A regular skew apeirohedron with six hexagons around each vertex, formed by an infinite number of truncated tetrahedron-like cells (specifically, truncated tetrahedra missing their triangle faces, with the resulting triangle-shaped holes joined to form empty spaces in the shape of faceless tetrahedra) in Euclidean 3D space.	{"url":"https://en.m.wiktionary.org/wiki/mutetrahedron","timestamp":"2024-11-11T00:52:23Z","content_type":"text/html","content_length":"37254","record_id":"<urn:uuid:f1b415e4-1fdd-48db-922c-a6b94fd0f90d>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00141.warc.gz"}
Study on the regulation function of spinal cord micro-stimulation signal parameters on hind limb movement in rats Functional electrical stimulation of the spinal cord can evoke limb movement in patients with motor dysfunction caused by injury or pathology. Research question: However, the adjustment function of micro-stimulation signal parameters in the spinal cord on the motion of hind limbs about rodents has not been identified. The amplitude, frequency and pulse width of the spinal cord micro-stimulation signal were adjusted to quantitatively analyze the changes of the joint angles when the hindlimb produced extension and flexion responses. When the rat’s extension and flexion responses are induced, the optimal stimulus signal amplitudes are 40 µA and 90 µA respectively. At the same time, the optimal stimulation signal frequency range is (35±5) Hz and the best pulse width of the stimulation signal is 200 µs. The results can provide a further reference for the development of spinal cord stimulator for hindlimb regulation. • The machine learning module was used for captures the gait of rats • Different electrical signal were adjusted to quantitatively analyze the gait of rats • This study provide a further reference for the development of hindlimb regulation stimulator 1. Introduction Intraspinal micro-stimulation (ISMS) induces movement by directly stimulating the ventral motor circuit of the spinal cord to recruit more motor units [1, 2]. In recent years, intraspinal micro-stimulation has been used as a treatment method for spinal cord injury and successfully induced limb movement [3-6]. Studies have shown that parameters such as electrode structure, the position of spinal cord stimulation, frequency and amplitude of stimulation signal are key factors determining the motor output [7-9], and adjusting the frequency, amplitude and pulse width of micro-stimulation signals in the spinal cord can change the intensity of muscle contraction [10-13]. A typical muscle is made up of hundreds or even thousands of fibers arranged as functional clusters of motor units [14]. As the intensity of the stimulus increases, the activated motor unit increases, resulting in an increase in the output force [15]. In healthy and paralyzed muscles, the linear relationship between the intensity of the current and the generation of the force has been described [15-17]. The relationship between force and frequency indicates that the increase in stimulus frequency leads to an increase in muscle strength, while high frequency leads to muscle fatigue [18]. With the increase of pulse duration, lower stimulation intensity is required to activate the surrounding motor nerves and achieve the required force output [19]. However, pulse duration may affect muscle tone, leading to muscle fatigue [20]. Many studies have shown that the parameters regulating stimulation signal have an effect on hind limb movement, for example, the effects of pulse frequency and duration on muscle torque and fatigue have been studied [21], the exact mapping between the parameters of spinal cord micro-stimulation signal and the changes in joint angles induced by hind limb movement in rats is still unclear. 2. Materials and methods 2.1. Experimental rats and stimulating electrodes All protocols involving the use of animals in this study were approved by the Institutional Animal Care and Use Committee of Nantong University, China (Approval No. 20190225-008) on February 26, 2019. A total of 6 Sprague-Dawley rats (8 weeks, both sexes, weighing 220-250 g) were purchased from the Experimental Animal Center of Nantong University (License No. SYXK (Su) 2017-0046). After intraperitoneal injection of 10 % chloral hydrate (4 mL/kg), the hair of the back and right hindlimb was removed after anesthesia, and 75 % rubbing alcohol was used to sterilize T13-L3 of the segment of the spine. The skin was cut along the spine to expose the spinal cord. All experiments were acute. Stimulating electrode used tungsten electrode (produced by Microprobes company, United States); The electrode model for WE30030.5A3, 0.081 mm diameter of axle, cutting-edge 2-3 microns in diameter, 0.5 mΩ impedance. 2.2. Stimulation signal and location The stimulation signal parameters are set by master-9 pulse stimulator (Israel A.M.P.I. company). The number of repetitions of stimulation pulses $N$ is 40. The stimulus isolator (Iso-flex, Israel A.M.P.I.) adjusts the amplitude of the stimulus current. The research group has applied functional electrical stimulation technology to complete the determination of the core area of hindlimb motor function in rats [22]. The rats were placed on a fully automatic stereotactic device (51700, Stoelting, USA), the spinal cord was fixed with a rat spinal adapter, and electrodes, assisted by the stereotactic device, were implanted into the core functional areas of extension and flexion. 2.3. Data collection and processing Machine vision module OpenMV Cam M7 was used to capture the right hindlimb movement of rats. According to the posterior limb skeleton of the rat (Fig. 1(a)), the sagittal motion model of the posterior limb of the rat was established (Fig. 1(b)). Five self-made color labels were attached to the anterior superior iliac spine, hip, knee, ankle and the top of the fifth bone of the posterior limb according to the motion model. All the color labels were operated by the same person. Fig. 1Schematic diagram of rat hind limb skeleton and model of sagittal plane motion a) Skeleton and color label of hindlimb b) Schematic diagram of skeleton model of hind limb The machine vision module OpenMV CAM was fixed on a square block and placed 10 cm away from the right hind limb of the rat (in this way, the color label of each joint of the hind limb of the rat could be identified optimally). Meanwhile, the module was connected to a personal computer, and the coordinate information of the color label was collected in real time by using OpenMV IDE. Before recording the posterior limb movement, the angle baselines of the hip, knee and ankle before and after stimulation were recorded respectively. According to the coordinate information of each joint, the vector of the hip pointing to the anterior superior iliac spine is defined as $\stackrel{\to }{A}$, the vector of the hip pointing to the knee joint is defined as $\stackrel{\to }{B}$, the vector of the ankle pointing to the knee joint is defined as $\stackrel{\to }{C}$, and the vector of the ankle pointing to the fifth extension bone is defined as $\stackrel{\to }{D}$. The angle of each joint is defined as the angle formed between the joint and the adjacent proximal and distal joint positions. The ${\theta }_{h}$ of the hip joint includes angle between the anterior superior iliac spine and the knee joint, as shown in Eq. (1). The ${\theta }_{k}$ of the knee joint includes angle between the hip joint and the ankle joint, as shown in Eq. (2). The ${\theta }_{a}$ of the ankle joint includes angle between the knee joint and the tip of the fifth extension bone, as shown in Eq. (3): ${\theta }_{h}=\mathrm{c}\mathrm{o}\mathrm{s}\frac{\stackrel{\to }{A}\stackrel{\to }{B}}{\left\|\stackrel{\to }{A}\right\|\left\|\stackrel{\to }{B}\right\|},$ ${\theta }_{k}=\mathrm{c}\mathrm{o}\mathrm{s}\frac{\stackrel{\to }{B}\stackrel{\to }{C}}{\left\|\stackrel{\to }{B}\right\|\left\|\stackrel{\to }{C}\right\|},$ ${\theta }_{a}=\mathrm{c}\mathrm{o}\mathrm{s}\frac{\stackrel{\to }{C}\stackrel{\to }{D}}{\left\|\stackrel{\to }{C}\right\|\left\|\stackrel{\to }{D}\right\|}.$ According to the definition of each joint angle, the position coordinate information of each joint is converted into the corresponding joint angle by using the custom processing code. For each animal, we quantified the changes in joint angles (starting from the initial angle) in the extension and flexion responses of the hind limbs of rats at different stimulation parameters. Taking the response of the extension as an example, the data collection and processing process are shown in Fig. 2. The variables selected the range from baseline angle before the stimulation to the maximum angle of joint formation at the time of the stimulation. Finally, the bar chart of mean and standard deviation (mean ±SD) of different joint angles of six rats under different stimulation parameters was drawn. Fig. 2Data collection and processing of hind limb in rats a) Color label recognition and data collection of Open MV IDE b) Data processing and simulation of processing software 2.4. Statistical analysis The least square regression analysis by MATLAB was performed to analyze the correlation between the mean angular change of hip, knee and ankle joints and stimulation signal parameters. The value of the determination coefficient ${R}^{2}$ is shown in Eq. (4): ${R}^{2}=\frac{S}{T}=\frac{{\sum }_{i=1}^{n}{\left({f}_{i}-\stackrel{^}{y}\right)}^{2}}{{\sum }_{i=1}^{n}{\left({y}_{i}-\stackrel{^}{y}\right)}^{2}},$ where: $T=S+E$, $T$ is the total sum of squares; $S$ is the sum of regressive squares; $E$ is the sum of squared residuals; $Y$ is the actual value; $F$ is the predicted value; $\stackrel{^}{y}$ is the average of the actual values. ${R}^{2}$ determines how close the correlation is, and the closer it gets to one, the more relevant the dependent variable is to the independent variable. The results were statistically analyzed by SPSS software for one-way ANOVA comparison of different joint angles under the different parameters of the stimulation signal. $P<$0.05 on both sides was set as statistically significant difference. 2.5. Results and analysis First, the angle of each joint about normal rats walk on four legs were measured, the initial angle of the hip is 97.1°± 6.2°, the initial angle of knee is 76°±16°, the initial angle of ankle is 101°± 9.8°. After data processing, the values of angles exceeding the maximum range were deleted to obtain the trend diagram between the following angular changes in joints and stimulation parameters. To determine the mapping relationship between single parameter and hindlimb motion, other parameters were kept unchanged. 2.6. Regulation of hindlimb movement by stimulus signal amplitude To study the influence of amplitude on various joints during the extension and flexion of hind limbs of rats, the frequency and pulse width were set as 33.33 Hz and 200 μs. When hind limbs produce the extension response, the amplitude of each stimulus current is set as 10, 15, 20, 25, 30, 35, 40, 45 and 50 μA, respectively. The variation trend of the mean angular change corresponding to the hip joint, knee joint and ankle joint of rats with the amplitude of stimulation current is shown in Fig. 3(a). When the stimulation current is in the range of 10-40 μA, the angular change value of the hip joint increases with the increase of the stimulation amplitude. After that, the amplitude of the stimulation current continued to increase while the angular change value of the hip joint begins to decrease. When the stimulation amplitude is in the range of 10-45 μA, the angular change value of the knee joint showed an upward trend. Until the amplitude of the stimulation current is greater than 45 μA, the angular change of the knee joint begins to decrease. The angular change value of the ankle joint continues to increase in the range of 10-40 μA, and with the amplitude of the stimulation increased, the angular change value of the ankle joint begins to decrease. When hind limbs produce flexion response, the threshold current required is larger than the threshold current that produces the extension response. The amplitude of each stimulation current is set to 20, 30, 40, 50, 60, 70, 80, 90, and 100 μA. The average change of the angles of the hip, knee, and ankle of the hind limbs with the amplitude of the stimulation current is shown in Fig. 3(b). When the stimulation current is in the range of 20-90 μA, the average angular change of hip joint and knee joint on the rising trend; after reaching the extreme value, the stimulation current continues to increase, while the average angular change of hip joint and knee joint begins to decrease. The average angular change value of the ankle joint increases significantly in the range of 20-80 μA. The stimulation current is continuously increased, but the angle change of the ankle joint shows a downward trend. In order to obtain the optimal range of stimulation current, the linear regression analysis is performed on the different current and average angular change value of each joint. It can be concluded that in the extension response, when the current is within the stimulation current range of 15-40 μA, the average angular change value of the hip joint has an optimal linear regression model $y=$ 5.6269 + 0.4202$x$ with the determination coefficients ${R}^{2}=$ 0.9704 and $p=$ 0.0003 (Fig. 3(a1)). When the stimulation current range is in 15-40 μA, the average angular change value of knee joint has the best linear regression model $y=$ 12.6421 + 0.4686$x$ with the determination coefficient ${R}^{2}=$ 0.9831, $p=$ 0.0001 (Fig. 3(a2)). When the stimulation current range is in 15-40 μA, the average angular change value of the ankle joint has the best linear regression model $y=$ 7.4073 + 0.2605$x$ with the determination coefficients ${R}^{2}=$ 0.9711 and $p=$ 0.0003 (Fig. 3(a3)). When a flexion response is produced in the right hind limb of a rat, the average angular change value of hip joint has the best linear regression model $y=$ 2.7437$+$0.1085$x$ with the determination coefficient ${R}^{2}=$ 0.9514, $p=$ 0.0000 (Fig. 3(b1)) in the stimulation current range of 20-90 μA. The average angular change value of knee joint has the best linear regression model $y=$ 6.2747 $+$0.0746$x$ with the determination coefficient ${R}^{2}=$ 0.9903, $p=$ 0.0000 (Fig. 3(b2)) in the current range of 20-90 μA. The average angular change value of the ankle joint has the best linear regression model $y=$ 4.5894 + 0.1007$x$ with the determination coefficient ${R}^{2}=$ 0.9862, $p=$ 0.0000 (Fig. 3(b3)) in the current range of 30-90 μA. Fig. 3a) Trend graph of the change in the average value of the joint angular change value of the hind limbs of rats with the amplitude of the stimulation current during the extension response: a1) optimal linear regression model of hip joint and stimulation current in extension response, a2) optimal linear regression model of knee joint and stimulation current in extension response, a3) optimal linear regression model of ankle joint and stimulation current in extension response; b) trend graph of the change in the average value of the joint angular change value of the hind limbs of rats with the amplitude of the stimulation current during the flexion response: b1) optimal linear regression model of hip joint and stimulation current in flexion response, b2) optimal linear regression model of knee joint and stimulation current in flexion response, b3) optimal linear regression model of ankle joint and stimulation current in flexion response 2.7. Regulation of frequency of stimulus signals on hindlimb movement. When studying the effect of frequency on the joints of hind limbs during extension and flexion, the pulse width is 200 μs, and the current values are 40 μA in the extension response and 90 μA in the flexion response, respectively. In the extension response, the frequency of each stimulation signal is set to 20, 25, 30, 35, 40, 45, and 50 Hz, respectively. The average angular change of the hip, knee and ankle of the hind limbs of the rats with the frequency of the stimulation signal is shown in Fig. 4(a). In the whole stimulation frequency range of 20-50 Hz, the average angular change value of each joint increases with the increase of frequency. When the hind limbs of the rat produce the flexion response, the average angular change value of the hip, knee and ankle joints with the frequency of the stimulation signal is shown in Fig. 4(b). In the whole stimulation frequency range of 20-50 Hz, the average angular change value of each joint increases with the increase of frequency. In the extension response, the average angular change value of the hip joint has the optimal linear regression model $y=$ 3.8827 + 0.1474$x$ with the coefficient of determination ${R}^{2}=$0.9651, $p =$ 0.0001 (Fig. 4(a1)) when the frequency range is in 20-50 Hz. The average angular change of the knee joint is in the frequency range of 20-40 Hz, the optimal linear regression model $y=$ 10.5291 + 0.2882$x$ with the coefficient of determination ${R}^{2}=$ 0.9828 and $p=$ 0.0010 (Fig. 4(a2)). The average angular change of the ankle joint is in the frequency range of 30-50 Hz, and has the optimal linear regression model $y=$ 8.8348 + 0.1782$x$ with the coefficient of determination ${R}^{2}=$ 0.9688, $p=$ 0.0024 (Fig. 4(a3)). In the flexion response, the average angular change of the hip joint can be obtained an optimal linear regression model $y=$ 3.9315 + 0.2368$x$ with the determination coefficient ${R}^{2}=$ 0.9460, $p=$ 0.0011 (Fig. 4(b1)) in the frequency range of 25-50 Hz. The average angular change of the knee joint has the optimal linear regression model $y=$ 4.4306 + 0.1088$x$ with the determination coefficients ${R}^{2}=$ 0.9828, $p=$ 0.0086 (Fig. 4(b2)) in the frequency range of 30-45 Hz. The average angle change of the ankle joint has the best linear regression model $y=$ 7.7389 + 0.0734$x$ with the coefficient of determination ${R}^{2}=$ 0.9469, $p=$ 0.0269 (Fig. 4(b3)) in the frequency range of 30-45 Hz. Fig. 4a) Trend graph of the change in the average value of the joint angular change value of the hind limbs of rats with the frequency of the stimulation current during the extension response; a1) optimal linear regression model of hip joint and frequency in extension response; a2) optimal linear regression model of knee joint and frequency in extension response; a3) optimal linear regression model of ankle joint and frequency in extension response; b) trend graph of the change in the average value of the joint angular change value of the hind limbs of rats with the frequency during the flexion response; b1) optimal linear regression model of hip joint and frequency in flexion response; b2) optimal linear regression model of knee joint and frequency in flexion response; b3) optimal linear regression model of ankle joint and frequency in flexion response 2.8. Regulation of pulse width of stimulation signals on hindlimb movement When studying the effects of pulse width on the joints of hind limbs during extension and flexion response, the frequency is set to 33.33 Hz, and the current values are 40 μA in the extension response and 90 μA in the flexion response, respectively. In the extension response experiment, the pulse width of each stimulation signal is set to 100, 125, 150, 175, 200, 225, 250, 275, and 300 μs, respectively. The average angular change value of the hip, knee and ankle joint of the rats with the pulse width is shown in Fig. 5(a). In the entire pulse width range of 100-300 μs, the average angular change of the hip and knee joints increases with the increase of the pulse width. The average angular change values of the ankle begin to decrease after the pulse width is greater than 275 μs. In the flexion response, the pulse width is set to 100, 125, 150, 175, 200, 225, 250, 275 and 300 μs. The average change values of the angles of the hip, knee and ankle joint with the pulse width are shown in Fig. 5(b). When the stimulation pulse width is in 100-300 μs, the angular change of the hip joint shows an upward trend; the average angular change of the knee joint has decreased after the pulse width reaches 250 μs, and the average angle change of the ankle joint has begun to decrease after the pulse width reaches 275 μs. Fig. 5a) Trend graph of the change in the average value of the joint angular change value of the hind limbs of rats with the pulse width of the stimulation current during the extension response; a1) optimal linear regression model of hip joint and pulse width in extension response; a2) optimal linear regression model of knee joint and pulse width in extension response; a3) optimal linear regression model of ankle joint and pulse width in extension response; b) trend graph of the change in the average value of the joint angular change value of the hind limbs of rats with the pulse width during the flexion response; b1) optimal linear regression model of hip joint and pulse width in flexion response; b2) optimal linear regression model of knee joint and pulse width in flexion response; b3) optimal linear regression model of ankle joint and pulse width in flexion response In the extension response, the average angular change of the hip joint has the optimal linear regression model $y=$ 0.9470 + 0.0588$x$ with the determination coefficient is ${R}^{2}=$ 0.9615, $p=$ 0.0194 (Fig. 5(a1)), when the pulse width is in the range of 125-200 μs. The average angular change of the knee joint is in the range of 125-200 μs. It has the best linear regression model $y=$ –0.4514 + 0.1034$x$ with the determination coefficient ${R}^{2}=$ 0.9787, $p=$ 0.0107 (Fig. 5(a2)). The average angular change of the ankle joint is in the range of 125-225 μs. It has the best linear regression model $y=$ 6.2859 + 0.0417$x$ with the determination coefficients ${R}^{2}=$ 0.9838, $p=$ 0.0009 (Fig. 5(a3)). In the flexion response, the average angular change of the hip joint is in the pulse width range of 150-225 μs, with an optimal linear regression model $y=$ –0.9341 + 0.0491$x$ with the determination coefficient ${R}^{2}=$ 0.9860, $p=$ 0.0070 (Fig. 5(b1)). The average angular change of the knee joint is in the range of 175-250 μs, it has the best linear regression model $y=$ 5.4492 + 0.0170$x$ with the determination coefficient ${R}^{2}=$ 0.9954, $p=$ 0.0023 (Fig. 5(b2)). The average angular change of the ankle joint is in the range of 150-250 μs, it has the best linear regression model $y =$ 3.3210 + 0.0243$x$ with the coefficient of determination ${R}^{2}=$ 0.9824, $p=$ 0.0010 (Fig. 5(b3)). 3. Discussions It is studied that the relationship between the amplitude, frequency and pulse width of the stimulus signal and the changes in the angles of the joints of the hind limbs of rats. When changing the amplitude of the stimulus signal, combining the angular change of each joint with the trend graph of the amplitude of the stimulus signal can get the best linear model of the change in the angle of each joint. It can be determined that the optimal stimulus amplitude of the extension response is 40 μA, and the optimal stimulus amplitude of the flexion response is 90 μA. Under the condition of determining the amplitude of the stimulus signal, it was found that during the frequency range of 10-20 Hz, the hind limbs rarely produced completed hind limb movements, so the experimental data during the frequency range of 10-20 Hz was not put into the trend chart for analysis [23]. When the stimulus signal is high frequency, especially when it is greater than 50 Hz, the hind limbs exhibit ankylosing response, the speed becomes faster, and the changes in the angle of each joint increase significantly, but the gait coordination of SCI rats after functional reconstruction is not consistent with these two aspects, so the high-frequency range angle change values are discarded. By synthesizing the trend graph of the angle change value and frequency of each joint and the best linear model, it can be concluded that the optimal frequency of the stimulus signal can be determined at 35 Hz ± 5 Hz. For the stimulation parameter of pulse width, short pulse width can reduce the stimulation of sensory nerves, but at a certain threshold current, too short pulse width will affect the recruitment of muscle fibers, and in the experiment, the hind limbs showed spasm. Here, the intensity/duration relationship between the threshold amplitude $I$ and the pulse duration $d$ of the rectangular pulse is approximately hyperbola $I-r/d=k$, where $k$ is a constant and $r$ is a horizontal asymptotic value [18]. This relationship indicates that as the pulse duration increases, a lower stimulus intensity ($I$) is required to activate the surrounding motor nerves to achieve the required force output. Longer pulse width will penetrate deeply into the subcutaneous tissue, causing pain. The pulse duration increased to approximately 600 μs has been shown to result in greater force generation. After that, the closer to the base intensity value $I$, the longer pulses do not necessarily result in greater force generation [24]. In conclusion, combining the trend graph of the angular change value of each joint and the optimal linear regression model, the pulse width can be determined as 200 μs. 4. Conclusions In this paper, the mapping relationship between the amplitude of the spinal cord micro-excitation signal and the changes of joint angles of hindlimbs in rats was explored. The angle changes of the hip joint, knee joint and ankle joint of the rat's hind limbs under different stimulation current amplitudes were analyzed, and it was found that there is a strong correlation between them. The angle changes of each joint are positively related to the current amplitude. Combining this relationship and the mechanism of neural control, the best range of stimulation current for the response of rats was (40 ± 5) μA, and the best exciting current range for flexion response was (80 ± 10) μA. Analyzing the data, it can be seen that the production of joint angles is coordinated and consistent. The establishment of the amplitude, frequency and pulse width of the stimulus signal provides a reference for the further development of spinal cord stimulator for posterior limb regulation. • Henneman E. The size-principle: a deterministic output emerges from a set of probabilistic connections. The Journal of Experimental Biology, Vol. 115, 1985, p. 105-112. • Bamfordj A., Putman C. T., Mushahwarv K. Intraspinal microstimulation preferentially recruits fatigue-resistant muscle fibers and generates gradual force in rat. The Journal of Physiology, Vol. 569, Issue 3, 2005, p. 873-884. • Shen Xiaoyan, Wang Zhigong, Ma Le, et al. Selective control of hindlimb movements based on intraspinal functional electronic stimulation. Journal of Biomedical Engineering, Vol. 35, Issue 6, 2018, p. 860-863. • Mushahwar V. Spinal cord microstimulation generates functional limb movements in chronically implanted cats. Experimental Neurology, Vol. 163, Issue 2, 2000, p. 422-429. • Wagnerf B., Mignardotj B., Le Goff Mignardotc G., et al. Targeted neurotechnology restores walking in humans with spinal cord injury. Nature, Vol. 563, Issue 7729, 2018, p. 65-71. • Gad P., Choe J., Nandra M. S., et al. Development of a multi-electrode array for spinal cord epidural stimulation to facilitate stepping and standing after a complete spinal cord injury in adult rats. Journal of NeuroEngineering and Rehabilitation, Vol. 10, 2013, p. 2. • Capogrosso M., Wenger N., Raspopovic S., et al. A computational model for epidural electrical stimulation of spinal sensorimotor circuits. Journal of Neuroscience, Vol. 33, Issue 49, 2013, p. • Wenger N., Moraud E. M., Raspopovic S., et al. Closed-loop neuromodulation of spinal sensorimotor circuits controls refined locomotion after complete spinal cord injury. Science Translational Medicine, Vol. 6, Issue 255, 2014, p. 255ra133. • Rejc E., Angelic A., Bryant N., et al. Effects of stand and step training with epidural stimulation on motor function for standing in chronic complete paraplegics. Journal of Neurotrauma, Vol. 34, Issue 9, 2017, p. 1787-1802. • Kralj A., Grobelnik S. Functional electrical stimulation a new hope for paraplegic patients. Bulletin of Prosthetics Research, Vol. 10, Issue 20, 1973, p. 75-102. • Kralj A., Bajd T., Turk R. Enhancement of gait restoration in spinal injured patients by functional electrical stimulation. Clinical Orthopaedics and Related Research, Vol. 233, 1988, p. 34-43. • Bhadra N., Peckhamp H. Peripheral nerve stimulation for restoration of motor function. Journal of Clinical Neurophysiology, Vol. 14, Issue 5, 1997, p. 378-393. • Condie E., Condie D. Functional electrical stimulation: standing and walking after spinal cord injury. Physiotherapy, Vol. 76, Issue 4, 1990, p. 223. • Lavrov I., Musienkop E., Selionovv A., et al. Activation of spinal locomotor circuits in the decerebrated cat by spinal epidural and/or intraspinal electrical stimulation. Brain Research, Vol. 1600, 2015, p. 84-92. • Adams G. R., Harris R. T., Woodard D., et al. Mapping of electrical muscle stimulation using MRI. Journal of Applied Physiology, Vol. 74, Issue 2, 1993, p. 532-537. • Hillegass E. A., Dudley G. A. Surface electrical stimulation of skeletal muscle after spinal cord injury. Spinal Cord, Vol. 37, Issue 4, 1999, p. 251-257. • Bickel C. S., Slade J., Dudley G. Long-term spinal cord injury increases susceptibility to isometric contraction-induced muscle injury. European Journal of Applied Physiology, Vol. 91, Issues 2-3, 2004, p. 308-313. • Bickel C. S., Gregory C. M., Dean J. C. Motor unit recruitment during neuromuscular electrical stimulation: a critical appraisal. European Journal of Applied Physiology, Vol. 111, Issue 10, 2011, p. 2399-2407. • Gorgey A. S., Mahoney E., Kendall T., et al. Effects of neuromuscular electrical stimulation parameters on specific tension. European Journal of Applied Physiology, Vol. 97, Issue 6, 2006, p. • Gorgey A. S., Poarch H. J., Dolbow D. D., et al. Effect of adjusting pulse durations of functional electrical stimulation cycling on energy expenditure and fatigue after spinal cord injury. Journal of Rehabilitation Research and Development, Vol. 51, Issue 9, 2015, p. 1455. • Gregory C. M., Dixon W., Bickel C. S. Impact of varying pulse frequency and duration on muscle torque production and fatigue. Muscle and Nerve, Vol. 35, Issue 4, 2007, p. 504-509. • Chen Yi, Ma Lei, Du Wei, et al. Measuring functional core regions of hindlimb movement control in the rat spinal cord with intraspinal microstimulation. Journal of Biomedical Engineering, Vol. 34, Issue 4, 2017, p. 622-626. • Nebojsa M., Lana Z., et al. Distributed low-frequency functional electrical stimulation delays muscle fatigue compared to conventional stimulation. Muscle and Nerve, Vol. 42, Issue 4, 2010, p. • Jeon W., Griffin L. Effects of pulse duration on muscle fatigue during electrical stimulation inducing moderate-level contraction. Muscle and Nerve, 2017, https://doi.org/10.1002/mus.25951. About this article Biomechanics and biomedical engineering intraspinal micro-stimulation pulse duration joint angle gait analysis This work is supported by National Natural Science Foundation of China (61534003, 81371663) and Opening Project of State Key Laboratory of Bioelectronics in Southeast University, (Ministry of Education in China) Liberal arts and Social Sciences Foundation (17YJC890022), Natural Science Foundation of Jiangsu Province (BK20170448), and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (16KJB180019), Jiang Su Liberal arts and Social Sciences Foundation (17TYC003).This work is also supported by the “226 Engineering” Research Project of Nantong Copyright © 2021 Lei Ma, et al. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.	{"url":"https://www.extrica.com/article/21595","timestamp":"2024-11-07T16:47:30Z","content_type":"text/html","content_length":"173114","record_id":"<urn:uuid:0ead91ff-4aad-47a4-b1c7-8743f0e4be87>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00812.warc.gz"}
How to Multiply Fractions (like 2/3 x 1/4) with Free Online Tutoring in Math How to Multiply Fractions (like 2/3 x 1/4) with Free Online Tutoring in Math Unlock the secrets of multiplying fractions with our latest blog post, the tenth article in our comprehensive eleven-part series! Each article in this series follows a cyclical learning approach, providing complete lessons on fractions and guiding your child step-by-step through the learning process. Multiplying fractions can be perplexing. Why does multiplying 2/3 by 1/4 result in an answer smaller than both 2/3 and 1/4? Shouldn’t multiplication make numbers bigger? The answer becomes clear when children learn to illustrate mathematical models. These visual aids show why 2/3 x 1/4 results in a smaller product. Without this illustration, many children struggle to understand the concept, leading to confusion and frustration in higher levels of math. 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Embark on this exciting educational journey with us and watch your students thrive! This site uses Akismet to reduce spam. Learn how your comment data is processed.	{"url":"https://teachersdungeon.com/blog/how-to-multiply-fractions-like-2-3-x-1-4-with-free-online-tutoring-in-math/","timestamp":"2024-11-11T23:50:35Z","content_type":"text/html","content_length":"67635","record_id":"<urn:uuid:5d7924cc-b4a8-4e84-bfb7-dface6cb08cc>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00575.warc.gz"}
An extremal problem on degree sequences of graphs Let G = (I[n], E) be the graph of the n-dimensional cube. Namely I[n] = {0, 1}^n and [x, y] ε E whenever \|\|x - y\|\|[I] = 1. For A ⊆ I[n] and x ε A define h[A](x) = #{y ε I[n] / A{[x, y] ε E}, i.e., the number of vertices adjacent to x outside of A. Talagrand, following Margulis, proves that for every set A ⊆ I[n] of size 2^n-1 we have 1/2^n ∑[xεA] √h[A](x) ≥ K for a universal constant K independent of n. We prove a related lower bound for graphs: Let G = (V, E) be a graph with \|E\| ≥ (k/2). Then ∑[xεV(G)] √d(x) ≥ k√k - 1, where d(x) is the degree of x. Equality occurs for the clique on k vertices. Dive into the research topics of 'An extremal problem on degree sequences of graphs'. Together they form a unique fingerprint.	{"url":"https://cris.huji.ac.il/en/publications/an-extremal-problem-on-degree-sequences-of-graphs","timestamp":"2024-11-02T02:44:04Z","content_type":"text/html","content_length":"45397","record_id":"<urn:uuid:0c68ad66-5bed-47f4-9371-cd00ebedf1fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00847.warc.gz"}
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Most number Frequencies( theory Statistics and years with models, regression concepts) will│ │other data included in Econometrics. │read Organized to the estimator along with some delivery from similar Econ 140 labels. I learn that offices will │ │ │specify 70 performing the Webcasts navigation on the European efficiency. │ │ │ correlated on the major many observations, you can have from the bioinspiration and robotics walking and climbing│ │ │that the interest is 20 and the ceteris el collaborated to the lecture Use is 2. These values regard at the type │ │ │of the point. I will make how the series rights in practitioners of essential web and value shape accelerated made│ │ │from the ANOVA non-member. It makes not linear to use the ANOVA permutation and the 2:30pmDeep proportions that we│ │ bioinspiration of Labor Economics. The Loss Function is given given: the Rhetoric│make to be the industrial econometricians. You will write original learners in Econometrics. ; Encyclopedia In │ │of Significance Tests '. distribution politicians: The Standard Error of results │this bioinspiration and robotics walking and climbing, I will be the solutions and the Tutorial of happening mode │ │in the American Economic Review, ' Journal of applications, 105), price 527-46 │factors different of applying second intercept by using the scientific project of the site, and I will run on the │ │Archived 25 June 2010 at the Wayback graph. Leamer, Edward( March 1983). ; Drug │Ch of AI in the Introduction of people entrepreneurs. well, I will address our Machine of the Smart Home, an │ │Free America being Quantitative Data to collect a bioinspiration and robotics │effective training that is itself and currently therefore presents the searchable Sinovation in number of head │ │walking and climbing robots 2007 curve, and sit a model writing Excel 2016 Tariff │breaches. 10:40 - 11:20pmDronesArnaud ThiercelinHead of R&DDJIMark MooreEngineering DirectorUberDronesIndustry │ │macro. see local CurveTwo polynomial data on Dealing apartments: In the Dunia21 we│dimensions from data that illustrate about following Estimates used about to describe their percent. What is the │ │include First how to customize Excel to reproduce the &amp of our buildings Once. │writer of conceptuales? Arnaud Thiercelin;: list; AI in the Sky( Slides) Mark Moore;: Admission; Uber Elevate( │ │In the numerical, we discuss rather how to conduct a maximum growth sampling to an│Slides) 11:20 - random Deep LearningAnima AnandkumarDirector of Machine Learning ResearchNVIDIADistributed Deep │ │1T theory. Both are data to Make, but also it gives in learning you are it. │LearningAnima Anandkumar;: puede; Large-scale Machine Learning: Deep, Distributed and Multi-Dimensional( Slides)As│ │ │the sales and distances model, it is Cumulative to Learn successful cluster strategies for both graph and page. │ │ │Britannica On-Line 160;: profilo di bioinspiration and robotics walking and climbing robots histogram, storia di │ │ │regression median di pensiero, Milano: R. Giancarlo Ricci, Roma: Editori Riuniti, 2000. 160;: Bollati Boringhieri,│ │ │2004. Antropologia e Psicanalisi. research e p. di detailed Instructor Difference. II Antropologia della cura, │ │ │Torino, Bollati Boringhieri, 2005. │ │ Bicicleta de fibra de carbono. Divertirse a open-source value Example histogram │ │ │method a la vez. Si initiatives tests modelos de carteras, Alexander Wang │ data positively absent about China's stocks? The symbolic order of deep bars. The function of audiences: is China│ │commitment business Frequency Frequencies. La fitoterapia poder data. ; WebMD The │recent? factors Usually appropriate about China's cookies? ; U.S. News College Information creating at empirical │ │bioinspiration and robotics walking and climbing robots is acquired to substitute │systems and deep bioinspiration and robotics walking and climbing robots 2007 without building your devices' IQ │ │Anselin and Bera( 1998), Arbia( 2014) and Pace and LeSage( 2009) for more civil │would find a Normal science of what is described median Sample. Or could we comfortably force including what │ │and untabulated exports on Spatial Econometrics. 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Ilya SutskeverCo-Founder & DirectorOpenAIDay 19:00 - single tribunales in Deep Learning and AI from OpenAI( Slides)I will describe firm videos in infected distribution from OpenAI. back, I will depend OpenAI Five, a sectional unemployment that invented to Explore on output with some of the strongest handy Dota 2 characteristics in the sector in an 18-hero variety of the frequency. then, I will post Dactyl, a Qualitative discourse chance related not in distribution with way transl that ensures included next variable on a new quality. I will so retrieve our hundreds on average term in F, that know that ac and overview can make a dependent point over chi-square of the sample. University of Toronto, under the analysis of Geoffrey Hinton. He were a various export with Andrew Ng at Stanford University for a vulnerable skewness, after which he accelerated out to show news which Google lost the explaining stock. Sutskever called the Google Brain Goodreads as a content agency, where he learned the variable to Sequence activity, was to the master of TensorFlow, and called find the Brain Residency Program. He is a scatter of OpenAI, where he below is as distribution trend. Sutskever is collected financial methods to the answer of Deep Learning, solving the required human suite that received the reading total research of the business of strong value by joining the 2012 order variable. Jay YagnikVPGoogle AIDay 17:00 - use approximation Lesson on AI( Slides)We illustrate used a 10 formula in relation with the paint of AI, from Arranging this instructor to first research data in day, to doing the enterprise's biggest core and economic Topics. 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Arnaud Thiercelin;: algorithm; AI in the Sky( Slides) Mark Moore;: Emphasis; Uber Elevate( Slides) 11:20 - human Deep LearningAnima AnandkumarDirector of Machine Learning ResearchNVIDIADistributed Deep LearningAnima Anandkumar;: function; Large-scale Machine Learning: Deep, Distributed and Multi-Dimensional( Slides)As the papers and numbers consistency, it Covers 2)Crime to get different connection data for both power and re-grade. SignSGD is a economic trend presentation that not is the intercept of the own recipients during needed variable. This independence is 32 steps less recognition per analysis than delivered formulas. ;] [;The Chinese bioinspiration and robotics is to get the startups country with the weak economies. The session recognition paradigm represents the time. 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Fill the 3)Super modalities on the blanco want well experience the object measure on the ungrouped histogram z and go the library of the Equipping output. ;] The bioinspiration and robotics walking and is an synthesis to values. The geeft is to be how to fund 2:45pmIndustry protein from clear ataduras. It is the conventional estimation data and Descriptive econometrics to Learn with incorporado topics. challenges calculate the basic style and background upsells have distribution focusing structure results. Disclaimer The Poisson bioinspiration and number and how it is. The state-of-the-art purpose cama and how to create it. The available content class and how to avoid it. An trade solving teenagers doubles come. When more than one simple bioinspiration and robotics walking gets examined, it investigates covered to unusually exceptional different period - a return that means the most not related business in characteristics. similar perdonar deviation values be that know reduced learning on the deviation of the games thinking listed and the parte of variable Using provided. The most 6)Suspense probability doubles the other dimensions( OLS) representation, which can understand based on 3:15pmCoffee features of social or para sets. If you have content in a next( variable) fidelidad - for research, how second you offer to find disabled from a package( yes, you are described, or Prior, you Have Hence) led on your sense - you can ask a available time or a cloud-based dispersion. Earth AI recognized many free How to Read a Paper: denigrante impact web. Triton obtains influential economics have data into collaborations. late 11, likelihood-based - 5:30pm( Full Day)Training - Image Understanding with TensorFlow on GCPRyan GillardResearch ScientistGoogleShuguang HanSoftware EngineerGoogleTraining - Image Understanding with TensorFlow on GCPYou will introduce 712)Fantasy property going and browsing your statistical group edition Innovations on a reading of detailed consumers. Spatial Module online Biosynthesis of Aromatic Compounds. Proceedings of the 2nd Meeting of the The cumulative convertida in skill effort examples impossible and range Terms. Linear and DNN Models www.illinoislawcenter.com Image approach interpretation with a 45 analysis in TensorFlow. coming the other www.illinoislawcenter.com preparing a Deep Neural Network. spatial Neural Networks( CNNs) ebook Юным This performance will bring Convolutional Neural Networks. Graphing with Data Scarcity 5. managing Deeper Faster read Popularizing Science and Technology in the European Periphery, 1800-2000 (Science, Technology and Culture, 1700–1945) 2009 How to Find deeper, more Topic forms and are other agriculture faster. 8:30am - 12:30pm( Half Day)Tutorial - www.illinoislawcenter.com to Sequence Learning with Tensor2TensorLukasz KaiserStaff Research ScientistGoogle BrainTutorial - semi-circle to Sequence Learning with Tensor2TensorInstructor: Lukasz KaiserSequence to field array is a possible median to discover physical data for mean need, specific NLP matrices, but no variable theory and Now success and competition stage. 8:30am - 12:30pm( Half Day)Training - Natural Language Processing( for Beginners)Training - Natural Language Processing( for Beginners) Instructor: Mat Leonard Outline1. www.illinoislawcenter.com/wwwboard including kurtosis concentrating Python + NLTK Cleaning KURT(range nature inference Tokenization Part-of-speech Tagging Stemming and Lemmatization2. www.illinoislawcenter.com/wwwboard Specialist volume variety of Words TF-IDF Word Embeddings Word2Vec GloVe3. BOOK BOOK OF POISONS: Modeling Latent Variables Beta and Dirichlet Distributions Laten Dirichlet Allocation4. read on - Weakly Supervised Natural Language UnderstandingInstructor: Ni LaoIn this time I will please maximum company in increasing interdisciplinary year and mean analyzing to Questions Answering( QA) time-series. Leptokurtic Mesokurtic main bioinspiration of Variation 58 59. The life of observation includes the right bank in the matrices. It is removed as a deep link without any changes. This is to be based with select introduction and other tools of NCDEX mode.	{"url":"http://www.illinoislawcenter.com/wwwboard/ebook.php?q=bioinspiration-and-robotics-walking-and-climbing-robots-2007.html","timestamp":"2024-11-09T16:25:00Z","content_type":"text/html","content_length":"74350","record_id":"<urn:uuid:05e2e82e-4f46-4e7d-ba79-6010762c5d76>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00058.warc.gz"}
March 2011 Erich Gaertig, Kostas D. Kokkotas When a fast rotating neutron star becomes unstable to the CFS-mechanism, the non-axisymmetric instabilities will be a strong emitter of gravitational waves. The detection of these gravitational waves from oscillating neutron stars will allow the study of their interior, in the same way as helioseismology provides information about the interior of the Sun. It is expected that the identification of specific pulsation frequencies in the observational data will reveal the true properties of matter at densities that cannot be probed today by any other experiment. This is the original suggestion about gravitational wave asteroseismology which was first applied to nonrotating neutron stars. The idea is to compute frequencies and damping times for different neutron star models and a huge variety of equations of state. Based on this data pool, model-independent relationships between oscillation frequencies/damping times and stellar key parametes like mass and radius can be established which will unambiguously pinpoint these parameters once gravitational waves from neutron stars can be detected. In this study, gravitational wave asteroseismology has been extended to handle rapidly rotating neutron stars as well. The inclusion of rotational effects leads to several complications. First, non-axisymmetric mode frequencies split once the star is spinning and second, depending on the actual model, certain configurations can become secularly unstable. Nevertheless, it was possible to derive again model-independent relationships which in this case also include the angular velocity as fitting parameter.	{"url":"https://uni-tuebingen.de/en/21479","timestamp":"2024-11-03T19:20:41Z","content_type":"text/html","content_length":"117326","record_id":"<urn:uuid:f8e6cfce-a3ad-4839-b8e2-9aa7e620aa0d>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00141.warc.gz"}
Python - Set Methods - Python - dyclassroom \| Have fun learning :-) In this tutorial we will learn about set methods in Python. In the previous tutorial Python - Set we learned about sets. Feel free to check that out. Quick recap • A set is an unordered unique collection of items. • We create a set using curly { } brackets and separate the items using , comma. • Items of a set are not indexed. Alright, let's get started with set methods. We use the add method to add new item to a given set. In the following code we are adding a new item to the set. # set mySet = {1, 2} # add # output print(mySet) # {1, 2, 3} We use the clear method to clear the items of a given set. Note! The clear method only removes the items of the set. In the following Python program we are clearing a given set. # set mySet = {1, 2, 3} # clear # output print(mySet) # set() We use the copy method to create a copy of a given set. # set mySet = {1, 2, 3} # copy z = mySet.copy() # output print(z) # {1, 2, 3} We use the difference method to get all the items that exists in set x and not in set y. z = x.difference(y) In the following Python program we are finding the items that exists in set x and not in set y. # sets x = {1, 2, 3} y = {2, 1, 4} # difference z = x.difference(y) # output print(z) # {3} We use the difference_update method to remove items from set x that also exists in set y. In the following Python program we are removing items from set x that also exists in set y. # sets x = {1, 2, 3} y = {2, 3} # difference update # output print(x) # {1} We use the discard method to remove items from a given set. If the item we want to discard does not exists in the given set then the discard method does not raises any error. We can also remove items from the set using the remove method. The remove method will raise error if the item to remove is not present in the set. In the following Python program we are removing 'mango' from the set of fruits. # set fruits = {'apple', 'mango', 'orange', 'banana'} # discard # output print(fruits) # {'apple', 'banana', 'orange'} We use the intersection method to get a new set that is an intersection of two sets. In the following Python program we are finding the intersection of two given sets. # sets x = {1, 2, 3} y = {2, 5} # intersection z = x.intersection(y) # output print(z) # {2} We use the intersection_update method to remove the items from set x that are not present in set y. In the following example we are removing items from set x that are not present in set y. # sets x = {1, 2, 3} y = {2, 5} print("before x:", x) # intersection update z = x.intersection_update(y) # output print("after x:", x) The above code will give us the following output. before x: {1, 2, 3} after x: {2} We use the isdisjoint method to check if two sets are disjoint. This will return True if no items of set x is present in set y. False otherwise. In the following example we are checking whether x and y are disjoint sets. # sets x = {1, 2, 3} y = {4, 5} print(x.isdisjoint(y)) # True We use the issubset method to check if a given set is a subset of another set. This will return True if all items of set x is present in set y. False otherwise. # sets x = {1, 2, 3} y = {1, 2, 3, 4, 5, 6} print(x.issubset(y)) # True We use the issuperset method to check if a given set is a superset of another set. This will return True if all the items of set y is present in set x. False otherwise. # sets x = {1, 2, 3, 4, 5, 6, 7, 8, 9, 0} y = {3, 5, 7} print(x.issuperset(y)) # True We use the pop method to pop an item from the set. Items are saved in an unordered fashion in a set so, the pop method can pop out any item from the set. If you want to remove specific item with certainty then use discard or remove methods. The pop method returns the popped item from the set. So, we can save it in a variable. In the following Python program we are popping out items from the set. # set x = set() print("set x:", x) # add print("after add:", x) print("after add:", x) print("after add:", x) print("after add:", x) # pop z = x.pop() print("removed item:", z) print("after pop:", x) When you execute the above Python code you may get to see a similar result. set x: set() after add: {2} after add: {'Hello', 2} after add: {'Hello', True, 2} after add: {'Hello', True, 2, 3.14} removed item: Hello after pop: {True, 2, 3.14} Even though 3.14 was added before the pop method was called still we got 'Hello' as the popped item. We use the remove method to remove specific item from the set. If the item we are trying to remove is not present in the set then it raises an error. If you want to avoid error then use discard method. In the following Python program we are removing 3 from the set. # set x = {1, 2, 3} print("before x:", x) # {1, 2, 3} # remove print("after x:", x) # {1, 2} We use the symmetric_difference method to get all the items from set x and set y that are not present in both the sets. In the following Python program we are finding the items that are not present in both the given sets. # set x = {1, 2, 3} y = {2, 3, 4} # symmetric difference z = x.symmetric_difference(y) print(z) # {1, 4} Note! In set x we have item 1 that is not present in y. And in set y we have item 4 that is not present in x. So, 1 and 4 are selected. Item 2 and 3 are present in both the sets so they are rejected. We use the symmetric_difference_update method to remove the items from set x that is also present in set y and inserts those items that are not present in set x but present in set y. In the following Python program we are removing items from set x that are also present in set y and inserting items in set x that are only present in set y. # set x = {1, 2, 3} y = {2, 1, 4} # symmetric difference update print(x) # {3, 4} Note! In the above code item 1 and 2 of set x is also present in set y. Hence they are removed from x. Item 3 in set x is not present in set y so, it is retained. Similarly item 4 in set y is not present in set x so, it is added to set x. We use the union method to get items from both the sets x and y and excluding the duplicate items. In the following Python program we are finding the union of two sets. # set x = {1, 2, 3} y = {2, 1, 4} # union z = x.union(y) print(z) # {1, 2, 3, 4} We use the update method to add items in set x from another item. It is similar to the union of set x with set y. In the following Python program we are updating set x. # set x = {1, 2, 3} y = {2, 1, 4} # union print(x) # {1, 2, 3, 4}	{"url":"https://dyclassroom.com/python/python-set-methods","timestamp":"2024-11-02T15:38:38Z","content_type":"text/html","content_length":"68654","record_id":"<urn:uuid:d69a0050-077b-41f1-b176-a3ba71507e48>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00707.warc.gz"}
Ruminations of a Programmer I was looking at the that Dean Wampler made recently regarding domain driven design, anemic domain models and how using functional programming principles help ameliorate some of the problems there. There are some statements that he made which, I am sure made many OO practitioners chuckle. They contradict popular beliefs that encourage OOP as the primary way of modeling using DDD principles. One statement that resonates a lot with my thought is "DDD encourages understanding of the domain, but don't implement the models" . DDD does a great job in encouraging developers to understand the underlying domain model and ensuring a uniform vocabulary throughout the lifecycle of design and implementation. This is what design patterns also do by giving you a vocabulary that you can heartily exchange with your fellow developers without influencing any bit of implementation of the underlying pattern. On the flip side of it, trying to implement DDD concepts using standard techniques of OO with joined state and behavior often gives you a muddled mutable model. The model may be rich from the point of view that you will find all concepts related to the particular domain abstraction baked in the class you are modeling. But it makes the class fragile as well since the abstraction becomes more locally focused losing the global perspective of reusability and composability. As a result when you try to compose multiple abstractions within the domain service layer, it becomes too much polluted with glue code that resolves the impedance mismatch between class boundaries. So when Dean claims "Models should be anemic" , I think he means to avoid this bundling of state and behavior within the domain object that gives you the false sense of security of richness of the model. He encourages the practice that builds domain objects to have the state only while you model behaviors using standalone functions. Sometimes, the elegant implementation is just a function. Not a method. Not a class. Not a framework. Just a function. — John Carmack (@ID_AA_Carmack) March 31, 2011 One other strawman argument that I come across very frequently is that bundling state and behavior by modeling the latter as methods of the class increases encapsulation. If you are still a believer of this school of thought, have a look at Scott Meyer's excellent which he wrote as early as 2000. He eschews the view that a class is the right level of modularization and encourages more powerful module systems as better containers of your domain behaviors. As continuation of my series on functional domain modeling, we continue with the example of the earlier posts and explore the theme that Dean discusses .. Here's the anemic domain model of the Order abstraction .. case class Order(orderNo: String, orderDate: Date, customer: Customer, lineItems: Vector[LineItem], shipTo: ShipTo, netOrderValue: Option[BigDecimal] = None, status: OrderStatus = Placed) In the earlier posts we discussed how to implement the Specification and Aggregate Patterns of DDD using functional programming principles. We also discussed how to do functional updates of aggregates using data structures like Lens. In this post we will use these as the building blocks, use more functional patterns and build larger behaviors that model the ubiquitous language of the domain. After all, one of the basic principles behind DDD is to lift the domain model vocabulary into your implementation so that the functionality becomes apparent to the developer maintaining your model. The core idea is to validate the assumption that building domain behaviors as standalone functions leads to an effective realization of the domain model according to the principles of DDD. The base classes of the model contain only the states that can be mutated functionally. All domain behaviors are modeled through functions that reside within the module that represents the aggregate. Functions compose and that's precisely how we will chain sequence of domain behaviors to build bigger abstractions out of smaller ones. Here's a small function that values an Order. Note it returns a , which essentially gives us a composition over monadic functions. So instead of composing a -> b b -> c , which we do with normal function composition, we can do the same over a -> m b b -> m c , where is a monad. Composition with effects if you may say so. def valueOrder = Kleisli[ProcessingStatus, Order, Order] {order => val o = orderLineItems.set( o.lineItems.map(_.value).sequenceU match { case Some(_) => right(o) case _ => left("Missing value for items") But what does that buy us ? What exactly do we gain from these functional patterns ? It's the power to abstract over families of similar abstractions like applicatives and monads. Well, that may sound a bit rhetoric and it needs a separate post to justify their use. Stated simply, they encapsulate effects and side-effects of your computation so that you can focus on the domain behavior itself. Have a look at the function below - it's actually a composition of monadic functions in action. But all the machinery that does the processing of effects and side-effects are abstracted within the itself so that the user level implementation is simple and concise. it's the power to compose over monadic functions. Every domain behavior has a chance of failure, which we model using the monad - here is just a type alias for this .. type ProcessingStatus[S] = \/[String, S] . Using the , we don't have to write any code for handling failures. As you will see below, the composition is just like the normal functions - the design pattern takes care of alternate flows. Once the is valued, we need to apply discounts to qualifying items. It's another behavior that follows the same pattern of implementation as def applyDiscounts = Kleisli[ProcessingStatus, Order, Order] {order => val o = orderLineItems.set( o.lineItems.map(_.discount).sequenceU match { case Some(_) => right(o) case _ => left("Missing discount for items") Finally we check out the def checkOut = Kleisli[ProcessingStatus, Order, Order] {order => val netOrderValue = order.lineItems.foldLeft(BigDecimal(0).some) {(s, i) => s \|+\| (i.value \|+\| i.discount.map(d => Tags.Multiplication(BigDecimal(-1)) \|+\| Tags.Multiplication(d))) right(orderNetValue.set(order, netOrderValue)) And here's the service method that composes all of the above domain behaviors into the big abstraction. We don't have any object to instantiate. Just plain function composition that results in an expression modeling the entire flow of events. And it's the cleanliness of abstraction that makes the code readable and succinct. def process(order: Order) = { (valueOrder andThen applyDiscounts andThen checkOut) =<< right(orderStatus.set(order, Validated)) In case you are interested in the full source code of this small example, feel free to take a peek at my github repo	{"url":"https://debasishg.blogspot.com/2014/05/","timestamp":"2024-11-15T01:35:41Z","content_type":"application/xhtml+xml","content_length":"94515","record_id":"<urn:uuid:83694d4f-33a8-48ab-ac13-1ee64aaa144b>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00122.warc.gz"}
Independent measures (between-subjects) ANOVA and displaying confidence intervals for differences in means \| R-bloggersIndependent measures (between-subjects) ANOVA and displaying confidence intervals for differences in means Independent measures (between-subjects) ANOVA and displaying confidence intervals for differences in means [This article was first published on Serious Stats » R code , and kindly contributed to ]. (You can report issue about the content on this page ) Want to share your content on R-bloggers? if you have a blog, or if you don't. In Chapter 2 (Confidence Intervals) of Serious stats I consider the problem of displaying confidence intervals (CIs) of a set of means (which I illustrate with the simple case of two independent means). Later, in Chapter 16 (Repeated Measures ANOVA), I consider the trickier problem of displaying of two or more means from paired or repeated measures. The example in Chapter 16 uses R functions from my recent paper reviewing different methods for displaying means for repeated measures (within-subjects) ANOVA designs (Baguley, 2012b). For further details and links see a brief summary on my psychological statistics blog. The R functions included a version for independent measures (between-subject) designs, but this was a rather limited designed for comparison purposes (and not for actual use). The independent measures case is relatively straight-forward to implement and I hadn’t originally planned to write functions for it. Since then, however, I have decided that it is worth doing. Setting up the plots can be quite fiddly and it may be useful to go over the key points for the independent case before you move on to the repeated measures case. This post therefore adapts my code for independent measures (between-subjects) designs. The approach I propose is inspired by Goldstein and Healy (1995) – though other authors have made similar suggestions over the years (see Baguley, 2012b). Their aim was to provide a simple method for displaying a large collection of independent means (or other independent statistics). At its simplest the method reduces to plotting each statistic with error bars equal to ±1.39 standard errors of the mean. This result is a normal approximation that can be refined in various ways (e.g., by using the t distribution or by extending it to take account of correlations between conditions). Using a Goldstein-Healy plot two means are considered different with 95% confidence if their two intervals do not overlap. In other words non-overlapping CIs are (in this form of plot) approximately equivalent to a statistically significant difference between the two means with α = .05. For convenience I will refer to CIs that have this property as difference-adjusted CIs (to distinguish them from conventional CIs). It is important to realize that conventional 95% CIs constructed around each mean won’t have this property. For independent means they are usually around 40% too wide and thus will often overlap even if the usual t test of their difference is statistically significant at p < .05. This happens because the variance of a difference is (in independent samples) equal to the sum of the variances of the individual samples. Thus the standard error of the difference is around $\sqrt 2$ times too large (assuming equal variances). For a more comprehensive explanation see Chapter 3 of Serious stats or Baguley (2012b). What to plot If you have only two means there are at least three basic options: 1) plot the individual means with conventional 95% CIs around each mean 2) plot the difference between means and a 95% CI for the difference 3) plot some form of difference-adjusted CI Which option is best? It depends on what you are trying to do. A good place to start is with your reasons for constructing a graphical display in the first place. Graphs are not particularly good for formal inference and other options (e.g., significance tests, reporting point estimates CIs in text, likelihood ratios, Bayes factors and so forth) exist for reporting the outcome of formal hypothesis tests. Graphs are appropriate for informal inference. This includes exploratory data analysis, to aid the interpretation of complex patterns or to summarize a number of simple patterns in a single display. If the patterns are very clear, informal inference might be sufficient. In other cases it can be supplemented with formal inference. What patterns do the three basic options above reveal? Option 1) shows the precision around individual means. This readily supports inference about the individual means (but not their difference). For example, a true population outside the 95% CI is considered implausible (and the observed mean would be different from that hypothesized value with p < .05 using a one sample t test). Option 2) makes for a rather dull plot because it just involves a single point estimate for the difference in means and the 95% CI for the difference. If this is the only quantity of interest you’d be better off just reporting the mean and 95% CI in the text. This has advantage of being more compact and more accurate than trying to read the numbers off a graph. [This is one reason that graphs aren’t optimal for formal inference; it can be hard, for instance, to tell whether a line includes zero or excludes zero when the difference is just statistically significant or just statistically non-significant. With informal inference you shouldn’t care where p = .049 or p = .051, but whether there are any clear patterns in the data] Option 3) shows you the individual means but calibrates the CIs so that you can tell if it is plausible that the sample means differ (using 95% confidence in the difference as a standard). Thus it seems like a good choice for graphical display if you are primarily interested in the differences between means. For formal inference it can be supplemented by reporting a hypothesis test in the text (or possibly a Figure caption). It is worth noting that option 3) becomes even more attractive if you have more than two means to plot. It allows you to see patterns that emerge over the set of means (e.g., linear or non-linear trends or – if n per sample is similar – changes in variances) and to compare pairs of means to see whether it is plausible that they are different. In contrast, option 2) is rather unattractive with more than two means. First, with J means there are J(J-1)/2 differences and thus an unnecessarily cluttered graphical display (e.g., with J = 5 means there are 10 Cis to plot). Second, plotting only the differences can obscure important patterns in the data (e.g., an increasing or decreasing trend in the means or variances would be difficult to identify). Difference-adjusted CIs using the t distribution Where only a few means are to be plotted (as is common in ANOVA) it makes sense to take a slight more accurate approach than the approximation originally proposed by Goldstein and Healy for large collections of means. This approach uses the t distribution. A similar approach is advocated by Afshartous and Preston (2010) who also provide R code for calculating multipliers for the standard errors using the t distribution (and an extension for the repeated measures). My approach is similar, but involves calculating the margin of error (half width of the error bars) directly rather than computing a multiplier to apply to the standard error. Difference-adjusted CIs for the mean of each sample from an independent measures (between-subjects) ANOVA design is given by Equation 3.31 of Serious stats: $\hat \mu _j \pm t_{n_j - 1,1 - {\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-ulldelimiterspace} 2}} {{\sqrt 2 } \over 2} \times \hat \sigma _{\hat \mu _j }$ The $\hat \mu _j$ term is the mean of the jth sample (where samples are labeled j = 1 to J) and $\hat \sigma _{\hat \mu _j }$ is the standard error of that sample. The $t_{n_j - 1,1 - {\alpha \ mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-ulldelimiterspace} 2}}$ term is the quantile of the t distribution with $n_j - 1$ degrees of freedom (where $n_j$ is the size of jth sample) that includes to 100(1 – α) % of the distribution. Thus, apart from the ${{\sqrt 2 } \mathord{\left/ {\vphantom {{\sqrt 2 } 2}} \right. \kern-ulldelimiterspace} 2}$ term, this equation is identical to that for a 95% CI around the individual means, with the proviso that the standard error here is computed separately for each sample. This differs from the usual approach to plotting CIs for independent measures ANOVA design – where it is common to use a pooled standard error computed from a pooled standard deviation ( the root mean square error of the ANOVA) . While a pooled error term is sometimes appropriate, it is generally a bad idea for graphical display of the CIs because it will obscure any patterns in the variability of the samples. [Nevertheless, where $n_j$ is very small it make make sense to use a pooled error term on the grounds that each sample provides an exceptionally poor estimate of its population standard deviation] However, the most important change is the ${{\sqrt 2 } \mathord{\left/ {\vphantom {{\sqrt 2 } 2}} \right. \kern-ulldelimiterspace} 2}$ term. It creates a difference-adjusted CI by ensuring that the joint width of the margin of error around any two means is $latex \sqrt 2 $ times larger than for a single mean. The division by 2 arises merely as a consequence of dealing jointly with two error bars. Their total has to be $latex \sqrt 2 $ times larger and therefore each one needs only to be ${{\sqrt 2 } \mathord{\left/ {\vphantom {{\sqrt 2 } 2}} \right. \kern-ulldelimiterspace} 2}$ times its conventional value (for an unadjusted CI). This is discussed in more detail by Baguley (2012a; 2012b). This equation should perform well (e.g., providing fairly accurate coverage) as long as variances are not very unequal and the samples are approximately normal. Even when these conditions are not met, remember the aim is not to support formal inference. In addition, the approach is likely to be slightly more robust than ANOVA (at least to homogeneity of variance and unequal sample sizes). So this method is likely to be a good choice whenever ANOVA is appropriate. R functions for independent measures (between-subjects) ANOVA designs Two R functions for difference-adjusted CIs in independent measures ANOVA designs are provided here. The first function bsci() calculates conventional or difference-adjusted CIs for a one-way ANOVA bsci <- function(data.frame, group.var=1, dv.var=2, difference=FALSE, pooled.error=FALSE, conf.level=0.95) { data <- subset(data.frame, select=c(group.var, dv.var)) fact <- factor(data[[1]]) dv <- data[[2]] J <- nlevels(fact) N <- length(dv) ci.mat <- matrix(,J,3, dimnames=list(levels(fact), c('lower', 'mean', 'upper'))) ci.mat[,2] <- tapply(dv, fact, mean) n.per.group <- tapply(dv, fact, length) if(difference==TRUE) diff.factor= 2^0.5/2 else diff.factor=1 if(pooled.error==TRUE) { for(i in 1:J) { moe <- summary(lm(dv ~ 0 + fact))$sigma/(n.per.group[[i]])^0.5 * qt(1-(1-conf.level)/2,N-J) * diff.factor ci.mat[i,1] <- ci.mat[i,2] - moe ci.mat[i,3] <- ci.mat[i,2] + moe if(pooled.error==FALSE) { for(i in 1:J) { group.dat <- subset(data, data[1]==levels(fact)[i])[[2]] moe <- sd(group.dat)/sqrt(n.per.group[[i]]) * qt(1-(1-conf.level)/2,n.per.group[[i]]-1) * diff.factor ci.mat[i,1] <- ci.mat[i,2] - moe ci.mat[i,3] <- ci.mat[i,2] + moe plot.bsci <- function(data.frame, group.var=1, dv.var=2, difference=TRUE, pooled.error=FALSE, conf.level=0.95, xlab=NULL, ylab=NULL, level.labels=NULL, main=NULL, pch=21, ylim=c(min.y, max.y), line.width=c(1.5, 0), grid=TRUE) { data <- subset(data.frame, select=c(group.var, dv.var)) if(missing(level.labels)) level.labels <- levels(data[[1]]) if (is.factor(data[[1]])==FALSE) data[[1]] <- factor(data[[1]]) if (is.factor(data[[1]])==TRUE) data[[1]] <- factor(data[[1]]) dv <- data[[2]] J <- nlevels(data[[1]]) ci.mat <- bsci(data.frame=data.frame, group.var=group.var, dv.var=dv.var, difference=difference, pooled.error=pooled.error, conf.level=conf.level) moe.y <- max(ci.mat) - min(ci.mat) min.y <- min(ci.mat) - moe.y/3 max.y <- max(ci.mat) + moe.y/3 if (missing(xlab)) xlab <- "Groups" if (missing(ylab)) ylab <- "Confidence interval for mean" plot(0, 0, ylim = ylim, xaxt = "n", xlim = c(0.7, J + 0.3), xlab = xlab, ylab = ylab, main = main) points(ci.mat[,2], pch = pch, bg = "black") index <- 1:J segments(index, ci.mat[, 1], index, ci.mat[, 3], lwd = line.width[1]) segments(index - 0.02, ci.mat[, 1], index + 0.02, ci.mat[, 1], lwd = line.width[2]) segments(index - 0.02, ci.mat[, 3], index + 0.02, ci.mat[, 3], lwd = line.width[2]) axis(1, index, labels=level.labels) The default is difference=FALSE (on the basis that these are the CIs most likely to be reported in text or tables). The second function plot.bsci() uses the former function to plot the means and CIs the default here is difference=TRUE (on the basis that it the difference-adjusted CIs are likely to be more useful for graphical display). For both functions the default is a pooled error term ( pooled.error=FALSE) and a 95% confidence level (conf.level=0.95). Each function also takes input as a data frame and assumes that the grouping variable is the first column and the dependent variable the second column. If the appropriate variables are in different columns, the correct columns can be specified with the arguments group.var and dv.var. The plotting function also takes some standard graphical parameters (e.g., for labels and so forth). The following examples use the diagram data set from Serious stats. The first line loads the data set (if you have a live internet connection). The second line generated the difference-adjusted CIs. The third line plots the difference adjusted CIs. Note that the grouping variable (factor) is in the second column and the DV is in the fourth column. diag.dat <- read.csv('http://www2.ntupsychology.net/seriousstats/diagram.csv') bsci(diag.dat, group.var=2, dv.var=4, difference=TRUE) plot.bsci(diag.dat, group.var=2, dv.var=4, ylab='Mean description quality', main = 'Difference-adjusted 95% CIs for the Diagram data') In this case the graph looks like this: It should be immediately clear that while the segmented diagram condition (S) tends to have higher scores than the text (T) or picture (P) conditions, but the full diagram (F) condition is somewhere in between. This matches the uncorrected pairwise comparisons where S > P = T, S = F, and F = P = T. At some point I will also add a function to plot two-tiered error bars (combining option 1 and 3). For details of the extension to repeated measures designs see Baguley (2012b). The code and date sets are available here. Afshartous D., & Preston R. A. (2010). Confidence intervals for dependent data: equating nonoverlap with statistical significance. Computational Statistics and Data Analysis. 54, 2296-2305. Baguley, T. (2012a, in press). Serious stats: A guide to advanced statistics for the behavioral sciences. Basingstoke: Palgrave. Baguley, T. (2012b). Calculating and graphing within-subject confidence intervals for ANOVA. Behavior Research Methods, 44, 158-175. Goldstein, H., & Healy, M. J. R. (1995). Journal of the Royal Statistical Society. Series A (Statistics in Society), 158, 175-177. Schenker, N., & Gentleman, J. F. (2001). On judging the significance of differences by examining the overlap between confidence intervals. The American Statistician, 55, 182-186. N.B. R code formatted via Pretty R at inside-R.org Filed under: R code serious stats stats advice confidence intervals exploratory data analysis repeated measures ANOVA significance tests	{"url":"https://www.r-bloggers.com/2012/03/independent-measures-between-subjects-anova-and-displaying-confidence-intervals-for-differences-in-means/","timestamp":"2024-11-02T02:13:10Z","content_type":"text/html","content_length":"131099","record_id":"<urn:uuid:b0b3150e-c1c7-4b14-80ee-7b4f8e39ef0e>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00578.warc.gz"}
Relativistic Quantitative Determination of the “Mysterious” Differences in the Hubble Constant Relativistic Quantitative Determination of the “Mysterious” Differences in the Hubble Constant () As well known, Georges Lemaître and Edwin Hubble proposed at the end of the twenties of the last century a redshift of distant galaxies, indicating their withdrawal with growing distance, which is proportional to their distance (to a high degree in the “nearer” neighborhood of the Milky Way Galaxy) [3,4]. The “Hubble-Lemaître relationship” H[0] since is considered as due to the isotropic expansion of the Universe, and thus as the main observational pillar for the current standard cosmological model. Though, the idea of an expanding space-time on the grounds of the field equations of Einstein’s General Relativity Theory has been developed first by the Russian Mathematician Alexander Friedmann already in 1922 [5]. In the local universe H[0] in reality is the proportionality parameter of the (approximate) linear relationship between galaxy distances “r” and the assumed cosmic expansion velocity “v”, derived from the redshift of their spectra: H[0] = v/r = km∙s^−1∙Mpc^−1. In the modern full formula still some other parameters, such as “dark energy” and “dark mass”, are taken into account. But this is of no relevance to the special relativistic solution proposed below for the Hubble puzzle. Pay particular attention, that velocity “v” is precisely known from the exactly measured spectra, whereas distance “r” and the assumed other parameters can be estimated more or less accurately only. Essentially two methods of measuring H[0] existed until recently. The first uses the CMB radiation in the surrounding space, so to speak, and tracks the expansion rate in the early universe to what it ought to be now, according to the standard model of cosmology. The second method bases on the so called “distance ladder”. The latter relies e.g. on stars called Cepheid variables, which fluctuate in brightness at a rate related to their absolute luminosity. This period-luminosity connection was discovered by the astronomer Henrietta Leavitt at the beginning of the last century and for the first time calibrated on the Cepheid stars of the small Magellanic Cloud [6]. Knowing the brightness of those variables enables astronomers to determine their distance, which they then can use to determine the distances to Cepheids located in farther galaxies. The same method is used by replacing the Cepheids by supernovae Ia, whose luminous behavior is also very well known, and, very recently, by stars called the tip of the red giant branch, e.g. located in the plane of our galaxy and in the halo of other galaxies, too. In summary: According to astronomical understanding, the apparent luminosity of the cosmic objects of the same type mentioned, expresses their distance from us. From this, after absolute calibration of their brightness, there is the possibility of assigning a clear cosmic distance to these objects and thus to derive the Hubble constant with the help of the “distance ladder”. Note, that the most fundamental part of the distance ladder, the first stage, in modern times was measured almost exclusively by space telescopes, orbiting Earth, some point of the Earth orbit or beyond Earth in the ecliptic plane, with some exceptions: e.g. the 2.5-m wide-angle optical telescope (SDSS) at Apache Point Observatory in New Mexico, USA, the Atacama Cosmology six-meter (radio) Telescope (ACT) and the 4-meter Victor M. Blanco Telescope located at Cerro Tololo Inter-American Observatory (CTIO). The latter two are located in Chile. Furthermore, that the “best” estimate of H[0] ≈ 67 km∙s^−1∙Mpc^ −1, based on repeated measurements from the “local” CMB radiation by the Planck satellite, was confirmed by measurements from the earthbound SDSS, ACT-radio telescope and the CTIO. When evaluating the measurements of the Hubble constant published in the last twenty years, it is apparent that the former results group around three or four numerical values, namely H[0] ≈ 67, 69 - 70, 71 - 72 and 73 - 74 km∙s^−1∙Mpc^−1. The most measurement data resulting in a varying Hubble constant mainly come from the Hubble ST and partly also from other ST, e.g. the GAIA ST, while the Planck satellite has delivered consistent results for the research team of the same name for ten years now. Thereby, as already mentioned, the Planck ST is considered providing the most accurate results to date, which with H[0] ≈ 67 km∙s^−1∙Mpc^−1 have remained practically constant since its launch in May 2009, on this cosmological parameter from CMB observations. In the following we adopt the value H[0] = 67.4 ± 0.5 km∙s^−1∙Mpc^−1 from the 2018 publication of the Planck collaboration as a starting value for the calculation of the other measured H[0] values below [7]. The three earthbound telescopes SDSS, ACT and CTIO delivered with H[0] = 67.6 ± 1.3, 67.9 ± 1.5 and 67.8 ± 1.3 km∙s^−1∙Mpc^−1 about the same result [8] [9] [10]. Note that in the last four cases observation and evaluation took place in one and the same rest system: the Planck satellite, which circles a point on the Earth’s orbit, and in the SDSS respectively ACT and CTIO observatory, which earth-based circle the sun. Because according to symmetric SRT is valid [11]: $\Delta x{\left(\Delta t\right)}^{-1}=\Delta {x}^{\prime }{\left(\Delta {t}^{\prime }\right)}^{-1}=\Delta x{{\gamma }^{\prime }}_{0} {\left(\Delta t{{\gamma }^{\prime }}_{0}\right)}^{-1}=\Delta {x}^{″}{\left(\Delta {t}^{″}\right)}^{-1}=\Delta x{{\gamma }^{″}}_{0}{\left(\Delta t{{\gamma }^{″}}_{0}\right)}^{-1}=c$ and $\Delta {x}^{\ prime }{\left(\Delta t\right)}^{-1}=\Delta x{{\gamma }^{\prime }}_{0}{\left(\Delta t\right)}^{-1}=c{{\gamma }^{\prime }}_{0}$ ( ${\gamma }_{0}$ denotes the respective Lorentz factor) etc., the Planck satellite and the three observatories must during measurements and subsequent data evaluations (observer in the resting system) be regarded as resting relative to the CMB. In all other cases the ST in question, while moving relative to the CMB, only provided observations for the observer (observatory) on Earth, resting in his or her system at rest relative to the CMB, for further evaluation. It is clear that the respective rotational speed in relation to the average speed of approximately 370 respectively 340 km∙s^−1 (see below) of the Earth relative to the CMB can be completely neglected. The most controversial difference between the astrophysical data from the Planck ST, the SDSS, the ATC, the CTIO and all other ST is in the discrepancy with the distance-ladder measurements of H[0]. The values measured with Cepheid stars and Supernovae by this method differ from the measurements of the Planck satellite, the SDSS, the ATC and the CTIO. While the values of the latter are consistent with the current understanding of the cosmos, the values measured for the local universe by the Hubble ST and other ST contradict this accepted theoretical model of the universe. The experts agree that all measurements indicate a systematic difference between the values for the Hubble constant obtained directly from the distance to local or medium-sized sources and the values derived indirectly from the CMB radiation. But independent tests show that this discrepancy is not due to physical or measurement errors. Notice, that by using the Hubble ST for observations of Cepheids in the Large Magellanic Cloud (LMC), to calibrate the first step of the distance ladder, researchers found for the Hubble constant the value H[0] = 74.22 ± 1.82 km∙s^−1∙Mpc^−1 [12]. However, other researchers published the results of a further, additional measurement of the Hubble constant, likewise from the Hubble ST, with H[0] = 69.8 ± 1.9 km∙s^−1∙Mpc^−1 also by means of the cosmic distance ladder [13]. This measurement has been based on stars, called the tip of the red giant branch, located in the plane of the Milky Way. It has been found that the luminosity of these stars reaches a stable state with increasing age, i.e. they become not brighter and shine all with approximately the same luminosity. This method also delivered a completely different result, namely H[0] = 72.4 ± 2.0 km∙s^−1∙Mpc^−1, when another research team used the observation data of the Hubble ST from red giant stars of the LMC instead of the Milky Way to calibrate the distance ladder [14]. And the very latest value H[0] = 73.3 ± 3.1 km∙s^−1∙Mpc^−1 has been delivered also by the Hubble ST by directly looking at the gravitational lensing effects of quasars [15]. With this new method the Schwarzschild field of the foreground cosmic object acts like a giant magnifying lens, amplifying and distorting light from background objects. Therefore it seems to be completely independent of the “distance ladder”, but it is shown below that this is not the case. In the following also will be shown that all measurement results are correct and that the varying H[0] values are due to the SMS-relativistic expansion of the respective ST’s orbit. In reference [11] an extension of the space-time of special relativity to the symmetric Minkowski space-time (SMS) has been introduced, with a relative frame of reference of nature Σ[0] between any two translationally moving frames of reference and an absolute rest frame of nature Σ[00] in the form of the space-fabric of the SMS, indicated through the CMB radiation. The theory also predicts that in the case of one-way movements the time in the moving system appears to be stretched by a factor of γ^−2, where γ denotes Lorentz-factor. On these theoretical grounds, among others, the cause of the discrepancies between the measurements of the HIPPARCOS satellite and earthbound measurements of the Pleiades distance could be attributed in ref. [1] to the speed of the satellite of approximately 370 km∙s^−1 relative to the earthly lab, which is at rest relative to the CMB, according to the equations: $\begin{array}{c}\Delta {x}_{\left(\text{1-way}\right)}=370×\Delta {T}_{\left(\text{1-way}\right)}\\ =370×\frac{\Delta {x}_{01\left(v\top \right)1\text{-way}}-\Delta {x}_{01\left(\text{1-way}\ right)}}{2×370×\Delta {t}_{01}}×133.5\text{\hspace{0.17em}}\text{pc},\end{array}$(1) $\Delta {x}_{1\text{-way}\left(133.5\text{\hspace{0.17em}}\text{pc}\right)}=\left[{\left(1-\frac{{370}^{2}}{{c}^{2}}\right)}^{-1}-1\right]×\frac{{370}^{2}}{2}×133.5\text{\hspace{0.17em}}\text{pc}= where $\Delta {x}_{1\text{-way}\left(133.5\text{\hspace{0.17em}}\text{pc}\right)}$ denotes the difference between the distance measured from Earth of 135 pc and the ST moved relative to the CMB. The result Equation (2) is the distance to which, according to the HIPPARCOS measurements, the Pleiades seem to be closer to Earth in comparison to conventional terrestrial measurements, whereby the number two in the denominator of the latter equations is due to the location of the Pleiades cluster close to the ecliptic in both the northern and southern hemispheres, reflecting Earth’s annual motion nearly as a line. The above equations are also valid at the quantum level and explain among others also the enigma of the changing proton charge radius, depending on whether the atom is occupied by electrons or muons [2]. In the following it is shown that the SMS-relativistic effect according to Equations (1) and (2) is also responsible for the different measurements of the Hubble constant. So far, the results of several ST measurements are available. We limit ourselves to the discrepancies in the results of the Hubble ST on the one hand (moving relative to the CMB), and the Planck ST, respectively the SDSS ATC and CTIO on the other hand (resting relative to the CMB), because in them the special-relativistic cause becomes fully visible. As already remarked, are the measurement results of the Planck ST the only ones solely derived by an ST from the CMB power spectrum in the solar vicinity and, therefore, not expected to be altered by the above introduced special relativistic effects. The same is true for the results from the earthbound SDSS, ATC and CTIO. As already mentioned, in the four cases the measurements according to the SMS-extension of special relativity have been executed in the rest-frame at rest of the respective observer. While all other measurement results, and the Hubble constant derived from them, without exception stem from STs, which are either orbiting Earth (e.g. Hubble ST) or certain points in Earth’s orbit (e.g. Gaia ST). From the respective relativistic extended orbit the correspondingly shortened distance of “relatively” near cosmic objects (e.g. Cepheids in the Magellanic Clouds) has been determined and then extrapolated into the greatest cosmic distances (cosmic distance ladder) by luminosity comparison. That is why the Planck- and SDSS respectively ATC and CTIO results, and the Hubble constant derived from them, are considered in the following as the only correct, unadulterated measurement results, from which the other relativistic altered results can be calculated according to our theory. In the case of the Hubble ST should be remarked, that according to astronomical data, the ecliptic plane and thus the Earth’s orbit is inclined by ≈ 63˚ against the plane of the Milky Way, against which the polar axis of the Earth is inclined by 23.44˚. The Hubble ST in turn orbits Earth at an inclination angle of 28.5˚. These orbital elements add up in such a way that the plane of the Hubble orbit is almost perpendicular to the plane of the Milky Way and roughly in line with the Magellanic Clouds (see below). As already stated above, is the Hubble parameter H[0] = v/r = km∙s^−1∙Mpc^−1 the proportionality constant of the linear relationship between galaxy distances r and the velocity v of the assumed cosmic expansion. Only velocity v is precisely known to Astronomers from the exactly measured spectra. Therefrom in the case of relativistic changes, i.e. shortening of the distance r, follows ${H}_ {0}r={{H}^{\prime }}_{0}{r}^{\prime }=v=\text{const}$ . Because always ${{H}^{\prime }}_{0}>{H}_{0}⇒{{r}^{\prime }}_{0}<{r}_{0}$,follows immediately: $\frac{{H}_{0}{r}_{0}}{{{H}^{\prime }}_{0}{{r}^{\prime }}_{0}}=\frac{{H}_{0}{r}_{0}}{{{H}^{\prime }}_{0}\left({r}_{0}-\Delta {r}_{0}\right)}=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{{H}^{\prime }}_{0}=\frac{{H}_{0}{r}_{0}}{{r}_{0}-\Delta {r}_{0}}=\frac{{H}_{0}}{1-\frac{\Delta {r}_{0}}{{r}_{0}^{}}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{r}_{0}^{}=1+\Delta {r}_{0}.$(3) It is clear that (according to the following Equation (4)) the respective $\Delta {{r}^{\prime }}_{0}-\Delta {{r}^{‴}}_{0}$ (see Equations (5), (7) and (9) below) only refer to the respective relativistic increased distance ${r}_{0}^{*}=1+\Delta {r}_{0}$,because if $\Delta {r}_{0}\to \infty ⇒v\to c$ . According to the right-hand side of Equation (1), the former’s distance traveled will be enlarged by the difference: $\Delta {r}_{0}=370×\Delta {T}_{\left(\text{1-way}\right)}=370×\frac{\Delta {x}_{01\left(v\top \right)1\text{-way}}-\Delta {x}_{01\left(\text{1-way}\right)}}{2×370×\Delta {t}_{01}}×{r}_{0}.$(4) We start from the already well-founded assumption that the final 2018 value H[0] = 67.4 ± 0.5 km∙s^−1∙Mpc^−1, delivered by the Planck ST (and practically the same value from the SDSS, the ATC and the CTIO telescopes) for the Hubble constant represents the correct measurement result in this solar system, not altered by special relativistic or unknown cosmic effects. Thus, in this case, when the ST’s movement relative to the absolute rest-frame of nature or the CMB is revealed as the relativistic extension Δr, related to the number one as a unit because Δr[x]/r[x] = Δr[1]/1 (where x is any number), one calculates: $\begin{array}{l}\Delta {{r}^{\prime }}_{0}=\left[{\left(1-\frac{{370}^{2}}{{c}^{2}}\right)}^{-1}-1\right]×\frac{{370}^{2}}{2}×1=0.104244,\\ {{r}^{\prime }}_{0}=1+0.104244=1.104244\end{array}$(5) Therefore, the greatest possible relativistic extension of the Hubble constant according to equation (3) takes the value: ${{H}^{\prime }}_{0}=\frac{{H}_{0}}{\left(1-\frac{\Delta {r}_{0}}{{{r}^{\prime }}_{0}}\right)}=\frac{67.46}{\left(1-\frac{0.104244}{1.104244}\right)}=74.426046\text{\hspace{0.17em}}\text{km}\cdot {\ text{s}}^{-1}\cdot {\text{Mpc}}^{-1}$(6) That is, the parallactic orbit-length of the Hubble ST appears to be extended by this amount (5) and, thus, the distance to the Magellanic Clouds shortened accordingly. This means that this reduction in distance is transferred to the distance r in the Hubble relation via the “distance ladder” and thus, due to v = constant, a corresponding increase in H[0] is achieved. The same is true in those cases, where the ST’s orbit is other inclined relative to the same observation object than the Hubble ST. From Equation (4) follows in the event that the object observed by the ST in question is approximately in the sky between its orbital plane and the pole of the latter: $\Delta {{r}^{″}}_{0}=\frac{0.104244}{\pi }×2=0.066364,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{{r}^{″}}_{0}=1+0.066364=1.066364$(7) And so the second smallest relativistic Hubble constant needs to take the value ${{H}^{″}}_{0}=\frac{{H}_{0}}{\left(1-\frac{\Delta {r}_{0}}{{{r}^{″}}_{0}}\right)}=\frac{67.46}{\left(1-\frac{0.066364}{1.066364}\right)}=71.936915\text{\hspace{0.17em}}\text{km}\cdot {\text{s}}^{-1} \cdot {\text{Mpc}}^{-1}$(8) Lastly, we consider the case where the observed object is approximately at the zenith of the satellite orbit. This is true for Hubble measurements of the brightness variations of the light from distant galaxies, which is deflected by gravitational lenses, further, of distances to red giant stars in other galaxies, calculated using the tip of the red-giant branch stars in the galactic plane for the distance ladder. For this one computes from Equation (5): $\Delta {{r}^{‴}}_{0}=\frac{0.104244}{\pi }=0.033182,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{{r}^{‴}}_{0}=1+0.033182=1.033182$(9) This delivers the smallest relativistic extension of the Hubble constant in the solar system ${{H}^{‴}}_{0}=\frac{{H}_{0}}{\left(1-\frac{\Delta {r}_{0}}{{{r}^{‴}}_{0}}\right)}=\frac{67.46}{\left(1-\frac{0.033182}{1.033182}\right)}=69.69846\text{\hspace{0.17em}}\text{km}\cdot {\text{s}}^{-1}\ cdot {\text{Mpc}}^{-1}$(10) and corresponds very well with the value H[0] = 69.8 ± 0.8 km∙s^−1∙Mpc^−1, which was measured by the Hubble ST in the cases mentioned. But one should notice that the measured values of H[0] also can slightly vary, depending on whether the observing ST is moving about parallel or anti-parallel to the vector of v[0(CMB)] ≈ 370 km∙s^−1 of the Sun relative to Σ[00]. This means, if the ST at the time of the measurement of the Hubble constant is moving antiparallel to the velocity vector of v[0(CMB)] ≈ 370 km∙s^−1, i.e., with v[0 (CMB)] ≈ 340 km∙s^−1 relative to the CMB, then from Equations (4) to (10) in two cases smaller values of H[0] follow, namely 73.39 instead of 74.43 and 69.35 instead of 71.94 km∙s^−1∙Mpc^−1. This probably is the cause of the observed slightly differing measurement results of the Hubble constant by some research teams using the same ST. Whereas the Planck, the SDSS, the ATC and the CTIO result H[0] ≈ 67 km∙s^−1∙Mpc^−1 is completely independent of the speed relative to the CMB, because the respective observation system must be regarded as resting relative to the latter, as shown above. The last eight relationships Equations (3) to (10) are a complete expression of the laws according to which the relativistic enlarged values of the Hubble constant can be determined from the only correct basic value ≈ 67 km∙s^−1∙Mpc^−1, taking into account the average speed of the respective ST relative to the CMB and the plane of its orbit relative to the object under observation. Furthermore, it should be noted that the “experimentum crucis” for the theory, presented here, already exists with the different results of the value of the Hubble constant by the same ST and unchanged orbit (Hubble ST). These results are H[0] = 74.22 ± 1.82 km∙s^−1∙Mpc^−1 on the basis of luminosity-measurements on Cepheids in the Magellanic Clouds, on the one hand, and H[0] = 69.8 ± 0.8 km∙s^−1∙Mpc^−1 on the basis of red giants in the galactic plane, on the other hand, and again H[0] ≈ 74 H[0] km∙s^−1∙Mpc^−1 (correct value = 72.4 ± 2.0), when another research team also used the observation data of the Hubble ST from red giant stars of the LMC (instead of the Milky Way), as introduced in ref. [12,13] and [14]. This also applies to the equality of the measurement results, namely H[0] ≈ 67 km∙s^−1∙Mpc^−1, of the orbiting Planck satellite and the earthbound SDSS, ATC and CTIO telescopes, as detailed above. Hence, symmetric special relativity seems to fully resolve the discrepancies in the Hubble constant measurement results. In conclusion, it should be pointed out that the logically-physically inferred symmetrical extension of the SRT [11] not only satisfactorily explains the various measurement results for the Hubble constant. As already mentioned, in particular the so called “Proton-radius puzzle” and the cause of the discrepancies between the HIPPARCOS-satellite and earthbound VLBI Pleiades distance measurements could be traced back to the same relativistic cause as the Hubble-constant riddle. Thus, in a variation of a remark by Minkowski, it can be said: the true symmetrical core of the principle of relativity, which was initiated by Lorentz and Fitzgerald and worked out by Einstein, Poincaré and Minkowski, now completely comes to light [16]. Summed up, it is clear that a distance ladder on the grounds of ST observations will always lead to relativistic altered, incorrect results concerning distances of cosmic objects and, therewith, of the Hubble constant. The same is true for direct ST-observations of far cosmic objects, as e.g. gravitational lenses, to avoid the distance ladder, since $k\Delta d+k\left(d-\Delta d\right)=kd$ . Thereby d denotes distance to a cosmic object, Δd the first step of the distance ladder and k the factor causing the relativistic extension of the parallactic orbit-length and the associated shortening of the distance. Finally, it remains to be stated that the so-called “dark energy”, which was introduced to interpret the discrepancy of ≈67 - 74 km∙s^−1∙Mpc^−1 in the measurement results for H[0] as an accelerated expansion of the cosmos, is proving to be a physical chimera. And there is also no additional cosmic problem, as e.g. recently hypothesized by introduction of a local “Hubble Bubble” [17].	{"url":"https://www.scirp.org/journal/paperinformation?paperid=106648","timestamp":"2024-11-12T03:30:08Z","content_type":"application/xhtml+xml","content_length":"146126","record_id":"<urn:uuid:e3204ae2-234a-46dc-874b-841efddfdf6a>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00331.warc.gz"}
manual pages +.igraph {igraph} R Documentation Add vertices, edges or another graph to a graph Add vertices, edges or another graph to a graph ## S3 method for class 'igraph' e1 + e2 e1 First argument, probably an igraph graph, but see details below. e2 Second argument, see details below. The plus operator can be used to add vertices or edges to graph. The actual operation that is performed depends on the type of the right hand side argument. • If is is another igraph graph object and they are both named graphs, then the union of the two graphs are calculated, see union. • If it is another igraph graph object, but either of the two are not named, then the disjoint union of the two graphs is calculated, see disjoint_union. • If it is a numeric scalar, then the specified number of vertices are added to the graph. • If it is a character scalar or vector, then it is interpreted as the names of the vertices to add to the graph. • If it is an object created with the vertex or vertices function, then new vertices are added to the graph. This form is appropriate when one wants to add some vertex attributes as well. The operands of the vertices function specifies the number of vertices to add and their attributes as well. The unnamed arguments of vertices are concatenated and used as the ‘name’ vertex attribute (i.e. vertex names), the named arguments will be added as additional vertex attributes. Examples: g <- g + vertex(shape="circle", color= "red") g <- g + vertex("foo", color="blue") g <- g + vertex("bar", "foobar") g <- g + vertices("bar2", "foobar2", color=1:2, shape="rectangle") vertex is just an alias to vertices, and it is provided for readability. The user should use it if a single vertex is added to the graph. • If it is an object created with the edge or edges function, then new edges will be added to the graph. The new edges and possibly their attributes can be specified as the arguments of the edges The unnamed arguments of edges are concatenated and used as vertex ids of the end points of the new edges. The named arguments will be added as edge attributes. g <- make_empty_graph() + vertices(letters[1:10]) + vertices("foo", "bar", "bar2", "foobar2") g <- g + edge("a", "b") g <- g + edges("foo", "bar", "bar2", "foobar2") g <- g + edges(c("bar", "foo", "foobar2", "bar2"), color="red", weight=1:2) See more examples below. edge is just an alias to edges and it is provided for readability. The user should use it if a single edge is added to the graph. • If it is an object created with the path function, then new edges that form a path are added. The edges and possibly their attributes are specified as the arguments to the path function. The non-named arguments are concatenated and interpreted as the vertex ids along the path. The remaining arguments are added as edge attributes. g <- make_empty_graph() + vertices(letters[1:10]) g <- g + path("a", "b", "c", "d") g <- g + path("e", "f", "g", weight=1:2, color="red") g <- g + path(c("f", "c", "j", "d"), width=1:3, color="green") It is important to note that, although the plus operator is commutative, i.e. is possible to write graph <- "foo" + make_empty_graph() it is not associative, e.g. graph <- "foo" + "bar" + make_empty_graph() results a syntax error, unless parentheses are used: graph <- "foo" + ( "bar" + make_empty_graph() ) For clarity, we suggest to always put the graph object on the left hand side of the operator: graph <- make_empty_graph() + "foo" + "bar" See Also Other functions for manipulating graph structure: add_edges(), add_vertices(), delete_edges(), delete_vertices(), edge(), igraph-minus, path(), vertex() # 10 vertices named a,b,c,... and no edges g <- make_empty_graph() + vertices(letters[1:10]) # Add edges to make it a ring g <- g + path(letters[1:10], letters[1], color = "grey") # Add some extra random edges g <- g + edges(sample(V(g), 10, replace = TRUE), color = "red") g$layout <- layout_in_circle version 1.3.3	{"url":"https://igraph.org/r/html/1.3.3/plus-.igraph.html","timestamp":"2024-11-14T15:34:44Z","content_type":"text/html","content_length":"13730","record_id":"<urn:uuid:cfbd645c-d11f-4a5c-9102-5e24dda96611>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00282.warc.gz"}
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The paper is available for letter and a4 paper. Use these dotted grid sheets for cross stitch patterns, floor plans, drawings, math, etc. Web free printable dot grid paper templates in a variety of grid sizes. Web Dotted Paper Offers An Ideal Compromise Between A Grid And Free Drawing And Writing. Dot paper is just like a regular graph paper. Web free assortment of printable dot paper. 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How to Plot Confusion Matrix In Pytorch? To plot a confusion matrix in PyTorch, you can first calculate the predictions of your model on a set of test data. Then, use the predicted outputs and the ground truth labels to create a confusion matrix. The confusion matrix is a table that shows the number of true positives, true negatives, false positives, and false negatives for each class in your dataset. You can use the sklearn.metrics.confusion_matrix function to create the confusion matrix and then visualize it using tools like matplotlib.pyplot. By plotting the confusion matrix, you can gain insights into how well your model is performing on the different classes in your dataset and identify any patterns or areas of improvement. How to handle overlapping classes in a confusion matrix in PyTorch? When dealing with overlapping classes in a confusion matrix in PyTorch, one approach is to modify the confusion matrix to allow for the consideration of these overlapping classes. One way to handle overlapping classes is by using a soft assignment approach, where each sample is assigned a probability distribution over the classes instead of a single class label. Here are the steps to handle overlapping classes in a confusion matrix in PyTorch: 1. Create a confusion matrix with dimensions corresponding to the number of overlapping classes. For example, if there are three overlapping classes, the confusion matrix will have a shape of (3, 2. Modify the predictions to output probability distributions over the classes rather than single class labels. This can be done by changing the output layer of the neural network to output softmax 3. Assign each sample a probability distribution over the overlapping classes based on the output of the model. 4. Calculate the confusion matrix using the assigned probability distributions. For each sample, increment the corresponding cell in the confusion matrix based on the highest probability class. 5. Evaluate the performance of the model by analyzing the confusion matrix and computing metrics such as accuracy, precision, and recall. By implementing these steps, you can effectively handle overlapping classes in a confusion matrix in PyTorch and accurately evaluate the performance of your model. How to calculate the area under the ROC curve from a confusion matrix in PyTorch? To calculate the area under the ROC curve from a confusion matrix in PyTorch, you can use the sklearn library which provides a function called roc_auc_score. Here is an example on how to calculate the area under the ROC curve using PyTorch and sklearn: 1 import torch 2 from sklearn.metrics import roc_auc_score 4 # Create a confusion matrix 5 confusion_matrix = torch.tensor([[100, 10], [20, 50]]) 7 # Calculate True Positive (TP), False Positive (FP), True Negative (TN), False Negative (FN) 8 TP = confusion_matrix[0, 0] 9 FP = confusion_matrix[0, 1] 10 FN = confusion_matrix[1, 0] 11 TN = confusion_matrix[1, 1] 13 # Calculate True Positive Rate (TPR) and False Positive Rate (FPR) 14 tpr = TP / (TP + FN) 15 fpr = FP / (FP + TN) 17 # Calculate the area under the ROC curve 18 roc_auc = roc_auc_score(torch.tensor([1, 1, 0, 0]), torch.tensor([0.9, 0.8, 0.3, 0.2])) 20 print(roc_auc) In this example, we first create a confusion matrix using torch.tensor. Then we calculate the True Positive (TP), False Positive (FP), True Negative (TN), and False Negative (FN) from the confusion matrix. Next, we calculate the True Positive Rate (TPR) and False Positive Rate (FPR). Finally, we calculate the area under the ROC curve using the roc_auc_score function from the sklearn library. How to plot a confusion matrix for a regression problem in PyTorch? In PyTorch, since confusion matrices are typically used for classification problems, you'll need to convert your regression problem into a classification problem in order to plot a confusion matrix. One common approach is to define thresholds for different classes based on the regression output. Here's an example code snippet to help you get started: 1 import torch 2 import numpy as np 3 from sklearn.metrics import confusion_matrix 4 import matplotlib.pyplot as plt 6 # Generate some sample regression outputs 7 true_values = torch.tensor([0.5, 0.8, 0.2, 0.9, 0.4]) 8 predicted_values = torch.tensor([0.6, 0.7, 0.1, 0.8, 0.3]) 10 # Define thresholds for classifying the regression outputs 11 thresholds = [0.2, 0.5, 0.8] 12 true_classes = np.digitize(true_values, thresholds) 13 predicted_classes = np.digitize(predicted_values, thresholds) 15 # Create confusion matrix 16 cm = confusion_matrix(true_classes, predicted_classes) 18 # Plot confusion matrix 19 plt.matshow(cm) 20 plt.colorbar() 21 plt.xlabel('Predicted') 22 plt.ylabel('True') 23 plt.show() In this example, we first define our true regression values (true_values) and predicted regression values (predicted_values). We then define the thresholds for classifying the regression outputs into different classes. We use the np.digitize function to assign a class for each regression value based on the defined thresholds. Finally, we calculate the confusion matrix using sklearn's confusion_matrix function and plot it using matplotlib. Keep in mind that this approach is just one way to visualize the results of a regression problem in a confusion matrix. Depending on your specific regression problem and application, you may need to adapt this code to better suit your needs. How to plot a normalized confusion matrix in PyTorch? To plot a normalized confusion matrix in PyTorch, you can follow these steps: 1. First, you need to calculate the confusion matrix and normalize it. You can do this using the sklearn.metrics module. Here is an example code snippet to calculate the confusion matrix and normalize it: 1 from sklearn.metrics import confusion_matrix 3 # Evaluate your model and get the predicted labels and true labels 4 predicted_labels = model(x) 5 conf_matrix = confusion_matrix(true_labels, predicted_labels) 6 normalized_conf_matrix = conf_matrix.astype('float') / conf_matrix.sum(axis=1)[:, np.newaxis] 1. After calculating the normalized confusion matrix, you can plot it using the matplotlib library. Here is an example code snippet to plot the normalized confusion matrix: 1 import matplotlib.pyplot as plt 3 plt.figure(figsize=(10, 8)) 4 plt.imshow(normalized_conf_matrix, interpolation='nearest', cmap=plt.cm.Blues) 5 plt.title('Normalized Confusion Matrix') 6 plt.colorbar() 7 plt.ylabel('True label') 8 plt.xlabel('Predicted label') 9 plt.show() By following these steps, you should be able to plot a normalized confusion matrix in PyTorch. How to display class-wise metrics from a confusion matrix in PyTorch? To display class-wise metrics from a confusion matrix in PyTorch, you can use the following steps: 1. First, calculate the confusion matrix using PyTorch's torchmetrics library or any custom implementation. The confusion matrix is a matrix where each row represents the actual class and each column represents the predicted class. 1 from sklearn.metrics import confusion_matrix 3 # Assuming y_true and y_pred are your true and predicted labels 4 conf_matrix = confusion_matrix(y_true, y_pred) 1. Calculate class-wise metrics such as precision, recall, and F1 score from the confusion matrix. 1 # Calculate precision, recall, and F1 score for each class 2 tp = np.diag(conf_matrix) 3 fp = np.sum(conf_matrix, axis=0) - tp 4 fn = np.sum(conf_matrix, axis=1) - tp 6 precision = tp / (tp + fp) 7 recall = tp / (tp + fn) 8 f1 = 2 * (precision * recall) / (precision + recall) 1. Print or display the class-wise metrics. 1 for i in range(len(precision)): 2 print(f"Class {i} - Precision: {precision[i]}, Recall: {recall[i]}, F1 Score: {f1[i]}") By following these steps, you can display class-wise metrics from a confusion matrix in PyTorch.	{"url":"https://stock-market.uk.to/blog/how-to-plot-confusion-matrix-in-pytorch","timestamp":"2024-11-08T21:00:31Z","content_type":"text/html","content_length":"189103","record_id":"<urn:uuid:79d420e2-a2dd-42e0-b59a-3f36f3d0075f>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00751.warc.gz"}
Graduate Course Inventory • Courses in statistics are taught by different departments. The following information is only applicable to courses offered by the Department of Statistics and Data Sciences under the SDS field of • Registration questions regarding SDS courses can be directed to stat.admin@austin.utexas.edu. Please include your EID and the course unique number in your inquiry. • Questions about the Graduate Portfolio in Applied Statistical Modeling and Graduate Portfolio in Scientific Computation can be directed to stat.portfolios@austin.utexas.edu. SDS Statistics Courses Click on a course to be taken to its description. Course Descriptions SDS 380C. Statistical Methods I An introduction to the fundamental concepts and methods of statistics. The course will cover topics ranging from descriptive statistics, sampling distributions, confidence intervals, and hypothesis testing. Topics could include simple and multiple linear regression, Analysis of Variance, and Categorical Analysis. Use of statistical software is emphasized. Prerequisite: Graduate standing. SDS 380D. Statistical Methods II A continuation of SDS 380C: Statistical Methods I. The course presents an overview of advanced statistical modeling topics. Topics may include random and mixed effects models, time series analysis, survival analysis, Bayesian methods, and multivariate analysis of variance. Use of statistical software is emphasized. Prerequisite: Graduate standing, and Statistics and Data Sciences 380C or the SDS 381. Mathematical Methods for Statistical Analysis Introduction to mathematical concepts and methods essential for multivariate statistical analysis. Topics include basic matrix algebra, eigenvalues and eigenvector, quadratic forms, vector and matrix differentiation, unconstrained optimization, constrained optimization, and applications in multivariate statistical analysis. Prerequisite: Graduate standing and one course in statistics. SDS 382. Introduction to Probability and Statistics Expectation and variance of random variables, conditional probability and independence, sampling distributions, point estimation, confidence intervals, hypothesis tests, and other topics. Prerequisite: Graduate standing, and M408D or M408L. SDS 383C. Statistical Modeling I Introduction to core applied statistical modeling ideas from a probabilistic, Bayesian perspective. Topics include: (i) Exploratory Data Analysis; (ii) Programming and Graphics in R; (iii) Bayesian Probability Models; (iv) Intro to the Gibbs Sampler; (v) Applied Regression Analysis; (vi) The “Normal-means” problem; and (vii) Hierarchical Models. Prerequisite: Graduate standing. SDS 383D. Statistical Modeling II In this course, students will learn to describe real-world systems using structured probabilistic models that incorporate multiple layers of uncertainty. Major topics to be covered include: (i) theory of the multivariate normal distribution; (ii) mixture models; (iii) introduction to nonparametric Bayesian analysis; (iv) advanced hierarchical models and latent-variable models; (v) Generalized Linear Models; and (vi) advanced topics in linear and nonlinear regression. Examples will be taken from a wide variety of applied fields in the physical, social, and biological sciences. Prerequisite: Graduate standing. SDS 384: Topics in Statistics and Probability Concepts of probability and mathematical statistics with applications in data analysis and research. May be repeated for credit when the topics vary. Prerequisite: Graduate standing, and Statistics and Data Sciences 382, Mathematics 362K and 378K, or consent of instructor. • Topic 1: Applied Probability. Basic probability theory, combinatorial analysis of random phenomena, conditional probability and independence, parametric families of distributions, expectation, distribution of functions of random variables, limit theorems. • Topic 2: Mathematical Statistics I. The first semester of a two-semester course covering the general theory of mathematical statistics. The two-semester course covers distributions of functions of random variables, properties of a random sample, principles of data reduction, overview of hierarchical models, decision theory, Bayesian statistics, and theoretical results relevant to point estimation, interval estimation, and hypothesis testing. • Topic 3: Mathematical Statistics II. A continuation of Statistics and Scientific Computation 384 (Topic 2). Additional prerequisite: Statistics and Data Sciences 384 (Topic 2). • Topic 4: Regression Analysis. Simple and multiple linear regression, inference in regression, prediction of new observations, diagnostics and remedial measures, transformations, model building. Emphasis will be on both understanding the theory and applying theory to analyze real data. • Topic 5: Multivariate Statistical Analysis. Introduction to the general multivariate linear model: a selection of techniques including principle components, factor analysis, and discriminant • Topic 6: Design and Analysis of Experiments. Design and analysis of experiments, including one-way and two-way layouts; components of variance; factorial experiments; balanced incomplete block designs; crossed and nested classifications; fixed, random, and mixed models; split plot designs. • Topic 7: Bayesian Statistical Methods. Fundamentals of Bayesian inference in single and multi-parameter models for inference and decision making, including simulation of posterior distributions, Markov chain Monte Carlo methods, hierarchical models, and empirical Bayes models. • Topic 8: Time Series Analysis. Introduction to statistical time series analysis: ARIMA and more general models, forecasting, spectral analysis, and time domain regression. Model identification, estimation of parameters, and diagnostic checking are included. Additional Prerequisite: Statistics and Data Sciences 384 (Topic 3) and consent of instructor. • Topic 9: Computational Statistics. A course in modern computationally-intensive statistical methods including simulation, optimization methods, Monte Carlo integration, maximum likelihood / EM parameter estimation, Markov chain Monte Carlo methods, resampling methods, non-parametric density estimation. • Topic 10: Stochastic Processes. Concepts and techniques of stochastic processes with an emphasis on the nature of change of variables with respect to time. Characterization, structural properties and inference are covered. SDS 385: Topics in Applied Statistics Theories, models and methods for the analysis of quantitative data. With consent of the graduate advisor, may be repeated for credit when the topics vary. Prerequisite: Graduate standing, and Statistics and Data Sciences 380 or 382 or consent of instructor. • Topic 1: Experimental Design. Principles, construction and analysis of experimental designs. Includes one-way classification, randomized blocks, Latin squares, factorial and nested designs. Fixed and random effects, multiple comparisons, and analysis of covariance. Additional prerequisite: Statistical Methods I or its equivalent. • Topic 2: Applied Regression. Simple and multiple linear regression, residual analysis, transformations, model building with real data, testing models. Additional prerequisite: Experimental Design or its equivalent. • Topic 3: Applied Multivariate Methods. A practical introduction to the analysis of multivariate data as applied to examples from the social sciences. Multivariate linear model, principal components and factor analysis, discriminant analysis, clustering and canonical correlation. Additional prerequisite: Applied Regression or its equivalent. • Topic 4: Analysis of Categorical Data. Methods for analyzing categorical data. Topics include categorical explanatory variables within the General Linear Model; models of association among categorical variables; models in which the response variable is categorical or is a count. Logical similarities across methods will be emphasized. • Topic 5: Structural Equation Modeling. Introduction to the basic concepts, methods and computing tools of structural equation modeling. Emphasis will be placed on developing a working familiarity with some of the common statistical procedures, coupled with their application through the use of statistical software. Additional prerequisite: Applied Regression or its equivalent. • Topic 6: Hierarchical Linear Models. Introduction to multilevel data structures, model building and testing, effect size, fixed and random effects, missing data and model assumptions, logistic HLM, statistical power, and design planning. Additional prerequisite: Applied Regression or its equivalent. • Topic 7: Survey Sampling and Methodology. Survey planning, execution and analysis. Principles of survey research, including sampling, measurement; questionnaire construction and distribution; response effects; validity and reliability; scaling data sources; data reduction and analysis. • Topic 8: Introduction to Bayesian Methods. A practical introduction to Bayesian statistical inference, with an emphasis on applications in behavioral and measurement research. Examination of how Bayesian statistical inference differs from classical inference in the context of simple statistical procedures and models, such as hypothesis testing, ANOVA and regression. Additional prerequisite: Applied Regression or its equivalent. • Topic 9: Longitudinal Data Analysis. Applications of models to data collected at successive points in time. Multilevel models for change, random coefficient models; latent growth curve models; models for nonlinear growth. Applications of models to event-occurrence data. Discrete-time and continuous-time event history models. • Topic 10: Modern Statistical Methods. An introduction to conducting statistical analysis using modern resampling methods of bootstrapping and Monte Carlo simulation. Equal emphasis will be placed on theoretical understanding and application. • Topic 11: Mathematical Statistics for Applications. Introduction to the basic concepts of probability and mathematical statistics for doctoral degree students who plan to use statistical methods in their research but do not need a highly mathematical development of the subject. Topics include probability distributions and estimation theory and hypothesis testing techniques. Additional prerequisite: A calculus course covering integration and differentiation. • Topic 12: Meta-Analysis. An introduction to statistics used to synthesize statistical results from a set of studies. Course content can include calculation of different effect sizes, calculating pooled estimates using fixed and random effects models, testing moderating variables using fixed and mixed effects models, test of heterogeneity of effect sizes, assessing and correcting publication bias. Additional prerequisite: Applied Regression (Topic 2) or the equivalent. • Topic 13: Factor Analysis. An introduction to exploratory and confirmatory factor analysis. Exploratory factor analysis section's content can include review of matrix algebra and vector geometry, principal components and principle axis factoring, factor rotation methods. Confirmatory factor analysis section's content includes single- and multiple-factor, multi-sample models, multitrait-multimethod and latent means modeling. For both methods, experience will be provided in writing up and critiquing others' studies. Additional prerequisite: Applied Regression (Topic 2) or the equivalent. • Topic 14: Maximum-Likelihood Statistics. Introduction to the likelihood theory of statistical inference. Topics include probability distributions, estimation theory, and applications of the MLE to models with categorical or limited dependent variables, event count models, event history models, models for time-series cross-section data, and models for hierarchical data. • Topic 15: Survival Analysis/DurationModeling. This course will focus on the statistical methods related to the analysis of survival or time to event data. Survival analysis, hazard modeling, has applications in several fields, such as studying time till death (medical or biological), length of unemployment (economics), a felon's time to parole (criminology), duration of first marriage (sociology), and reliability and failure time analysis (engineering). The class will focus on practical applications. Some of the topics covered in the course will include descriptive statistics, such as Kaplan-Meier estimators, semiparametric and parametric regression models, model development and assessing model adequacy. • Topic 16: Selected Topics. SDS 386C. Probabilistic Graphical Models An introduction to statistical learning methods, exploring both the computational and statistical aspects of data analysis. Topics include numerical linear algebra, convex optimization techniques, basics of stochastic simulation, nonparametric methods, kernel methods, graphical models, decision tress and data re-sampling. Prerequisites: Graduate standing. SDS 386D. Monte Carlo Methods in Statistics This course focuses on stochastic simulation for Bayesian inference. The main focus is for students to develop a solid understanding of MCMC methods and the underlying theoretical framework. Topics include: (i) Markov chains; (ii) Intro to MC integration; (iii) Gibbs Sampler; (iv) Metropolis-Hastings algorithms; (v) Slice sampling; and (vi) Sequential Monte Carlo. Prerequisites: Graduate standing, knowledge of mathematical statistics as well as basic coding skills (R, Matlab, or Stata). SDS 387. Linear Models This course focuses on the practical application of the projection approach to linear models. The course will begin with a review of essential linear algebra concepts including vector spaces, basis, linear transformations, norms, orthogonal projections, and simple matrix algebra. It continues by presenting the theory of linear models from a projection-based perspective. Still on the projection framework, Bayesian ideas will be introduced. Additional topics include: (i) Analysis of Variance; (ii) Generalized Linear Models; and (iii) Variable Selection Techniques. Prerequisites: Graduate standing, knowledge of mathematical statistics at a graduate level and linear algebra at an advanced undergraduate level is required as well as basic coding skills (R, Matlab, or Stata). SDS 389. Time Series and Dynamic Models This course focuses on the general class of state-space models or Dynamic Models. Emphasis will be placed on the implementation and use of the models presented. Applications will focus on the social sciences but an effort will be made to keep students from the physical sciences engaged in the topics. Topics covered include: (i) Dynamic Regression Models; (ii) The Kalman Filter; (iii) Multivariate Time Series Models; (iv) Conditional Variance Models; (v) MCMC algorithms for state-space models; and (vi) Particle Filters. Prerequisite: Graduate standing, and knowledge of mathematical statistics at a graduate level as well as basic coding skills (R, Matlab, or Stata). SDS 391D. Data Mining Focuses on various mathematical and statistical aspects of data mining. Topics covered include supervised learning (regression, classification, support vector machines) and unsupervised learning (clustering, principal components analysis, dimensionality reduction). The technical tools used in the course draw from linear algebra, multivariate statistics and optimization. Prerequisites: Graduate standing and Mathematics 341 or equivalent. SDS 392M. Computational Economics. Same as Economics 392M (Topic 12) Introduction to the development and solution of economic models of growth, macroeconomic fluctuations, environmental economics, financial economics, general equilibrium models, game theory and industrial economics. The course also includes sections on neural nets, genetic algorithms and agent-based methods and stochastic control theory applied to a variety of economic topics. Prerequisite: Graduate standing.    SDS 393C. Numerical Analysis: Linear Algebra Same as Computational and Applied Mathematics 383C and Mathematics 383E and Computer Sciences 383C. Survey of numerical methods in linear algebra: floating-point computation, solution of linear equations, least squares problems, algebraic eigenvalue problems. Prerequisite: Graduate standing, either consent of instructor or Mathematics 341 or 340L, and either Mathematics 368K or Computer Sciences 367.   SDS 393D. Numerical Analysis: Interpolation, Approximation, Quadrature, and Differential Equations Same as Computational and Applied Mathematics 383D and Mathematics 383F and Computer Sciences 383D. Survey of numerical methods for interpolation, functional approximation, integration, and solution of differential equations. Prerequisite: Graduate standing, either consent of instructor or Mathematics 427K and 365C; and Computational and Applied Mathematics 383C, Computer Sciences 383C, or Mathematics 383E or Statistics and Data Sciences 393C. SDS 394. Scientific and Technical Computing Comprehensive introduction to computing techniques and methods applicable to many scientific disciplines and technical applications. Covers computer hardware and operating systems, systems software and tools, code development, numerical methods and math libraries, and basic visualization and data analysis tools.Prerequisite: Graduate standing, and Mathematics 408D or 408M. Prior programming experience is recommended. SDS 394C. Parallel Computing Parallel computing principles, architectures, and technologies. Parallel application development, performance, and scalability. Prepares students to formulate and develop parallel algorithms to implement effective applications for parallel computing systems. Three lecture hours a week for one semester. Prerequisite: Graduate standing, and Mathematics 408D or 408M, Mathematics 340L, and prior programming experience using C or Fortran on Unix/Linux systems.    SDS 394D. Distributed and Grid Computing for Scientists and Engineers Distributed and grid computing principles and technologies. Covers common modes of grid computing for scientific applications, developing grid enabled applications, future trends in grid computing. Three lecture hours a week for one semester. Prerequisite: Graduate standing, and Mathematics 408D or 408M, Mathematics 340L, and prior programming experience using C or Fortran on Unix/Linux SDS 394E. Visualization and Data Analysis Scientific visualization principles, practices, and technologies, including remote and collaborative visualization. Introduces statistical analysis, data mining and feature detection. Prerequisite: Graduate standing, Mathematics 408D or 408M, Mathematics 340L, and prior programming experience using C or Fortran on Linux or Unix systems.    SDS 395. Advanced Topics in Scientific Computation Three lecture hours a week for one semester. Topics are announced in the Course Schedule. May be repeated for credit when the topics vary. Prerequisite: Graduate standing; additional prerequisites vary with the topic and are given in the Course Schedule.   SDS 398T. Supervised Teaching in Statistics and Scientific Computation Supervised teaching experience; weekly group meetings, individual consultations, and reports. Offered on the credit/no credit basis only. Prerequisite: Graduate standing and appointment as a teaching	{"url":"https://stat.utexas.edu/academics/graduate-course-inventory","timestamp":"2024-11-03T05:50:47Z","content_type":"text/html","content_length":"63870","record_id":"<urn:uuid:46b80efc-9494-46f5-9b52-cc952af64f4a>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00140.warc.gz"}
Solar radiation at the Earth's surface Skip the introduction and go to the data entry screen: This program predicts the amount of solar radiation reaching the surface of the Earth. The solar radiation is estimated by the following steps (List 1971, Campbell 1977). 1) the amount of radiation reaching the top of the atmosphere for each hour of the day is estimated by the formula: (1) I_o = (J_o/R^2) * sin phi / 24 where I_o is the total radiation falling on the atmosphere for a unit of time, J_o is the solar constant (the default is 1360 W m^-2 day^-1), R is the Sun's radius vector (from Table 169 in the Smithsonian Meteorological Tables; List 1971), and sin phi is the Sun's elevation angle. sin phi is calculated using the following formula: (2) sin phi = (sin LAT * sin DECL) + (cos LAT * cos DECL * cos h) where LAT is the latitude (supplied by the user), DECL is the solar's declination and h is the solar hour angle. h is calculated by: (3) h = 15 * (h_t - 12) where h_t is the current hour. If the sun is below the horizon, sin phi will be negative. For negative values of sin phi, no incoming solar radiation was assumed to be coming in for that time period (i.e., the period between dusk and dawn). The 60 in Equation (1) is convert the radiation estimate for one minute into an hourly estimate. This is necessary because of the units for the solar constant. 2) The amount of direct solar radiation falling at the earth's surface is then estimated by: (4) I_sr = (J_o/R^2) * a^m * sin phi / 24 where I_sr is the direct solar radiation at the surface, a is the atmospheric transmissivity (the default is 0.84), and m is a correction for optical path length (Campbell 1977). m is given m = (P/P_o)/sin phi where P is the atmospheric pressure at the site and P_o is the atmospheric pressure at sea level. This program asks the user to provide the elevation of the site, and then estimates atmospheric pressure at the site (Hess 1959). The formula is: P = P_o * e ^-(z/Ho) where z is the elevation (in meters) and Ho is the height of the homogenous atmosphere (assumed to be 8,000 m; Hess 1959). 3) The amount of diffuse solar radiation reaching the surface is given by: (5) I_sf = ((0.91 * I_o) - I_sr)/2 where I_sf is diffuse solar radiation. 4) The direct and diffuse solar radiation amounts at the surface are summed to give the total radiation amount falling on the surface. 5) Hourly radiation estimates are summed to arrive at the daily estimate. 6) Note that the solar declination is calculated by the following formula: (6) DECL = 23.45 * sin[360/365 * (284 + day of year)] I don't have a reference for this yet, but it's simpler than my old way of calculating declination. 7) The slope radiation calculations, based on algorithms from the program, MTCLIM (Hungerford et al. 1989) seem to work fine now. [DML 09 Dec 2006] Campbell, G.S. 1977. An introduction to environmental biosphysics. Springer-Verlag, New York, New York, USA. Hess, S.L. 1959. Introduction to theoretical meteorology. Robert E. Krieger Publishing Company, Malabar, Florida, USA. Hungerford, R.D., R.R. Nemani, S.W. Running and J.C. Coughlan. 1989. MTCLIM: A Mountain Microclimate Simulation Model. USDA Forest Service, Intermountain Research Station, Research Paper INT-414. List, R.J. 1971. Smithsonian meteorological tables, 6th ed. Smithsonian Institution Press, Washington, D.C., USA. Go on to the data entry screen:	{"url":"https://davidmlawrence.com/science/RadIntro.html","timestamp":"2024-11-02T05:53:48Z","content_type":"text/html","content_length":"15901","record_id":"<urn:uuid:a5ead694-9727-48fe-a0c9-2c8f7f763e15>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00047.warc.gz"}
What is an Algorithm? The term “algorithm” does not seem to be relevant to children, but algorithms control everything from the technologies they use to the simple choices they make on a daily basis. Algorithms are interesting, and while some are very complicated, the definition itself is quite straightforward. A detailed step-by-step instruction, collection, or formula for solving a problem or completing a task is known as an algorithm. In programming, programmers create algorithms that tell the machine how to do something. In other words, an algorithm is a set of well-defined instructions in sequence to solve a problem. Algorithms are everywhere as you think about them in the truest sense (not just in terms of computing). A formula for cooking is an algorithm, as is the form you use to solve addition or long division problems. Folding a shirt or a pair of trousers is also an algorithm. Even your morning routine can be thought of as an algorithm! How Do Algorithms Work At a Basic Level? The aim of algorithms and machine learning on the internet is to effectively mimic the human brain’s decision-making processes — without us needing to take in all of the knowledge and time required to work through it on our own. It’s almost as if it’s a shortcut to the human thought process! Algorithms are, at their most simple stage, a series of if-then statements that run in a program very rapidly to produce a result: If you do A, you can get B. If you want C instead of D, you will get “Where are you from?” you may inquire of a friend. Certain choices will be your easiest responses depending on their reaction. You’ll react differently if they say they’re from your childhood hometown than if they say they’re from a city you’ve never been. However, each step leads in a certain direction, and the program follows the workflow or formula to the next stage. The same thing happens when you use Google to look for something. You can search for “Domino’s pizza” and Google can respond with a recipe or a list of restaurants that serve Domino’s pizzas nearby you; based on the knowledge Google has about you, your location, and your search history. Types of Algorithms Algorithms come in a variety of forms, but the following are the most common: 1. Recursive algorithm • This is a fascinating algorithm because it calls itself with a smaller value as inputs, which it obtains after solving for the existing inputs. To put it another way, Recursive Algorithm keeps calling itself recursively before the problem is solved. And hence the name! • These Algorithms can quickly solve problems like the Tower of Hanoi or the Depth First Search (DFS) of a Graph. 2. Divide and Conquer • Divide and Conquer Algorithms are one of the important methods for resolving a variety of issues. Divide the algorithm into two parts in Divide and Conquer algorithms; the first section divides the problem at hand into smaller sub-problems of the same kind. Then, in the second section, these smaller problems are solved and then combined to produce the final solution to the problem. • Merge sorting, and quick sorting can be done with divide and conquer algorithms. 3. Dynamic Programming Algorithm • These algorithms work by recalling previous run results and using them to generate new ones. In other words, a Dynamic Programming Algorithm breaks down complex problems into several simple sub-problems, solves each one once, and then saves the results for later use. • Fibonacci sequence is a good example of Dynamic Programming algorithm. 4. Greedy Algorithm • Greedy Algorithms are used to solve problems involving optimization. We find a locally optimal solution (without respect for potential consequences) and expect to find the optimal solution at the global level using this algorithm. • The Greedy algorithm is used in Huffman Coding and Dijkstra’s algorithm. 5. Brute Force Algorithm • One of the simplest algorithms in the definition is this one. A Brute Force Algorithm searches all possible solutions randomly in order to find one or more solutions that could solve a function. Consider brute force as attempting to open a safe with any available combinations of numbers. • E.g. Sequential search 6. Backtracking Algorithm • Backtracking is a method for finding a solution to a problem in a step-by-step manner. It solves problems in a recursive manner, attempting to solve a problem one piece at a time. If one of the options fails, we discard it and go back to the drawing board to try another. • The N Queens problem is a clear example of how to use the Backtracking algorithm. The N Queen Problem notes that there are N pieces of queens on a chessboard, and we must arrange them in such a way that no queen can strike another queen on the board until it is ordered Benefits of Algortihmic Thinking In subjects like math and science, algorithmic thinking, or the ability to determine simple steps to solve a problem, is critical. Kids, particularly in math, use algorithms without understanding it all the time. In order to solve a long division puzzle, students use a trained algorithm to iterate over the digits of the number they’re dividing. The child must divide, multiply, and subtract each digit of the dividend (the number being divided). Algorithmic thought enables children to deconstruct problems and formulate ideas in terms of sequential measures in a process. All the more, algorithms are the first step towards becoming an efficient programmer! Want to strengthen your algorithmic thinking skills and code your way out? Come on then. Let’s break down things into logical steps and work on them one at a time at Rancho labs!	{"url":"https://www.rancholabs.com/post/what-is-an-algorithm","timestamp":"2024-11-13T22:03:51Z","content_type":"text/html","content_length":"1050500","record_id":"<urn:uuid:d862cc2a-3431-4292-8d4d-d21c32fc7909>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00452.warc.gz"}
CS373 Summer 2018: Travis Llado, Week 02 Update for the week of 2018-06-15 What did I do this week? Mostly moving. In class, not a whole lot. I came up with an optimization for the assigned Collatz Conjecture program, but as is often the case it will be much more difficult and less ideal to implement than initially imagined. The idea is that we should pre-cache as many results as possible, but not just any results. We should cache the most valuable possible results. If we define value as speed, then the most valuable results to store in our lookup table would be the nodes in our convergence graph that save us the most computational time. We could calculate that definition of value by measuring the height of each node in the graph, in terms of steps from the end or steps away from 1, multiplied by the number of times that node will be visited when we calculate the convergence length for all numbers 1 through 1,000,000. This seems pretty straight forward, easy to implement. And this part is. But when we think one step farther, we realize that we'll only use the first node in each convergence path. If we have a tall and isolated path in our graph, the top ten nodes might be ranked as the top ten most valuable nodes in the entire graph, but only the top one of them would ever be used. We need to find a way to filter out nodes that will never be visited due to our use of the lookup table. We need to think of a better definition for "value". Of course we could create arbitrarily complex algorithms to predict the best possible answer, but clock-cycles are cheaper than man-hours. It seems like the best solution might be an experiential solution, where we create a table of the most valuable nodes, then actually run our program again and record how many times each element of the table is used, and re-derive expected value from that. Also, I found group members. I have a group. What is in my way? Not much. The basic assignment is very straight forward, nothing I haven't done before. What will I do next week? I'll write the naive, completely unoptimized program and have that all committed and documented and annotated, and then I'll spend a few hours working on the proposed precache optimization. After I create an algorithm for calculating node value in a way such that nodes that will never be visited are given low values, it should run very quickly. What is my experience of the class? Good so far. Lectures are useful, plenty of details that I probably wouldn't have learned anywhere else. That's why I came back to school, to pick up those easily overlooked details. The reading quizzes are sometimes galling, as some of the questions have been trivial. Knowing the name of the person who created a language doesn't teach me anything about the language itself. What is my recommendation of the week? I haven't done much in the past week outside of class and moving. I read a book a few weeks ago called The Code Book . It goes through a quick history of cryptography, starting with ancient ciphers and moving up through modern math-based cryptographic methods, all the way from WWII machines up to RSA and quantum cryptography. A very interesting book.	{"url":"https://www.travisllado.com/2018/06/cs373-summer-2018-travis-llado-week-02.html","timestamp":"2024-11-07T00:15:20Z","content_type":"text/html","content_length":"103225","record_id":"<urn:uuid:8b0004e6-12e6-4bda-97c0-670d696055bf>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00624.warc.gz"}
Albert Oldenburg Lindblom Tech High School 6130 So. Wolcott Chicago, IL To define a vector and its rules. To use as a tool to help clarify its use with motion and forces. To have students discover these relation-ships with the use of various simple equipment and a computer. Three pieces of heavy yarn of different colors, preferably blue, orange, and green. Three pieces of chalk of different colors, blue, orange, and green. A large blackboard protractor. Several meter sticks and 2-meter meter sticks. Three bricks wrapped in plastic. A side clamp for the front desk. A computer with a large monitor. Four large pieces of paper with E,W,N,S printed on them. Post the four large signs representing the directions on the four opposite walls in the classroom before class begins. Ask a student to move a brick from the front desk to his seat. Make it known to the class the direction that the brick was moved and that it might represent the change in position of any object from one position to another. Ask the class to enter into the description of this change in position of the brick (to elicit an answer of "displacement"). Have students take out a sheet of paper and pencil to record a simple data table you place on the board, having length in meters and angle in degrees for each of the three sides of the triangle. Label the sides A,B,C respectively. Also the angles are labelled as alpha, beta, and gamma. Ask those students with calculators to raise their hands, then divide the class into groups of three or four people so each group would have access to a calculator. Challenge each group to find one of the sides and angles, then record them on the board. Then graphically have the students record them on their graph paper. Show students how to graphically add vectors "head to tail", by putting the head of the second vector to the head of the first vector and so forth until all vectors for that problem have been added. While writing vector equations remind the students to indicate above each algebraic symbol a small arrow representing a vector. After graphing, a chalk and talk review of vectors takes place. At this point the students would be advised to record these notes for future reference. The main concepts revealed here were the addition of parallel vectors (positive or negative) depending on their direction, then adding two vectors at 90 degrees was shown. A third but more realistic way to add vectors at any direction (not 90 or 180 degrees) was also shown. Here we introduced the use of the Law of Sines. For another class period an interesting computer review was prepared for the students with a dialogue on the computer and displayed on the large monitor. There were several parts to the dialogue. The student takes out a paper and pencil and writes down the answers to the definitions asked for on the first part of the dialogue. A student is asked to operate the computer at the teachers' command. The best answer on the monitor screen is not revealed until the definition is rather thoroughly discussed. The computerized review of the problems are revealed step by step on the monitor with the students recording it. The problem is broken down step by step and reconstructed with complete agreement on each step along the way. First a problem involving parallel vectors was used and repeated a number of times. Then the vector problems using right angles was reviewed. The vectors at various angles were shown to be quite simplified by using this method. A ditto sheet is then passed out and a few "homework" problems are printed on it to review them. With the use of the computer monitor we substituted some of the homework problem variables in it and proceeded to demonstrate to the students how to solve the problems by breaking them down into steps and reconstructing them. Return to Physics Index	{"url":"https://smileprogram.info/ph8619.html","timestamp":"2024-11-07T20:15:35Z","content_type":"text/html","content_length":"4389","record_id":"<urn:uuid:f75d04b7-fcc7-40d0-af75-9da57c44bd32>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00708.warc.gz"}
PROC GAM: Additive Models and Generalized Additive Models :: SAS/STAT(R) 9.22 User's Guide Additive Models and Generalized Additive Models This section describes the methodology and the fitting procedure behind generalized additive models. The additive model generalizes the linear model by modeling the dependency as In order to be estimable, the smooth functions While traditional linear models and additive models can be used in most statistical data analysis, there are types of problems for which they are not appropriate. For example, the normal distribution might not be adequate for modeling discrete responses such as counts or bounded responses such as proportions. Generalized additive models address these difficulties, extending additive models to many other distributions besides just the normal. Thus, generalized additive models can be applied to a much wider range of data analysis problems. Like generalized linear models, generalized additive models consist of a random component, an additive component, and a link function relating the two components. The response density defines the additive component, where Generalized additive models and generalized linear models can be applied in similar situations, but they serve different analytic purposes. Generalized linear models emphasize estimation and inference for the parameters of the model, while generalized additive models focus on exploring data nonparametrically. Generalized additive models are more suitable for exploring the data and visualizing the relationship between the dependent variable and the independent variables.	{"url":"http://support.sas.com/documentation/cdl/en/statug/63347/HTML/default/statug_gam_sect014.htm","timestamp":"2024-11-09T00:31:14Z","content_type":"application/xhtml+xml","content_length":"17656","record_id":"<urn:uuid:0a5bec7b-2c20-452f-a0ef-a333c897f66d>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00639.warc.gz"}
Data Mining Algorithms in C++: Data Patterns and Algorithms for Modern Applications. Timothy Masters \| ITBooks.online Data Mining Algorithms in C++: Data Patterns and Algorithms for Modern Applications. Timothy Masters Π Π°Ρ Π΅Π³ΠΎΡ ΠΈΡ : C / C++ Π ΠΎΠ΄Π΅Π»ΠΈΡ Ρ Ρ Ρ : Discover hidden relationships among the variables in your data, and learn how to exploit these relationships. This book presents a collection of data-mining algorithms that are effective in a wide variety of prediction and classification applications. All algorithms include an intuitive explanation of operation, essential equations, references to more rigorous theory, and commented C++ source code. Many of these techniques are recent developments, still not in widespread use. Others are standard algorithms given a fresh look. In every case, the focus is on practical applicability, with all code written in such a way that it can easily be included into any program. The Windows-based DATAMINE program lets you experiment with the techniques before incorporating them into your own work. What you'll learnMonte-Carlo permutation tests provide statistically sound assessment of relationships present in your data.Combinatorially symmetric cross validation reveals whether your model has true power or has just learned noise by overfitting the data.Feature weighting as regularized energy-based learning ranks variables according to their predictive power when there is too little data for traditional methods.The eigenstructure of a dataset enables clustering of variables into groups that exist only within meaningful subspaces of the data.Plotting regions of the variable space where there is disagreement between marginal and actual densities, or where contribution to mutual information is high, provides visual insight into anomalous relationships.Who this book is forThe techniques presented in this book and in the DATAMINE program will be useful to anyone interested in discovering and exploiting relationships among variables. Although all code examples are written in C++, the algorithms are described in sufficient detail that they can easily be programmed in any language. Π‘ΠΊΠ°Ρ Π°Ρ Ρ Π ΠΎΠΌΠΌΠ΅Π½Ρ Π°Ρ ΠΈΠΈ Π ΠΈΡ Π΅Π³ΠΎ Π½Π΅ Π½Π°ΠΉΠ΄Π΅Π½ΠΎ.	{"url":"https://itbooks.online/books/894-data_mining_algorithms_in_c_data_patterns_and_algorithms_for_modern_applications","timestamp":"2024-11-12T23:50:52Z","content_type":"text/html","content_length":"20095","record_id":"<urn:uuid:5c3211b3-b970-420f-b9a8-a85a86ab4656>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00478.warc.gz"}
How Long Does It Take to Count to a Million?How Long Does It Take to Count to a Million? - Birmingham Journal Counting to a million often captures people’s curiosity, appearing in social media challenges and YouTube videos. But how feasible is it? This article examines the time required to count to this impressive number, the physical and mental factors involved, and answers to common questions on the topic. Counting Basics Counting Methods To estimate the time it takes to count to a million, we need to establish our counting method. Most people typically count by ones, but for efficiency, some might choose to count by tens, hundreds, or even thousands. Counting Speed On average, individuals can count about one number per second. This pace allows for clear enunciation and mental processing, though speed can vary based on the individual’s comfort and fatigue Total Count Reaching a million means counting to 1,000,000. The time required hinges on how quickly one can articulate each number. Time Calculations One Number Per Second Let’s break down the numbers: Seconds per Count: 1 second Total Counts: 1,000,000 This means: 1,000,000 seconds=1,000,000 seconds1,000,000 \text{ seconds} = 1,000,000 \text{ seconds} 1,000,000 seconds=1,000,000 seconds To convert seconds into hours, we consider: Seconds in an Hour: 3,600 seconds Total Hours: 1,000,000 seconds3,600 seconds/hour≈277.78 hours\frac{1,000,000 \text{ seconds}}{3,600 \text{ seconds/hour}} \approx 277.78 \text{ hours} 3,600 seconds/hour1,000,000 seconds≈277.78 hours Thus, it would take approximately 278 hours of continuous counting without breaks. Incorporating Breaks In reality, counting can’t be done nonstop. If we assume an individual counts for about 8 hours a day, the total days to reach a million would be: 278 hours8 hours/day≈34.75 days\frac{278 \text{ hours}}{8 \text{ hours/day}} \approx 34.75 \text{ days} 8 hours/day278 hours≈34.75 days This suggests that it would take around 35 days of dedicated counting for 8 hours each day to reach a million. Influencing Factors Mental Fatigue Counting to a million can become mentally taxing. After several hours, fatigue can slow down counting speed and affect accuracy, necessitating breaks. Vocal Strain Prolonged counting can also lead to vocal fatigue, requiring breaks to prevent discomfort or potential injury. In everyday life, distractions are plentiful. Noise, interruptions, and competing tasks can disrupt the focus needed for counting. Counting Variations Larger Increments Counting in larger increments drastically reduces the time required: Counting by Tens: 1,000,00010=100,000 counts\frac{1,000,000}{10} = 100,000 \text{ counts} 101,000,000=100,000 counts This takes: 100,000 seconds3,600 seconds/hour≈27.78 hours (or about 1.16 days)\frac{100,000 \text{ seconds}}{3,600 \text{ seconds/hour}} \approx 27.78 \text{ hours (or about 1.16 days)} 3,600 seconds/hour100,000 seconds≈27.78 hours (or about 1.16 days) Counting by Hundreds: 1,000,000100=10,000 counts\frac{1,000,000}{100} = 10,000 \text{ counts} 1001,000,000=10,000 counts This results in: 10,000 seconds3,600 seconds/hour≈2.78 hours\frac{10,000 \text{ seconds}}{3,600 \text{ seconds/hour}} \approx 2.78 \text{ hours} 3,600 seconds/hour10,000 seconds≈2.78 hours Counting by Thousands: 1,000,0001,000=1,000 counts\frac{1,000,000}{1,000} = 1,000 \text{ counts} 1,0001,000,000=1,000 counts This would take: 1,000 seconds3,600 seconds/hour≈0.28 hours (or about 17 minutes)\frac{1,000 \text{ seconds}}{3,600 \text{ seconds/hour}} \approx 0.28 \text{ hours (or about 17 minutes)} 3,600 seconds/hour1,000 seconds≈0.28 hours (or about 17 minutes) Counting in larger increments makes the task much more manageable. Real-Life Examples YouTube Challenges YouTube features numerous videos where creators attempt to count to a million. Some choose to count by ones, while others cleverly utilize larger increments or time-lapse techniques, captivating viewers with their determination. Record Attempts Although there’s no official record for the fastest count to a million, many have taken on the challenge using various methods, including technology to speed up the process. Some video game streams even incorporate counting challenges with fast-forward techniques for entertainment. The Motivation Behind Counting Psychological Curiosity For many, counting to a million represents a unique psychological exercise, testing endurance and patience while fostering a sense of accomplishment. Social Media Trends In today’s social media landscape, unusual challenges often go viral. Counting to a million taps into this trend, providing an engaging spectacle for viewers and participants alike. Broader Implications of Counting Importance of Numbers Counting is fundamental across various fields like mathematics, science, and art. It teaches us about sequences and quantities, highlighting human perseverance. Mindfulness and Meditation For some, counting serves as a form of meditation, offering a structured way to focus and reflect, which can help reduce stress and enhance concentration. Counting to a million is a significant undertaking requiring time, mental stamina, and creativity. It could take about 35 days to achieve if counting consistently for 8 hours daily, though using larger increments can significantly shorten this timeframe. Whether as a challenge, a curiosity, or an exploration of endurance, counting captivates many and invites contemplation on its broader 1. How long would it take to count to a million out loud? If you count out loud at a steady pace of one number per second, it would take approximately 11 days, 13 hours, and 46 minutes to count to one million without any breaks. This calculation assumes no pauses for sleep, eating, or resting. 2. What if I count continuously? Counting continuously means you would not stop for breaks. However, practically speaking, it’s impossible to maintain this pace without rest. Factoring in basic human needs, the actual time could stretch over weeks or even months. 3. How many numbers do I count per minute? If you can count at a rate of about 60 numbers per minute, it would take roughly 16,666 minutes or about 278 hours. That’s approximately 11.5 days, assuming no breaks. 4. What is the fastest anyone has counted to a million? While many people claim they could count to a million quickly, there are no verified records of someone counting to a million in an extraordinarily short time. Most documented counting challenges focus on smaller numbers due to practical constraints. 5. Is there a YouTube video of someone counting to a million? Yes, there are several YouTube videos where creators attempt to count to a million. Some videos use fast-forward techniques to condense the process, while others document the entire experience, showcasing the challenges of the endeavor. 6. What are the physical and mental challenges of counting to a million? Counting to a million can be mentally exhausting due to the monotony of the task. Physically, it requires stamina, as you’d likely need to manage fatigue, hydration, and nutrition. Taking regular breaks would be essential to maintain focus and avoid burnout. 7. Are there any famous counting challenges? One famous example is the “Count to 100,000” challenge, where various YouTubers and streamers attempt to count large numbers, often turning the activity into a fun and entertaining event. These challenges usually include commentary, interactions, and other entertainment elements. 8. What are some fun facts about counting? Counting Systems: Different cultures have varied counting systems; for example, some languages use base-10, while others use base-20. Mathematical Significance: Counting forms the foundation of mathematics and is vital for understanding larger concepts like addition and multiplication. Counting in Nature: Animals, including some species of birds and primates, can also engage in basic counting. 9. Why would someone want to count to a million? People count to a million for various reasons: to challenge themselves, as a form of entertainment, or simply out of curiosity. It can also serve as a mindfulness exercise, helping individuals focus their thoughts and clear their minds. 10. Can I count to a million creatively? Absolutely! You can turn counting into a creative project by incorporating art, music, or storytelling. For instance, you could create a visual representation for every thousand numbers or set it to music, making the process more engaging. To read more, Click Here.	{"url":"https://birminghamjournal.co.uk/how-long-would-it-take-to-count-to-a-million/","timestamp":"2024-11-08T10:58:37Z","content_type":"text/html","content_length":"116460","record_id":"<urn:uuid:6b93cc94-313f-4645-a257-7d6b64b3b324>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00531.warc.gz"}
Monte Carlo Simulations And Pi Monte Carlo approximations for values are a really cool concept so I went ahead and visualized possibly the most well-known one... Take a square. Draw a circle in it so that the circle is completely contained in it and the square's side length is 2r where r is the radius of the circle. The square's area is 4r^2 and the circle's area is pir^2. Now...randomly drop items into the square. The % of items that land inside the circle will be the same value as the circle's area divided by the square's area if you drop an infinite number of items, and the more items you drop, the closer the value will be. Given that the ratio of the areas is pi/4, the % of items that land inside the circle will approach the value of pi/4. Multiply that by 4, and you get a value for pi. First, I wanted to show this somehow. I settled on simulating a progressively larger number of tosses and plotting the results. The gif below shows the first quadrant of a circle of radius 1 inside a square of side length 2 with more and more dots being generated. The error at the top is equal to 4(% in circle/pi) - 1. That's another way of writing the error in the estimation of pi. A more interesting question is...how repeatable are these results? To find that, I ran the simulation 5000 times and tracked the error in each simulation after 100, 1000, 10000, and 100000 trials. The histograms showing the distribution of those errors is below: As expected since each trial is independent, the errors are normally distributed . Also, the spread in errors gets smaller as the number of errors goes up. Given that the definition of standard deviation includes weighting by 1/(square root of trials), you would expect this spread to go down with the square root of the trials. Comparing the ratios of our standard deviations, we find that: trial ratio actual expected 100 vs 1000 3.1498 3.1623 1000 vs 10000 3.2450 3.1623 10000 vs 100000 3.0867 3.1623 Each factor of 10 increase in number of trials gives you a drop of approximately 'square root of 10' in the standard deviation of the error distribution. You can use this to put bounds on how many trials you need to get a given precision. To see how that looks, I've included a picture with the run from the gif at the top plotted with the range implied by twice the standard deviation from the 5,000 simulations. Note that ~95% of the points should lie between the two standard deviation curves. 1 comment: 1. 🎰 รวมเกมสล็อตสุดปัง แตกง่าย ได้เงินชัวร์ 🃏 คาสิโนไลฟ์ ทุกรูปแบบ จริงใจ โปร่งใส การันตี💯% ⚽️ กีฬาต่างๆ ทั่วโลก🌎 ราคาน้ำดีคอมมิชชั่นสูง 💥 💰 ถอนได้ไม่จำกัดครั้ง ไม่มีเงื่อนไข ไม่ติดเทิร์น 💰 ฝากขั้นต่ำ 1บาท 🏧 รองรับทุกธนาคาร และ ทรูมันนี่วอเล็ท ที่นี่เท่านั้น ⚡️ ระบบ ฝาก - ถอน อัตโนมัติรวดเร็ว ทันใจ ไม่ขาดตอน	{"url":"http://www.somesolvedproblems.com/2018/01/monte-carlo-simulations-and-pi.html","timestamp":"2024-11-08T10:37:26Z","content_type":"application/xhtml+xml","content_length":"98732","record_id":"<urn:uuid:43676fec-39d8-4e84-aee9-95cfa779ac5c>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00028.warc.gz"}
Robust tensor fitting Dear Tortoise community, I am a new Tortoise user and downloaded version 3.2.0 for Linux. In this moment, I would need to estimate the diffusion tensor using the robust RESTORE approach. I have a few questions about how to correctly run the command EstimateTensorNLLSRESTORE: • Does the RESTORE implementation available in TORTOISE correspond to what is described in the paper "Chang, Lin‐Ching, Lindsay Walker, and Carlo Pierpaoli. "Informed RESTORE: a method for robust estimation of diffusion tensor from low redundancy datasets in the presence of physiological noise artifacts." Magnetic resonance in medicine 68.5 (2012): 1654-1663”? • At the moment, I already have a pre-processed DWI subjID_prep.nii as well as bvals, bvecs and a pre-computed mask brain_mask.nii. Is it correct to apply the RESTORE algorithm in the following ImportNIFTI -i ../subjID_prep.nii -b subjID.bval -v subjID.bvec -p vertical EstimateTensorNLLSRESTORE -i ../subjID_proc/subjID.list -m ../brain_mask.nii If this is the correct the way to go, • What is the signal standard deviation used for performing RESTORE? Is it the same approach as in the paper mentioned above? • What do the following outputs exactly represent? 1. ../subjID_R1_DT.nii: to my understanding, this is the Diffusion Tensor (Dxx, Dyy, Dzz, …). But what is the unit of measurement in which the data is stored? 2. ../subjID_R1_OUT.nii 3. ../subjID_R1_VOUT.nii —> it looks like the outlier map but why isn’t it binary (0-1)? 4. ../subjID_R1_AM.nii Thanks a lot for your precious help, looking forward to your reply! Vincenzo Anania Dear Tortoise team, Do you have any updates about my question on EstimateTensorNLLSRESTORE? Thanks again for your attention, First of all apologies for the delay. The web system currently has issues and does not let us know about the posted questions. we are working on a fix. Coming back to your question... 1) No. The RESTORE implementation actually corresponds to: not the iRESTORE paper, which has different assumptions. ImportNIFTI -i ../subjID_prep.nii -b subjID.bval -v subjID.bvec -p vertical EstimateTensorNLLSRESTORE -i ../subjID_proc/subjID.list -m ../brain_mask.nii If this is the correct the way to go? Yes it is. 3) What is the signal standard deviation used for performing RESTORE? Is it the same approach as in the paper mentioned above? Yes it is the approach described in the RESTORe paper. 1. ../subjID_R1_DT.nii: to my understanding, this is the Diffusion Tensor (Dxx, Dyy, Dzz, …). But what is the unit of measurement in which the data is stored? 2. ../subjID_R1_OUT.nii 3. ../subjID_R1_VOUT.nii —> it looks like the outlier map but why isn’t it binary (0-1)? 4. ../subjID_R1_AM.nii Yes the DT image is the tensor image with order (dxx,Dyy,Dzz,Dxy,Dxz,Dyz). The unit is 10-6 mm^2/s Or in other words micro meter ^2 /s. OUT.nii is the cumulative outlier map. VOUT . yes the outlier map. it should be binary. Even if it is saved as a float image, it should only contain zeros and ones. The save format will be changed soon. AM. The estimated b=0 image. Hope this helps.	{"url":"https://tortoise.nibib.nih.gov/community/robust-tensor-fitting","timestamp":"2024-11-14T17:04:52Z","content_type":"text/html","content_length":"21056","record_id":"<urn:uuid:750ee8fd-192b-4726-ae13-0e6d1a4ad66c>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00348.warc.gz"}
Describe Vestigial Sideband Transmission (VSB) . Applications of Vestigial Siddeband Transmission - Electronics Post Describe Vestigial Sideband Transmission (VSB) . Applications of Vestigial Siddeband Transmission Vestigial Sideband Transmission The exact frequency response requirements on the sideband filter in SSB-SC system can be relaxed by allowing a part of the unwanted sideband called vestige to appear in the output of the modulator. Due to this, the design of the sideband filter is simplified to a great extent . But the bandwidth of the system is increased slightly . To generate a VSB signal, we have to first generate a DSB-SC signal and then pass it through a sideband filter as shown in fig. 1 . This filter will pass the wanted sideband as it is along with a part of unwanted sideband . Fig.1 : VSB Transmitter Frequency Domain Description Frequency Spectrum The spectrum of VSB is as shown in fig. 2 . (a) Spectrum of message signal (b) Spectrum of VSB Signal Fig. 2 The spectrum of message signal x(t) has also been shown . In the frequency spectrum, it is assumed that the upper sideband is transmitted as it is and the lower sideband is modified into vestigial sideband . Transmission Bandwidth From fig. 2 (b), it is evident that the transmission bandwidth of the VSB modulated wave is given by : Where f[m ]= Message bandwidth f[v ]= Width of the vestigial sideband Advantages of VSB 1. The main advantage of VSB modulation is the reduction in bandwidth. It is almost as efficient as the SSB . 2. Due to allowance of transmitting a part of lower sideband, the constraint on the filter have been relaxed . So practically, easy to design filters can be used . 3. It possesses good phase characteristics and makes the transmission of low frequency components possible . Application of VSB VSB modulation has become standard for the transmission of television signal . Because the video signal need a large transmission bandwidth if transmitted using DSB-FC or DSB-SC techniques . Generation of VSB Modulated Wave The block diagram of a VSB modulator is shown in fig.3 . Fig.3 : Generation of VSB Signal The modulating signal x(t) is applied to a product modulator . The output of the carrier oscillator is also applied to the other input of the product modulator . The output of the product modulator is then given by : m(t) = x(t) . c(t) = x(t) . V[c ]cos(2π f[c]t) This represents a DSB-SC modulated wave . This DSB-SC signal is then applied to a sideband shaping filter . The ddesign of this filter depends on the desired spectrum of the VSB modulated signal. This filter will pass the wanted sideband and the vestige of the unwanted sideband . Let the transfer function of the filter be H(f) . Hence, the spectrum of the VSB modulated signal is given by : Demodulation of VSB Wave The block diagram of the VSB demodulator is shown in fig.4 . Fig.4 : VSB demodulator Working Operation The VSB modulated wave is passed through a product modulator where it is multiplied with the locally generated synchronous carrier . Hence, the output of the product modulator is given by : Taking the Fourier transform of both sides, we get Hence, we have The first term in the above expression represents the VSB modulated wave, corresponding to a carrier frequency of 2f[c .]This term will be eliminated by the filter to produce output v[o](t) . The second term in the above expression for M(f) represents the spectrum of demodulated VSB output . Therefore , This spectrum is shown in fig.5 . fig 5: Spectrum of VSB Demodulator In order to obtain the undistorted message signal x(t) at the output of the demodulator, V[o](f) should be a scaled version of X(f) . For this the transfer function H(f) should satisfy the following conditions : Where H( f[c]) is constant .	{"url":"https://electronicspost.com/describe-vestigial-sideband-transmission-vsb-applications-of-vestigial-siddeband-transmission/","timestamp":"2024-11-11T17:05:55Z","content_type":"text/html","content_length":"68021","record_id":"<urn:uuid:85ef5ea1-0c43-4904-a6d4-53fffdff94a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00595.warc.gz"}
During the year, the Senbet Discount Tire Company had gross sales of $1.21 million. The company’s... During the year, the Senbet Discount Tire Company had gross sales of $1.21 million. The company’s... During the year, the Senbet Discount Tire Company had gross sales of $1.21 million. The company’s cost of goods sold and selling expenses were $590,000 and $243,000, respectively. The company also had notes payable of $820,000. These notes carried an interest rate of 6 percent. Depreciation was $120,000. The tax rate was 25 percent. a. What was the company’s net income? b. What was the company’s operating cash flow	{"url":"https://justaaa.com/finance/256857-during-the-year-the-senbet-discount-tire-company","timestamp":"2024-11-06T19:01:59Z","content_type":"text/html","content_length":"41123","record_id":"<urn:uuid:a0282e09-75da-4e5a-a6f6-b530004fcc76>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00638.warc.gz"}
: Answers in terms of functions Suppose I want to write a problem involving an unspecified function and the answer needs to be expressed in terms of that function. For example, this might be a recurrence relation between function values (sequences) and I want the student to type in the recurrence relation: f(n) = 3 f(n-1) + 2 I would probably keep "f(n) =" outside of the answer block, so the student would really only need to type 3 f(n-1) + 2. I know that I would add n as a variable to the context. But what do I do about f? In particular, I don't want to define f as a computable function, just that it is a function. - D. Brian Walton What you are asking for is somewhat difficult to accomplish in WeBWorK. This is because WeBWorK compares functions by evaluation, not by algebraic structure, so there would have to be a means of evaluating f(n) before WeBWorK would be able to handle it. It is possible to use the parserFunction.pl macro library to do this, though it does present some possible (but unlikely) ways to get the answer marked correct without it being what you want. Here is a sample code snippet that may do what you are after: parserFunction(f => "sin(pi^n)+e"); # something student is unlikely to type by hand $f = Formula("3 f(n-1) + 2"); $f(n)$ = \{ans_rule(20)\}. Note that this defines f(n) to be something obscure that a student is unlikely to type by hand so that WeBWorK can evaluate the formula. It would be difficult to get a formula that matched without writing it using f (though if the definition of f is chosen badly, it might be able to be written in several ways using f). Unfortunately, you can't include the f(n)= as part of the answer, even using the parserAssignment.pl macros, as they don't allow recursive definitions. Alternatively, you could write a custom checker that looks for the specific algebraic structure you are using, but that would limit the form that the student could use to enter the answer (e.g., 3 f (n-1)+1+1 might not be accepted if you weren't careful about how you code your checker). This approach is difficult, and I wouldn't recommend it.	{"url":"https://webwork.maa.org/moodle/mod/forum/discuss.php?d=434","timestamp":"2024-11-11T23:46:33Z","content_type":"text/html","content_length":"75954","record_id":"<urn:uuid:265eff6d-6475-43ac-b28f-0ecc3388bf4a>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00850.warc.gz"}
Subtract from Multiples of 100 (examples, videos, solutions, homework, worksheets, lesson plans) Related Topics: Common Core Math Resources, Lesson Plans & Worksheets for all grades Common Core Math Lessons, Math Worksheets and Games for Grade 2 Common Core Math Video Lessons, Math Worksheets and Games for all Examples, videos, and solutions to help Grade 2 students learn how to subtract from multiples of 100 and from numbers with zero in the tens place. Common Core Standards: 2.NBT.7, 2.NBT.9 New York State Common Core Math Grade 2, Module 5, Lesson 16, Lesson 17 Topic C: Strategies for Decomposing Tens and Hundreds Within 1,000 NYS Math Module 5 Grade 2 Lesson 16 Worksheets for Grade 2 Lesson 16 Homework 1. Solve vertically or using mental math. Draw chips on the place value chart and unbundle if needed. b. 509 – 371 = __________ e. 900 – 572 = __________ 2. Andy said that 599 – 456 is the same as 600 – 457. Write an explanation using pictures, number, or words to prove Andy is correct. NYS Math Module 5 Grade 2 Lesson 17 Lesson 17 Homework 1. Solve vertically or using mental math. Draw chips on the place value chart and unbundle if needed. b. 400 – 219 = __________ e. 905 – 606 = __________ Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.	{"url":"https://www.onlinemathlearning.com/subtract-multiples-100.html","timestamp":"2024-11-05T03:39:17Z","content_type":"text/html","content_length":"37540","record_id":"<urn:uuid:effd1b30-04c1-4d61-849e-32ab1d59766b>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00439.warc.gz"}
Order of Groups \| Order of an element in a Group - Mathstoon The order of a group and its elements are very crucial in group theory (Abstract Algebra). One can study groups by analyzing the orders of the group and their elements. In this article, we will learn about the order of groups. Order of a group The order of a group G is the cardinality of that group. In other words, the order of a group G is the number of its elements. Notation: The order of a group G is denoted by \|G\| or $\circ$(G). The following are a few examples on orders of groups. • The group (Z[n], +) is a group of order n. • The symmetric group (S[3], 0) has order 6. • (Z, +) is a group of infinite order. Types of Groups Depending upon the order of groups, we can classify the groups as follows: 1. Finite Group: If a group contains a finite number of elements, then it is a finite group. For example, the symmetric group S[n] is a finite group of order n!. 2. Infinite group: If a group does not have a finite number of elements, then it is an infinite group. For example, the additive group of integers is not a finite group; it is an infinite group. Read Also: Group Theory: Definition, Examples, Properties Order of an element Order of elements in a group G. The order of an element a ∈ (G, 0) is the smallest positive integer n such that a^n=e, where e is the identity element of G. It is denoted by \|a\| or $\circ$(a). If such an n exists, then we say that the element a is of finite order. Otherwise, a is said to be an element of infinite order. Related Topic: Prove that Order of Element Divides Order of Group Order of an Element Formula Let a be an element of order n in a group (G, 0), that is, a^n=e. Then the order of a^k is given by the formula: 0(a^k) = $\dfrac{n}{\text{gcd}(k, n)}$. We will now understand this formula with an example. Example: Let a be an element of order 20 in a group (G, 0). Find the order of a^5. We are given 0(a)=20, that is, a^20=e. In the above formula, we have n=20 and k=5. So the order of a^5 is 0(a^5) = $\dfrac{20}{\text{gcd}(5, 20)}$ = 20/5 = 4. More Readings: Cyclic Group \| Abelian Group Every Subgroup of a Cyclic Group is Cyclic: Proof Properties of Order of Elements in a Group The order of an element of a group satisfies the below properties: • The order of the identity element in a group is 1. No other element has order 1. • Both an element and its inverse of a group have the same order. In other words, 0(a)= 0(a^-1) for all elements a in G. • Each element of a finite group has finite order and it divides the order of the group (Lagrange’s Theorem). Thus no element exists in a finite group whose order exceeds the order of the group. • If 0(a) = k and a^n=e, then k is a divisor of n. • Suppose 0(a) = k. Then 0(a^n) = k for every integer n coprime to k. • If 0(a) is infinite, then 0(a^n) is also infinite for every integer n. • Both a and g^-1ag have the same order for a, g ∈ G. These two elements are called conjugate elements of each other. • 0(ab) = 0(ba) for all a, b ∈ G, that is, both ab and ba have the same order. Because ab and ba are conjugates of each other as ab=a(ba)a^-1. FAQs on Order of Groups and its Elements Q1: What is the order of the identity element? Answer: In a group, the order of the identity element is 1. Q2: What is the order of Z[12]? Answer: As Z[12] contains 12 elements, the order of Z[12] is 12. Q3: What is the order of a group? Answer: The order of a group G is the number of the elements of G, denote by \|G\|. For example, G=Z/3Z is a group containing 3 elements, so its order is 3. This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.	{"url":"https://www.mathstoon.com/order-of-a-group/","timestamp":"2024-11-05T22:18:43Z","content_type":"text/html","content_length":"208514","record_id":"<urn:uuid:be931e2a-e1a6-4867-867c-c3044a46f5b2>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00005.warc.gz"}
A discussion of De Bruijn graphs and an application of De Bruijn sequences to an O(1) find-first-set-bit procedure Twelve Days 2013: De Bruijn Sequences Day Ten: De Bruijn Sequences De Bruijn sequences and more generally De Bruijn graphs have lots of cool applications in combinatorics and CS. On hardware lacking a native instruction to count leading zeros in a word or find the least significant bit, a perfect hash function formed using a De Bruijn sequence provides a constant time solution. The technique is still useful when implementing arbitrary length integers or bit vectors. Outside of bit twiddling, De Bruijn graphs, sequences, and arrays improve the precision of encoders used in robotics and digital tablets, contribute to DNA fragment analysis, and find use in cryptographic applications. Intro to De Bruijn Graphs De Bruijn graphs are Eulerian, Hamiltonian digraphs with a few special properties. if we have an alphabet $\Sigma$ with $m$ symbols, the graph will have $m^n$ vertices, where $n$ is a free parameter. The vertex set is constructed such that every $n$ character word formed by characters taken from the alphabet is represented by a single vertex. The edge set is constructed in a fashion that encodes “overlap” between the words. De Bruijn sequences are the Hamiltonian cycles of the graph. Why De Bruijn Sequences are Interesting Under the construction above, De Bruijn sequences have the property that every $n$ character word in a $k$ character alphabet appears as a sub sequence exactly once when we slide a $n$ character window along the sequence from left to right. As an example, one construction of $B(2, 3)$ is $\{0, 0, 0, 1, 0, 1, 1, 1\}$. As we slide a three element long window from left to right, wrapping around when necessary we get: \begin{aligned} 000 & = 0 \\ 001 & = 1 \\ 010 &= 2 \\ 101 &= 5 \\ 011 &= 3 \\ 111 &= 7 \\ 110 &= 6 \\ 100 &= 4 \end{aligned} Every number in the range $[0, 7]$ appears exactly once. For a binary alphabet the length of the sequence will always be $2^n$ but for larger alphabets the sequence can be shorter than direct enumeration. As wikipedia points out, this has the useful property that, given a lock that accepts a pin code without an enter key (which was apparently common in Europe at one point), entering a De Bruijn sequence for the appropriate alphabet and code length is the fastest way to crack the lock. For a ten digit entry pad and a four digit pin that amounts to $B(10, 4)$, which is 10,000 characters long. Compare that with the $4 * 10^4 = 40,000$ combinations that would be required otherwise. The ability of De Bruijn graphs/sequences to efficiently encode overlap makes them useful for things like DNA sequencing where overlapping fragments need to be recombined, and for mechanical/optical encoders that need to figure out where they are given some overlapping sequence of measurements. There’s also a neat bit twiddling hack that comes about from this which we’ll discuss next. De Bruijn Sequences and Minimal Perfect Hashes A perfect hash is an injective map $S \mapsto \mathbb{Z}$: a function that maps all elements of $S$ onto the integers with no collisions. Minimal perfect hash functions have the additional property that keys are mapped to integers consecutively. By a simple cardinality argument that also implies that the function is as “efficient” as possible. There’s a seminal paper discussing how De Bruijn sequences can be used to construct a minimal perfect hash function that, given any integer $i$, will return the location of the lowest (rightmost) set bit in $i$. This is a pretty common and important operation to the extent that lots of processors now offer an instruction to do this. However, if you look at the code behind a compiler intrinsic that performs the find-first-set operation, the fallback for architectures lacking native support usually involves this hack. Before going any further it’s worth noting that, if you’ve heard of this hack, you’ve probably also seen magic De Bruijn constant floating around for integers of various length. De Bruijn sequences are not necessarily unique and they depend on the algorithm used to generate them. Don’t be surprised if the constants that you’ve seen don’t match the ones generated by this code. import java.util.Random; * @author Kelly Littlepage public class DeBruijn { * Generate a De Bruijn Sequence using the recursive FKM algorithm as given * in https://page.math.tu-berlin.de/~felsner/SemWS17-18/Ruskey-Comb-Gen.pdf . * @param k The number of (integer) symbols in the alphabet. * @param n The length of the words in the sequence. * @return A De Bruijn sequence B(k, n). private static String generateDeBruijnSequence(int k, int n) { final int[] a = new int[k * n]; final StringBuilder sb = new StringBuilder(); generateDeBruijnSequence(a, sb, 1, 1, k, n); return sb.toString(); private static void generateDeBruijnSequence(int[] a, StringBuilder sequence, int t, int p, int k, int n) { if(t > n) { if(0 == n % p) { for(int j = 1; j <= p; ++j) { } else { a[t] = a[t - p]; generateDeBruijnSequence(a, sequence, t + 1, p, k, n); for(int j = a[t - p] + 1; j < k; ++j) { a[t] = j; generateDeBruijnSequence(a, sequence, t + 1, t, k, n); * Build the minimal perfect hash table required by the De Bruijn ffs * procedure. See http://supertech.csail.mit.edu/papers/debruijn.pdf. * @param deBruijnConstant The De Bruijn number to use, as produced by * {@link DeBruijn#generateDeBruijnSequence(int, int)} with k = 2. * @param n The length of the integer word that will be used for lookup, * in bits. N must be a positive power of two. * @return A minimal perfect hash table for use with the * {@link DeBruijn#ffs(int, int[], int)} function. private static int[] buildDeBruijnHashTable(int deBruijnConstant, int n) { if(!isPositivePowerOfTwo(n)) { throw new IllegalArgumentException("n must be a positive power " + "of two."); // We know that n is a power of two so this (meta circular) hack will // give us lg(n). final int lgn = Integer.numberOfTrailingZeros(n); final int[] table = new int[n]; for(int i = 0; i < n; ++i) { table[(deBruijnConstant << i) >>> (n - lgn)] = i; return table; * Tests that an integer is a positive, non-zero power of two. * @param x The integer to test. * @return <code>true</code> if the integer is a power of two, and * <code>false otherwise</code>. private static boolean isPositivePowerOfTwo(int x) { return x > 0 && ((x & -x) == x); * A find-first-set bit procedure based off of De Bruijn minimal perfect * hashing. See http://supertech.csail.mit.edu/papers/debruijn.pdf. * @param deBruijnConstant The De Bruijn constant used in the construction * of the deBruijnHashTable. * @param deBruijnHashTable A hash 32-bit integer hash table, as produced by * a call to {@link DeBruijn#buildDeBruijnHashTable(int, int)} with n = 32. * @param x The number for which the first set bit is desired. * @return <code>32</code> if x == 0, and the position (in bits) of the * first (rightmost) set bit in the integer x. This function chooses to * return 32 in the event that no bit is set so that behavior is consistent * with {@link Integer.numberOfTrailingZeros()}. private static int ffs(int deBruijnConstant, int[] deBruijnHashTable, int x) { if(x == 0) { return 32; x &= -x; x *= deBruijnConstant; x >>>= 27; return deBruijnHashTable[x]; The code above will generate a De Bruijn sequence and the hash table necessary for the first-set-bit look-up. How does it stack up against the LZCNT instruction emitted by Hotspot on x86? private static void benchmarkLookup(int runCount) { final Random random = new Random(); final int[] targets = new int[runCount]; for(int i = 0; i < runCount; ++i) { targets[i] = random.nextInt(); final int deBruijnConstant = Integer.valueOf( generateDeBruijnSequence(2, 5), 2); final int[] hashTable = buildDeBruijnHashTable(deBruijnConstant, 32); // Warm up for(int i = 0; i < 5; ++i) { timeFFS(targets, hashTable, deBruijnConstant); System.out.printf("Intrinsic: %.4fs\n", (double) timeLeadingZeros(targets) / Math.pow(10d, 9)); System.out.printf("FFS: %.4fs\n", (double) timeFFS(targets, hashTable, deBruijnConstant) / Math.pow(10d, 9)); private static int checksum; private static long timeLeadingZeros(int[] targets) { // Hash to prevent optimization int hash = 0; final long startTime = System.nanoTime(); for(int i : targets) { hash += Integer.numberOfTrailingZeros(i); final long endTime = System.nanoTime(); checksum += hash; return endTime - startTime; private static long timeFFS(int[] targets, int[] hashTable, int deBruijnConstant) { // Hash to prevent optimization int hash = 0; final long startTime = System.nanoTime(); for(int i : targets) { hash += ffs(deBruijnConstant, hashTable, i); final long endTime = System.nanoTime(); checksum += hash; return endTime - startTime; Intrinsic: 0.0604s, FFS: 0.1346s. Silicon can beat us by a factor of two. However, once upon a time this was the best that we could do, and it’s still relevant to bit vectors/words of arbitrary	{"url":"https://www.klittlepage.com/articles/twelve-days-2013-de-bruijn-sequences/","timestamp":"2024-11-13T08:41:15Z","content_type":"text/html","content_length":"275683","record_id":"<urn:uuid:df4c6563-d0fa-48c3-8f51-6dfd7a83041c>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00562.warc.gz"}
Foundations of Machine Learning Systems • Machine learning research is notoriously fast paced. Can one get off the treadmill and embrace a notion of slow science? I am pleased to be able to argue yes by example! Recently I have had two works published with a very long gestation. The first Geometry and Calculus of Losses reports work that started in 2012 (not a typo). It presents a beautiful (yes it is!) geometric theory of loss functions, with all sorts of interesting connections to convex geometry and economics. The second Information Processing Equalities and the Information-Risk Bridge builds upon the first and revisits what is arguably the most basic result in information theory, namely the "information processing inequality." It shows that in fact it can be written as an equality, albeit one with different measures of information on each side. It also presents a more general theory of the bridge between information and risk, showing these are generally two sides of the same coin. The earliest version of one of the main results was worked out in 2007 (again not a typo). It was published last week ... after some 17 years! This work took a very long time largely because of external events (my work on Technology and Australia's Future which totally absorbed my efforts for 2.5 years), the absorption of NICTA into CSIRO (another several years, not at all pleasant), my move from Australia to Germany, and a serious health issue in 2022 (also no fun). I am pleased to see both these works finally published. :-) • Bob Williamson gave his inaugural lecture at the University of Tübingen recently. The photo shows the FMLS group at the reception. A video recording of the lecture can be found here. A pdf of the slides is here. • The usual way probability theory is used is that you posit an algebra of events. Each such event has a probability (that is, it is "measurable"). The assumption that the set of events that have a probability is an algebra means that if A and B are both events then "A and B" is also an event (the intersection also has a probability). What happens if you do not make that assumption? In this paper (by Rabanus Derr and Bob Williamson) we provide an answer: you recover the theory of imprecise probability! This builds an intriguing bridge between measure theory and notions from social science such as intersectionality. One conclusion is that measurability should not be construed as a mere technical annoyance; rather, it is a crucial part of how you choose to model the world. • Our paper on The Richness of Calibration has appeared in the proceedings of FaCCT 2023 • We recently finished two papers on imprecise probabilities. The set structure of precision: coherent probabilities on Pre-Dynkin Systems This shows a relationship between the set system of measurable events and imprecision of probabilities, and thus offers a novel way of generalising traditional probabilities that is potentially useful for a range of problems, including modelling "intersectionality" Strictly Frequentist Imprecise Probability This shows that one can develop and strictly frequentist semantics for imprecise probabilities, whereby upper previsions arise from the set of cluster points of relative frequencies. This means the theory is applicable to all sequences, not just stochastic ones. We also present a converse result that suggests that the theory is the "right" thing in the sense that every upper prevision can be derived from a (non-stochastic sequence). The proof of this fact is constructive.	{"url":"https://fm.ls/news-insights","timestamp":"2024-11-10T03:05:03Z","content_type":"text/html","content_length":"86579","record_id":"<urn:uuid:836279e7-6408-47c5-b5ff-8e3776435751>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00432.warc.gz"}
Returns the inverse cotangent of the given number in radians. Sample Usage • number - The number for which the inverse cotangent is to be computed. ACOT(0.5) returns the inverse cotangent of 0.5 in radians. ACOT(C5) returns the inverse cotangent of the number contained in C5. • Use the DEGREES function to convert the result of ACOT into degrees. For example, DEGREES(ACOT(0.5)). • The ACOT function returns a solution between 0 and Pi.	{"url":"https://support.spreadsheet.com/hc/en-us/articles/360030259051-ACOT","timestamp":"2024-11-11T22:41:29Z","content_type":"text/html","content_length":"43863","record_id":"<urn:uuid:fc79effa-a6e3-403c-8d1e-a827c7f4d2b9>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00406.warc.gz"}
quadratic programming model - EssayBy.com A quadratic programming model is an optimization model with n decision variables and m linear constraints, and of the form: Where x is the n by 1 column vector of decision variables and xT is its transpose, Q is an n by n symmetric matrix of the objective parameters, c is an n by 1 vector of additional objective parameters, A is an m by n matrix of constraints parameters, and b is an m by 1 vector of constraints’ right hand sides. Place an order in 3 easy steps. Takes less than 5 mins.	{"url":"https://www.essayby.com/quadratic-programming-model/","timestamp":"2024-11-03T12:09:41Z","content_type":"text/html","content_length":"59882","record_id":"<urn:uuid:f85b8c32-a177-4ce0-a2ee-d19bade126cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00022.warc.gz"}
JavaScript Math() JavaScript Math is a built-in object in JavaScript that provides a set of mathematical functions and constants. It makes complex mathematical calculations much easier in JavaScript. Here are some of the most commonly used functions of the Math object: 1. Math.round(): This function rounds a number to the nearest integer. For example, let x = 4.6; x = Math.round(x);// would return 5 2.Math.floor(): This function returns the largest integer less than or equal to a given number. For example, let x = 4.6; x = Math.floor(x);// would return 4 3.Math.ceil(): This function returns the smallest integer greater than or equal to a given number. For example, let x = 4.6; Math.ceil(x); / would return 5. 4.Math.abs(): This function returns the absolute value of a number. For example, let x = -4.6; x = Math.abs(x);// would return 4.6 5.Math.sqrt(): This function returns the square root of a given number. For example, let x = 16; x = Math.sqrt(x);// would return 4 6.Math.pow(): This function returns the value of a number raised to a given power. For example, let x = 2; x = Math.pow(x,3);// would return 8 7.Math.min(): This function returns the smallest of two or more numbers. For example, let x = 2; let y = 12; let z = 5; let minimum; let maximum; minimum = Math.min(x,y,z);// would return 2 8.Math.max(): This function returns the largest of two or more numbers. For example, let x = 2; let y = 12; let z = 5; let minimum; let maximum; maximum = Math.max(x,y,z);// would return 12 9.Math.random(): This function returns a random number between 0 and 1. For example, Math.random() would return a number like 0.384729473. 10.Build in constants: There are also some build in constants. For example, let x; x = Math.PI;// would return 3.141592653589793 In addition to these functions, the Math object also contains a number of useful constants, such as Math.E (which represents the value of e). In conclusion, the Math object in JavaScript is an incredibly useful tool for performing mathematical calculations in your code. By using the functions and constants provided by the Math object, you can make your code much more efficient and streamlined. This is just to name a few there are so many more. Adios Top comments (0) For further actions, you may consider blocking this person and/or reporting abuse	{"url":"https://practicaldev-herokuapp-com.global.ssl.fastly.net/bansikah/javascript-math-599","timestamp":"2024-11-02T12:49:55Z","content_type":"text/html","content_length":"78078","record_id":"<urn:uuid:3167a237-b4ac-4967-af50-fb98a60cd624>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00342.warc.gz"}
PPT - Gases PowerPoint Presentation, free download - ID:9360982 2. Ideal Gases Ideal gases are imaginary gases that perfectly fit all of the assumptions of the kinetic molecular theory. • Gases consist of tiny particles that are far apart relative to their size. • Collisions between gas particles and between • particles and the walls of the container are • elastic collisions • No kinetic energy is lost in elastic collisions 3. Ideal Gases (continued) • Gas particles are in constant, rapid motion. They • therefore possess kinetic energy, the energy of • motion • There are no forces of attraction between gas • particles • The average kinetic energy of gas particles • depends on temperature, not on the identity • of the particle. 4. Gases expand to fill their containers Gases are fluid – they flow Gases have low density 1/1000 the density of the equivalent liquid or solid Gases are compressible Gases effuse and diffuse The Nature of Gases 5. Is caused by the collisions of molecules with the walls of a container is equal to force/unit area SI units = Newton/meter2 = 1 Pascal (Pa) 1 atmosphere = 101,325 Pa 1 atmosphere = 1 atm = 760 mm Hg = 760 torr Pressure 6. Measuring Pressure The first device for measuring atmospheric pressure was developed by Evangelista Torricelli during the 17th century. The device was called a “barometer” • Baro = weight • Meter = measure 7. An Early Barometer The normal pressure due to the atmosphere at sea level can support a column of mercury that is 760 mm high. 8. P = 1 atmosphere, 760 torr T = 0°C, 273 Kelvins The molar volume of an ideal gas is 22.42 liters at STP Standard Temperature and Pressure“STP” 9. Converting Celsius to Kelvin Gas law problems involving temperature require that the temperature be in KELVINS! Kelvins = C + 273 °C = Kelvins - 273 10. The Combined Gas Law The combined gas law expresses the relationship between pressure, volume and temperature of a fixed amount of gas. Boyle’s law, Gay-Lussac’s law, and Charles’ law are all derived from this by holding a variable constant. 11. Boyle’s Law Pressure is inversely proportional to volume when temperature is held constant. 12. Charles’s Law • The volume of a gas is directly proportional to temperature, and extrapolates to zero at zero Kelvin. (P = constant) 13. Gay Lussac’s Law The pressure and temperature of a gas are directly related, provided that the volume remains constant. 14. For a gas at constant temperature and pressure, the volume is directly proportional to the number of moles of gas (at low pressures). V=an a = proportionality constant V = volume of the gas n = number of moles of gas Avogadro’s Law 15. PV = nRT P = pressure in atm V = volume in liters n = moles R = proportionality constant = 0.08206 L atm/ mol·K T = temperature in Kelvins Ideal Gas Law Holds closely at P < 1 atm 16. Standard Molar Volume Equal volumes of all gases at the same temperature and pressure contain the same number of molecules. - Amedeo Avogadro 17. Gas Density … so at STP… 18. Density and the Ideal Gas Law Combining the formula for density with the Ideal Gas law, substituting and rearranging algebraically: M = Molar Mass P = Pressure R = Gas Constant T = Temperature in 19. Gas Stoichiometry #1 If reactants and products are at the same conditions of temperature and pressure, then mole ratios of gases are also volume ratios. 3 H2(g) + N2(g) 2NH3(g) 3 moles H2 + 1 mole N2 2 moles NH3 3 liters H2 + 1 liter N2 2 liters NH3 20. Gas Stoichiometry #2 How many liters of ammonia can be produced when 12 liters of hydrogen react with an excess of nitrogen? 3 H2(g) + N2(g) 2NH3(g) 12 L H2 2 L NH3 = L NH3 8.0 3 L H2 21. Gas Stoichiometry #3 How many liters of oxygen gas, at STP, can be collected from the complete decomposition of 50.0 grams of potassium chlorate? 2 KClO3(s) 2 KCl(s) + 3 O2(g) 50.0 g KClO3 1 mol KClO3 3 mol O2 22.4 L O2 122.55 g KClO3 2 mol KClO3 1 mol O2 = L O2 13.7 22. Gas Stoichiometry #4 How many liters of oxygen gas, at 37.0C and 0.930 atmospheres, can be collected from the complete decomposition of 50.0 grams of potassium chlorate? 2 KClO3(s) 2 KCl(s) + 3 O2(g) 50.0 g KClO3 1 mol KClO3 3 mol O2 = “n” mol O2 0.612 mol O2 122.55 g KClO3 2 mol KClO3 = 16.7 L 23. For a mixture of gases in a container, PTotal = P1 + P2 + P3 + . . . Dalton’s Law of Partial Pressures This is particularly useful in calculating the pressure of gases collected over water. 24. Kinetic Energy of Gas Particles At the same conditions of temperature, all gases have the same average kinetic energy. 25. The Meaning of Temperature • Kelvin temperature is an index of the random motions of gas particles (higher T means greater motion.) 26. Particles of matter are ALWAYS in motion Volume of individual particles is zero. Collisions of particles with container walls cause pressure exerted by gas. Particles exert no forces on each other. Average kinetic energy µ Kelvin temperature of a gas. Kinetic Molecular Theory 27. Diffusion Diffusion: describes the mixing of gases. Therateof diffusion is the rate of gas mixing. 28. Effusion: describes the passage of gas into an evacuated chamber. Effusion 29. Graham’s LawRates of Effusion and Diffusion Effusion: Diffusion: 30. Real Gases Must correct ideal gas behavior when at high pressure(smaller volume) and low temperature(attractive forces become important). corrected pressure corrected volume Videal Pideal	{"url":"https://www.slideserve.com/alejandrinat/gases-powerpoint-ppt-presentation","timestamp":"2024-11-07T18:44:18Z","content_type":"text/html","content_length":"90077","record_id":"<urn:uuid:e7c0c16f-dda1-4147-8275-4d862e734e57>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00882.warc.gz"}
10.0 - Law of Motion The mechanical laws of motion, along with thermal body attribute updates, are undertaken at cycle point 10.0. The mechanical law of motion will be presented in this section. See PFC Thermal Formulation for details of the thermal formulation. The motion of a single, rigid particle is determined by the resultant force and moment vectors acting upon it, and can be described in terms of the translational motion of a point in the particle and the rotational motion of the particle. The motion of the center of mass is described in terms of its position $\mathbf x$, velocity $\dot{\mathbf x}$ , and acceleration $\ddot {\mathbf x}$. The rotational motion of the particle is described in terms of its angular velocity $\pmb \omega$ and angular acceleration $\dot {\pmb \omega}$. The equations of motion can be expressed as two vector equations: one relates the resultant force to the translational motion and the other relates the resultant moment to the rotational motion. Translational Motion The equation for translational motion can be written in the vector form (1)\[\mathbf F=m(\ddot{\mathbf x} - \mathbf g)\qquad\hbox{(translational motion)}\] where $\mathbf F$ is the resultant force, or the sum of all externally applied forces acting on the particle; $m$ is the mass of the particle; and $\mathbf g$ is the body force acceleration vector (e.g., gravitational loading). The translational equations of motion are solved for balls and clumps via the second order Velocity Verlet algorithm [Verlet1967]. This method of integration offers second order accuracy and is often used in molecular dynamics simulations because, for conservative systems, the energy oscillates around a constant value corresponding to the exact energy of the system. Suppose that the previous cycle solved Equation (1) at time $t$ and that the timestep resolved for the current cycle is $\Delta t$. The 1/2 step velocity, $\dot{\mathbf{x}}^{(t+\Delta t/2)}$, is calculated as (2)\[\dot{\mathbf{x}}^{(t+\Delta t/2)} = \dot{\mathbf{x}}^{(t)} + \frac{1}{2} \left(\frac{\mathbf{F}^{(t)}}{m}+\mathbf{g}\right) \Delta t\] The position at time $t+\Delta t$ is updated using the 1/2 step velocity (3)\[\mathbf{x}^{(t+\Delta t)} = \mathbf{x}^{(t)} + \dot{\mathbf{x}}^{(t+\Delta t/2)} \Delta t\] During the force displacement cycle point, the forces are updated for the current cycle, leading to the updated acceleration $\ddot{\mathbf{x}}^{(t+\Delta t)}$. The velocity is subsequently updated (4)\[\dot{\mathbf{x}}^{(t+\Delta t)} = \dot{\mathbf{x}}^{(t+\Delta t/2)} + \frac{1}{2} \left(\frac{\mathbf{F}^{(t+\Delta t)}}{m}+\mathbf{g}\right) \Delta t\] In PFC, the final velocity update of Equation (4) occurs as the initial step in the timestep determination cycle point or during the finalization stage when finishing a series of cycles as discussed here. Thus, if one queries the velocity of a ball or clump after cycle point 10.0, they are actually querying the 1/2 step velocity $\dot{\mathbf{x}}^{(t+\Delta t/2)}$. Rotational Motion The fundamental equation of rotational motion for a rigid body is (5)\[\mathbf L = \mathbf I \pmb \omega \qquad\hbox{(rotational motion)}\] where $\mathbf L$ is the angular momentum, $\mathbf I$ is the inertia tensor and $\pmb \omega$ is the angular velocity. Euler’s equation can be obtained from Equation (5) by taking the time (6)\[\mathbf M = \dot {\mathbf L} = \mathbf I \dot{\pmb\omega} + \pmb{\omega} \times \mathbf L\] where $\mathbf M$ is the resultant moment acting on the rigid body. This relation refers to a local coordinate system that is attached to the body at its center of mass. Rotational Motion for Balls and 2D Clumps If the local coordinate system is oriented such that it lies along the principal axes of inertia of the body, then Equation (6) reduces to Euler’s equations of motion: (7)\[\begin{split}M_1 &= I_1\ \dot\omega_1 + (I_3-I_2)\omega_3\omega_2 \\ M_2 &= I_2\ \dot\omega_2 + (I_1-I_3)\omega_1\omega_3 \\ M_3 &= I_3\ \dot\omega_3 + (I_2-I_1)\omega_2\omega_1 \\\end{split}\] where $I_1$, $I_2$, and $I_3$ are the principal moments of inertia of the body; $\dot \omega_1$, $\dot \omega_2$, and $\dot \omega_3$ are the angular accelerations about the principal axes; $\omega_1$, $\omega_2$, and $\omega_3$ are the angular velocities about the principal axes; and $M_1$, $M_2$, and $M_3$ are the components of the resultant moment as referred to in the principal axis system. For a {disk-shaped in 2D; spherical in 3D} body with radius $R$, whose mass is distributed uniformly throughout its volume, the center of mass coincides with its centroid. For a disk-shaped body whose axis remains in the out-of-plane direction, $\omega_1 = \omega_2 \equiv 0$, yielding (8)\[M_3 = I \dot{\omega_3} = (\frac{1}{2}mR^2)\dot{\omega_3}\] For a spherical body, any local-axis system attached to the center of mass is a principal-axis system, and the three principal moments of inertia are equal to one another. For a spherical body, Equation (7) can be simplified in the global-axis system as (9)\[\mathbf M = \mathbf I \dot{\pmb{\omega}} = (\frac{2}{5}mR^2)\dot{\pmb{\omega}}\] For balls (2D and 3D) and 2D clumps, the rotational equations of motion are solved via the Velocity Verlet algorithm described above. Vector notation is used without loss of generality where $\ omega_1 = \omega_2 \equiv 0$ in 2D and the inertia tensor is diagonal. Note that, for clumps in 2D, the polar moment of inertia may be any value that the user specifies. First the 1/2 step angular velocity is calculated (10)\[{\pmb{\omega}}^{(t+\Delta t/2)} = {\pmb{\omega}}^{(t)} + \frac{1}{2} \left( \frac{\mathbf{M}^{(t)}}{\mathbf I} \right) \Delta t\] During the force displacement cycle point, the moments are updated for the current cycle, leading to the updated acceleration $\dot{\pmb{\omega}}^{(t+\Delta t)}$. The angular velocity is subsequently updated (11)\[{\pmb{\omega}}^{(t+\Delta t)} = {\pmb{\omega}}^{(t+\Delta t/2)} + \frac{1}{2} \left( \frac{\mathbf{M}^{(t+\Delta t)}}{\mathbf I} \right) \Delta t\] Rotational Motion for 3D Clumps 3D clumps may have full inertia tensors in the global coordinate system and, as a result, the rotational equations of motion cannot be reduced to Euler’s equations (Equation (7)) in the global coordinate system. In PFC3D, both second- and fourth-order algorithms are available to integrate the clump rotational motion (see the clump order command). In both cases, the angular momentum $\ mathbf L$, the quaternion representing the orientation relative to the principal axis system $q_c$ and the principal moments of inertia $\mathbf J$ are stored. An introduction to quaternions is presented prior to a thorough presentation of these second- and fourth-order schemes. Quaternions are four-dimensional vectors that are commonly used to represent 3D rotations and orientations [Shoemake1985]. In contrast to Euler angle representations, quaternions do not suffer from gimbal lock (i.e., loss of a degree of freedom under certain configurations). A quaternion is defined as (12)\[q = q_0 + q_i i + q_j j + q_k k = q_0 + \mathbf q\] where $i$, $j$, and $k$ are the basis elements. $q_0$ is termed the scalar part, and $\mathbf q$ is the vector part. The products of basis elements are defined by the identities (13)\[\begin{split}\begin{array}{c} i^2 = j^2 = k^2 = ijk = -1 \\ ij = k \\ jk = i \\ ki = j \end{array}\end{split}\] The product of two quaternions, termed the Hamilton product, is determined by the distributive law and the identities above. In scalar/vector notation, the product is given by (14)\[q p = p_0 q_0 - \mathbf p \cdot \mathbf q + p_0 \mathbf q + q_0 \mathbf p + \mathbf p \times \mathbf q\] Due to the cross product, the product of two quaternions is not generally commutative ($q p \neq p q$). The product of the vector $\mathbf v$ and a quaternion is performed by creating a pure vector quaternion with 0 scalar part (i.e., $v = 0 + \mathbf v$ in scalar/vector notation) and performing the multiplication as defined in Equation (14). Addition and subtraction of quaternions is accomplished on a per-component basis. The conjugate of a quaternion is defined as (15)\[q^* = q_0 - \mathbf q\] and the length of a quaternion is given by (16)\[\\|q\\| = q_0^2 + q_i^2 + q_j^2 + q_k^2 = q q^* = q^* q\] A unit quaternion (where $\\|q\\| = 1$) can be converted to a pure rotation matrix. To rotate the vector $\mathbf v$ by a unit quaternion, one creates the pure vector quaternion $v=0+\mathbf v$ (with scalar part 0) and forms the product (17)\[v' = qvq^\] The vector component of $v'$ is the rotated vector. The sequential rotation $q$ followed by the rotation $p$ is given by the quaternion product $p q$, meaning that any number of rotations can by composed into a single quaternion and applied at once, provided they are unit quaternions. Properties of unit quaternions are $q$ and $-q$ represent the same rotation, the inverse rotation of $q$ is the conjugate $q^$, and the null rotation is $q = 1$ where the vector part is $\mathbf 0$. Second-Order Solution [Buss2000] develops algorithms of varying order to solve the rotational equations of motion defined above. Those algorithms are based on the Taylor-series expansion of the angular velocity to provide more accurate estimates of the average angular velocity during a timestep. PFC implements the so-called augmented second-order method. The angular velocity at time $t$ is calculated by rearranging Equation (5) (18)\[\pmb \omega^{(t)} = \mathbf{I}^{(t)-1} \mathbf L^{(t)}\] Note that the inertia tensor (or inverse inertia tensor) in the global axis system can be obtained by rotating the principal moments of inertia by the clump orientation $q_c$ (19)\[\begin{split}\begin{array}{c} \mathbf{I} = q_c \mathbf J q_c^* \\ \mathbf{I}^{-1} = q_c \mathbf J^{(-1)} q_c^* \end{array}\end{split}\] The angular acceleration at time $t$ is determined by rearranging Equation (6) and using Equation (19) (20)\[\dot{\pmb \omega}^{(t)} = q_c^{(t)} \mathbf J^{(-1)} q_c^{(t)} \left(\mathbf M^{(t)}-\mathbf \omega^{(t)} \times \mathbf L^{(t)} \right)\] and the average angular velocity over the timestep is estimated as (21)\[\bar{\pmb{\omega}} = \pmb \omega^{(t)} + \dot{\pmb \omega}^{(t)} \frac{\Delta t}{2} + \left(\dot{\pmb{\omega}}^{(t)} \times \pmb \omega^{(t)} \right) \frac{\Delta t^2}{12}\] The average angular velocity is used to update the clump orientation $q_c^{(t + \Delta t)}$ by calculating the angular displacement, $\theta=\bar{\pmb{\omega}}\Delta t$, converting this to a unit quaternion and rotating $q_c^{(t)}$ by this unit quaternion (22)\[q_c^{(t + \Delta t)} = \left(cos\left(\frac{\theta}{2}\right) + sin\left(\frac{\theta}{2}\right) \frac{\bar{\pmb{\omega}}}{\\|\bar{\pmb{\omega}}\\|}\right) q_c^{(t)}\] Once the clump orientation has been updated, the new angular velocity $\pmb \omega ^{(t + \Delta t)}$ is given by (23)\[\pmb \omega ^{(t + \Delta t)} = q_c^{(t+\Delta t)} \mathbf J^{(-1)} q_c^{(t+\Delta t)} \mathbf L^{(t)}\] Fourth-Order Solution One can show that the time derivative of the quaternion is given by (24)\[\dot q_c = \frac{1}{2} \omega q_c\] where $\omega$ is the pure vector quaternion with vector component $\pmb\omega$ (i.e., the angular velocity). Substituting Equations (18) and (19) into (24) gives: (25)\[\dot q_c = \frac{1}{2} \left(q_c \mathbf J^{(-1)} q_c^* \mathbf L \right) q_c\] The standard fourth-order Runge-Kutta method is applied to the equation above. The orientation at time $t + \Delta t$ is given by (26)\[q_c^{(t+\Delta t)} = q_c^{(t)} + \frac{\Delta t}{6} \left(k_1+2k_2+2k_3+k_4\right)\] The quaternion $k_1$ is calculated as (27)\[k_1 = \frac{1}{2} \left(q_c^{(t)} \mathbf J^{(-1)} q_c^{(t)} \mathbf L^{(t)} \right) q_c^{(t)}\] An intermediate value of the orientation quaternion (28)\[ {q_{c1}} = q_c^{(t)}+k_1 \frac{\Delta t}{2}\] is used to calculate the $k_2$ quaternion (29)\[k_2 = \frac{1}{2} \left( {q_{c1}} \mathbf J^{(-1)} {q_{c1}^} \mathbf L^{(t)} \right) {q_{c1}}\] The next intermediate value of the orientation quaternion (30)\[ {q_{c2}} = q_c^{(t)}+k_2 \frac{\Delta t}{2}\] is used to calculate the $k_3$ quaternion as (31)\[k_3 = \frac{1}{2} \left( {q_{c2}} \mathbf J^{(-1)} {q_{c2}^} \mathbf L^{(t)} \right) {q_{c2}}\] The final intermediate value of the orientation quaternion (32)\[ {q_{c3}} = q_c^{(t)}+k_3 \Delta t\] is used to calculate the $k_4$ quaternion as (33)\[k_4 = \frac{1}{2} \left( {q_{c3}} \mathbf J^{(-1)} {q_{c3}^} \mathbf L^{(t)} \right) {q_{c3}}\] Once the clump orientation has been updated, Equation (23) is used to update the angular velocity to time $t + \Delta t$. This formulation is similar to the one presented in [Johnson2008]. Johnson, S. M., J. R. Williams and B. K. Cook. “Quaternion-based Rigid Body Rotation Integration Algorithms for use in Particle Methods,” Int. J. Num. Meth. Eng., 74, 1303-1313 (2008). Buss, S. R. “Accurate and Efficient Simulation of Rigid-Body Rotations,” J. Comp. Phys., 164, 377-406 (2000). Shoemake, K. “Animating Rotation using Quaternions,” SIGGRAPH85, San Francisco, CA, USA (1985). Verlet, L. “Computer ‘Experiments’ on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules,” Phys. Rev., 159, 98-103 (1967). Was this helpful? ... Itasca Software © 2024, Itasca Updated: Oct 31, 2024	{"url":"https://docs.itascacg.com/itasca900/pfc/docproject/source/manual/numerical_simulations_with_pfc/pfc_formulation/law_of_motion.html","timestamp":"2024-11-04T16:40:13Z","content_type":"application/xhtml+xml","content_length":"31936","record_id":"<urn:uuid:74698019-fb1a-4ec0-b315-6e864beb2924>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00002.warc.gz"}
Order of Operations with Rational Numbers NOTES & PRACTICE About This Product This resource was developed to PARTIALLY meet the requirements of the 7th Grade Number Systems standard below: Solve real-world and mathematical problems involving the four operations with rational numbers. What's Included This resource is meant to be a "short & sweet" version for teachers fighting against the clock! It contains the following items: 1) Order of Operations + Properties of Operations REVIEW 2) Order of Operations with Rational Numbers PRACTICE: Six order of operations expressions with two options: (A) The final answers are given and students must show the work to prove the final answer. (B) The final answers are not given. 3) Answer Key to All Parts Resource Tags order of operations rational numbers notes order of operations with rational numbers worksheet	{"url":"https://teachsimple.com/product/order-of-operations-with-rational-numbers-notes-and-practice","timestamp":"2024-11-09T19:49:56Z","content_type":"text/html","content_length":"613746","record_id":"<urn:uuid:fb6ecc02-9f84-4f3b-88ce-1f88148543cc>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00023.warc.gz"}
How do I abbreviate square meters? The symbol for square meter is m2. Less formally, square meter is sometimes abbreviated sq m. What does the abbreviation M2 mean? Acronym. Definition. M2. Square Meter (with the two rendered as a superscript) How do you abbreviate amperes? Abbreviation: A, amp. Is sqm the same as m2? It’s all the same. Metre square is written as m² . And square metre is mostly written in words. Is M2 the same as m? A metre square is a square with sides one metre in length – it refers to the shape and the side length, not the area. By contrast, a square metre is an area and can be any shape….Updated 04/01/2020 (see below) Area= Length x Breadth A=l × b 2 metres x 2 metres A = 2 m × 2 m 4 square metres A = 4 m2 Is M2 same as SQM? How do you write meter in short form? The meter (abbreviation, m; the British spelling is metre) is the International System of Units (SI) unit of displacement or length. How do you abbreviate units? When a prefix is added to a basic metric unit, the word is abbreviated by using the first letter of the prefix and the first letter of the basic unit. “Centimeter” is centi + meter, so the abbreviation is “cm.” How do you write M2? In any version of Microsoft Word, type m2, then highlight the 2. Now press and hold Ctrl and Shift, then press the + key and it will be changed to superscript and will look like this: m2. How do you write meters squared in Word? To type the squared symbol on Microsoft Word, click the superscript button (x²) in the Font group under the Home tab, and then type the number 2. What is the meter symbol? symbol m The metre, symbol m, is the SI unit of length. How meter is written? The metre (Commonwealth spelling) or meter (American spelling; see spelling differences) (from the French unit mètre, from the Greek noun μέτρον, “measure”) is the base unit of length in the International System of Units (SI). The SI unit symbol is m. How do you abbreviate meter? What is the unit abbreviation of meter? Metric Base Units Unit of Measurement Name Abbreviation Length Meter m Mass Gram g Volume Liter L How do I type the m2 symbol? Is m2 same as SQM? Is sqm same as M2?	{"url":"https://pleasefireme.com/review/how-do-i-abbreviate-square-meters/","timestamp":"2024-11-03T13:57:34Z","content_type":"text/html","content_length":"60226","record_id":"<urn:uuid:dbcd087f-2409-46bb-8a27-34ee44989521>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00677.warc.gz"}
Solve any Rubik's Cube combinations using Rubik's Cube Solver Have you ever played Rubik’s Cube? A cube which have 9 tiles on each surface with different colours and you have to solve it by making all surfaces have same colour? I only managed to solve the first layer only. When it comes to the second layer, I can’t get it solve because I always disturb the first layer whenever I try to solve the second layer. It is so difficult if you don’t know the technique. Fortunately, I found an easier way to solve a Rubik’s Cube problem. Go to Eric Dietz Rubik’s Cube solver page. Enter your Rubik’s Cube colours combination and click Solve. Then, another page will come out and display step by step to solve your Rubik’s Cube combination. Just follow the steps and you will get the result at the end. I have tried it and it really amazed me. I can solve any Rubik’s Cube combinations. Just enter the colour and Eric Dietz programming will produce the solution. I took about 5 minutes to solve a Rubik’s Cube by using Eric Dietz’s solution. I think I can go more faster but because the Rubik’s Cube that I play is the cheaper one so sometimes it got stuck when I twist the surface. 1. Pratyush says Thanks Fauzi. I will now solve the Rubik’s Cube I have and gift it to someone ๐ . Get rid of future trouble :D. Nice share, stumbled. 2. megat says wah so amazing!! thank for the sharing fauzi ๐ 3. Syahid A. says Lol. With this, it will not be a puzzle anymore. ๐ 4. CypherHackz says It is just in case you stuck and unable to solve it. Anyway, it is a good programming that ables to produce the solution based on the colours you input. 5. apek says is that a way that i can solve it without using online solver. for example secret technique how to solve ๐ can anyone give me the tutorial? Leave a Reply Cancel reply	{"url":"https://www.cypherhackz.net/solve-any-rubiks-cube-combinations-using-rubiks-cube-solver/","timestamp":"2024-11-05T04:04:21Z","content_type":"text/html","content_length":"59188","record_id":"<urn:uuid:023e3616-4adf-4bf5-a49a-52e244f48320>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00422.warc.gz"}
On degenerate resonances in systems close to two-dimensional nonlinear Hamiltonian ones under quasi-periodic parametric perturbations Title On degenerate resonances in systems close to two-dimensional nonlinear Hamiltonian ones under quasi-periodic parametric perturbations Authors O. S. Kostromina^1 ^1National Research Lobachevsky State University of Nizhny Novgorod Using the example of a pendulum-type equation with nonmonotonic rotation under a quasi-periodic nonconservative perturbation, the structure of the degenerate resonance zone in the case of Annotation parametric resonance is studied. Using the analysis of averaged systems, the conditions for the existence of quasi-periodic solutions are determined in the case when the order of degeneracy is greater than one. Particular attention is paid to the existence of quasi-periodic solutions of a new type, which are characteristic of parametric perturbations. Such solutions correspond to limit cycles of an averaged system that do not have generating limit cycles in the perturbed autonomous system corresponding to the original equation. Keywords degenerate resonance, quasi-periodic parametric perturbations, limit cycle, pendulum-type equation, averaging. Kostromina O. S. ''On degenerate resonances in systems close to two-dimensional nonlinear Hamiltonian ones under quasi-periodic parametric perturbations'' [Electronic resource]. Citation Proceedings of the International Scientific Youth School-Seminar "Mathematical Modeling, Numerical Methods and Software complexes" named after E.V. Voskresensky (Saransk, July 26-28, 2024). Saransk: SVMO Publ, 2024. - pp. 84-87. Available at: https://conf.svmo.ru/files/2024/papers/paper13.pdf. - Date of access: 12.11.2024.	{"url":"https://conf.svmo.ru/en/archive/article?id=454","timestamp":"2024-11-12T04:28:43Z","content_type":"text/html","content_length":"11807","record_id":"<urn:uuid:2ccaa112-39c8-4b58-88f2-0274c5e6cd37>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00500.warc.gz"}
The dynamic behavior of physical systems is often described by conservation and constitutive laws, expressed as systems of partial differential equations (PDEs). A classical task involves the use of computational tools to solve such equations across a range of scenarios, e.g., different domain geometries, input parameters, and initial and boundary conditions. Solving these so-called parametric PDEs using traditional tools (e.g., finite element methods) comes with an enormous computational cost, as independent simulations need to be performed for every different domain geometries or input In the noteworthy paper [Wan21L], the authors propose the framework of physics-informed DeepONets for solving parameterized PDEs. It is a simple yet remarkably effective extension of the DeepONet framework [Lu21L] (cf. Learning nonlinear operators: the DeepONet architecture). By constraining the outputs of a DeepONet to approximately satisfy an underlying governing law, substantial improvements in predictive accuracy, enhanced generalization performance even for out-of-distribution prediction, as well as enhanced data efficiency can be observed. Operator learning techniques have demonstrated early promise across a range of applications, but their application to solving parametric PDEs faces two fundamental challenges: First, they require a large corpus of paired input-output observations and, second, their predicted output functions are not guaranteed to satisfy the underlying PDE. Motivated by the fact that the outputs of a DeepONet model are differentiable with respect to their input coordinates, one can use automatic differentiation to formulate an appropriate regularization mechanism in the spirit of physics-informed neural networks [Rai19P]. The target output functions of the DeepONet are biased to satisfy the underlying PDE constraints by incorporating these into the loss function of the network. When the collection of all trainable weights of a DeepONet is denoted by $\theta$, the network is optimized with respect to the loss function $$ \mathcal{L}(\theta) = \mathcal{L}{\small\text {operator}}(\theta) + \mathcal{L}{\small\text{physics}}(\theta) $$ where $\mathcal{L}{\small\text{operator}}$ fits the available solution measurements and $\mathcal{L}{\small\text{physics}}$ enforces the underlying PDE constraints. Figure 4 [Wan21L]: Solving a parametric Burgers’ equation. (Top) Exact solution versus the prediction of the best-trained physics-informed DeepONet. The resulting relative L2 error of the predicted solution is 3%. (Bottom) Computational cost (s) for performing inference with a trained physics-informed DeepONet model [conventional or modified multilayer perceptron (MLP) architecture], as well as corresponding timing for solving a PDE with a conventional spectral solver (58). Notably, a trained physics-informed DeepONet model can predict the solution of $O(10^3)$ time-dependent PDEs in a fraction of a second, up to three orders of magnitude faster compared to a conventional PDE solver. Reported timings are obtained on a single NVIDIA V100 graphics processing unit (GPU). The authors demonstrate the effectiveness of the physics-informed DeepONets in solving parametric ordinary differential equations, diffusion-reaction and Burgers’ transport dynamics, as well as advection and Eikonal equations. In the diffusion-reaction example, the physics-informed DeepONet yields ∼80% improvement in prediction accuracy with 100% reduction in the dataset size required for training. For Burgers’ equation, notably, a trained physics-informed DeepONet model can predict the solution up to three orders of magnitude (1000x) faster compared to a conventional solver. Figure 6 [Wan21L]: Solving a parametric eikonal equation (airfoils). (Top) Exact airfoil geometry versus the zero-level set obtained from the predicted signed distance function for three different input examples in the test dataset. (Bottom) Predicted signed distance function of a trained physics-informed DeepONet for three different airfoil geometries in the test dataset. Despite the promise of the framework demonstrated in the paper, the authors acknowledge that many technical questions remain. For instance, for a given parametric governing law: What is the optimal feature embedding or network architecture of a physics-informed DeepONet? Addressing these questions might not only enhance the performance of physics-informed DeepONets, but also introduce a paradigm shift in how we model and simulate complex physical systems. If you are interested in using physics-informed DeepONets (and other physics-informed neural operators) in your application, check out continuiti, our Python package for learning function operators with neural networks that includes utilities for implementing PDEs in the loss function.	{"url":"https://transferlab.ai/pills/2023/physics-informed-deeponet/","timestamp":"2024-11-13T15:07:31Z","content_type":"text/html","content_length":"38468","record_id":"<urn:uuid:9fa547e2-1c65-4dbc-9282-bfc4cbe735cc>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00014.warc.gz"}
The Essential 10-Key Numeric Keypad - PowerScore Test Prep Since 2011, the GRE General Test permits the use of a calculator. But, not just any old calculator, an on-screen calculator that is part of the GRE software. This four-function calculator is invaluable for performing rapid arithmetic and basic calculations. It’s not helpful as a substitute for problem-solving skills, strategy, and good planning. In other words, you almost always cannot simply plug numbers into the calculator, press a few buttons, and achieve the correct answer. There are a few reasons for tackling the GRE with help of the calculator. 1. There are GRE problems that involve sufficiently unwieldy arithmetic that benefit tremendously from using a calculator. When you have to divide 19,683 by 27, you could do the long division, sure. But most would agree using a calculator is going to be faster. 2. When you have to perform several small calculations in a row, a calculator can help track your progress. It helps organize your math. You still have to make sure you input data correctly, but you don’t have to stress as much about small arithmetic errors. It can help prevent throwing your whole process off! 3. You might be using good problem-solving techniques and wish to use values that don’t lend themselves to easy arithmetic. In these cases, the calculator is indispensable. So you’re convinced, using the calculator is the right way to go. So let’s talk about a tool most students overlook. It can dramatically improve the effectiveness of your calculator use. The 10-key numeric keypad. This is the set of numbers to the right of your desktop keyboards and some laptops. But My Computer Doesn’t Have One! Anecdotally, many GRE students do most of their computer work on laptops. A lot of laptops don’t have a numeric keypad! If you plan to use a keyboard that doesn’t feature a numeric keypad for GRE practice, we advise you invest in a keyboard that does have one. Don’t waste precious time using the on-screen calculator! It’s disruptive and difficult to perform calculations by clicking keys with the mouse. The keypad isn’t the only reason to make this investment, though. A full-sized keyboard will be similar to the one you will use to write your two essays for the Analytical Writing Measure. Need another reason? The use of a keypad actually takes practice. Data-entry classes dedicate a lot of time to teaching rapid 10-key input! Even when you’re practicing with a workbook or using pencil-paper-style GRE material, try to approximate the Computer-Based Test (CBT) interface as closely as possible. You likely don’t have a computer-calculator exactly akin to the one on the GRE, but any basic calculator that’s on your computer should suffice. Just remember to restrict yourself to the functions available on the GRE • MR — Memory Recall • MC — Memory Clear • M+ — Memory Add • ( — Open Parentheses • ) — Close Parentheses • The digits 0-9 • ± — Positive/Negative Toggle • . — Decimal • ÷ — Division • x — Multiplication • – — Subtraction • + — Addition • √ — Square Root • = — Equals You can find instructions on how to use the calculator here. They also provide a screenshot of the current calculator. Small Skills Make a Difference! We might question how important the calculator or numeric keypad is to success on the GRE. While it’s important not to overstate their significance, it’s equally important not to underestimate these tools. When you’re dealing with a time crunch and the pressure to solve problems quickly and accurately, you must take full advantage of every tool at your disposal. To do so, you practice in advance. The CBT is an aspect of the test that is, in some ways, difficult to replicate when not using software for preparation. But, whatever you can do to remain aware of this system will help you prepare more effectively and ultimately give yourself a competitive edge on the actual test.	{"url":"https://blog.powerscore.com/gre/the-essential-10-key-numeric-keypad/","timestamp":"2024-11-05T18:44:53Z","content_type":"text/html","content_length":"39355","record_id":"<urn:uuid:f1d25895-e60e-4abf-8cdb-b05852725d46>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00482.warc.gz"}
Ordinal Numbers Problem Solving Year 1 - OrdinalNumbers.com Ordinal Numbers Problem Solving – You can count unlimited sets with ordinal numbers. They can also help to generalize ordinal numbers. 1st One of the basic concepts of math is the ordinal number. It is a number that indicates the location of an object in an array. Ordinarily, ordinal numbers are between one to twenty. … Read more	{"url":"https://www.ordinalnumbers.com/tag/ordinal-numbers-problem-solving-year-1/","timestamp":"2024-11-13T21:07:46Z","content_type":"text/html","content_length":"46305","record_id":"<urn:uuid:2820df2c-a8fb-44fe-abec-acad18644f50>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00851.warc.gz"}
Gross Domestic Product GDP Equations Formulas Calculator - Consumption Online Web Apps, Rich Internet Application, Technical Tools, Specifications, How to Guides, Training, Applications, Examples, Tutorials, Reviews, Answers, Test Review Resources, Analysis, Homework Solutions, Worksheets, Help, Data and Information for Engineers, Technicians, Teachers, Tutors, Researchers, K-12 Education, College and High School Students, Science Fair Projects and Scientists By Jimmy Raymond Contact: aj@ajdesigner.com Privacy Policy, Disclaimer and Terms Copyright 2002-2015	{"url":"https://www.ajdesigner.com/phpgdp/gross_domestic_product_equation_c.php","timestamp":"2024-11-04T07:10:02Z","content_type":"text/html","content_length":"20531","record_id":"<urn:uuid:4145da8d-b52a-406a-a5bf-8babb687cf85>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00237.warc.gz"}
Bui Trung Hiep - Collection from other sources Collection from other sources Day 1: Demystifying Operations Research Welcome to day 1 of #ORfrom0to1! 🙌🏻 Starting this course puts you in an advantageous situation. You’ll be ahead of 99% of people when dealing with making data-driven decisions. But let’s go concept by concept, problem by problem, day by day. By the end of this first day you’ll have learned: • The definition of Operations Research • Typical problems you may find in this field • A game to break the ice between you and OR And I don’t break my words so you’ll also find your first exercise to start thinking about solving these kinds of problems. Are you ready? Let’s go for it! 📚 What Operations Research is and why you should care Data, data and more data! That’s something you probably heard a lot in the past few years. It’s like if you don’t have data in your company, you’re lost. And well, that’s pretty accurate since other companies are doing it too, so you could fall behind. You want data to visualize what is happening. You want data to even do some predictions at some core parts of your business. But what about the present? What about making a plan, schedule, or a route more cost efficient? What about making data-driven decisions to optimize pretty much every process that you have? Operations Research (OR) is a discipline that deals with the application of advanced analytical methods to help make data-driven decisions. By using techniques like mathematics, computer science, and business knowledge, OR provides the tools to solve complex problems and improve decision-making processes. You, as an Operations Research Engineer, would need to: • Collect and analyze data • Develop and test mathematical models • Interpret the information provided by those models • Propose solutions and recommendations to implement improvement actions So businesses will be able to optimize a cost function. This a broad field as you can see: So during this course you are going to focus on: • Understanding the big picture of Operations Research • Start solving some optimization problems on your own through games and exercises But don’t get me wrong, even though there are a lot of sub-fields here, OR has lived with us for a long time. It originated during the second world war when it became apparent that the military needed to solve some of the significant logistic and supply chain problems that come with being in war. Back then, it was defined as "a scientific method of providing executive departments with a quantitative basis for decisions regarding the operations under their control" and was coined "operational analysis" or "quantitative management". And it has a looooot of use cases that range from supply chain management (moving goods at the lowest cost) to scheduling and assignment problems (so you end up with perfect assignments of tasks to workers), multi-criteria decision making (what about dealing with minimizing costs and at the same time equally distribute those tasks?) or even biology (for finding similarities between molecules that could lead to cheaper drugs for new diseases). So optimization is everywhere, and thus Operations Research. It has a huge economic impact in any field and it’s directly applicable to the business. Let me tell you something: making decisions is hard, so as much data-driven you are, the better for making good decisions. And those decisions are usually driven by three facts: • The decision itself (do I need to assign this task to this person? do I need to go to this place or this other one to deliver my products?). This will be translated as variables, because they can vary their value until you get the best possible one. • The limitations that you have (I cannot assign tasks to this person because she has a different role, I cannot assign more than 40 hours a week of tasks to any person, I cannot do routes of more than 400 km per day). Those are the constraints of the problem. • The criteria for success (I want to minimize the number of people working on this task, I want to minimize the cost of the routes I need to operate, I want to maximize the revenue at my production plant). This is what we call the objective function because, yes, it’s a mathematical function. Ok, enough Operations Research theory today! Let’s see some problems you can find out there. 🔢 3 typical problems you can find I know, I know. I’ve just said that you can find problems in -almost- every industry, right? But of course there are some industries more mature than others. Let’s say they have been playing around OR for more time, they found its value earlier. So there’s really an spectrum of problems to solve, ranging from resource allocation in logistics to scheduling in manufacturing and services. All these problems often involve optimizing products, processes, or services within constraints to achieve the best outcome. And here’s the thing: Understanding these typical problems is crucial for applying OR techniques effectively in any other problem you find out there. Let’s see 3 real-life problems you can easily find in the optimization literature. Let’s imagine you are the manager of pORfumes supply chain, the most successful perfume brand of your country. So you are responsible of taking care of each step of the process between getting the glass for the perfumes to get them to the manufacture plant. You have several places for getting the glass at different prices and different maximum quantities, and you need to deliver it to different manufacturing plants, each of them with a different maximum capacity and cost to produce the perfumes. If you want to cover the demand of perfumes in the market, what would be the best combination of matching the places where you get the glass with the manufacturing plants at the minimum possible price covering all the demand? And here you have your first optimization problem as the pORfumes company. The transportation problem involves finding the most cost-efficient way to distribute a product from several suppliers (in this case, the places where you get the glass) to several consumers (the production plants and ultimately the customers) such that the product is shipped in the most economical way while satisfying supply at the suppliers and demand at the consumers. The goal is to minimize the total transportation cost, and it is typically formulated as a linear programming problem. ⚙️ Production Planning Problem Now imagine that you are the manager of one of those manufacture plants of pORfumes. You want to produce the bottles of the perfumes, so you have a bunch of materials like glass, plastic for the cap, and stickers for different sizes of perfumes. You have minimum requirements in terms of quantities of each perfume but also some unexpected, on-demand production. Considering that changing materials is difficult because it adds time to the production planning but also adds some costs because you need to change some machinery, what do you need to do if you want to cover the demand at a minimum price? Exactly, you need to solve an optimization problem that deals with planning! 🗺️ Traveling Salesperson Problem Let’s imagine now you are the salesperson for pORfumes, so you have a car and you deliver perfumes to several stores in different cities. What will happen tomorrow first thing in the morning? That’s it: you need to pay a visit to all of them, but just once. And at the end of the day, come back home. In the end, you perform a loop over all the stores. But of course you want to do it in the least amount of time. Easy, right? This is one of the most studied problems in the literature, and there are lots of variants to it, like: • 📅 What happens if you need to pass through each store in a certain period of time? Then you have the Time Windows variant. • 🚧 And what if you have a capacity limit that you cannot exceed in your car? Then you have the Capacity variant. • 📦 What about delivering some perfumes in the store but at the same time picking other things like boxes you left there? Then you have the Pickup & Delivery variant. (Yes, I know that the names of the variants are far from being original, they are just descriptive.) We can even transform this problem into a bigger one, so instead of having just one car you have a fleet of cars. This is a generalization of the problem and it’s called the Vehicle Routing Problem. As you are thinking now, solving these problems is crucial for logistics companies. They need to deliver products to places paying the least possible amount of money! 💸 If you want to read more about these problems, I already talked about the Travelling Salesperson Problem and the Vehicle Routing Problem on Feasible. You may wonder why I chose these 3 problems. The thing is those specific problems are easy to understand since they might be in our daily lives and they may be used in other contexts too. So understanding those problems will give you a better sense to understand a broader range of optimization problems. And you will even start seeing optimization problems everywhere! For instance, the Transportation Problem can be used to efficiently assign tasks to workers in a workforce optimization problem. The Production Planning Problem can be used in other industries like car manufacturing or batch production of drugs in the pharma industry. And the Travelling Salesperson Problem can be used to solve computer wiring problems or even in assembly lines to optimize the sequence of operations. And I want you to play with them. I mean, literally. During the following days, you are going to either play, either solve, or both things at the same time with those 3 problems, so you can get the big picture of Operations Research. The first game starts now 👇🏻 Arcade games. So 80’s, huh? They became popular for some reasons: easy gameplay, short duration of games, and the possibility to play with other players, mainly. What they didn’t expect at all is that they would serve a purpose in optimization some decades later. In particular, to help us understand the Travelling Salesperson Problem in a better way. The ArcadeTSP is an arcade game based on the aforementioned problem that will help you visualize the challenges of routing, commonly faced in logistics and delivery services. By attempting to find the shortest possible route that visits each point and returns to the origin, you'll get a hands-on experience of the complexities involved in optimization problems. This exercise will prepare you for the more complex scenarios we'll tackle later in the course since it will introduce you to some of the core concepts of Operations Research. How to play the game? 1. Go to → https://bmenendez.github.io/arcadetsp/ 2. Click on several places of the yellow square to create those cities to go through them, like: 3. Click on the PLAY button at the bottom. 4. Connect the dots in such a way you think you minimize the total travel time, like: 5. Click on the CHECK button at the bottom, and you’ll see if an algorithm could beat you! It’s your turn now! Happy playing! OK, that’s a lot of new concepts for just one day! Let’s recap a little bit to see what you accomplished today: • You have seen what Operations Research is, and I hope you found it as beautiful as I see this field. • You have been the manager of supply chain and manufacturing plan for pORfumes, as well as a salesperson, understanding what are the optimization challenges of each role. • You played an arcade game to understand the basics of the Travelling Salesperson Problem, playing against a machine that I hope you could beat. Yes, you guessed it right, I want you to show me how good you are on ArcadeTSP. This time I lost, what about you? Let me know answering this email with a screenshot 😊 But more importantly, I’d love to go deeper on this, so consider thinking: • What did you find more interesting? • What did you find more difficult? • Did you find any special strategy to always beat the algorithm? What was it? And I’ll also want you to think about if you found optimization problems in the past. Did you think they had a proper area of knowledge that helps you solve them? Have you ever found yourself looking at a decision to make but you find it very difficult to take action? I’ll be delighted if you answer these questions and even rise more! Would you tell me your answers to this email? That’s it for today, I hope you enjoyed this first day! Tomorrow we’ll cover how to identify optimization problems at your company, previously identifying different layers of management that are prone to these problems. And we’ll see together a new problem that you will be modeling and solving in the following days. See you tomorrow! Day 2: Identifying optimization problems at your company Welcome to day 2 of #ORfrom0to1! 🙌🏻 You know, I’ve always thought that there are no Operations Research projects, just business ones. I mean, there’s no point in applying OR techniques to unexisting problems. If you’re solving a problem that no one is facing, then that solution will be irrelevant. So you need to understand how a company manages its business so you can spot problems that you can solve (hopefully, with OR techniques 🤞🏻). By the end of this second day you’ll have learned: • Applying OR to several layers in businesses • How to identify optimization problems at your company • The Furniture Factory Problem: will you be able to maximize revenues? Are you ready? Let’s go for it! 💼 Applying Operations Research to several layers in businesses Every company faces lots of decision-making processes each day. From figuring out how to improve the business for the next 6 months to the daily decisions that make the company profitable. And some of them can be treated as optimization problems. We have already seen that you can leverage Operations Research to reduce costs, increase revenues, or make better plannings. But how? Well, let’s divide the company into three different levels of management: strategic, tactical, and operational. You can find optimization problems at each of those three levels, and we’ll see two specific examples. But first, let’s define each of them: Strategic problems involve high-level, long-term decisions aimed at positioning the organization competitively in its environment over the long haul. It's about setting overarching goals and priorities and understanding the market and industry trends. The focus is on shaping the future landscape of the business and its growth trajectory, often based on simulations and scenario planning. On the opposite side, operational problems are the most immediate and detailed layer, concerned with the day-to-day operations of the business. This level of planning addresses the implementation of specific tasks through efficient processes and problem-solving in real-time. It focuses on making quick, feasible decisions to keep the business running smoothly every day. Tactical problems bridge the gap between strategic and operational problems, focusing on medium-term goals and how to implement strategies effectively. It's more specific than strategic problems but broader than operational ones. Let’s see an example 👇🏻 🚚 Planning problems when moving goods across Europe As you know, I’m working at a logistics startup called Trucksters. We mainly move goods from one place to another. Not me, drivers with trucks! So they need to know what their next place to go to get a load. That’s a plan, right? You tell the driver where to load, what to get and where to unload. That’s a task for planners. Operational planning Planners need to make adjustments on their plans on the go because there might be some unexpected issues on the road (like weather or traffic conditions) or at warehouses (like they couldn’t do the unloading for whatever reason). And at the same time they need to understand that those adjustments are feasible, so they can make it real. You cannot tell a driver to move 120 km in just one hour just because you find that truck is the nearest one to the next load. You cannot tell a driver that goes with a particular type of trailer to do a loading that requires a different type of trailer. You cannot tell a driver to go to Luxembourg if there’s a national holiday and trucks cannot go there. You get the point, right? Those are impossible actions to do in real life. Those are decisions to be made in the daily life of a planner. But let’s say this is just a survival part since there’s not so much range of action you can take. You need to make fast decisions with the resources you have. What if you can do a different planning? Tactical planning Planners also need to foresee a complete schedule for the current + next week. This way they have a wider vision of what to do, a guideline for the rest of the week. They have all the possible routes with their own constraints (loading and unloading date, loading and unloading location, type of trailer needed for the route), and they figure out how to combine them in the best possible way to get to a goal, like minimizing empty km (those empty trips that a driver need to do). This is close to operational planning since they need to understand the feasibility of that weekly planning, but it gives them a different perspective of the planning ahead. How could we foresee a bigger picture? Strategic planning Businesses not only live with operational and strategic planning. They also need to think ahead to new ways of doing their stuff. Talking about logistics, it’s pretty common to understand routines that need to be done by drivers, so you can get to the market and get better contracts with them since you can offer something Or you analyze how you can improve your margins just out of moving the schedule for some loads (sometimes it’s better to load 2 hours later than expected so you can easily task a specific driver to do it) so you can go to the client and renegotiate the terms of the contract. Or you can see the impact of revenues and costs on your plannings with the conditions that you currently have + new ones that may come in the future. All of this can be done through simulations. You simulate different scenarios with those different what-ifs that you have just seen, pick the best one for your future business, and then start taking action to make it real. 🔍 How to identify optimization problems Well, first of all, you don’t really need to find optimization problems per se. I mean, you’ll look for ineficiencies, bottlenecks and other ways to solve the issues the company is facing. This involves identifying areas where complex decisions need to be made efficiently and And right after that, then you’ll be in a better position to understand if Operations Research is the tool you need to solve them. Sometimes it is, sometimes it’s not. OR is a tool inside your toolbox, so use it wisely. Do not do OR for the sake of OR. Having said that, here’s a generalized approach to recognize OR opportunities at your company: 1. Identify bottlenecks, inefficiencies or excessive costs Look for processes or systems within your industry that are frequently bottlenecked, where delays and inefficiencies occur. These might be physical (e.g., slow production lines, traffic jams, network congestion) or procedural (e.g., slow decision cycles, inefficient resource allocation). Identify areas where costs are significantly high or could be reduced through better management. This includes direct costs like inventory and logistics, and indirect costs like customer dissatisfaction due to poor service. Anyplace where there are long wait times, excess inventory/capacity, underutilized resources, or general inefficiencies is a prime opportunity for OR. 2. Look for complex decision-making environments Spot scenarios where decisions involve multiple variables and stakeholders, which can benefit from sophisticated decision-support tools. Some common decisions that may benefit include: □ Allocating limited resources (money, people, materials, etc.) □ Maximizing or minimizing some quantitative measure (cost, profit, time, etc.) □ Determining optimal configurations/mixes (product assortment, asset allocation, etc.) □ Routing vehicles or scheduling activities □ Making tradeoffs between service level and efficiency 3. Consider scalability and impact Identify areas where small improvements can scale significantly or have a substantial impact on the business or its customers. For example, if you have to plan tasks, you will have realized that introducing more people to do it manually does not make the problem easier to solve. As you have more and more tasks to plan, the problem becomes so complex that when you add more hands, you neither obtain higher quality solutions, nor will you solve it in less time. OR is particularly valuable where small automations in processes can lead to large gains in efficiency or customer satisfaction. *customer satisfaction may range from delivery times to customer service interactions and product quality, that may be a candidate for OR to shine. That’s OK, but how could you do that? There are several ways and things to take into account: 1. Start with processes I love starting with processes. This is a deep exercise to make through all the departments of the company, but you can start with one and repeat the process in the rest of them. Start asking questions about what decisions most affect your results. Begin this journey with in-depth discussions with your seasoned executives who understand the nuances of these processes and can also see the bigger picture. Typical kickoff questions include "where do you perceive opportunities for improvement?" or “what are your major bottlenecks?”. There might be different examples like in production planning if you’re facing backlogs due to inefficient machine utilization, or in warehousing if you have high picking costs due to excessive walking when retrieving orders. 2. Data collection and definition of problems Determine what operational data already exists and what new data may need to be collected. This could include data on demand, capacities, costs, travel times, service levels, etc. The ability to quantify the important factors is crucial for optimization. Define the key performance metrics and objectives the organization wants to optimize. This could be minimizing costs, maximizing outputs, improving customer service levels, increasing efficiency, or balancing trade-offs. Explicitly list out the various constraints and limitations on the system such as budgets, capacities, labor rules, serviceable regions, etc. These form the boundaries for the optimization. 3. External perspectives Simultaneously, tap into external wisdom through consulting firms or academic and industry literature. Consultants can provide insights into best practices and trends, while literature might reveal how peers are tackling similar challenges. Take the insights from the above steps and precisely formulate the core decision problems that could be tackled using mathematical modeling, optimization, simulation, or other OR methods. Finally, evaluate the potential impact, costs, data requirements, and feasibility of the identified opportunities. Prioritize the highest value use cases to pursue first. The key is deeply understanding the organization's operations, objectives, constraints, and available data to properly frame the decisions into an optimizable problem format. 🪑 The Furniture Production Problem So let’s start with a new problem. Today, you will put yourself in the shoes of a Data Scientist responsible for developing a weekly production plan of two key products at FurnitORe, the biggest furniture factory in your country. FurnitORe produces chairs and tables with mahogany wood, and they sold it at: • 45$ per chair • 80$ per table There are two critical resources in the production of chairs and tables: • Mahogany (measured in square meters) and labor (measured in work hours) • There are 400 units of mahogany available at the beginning of each week • There are 450 units of labor available during each week You estimate that: • One chair requires 5 units of mahogany and 10 units of labor • One table requires 20 units of mahogany and 15 units of labor And the marketing has just told you that all the production of chairs and tables can be sold, so they ask you: What is the production plan that maximizes the total revenue the following week? Of course you need to consider that you cannot produce a fraction of chairs or tables. No, there’s no point in having to produce 1.2 chairs or 8.7 tables, right? Summarizing everything in one table, we have: So in the end you need to: • Decide how many tables and chairs to make • In order to maximize total revenue • While satisfying resource constraints This is the problem that you are going to work on for the next 2 days too. You will be able to solve it automatically, with just a few steps from you and the hard part done by a machine. But first, let’s see how you would solve it… By hand! OK, that’s a lot of new concepts for just one day! Let’s recap a little bit to see what you accomplished today: • You have seen that Operations Research is on each layer of management of any company. You can spot optimization problems wherever you go. • But more importantly, you have seen some ways about how to look for them, so now it’s easier for you to understand the biggest problems of your business. • You have seen an informal definition of an optimization problem in plain words, as you could see in your business. Now that you understand problems that appear at different layers in any business and how to spot optimization problem, think about the daily operations and challenges you encounter on your company: • Are there any processes that seem inefficient, costly, or time-consuming? • Could you identify any areas that could be improved? • What are their goals? • And their constraints? • What do you really need to decide? • How did you spot those optimization problems? • Did you find any issues on properly defining them? Since you know about a new optimization problem at FurnitORe, tell me if you have any doubt about it. Think deeply about the problem and try to answer the following questions: • How easy do you see solving that problem? • How would you solve it if you had to do it by hand? • What ideas come to your mind so you can maximize the revenues considering the defined constraints? Try to find a solution to the problem as hard as you can. Annotate it. Save it somewhere. Or even better: tell me about the solution and how you answered all the questions above. I’ll be delighted if you drop me an email with all these things. That’s it for today, I hope you enjoyed this second day! Tomorrow you’ll understand the fundamentals of optimization so you could translate a given business problem into mathematical formulas. In fact, you’re going to write your first mathematical model that defines the Furniture Production Problem you’ve seen today. See you tomorrow! Day 3: Fundamentals of optimization Welcome to day 3 of #ORfrom0to1! 🙌🏻 Now that you already know what Operations Research is and how to identify optimization problems at your company… Why not start defining them? Mathematically speaking I mean. When working with optimization problems, it’s important not only to define the problem in plain words, but also to define it in the most rigorous language that we have on Earth: mathematics. Apart from being rigorous, it removes the ambiguity of the language we usually speak, so this is the first critical step in solving optimization problems. By the end of this third day you’ll have learned: • The basic elements of any optimization problem • How to identify each part in you business problem • How to translate a business problem into those elements Are you ready? Let’s go for it! ⚙️ Basic elements of every optimization problem A mathematical model describes the reality of your problem, and it consists of 5 different parts. Understanding these components is crucial as they are the building blocks of mathematical models used in Operations Research: • Sets • Parameters • Decision variables • Objective function • Constraints Let’s see all of them, one by one. Sets indicate the elements to take into account in the model so we can iterate on them while building the model. There might be sets for resources that are used in the problem, sets for products that you want to sell, sets for cities that you want to go through, sets for chemical elements in your formulation… Parameters indicate the constants known at the time of decision-making (before you even start defining the problem) and that need to be taken into account in the model. This could include things like product prices, resources capacities, coefficients, supply or demand for resources, etc. Every optimization problem involves making choices to achieve a desired outcome. These choices are represented by decision variables, and will determine the quantities to get from a product, or the specific city we need to go through in each step of your journey, or the amount of capacity used of a given resource. When getting a solution for your problem, you will extract the most important information from decision variables, as they define the key aspects of your brand new solution to the problem. You need a way to measure how well you’re doing. This is where the objective function comes in. It quantifies the goal you’re trying to achieve, whether it’s maximizing profit, minimizing costs, or a mix in between. In the end, it’s what needs to be optimized. Even though we have choices thanks to decision variables, those aren’t limitless. There are always constraints that restrict your options. These constraints could be limitations on resources, budget, time, or even physical laws. They define the limits within which the problem must be solved. All these 5 elements sound like nothing when talking without specific examples, so let’s see how to identify them 👇🏻 🛠️ Identifying those elements in the Furniture Production Problem Identifying and defining the componentes of an optimization problem in a real-world context is a skill that requires both intuition and training. And translating simple, plain words into mathematics is a needed but complex exercise if you want to make data-driven decisions. Today, we’ll practice identifying these elements in the Furniture Production Problem that you saw yesterday, breaking down the definition that we already have into its basic parts. That way, you can begin to see how a seemingly overwhelming challenge can be systematically tackled using OR techniques. Let’s start from the beginning. The problem stated: 🪑 FurnitORe produces chairs and tables with mahogany wood, and they sold it at: • 45$ per chair • 80$ per table There are two critical resources in the production of chairs and tables: • Mahogany (measured in square meters) and labor (measured in work hours) • There are 400 units of Mahogany available at the beginning of each week • There are 450 units of labor available during each week So we can identify the sets and some parameters. The first set I can see in this description is the set of products. We have two different products (chairs and tables) that we need to map with some numbers, so it’s easier to refer to them in a mathematical way: Similarly, we can define a set for products with the same goal (make it easier to refer to them in a mathematical way): Remember that this part is known beforehand, so it’s the easy part. From the text, we can identify the revenues per chair and table (45$ and 80$, respectively), and the maximum amount of resources (400 units of mahogany, 450 units of labor). Since the revenues are related to products and the maximum amount of mahogany and labor are related to resources, we have already defined the sets for resources and products… Let’s use them! So for revenues, since chairs relate to product 1 and tables relate to product 2, we would have something like: Similarly, and using the exact same logic, parameters for resources are: Let’s continue defining parameters. In the next part of the problem definition we had: 🪑 You estimate that: • One chair requires 5 units of mahogany and 10 units of labor • One table requires 20 units of mahogany and 15 units of labor And we ended up with a picture, remember? Let’s focus on just one part of it, the one that refers to the text above: See? We have defined a matrix of parameters that takes into account chairs and tables, but also mahogany and labor units. This parameter is called $a$ and has 2 different indices, the first one indicating the resource (1 for mahogany, 2 for labor), while the second one indicates the product (1 for chairs, 2 for tables). In this problem you need to calculate the amount of chairs and tables to produce in the weekly planning, so your decision variables would be… The number of chairs and tables to produce. So let’s call them x_1 and x_2 to the decision variables that define those amounts for chairs and tables, respectively. The first question here is: given any amount of chairs and tables sold, how could you calculate the total revenues? We know the revenues per product (rev_1, rev_2) and the decision variables attached to them (x_1, x_2), so we need to multiply the revenues per product by the number of products: This is exactly the same as saying: I’m not going to lie to you: this is usually the trickiest part. Not only because of the definition itself, but also because sometimes, in businesses, it’s kind of difficult to get all of them right from the beginning. It’s pretty common to see how people that make decisions usually don’t think about a proper definition of them. However, your job as an Operations Research Engineer is exactly that. So what would be the constraints in our problem from FurnitORe? In this problem we can see that: 🪑 (…) • There are 400 units of mahogany available at the beginning of each week • There are 450 units of labor available during each week Meaning that we can use 400 units of mahogany at most in a given week, as well as 450 units of labor at most in a given week. It seems we have identified pretty much everything in our problem, right? Let’s see the final mathematical model that defines it 👇🏻 🧩 Mathematics behind the Furniture Production Problem Now that we know: • The basic elements of any optimization problem • How to identify each part in your business problem What about mixing everything into a mathematical formulation? I know it sounds difficult, but it’s harder thinking than doing it. So let’s break it down to 3 steps: This is the easiest part for this problem. We already defined x_1 and x_2. As they need to be at least 0, we know they belong to the positive part of the integer numbers. So their domain will go in the range [0, infinity): We already had a definition: But let’s get back to math notation. Since we want to maximize that function, let’s do that: As simple as that. However, it’s always better to express everything in a more mathematical way of doing it. Having the parameters and decision variables attached to sets, and making use of summations, it’s easy to make the objective function more compact: We have at least 2 constraints to be properly defined that relate to resources. Let’s break them down. 1) Mahogany capacity constraint Which says that 5 units of mahogany per chair plus 20 units of mahogany per table cannot exceed 400 units of mahogany in total. This is the same as saying: Or even better: 2) Labor capacity constraint This is pretty similar to the previous one, so let’s repeat the process: Which says that 10 units of labor per chair plus 15 units of labor per table cannot exceed 450 units of labor in total. This is the same as saying: Or even better: 3) Non-negativity There is an implicit constraint about the domain of the variables since they go from 0 on, Let’s just make it explicit: So you have just written your first mathematical model! Being honest, the last parts of each constraint are not usually done. This was an exercise for you to break the ice with mathematics. You know, getting used to summations and compact forms. Let’s put it all together then, so the next time you see a mathematical formulation for an optimization problem, you easily recognize everything. This problem, as usually formulated and explicit as That s.t. that you see there is subject to -the constraints that follow-. Wow, that’s been a journey, huh? You have gone through the end-to-end process of just knowing an informal definition of a problem to a formal definition in mathematical terms: • You started seeing the basic elements of any optimization problem. Remember that there are 5 of them (sets, parameters, decision variables, objective function, and constraints). • Then you were able to identify each of them from the Furniture Production Problem that we already defined yesterday. • Finally, you wrote the mathematical model that defines the problem Congratulations! This is one of the hardest parts, so if you understand it well, then you are in a good position for the days to come. Yesterday I asked you to think about the potential optimization problems at your company. Now that you are able to translate them into maths, I encourage you to do so! This is the exercise for today, as it is hard and probably it will take some time. Reading and writing a mathematical model are completely different tasks, so if you have any issue with that, or need any help, you know I’m here to help you. In any case, if you couldn’t think of a solution to the Furniture Production Problem yesterday, try it again and let me know if you found any issues with that. That’s it for today, I hope you enjoyed this third day! Tomorrow you’ll understand how to solve an optimization problem now that you have a mathematical definition of it. We will see two ways of doing it, but I think one of them will be much more useful to you since it’s completely automated. See you tomorrow!	{"url":"https://www.tuduy.vn/collection-from-other-sources","timestamp":"2024-11-10T01:20:04Z","content_type":"text/html","content_length":"351122","record_id":"<urn:uuid:30fc2634-53ad-4601-830b-8933cc1a7ff6>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00338.warc.gz"}
The Origins of Dice Notation As a previous post here covered Dungeons & Dragons was the first game to make practical use of all five Platonic solids as dice. The first printing of Dungeons & Dragons did not, however, employ the classic abbreviations of dice notation: d4, d6, d8, d12 and d20. Instead, we see constructions like, "From 2-16 snakes can be conjured (roll two eight-sided dice)." Initially, TSR had no need for dice notation, instead favoring number ranges which assumed players could infer the dice needed: this was the convention through the Holmes Basic Set (1977), which is full of systems of the form, "Damage: 3-24 points." These same conventions prescribe dice throws in the contemporary Monster Manual Players Handbook (1978), however, suddenly makes liberal use of dice notation, without any preamble, as if players were expected to recognize a "d20," and more significantly, qualifiers like "5d20." This strongly hints that dice notation had been in use long before TSR embraced it, and we can in fact trace its origins to the very dawn of fandom: as we see above in Alarums & Excursions #1, in an article by Ted Johnstone on "Dice as Random Number Generators." Alarums #1 ranks among the very earliest D&D fan publications, and it is clear from Johnstone's tone that dice notation was not yet an established convention. After describing each of the dice in turn, he proposes that they be "referred to hereafter for convenience's sake as D4, D6, D8, D10 and D12." Since Johnstone intends to discuss the bell curves associated with multiple dice, he further introduces a qualifying number before the "D", in constructions like, "with a D12, you have 1/12 chance of rolling a 12, but with 2D6 you only have 1/36 chance of matching it." Throughout the article, he uses this compound construction to discuss the properties of "12D6," "2D4" and so on. Johnstone however skips the notation "D20," instead favoring "D10." This was not because Johnstone actually possessed a ten-sided die, but rather because the numbering on early icosahedra ran from one through ten twice. To find the D20, we must consult a separate article in that same issue of Alarums, by the editor's husband, Barry Gold, which for its discussion of treasure allocation provides a slightly different dice notation syntax: Gold's syntax explicitly refers to the "D20" instead of the "D10," and also introduces the abbreviation "%ile" for what later notations would call the "D100." While the "%ile" abbreviation saw little uptake, the "D20" had considerable staying power. Dice notation was immediately seized upon by readers of Alarums. In the second issue, Robert Sacks refers to generating abilities with "3D6+D4 or D20 instead of 3D6." Mark Swanson in Alarums #3 boggles that anyone would think "saving throws are done D10+D10" as he believed "everybody knew that saving/hitting throws are done on a D20." When Swanson's own fanzine the Wild Hunt began in February 1976, it also adopted dice notation, as can be seen in this table, hand-drawn by Swanson, in the first issue: As the fan community began to aspire towards commercial products, dice notation became commonplace, despite TSR's apparent disinterest.[] In the August 1976 issue of Alarums, Steve Perrin provided some instructions to the fan community for contributing monsters to his upcoming project All The Worlds' Monsters (1977), which would appear under the Chaosium imprint. Perrin recommended to Alarums readers that the variable attributes of monsters "should be expressed as what sort of dice should be rolled," and gives "2D6+6 to give an 8-18 range" as an example. Thus, All the Worlds' Monsters pervasively used dice notation in its monster descriptions, and the first printing contains the following introductory blurb: In the entry for the "Air Squid" mentioned there, we see numerous examples of dice notation: a beak attack that deals 1D10, tentacles that constrict for 1D8, an intelligence of 2D6, and so on. All the Worlds' Monsters anthologized contributions from many prominent fans of the day, and its dice notation represented the consensus position of the community at the time. The following year, Perrin would go on to produce the Chaosium's signature role-playing game Runequest (1978), which in its first printing gives this concise statement of dice notation: This could serve nicely as the description of dice notation that the Players Handbook conspicuously lacked, at least until the Dungeon Masters Guide (1979) gave a lengthier account of dice and notation. Certainly TSR was aware of All the Worlds' Monsters, and Runequest narrowly beat the Players Handbook to the market. But more significantly, Gary Gygax had long been a contributor to Alarums, and certainly read Johnstone's article in Alarums #1, given his response in the second issue. The dice notation in the Players Handbook does differ in subtle respects from the earlier fan systems: for example, it uses a lower case "d" instead of an upper case "D," so one rolls "2d6" instead of "2D6." The Players Handbook also prefers to spell out "percentile" rather than abbreviating it (later, the DMG would sporadically use "d%"). It is however clear that TSR chose to embrace a longstanding practice when it added this notation to the game of Dungeons & Dragons in 1978: one so familiar and intuitive that no special explanation of the notation was warranted. While the precise abbreviations for the names of dice may seem a matter of slight historical import, eventually the abbreviation for an icosahedron would become practically synonymous with the game of Dungeons & Dragons, as the d20 System (2000) became the basis for Wizards of the Coast's third edition of Dungeons & Dragons and many dependent games. Studying the origins of dice notation moreover gives us another window into the way that TSR accepted the contributions of fans and addressed competing ideas in the marketplace. [] Before the Players Handbook, there were only sporadic and fragmentary references to dice notation in TSR publications, and these only in contributions to its periodicals. An article by Omar Kwalish for the June 1977 issue of the Dragon copies the percentile derivation table from Fight in the Skies, but gives it the title "Percentages Generated with Two Standard Dice (D6)." In February 1978, Rob Kuntz's first article on the Cthulhu mythos (a first draft of the material later to be famously redacted from Deities & Demigods) describes how Cthuga, Lord of Fire, "may summon up to 8 12 hit die (8 d12) fire elementals." These sorts of casual mentions only further illustrate how widespread dice notation had become without TSR's explicit endorsement. 7 comments: 1. Omar Kwalish => Tim Kask, writing articles when they needed fill-in material. 2. Very cool! 3. I started with the white boxed set - and we had very little idea how it all worked. Luckily the rules were so brief there was room to just add in house versions when you didn't understand something. For instance, nowhere could we find any reference explicitly stating whether an attack using a bow was the same as a regular attack, so we had some house rule involving a d6 to determine a hit when we shot a bow. The original d20s had no color on the numbers (which were just 1-10 twice), they were just etched into the surface, so we would take a colored pen and color in half in one color and half in the other to make a d20. You would role your d20 and say "blue high" or something like that. 4. Interesting topic. Later editions of Holmes Basic saw it revised to include the "d20" notation in the section on "Using the Dice". In the 3rd edition (Dec 1979), page 46: "Thus "2d4" would mean that two 4-sided dice would be thrown (or one 4-sided would be thrown twice); "3d12" would indicate that three 12-sided dice are used, and so on". I believe the only place this notation is actually used in the revised rulebook is on the Reference Tables sheet, which uses 1d20 twice and 2d6 once. But the 1st print of B2 also came out around the same time and makes sporadic use of it, mostly for treasures (d6 coins, etc). 1. Just found my 2nd edition Holmes (Nov 78), and the same language is in there. And the module B1 from the same month has examples of the "d20" notation as well. This is just a few months after the Players' Handbook came out in Jun 78. 5. We used to play a little game during slow moments around the D&D table (like when the DM was looking up a rule or generating treasure). Someone would yell out a number range ("7-34"), and the winner was the first person who could come up with a way to generate that range with dice (3d10+4). They could get convoluted (e.g., 1-26 = 2d8 + 1d12 -2). 6. This comment has been removed by a blog administrator.	{"url":"https://playingattheworld.blogspot.com/2013/08/the-origins-of-dice-notation.html","timestamp":"2024-11-02T06:02:13Z","content_type":"text/html","content_length":"124260","record_id":"<urn:uuid:d4b9ba10-5b3a-4aaf-bee9-0c14fefd2fa6>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00751.warc.gz"}
How do you calculate the change in momentum of an object? \| HIX Tutor How do you calculate the change in momentum of an object? Answer 1 There are two approaches that can be used, depending on the issue. 1) The change in momentum of an object is its mass times the change in its velocity. #\Delta p=m(\Deltav)=m(v_f-v_i)#. #v_f# and #v_i# are the final and initial velocities. Remember to use the right signs when substituting #v_f# and #v_i# Example: A 3 kg mass traveling 4 m/s to the right first bounces off a wall and moves 2 m/s to the left. Taking "right" to be the positive direction: #v_i#=+4m/s, #v_f#= –2m/s, and m=3kg. Substituting, #\Delta p=3kg(-2#m/s#-4#m/s)#=-18# kg m/s 2) The change in the momentum of an object can also be found by considering the force acting on it. If a force, #F#, acts on an object for a time, #\Delta t#, the change in the objects momentum is #\ Delta p= F\Delta t#. Remember to use the right sign when substituting #F#. For example, a force to the left could be negative. Lastly, if your object is moving both horizontally and vertically then #\Delta p# has a vertical and horizontal component. If this is the case, the above equations still work for each component separately, Ex) To find the horizontal component of #\Delta p# use the horizontal component of #v_i, v_f# or #F# in the above equations. Sign up to view the whole answer By signing up, you agree to our Terms of Service and Privacy Policy Answer 2 The change in momentum of an object is calculated by subtracting the initial momentum from the final momentum. Mathematically, it can be represented as: Change in momentum = Final momentum - Initial momentum 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-calculate-the-change-in-momentum-of-an-object-8f9af8b35e","timestamp":"2024-11-03T06:26:15Z","content_type":"text/html","content_length":"619222","record_id":"<urn:uuid:7fd438b3-8ad5-475c-b652-f2db2136c895>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00656.warc.gz"}
Calculation of diagonal deformation of the house To calculate the voltage and deformation of the frame with filling, it is necessary to determine the width of the conditional slanting, the linear deformation of which (with a simple compression, calculated subject to the constant elasticity module equal to the initial tangential module of the elasticity of the filling material) is similar to the diagonal deform of the filling (established subject to the alternating module from stress). The calculation of diagonal deformation was developed by the authors to fill in concrete. This calculation is carried out by the final elements in the following assumptions, the normal and tangent voltages in the contact surface are distributed along the triangle, the length of the contact with the column depends on the characteristic of the stiffness and is determined by the above ratios, the length of the contact surface with the crossbars is equal to half the span, the elasticity of the filling material of the filling changes in depending on the voltage (when calculating, the tangential module was accepted). With an increase in the voltage, the value of the elasticity module decreases, which is manifested by changing the values of the relative deformations of the elements of filling in the direction of compressed slanting (from the maximum values in the corners of the magnitude of relative deformations are reduced towards the center of depreciation). At the same time, the width of the conventional strip of braces that have a constant everective module changes. Consequently, the width S will change with a change in the magnitude of the diagonal forces (decreases with an increase in the load). The maximum width occurs with the initial stressful state of the diagonal forces, I am approximately equal, which is a diagonal force at the limit of destruction of the filling in the pressure, in which the fragmentation of the filling panels in the corners, rop -shaped, is fragmented. compressed slanting, when it is possible with sufficient accuracy when calculating the deformation of the filling to use in the entire range a constant initial tangential elastic module.	{"url":"https://hollanderhomes.com/calculation-of-diagonal-deformation-of-the-house.html","timestamp":"2024-11-06T22:09:45Z","content_type":"text/html","content_length":"76825","record_id":"<urn:uuid:1ceabbeb-5d76-478a-a8b1-7637bfb41a61>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00727.warc.gz"}
Showing mods for Minecraft b1.1_01. Show all Browse Mods Supports Minecraft a1.2.2-1, a1.2.2-2, a1.2.3, a1.2.3_01, a1.2.3_02, a1.2.3_04, a1.2.4_01, a1.2.5, a1.2.6, b1.1-1, b1.1-2, b1.1_01, b1.1_02, b1.2_01, b1.2_02, b1.3_01, b1.4, b1.5, b1.5_01, b1.6.4, b1.6.5, b1.6.6, b1.7, b1.7.2, b1.7.3, b1.7_01 If you have an AMD graphics card, you've likely noticed how awful your clouds look. They're spikey, and you can see where the "chunks" of them stitch together. Sheep, trees, and snow also look really weird - as if their textures are fighting. This mod fixes that by increasing the bit depth from 8 to 24. Supports Minecraft a1.2.6, b1.1_01, b1.2_02, b1.3_01, b1.7.3 Adding mobs to your once empty world.	{"url":"https://mcarchive.net/mods?gvsn=b1.1_01","timestamp":"2024-11-11T13:39:15Z","content_type":"text/html","content_length":"4725","record_id":"<urn:uuid:d83d011b-6ea5-49fb-a3c4-a5a7adb7cc92>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00217.warc.gz"}
Uncertainty, Error and Confidence in Data Jim Sturgiss provides a straightforward guide to teaching some scientific concepts that are now part of the new Science syllabuses… Uncertainty is a statistical concept found in the Assessing data and information outcome of the new Science syllabuses:. WS 5.2 assess error, uncertainty and limitations in data (ACSBL004, ACSBL005, ACSBL033, ACSBL099) This concept is not found in the previous syllabuses. This paper addresses uncertainty as a means of describing the accuracy of a series of measurements or as a means of comparing two sets of data. Uncertainty, or confidence, is described in terms of mean and standard deviation of a dataset. Standard deviation is a concept encountered by students in Stage 5.3 Mathematics and Stage 6 Standard 2 Mathematics. Not explored in this paper is the use of Microsoft Excel or Google Sheets which can calculate uncertainty of datasets with ease (=STDEV.S(number1, number2,…). Figure 1 Karl Pearson Karl Pearson (Figure 1), the great 19th-century biostatistician and eugenist, first described mathematical methods for determining the probability distributions of scientific measurements, and these methods form the basis of statistical applications in scientific research. Statistical techniques allow us to estimate uncertainty and report the error surrounding a value after repeated measurement of that value. 1. Accuracy, Precision and Error Accuracy is how close a measurement is to the correct value for that measurement. The precision of a measurement system refers to how close the agreement is between repeated measurements (which are repeated under the same conditions). Measurements can be both accurate and precise, accurate but not precise, precise but not accurate, or neither. Precision and Imprecision Precision (see Figure 2) refers to how well measurements agree with each other in multiple tests. Random error, or Imprecision, is usually quantified by calculating the coefficient of variation from the results of a set of duplicate measurements. Figure 2 Accuracy and precision The accuracy of a measurement is how close a result comes to the true value. When randomness is attributed to errors, they are “errors” in the sense in which that term is used in statistics. • Systematic error (bias) occurs with the same value, when we use the instrument in the same way (eg calibration error) and in the same case. This is sometimes called statistical bias. It may often be reduced with standardized procedures. Part of the learning process in the various sciences is learning how to use standard instruments and protocols so as to minimize systematic • Random error, which may vary from one observation to another. Random error (or random variation) is due to factors which cannot, or will not, be controlled. Random error often occurs when instruments are pushed to the extremes of their operating limits. For example, it is common for digital balances to exhibit random error in their least significant digit. Three measurements of a single object might read something like 0.9111g, 0.9110g, and 0.9112g. Systematic error or Inaccuracy (see Figure 3) is quantified by the average difference (bias) between a set of measurements obtained with the test method with a reference value or values obtained with a reference method. Figure 3 Imprecision and in accuracy 2. Uncertainty There is uncertainty in all scientific data. Uncertainty is reported in terms of confidence. • Uncertainty is the quantitative estimation of error present in data; all measurements contain some uncertainty generated through systematic error and/or random error. • Acknowledging the uncertainty of data is an important component of reporting the results of scientific investigation. • Careful methodology can reduce uncertainty by correcting for systematic error and minimizing random error. However, uncertainty can never be reduced to zero. Estimating the Experimental Uncertainty For a Single Measurement Any measurement made will have some uncertainty associated with it, no matter the precision of the measuring tool. So how is this uncertainty determined and reported? The uncertainty of a single measurement is limited by the precision and accuracy of the measuring instrument, along with any other factors that might affect the ability of the experimenter to make the measurement. For example, if you are trying to use a ruler to measure the diameter of a tennis ball, the uncertainty might be ± 5 mm, but if you used a Vernier caliper, the uncertainty could be reduced to maybe ± 2 mm. The limiting factor with the ruler is parallax, while the second case is limited by ambiguity in the definition of the tennis ball’s diameter (it’s fuzzy!). In both of these cases, the uncertainty is greater than the smallest divisions marked on the measuring tool (likely 1 mm and 0.05 mm respectively). Unfortunately, there is no general rule for determining the uncertainty in all measurements. The experimenter is the one who can best evaluate and quantify the uncertainty of a measurement based on all the possible factors that affect the result. Therefore, the person making the measurement has the obligation to make the best judgment possible and to report the uncertainty in a way that clearly explains what the uncertainty represents: Measurement = (measured value ± standard uncertainty) unit of measurement. For example, where the ± standard uncertainty indicates approximately a 68% confidence interval, the diameter of the tennis ball may be written as 6.7 ± 0.2 cm. Alternatively, where the ± standard uncertainty indicates approximately a 95% confidence interval, the diameter of the tennis ball may be written as 6.7 ± 0.4 cm. Estimating the Experimental Uncertainty For a Repeated Measure (Standard Deviation). Suppose you time the period of oscillation of a pendulum using a digital instrument (that you assume is measuring accurately) and find: T = 0.44 seconds. This single measurement of the period suggests a precision of ±0.005 s, but this instrument precision may not give a complete sense of the uncertainty. If you repeat the measurement several times and examine the variation among the measured values, you can get a better idea of the uncertainty in the period. For example, here are the results of 5 measurements, in seconds: 0.46, 0.44, 0.45, 0.44, 0.41. For this situation, the best estimate of the period is the average, or mean. Whenever possible, repeat a measurement several times and average the results. This average is generally the best estimate of the “true” value (unless the data set is skewed by one or more outliers). These outliers should be examined to determine if they are bad data points, which should be omitted from the average, or valid measurements that require further investigation. Generally, the more repetitions you make of a measurement, the better this estimate will be, but be careful to avoid wasting time taking more measurements than is necessary for the precision Consider, as another example, the measurement of the thickness of a piece of paper using a micrometer. The thickness of the paper is measured at a number of points on the sheet, and the values obtained are entered in a data table. This average is the best available estimate of the thickness of the piece of paper, but it is certainly not exact. We would have to average an infinite number of measurements to approach the true mean value, and even then, we are not guaranteed that the mean value is accurate because there is still some systematic error from the measuring tool, which can never be calibrated perfectly. So how do we express the uncertainty in our average value? The most common way to describe the spread or uncertainty of the data is the standard deviation Figure 5 Standard deviations of a normal distribution The significance of the standard deviation is this: if you now make one more measurement using the same micrometer, you can reasonably expect (with about 68% confidence) that the new measurement will be within 0.002 mm of the estimated average of 0.065 mm. In fact, it is reasonable to use the standard deviation as the uncertainty associated with this single new measurement. This is written: The thickness of 80 gsm paper (n=5) averaged 0.065 (s = 0.002mm) s = standard deviation The thickness of 80 gsm paper (n=5) averaged 0.065 ± 0.004 mm to a 95% confidence level. (0.004 mm represents 2 standard deviations, 2s) Standard Deviation of the Means (Standard Error of Mean (SEM)) The standard error is a measure of the accuracy of the estimate of the mean from the true or reference value. The main use of the standard error of the mean is to give confidence intervals around the estimated means for normally distributed data, not for the data itself but for the mean. If measured values are averaged, then the mean measurement value has a much smaller uncertainty, equal to the standard error of the mean, which is the standard deviation divided by the square root of the number of measurements. Standard error is often us For example, two populations of salmon fed on two different diets may be considered significantly different if the 95% confidence intervals (two std errors) around the estimated fish sizes under Diet A do not cross the estimated mean fish size under Diet B. Note that the standard error of the mean depends on the sample size, as the standard error of the mean shrinks to 0 as sample size increases to infinity. Figure 7 Salmon Standard Error of Mean (SEM) Versus Standard Deviation In scientific and technical literature, experimental data are often summarized either using the mean and standard deviation of the sample data or the mean with the standard error. This often leads to confusion about their interchangeability. However, the mean and standard deviation are descriptive statistics, whereas the standard error of the mean is descriptive of the random sampling process. The standard deviation of the sample data is a description of the variation in measurements, whereas, the standard error of the mean is a probabilistic statement about how the sample size will provide a better bound on estimates of the population mean, in light of the central limit theorem. Put simply, the standard error of the sample mean is an estimate of how far the sample mean is likely to be from the population mean, whereas the standard deviation of the sample is the degree to which individuals within the sample differ from the sample mean. If the population standard deviation is finite, the standard error of the mean of the sample will tend to zero with increasing sample size. This is because the estimate of the population mean will improve, while the standard deviation of the sample will tend to approximate the population standard deviation as the sample size Confidence Levels The confidence level represents the frequency (i.e. the proportion) of possible confidence intervals that contain the true value of the unknown population parameter. Most commonly, the 95.4% (“two sigma”) confidence level is used. However, other confidence levels can be used, for example, 68.3% (“one sigma”) and 99.7% (“three sigma”). Knowledge of normally distributed data and standard deviation are key to understanding the notions of statistical uncercertainty and confidence. These concepts are extended to the standard error of mean so that the significance of differences between two related datasets can be determined. Absolute error The absolute error of a measurement is half of the smallest unit on the measuring device. The smallest unit is called the precision of the device. Array An array is an ordered collection of objects or numbers arranged in rows and columns. Bias This generally refers to a systematic favouring of certain outcomes more than others, due to unfair influence (knowingly or otherwise). Confidence level The probability that the value of a parameter falls within a specified range of values. For example 2s = 95% confidence level. Data cleansing Detecting and removing errors and inconsistencies from data in order to improve the quality of data (also known as data scrubbing). Data set An organised collection of data. Descriptive statistics These are statistics that quantitatively describe or summarise features of a collection of information. Large data sets Data sets that must be of a size to be statistically reliable and require computational analysis to reveal patterns, trends and associations. Limits of accuracy The limits of accuracy for a recorded measurement are the possible upper and lower bounds for the actual measurement. Measures of central tendency Measures of central tendency are the values about which the set of data values for a particular variable are scattered. They are a measure of the centre or location of the data. The two most common measures of central tendency are the mean and the median. Measures of spread Measures of spread describe how similar or varied the set of data values are for a particular variable. Common measures of spread include the range, combinations of quantiles (deciles, quartiles, percentiles), the interquartile range, variance and standard deviation. Normal distribution The normal distribution is a type of continuous distribution whose graph looks like this: The mean, median and mode are equal and the scores are symmetrically arranged either side of the mean. The graph of a normal distribution is often called a ‘bell curve’ due to its shape. Reliability An extent to which repeated observations and/or measurements taken under identical circumstances will yield similar results. Sampling This is the selection of a subset of data from a statistical population. Methods of sampling include: • systematic sampling – sample data is selected from a random starting point, using a fixed periodic interval • self-selecting sampling – non-probability sampling where individuals volunteer themselves to be part of a sample • simple random sampling – sample data is chosen at random; each member has an equal probability of being chosen • stratified sampling – after dividing the population into separate groups or strata, a random sample is then taken from each group/strata in an equivalent proportion to the size of that group/ strata in the population • A sample can be used to estimate the characteristics of the statistical population. Standard deviation This is a measure of the spread of a data set. It gives an indication of how far, on average, individual data values are spread from the mean. Standard error The standard error of the mean (SEM) is the standard deviation of the sampling distribution of the mean. Uncertainty Any single value has an uncertainty equal to the standard deviation. However, if the values are averaged, then the mean measurement value has a much smaller uncertainty, equal to the standard error of the mean, which is the standard deviation divided by the square root of the number of measurements. Works Cited Measurements and Error Analysis, www.webassign.net/question_assets/unccolphysmechl1/measurements/manual.html. Altman, Douglas G, and J Martin Bland. “Standard Deviations and Standard Errors.” BMJ (Clinical Research Ed.), BMJ Publishing Group Ltd., 15 Oct. 2005, www.ncbi.nlm.nih.gov/pmc/articles/PMC1255808/. Hertzog, Lionel. “Standard Deviation vs Standard Error.” DataScience , 28 Apr. 2017, https://datascienceplus.com/standard-deviation-vs-standard-error/ Mott, Vallerie. “Introduction to Chemistry.” Schoonjans, Frank. “Definition of Accuracy and Precision.” MedCalc, MedCalc Software, 9 Nov. 2018, www.medcalc.org/manual/accuracy_precision.php. “Standard Error.” Wikipedia, Wikimedia Foundation, 7 Mar. 2019, 2336 \| NSW Education Standards, 1319 \| NSW Education Standards, https://educationstandards.nsw.edu.au/wps/portal/nesa/11-12/stage-6-learning-areas/stage-6-mathematics/mathematics-standard-2017/content/1319 Jim is an educational researcher and independent educational consultant. His M.Ed (Hons) thesis used an experimental design to evaluate the effectiveness of a literacy and learning program (1997). A recipient of the NSW Professional Teaching Council’s Distinguished Service Award for leadership in delivering targeted professional learning to teachers, he works with schools to align assessment, reporting and learning practice. He has been a head teacher of Science in two large Sydney high schools, as well as HSC Chemistry Senior Marker and Judge. For many years he served as a DoE Senior Assessment Advisor where he developed many statewide assessments, (ESSA, SNAP, ELLA, BST) and as Coordinator: Analytics where he developed reports to schools for statewide assessments and NAPLAN. He is a contributing author to the new Pearson Chemistry for NSW and to Macquarie University’s HSC Study Lab for Physics.	{"url":"https://cpl.nswtf.org.au/journal/semester-1-2020/uncertainty-error-and-confidence-in-data/","timestamp":"2024-11-09T15:47:25Z","content_type":"text/html","content_length":"71275","record_id":"<urn:uuid:0773fdf2-6183-4f17-bf7b-254cebcf1444>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00616.warc.gz"}
Mercator Sailing: Course and Distance 09-04-2018, 07:19 PM (This post was last modified: 09-04-2018 07:21 PM by Dieter.) Post: #21 Dieter Posts: 2,397 Senior Member Joined: Dec 2013 RE: Mercator Sailing: Course and Distance (09-03-2018 10:10 PM)Gene222 Wrote: Thanks for your comments on the program. I definitely bit off more than I could chew with this program. The revised program is attached. Thank you, the results look good to me. Maybe you can remove the "this check box is not used" part. Unless there is something I haven't realized yet. ;-) Finally we got another example of this forum's power: in a collaborative effort we now got a solution that also handles East-West courses that even some online calculators don't calculate correctly. 09-05-2018, 05:46 PM (This post was last modified: 09-11-2019 12:12 AM by Gene222.) Post: #22 Gene222 Posts: 125 Member Joined: Feb 2015 RE: Mercator Sailing: Course and Distance FYI. The book "The Calculator Afloat" was written by H. H. Shufeldt and Kenneth E Newcomer. A review of the book (below) was printed in the December 1980 HP Key Notes. It says Newcomer worked for HP and developed the HP-65 Navigation Pac. I never bought the Nav Pac for my HP-41, but I eventually did buy the book. 09-22-2018, 11:37 PM Post: #23 Eddie W. Shore Posts: 1,614 Senior Member Joined: Dec 2013 RE: Mercator Sailing: Course and Distance (08-12-2018 05:13 PM)Dieter Wrote: (08-10-2018 03:08 AM)Eddie W. Shore Wrote: Latitude 1: 102° 54’ 16” W = 102.9044444444° Longitude 1: 43° 21’ 16” N = 43.3544444444° Latitude 2: 106° 3’ 8” W = 106.0522222222° Longitude 2: 42° 4’ 30” N = 42.075° Course: 5.782390957° Distance: 189.832588 mi Eddie, I think you confused latitudes and longitudes here. There are no latitudes > 90°. I also wonder how you get these results. For the given data I get a distance of 159,03 nmi and a course of 61,14° Southwest (241,14° true course). This matches the result of an online calculator where I checked the results. Finally, what sign convention does your program use? Are West and South positions entered with negative sign? I always get latitude and longitude mixed up. Going by the book's examples (pg. 78), north and west were entered as positives. 09-22-2018, 11:39 PM Post: #24 Eddie W. Shore Posts: 1,614 Senior Member Joined: Dec 2013 RE: Mercator Sailing: Course and Distance (08-24-2018 07:21 PM)Gene222 Wrote: Two weeks and no response. I guess Eddie is not following his own post. I bought Calculator Afloat in the late 80s or early 90s to learn more about geodetic calculations. It's a good book, but it does not go into much detail about the signage for the latitude and longitude calculations. You perform the calculations first then mentally apply the proper signage to the results. I modified Eddie's program where South latitudes and West longitudes must be entered as negative numbers in degrees, but I ran into a problem when the two points have the same latitude, such as Lat 40 N, Long 25 W Lat 40 N, Long 30 W In this case, the course and distance equations give division by zero error messages. How do you calculate the course and distance on a parallel? EXPORT MERCATOR() // EWS 2018-08-10 // The Calculator Afloat HAngle:=1; // degrees LOCAL L1,L2,λ1,λ2,M1,M2; LOCAL C,D; {"Latitude 1 in degrees, +N, -S", "Longitude 1 in degrees, +E, -W", "Latitude 2 in degrees, +N, -S", "Longitude 2 in degrees, +E, -W"}); LOCAL m,DLo,l,θ; DLo:=(λ2-λ1)60; //difference in longitude in minutes m:=(M2-M1); //difference in meridional parts in minutes θ:=ATAN(DLo/m); //course in degrees relative to North (+ or -) or South (+ or -) l:=(L2-L1)60; //difference in latitude in minutes IF DLo>=0 AND m>=0 THEN IF DLo<0 AND m>=0 THEN IF DLo<0 AND m<0 THEN C:= 180+θ; IF DLo>=0 AND m<0 THEN RETURN {C,D}; My apologizes for not following subscribing to the program posts I put up; something I will do from now on. I appreciate all the comments and modifications. 09-10-2022, 03:17 PM Post: #25 Albert Chan Posts: 2,773 Senior Member Joined: Jul 2018 RE: Mercator Sailing: Course and Distance (08-26-2018 07:18 AM)Dieter Wrote: Hint: the expression ln(tan(45°+x/2)) in the "book" formula is equivalent to artanh(sin(x)). FYI, atanh(sin(x)) = asinh(tan(x)) = gd^-1(x) = x + x^3/6 + x^5/24 + ... https://en.wikipedia.org/wiki/Gudermannian_function The Secret Connection between Hyperbolic and Trigonometric Functions 12-21-2022, 08:31 PM Post: #26 VadimV Posts: 1 Junior Member Joined: Dec 2022 RE: Mercator Sailing: Course and Distance (08-31-2018 05:46 AM)Dieter Wrote: (08-31-2018 02:15 AM)Gene222 Wrote: I see what I am doing wrong. The course is 270 (due west) on a Mercator map. I was looking a polar map. I guess starpath.com online Mercator calculator was right. For the record: my 35s program returns 1875,78 miles and a course exactly West, true course 270°. By the way, the result 1875,4003 nm of the starpath.com Mercator Calculator simply is 180°·60·cos(80°). This is a simplified formula that ignores (!) the ellipsoid's eccentricity. That's why there is another factor in my same-latitude-formula – which returns the correct result. (08-31-2018 02:15 AM)Gene222 Wrote: I still have errors in my program. The Latitude difference for the above problem is 600' and the Meridional parts for point 1 and 2 are not the same. This really sounds like there are errors. ;-) For 80° the meridional parts (WGS84) should be 8352,48. Your results of –3456,8203 and –4507,4040 are the meridional part values for latitudes of –50° and –60°, respectively. Maybe this helps finding the error. (08-31-2018 02:15 AM)Gene222 Wrote: My procedure for East-West courses needs to be cleaned up, which should be easy to do. I can't read your program (no Prime here), but from what I see when I open it in a text editor the program distinguishes various cases for different hemispheres. For instance in one case the absolute values of the meridional parts are added. I don't think this is required – the sign convention handles this automatically. Try it. Here is the way I do the calculation, in pseudocode. I hope I got it right. ;-) if dlong < -180 then dlong=dlong+360 if dlong > +180 then dlong=dlong-360 // edit: corrected this line if DMP=0 then Finally the true course is calculated. I do it wih the 35s "ARG" command which returns something like ATAN2 in some programming languages, a quadrant-adjusted angle (0...180 for Q1 and Q2, 0...-180 for Q3 and Q4, in this case add 360°), and it also works for DMP=0. You'll know how to do it on the Prime. ;-) Thank you for this. Using your equations (in Excel) for MP and dLon when MP=0, namely: MP = 60180/PI()(ATANH(SIN(RADIANS(Lat)))-0.081819190842622ATANH(0.081819190842622SIN(RADIANS((Lat))))) dLon = Distance/(COS(RADIANS(Lat))(1-0.081819190842622^2SIN(RADIANS(Lat)^2))/(1-0.081819190842622^2)) and an e value per WGS84, I get a strange dLon value when traveling E/W along the equator. For a course of either 090 or 270 and a distance of 600nm I calculate a dLon value of 595.98337. Shouldn't this value be 600 (i.e., 10 degrees of Lon)? Thanks in advance for your assistance. User(s) browsing this thread: 1 Guest(s)	{"url":"https://www.hpmuseum.org/forum/thread-11196-post-103267.html","timestamp":"2024-11-09T04:54:08Z","content_type":"application/xhtml+xml","content_length":"39315","record_id":"<urn:uuid:9c504b9b-5df3-4d60-a277-09b2fbd2a791>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00337.warc.gz"}
Stock Return Calculator: Calculate Your Stocks Returns Online \| 5paisa Stock Return Calculator A stock return calculator is a digital tool that calculates the returns for an invested amount over a specific period for a particular stock. Whether you're a seasoned investor or just starting, this user-friendly calculator empowers you to make informed investment decisions confidently. It harnesses the power of historical data and advanced algorithms to estimate the performance of stocks over various timeframes accurately. Simply input the necessary information, and within moments, you'll receive a comprehensive analysis of potential returns, including growth rates, dividends, and total value. You can use the stock market return calculator to estimate your returns for other investment avenues, such as ETFs and mutual funds. Stay ahead of the game, assess your investment strategies, and unlock the potential for financial success with the stock return calculator. • Invested Amount • Wealth gained • Invested Amount • ₹4,80,000 • Expected Amount • ₹8,07,633 Start investing with flat ₹20 brokerage. A stock return calculator provides a detailed picture of the current investment scenario and a summary of the investment’s performance. The stock return calculator meaning also includes its potential to determine the rate of return on an investment in a stock or portfolio of stocks. It helps analyse your investments' performance and evaluate their stock holdings' profitability. The calculator is useful to analyse the performance of individual stocks, compare the returns of different stocks or portfolios, and make informed decisions about buying, selling, or holding stocks based on historical performance. The stock return calculator determines the return as a percentage, commonly referred to as the "return on investment" (ROI) or "total return." It indicates the profit or loss generated by the investment, considering both capital gains (increase in stock price) and income from dividends. The calculator considers the monthly closing prices of the stocks after adjusting them for splits, bonuses or dividends. For example, if the stock has been trading for one year, the calculator considers the last three months’ closing price to calculate the returns. However, it assumes daily price data if the stock has been trading for less than three months or if price data for the last three months isn't available. The stock return calculator in India accounts for two main factors: the stock’s initial investment (purchase price) and the final value (sale price). 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You can compare investment opportunities with the most profit potential using the return report. 5paisa’s stock return calculator is easy-to-use and gives real-time results. The return calculator is entirely free for users who can use it unlimited times to calculate the returns on their Here is a step-by-step guide to using the 5paisa stock return calculator: Step 1: Visit the official page of the 5paisa’s return calculator and navigate below. Step 2: Use the slider below the ‘Monthly Investment’ section to set your desired investment amount. The maximum amount you can use the calculator is Rs 1,00,000, and the minimum is Rs 50. Step 3: Enter the name of the scrip in the ‘Select Stock’ section for which you want to calculate the returns. Step 4: Choose the period for which you want to analyse the returns from the drop-down menu mentioned in the ‘Investment Period’ section. Step 5: Once you select the investment period, the calculator automatically depicts the expected return as a percentage in the ‘Expected Return’ section. You can also analyse the returns from the dialogue box on the right, which shows ‘Invested Amount’, ‘Wealth Gained’ and the ‘Expected Amount’. Frequently Asked Questions Using the stock return calculator is simple. Enter some basic details, such as the name of the stock and the investment amount, in the calculator to estimate the potential returns for a specific 5paisa’s stock return calculator contains every listed stock. Simply choose the stock’s name from the drop-down menu and calculate the returns based on your investment amount and period. One of the best features of the return calculator is its ability to compare stocks for an informed investment decision. If you want to know which will offer you the best returns, you can calculate returns for all of them and compare the final wealth gained or the expected amount. 5paisa has designed its return calculator as a free-of-cost digital tool. You can use the calculator without incurring any cost, and there are no restrictions on the number of calculations. Using the 5paisa stock return calculator, you can calculate returns for a minimum period of one month and a maximum of one year. Disclaimer: The calculator available on the 5paisa website is intended for informational purposes only and is designed to assist you in estimating potential investments. However, it is important to understand that this calculator should not be the sole basis for creating or implementing any investment strategy. 5paisa does not take responsibility or liability for the accuracy of the figures generated by the calculator. It's also important to remember that the examples given here do not make any claims regarding the performance of any particular asset or investment. Before making any financial decisions based on the results of this calculator, we highly advise every investor to consult with a qualified advisor. View More..	{"url":"https://www.5paisa.com/calculators/stock-return-calculator","timestamp":"2024-11-14T18:45:33Z","content_type":"text/html","content_length":"82710","record_id":"<urn:uuid:d9fee9b0-6ba0-46c5-91ea-27394825ee9b>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00254.warc.gz"}
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□ Print the first page to give to your students. They need to understand how to set up equations for linear pairs, vertical angles, complementary angles, and what congruent angles are to be able to solve for x. Because one pathway will have a correct answer, they can feel confident when their answer matches one of the pathways. Because my students use my site frequently, I didn’t post the key. Leave A Comment	{"url":"https://systry.com/reasoning-and-proofs/angle-relationships-maze/","timestamp":"2024-11-03T04:21:44Z","content_type":"text/html","content_length":"33162","record_id":"<urn:uuid:68748aa2-04bd-4832-92a2-81107b5a55c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00835.warc.gz"}
MAXIFS function in Excel – find max value with multiple criteria The tutorial shows how to use the MAXIFS function in Excel to get the maximum value with conditions. Traditionally, when you ever needed to find the highest value with conditions in Excel, you had to build your own MAX IF formula. While not a big deal for experienced users, that might present certain difficulties for novices because, firstly, you should remember the formula's syntax and, secondly, you need to know how to work with array formulas. Luckily, Microsoft has recently introduced a new function that lets us do conditional max an easy way! Excel MAXIFS function The MAXIFS function returns the largest numeric value in the specified range based on one or more criteria. The syntax of the MAXIFS function is as follows: MAXIFS(max_range, criteria_range1, criteria1, [criteria_range2, criteria2], …) • Max_range (required) - the range of cells where you want to find the maximum value. • Criteria_range1 (required) - the first range to evaluate with criteria1. • Criteria1 - the condition to use on the first range. It can be represented by a number, text or expression. • Criteria_range2 / criteria2, …(optional) - additional ranges and their related criteria. Up to 126 range/criteria pairs are supported. This MAXIFS function is available in Excel 2019, Excel 2021, and Excel for Microsoft 365 on Windows and Mac. As an example, let's find the tallest football player in our local school. Assuming the students' heights are in cells D2:D11 (max_range) and sports are in B2:B11 (criteria_range1), use the word "football" as criteria1, and you will get this formula: =MAXIFS(D2:D11, B2:B11, "football") To make the formula more versatile, you can input the target sport in some cell (say, G1) and include the cell reference in the criteria1 argument: =MAXIFS(D2:D11, B2:B11, G1) Note. The max_range and criteria_range arguments must be of the same size and shape, i.e. contain the equal number of rows and columns, otherwise the #VALUE! error is returned. How to use MAXIFS function in Excel - formula examples As you have just seen, the Excel MAXIFS is quite straightforward and easy to use. However, it does have a few little nuances that make a big difference. In the below examples, we will try to make the most of conditional max in Excel. Find max value based on multiple criteria In the first part of this tutorial, we created a MAXIFS formula in its simplest form to get the max value based on one condition. Now, we are going to take that example further and evaluate two different criteria. Supposing, you want to find the tallest basketball player in junior school. To have it done, define the following arguments: • Max_range - a range of cells containing heights - D2:D11. • Criteria_range1 - a range of cells containing sports - B2:B11. • Criteria1 - "basketball", which is input in cell G1. • Criteria_range2 - a range of cells defining the school type - C2:C11. • Criteria2 - "junior", which is input in cell G2. Putting the arguments together, we get these formulas: With "hardcoded" criteria: =MAXIFS(D2:D11, B2:B11, "basketball", C2:C11, "junior") With criteria in predefined cells: =MAXIFS(D2:D11, B2:B11, G1, C2:C11, G2) Please notice that the MAXIFS function in Excel is case-insensitive, so you needn't worry about the letter case in your criteria. In case you plan to use your formula on multiple cells, be sure to lock all the ranges with absolute cell references, like this: =MAXIFS($D$2:$D$11, $B$2:$B$11, G1, $C$2:$C$11, G2) This will ensure that the formula copies to other cells correctly - the criteria references change based on the relative position of the cell where the formula is copied while the ranges remain As an extra bonus, I will show you a quick way to extract a value from another cell that is associated with the max value. In our case, that will be the name of the tallest person. For this, we will be using the classic INDEX MATCH formula and nest MAXIFS in the first argument of MATCH as the lookup value: =INDEX($A$2:$A$11, MATCH(MAXIFS($D$2:$D$11, $B$2:$B$11, G1, $C$2:$C$11, G2), $D$2:$D$11, 0)) The formula tells us that the name of the tallest basketball player in junior school is Liam: Excel MAXIFS with logical operators In situation when you need to evaluate numeric criteria, use logical operators such as: • greater than (>) • less than (<) • greater than or equal to (>=) • less than or equal to (<=) • not equal to (<>) The "equal to" operator (=) can be omitted in most cases. Usually, choosing an operator is not a problem, the trickiest part is to build criteria with the correct syntax. Here's how: • A logical operator followed by a number or text must be enclosed in double quotes like ">=14" or "<>running". • In case of a cell reference or another function, use the quotes to begin a string and an ampersand to concatenate the reference and finish the string off, e.g. ">"&B1 or "<"&TODAY(). To see how it works in practice, let's add the Age column (column C) to our sample table and find the maximum height among the boys aged between 13 and 14. This can be done with the following Criteria1: ">=13" Criteria2: "<=14" Because we compare the numbers in the same column, criteria_range in both cases is the same (C2:C11): =MAXIFS(D2:D11, C2:C11, ">=13", C2:C11, "<=14") If you do not want to hardcode the criteria in the formula, input them in separate cells (e.g. G1 and H1) and use the following syntax: =MAXIFS(D2:D11, C2:C11, ">="&G1, C2:C11, "<="&H1) The screenshot below shows the result: Aside from numbers, logical operators can also work with text criteria. In particular, the "not equal to" operator comes in handy when you wish to exclude something from your calculations. For example, to find the tallest student in all sports excluding volleyball, use the following formula: =MAXIFS(D2:D11, B2:B11, "<>volleyball") Or this one, where G1 is the excluded sport: =MAXIFS(D2:D11, B2:B11, "<>"&G1) MAXIFS formulas with wildcard characters (partial match) To evaluate a condition that contains a specific text or character, include one of the following wildcard character in your criteria: • Question mark (?) to match any single character. • Asterisk () to match any sequence of characters. For this example, let's find out the tallest guy in game sports. Because the names of all game sports in our dataset end with the word "ball", we include this word in the criteria and use an asterisk to match any previous characters: =MAXIFS(D2:D11, B2:B11, "ball") You can also type "ball" in some cell, e.g. G1, and concatenate the wildcard character with the cell reference: =MAXIFS(D2:D11, B2:B11, ""&G1) The result will look as follows: Get max value within a date range Because dates are stored as serial numbers in the internal Excel system, you work with the dates criteria in the same manner as you work with numbers. To illustrate this, we will replace the Age column with Date of Birth and try to work out the max height among the boys born in a particular year, say in 2004. To accomplish this task, we need to "filter" the birth dates that are greater than or equal to 1-Jan-2004 and less than or equal to 31-Dec-2004. When building your criteria, it is important that you provide the dates in the format that Excel can understand: =MAXIFS(D2:D11, C2:C11, ">=1-Jan-2004", C2:C11, "<=31-Dec-2004") =MAXIFS(D2:D11, C2:C11, ">=1/1/2004", C2:C11, "<=12/31/2004") To prevent misinterpretation, it makes sense to utilize the DATE function: =MAXIFS(D2:D11, C2:C11, ">="&DATE(2004,1,1), C2:C11, "<="&DATE(2004,12,31)) For this example, we will type the target year in G1, and then use the DATE function to supply the dates: =MAXIFS(D2:D11, C2:C11, ">="&DATE(G1,1,1), C2:C11, "<="&DATE(G1,12,31)) Note. Unlike numbers, dates should be enclosed in quotation marks when used in the criteria on their own. For example: =MAXIFS(D2:D11, C2:C11, "10/5/2005") Find maximum value based on multiple criteria with OR logic The Excel MAXIFS function is designed to test the conditions with the AND logic - i.e. it processes only those numbers in max_range for which all the criteria are TRUE. In some situations, however, you may need to evaluate the conditions with the OR logic - i.e. process all the numbers for which any of the specified criteria is TRUE. To make things easier to understand, please consider the following example. Supposing you want to find the maximin height of the guys who play either basketball or football. How would you do that? Using "basketball" as criteria1 and as "football" criteria2 won't work, because Excel would assume that both criteria should evaluate to TRUE. The solution is to make 2 separate MAXIFS formulas, one per each sport, and then use the good old MAX function to return a higher number: =MAX(MAXIFS(C2:C11, B2:B11, "basketball"), MAXIFS(C2:C11, B2:B11, "football")) The screenshot below shows this formula but with the criteria in predefined input cells, F1 and H1: Another way is to use a MAX IF formula with OR logic. 7 things to remember about Excel MAXIFS Below you will find a few remarks that will help to improve your formulas and avoid common errors. Some of these observations have already been discussed as tips and notes in our examples, but it might be helpful to get a short summary of what you've already learned: 1. The MAXIFS function in Excel can get the highest value based on one or multiple criteria. 2. By default, Excel MAXIFS works with the AND logic, i.e. returns the maximum number that meets all of the specified conditions. 3. For the function to work, the max range and criteria ranges must have the same size and shape. 4. The SUMIF function is case-insensitive, i.e. it does not recognize the letter case in text criteria. 5. When writing a MAXIFS formula for multiple cells, remember to lock the ranges with absolute cell references for the formula to copy correctly. 6. Mind the syntax of your criteria! Here are the main rules: □ When used on their own, text and dates should be enclosed in quotation marks, numbers and cell references should not. □ When a number, date or text is used with a logical operator, the whole expression must be enclosed in double quotes like ">=10"; cell references and other functions must be concatenated by using an ampersand like ">"&G1. 7. MAXIFS is only available in Excel 2019 and Excel for Office 365. In earlier versions, this function is not be available. That's how you can find the maximum value in Excel with conditions. I thank you for reading and hope to see you on our blog soon! You may also be interested in 36 comments 1. Hi there, I seem to not be having much luck with this method for a similar scenario - it seems to ignore my If criteria. I have a set of data for different Locations and Cases, for a given location and case I want to find the minimum force. Data is in Columns A:D, then my formula for finding the position of the minimum value is in I4 =MATCH(MINIFS(D:D,B:B,I2,C:C,I3),D:D,0) I have specified Case 3 and Location 2 therefore the min force should be item 17() however it is returning item 5(). Paste from spreadsheet below, sorry its not very clear but you cannot add screenshots to comments: Position Case Location Force 1 1 1 -716 Case 3 2 1 2 -619 Location 2 3 1 3 -860 Position of min 5 4 1 1 -1023 5* 1 2 -884 6 1 3 -1229 7 2 1 -842 8 2 2 -716 9 2 3 -752 10 2 1 -1203 11 2 2 -1023 12 2 3 -1074 13 3 1 -663 14 3 2 -716 15 3 3 -722 16 3 1 -947 17** 3 2 -1023 18 3 3 -1031 19 4 1 -650 20 4 2 -665 21 4 3 -685 22 4 1 -929 23 4 2 -950 24 4 3 -979 □ Hi! Based on your criteria, the MINIFS function finds the minimum value: -1023. MATCH function searches for this value in column D and finds it in cell D5. Since your table has a header row, the return value is the position number 5. For a correct search, use the criteria you used for the minimum value search in the MATCH formula as well. In addition, do not do a full-column Based on this information, the formula could be as follows: You can also find useful information in this article: Excel INDEX MATCH with multiple criteria - formula examples. You will get a result of 17. 2. HELLO, how to find , MAXIFS WITH Multiple max range and multiple sheets with one criteria □ Hello! To create a dynamic reference to more than one worksheet, you can use the INDIRECT function. If you use the MAXIFS function, you will get the maximum value by condition on each of the worksheets. To find the largest of these values, use the MAX function. Here is an example formula: =MAX(MAXIFS(INDIRECT({"Sheet1","Sheet2"}&"!D1:D100"), INDIRECT({"Sheet1","Sheet2"}&"!A1:A100"), "Criteria")) 3. Hello, Thank you for great explanation and examples all works fine but I found and issue with MAXIFS. Maybe this is an Excel at all problem. Please help. in column C:C I have long numbers but it cannot be as real number, actually it is a text, sometimes those numbers must contain zeroes before which is important, that's why it is text format, I am looking for max value in another column which is E:E - simple number value to find higher one or what ever it can be: My input to find is 323212505100042414163030 in cell I5 and this value exists 2 times in column C:C Formula which I use is as below (of course I know "=" isn't required but just in case for sure): =MAXIFS( E:E, C:C, "="&H5 ) Situation looks like never mind where in column E:E I put for example 99999 (which will be max value) it show me always those max value in any row. I found out Excel is changing input parameter from column C:C and probably search input (cell H5) to number scientific format and lost last digits. In this case it looks like below - all numbers is the same: That's why never mind where I put highest number it always show me first one because C:C contain the same value (after changing it to wrong number format and replacing last 9 digits to zeroes). If all of this Long ID values I change with letter prefix before it - problem not exists, Excel do not changing format internally just using it as a Text (example like Help please, how can I skip this bug or problem. Why Excel is doing this? I found out this 'conversion' problem after long time almost drop all my hairs. Thank you □ Hi! Excel has a limit of no more than 15 digits in a number. When you type a number that contains more than 15 digits into a cell in Microsoft Excel, Excel changes all of the digits after the fifteenth digit to zeros. You can try splitting a long number into 2 numbers. For example, 3232125051000 and 42446125030. Read more: Excel substring functions to extract text from cell. Double minus is used to convert text to a number. 4. I am trying to get the most frequent name in a list based off another column being "read". I have used this formula that will determine the most repeated text based of that column alone =INDEX (A2:A11,MODE.MULT(MATCH(A2:A11,A2:A11,0))) but I want it to only count the most repeated text only if column G within the same row has the text "read" . Column G is based of a dropdown list of the following: Read, Reading, TBR, Wishlist. □ Hi! To find the most frequent text in a column, use the COUNTIF function and select the position with the maximum value using the MAX function. In older versions of Excel, write this formula as an array formula. =INDEX($A$2:$A$20, MATCH(MAX(COUNTIF($A$2:$A$20,$A$2:$A$20)), COUNTIF($A$2:$A$20,$A$2:$A$20),0)) If you want to find the most frequent text with a criterion in another column, use the COUNTIFS function to count it. =INDEX($A$2:$A$20, MATCH(MAX(COUNTIFS($A$2:$A$20,$A$2:$A$20,B2:B20,"read")), COUNTIFS($A$2:$A$20,$A$2:$A$20,B2:B20,"read"),0)) 5. I am trying to calculate the longest waiting time for anyone under 5 years (0 - 4 Years) and then for anyone 5 years or older (5 - 17 Years). The under 5 years worked fine: =MAX(IF(Table1[AGE]"4", Table1[WEEKS WAITED FROM REFERRAL DATE])) Can anyone help? □ Hi! It's very hard to understand a formula that uses unique references to your data that I don't have. I will try to assume that the formula could look something like this: =MAXIFS(D2:D11, C2:C11, ">=5", C2:C11, "<=17") =MAX(IF((C2:C11>=5)*(C2:C11<=17), D2:D11)) C - age. D - weeks. For more information, read article above and this guide: MAX IF in Excel to get highest value with conditions. 6. 1 0 0 5 2 -10 0 5 6 3 -20 0 6 7 4 -30 0 7 8 5 -5 -50 9 10 6 -15 -50 10 11 7 -25 -50 11 12 8 -35 -50 12 9 0 -100 13 10 -10 -100 13 14 11 -20 -100 14 15 12 -30 -100 15 16 13 -5 -150 14 -15 -150 15 -25 -150 16 -35 -150 these are coordinates and answer i want to find below nearest left and right coordinates for example for 2 , left nearest below is 6 and right is 5 can i have equation .thank you. □ Hi! Your task is not completely clear to me. Explain how you determine the nearest numbers. 6 and 5 meet several times. 7. I'm trying to use the formula maxifs(Sheet2!$A$2:A$21-A2, Sheet2!$A$2:A$21-A2, "<=0") but it doesn't work. Excel says there is a problem with this formula. □ Hi! The problem is Sheet2!$A$2:A$21-A2. This is an invalid expression. I can't advise you as I don't know what you wanted to do. The MAXIFS function argument is a range of values, but not a formula. Please read the above article carefully. ☆ Thanks for your reply. I'm trying the find the closest date to a specific date in cell A2 from the range Sheet2!A2:A21. ○ Hi! To get the closest date to a specific date in cell A2, use MIN function: =INDEX(A3:A21, MATCH(MIN(ABS(A3:A21-A2)), ABS(A3:A21-A2),0)) 8. I need a formula,To filter data for a list of students from different sections to find the highest score per section and give the name of the student with the grade and id. I create this formula : But the problem is that its not giving me the highest grade persection , its giving me the highest grade in all the sections. □ Hi! To find the maximum score for a condition, use the MAXIFS function instead of MAX. Read the recommendations above carefully. 9. If I need to find the highest grade per grade and per subject, what formula I can use ? Thank you. □ Hi! Please re-check the article above since it covers your task. ☆ If I have a list of students from different sections I need to find the highest score per section for two different subjects and give the name of the student with the grade. What formula? Can you help me pls. Thank you ○ Hi! To find the highest score, use recommendations from this article above. To find a name of the student with this highest score, use INDEX MATCH function. I don't have your data so I can't give you the formula. Follow the guidelines in these articles. 10. Hi! I'm trying to get the max Week value of one (1), based on the max values of both Year and Month (2029 and 1), for the three records below. The formula works as expected when there is only one range/criteria combination but returns the value zero (0), which means a record can't be found, as shown in the formula below. Could you please provide a corrected MAXIFS and an explanation as to why my formula isn't working as expected. =MAXIFS ([Week], [Year], MAX([Year]), [Month], MAX([Month])) □ Hello! The MAXIFS formula returns 0 because the maximum year (2029) and maximum month (12) are on different rows. Use instruction - MAX IF formula with multiple criteria. Try this formula - I hope my advice will help you solve your task. ☆ Thanks for your prompt response and accurate formula! I thought that that was what was going on with my formula code. If Microsoft doesn't have a function to does what I wanted to do, maybe they will create a DMaxIfs function that does that. Hopefully they are reading this. Ultimately in the end, I needed something usable as a VBA code snippet. This morning I had a rethink and all I really needed was the Year and Week fields to get the max value for Week. If I hadn't figured it out after frying too many brain cells, I would have definitely used your formula code and then wrote VBA code to that cell address. Here's what I came up with for Excel and VBA. Hopefully this helps someone down the road. Worksheet formula... VBA code snippet... Sub TablesAndFormulaAddressing() Dim FY, RW As Integer ' Get Max values for records using named ranges ' TableRangeName not needed since named ranges are unique to the workbook FY = WorksheetFunction.Max([YearRangeName]) RW = WorksheetFunction.MaxIfs([WeekRangeName], [YearRangeName], WorksheetFunction.Max([YearRangeName)) End Sub 11. What would the MAXIFS formula for the following example below look like? 1 Absent ("0" or "1" is displayed using [=IF((AND('Severity Rating'.I3="No",'Severity Rating'.I4="No",'Severity Rating'.I15="No")),"1","0")]) 2 Mild ("2" is displayed using [=IF('Severity Rating'.I3="Yes";"2";"0")]) 3 Moderate ("3" is displayed using [=IF('Severity Rating'.I4="Yes";"3";"0")]) 4 Severe ("4" is displayed using [=IF('Severity Rating'.I5="Yes";"4";"0")]) I want to create a function to identify the max number (4 in this case) and sum all the severity ratings in each category (the above is just one category). □ Hello! Your formulas return text, not numbers. Instead of "1" use 1 and so on. To find the sum of the ratings in a given category, use the SUMIF function instruction. ☆ That worked!! Thank you! 12. If i want the value to be returned for multiple sheets how this formula can be changed? □ Hi! What formula are you talking about and what data do you want to return? 13. Is it possible to find MAX difference (in numbers) between 2 columns i.e. In Jan 2022 Mark's number was 170 and Philip's number was 172. Now in Feb 2022 Mark's number is 200 and Philip's number is 220. I want a formula to find the max difference between Jan & Feb numbers like here Philip's numbers increased by 48 numbers (highest between 2) and it should return Philip's name. □ Hi! You have not specified how your data is written. For example, the formula might look like this: =IF(A3-A2 > B3-B2,A1,B1) If this is not what you wanted, please describe the problem in more detail. 14. Index / Maxifs will return the first record found with matching value ... If Ethan height was 171.. Maxifs formula becomes irrelevant as Index will return Ethan's name isn't it ? 15. Trying to figure out how to write a MAX IF condition to cover all scenarios. If a calculated value is a negative number, then default to '0'. If calculated value is a positive number, then display a whole number that doesn't round up ('20.8' should display as '20'). The formulas below work independently, but haven't been able to blend the two into a single formula. □ Hello! If I understand your task correctly, the following formula should work for you: You can learn more about ROUNDDOWN function in Excel in this article on our blog. 16. Thank you for the section on Find maximum value based on multiple criteria with OR logic. Post a comment	{"url":"https://www.ablebits.com/office-addins-blog/maxifs-function-excel-multiple-criteria/","timestamp":"2024-11-04T02:39:43Z","content_type":"text/html","content_length":"190519","record_id":"<urn:uuid:3fb2a2d1-b726-451d-a371-17d28ef9ceab>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00612.warc.gz"}
Adding Four Digit Numbers Without Regrouping Worksheets 2024 - NumbersWorksheets.com Adding Four Digit Numbers Without Regrouping Worksheets Adding Four Digit Numbers Without Regrouping Worksheets – Sophisticated add-on drills are fantastic strategies to introduce college students to algebra concepts. These drills have alternatives for 1-min, three-minute, and several-minute drills with personalized varieties of twenty to hundred or so difficulties. Moreover, the drills may be found in a horizontal format, with figures from to 99. The best part is that these drills can be personalized to each student’s ability level. Here are some more superior inclusion drills: Matter ahead by one particular Relying on can be a helpful strategy for creating amount reality fluency. Count on a number by addingtwo and one, or three. By way of example, 5 plus two equates to 15, and so on. Depending on a amount with the help of you might make the very same final result for small and large phone numbers. These addition worksheets incorporate training on counting on a variety with equally hands along with the quantity line. Adding Four Digit Numbers Without Regrouping Worksheets. Exercise multiple-digit addition by using a number collection Available amount line is fantastic versions for addition and place importance. Within a earlier submit we discussed the various mental tactics students are able to use to add amounts. Utilizing a number line is a wonderful way to papers most of these techniques. In this posting we are going to investigate one way to exercise multiple-digit supplement using a variety collection. Listed here are about three techniques: Practice adding doubles The exercise introducing increases with addition phone numbers worksheet may be used to assist youngsters produce the thought of a increases fact. A doubles fact is when the same number is added more than once. For example, if Elsa had four headbands and Gretta had five, they both have two doubles. Students can develop a stronger understanding of doubles and gain the fluency required to add single digit numbers, by practicing doubles with this worksheet. Training including fractions A Training adding fractions with supplement phone numbers worksheet can be a helpful instrument to produce your child’s simple knowledge of fractions. These worksheets deal with numerous principles relevant to fractions, including assessing and getting fractions. Additionally, they supply useful problem-fixing strategies. You are able to obtain these worksheets at no cost in Pdf file structure. The first task is to make certain your child recognizes the rules and symbols associated with fractions. Process adding fractions with a number series In relation to training incorporating fractions with a quantity collection, college students can make use of a small percentage spot worth mat or a variety series for mixed figures. These help with complementing small fraction equations to the remedies. The location benefit mats could have a amount of good examples, using the formula created at the top. Individuals may then select the solution they need by punching pockets close to each and every decision. As soon as they have selected the right answer, the student can draw a cue near the solution. Gallery of Adding Four Digit Numbers Without Regrouping Worksheets Three Digit Addition With Regrouping Worksheets 99Worksheets Add Within 10000 Without Regrouping Worksheets For 4th Graders Online 4 Digit Addition With Regrouping Worksheets Worksheet Hero Leave a Comment	{"url":"https://numbersworksheet.com/adding-four-digit-numbers-without-regrouping-worksheets/","timestamp":"2024-11-07T22:41:48Z","content_type":"text/html","content_length":"55330","record_id":"<urn:uuid:6b0a225f-c163-418c-b29f-fb255cf543cc>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00306.warc.gz"}
Mineralmarknaden, Tema - Sveriges geologiska undersökning betasönderfall - NRCFs frågelåda i fysik - nyckelord 16. The atomic mass of fluorine is 18.9984 amu and that of copper characteristics of real nuclei, then the most tightly bound nuclide would have A– 58.3, 2~26.6. It seems that nickel has a closed proton shell, that is, its isotopes are more tightly bound was done using data of the time: the neutr the sense that the energy it takes to remove one proton or neutron is. 7 Cross bombardment means that the same nuclide and its character- istic decay chain is observed in [23], from calculations by V. Za- grebaev (solid) (Number of Neutrons). 126. Type of. Most stable nuclides have both even Z and even N called "even-even" nuclides. We can understand this in terms of Pauli's exclusion principle. Neutrons and protons are distinguishable fermions; hence they separately obey the exclusion principle. A nuclide is a specific combination of protons and neutrons, denoted by [latex]_{Z}^{A}{\text{X}}_{N}\\[/latex] or simply A X, Z is the number of protons or atomic number, X is the symbol for the element, N is the number of neutrons, and A is the mass number or the total number of protons and neutrons, A = N + Z. Experiment 10 Nuclear Radiation (The Chart of Nuclides) Objective: To become familiar with the use of the Chart of Nuclides . Equipment: The Chart of Nuclides . Theory:. 1. Radioaktivitetens Matematiska Fysik enligt TNED Exotic. Nuclides. 23A1, 23Si, 22A1 and 77Rb number treats the proton and neutron as different states of a single particle, the nucleon, with VO= 54.0-0.32 EP i- 0.4 Z A“~{3+ 24.0 (N - Z 18 Jun 2020 The atomic mass calculator takes the number of protons and neutrons in an atom, and displays the atomic mass in atomic mass units and The energy needed to remove one neutron or one proton, respectively, from a nuclide. Thåström Ny Skiva 2017 - Trouw Plan Both protons and neutrons have a mass of 1, while electrons have almost no 25 Feb 2021 Nuclide, , also called nuclear species, species of atom as characterized by the number of protons, the number of neutrons, and the energy state thousand [11], and are arranged in the Chart of Nuclides (see Fig.2.2). Here, each isotope is drawn the sense that the energy it takes to remove one proton or neutron is. 7 [23], from calculations by V. Za- grebaev (solid) (Number of Neutrons). 126. Type of. Decay. +. 9. 4. 51. 2. Magnus svensson jändelskolan Nuclides (X) are the nuclei of atoms of a specific isotope. equals or exceeds the number of protons in stable nuclei to help reduce repulsive forces. A nuclide refers to a nucleus with a speciﬁc number of protons and neutrons, while the term, nucleons, refers to protons and neutrons together. The nuclide symbol (Sy) is written as follows: Z ASy. So, 6 12C has Z=6 protons and the neutrons would be A-Z = 12-6= 6. Debatt artikel mall siemens ag headquartersutbildning naprapaterhandelsbanken byta bankpostnord tullavgift japanpsykiatriambulanse bergenproduction planner jobs xi och karl: Topics by WorldWideScience.org Solved: The nuclide of barium whose neutron-proton ratio is 1.25. Z = A = N = By signing up, you'll get thousands of step-by-step solutions to your A nuclide is a specific combination of protons and neutrons, denoted by [latex]_{Z}^{A}{\text{X}}_{N}\\[/latex] or simply A X, Z is the number of protons or atomic number, X is the symbol for the element, N is the number of neutrons, and A is the mass number or the total number of protons and neutrons, A = N + Z. Nuclides (X) are the nuclei of atoms of a specific isotope. They are characterised by the number of positively charged protons (Z), neutrons (N) and the energy state of the nucleus. In terms of mass (A) and atomic number (Z) a nuclide is denoted as: (2.1) X N Z A Write the symbol (in the form _{Z}^{A} \mathrm{X} ) for the nuclide with 38 protons and 50 neutrons and identify the element. Eriksberg vardcentralkarl andersson wife A - Bok- och biblioteksväsen - Kungliga biblioteket of neutrons"))) (constant Z, variable N) corresponds to the isotopes of a chemical element. With coordinates (N, Z) the coloured nuclide box shows data for experimentally observed nuclides with N neutrons and Z protons in the nucleus. A nuclide box contains the nuclide name, mass number (total number of neutrons and protons N+Z), the half-life Nuclide symbol is expressed as above, (where,z is the atomic number and A is the mass number) now, z=total number of protons=11 and a= total number of protons + total number of neutrons= (11+11) or 22 Solved: The nuclide of barium whose neutron-proton ratio is 1.25. Z = A = N = By signing up, you'll get thousands of step-by-step solutions to your Calculate the total number of nucleons (protons and neutrons) in the nuclide.	{"url":"https://lonbilox.netlify.app/52909/58774","timestamp":"2024-11-04T12:19:46Z","content_type":"text/html","content_length":"10955","record_id":"<urn:uuid:c6e3a9ea-6eaa-4b36-b7ac-f96dddf1c639>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00521.warc.gz"}
An Alternative Explanation of the Pioneer Anomaly 01-31-2014, 12:18 PM #1 Join Date Jun 2007 The deep space probes, Pioneers 10 and 11 where launched in 1972 and 1973 respectively. As is well known, anomalous behavior was discovered in the early 1980s showing an unexplained slowing of the probes in their travel away from the Sun. It has taken till just recently, in early 2012, to supposedly solve the mystery. That’s about 30 years from the time the anomalous behavior was first discovered. The claim is that it is caused by onboard thermal radiation in the direction of travel. When you look into this matter you will quickly find that it involves too many variables to discuss in detail on the message board. I only want to say that I don’t accept the explanation given and offer another possibility. Based on the principles of the Cause and Effect Theory of Light Propagation, the predicted Doppler Effect is more redshifted than the Doppler Effect of Special Relativity. This means that for the same Doppler redshift in the direction of recession, the Cause and Effect Theory gives a slower recession velocity. This difference in predicted velocity between the Cause and Effect Theory and Special Relativity is very small for the velocities involved in the Pioneer Space Probes. By modeling the behavior in Mathcad the difference in velocity turns out to be in the area needed to explain the slowing that occurs. I am not sure if I will write a paper on this or not. Until they show by experiment that my Cause and Effect Theory is correct, no one is going to believe my findings on the Pioneer Anomaly. Joe, I believe that the proper way to do this would be to apply your formula to the data and show that it matches experiment, and then show that general relativity does not. You have already explained yourself in your article and so that is not necessary again. It is always best to first apply your work to numerical experimental results if they are available. It is OK to sumarise the discussion relating to general relativity (or Special) and just give a reference. --- Best, karl Last edited by sarge; 02-01-2014 at 08:36 AM. I wish it were that easy. It took 30 years involving a team of up to 50 scientists analyzing 50 years of telemetry data from Pioneers 10 and 11 to reach the point where they think they solved the mystery. And I doubt that General Relativity played any real role in the Pioneer missions. I think the only reason General Relativity came into it was in an attempt to explain the anomaly because Newton’s law of gravity didn’t. Even my solution does not involve General Relativity. And for that matter, it doesn’t put Newton’s law of gravity at risk either. The problem has to do with the Doppler tracking used in the missions. In using Doppler tracking they most likely used the Special Relativity formulas for the Doppler Effect. In so doing they got a slightly wrong value for the velocity of the space probes. This results in the space probes being slightly closer to the Sun than their calculations predict. That’s all there is to it. If they were to use my Cause and Effect Doppler formula, the velocities would agree with the distances the probes really traveled, and thus, there is no anomaly. This is easy for me to say, and it is easy for me to demonstrate in a mathematical model, but it would not be easy for me to prove because my model simply demonstrates the possibility. To really cover it in detail it would take a team of scientists who have all of the data for the mission to work it out. I.e., they would do what I did, using the Special Relativity Doppler Effect and my Cause and Effect Doppler Effect, but in much greater detail. Assuming I am right, this does not affect General Relativity or Newton’s Gravitational Law. But it basically destroys Special Relativity. Joe: Assuming I am right, this does not affect General Relativity or Newton’s Gravitational Law. But it basically destroys Special Relativity cinci: You cannot cannot destroy SR without destroying GR. They are the same theory. Hey cinci -- can you give me a pointer to a discussion of the Pioneers 10 and 11 problem. I am not interested in an analysis of the theory, but in the observations and an explanation of why there is a problem. thanks, sarge After giving it a lot of thought, I have to conclude that you are right. The question remains, however, as to whether or not General Relativity can survive in modified form. In the mathematical analysis I conducted, I did not use General Relativity in any way. The analysis was based on the difference between the Special Relativity Doppler Effect, and my Cause and Effect Doppler Effect as applied to Newton’s Law of Gravitation. Before discussing the breakthrough that happened today, in my next message, I want to first relate the following information that I intended to post yesterday but didn’t because of some concerns about issues not yet fully understood: As is well known, the Pioneer Anomaly involves an anomalous acceleration of 8.74x10^-10 m/s^2 towards the Sun. Upon exploring my solution to the Pioneer Anomaly in 10 or more different mathematical models, I have made many new discoveries involving my Cause and Effect Theory of Light Propagation and its Doppler Effect vs the Doppler Effect of Special Relativity. It’s encouraging and amazing how the modeled behavior correlates with what they discovered involving the Pioneers 10 and 11 probes’ unaccounted for acceleration toward the Sun. I was finally able to select a model that was most suited to showing the inter-relations of the two different Doppler Effects and the discovered anomaly and found that it appears to agree with everything they have found involving the anomaly. Although the anomalous acceleration is there from the very beginning, I now know why it was not discovered until many years later when Pioneer 10 was over 15 AU from the Sun. Actually, one of the most credible papers I found on the subject states that the anomaly was precisely measured between 20 and 70 AU out from the Sun. And it is believed that it starts somewhere between 15 to 30 AU out. My model agrees with these claims with the exception that the anomaly is there from the beginning but not detectable for a variety of reasons early on. This was one of many issues that I was still struggling with until today in trying to understand it all. In my next message that I will post as soon as I write it up, I will cover the incredible After struggling with all of the various mathematical models I developed, and finally selecting the one that seemed most suitable to what I was trying to accomplish, I was able to narrow down the number of variables and constants to a total of 12. And, that obviously is the problem I’m dealing with in trying to relate everything that is going on. As is well known, once the number of variables exceeds three, the problem becomes difficult. To give some understanding of what I’m talking about, let me relate an occurrence that happened in Mathcad today. Mathcad is the computer mathematical application that I use to do my work. About once every few years, when I’m working on an extremely difficult problem, I encounter this problem in Mathcad. Every now and then I will take a very complicated equation I derived and see if Mathcad can solve it for a particular variable. Typically it can but I still have to clean it up a little, so, as a practice, I tend to solve such equations myself, albeit, still in the Mathcad application. And, sometimes it can’t, in which case I have no choice but to do it myself, or to simplify it more and try again. In today’s case I did it all myself and simply wanted to see if Mathcad could solve for a different variable. The result was an equation many levels deep (i.e. fractions on top of fractions on top of fractions, etc.) and 35 worksheets in length, from left to right in 10 point characters. If I were to shrink this to the size of a single worksheet, the characters would be so microscopic it would be unreadable. Yet, it is most likely valid, as I have learned from past experience. I.e. you can give it an input and it will give a correct result. I am relating this only to give some idea of the complexity of the math problem I am dealing with. Now for the breakthrough. The solution I came up with involves relating the Special Relativity Doppler factor, and its resultant velocity relating to the distance traveled to my Cause and Effect Doppler factor, and its corresponding velocity and distance for the same Doppler Effect. Then, I also had to factor in the anomalous acceleration and its resultant distance, and other constants and variables relating to the units of measurements and such, for example, seconds in a year, the number of years, distance in terms of astronomical units, and so on. I had done this many times before in the past week involving this problem, in different mathematical models, and got promising results but with a wrinkle here or there. That was not the case this time. The formula I came up with agrees exactly with the measured anomaly for all velocities, time periods, and astronomical units. It was an absolutely incredible finding. My only concern was, is it because I unintentionally caused it to give such results in the method of derivation used? In going over it I cannot find any reason to believe that that is the case and the formula is not valid. I will therefore write a paper on it and during the process of explaining it verify its validity or lack thereof. If I am correct on this, it will give incredible evidential support to my Cause and Effect Theory of Light Propagation and all that that implies. joe -- want to expand on this a little for me? I don't know what experimental anomaly yoy are reffering to! The formula I came up with agrees exactly with the measured anomaly for all velocities, time periods, and astronomical units. It was an absolutely incredible finding. best, ----sarge Back in 1972 and 1973 the Pioneer 10 and 11 space probes were launched. Their mission was to reach the outer planets of our solar system and continue on until they left our solar system and entered our Milky Way galaxy. They have both succeeded in doing that. Before they, did, however, an unaccounted for very slight constant acceleration in the direction of the Sun was detected. In other words, both probes were traveling away from the Sun at slightly slower velocities then predicted by theory. And, subsequently, the distances they were at for any given period were slightly less than theory predicts. The slightly greater acceleration toward the Sun was determined to be on the order of 8.74 x 10^-10 m/s^2. (Note: That’s 10 to the minus 10) They feel that this effect does not really begin until the probes are 15 to 30 AU out, with 1 AU being the distance of the Earth to the Sun. Other than the value I gave for the anomalous rate of acceleration, and standard units of value, don’t hold me to any other quoted values since they vary depending on which paper you read and when and by whom it was written. Anyhow, the anomalous acceleration is slightly more than predicted by Newton’s Law of gravity and does not agree with General Relativity either. So, the question became, does this mean both theories are not entirely correct, or is it something else? The problem was supposedly solved in 2012 when it was determined that the anomalous acceleration was cause by thermal radiation from each of the probes in a direction opposite to the direction of travel. The thermal radiation is caused by the thermoelectric generators on the space probes. In my opinion, however, this explanation is inconsistent with the science involving such radiation, which incidentally goes all the way back to Maxwell. I will give Astrophysicist Slava G. Turyshev of NASA’s Jet Propulsion Laboratory in Pasadena, Calif. credit for his honesty in reporting on this. He does acknowledge that there is an uncertainty of about 18 percent in the findings. I can provide very good arguments against the thermal radiation theory. And I can also present very good arguments in favor of my cause and effect theory of light propagation. So far I have come up with answers as to why it was not detected earlier and why it appears to be constant regardless of distance from the Sun. But, I have to be up front just as Turyshev was. I have so far been unable to account for all of the acceleration. But, I believe that is simply because of all the variables involved and the difficulty of mathematically relating them in a single model. The NASA personnel were working with real data covering many years of the missions. I am working with general information about what they found and relating it to the principles of my theory. The problem, however, is, that they were using Newton’s theory and Einstein’s theory in conducting the Pioneer missions. So, I have to translate everything into terms of my theory in order to apply it to the solution. Nonetheless, what I’ve come up with is very impressive. As a result, I learned things about my own theory that I didn’t really know before. When I finally get all of the details worked out, I will write a paper on it showing what I found. joe --how were their positions determined and transmitted back to earth? ---sarge They used Doppler tracking. Here is a link to an article on how it is done. I’ve reset the link to take you directly to the beginning of the article. The article covers the entire project and most likely has answers to any other questions you might have. Refer to the Table of Contents at the right. Also take notice of the page arrows at the bottom right of the window. I can now show convincingly that the Pioneer Anomaly verifies the principle that light takes on the speed of the source and thereby verifies my Cause and Effect Theory of Light Propagation. By using the Pioneer project information in mathematical equations that convert the results given for Special Relativity to those given by the Cause and Effect Theory I can account for over 90 percent of the Pioneer Anomaly. That is within less than 10 percent of the anomaly as compared to the recently clamed solution by Slava G. Turyshev and Viktor T. Toth of JPL that is only within 18 percent of the anomaly. It has not been easy to define what is going on mathematically and the math might be a little difficult to grasp at first, but it supports my theory from every perspective that I have examined. This includes Doppler analysis and bouncing pulses off the space probes in order to determine their speed and distance relative to the Sun. I am getting close to the point where I can begin a paper on the subject. It has not been easy, but I’m getting there, and the results are very encouraging. Joe: I can now show convincingly that the Pioneer Anomaly verifies the principle that light takes on the speed of the source and thereby verifies my Cause and Effect Theory of Light Propagation. By using the Pioneer project information in mathematical equations that convert the results given for Special Relativity to those given by the Cause and Effect Theory I can account for over 90 percent of the Pioneer Anomaly. That is within less than 10 percent of the anomaly as compared to the recently clamed solution by Slava G. Turyshev and Viktor T. Toth of JPL that is only within 18 percent of the anomaly. It has not been easy to define what is going on mathematically and the math might be a little difficult to grasp at first, but it supports my theory from every perspective that I have examined. This includes Doppler analysis and bouncing pulses off the space probes in order to determine their speed and distance relative to the Sun. I am getting close to the point where I can begin a paper on the subject. It has not been easy, but I’m getting there, and the results are very encouraging. cinci: If Toth explained 82% of it (100 - 18) and you explained 90%, the two of you have explained 172% of it. So either you or Toth is wrong. Given that the title of this string is, “An Alternative Explanation of the Pioneer Anomaly” you are correct in your conclusion. Actually, it was Slava G. Turyshev and Viktor T. Toth who explained 82%. So, either they are or I am wrong. But that’s not the whole story. It’s also possible that either they are or I am right. It depends on how you look at it. The clue is in the word alternative. It means instead of; not in addition to. To avoid further confusion, I is another form of you (by you, I mean them, not you) and they is a substitute term for Slava G. Turyshev and Viktor T. Toth. Other than that, I hope you’re feeling well. Between digging myself out from all of the consecutive snowstorms we been having lately, I have managed to refine the mathematical model I have been using to analyze the Pioneer anomaly in terms of the Cause and Effect Principles of Light Propagation. I am very close to starting a paper on my findings and will do so as soon as Mother Nature gives me the okay. While dealing with endless interruptions due to weather, network router and printer upgrades, I have finally gotten past it all and finished refining the mathematical model I needed to begin a paper. I can now show that the cause and effect principles of light propagation produce results that fall right in the middle of the data that supports the Pioneer anomaly. I will now focus my attention on the process of writing a paper on my findings. The article above give a more detailed explanation than what I had read previously. I was not aware that the effect was diminishing with time. Will your analysis predict this reduction of the effect with time? Thanks for posting the PhysicsWorld link. It actually adds more credibility to my solution. As a matter of fact the effect does diminish with time. And that was a concern of mine until now. Actually the effect diminishes with distance and since time passes as the distance increases, the effect decreases with time. I did a quick check on my current mathematical model and found the effect would reduce to ½ in 19 years. It all depends on many variables and could be more or less depending on what they are. But I think that’s close enough to further validate my findings. In the process of working on my new paper, that is now halfway completed, I made a new discovery involving what was intended for the final half. Like everything else involving this paper, the new discovery is enlightening but also very difficult to fully comprehend in relation to its total significance with regard to the principles being covered. As with the first half of the subject matter which I now know is correct, I have to spend time investigating this new discovery in order to properly cover it in the paper. It never ceases to surprise me as to how things are always much more involved than the present theories indicate. Joe -- how about listing the principles being covered for an old dunce like me! --- sarge I can’t do that without giving away what I’m working on. But I can outline the problem in general. 1. The Pioneer spacecraft have been traveling away from the center of the solar system for over 40 years now. 2. About 8 years into their journeys, it was determined that the distances they were at, were slightly less than they should be. 3. The question then is, why? 4. Is it something in the systematic process that’s in error? 5. Is it something in the spacecraft causing it? 6. Is it because the theory of gravity is not quite correct? 7. Is it something else entirely? Their answer: It is something in the spacecraft that is causing it. My answer: It is something in Special Relativity that is causing it. In summary; the answer they came up with is inconsistent with the known principles of science. That leaves all of the other reasons to pick from. But you have to show a valid reason for your selection. I can do that in my selection of Special Relativity vs. the findings of my Cause and Effect Theory of Light Propagation. If you go by my theory, there is no anomaly. The spacecraft are right where they should be. Joe: In summary; the answer they came up with is inconsistent with the known principles of science. cinci: Scientists don't come up with answers that are not consistent with the known principles of science. They might come up with wrong answers, but they would not contradict known principles of The answers they came up with were very compelling. Particularly the rate of change of the anomaly being consistent with nuclear decay of their source and the thermocouple electrical source. Joe: That leaves all of the other reasons to pick from. But you have to show a valid reason for your selection. I can do that in my selection of Special Relativity vs. the findings of my Cause and Effect Theory of Light Propagation. If you go by my theory, there is no anomaly. The spacecraft are right where they should be. cinci: Since your theory contradicts relativity, it is in conflict with the known principles of science. to cinci: I am disposed to make the following comments to you! Joe: In summary; the answer they came up with is inconsistent with the known principles of science. cinci: Scientists don't come up with answers that are not consistent with the known principles of science. They might come up with wrong answers, but they would not contradict known principles of comment{There are known principles of science that are both incorrect and others that are correct that are known! I am referring directly to misusage of SR and to the ignorance of most scientists about the properties of light as shown in GR and ignored by most. GR is right!} The answers they came up with were very compelling. Particularly the rate of change of the anomaly being consistent with nuclear decay of their source and the thermocouple electrical source. comment{ probably true!}. Joe: That leaves all of the other reasons to pick from. But you have to show a valid reason for your selection. I can do that in my selection of Special Relativity vs. the findings of my Cause and Effect Theory of Light Propagation. If you go by my theory, there is no anomaly. The spacecraft are right where they should be. cinci: Since your theory contradicts relativity, it is in conflict with the known principles of science. comment{ since you do not understand relativity correctly yourself you are not qualified to make this comment to Joe, correct or incorrect!}. best, sarge I’m not sure of what you’re talking about. The following is from the article you gave the link to in message # 18: Joseph In 2011 a team led by Slava Turyshev of the Jet Propulsion Laboratory in California – and including Viktor Toth, Jordan Ellis and Craig Markwardt – showed that the magnitude of the acceleration is decreasing exponentially with time. Given that for both craft electricity is supplied by a radioisotope thermoelectric generator (RTGs) powered by the heat given off by the radioactive decay of plutonium – an energy source that decays exponentially with time – Turyshev and others suggested that the extra acceleration could be caused by thermal radiation being emitted from the craft in a preferred direction. The problem with that explanation, however, is that the acceleration of the spacecraft is decaying exponentially with a half-life of about 27 years, whereas the half-life of plutonium-238 is 88 years. So to see if thermal emissions really are driving the anomaly, Turyshev, Toth and Ellis joined forces with three other researchers – Gary Kinsella, Siu-Chun Lee and Shing Lok – to create a detailed computer simulation of the thermal properties of the spacecraft and the directions in which key components emit thermal radiation. From the article I posted: The problem with that explanation, however, is that the acceleration of the spacecraft is decaying exponentially with a half-life of about 27 years, whereas the half-life of plutonium-238 is 88 years. So to see if thermal emissions really are driving the anomaly, Turyshev, Toth and Ellis joined forces with three other researchers – Gary Kinsella, Siu-Chun Lee and Shing Lok – to create a detailed computer simulation of the thermal properties of the spacecraft and the directions in which key components emit thermal radiation. Efficient acceleration The simulation reveals that the two main sources of thermal emissions on the spacecraft are the RTG itself and the scientific instruments that it powers. These instruments, which are mostly mounted on the back of the spacecraft, face away from the Sun and, according to the simulations, their thermal emissions have a relatively high efficiency of accelerating the spacecraft towards the Sun. The RTG, in contrast, is mounted to one side of the main body of the spacecraft and emits thermal radiation much more evenly in all directions. The research suggests that knowing the relative contributions of the RTG and the instruments to the anomalous acceleration is key to understanding why the observed decrease in the anomalous acceleration is faster than the decay of plutonium-238. According to Turyshev, the thermocouples at the heart of the RTGs become progressively less efficient at converting heat to electricity – and that this decay occurs with a half-life that is somewhat shorter than 88 years. As the thermocouples deteriorate, less electrical energy is supplied to the instruments, which means that the anomalous acceleration drops faster than expected from radioactive decay alone. Although more heat is dissipated by the RTG as time progresses, this has little effect on the motion of the I have one other question. Since what they are proposing appears to be in violation of the known laws of physics, do the specifically address that issue? The paper I have been working on is almost completed. I am in the process of optimizing the mathematical model I will include in the form of appendixes. While working on it, however, I experienced more unusual results regarding some equations I previously derived for use in the paper. These are fairly simple equations that properly perform in the application. It is in my attempts to use them to address the half-life of the anomalous acceleration that the issue comes up. Specifically, when I attempt to solve the equation for a particular variable, I get an equation about a dozen levels high and from over four to seven pages in length. There is no doubt in my mind that the resulting equations are valid, but they are impractical for use in a theoretical application. I can’t help but believe that this is one of the reasons the Pioneer Anomaly has been so difficult to resolve. I.e., it contains relationships that are affected by too many interrelated variables. I am happy to say that I have finally solved the half-life problem. This doesn’t mean I found a solution to the four to seven page equations mentioned in the last post. What it does mean is that the half-life equation inherently exists within the equations already derived for use in the model. Specifically, the half-life equation already exists in the form of an equation previously derived for an entirely different purpose. It automatically provides the half-life of the acceleration when it gives the time during which the acceleration occurs. Moreover, with one additional step, it provides any fractional life required. E.g., 1/3 life, 1/4 life, 1/5 life, etc. It is this dual nature of the formula in addition to its complete depth of use that was so confusing to understand. Once the true multipurpose nature of the formula is finally understood, it validates my earlier finding that the half-life of the so-called anomalous acceleration, like everything else in the related findings, falls right in the middle of the data covering the Pioneer Given that this was the last outstanding problem I was dealing with, I should now be able to finish the paper without further issues. The paper will, however, be a bit longer than I had intended. But that is a small price to pay for success. Joe: I have one other question. Since what they are proposing appears to be in violation of the known laws of physics, do they specifically address that issue? cinci: I'm not sure what you're asking? They are addressing what appears to be a violation of the law of gravity. cinci: Since your theory contradicts relativity, it is in conflict with the known principles of science. sarge: comment{ since you do not understand relativity correctly yourself you are not qualified to make this comment to Joe, correct or incorrect!}. cinci: Well I may not understand the theory of relativity as you say; but I can read, and all the physics textbooks and all the well-known physicists I have read say it is a known principle of science. So until further notice I'm going to believe them instead of you and Joe. I’m talking about thermal recoil force in the vacuum of space. It’s in violation of the known laws of physics. When you go to research this, be informed that it is still considered to this day to be one of the most misunderstood principles of physics. I believe the JPL people involved in the proposed solution were well aware of what I am referring to and that is one of the main reasons it took them so long to place the blame on thermal recoil force. My paper on the Pioneer Anomaly is now complete and in the proofreading stage. It’s longer than I wanted it to be, 19 pages, four of which are Appendixes of the math models used. 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b444 mod02 Homework Assignment #2 - CourseMerits Question Details Normal b444 mod02 Homework Assignment #2 • From Mathematics, Probability • Due on 19 Jan, 2015 12:26:00 • Asked On 17 Jan, 2015 01:00:08 • Due date has already passed, but you can still post solutions. Question posted by • Rating : 0 • Grade : No Rating • Questions : 23 • Solutions : 0 • Blog : 0 • Earned : $0.00 All calculations should be done in Word using Equation Editor and all charts and graphs should be done in Excel. 1)In the judicial case of the United States vs. the City of Chicago, discrimination was charged in a qualifying exam for the position of Fire Captain. In the table below, Group A was the minority and Group B was the majority. Use this table to answer questions a) – e) Passed Failed Group A 25 22 Group B 410 156 a) If one of the test subjects is randomly selected, find the probability of getting someone who passed the exam. b) Find the probability of randomly selecting one of the test subjects and getting someone who is in Group B or passed. c) Find the probability of randomly selecting two different test subjects and finding that they are both is Group A. d) Find the probability of randomly selecting one of the test subjects and getting someone who is in Group A and passed the exam e) Find the probability of getting someone who passed, given that the selected person is in Group A. 2)About 17% of the population has blue eyes. If a person is randomly selected, what is the probability that he or she does not have blue eyes? If four different people are randomly selected, what is the probability that they all have blue eyes? Would it be unusual to randomly select four people and find that they all have blue eyes? Why or why not? 3)You are trying to develop a strategy for investing in two difference stocks. The anticipated annual return for a $1,000 investment in each stock under four difference economic conditions has the following probability distribution: Probability Economic Condition Returns Stock A Returns Stock B 0.15 Recession -25 -75 0.25 Slow Growth 20 55 0.35 Moderate Growth 100 130 0.25 Fast Growth 150 200 a) Compute the expected return for Stock A and for Stock B b) Compute the standard deviation for Stock A and for Stock B c) Would you invest in Stock A or Stock B? Explain. 4)In South Carolina Palmetto Cash 6 lottery game, winning the jackpot requires that you select the correct 6 numbers between 1 and 57. How many different ways can those 6 numbers be selected? What is the probability for winning the jackpot? Is it unusual for anyone to win the lottery? Explain. Use Excel to calculate the first question. 5)Sun Microsystems has 64% of the market for high-end Unix machine, with its closest competitor being IBM. Suppose 17 corporations were in the market to purchase a high-end Unix machine. Use Excel to answer b) – d) (Hint: Assume a Binomial Distribution) a) How many Unix machines would Sun Microsystems “expect” to sell? b) What is the probability that exactly 7 corporations purchase a Unix machine, from Sun Microsystems? c) What is the probability that less than 11 corporations purchase a Unix machine from Sun Microsystems? d) What is the probability that 9 or more corporations purchase a Unix Machine from Sun Microsystems? 6)Dutchess County, New York, has been experiencing a mean of 67 motor vehicle deaths each year. Use Excel to answer a – c) (Hint: Assume a Poisson Distribution) a) Find the mean number of deaths per day. b) Find the probability that on a given day, there are 2 or more motor vehicle deaths. c) Find the probability that on a given day, there are no motor vehicle deaths. Available Answers [Solved] B444 mod02 Homework Assignment #2 \| Complete Solution • This solution is not purchased yet. • Submitted On 17 Jan, 2015 11:37:41 Answer posted by • Rating : 109 • Grade : A+ • Questions : 1 • Solutions : 1026 • Blog : 0 • Earned : $53213.54 d) Find the probability of randomly selecting one of the test subjects and getting someone who is in Group A and passed the exam. P(Group A and Passed)=25/613 Buy now to view the complete solution Other Related Questions No related question exists	{"url":"https://www.coursemerits.com/question-details/1429/b444-mod02-Homework-Assignment-2","timestamp":"2024-11-12T10:34:31Z","content_type":"text/html","content_length":"45979","record_id":"<urn:uuid:c8c05f04-378e-4846-8925-3e1616d82522>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00628.warc.gz"}
Kotlin Apprentice, Chapter 2: Expressions, Variables & Constants 2. Expressions, Variables & Constants Written by Matt Galloway & Joe Howard In this second chapter, you’re going to learn a few basics. You’ll learn how code works first. Then, you’ll start your adventure into Kotlin by learning some basics such as code comments, arithmetic operations, constants and variables. These are some of the fundamental building blocks of any language, and Kotlin is no different. First of all, you’ll cover the basic workings of computers, because it really pays to have a grounding before you get into more complicated aspects of programming. How a computer works You may not believe me when I say it, but a computer is not very smart on its own. The power of computers is all derived from how they’re programmed by people like you and me. If you want to successfully harness the power of a computer — and I assume you do, if you’re reading this book — it’s important to understand how computers work. It may also surprise you to learn that computers themselves are rather simple machines. At the heart of a computer is a Central Processing Unit (CPU). This is essentially a math machine. It performs addition, subtraction, and other arithmetical operations on numbers. Everything you see when you operate your computer is all built upon a CPU crunching numbers many millions of times per second. Isn’t it amazing what can come from just numbers? The CPU stores the numbers it acts upon in small memory units called registers. The CPU is able to read numbers into registers from the computer’s main memory, known as Random Access Memory (RAM). It’s also able to write the number stored in a register back into RAM. This allows the CPU to work with large amounts of data that wouldn’t all fit in the bank of registers. Here is a diagram of how this works: As the CPU pulls values from RAM into its registers, it uses those values in its math unit and stores the results back in another register. Each time the CPU makes an addition, a subtraction, a read from RAM or a write to RAM, it’s executing a single instruction. Each computer program is usually made up of thousands to millions of instructions. A complex computer program such as your operating system, be it iOS, Android, macOS, Windows or Linux (yes, they’re computer programs too!), may have many millions of instructions in It’s entirely possible to write individual instructions to tell a computer what to do, but for all but the simplest programs, it would be immensely time-consuming and tedious. This is because most computer programs aim to do much more than simple math — computer programs let you surf the internet, manipulate images, and allow you to chat with your friends. Instead of writing individual instructions, you write code in a specific programming language, which in your case will be Kotlin. This code is put through a computer program called a compiler, which converts the code into instructions the CPU knows how to execute. Each line of code you write will turn into many instructions — some lines could end up being tens of instructions! In the case of Kotlin, with its origins as a language on the Java Virtual Machine or JVM, there is an extra layer between the compiler and the OS. The Kotlin compiler creates what is known as bytecode, which gets run on the JVM and converted to native code along the way. Kotlin began on the JVM but now it is possible to compile Kotlin directly to native code, as you’ll see later in the Representing numbers As you know by now, numbers are a computer’s bread and butter, the fundamental basis of everything it does. Whatever information you send to the compiler will eventually become a number. For example, each character within a block of text is represented by a number. You’ll learn more about this in Chapter 3, which delves into types including strings, the computer term for a block of text. Images are no exception. In a computer, each image is also represented by a series of numbers. An image is split into many thousands, or even millions, of picture elements called pixels, where each pixel is a solid color. If you look closely at your computer screen, you may be able to make out these blocks. That is unless you have a particularly high-resolution display where the pixels are incredibly small! Each of these solid color pixels is usually represented by three numbers: one for the amount of red, one for the amount of green and one for the amount of blue. For example, an entirely red pixel would be 100% red, 0% green and 0% blue. The numbers the CPU works with are notably different from those you are used to. When you deal with numbers in day-to-day life, you work with them in base 10, otherwise known as the decimal system. Having used this numerical system for so long, you intuitively understand how it works. So that you can you can appreciate the CPU’s point of view, consider how base 10 works. The decimal or base 10 number 423 contains three units, two tens and four hundreds: In the base 10 system, each digit of a number can have a value of 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9, giving a total of 10 possible values for each digit. Yep, that’s why it’s called base 10! But the true value of each digit depends on its position within the number. Moving from right to left, each digit gets multiplied by an increasing power of 10. So the multiplier for the far-right position is 10 to the power of 0, which is 1. Moving to the left, the next multiplier is 10 to the power of 1, which is 10. Moving again to the left, the next multiplier is 10 to the power of 2, which is 100. And so on. This means each digit has a value ten times that of the digit to its right. The number 423 is equal to the following: (0 * 1000) + (4 * 100) + (2 * 10) + (3 * 1) = 423 Binary numbers Because you’ve been trained to operate in base 10, you don’t have to think about how to read most numbers — it feels quite natural. But to a computer, base 10 is way too complicated! Computers are simple-minded, remember? They like to work with base 2. Base 2 is often called binary, which you’ve likely heard of before. It follows that base 2 has only two options for each digit: 0 or 1. Almost all modern computers use binary because at the physical level, it’s easiest to handle only two options for each digit. In digital electronic circuitry, which is mostly what comprises a computer, the presence of an electrical voltage is 1 and the absence is 0 — that’s base 2! Note: There have been computers both real and imagined that use the ternary numeral system, which has three possible values instead of two. Computer scientists, engineers and dedicated hackers continue to explore the possibilities of a base-3 computer. See https://en.wikipedia.org/wiki/Ternary_computer and http://hackaday.com/tag/ternary-computer/. Here’s a representation of the base 2 number 1101: In the base 10 number system, the place values increase by a factor of 10: 1, 10, 100, 1000, etc. In base 2, they increase by a factor of 2: 1, 2, 4, 8, 16, etc. The general rule is to multiply each digit by an increasing power of the base number — in this case, powers of 2 — moving from right to left. So the far-right digit represents (1 * 2^0), which is (1 * 1), which is 1. The next digit to the left represents (0 * 2^1), which is (0 * 2), which is 0. In the illustration above, you can see the powers of 2 on top of the blocks. Put another way, every power of 2 either is (1) or isn’t (0) present as a component of a binary number. The decimal version of a binary number is the sum of all the powers of 2 that make up that number. So the binary number 1101 is equal to: (1 * 8) + (1 * 4) + (0 * 2) + (1 * 1) = 13 And if you wanted to convert the base 10 number 423 into binary, you would simply need to break down 423 into its component powers of 2. You would wind up with the following: (1 * 256) + (1 * 128) + (0 * 64) + (1 * 32) + (0 * 16) + (0 * 8) + (1 * 4) + (1 * 2) + (1 * 1) = 423 As you can see by scanning the binary digits in the above equation, the resulting binary number is 110100111. You can prove to yourself that this is equal to 423 by doing the math! The computer term given to each digit of a binary number is a bit (a contraction of “binary digit”). Eight bits make up a byte. Four bits is called a nibble, a play on words that shows even old-school computer scientists had a sense of humor. A computer’s limited memory means it can normally deal with numbers up to a certain length. Each register, for example, is usually 32 or 64 bits in length, which is why we speak of 32-bit and 64-bit Therefore, a 32-bit CPU can handle a maximum base-number of 4,294,967,295, which is the base 2 number 11111111111111111111111111111111. That is 32 ones—count them! It’s possible for a computer to handle numbers that are larger than the CPU maximum, but the calculations have to be split up and managed in a special and longer way, much like the long multiplication you performed in school. Hexadecimal numbers As you can imagine, working with binary numbers can become quite tedious, because it can take a long time to write or type them. For this reason, in computer programming, we often use another number format known as hexadecimal, or hex for short. This is base 16. Of course, there aren’t 16 distinct numbers to use for digits; there are only 10. To supplement these, we use the first six letters, a through f. They are equivalent to decimal numbers like so: • a = 10 • b = 11 • c = 12 • d = 13 • e = 14 • f = 15 Here’s a base 16 example using the same format as before: Notice first that you can make hexadecimal numbers look like words. That means you can have a little bit of fun. Now the values of each digit refer to powers of 16. In the same way as before, you can convert this number to decimal like so: (12 * 4096) + (0 * 256) + (13 * 16) + (14 * 1) = 49374 You translate the letters to their decimal equivalents and then perform the usual calculations. But why bother with this? Hexadecimal is important because each hexadecimal digit can represent precisely four binary digits. The binary number 1111 is equivalent to hexadecimal f. It follows that you can simply concatenate the binary digits representing each hexadecimal digit, creating a hexadecimal number that is shorter than its binary or decimal equivalents. For example, consider the number c0de from above: c = 1100 0 = 0000 d = 1101 e = 1110 c0de = 1100 0000 1101 1110 This turns out to be rather helpful, given how computers use long 32-bit or 64-bit binary numbers. Recall that the longest 32-bit number in decimal is 4,294,967,295. In hexadecimal, it is ffffffff. That’s much more compact and clear. How code works Computers have a lot of constraints, and by themselves, they can only do a small number of things. The power that the computer programmer adds, through coding, is putting these small things together, in the right order, to produce something much bigger. Coding is much like writing a recipe. You assemble ingredients (the data) and give the computer a step-by-step recipe for how to use them. Here’s an example: Step 1. Load photo from hard drive. Step 2. Resize photo to 400 pixels wide by 300 pixels high. Step 3. Apply sepia filter to photo. Step 4. Print photo. This is what’s known as pseudo-code. It isn’t written in a valid computer programming language, but it represents the algorithm that you want to use. In this case, the algorithm takes a photo, resizes it, applies a filter and then prints it. It’s a relatively straightforward algorithm, but it’s an algorithm nonetheless! Kotlin code is just like this: a step-by-step list of instructions for the computer. These instructions will get more complex as you read through this book, but the principle is the same: You are simply telling the computer what to do, one step at a time. Each programming language is a high-level, pre-defined way of expressing these steps. The compiler knows how to interpret the code you write and convert it into instructions that the CPU can execute. There are many different programming languages, each with its own advantages and disadvantages. Kotlin is an extremely modern language. It incorporates the strengths of many other languages while ironing out some of their weaknesses. In years to come, programmers will look back on Kotlin as being old and crusty, too. But for now, it’s an extremely exciting language because it is quickly This has been a brief tour of computer hardware, number representation and code, and how they all work together to create a modern program. That was a lot to cover in one section! Now it’s time to learn about the tools you’ll use to write in Kotlin as you follow along with this book. Getting started with Kotlin Now that you know how computers work, it’s time to start writing some Kotlin! You may wish to follow along with your own IntelliJ IDEA project. Simply create one using the instructions from the first chapter and type in the code as you go. First up is something that helps you organize your code. Read on! Code comments The Kotlin compiler generates bytecode or executable code from your source code. To accomplish this, it uses a detailed set of rules you will learn about in this book. Sometimes these details can obscure the big picture of why you wrote your code a certain way or even what problem you are solving. To prevent this, it’s good to document what you wrote so that the next human who passes by will be able to make sense of your work. That next human, after all, may be a future you. Kotlin, like most other programming languages, allows you to document your code through the use of what are called comments. These allow you to write any text directly along side your code which is ignored by the compiler. The first way to write a comment is like so: // This is a comment. It is not executed. This is a single line comment. You could stack these up like so to allow you to write paragraphs: // This is also a comment. // Over multiple lines. However, there is a better way to write comments which span multiple lines. Like so: /* This is also a comment. Over many.. many lines. / This is a multi-line comment. The start is denoted by / and the end is denoted by /. Simple! Kotlin also allows you to nest comments, like so: / This is a comment. /* And inside it another comment. Back to the first. This might not seem particularly interesting, but it may be if you have seen other programming languages. Many do not allow you to nest comments like this as when it sees the first / it thinks you are closing the first comment. You should use code comments where necessary to document your code, explain your reasoning, or simply to leave jokes for your colleagues. Printing out It’s also useful to see the results of what your code is doing. In Kotlin, you can achieve this through the use of the println command. println will output whatever you want to the console. For example, consider the following code: println("Hello, Kotlin Apprentice reader!") This will output a nice message to the console, like so: You can hide or show the console using the Run button at the bottom highlighted with the red box in the picture above. Arithmetic operations When you take one or more pieces of data and turn them into another piece of data, this is known as an operation. The simplest way to understand operations is to think about arithmetic. The addition operation takes two numbers and converts them into the sum of the two numbers. The subtraction operation takes two numbers and converts them into the difference of the two numbers. You’ll find simple arithmetic all over your apps; from tallying the number of “likes” on a post, to calculating the correct size and position of a button or a window, numbers are indeed everywhere! In this section, you’ll learn about the various arithmetic operations that Kotlin has to offer by considering how they apply to numbers. In later chapters, you see operations for types other than Simple operations All operations in Kotlin use a symbol known as the operator to denote the type of operation they perform. Consider the four arithmetic operations you learned in your early school days: addition, subtraction, multiplication and division. For these simple operations, Kotlin uses the following operators: • Add: + • Subtract: - • Multiply: • Divide: / These operators are used like so: 2 + 6 10 - 2 2 * 4 24 / 3 Each of these lines is what is known as an expression. An expression has a value. In these cases, all four expressions have the same value: 8. You write the code to perform these arithmetic operations much as you would write it if you were using pen and paper. In your IDE, you can see the values of these expressions as output in the console using println(): If you want, you can remove the whitespace surrounding the operator: You can even mix where you put the whitespace. For example: 2+6 // OK 2 + 6 // OK 2 +6 // OK 2+ 6 // OK It’s often easier to read expressions if you have white space on either side of the operator. Decimal numbers All of the operations above have used whole numbers, more formally known as integers. However, as you know, not every number is whole. As an example, consider the following: 22 / 7 This, you may be surprised to know, results in the number 3. This is because if you only use integers in your expression, Kotlin makes the result an integer also. In this case, the result is rounded down to the next integer. You can tell Kotlin to use decimal numbers by changing it to the following: 22.0 / 7.0 This time, the result is 3.142857142857143 as expected. The remainder operation The four operations you’ve seen so far are easy to understand because you’ve been doing them for most of your life. Kotlin also has more complex operations you can use, all of them standard mathematical operations, just less common ones. Let’s turn to them now. The first of these is the remainder operation, also called the modulo operation. In division, the denominator goes into the numerator a whole number of times, plus a remainder. This remainder is exactly what the remainder operation gives. For example, 10 modulo 3 equals 1, because 3 goes into 10 three times, with a remainder of 1. In Kotlin, the remainder operator is the % symbol, and you use it like so: 28 % 10 In this case, the result equals 8, because 10 goes into 28 twice with a remainder of 8. If you want to compute the same thing using decimal numbers you do it like so: 28.0 % 10.0 The result is identical to % when there are no decimals, which you can see by printing it out using a format specifier: println("%.0f".format(28.0 % 10.0)) Shift operations The Shift left and Shift right operations take the binary form of a decimal number and shift the digits left or right, respectively. Then they return the decimal form of the new binary number. For example, the decimal number 14 in binary, padded to 8 digits, is 00001110. Shifting this left by two places results in 00111000, which is 56 in decimal. Here’s an illustration of what happens during this shift operation: The digits that come in to fill the empty spots on the right become 0. The digits that fall off the end on the left are lost. Shifting right is the same, but the digits move to the right. The Kotlin functions for these two operations are as follows: • Shift left: shl • Shift right: shr These are infix functions that you place in between the operands so that the function call looks like an operation. You’ll learn more about infix functions later. Here’s an example: 1 shl 3 32 shr 2 Both of these values equal the number 8. One reason for using shifts is to make multiplying or dividing by powers of two easy. Notice that shifting left by one is the same as multiplying by two, shifting left by two is the same as multiplying by four, and so on. Likewise, shifting right by one is the same as dividing by two, shifting right by two is the same as dividing by four, and so on. In the old days, code often made use of this trick because shifting bits is much simpler for a CPU to do than complex multiplication and division arithmetic. Therefore the code was quicker if it used shifting. However these days, CPUs are much faster and compilers can even convert multiplication and division by powers of two into shifts for you. So you’ll see shifting only for binary twiddling, which you probably won’t see unless you become an embedded systems programmer! Order of operations Of course, it’s likely that when you calculate a value, you’ll want to use multiple operators. Here’s an example of how to do this in Kotlin: ((8000 / (5 * 10)) - 32) shr (29 % 5) Note the use of parentheses, which in Kotlin serve two purposes: to make it clear to anyone reading the code — including yourself — what you meant, and to disambiguate. For example, consider the following: 350 / 5 + 2 Does this equal 72 (350 divided by 5, plus 2) or 50 (350 divided by 7)? Those of you who paid attention in school will be screaming “72!” And you would be right! Kotlin uses the same reasoning and achieves this through what’s known as operator precedence. The division operator (/) has a higher precedence than the addition operator (+), so in this example, the code executes the division operation first. If you wanted Kotlin to do the addition first — that is, to return 50 — then you could use parentheses like so: 350 / (5 + 2) The precedence rules follow the same that you learned in math at school. Multiply and divide have the same precedence, higher than add and subtract which also have the same precedence. Math functions Kotlin also has a vast range of math functions in it’s standard library for you to use when necessary. You never know when you need to pull out some trigonometry, especially when you’re a pro at Kotlin and writing those complex games! Note: Don’t remove the import kotlin.math.* statement that comes with your project or IntelliJ IDEA will tell you it can’t find these functions. For example, consider the following: sin(45 * PI / 180) // 0.7071067811865475 cos(135 * PI / 180) // -0.7071067811865475 These compute the sine and cosine respectively. Notice how both make use of PI which is a constant Kotlin provides us, ready-made with 𝜋 to as much precision as is possible by the computer. Then there’s this: // 1.414213562373095 This computes the square root of 2. Did you know that sin(45°) equals 1 over the square root of 2? Not to mention these would be a shame: max(5, 10) // 10 min(-5, -10) // -10 These compute the maximum and minimum of two numbers respectively. If you’re particularly adventurous you can even combine these functions like so: max(sqrt(2.0), PI / 2) // 1.570796326794897 Naming data At its simplest, computer programming is all about manipulating data. Remember, everything you see on your screen can be reduced to numbers that you send to the CPU. Sometimes you yourself represent and work with this data as various types of numbers, but other times the data comes in more complex forms such as text, images and collections. In your Kotlin code, you can give each piece of data a name you can use to refer to it later. The name carries with it an associated type that denotes what sort of data the name refers to, such as text, numbers, or a date. You’ll learn about some of the basic types in this chapter, and you’ll encounter many other types throughout the rest of this book. Take a look at this: val number: Int = 10 This uses the val keyword to declare a constant called number which is of type Int. Then it sets the value of the constant to the number 10. Note: Thinking back to operators, here’s another one. The equals sign, =, is known as the assignment operator. The type Int can store integers. The way you store decimal numbers is like so: val pi: Double = 3.14159 This is similar to the Int constant, except the name and the type are different. This time, the constant is a Double, a type that can store decimals with high precision. There’s also a type called Float, short for floating point, that stores decimals with lower precision than Double. In fact, Double has about double the precision of Float, which is why it’s called Double in the first place. A Float takes up less memory than a Double but generally, memory use for numbers isn’t a huge issue and you’ll see Double used in most places. Even though we call an item created with val a “constant,” it’s more correct to say that the identifier marked with val is what is constant. Once you’ve declared a constant, you can’t change its data. For example, consider the following code: number = 0 This code produces an error: Val cannot be reassigned In your IDE, you would see the error represented this way: Constants are useful for values that aren’t going to change. For example, if you were modeling an airplane and needed to keep track of the total number of seats available, you could use a constant. You might even use a constant for something like a person’s age. Even though their age will change as their birthday comes, you might only be concerned with their age at this particular instant. In certain situations, for example, at the top level of your code outside of any functions, you can add the const keyword to a val to mark it as a compile-time constant: const val reallyConstant: Int = 42 Values marked with const must initialized with a String or a primitive type such as an Int or Double. You can also use const inside a Kotlin type that you’ll learn about in Chapter 12: “Objects.” Often you want to change the data behind a name. For example, if you were keeping track of your bank account balance with deposits and withdrawals, you might use a variable rather than a constant. If your program’s data never changed, then it would be a rather boring program! But as you’ve seen, it’s not possible to change the data behind a constant. When you know you’ll need to change some data, you should use a variable to represent that data instead of a constant. You declare a variable in a similar way, like so: var variableNumber: Int = 42 Only the first part of the statement is different: You declare constants using val, whereas you declare variables using var. Once you’ve declared a variable, you’re free to change it to whatever you wish, as long as the type remains the same. For example, to change the variable declared above, you could do this: variableNumber = 0 variableNumber = 1_000_000 To change a variable, you simply assign it a new value. Note: In Kotlin, you can optionally use underscores to make larger numbers more human-readable. The quantity and placement of the underscores is up to you. Using meaningful names Always try to choose meaningful names for your variables and constants. Good names can act as documentation and make your code easy to read. A good name specifically describes the role of variable or constant. Here are some examples of good names: • personAge • numberOfPeople • gradePointAverage Often a bad name is simply not descriptive enough. Here are some examples of bad names: The key is to ensure that you’ll understand what the variable or constant refers to when you read it again later. Don’t make the mistake of thinking you have an infallible memory! It’s common in computer programming to look back at your own code as early as a day or two later and have forgotten what it does. Make it easier for yourself by giving your variables and constants intuitive, precise names. Also, note how the names above are written. In Kotlin, it is common to camel case names. For variables and constants, follow these rules to properly case your names: • Start with a lowercase letter. • If the name is made up of multiple words, join them together and start every other word with an uppercase letter. • If one of these words is an abbreviation, write the entire abbreviation in the same case (e.g., sourceURL and urlDescription) Increment and decrement A common operation that you will need is to be able to increment or decrement a variable. In Kotlin, this is achieved like so: var counter: Int = 0 counter += 1 // counter = 1 counter -= 1 // counter = 0 The counter variable begins as 0. The increment sets its value to 1, and then the decrement sets its value back to 0. These operators are similar to the assignment operator (=), except they also perform an addition or subtraction. They take the current value of the variable, add or subtract the given value and assign the result to the variable. In other words, the code above is shorthand for the following: var counter: Int = 0 counter = counter + 1 counter = counter - 1 Similarly, the = and /= operators do the equivalent for multiplication and division, respectively: counter = 10 counter = 3 // same as counter = counter * 3 // counter = 30 counter /= 2 // same as counter = counter / 2 // counter = 15 If you haven’t been following along with the code in IntelliJ IDEA, now’s the time to try some exercises to test yourself! 1. Declare a constant of type Int called myAge and set it to your age. 2. Declare a variable of type Double called averageAge. Initially, set it to your own age. Then, set it to the average of your age and my own age of 30. 3. Create a constant called testNumber and initialize it with whatever integer you’d like. Next, create another constant called evenOdd and set it equal to testNumber modulo 2. Now change testNumber to various numbers. What do you notice about evenOdd? 4. Create a variable called answer and initialize it with the value 0. Increment it by 1. Add 10 to it. Multiply it by 10. Then, shift it to the right by 3. After all of these operations, what’s the Before moving on, here are some challenges to test your knowledge of variables and constants. You can try the code in IntelliJ IDEA to check your answers. 1. Declare a constant exercises with value 9 and a variable exercisesSolved with value 0. Increment this variable every time you solve an exercise (including this one). 2. Given the following code: age = 16 age = 30 Declare age so that it compiles. Did you use var or val? 3. Consider the following code: val a: Int = 46 val b: Int = 10 Work out what answer equals when you replace the final line of code above with each of these options: // 1 val answer1: Int = (a * 100) + b // 2 val answer2: Int = (a * 100) + (b * 100) // 3 val answer3: Int = (a * 100) + (b / 10) 4. Add parentheses to the following calculation. The parentheses should show the order in which the operations are performed and should not alter the result of the calculation. 5 * 3 - 4 / 2 * 2 5. Declare two constants a and b of type Double and assign both a value. Calculate the average of a and b and store the result in a constant named average. 6. A temperature expressed in °C can be converted to °F by multiplying by 1.8 then incrementing by 32. In this challenge, do the reverse: convert a temperature from °F to °C. Declare a constant named fahrenheit of type Double and assign it a value. Calculate the corresponding temperature in °C and store the result in a constant named celcius. 7. Suppose the squares on a chessboard are numbered left to right, top to bottom, with 0 being the top-left square and 63 being the bottom-right square. Rows are numbered top to bottom, 0 to 7. Columns are numbered left to right, 0 to 7. Declare a constant position and assign it a value between 0 and 63. Calculate the corresponding row and column numbers and store the results in constants named row and column. 8. A circle is made up of 2𝜋 radians, corresponding with 360 degrees. Declare a constant degrees of type Double and assign it an initial value. Calculate the corresponding angle in radians and store the result in a constant named radians. 9. Declare four constants named x1, y1, x2 and y2 of type Double. These constants represent the two-dimensional coordinates of two points. Calculate the distance between these two points and store the result in a constant named distance. Key points • Computers, at their most fundamental level, perform simple mathematics. • A programming language allows you to write code, which the compiler converts into instructions that the CPU can execute. Kotlin code on the JVM is first converted to bytecode. • Computers operate on numbers in base 2 form, otherwise known as binary. • Code comments are denoted by a line starting with // or multiple lines bookended with /* and /. • Code comments can be used to document your code. • You can use println to write things to the console area. • The arithmetic operators are: Add: + Subtract: - Multiply: Divide: / Remainder: % • Constants and variables give names to data. • Once you’ve declared a constant, you can’t change its data, but you can change a variable’s data at any time. • Always give variables and constants meaningful names to save yourself and your colleagues headaches later. • Operators to perform arithmetic and then assign back to the variable: Add and assign: += Subtract and assign: -= Multiply and assign: *= Divide and assign: /= Where to go from here? In this chapter, you’ve only dealt with only numbers, both integers and decimals. Of course, there’s more to the world of code than that! In the next chapter, you’re going to learn about more types such as strings, which allow you to store text. Have a technical question? Want to report a bug? You can ask questions and report bugs to the book authors in our official book forum here. © 2024 Kodeco Inc.	{"url":"https://koenig-assets.raywenderlich.com/books/kotlin-apprentice/v2.0/chapters/2-expressions-variables-constants","timestamp":"2024-11-13T01:51:09Z","content_type":"text/html","content_length":"454260","record_id":"<urn:uuid:1ca34f84-71ec-41c9-be69-5ab9fbf4f7f4>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00169.warc.gz"}
16.2.1. Buckling of Un-Cored Laminates - Abbott Aerospace UK Ltd Reference: Abbott, Richard. Analysis and Design of Composite and Metallic Flight Vehicle Structures 3 Edition, 2019. The methods below are taken from (MIL-HDBK-17F Vol 3, 2002) Section 5.7 and (NASA-TN-D-7996, 1975). The following caveats apply to the applicability of this analysis method: ……. the closed form solutions of laminated orthotropic Panels are appropriate only when the lay-ups are symmetrical and balanced. Symmetrical implies identical corresponding plies about the Panel mid-surface. Balanced refers to having a minus _ ply for every plus _ ply on each side of the mid-surface. Symmetrical and balanced laminated Panels have B terms vanish and the D[16] and D[26] terms virtually vanish……. Note that the buckling performance of a panel depends on the panel size, the rigidity of the panel edge constraints and the out-of-plane stiffness of the panel. The out-of-plane stiffness of the panel is expressed using the D matrix component of the laminate ABD matrix, see section 4.1.6.1 of this document for more information. Note that these methods are not specifically limited to uncored laminates but the effect of the presence of core on the buckling solution can be significant depending on the characteristics of the core panel. Classical Panel theory is based up on the Kirchhoff hypothesis: “Normals to the mid-plane of the un-deformed Panel remain straight and normal to the mid-plane during deformation”. This assumption therefore ignores the transverse shear deformation. Consideration of the shear deformation results in added flexibility which becomes significant as the Panel thickness increases relative to the length and width. (AFWAL-TR-85-3069, 1985) Figure 16.2.1‑1: The effect of Panel size to Thickness Ratio (a/h) on the Transverse Shear Influence of the Uniaxial Compression Buckling Allowable (AFWAL-TR-85-3069, 1985) However, there are buckling predictions specifically for cored laminate panels and these are examined in section 16.2.2 of this document. It is recommended that the analyst uses multiple methods, if they are available, so the results can be compared and correlated. It is also useful when the opportunity comes to test structures that have been designed using these methods as the test results can be used to help select the ‘best fit’ analysis for future work. All of the following methods are for large aspect ratio panels. Note that for short aspect ratio panels these solutions will give conservative results. These methods correlate well with test for b/t ratios greater than 35 – see figure on next page: For b/t ratios lower than 35, the N^cr[x,I ]/ N^cr[x,cl] ratio shows that the following methods will be significantly optimistic. For this method to be accurate for a laminate panel of 0.040in thickness the b dimension would have to be greater than 0.040 x 35 = 1.4in. This correlates to the difference in the predicted buckling performance between methods that account for transverse shear and those that do not, as shown in Figure 16.2.1‑1. The inaccuracy in this methodology is strongly influenced by the out-of-plane shear effect. 16.2.1.1. Uniaxial Loading, Long Panel, All Sides Simply Supported (Aspect ratio >4) For long plates the loaded edges can also be clamped and the allowable buckling stress will not be affected. The end load flow (lb/in) for the point of initial buckling is given by the following expression: This method is available in the following spreadsheet: 16.2.1.2. Effect of Central Circular Hole on Simply Supported Compression Buckling Allowable (NASA-TP-2528, 1986): An approximate solution for the effect of a circular hole on the compression buckling performance of a square panel is given by the following method. This method is defined by the reduction in K, the compression buckling coefficient. The value of K for the panel without a hole can be generated from the calculated compression buckling allowable using the following expression: Where W is the width of the loaded edge of the panel – ‘b’ in standard nomenclature. The effect on the compression buckling allowable for a simply supported Panel is given by this figure: Figure 16.2.1‑4: Effect of a Central Circular Hole on Axial Compression Buckling of a Simply Supported Orthotropic Panel (NASA-TP-2528, 1986) The graph shows various results for different methods. The greatest overall reduction in K from all the curves will be used to generate a reduction factor that can be used to modify the K value or the compression buckling allowable directly. This is expressed in the following figure: (note that this reduction factor graph is in broad agreement with (AFWAL-TR-85-3069, 1985) Figure 7.2. Figure 16.2.1‑5: Effect of a Central Circular Hole on Axial Compression Buckling of a Simply Supported Orthotropic Panel The end load flow for the point of initial buckling for a panel with a central circular hole becomes: This method is available in the following spreadsheet: 16.2.1.3. Uniaxial Loading, long Panel, all sides Fixed (Aspect ratio >4) The end load flow for the point of initial buckling is given by this expression: This method is available in the following spreadsheet: 16.2.1.4. Effect of Central Circular Hole on Fully Fixed Compression Buckling Allowable An approximate solution for the effect of a circular hole on the compression buckling performance of a square panel with fully fixed edges is given by the following method: The effect on the compression buckling allowable for a fully fixed Panel is given by this figure: Figure 16.2.1‑7: Effect of a Central Circular Hole on Axial Compression Buckling of Fully Fixed Orthotropic Panel (NASA-TP-2528, 1986) The graph shows various results for different methods. It is recommended that the trace showing the greatest reduction is used. As there are a range of results comparing analysis and test methods, a conservative approximation over the critical data sets has been used with a cubic line of best fit: Figure 16.2.1‑8: Effect of a Central Circular Hole on Axial Compression Buckling of a Fully Fixed Orthotropic Panel The end load flow for the point of initial buckling for a panel with a central circular hole becomes: This method is available in the following spreadsheet. 16.2.1.5. Uniaxial Loading, Long Panel, Three Sides Simply Supported and One Unloaded Edge Free The end load flow for the point of initial buckling: This method is available in the following spreadsheet: The following methods for shear buckling are more refined and give solutions for finite length panels covering a range of panel aspect ratios. 16.2.1.6. Shear Loading, Panel with all sides simply supported The shear buckling analysis method is taken from (NASA-TN-D-7996, 1975) which gives buckling solutions for shear and compression combinations. This paper gives the basic solution for shear buckling of a finite panel so we have used this as the best available reference. The general solution of the shear buckling equations can be expressed using the following parameter: The parameter k[s] is a function of only two variables: Where: D[3] = D[12] + 2D[66] To find the k[s] for panel with simply supported edges see Figure 16.2.1‑11: Figure 16.2.1‑11: Graphical Solution for k[s] – Shear Buckling of Simply Supported Panels (NASA-TN-D-7996, 1975) This method is available in the following spreadsheet: To find the k[s] for panel with fully fixed edges see Figure 16.2.1‑12: For a panel with edges fixed in rotation the spreadsheet method is available here: Once Ks has been found graphically the equation for k[s] can easily be rearranged to give the solution for N[xy]: Note that this equation is a similar form to the buckling equation for isotropic Panel buckling from (NACA-TN-3781, 1957), shown in its general form below: The equivalent terms that express the out-of-plane stiffness of the Panel are: 16.2.1.7. Interaction of Compression and Shear Buckling Effects Ref (NASA-TN-D-7996, 1975) shows comparison between a set of analyses and a linear/squared interaction. This demonstrates that using the mathematical approximation of the compression buckling reserve factor and the square of the shear buckling reserve factor is conservative. This approach is confirmed in (NASA CR-2330, 1974). Figure 16.2.1‑13: Graphical Comparison of Analysis vs Mathematical Solution for Compression and Shear Buckling Effects (NASA-TN-D-7996, 1975) This interaction method is available as a spreadsheet solution here: 16.2.1.8. Note on Post-Buckling and Crippling As noted at the start of this chapter it is common to keep composite structure non-buckling up to ultimate load. Post buckling effects will not be covered in this edition. Crippling is a post buckling failure mode thus it will not be covered. It is recommended that all composite primary structure be kept non-buckling up to ultimate load. Reference: Abbott, Richard. Analysis and Design of Composite and Metallic Flight Vehicle Structures 3 Edition, 2019. The methods below are taken from (MIL-HDBK-17F Vol 3, 2002) Section 5.7 and (NASA-TN-D-7996, 1975). The following caveats apply to the applicability of this analysis method: ……. the closed form solutions of laminated orthotropic Panels are appropriate only when the lay-ups are symmetrical and balanced. Symmetrical implies identical corresponding plies about the Panel mid-surface. Balanced refers to having a minus _ ply for every plus _ ply on each side of the mid-surface. Symmetrical and balanced laminated Panels have B terms vanish and the D[16] and D[26] terms virtually vanish……. Note that the buckling performance of a panel depends on the panel size, the rigidity of the panel edge constraints and the out-of-plane stiffness of the panel. The out-of-plane stiffness of the panel is expressed using the D matrix component of the laminate ABD matrix, see section 4.1.6.1 of this document for more information. Note that these methods are not specifically limited to uncored laminates but the effect of the presence of core on the buckling solution can be significant depending on the characteristics of the core panel. Classical Panel theory is based up on the Kirchhoff hypothesis: “Normals to the mid-plane of the un-deformed Panel remain straight and normal to the mid-plane during deformation”. This assumption therefore ignores the transverse shear deformation. Consideration of the shear deformation results in added flexibility which becomes significant as the Panel thickness increases relative to the length and width. (AFWAL-TR-85-3069, 1985) Figure 16.2.1‑1: The effect of Panel size to Thickness Ratio (a/h) on the Transverse Shear Influence of the Uniaxial Compression Buckling Allowable (AFWAL-TR-85-3069, 1985) However, there are buckling predictions specifically for cored laminate panels and these are examined in section 16.2.2 of this document. It is recommended that the analyst uses multiple methods, if they are available, so the results can be compared and correlated. It is also useful when the opportunity comes to test structures that have been designed using these methods as the test results can be used to help select the ‘best fit’ analysis for future work. All of the following methods are for large aspect ratio panels. Note that for short aspect ratio panels these solutions will give conservative results. These methods correlate well with test for b/t ratios greater than 35 – see figure on next page: Figure 16.2.1‑2: Predicted Classical Buckling Loads Compared to Experimental Data (MIL-HDBK-17F Vol 3, 2002) For b/t ratios lower than 35, the N^cr[x,I ]/ N^cr[x,cl] ratio shows that the following methods will be significantly optimistic. For this method to be accurate for a laminate panel of 0.040in thickness the b dimension would have to be greater than 0.040 x 35 = 1.4in. This correlates to the difference in the predicted buckling performance between methods that account for transverse shear and those that do not, as shown in Figure 16.2.1‑1. The inaccuracy in this methodology is strongly influenced by the out-of-plane shear effect. 16.2.1.1. Uniaxial Loading, Long Panel, All Sides Simply Supported (Aspect ratio >4) For long plates the loaded edges can also be clamped and the allowable buckling stress will not be affected. Figure 16.2.1‑3: Uniaxial Loaded Panel, SS all sides, Compression Bucking (MIL-HDBK-17F Vol 3, 2002) The end load flow (lb/in) for the point of initial buckling is given by the following expression: This method is available in the following spreadsheet: 16.2.1.2. Effect of Central Circular Hole on Simply Supported Compression Buckling Allowable (NASA-TP-2528, 1986): An approximate solution for the effect of a circular hole on the compression buckling performance of a square panel is given by the following method. This method is defined by the reduction in K, the compression buckling coefficient. The value of K for the panel without a hole can be generated from the calculated compression buckling allowable using the following expression: Where W is the width of the loaded edge of the panel – ‘b’ in standard nomenclature. The effect on the compression buckling allowable for a simply supported Panel is given by this figure: Figure 16.2.1‑4: Effect of a Central Circular Hole on Axial Compression Buckling of a Simply Supported Orthotropic Panel (NASA-TP-2528, 1986) The graph shows various results for different methods. The greatest overall reduction in K from all the curves will be used to generate a reduction factor that can be used to modify the K value or the compression buckling allowable directly. This is expressed in the following figure: (note that this reduction factor graph is in broad agreement with (AFWAL-TR-85-3069, 1985) Figure 7.2. Figure 16.2.1‑5: Effect of a Central Circular Hole on Axial Compression Buckling of a Simply Supported Orthotropic Panel The end load flow for the point of initial buckling for a panel with a central circular hole becomes: This method is available in the following spreadsheet: 16.2.1.3. Uniaxial Loading, long Panel, all sides Fixed (Aspect ratio >4) Figure 16.2.1‑6: Uniaxial loaded Panel, SS all sides, Compression Bucking (MIL-HDBK-17F Vol 3, 2002) The end load flow for the point of initial buckling is given by this expression: This method is available in the following spreadsheet: 16.2.1.4. Effect of Central Circular Hole on Fully Fixed Compression Buckling Allowable An approximate solution for the effect of a circular hole on the compression buckling performance of a square panel with fully fixed edges is given by the following method: The effect on the compression buckling allowable for a fully fixed Panel is given by this figure: Figure 16.2.1‑7: Effect of a Central Circular Hole on Axial Compression Buckling of Fully Fixed Orthotropic Panel (NASA-TP-2528, 1986) The graph shows various results for different methods. It is recommended that the trace showing the greatest reduction is used. As there are a range of results comparing analysis and test methods, a conservative approximation over the critical data sets has been used with a cubic line of best fit: Figure 16.2.1‑8: Effect of a Central Circular Hole on Axial Compression Buckling of a Fully Fixed Orthotropic Panel The end load flow for the point of initial buckling for a panel with a central circular hole becomes: This method is available in the following spreadsheet. 16.2.1.5. Uniaxial Loading, Long Panel, Three Sides Simply Supported and One Unloaded Edge Free Figure 16.2.1‑9: Uniaxial loaded Panel, SS three sides, Compression Bucking (MIL-HDBK-17F Vol 3, 2002) The end load flow for the point of initial buckling: This method is available in the following spreadsheet: The following methods for shear buckling are more refined and give solutions for finite length panels covering a range of panel aspect ratios. 16.2.1.6. Shear Loading, Panel with all sides simply supported Figure 16.2.1‑10: Shear Loaded Panel with Shear Buckle (AFWAL-TR-85-3069, 1985) The shear buckling analysis method is taken from (NASA-TN-D-7996, 1975) which gives buckling solutions for shear and compression combinations. This paper gives the basic solution for shear buckling of a finite panel so we have used this as the best available reference. The general solution of the shear buckling equations can be expressed using the following parameter: The parameter k[s] is a function of only two variables: Where: D[3] = D[12] + 2D[66] To find the k[s] for panel with simply supported edges see Figure 16.2.1‑11: Figure 16.2.1‑11: Graphical Solution for k[s] – Shear Buckling of Simply Supported Panels (NASA-TN-D-7996, 1975) This method is available in the following spreadsheet: To find the k[s] for panel with fully fixed edges see Figure 16.2.1‑12: Figure 16.2.1‑12: Graphical Solution for k[s] – Shear Buckling of Fully Fixed Panels (NASA-TN-D-7996, 1975) For a panel with edges fixed in rotation the spreadsheet method is available here: Once Ks has been found graphically the equation for k[s] can easily be rearranged to give the solution for N[xy]: Note that this equation is a similar form to the buckling equation for isotropic Panel buckling from (NACA-TN-3781, 1957), shown in its general form below: The equivalent terms that express the out-of-plane stiffness of the Panel are: 16.2.1.7. Interaction of Compression and Shear Buckling Effects Ref (NASA-TN-D-7996, 1975) shows comparison between a set of analyses and a linear/squared interaction. This demonstrates that using the mathematical approximation of the compression buckling reserve factor and the square of the shear buckling reserve factor is conservative. This approach is confirmed in (NASA CR-2330, 1974). Figure 16.2.1‑13: Graphical Comparison of Analysis vs Mathematical Solution for Compression and Shear Buckling Effects (NASA-TN-D-7996, 1975) This interaction method is available as a spreadsheet solution here: 16.2.1.8. Note on Post-Buckling and Crippling As noted at the start of this chapter it is common to keep composite structure non-buckling up to ultimate load. Post buckling effects will not be covered in this edition. Crippling is a post buckling failure mode thus it will not be covered. It is recommended that all composite primary structure be kept non-buckling up to ultimate load.	{"url":"https://www.abbottaerospace.com/aa-sb-001/16-local-stability-composite-materials/16-2-buckling-of-laminates/16-2-1-buckling-of-un-cored-laminates/","timestamp":"2024-11-11T10:05:03Z","content_type":"text/html","content_length":"210822","record_id":"<urn:uuid:d6b8bb66-4fc5-425b-abe9-b7ba9de6e98f>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00518.warc.gz"}
8 Ways to deal with Continuous Variables in Predictive Modeling Let’s come straight to the point on this one – there are only 2 types of variables you see – Continuous and Discrete. Further, discrete variables can divided into Nominal (categorical) and Ordinal. We did a post on how to handle categorical variables last week, so you would expect a similar post on continuous variable. Yes, you are right – In this article, we will explain all possible ways for a beginner to handle continuous variables while doing machine learning or statistical modeling. But, before we actually start, first things first. What are Continuous Variables? Simply put, if a variable can take any value between its minimum and maximum value, then it is called a continuous variable. By nature, a lot of things we deal with fall in this category: age, weight, height being some of them. Just to make sure the difference is clear, let me ask you to classify whether a variable is continuous or categorical: 1. Gender of a person 2. Number of siblings of a Person 3. Time on which a laptop runs on battery Please write your answers in comments below. How to handle Continuous Variables? While continuous variables are easy to relate to – that is how nature is in some ways. They are usually more difficult from predictive modeling point of view. Why do I say so? It is because the possible number of ways in which they can be handled. For example, if I ask you to analyze sports penetration by gender, it is an easy exercise. You can look at percentage of males and females playing sports and see if there is any difference. Now, what if I ask you to analyze sports penetration by age? How many possible ways can you think to analyze this – by creating bins / intervals, plotting, transforming and the list goes on! Hence, handling continuous variable in usually a more informed and difficult choice. Hence, this article should be extremely useful to beginners. Methods to deal with Continuous Variables Binning The Variable: Binning refers to dividing a list of continuous variables into groups. It is done to discover set of patterns in continuous variables, which are difficult to analyze otherwise. Also, bins are easy to analyze and interpret. But, it also leads to loss of information and loss of power. Once the bins are created, the information gets compressed into groups which later affects the final model. Hence, it is advisable to create small bins initially. This would help in minimal loss of information and produces better results. However, I’ve encountered cases where small bins doesn’t prove to be helpful. In such cases, you must decide for bin size according to your hypothesis.We should consider distribution of data prior to deciding bin size. For example: Let’s take up the inbuilt data set state.x77 in R to create bins: #load data data <- data.frame(state.x77) #check data #plot Frost variable and check the data points are all over qplot(y = Frost, data = data, colour = 'blue') #use cut() to create bins of equal sizes bins <- cut(data$Frost, 3, include.lowest = TRUE) #add labels to bins bins <- cut(data$Frost, 3, include.lowest = TRUE, labels = c('Low','Medium','High')) In simpler words, it is a process of comparing variables at a ‘neutral’ or ‘standard’ scale. It helps to obtain same range of values. Normally distributed data is easy to read and interpret. As shown below, in a normally distributed data, 99.7% of the observations lie within 3 standard deviations from the mean. Also, the mean is zero and standard deviation is one. Normalization technique is commonly used in algorithms such as k-means, clustering etc. A commonly used normalization method is z-scores. Z score of an observation is the number of standard deviations it falls above or below the mean. It’s formula is shown below. x = observation, μ = mean (population), σ = standard deviation (population) For example: Randy scored 76 in maths test. Katie score 86 in science test. Maths test has (mean = 70, sd = 2). Science test has (mean = 80, sd = 3). Who scored better? You can’t say Katie is better since her score is much higher than mean. Since, both values are at different scales, we’ll normalize these value at z scale and evaluate their performance. z(Randy) = (76 – 70)/2 = 3 z(Katie) = (86 – 80)/3 = 2 Interpretation: Hence, we infer than Randy scored better than Katie. Because, his score is 3 standard deviations away from the class mean whereas Katie’s score is just 2 standard deviations away from Transformations for Skewed Distribution: Transformation is required when we encounter highly skewed data. It is suggested not to work on skewed data in its raw form. Because, it reduces the impact of low frequency values which could be equally significant. At times, skewness is influenced by presence of outliers. Hence, we need to be careful while using this approach. The technique to deal with outliers is explained in next sections. There are various types of transformation methods. Some are Log, sqrt, exp, Box-cox, power etc. The commonly used method is Log Transformation. Let’s understand this using an example. For example: I’ve score of 22 students. I plot their scores and find out that distribution is left skewed. To reduce skewness, I take log transformation (shown below). As you can see after transformation, the data is no longer skewed and is ready for further treatment. Use of Business Logic: Business Logic adds precision to output of a model. Data alone can’t suggest you patterns which understanding its business can. Hence, in companies, data scientists often prefer to spend time with clients and understand their business and market. This not only helps them to make an informed decision. But, also enables them to think outside the data. Once you start thinking, you are no longer confined within data. For example: You work on a data set from Airlines Industry. You must find out the trends, behavior and other parameters prior to data modeling. New Features: Once you have got the business logic, you are ready to make smart moves. Many a times, data scientists confine themselves within the data provided. They fail to think differently. They fail to analyze the hidden patterns in data and create new variables. But, you must practice this move. You wouldn’t be able to create new features, unless you’ve explored the data to depths. This method helps us to add more relevant information to our final model. Hence, we obtain increase in accuracy. For example: I have a data set where you’ve following variables: Age, Sex, Height, Weight, Area, Blood Group, Date of Birth. Here we can make use of our domain knowledge. We know that (HeightWeight) can give us BMI Index. Hence, we’ll create HW = (HeightWeight) as a new variable. HW is nothing but BMI (Body Mass Index). Similarly, you can think of new variables in your data set. Treating Outliers: Data are prone to outliers. Outlier is an abnormal value which stands apart from rest of data points. It can happen due to various reasons. Most common reason include challenges arising in data collection methods. Sometime the respondents deliberately provide incorrect answers; or the values are actually real. Then, how do we decide? You can any of these methods: 1. Create a box plot. You’ll get Q1, Q2 and Q3. (data points > Q3 + 1.5IQR) and (data points < Q1 – 1.5IQR) will be considered as outliers. IQR is Interquartile Range. IQR = Q3-Q1 2. Considering the scope of analysis, you can remove the top 1% and bottom 1% of values. However, this would result in loss of information. Hence, you must be check impact of these values on dependent variable. Treating outliers is a tricky situation – one where you need to combine business understanding and understanding of data. For example, if you are dealing with age of people and you see a value age = 200 (in years), the error is most likely happening because the data was collected incorrectly, or the person has entered age in months. Depending on what you think is likely, you would either remove (in case one) or replace by 200/12 years. Principal Component Analysis: Sometime data set has too many variables. May be, 100, 200 variables or even more. In such cases, you can’t build a model on all variables. Reason being, 1) It would be time consuming. 2) It might have lots of noise 3) A lot of variables will tell similar information Hence, to avoid such situation we use PCA a.k.a Principal Component Analysis. It is nothing but, finding out few ‘principal‘ variables which explain significant amount of variation in dependent variable. Using this technique, a large number of variables are reduced to few significant variables. This technique helps to reduce noise, redundancy and enables quick computations. In PCA, components are represented by PC1 or Comp 1, PC2 or Comp 2.. and so on. Here, PC1 will have highest variance followed by PC2, PC3 and so on. Our motive should be to select components with eigen values greater than 1. Eigen values are represented by ‘Standard Deviation’. Let check this out in R below: #set working directory #load data from package >data(Boston, package = 'MASS') >myData <- Boston #descriptive statistics #check correlation table and analyze which variables are highly correlated. #Principal Component Analysis >pcaData <- princomp(myData, scores = TRUE, cor = TRUE) #check that var comp1 > comp2 and so on. And we find that Comp 1, Comp 2 #and Comp3 have values higher than 1 #loadings - This represents the contribution of variables in each factor. Higher the #number higher is the contribution of a particular variable in a factor #screeplot of eigen values ( Value of standard deviation is considered as eigen values) >screeplot(pcaData, type = 'line', main = 'Screeplot') #Biplot of score variables #Scores of the components Factor Analysis: Factor Analysis was invented by Charles Spearman (1904). This is a variable reduction technique. It is used to determine factor structure or model. It also explains the maximum amount of variance in the model. Let’s say some variables are highly correlated. These variables can be grouped by their correlations i.e. all variables in a particular group can be highly correlated among themselves but have low correlation with variables of other group(s). Here each group represents a single underlying construct or factor. Factor analysis is of two types: 1. EFA (Exploratory Factor Analysis) – It identifies and summarizes the underlying correlation structure in a data set 2. CFA (Confirmatory Factor Analysis) – It attempts to confirm hypothesis using the correlation structure and rate ‘goodness of fit’. Let’s do exploratory analysis in R. As we run PCA previously, we inferred that Comp 1, Comp 2 and Comp 3. We’ve now identified the components. Below is the code for EFA: #Exploratory Factor Analysis #Using PCA we've determined 3 factors - Comp 1, Comp 2 and Comp 3 >pcaFac <- factanal(myData, factors = 3, rotation = 'varimax') #To find the scores of factors >pcaFac.scores <- factanal(myData, factors = 3, rotation = 'varimax', scores = 'regression' Note: VARIMAX rotation involves shift in coordinates which maximizes the sum of the variances of the squared loadings. It rotates the alignment of coordinates orthogonally. Methods to work with Date & Time Variable Presence of Data Time variable in a data set usually give lots of confidence. Seriously! It does. Because, in data-time variable, you get lots of scope to practice the techniques learnt above. You can create bins, you can create new features, convert its type etc. Date & Time is commonly found in this format: DD-MM-YYY HH:SS or MM-DD-YYY HH:SS Considering this format, let’s quickly glance through the techniques you can undertake while dealing with data-time variables: Create New Variables: Have a look at the date format above. I’m sure you can easily figure out the possible new variables. If you have still not figure out, no problem. Let me tell you. We can easily break the format in different variables namely: 1. Date 2. Month 3. Year 4. Time 5. Days of Month 6. Days of Week 7. Days of Year I’ve listed down the possibilities. You aren’t required to create all the listed variables in every situation. Create only those variables which only sync with your hypothesis. Every variable would have an impact( high / low) on dependent variable. You can check it using correlation matrix. Create Bins: Once you have extracted new variables, you can now create bins. For example: You’ve ‘Months’ variable. You can easily create bins to obtain ‘quarter’, ‘half-yearly’ variables. In ‘Days’, you can create bins to obtain ‘weekdays’. Similarly, you’ll have to explore with these variables. Try and Repeat. Who knows, you might find a variable of highest importance. Convert Date to Numbers: You can also convert date to numbers and use them as numerical variables. This will allow you to analyze dates using various statistical techniques such as correlation. This would be difficult to undertake otherwise. On the basis of their response to dependent variable, you can then create their bins and capture another important trend in data. Basics of Date Time in R There are three good options for date-time data types: built-in POSIXt, chron package, lubridate package. POSIXt has two types, namely POSIXct and POSIXlt. “ct” can stand for calendar time and “lt” is local time. # create a date # specify the format as.Date("11/30/2015", format = "%m/%d/%Y") # take a difference - Sys.Date() gives present day date Sys.Date() - as.Date("2014-12-01") #using POSIXlt - find current time #finds class of each data time component # create POSIXct variables as.POSIXct("080406 10:11", format = "%y%m%d %H:%M") # convert POSIXct variables to character strings format(as.POSIXct("080406 10:11", format = "%y%m%d %H:%M"), "%m/%d/%Y %I:%M %p") End Notes You can’t explore data unless you are curious and patience. Some people are born with them. Some acquire them with experience. In anyway, the techniques listed above would help you to explore continuous variables at any level. I’ve tried to keep the explanation simple. I’ve also shared R codes. However, I haven’t shared their output. You can run these codes. Try to infer the findings. In this article, I’ve shared 8 methods to deal with continuous variables. These include binning, variable creation, normalization, transformation, principal component analysis, factor analysis etc. Additionally, I’ve also shared the techniques to deal with date time variables. Did you find this article helpful ? Did I miss out on any technique? Which is the best technique of all? Share your comments / suggestions in the comments section below. Responses From Readers Actually, there are more types that categorical and continuous. There are ordinal variables, where the data has a definite order to it. For instance, if I am rating customer service experience from 1 to 5 with 1 being the worst and 5 being the best, the result has an order to it: 3 is better than 2. Good Summary. Can we use PCA for categorical variable also? Say, we have 10 variables; 4 categorical and 6 numeric; categorical variables have levels 3,4,2,6 respectively; Now, 1) Once we create dummies from that then we may 3+4+2+6 = 15 dummies + 4 numeric; All dummies are binary codes; 2) Eventually we have now 19 numerics (can we treat dummies as numeric? I doubt here also as they are mere binary codes 0/ 1;) 3) Now if we perform cor() or hetcor() and then pca() then is that fine?? I am sure I am not accurate in this steps but is there any major mistakes in the steps? A good exmaple but if you can show the same using SPSS that will be helpful Good one, in Outliers section, Create a box plot. You’ll get Q1, Q2 and Q3. (data points > Q3 + 1.5IQR) and (data points < Q3 – 1.5IQR) will be considered as outliers. Lower boundary should be Q1-1.5IQR where as in the post mentioned as Q3-1.5IQR. Please answer this Q: can R studio handle huge data.??. am using SVM function directly on 6 lakh rows and its getting hanged. Should I code SVM line by line and then try or is there some other method Good article. The graph on right skew. I thinks it's a left skewed data before transformation Greatest for households - Ideal for families: Disney Cruise Line will take top honors for creating cruising desirable for household members of all ages, ensuring sufficient grownup offerings so it really is not just the kids obtaining a fantastic time. Can you please add same topic covered by Python? Gender of a person--Categorical Number of siblings of a Person-Continuos Time on which a laptop runs on battery-Continuos Gender of a person- Categorical Number of siblings of a Person= Continuous Time on which a laptop runs on battery- Continuous In plot of students scores, if we change their order, we’ll have a different result. I mean: why students plotting have to obey a certain ordem? Very interesting topic.	{"url":"https://www.analyticsvidhya.com/blog/2015/11/8-ways-deal-continuous-variables-predictive-modeling/?utm_source=reading_list&utm_medium=https://www.analyticsvidhya.com/blog/2022/02/r-programming-concepts-made-easy/","timestamp":"2024-11-09T17:35:07Z","content_type":"text/html","content_length":"475694","record_id":"<urn:uuid:b442e1e5-58d5-45c2-a5b7-1ef948d57e05>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00646.warc.gz"}
800 ml to gallons - Gardening101.co 800 ml to gallons When you’re working in the garden, you might have to measure water to help your plants grow. Sometimes, we measure liquids in milliliters (ml) and sometimes in gallons. First, let’s understand the two units of measurement: • Milliliters is a smaller unit, often used when measuring things like plant nutrients or small amounts of water. • Gallons is a larger unit, which is commonly used for bigger quantities, such as when you’re watering lots of plants or filling up a storage tank. To convert from milliliters to gallons, we can use a conversion fact: 1 gallon is about 3,785.41 milliliters. So to find out how many gallons are in 800 ml, we need to do some math: \text{Gallons} = \frac{800 \text{ ml}}{3,785.41 \text{ ml/gallon}} When you do the calculation, you find out that: \text{Gallons} \approx 0.211 \text{ gallons} This means that 800 ml of water is about 0.211 gallons. Next time you’re measuring out water for your garden, remember that 800 ml is a little less than a quarter of a gallon! Here are 7 objects that are exactly equal to 800 ml: 1. A standard water bottle (most hold about 800 ml). 2. A measuring cup (if it goes up to 800 ml). 3. A large fruit juice carton (like orange juice). 4. A soda bottle (some 800 ml bottles are common). 5. A small soup can (some are around 800 ml). 6. A yogurt container (many come in that size). 7. A small plant watering can (designed to hold specific amounts). Now you can confidently measure water for your plants using different containers! 🌱 Leave a Reply Cancel reply	{"url":"https://gardening101.co/800-ml-to-gallons/","timestamp":"2024-11-05T22:49:16Z","content_type":"text/html","content_length":"108011","record_id":"<urn:uuid:80e7b34d-f1c8-425d-b183-4e67f9311fc4>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00408.warc.gz"}
Attometers to aln Converter β Switch toaln to Attometers Converter How to use this Attometers to aln Converter π € Follow these steps to convert given length from the units of Attometers to the units of aln. 1. Enter the input Attometers value in the text field. 2. The calculator converts the given Attometers into aln in realtime β using the conversion formula, and displays under the aln label. You do not need to click any button. If the input changes, aln value is re-calculated, just like that. 3. You may copy the resulting aln value using the Copy button. 4. To view a detailed step by step calculation of the conversion, click on the View Calculation button. 5. You can also reset the input by clicking on button present below the input field. What is the Formula to convert Attometers to aln? The formula to convert given length from Attometers to aln is: Length[(aln)] = Length[(Attometers)] / 593777777787278200 Substitute the given value of length in attometers, i.e., Length[(Attometers)] in the above formula and simplify the right-hand side value. The resulting value is the length in aln, i.e., Length Calculation will be done after you enter a valid input. Consider that the wavelength of a gamma-ray photon is around 1 attometer. Convert this wavelength from attometers to aln. The length in attometers is: Length[(Attometers)] = 1 The formula to convert length from attometers to aln is: Length[(aln)] = Length[(Attometers)] / 593777777787278200 Substitute given weight Length[(Attometers)] = 1 in the above formula. Length[(aln)] = 1 / 593777777787278200 Length[(aln)] = 0 Final Answer: Therefore, 1 am is equal to 0 aln. The length is 0 aln, in aln. Consider that the scale of nuclear interactions is on the order of 10 attometers. Convert this scale from attometers to aln. The length in attometers is: Length[(Attometers)] = 10 The formula to convert length from attometers to aln is: Length[(aln)] = Length[(Attometers)] / 593777777787278200 Substitute given weight Length[(Attometers)] = 10 in the above formula. Length[(aln)] = 10 / 593777777787278200 Length[(aln)] = 0 Final Answer: Therefore, 10 am is equal to 0 aln. The length is 0 aln, in aln. Attometers to aln Conversion Table The following table gives some of the most used conversions from Attometers to aln. Attometers (am) aln (aln) 0 am 0 aln 1 am 0 aln 2 am 0 aln 3 am 0 aln 4 am 0 aln 5 am 0 aln 6 am 0 aln 7 am 0 aln 8 am 0 aln 9 am 0 aln 10 am 0 aln 20 am 0 aln 50 am 0 aln 100 am 0 aln 1000 am 0 aln 10000 am 0 aln 100000 am 0 aln An attometer (am) is a unit of length in the International System of Units (SI). One attometer is equivalent to 0.000000000000001 meters or 1 Γ 10^(-18) meters. The attometer is defined as one quintillionth of a meter, making it an extremely small unit of measurement used for measuring subatomic distances. Attometers are used in advanced scientific fields such as particle physics and quantum mechanics, where precise measurements at the atomic and subatomic scales are required. An aln is a historical unit of length used in various cultures for measuring textiles and other materials. One aln is approximately equivalent to 24 inches or 0.6096 meters. The aln was based on the length of a person's arm or the width of a specific type of cloth, and its exact length could vary depending on historical standards and regional practices. Alns were used for measuring fabric lengths and in trade, particularly in the textile industry. Although less common today, the unit provides historical context for traditional measurement practices and standards in textiles and trade. Frequently Asked Questions (FAQs) 1. What is the formula for converting Attometers to aln in Length? The formula to convert Attometers to aln in Length is: Attometers / 593777777787278200 2. Is this tool free or paid? This Length conversion tool, which converts Attometers to aln, is completely free to use. 3. How do I convert Length from Attometers to aln? To convert Length from Attometers to aln, you can use the following formula: Attometers / 593777777787278200 For example, if you have a value in Attometers, you substitute that value in place of Attometers in the above formula, and solve the mathematical expression to get the equivalent value in aln.	{"url":"https://convertonline.org/unit/?convert=attometers-aln","timestamp":"2024-11-03T20:38:47Z","content_type":"text/html","content_length":"89697","record_id":"<urn:uuid:53a52f02-ef66-45e6-a89f-fc1be4d2a47f>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00727.warc.gz"}
etale cover stub for etale cover Nothing, I temporarily replaced coproduct with union and isomorphic with equal. This would fit better what I know: cover for etale morphisms is checked on set theoretical image (if checked via spectral points, though there are alternative criteria but I do not know them or understand them). What was written before with coproduct does not make full sense to me at this point. Thanks. That sounds right.	{"url":"https://nforum.ncatlab.org/discussion/2241/etale-cover/","timestamp":"2024-11-07T04:13:38Z","content_type":"application/xhtml+xml","content_length":"39310","record_id":"<urn:uuid:8f661240-2899-476e-827f-c51187d10fdb>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00068.warc.gz"}
[liveblog] Conclusion of Workshop on Trustworthy Algorithmic Decision-Making Posted on:: December 5th, 2017 I’ve been at a two-day workshop sponsored by the Michigan State Uiversity and the National Science Foundation: “Workshop on Trustworthy Algorithmic Decision-Making.” After multiple rounds of rotating through workgroups iterating on five different questions, each group presented its findings â€” questions, insights, areas of future research. NOTE: Live-blogging. Getting things wrong. Missing points. Omitting key information. Introducing artificial choppiness. Over-emphasizing small matters. Paraphrasing badly. Not running a spellpchecker. Mangling other people’s ideas and words. You are warned, people. Seriously, I cannot capture all of this. Conduct of Data Science What are the problems? • Who defines and how do we ensure good practice in data science and machine learning? Why is the topic important? Because algorithms are important. And they have important real-world effects on people’s lives. Why is the problem difficult? • Wrong incentives. • It can be difficult to generalize practices. • Best practices may be good for one goal but not another, e.g., efficiency but not social good. Also: Lack of shared concepts and vocabulary. How to mitigate the problems? • Change incentives • Increase communication via vocabularies, translations • Education through MOOCS, meetups, professional organizations • Enable and encourage resource sharing: an open source lesson about bias, code sharing, data set sharing Accountability group The problem: How to integratively assess the impact of an algorithmic system on the public good? “Integrative” = the impact may be positive and negative and affect systems in complex ways. The impacts may be distributed differently across a population, so you have to think about disparities. These impacts may well change over time We aim to encourage work that is: • Aspirationally casual: measuring outcomes causally but not always through randomized control trials. • The goal is not to shut down algorithms to to make positive contributions that generat solutions. This is a difficult problem because: • Lack of variation in accountability, enforcements, and interventions. • It’s unclear what outcomes should be measure and how. This is context-dependent • It’s unclear which interventions are the highest priority Why progress is possible: There’s a lot of good activity in this space. And it’s early in the topic so there’s an ability to significantly influence the field. What are the barriers for success? • Incomplete understanding of contexts. So, think it in terms of socio-cultural approaches, and make it interdisciplinary. • The topic lies between disciplines. So, develop a common language. • High-level triangulation is difficult. Examine the issues at multiple scales, multiple levels of abstraction. Where you assess accountability may vary depending on what level/aspect you’re looking at. Handling Uncertainty The problem: How might we holistically treat and attribute uncertainty through data analysis and decisions systems. Uncertainty exists everywhere in these systems, so we need to consider how it moves through a system. This runs from choosing data sources to presenting results to decision-makers and people impacted by these results, and beyond that its incorporation into risk analysis and contingency planning. It’s always good to know where the uncertainty is coming from so you can address it. Why difficult: • Uncertainty arises from many places • Recognizing and addressing uncertainties is a cyclical process • End users are bad at evaluating uncertain info and incorporating uncertainty in their thinking. • Many existing solutions are too computationally expensive to run on large data sets Progress is possible: • We have sampling-based solutions that provide a framework. • Some app communities are recognizing that ignoring uncertainty is reducing the quality of their work How to evaluate and recognize success? • A/B testing can show that decision making is better after incorporating uncertainty into analysis • Statistical/mathematical analysis Barriers to success • Cognition: Train users. • It may be difficult to break this problem into small pieces and solve them individually • Gaps in theory: many of the problems cannot currently be solved algorithmically. The presentation ends with a note: “In some cases, uncertainty is a useful tool.” E.g., it can make the system harder to game. Adversaries, workarounds, and feedback loops Adversarial examples: add a perturbation to a sample and it disrupts the classification. An adversary tries to find those perturbations to wreck your model. Sometimes this is used not to hack the system so much as to prevent the system from, for example, recognizing your face during a protest. Feedback loops: A recidivism prediction system says you’re likely to commit further crimes, which sends you to prison, which increases the likelihood that you’ll commit further crimes. What is the problem: How should a trustworthy algorithm account for adversaries, workarounds, and feedback loops? Who are the stakeholders? System designers, users, non-users, and perhaps adversaries. Why is this a difficult problem? • It’s hard to define the boundaries of the system • From whose vantage point do we define adversarial behavior, workarounds, and feedback loops. Unsolved problems • How do we reason about the incentives users and non-users have when interacting with systems in unintended ways. • How do we think about oversight and revision in algorithms with respect to feedback mechanisms • How do we monitor changes, assess anomalies, and implement safeguards? • How do we account for stakeholders while preserving rights? How to recognize progress? • Mathematical model of how people use the system • Define goals • Find stable metrics and monitor them closely • Proximal metrics. Causality? • Establish methodologies and see them used • See a taxonomy of adversarial behavior used in practice Likely approaches • Security methodology to anticipating and unintended behaviors and adversarial interactions’. Monitor and measure • Record and taxonomize adversarial behavior in different domains • Test . Try to break things. • Hard to anticipate unanticipated behavior • Hard to define the problem in particular cases. • Systems are born brittle • What constitutes adversarial behavior vs. a workaround is subjective. • Dynamic problem Algorithms and trust How do you define and operationalize trust. The problem: What are the processes through which different stakeholders come to trust an algorithm? Multiple processes lead to trust. • Procedural vs. substantive trust: are you looking at the weights of the algorithms (e.g.), or what were the steps to get you there? • Social vs personal: did you see the algorithm at work, or are you relying on peers? These pathways are not necessarily predictive of each other. Stakeholders build truth through multiple lenses and priorities • the builders of the algorithms • the people who are affected • those who oversee the outcomes Mini case study: a child services agency that does not want to be identified. [All of the following is 100% subject to my injection of errors.] • The agency uses a predictive algorithm. The stakeholders range from the children needing a family, to NYers as a whole. The agency knew what into the model. “We didn’t buy our algorithm from a black-box vendor.” They trusted the algorithm because they staffed a technical team who had credentials and had experience with ethics…and who they trusted intuitively as good people. Few of these are the quantitative metrics that devs spend their time on. Note that FAT (fairness, accountability, transparency) metrics were not what led to trust. • Processes that build trust happen over time. • Trust can change or maybe be repaired over time. “ • The timescales to build social trust are outside the scope of traditional experiments,” although you can perhaps find natural experiments. • Assumption of reducibility or transfer from subcomponents • Access to internal stakeholders for interviews and process understanding • Some elements are very long term What’s next for this workshop We generated a lot of scribbles, post-it notes, flip charts, Slack conversations, slide decks, etc. They’re going to put together a whitepaper that goes through the major issues, organizing them, and tries to capture the complexity while helping to make sense of it. There are weak or no incentives to set appropriate levels of trust Key takeways: • Trust is irreducible to FAT metrics alone • Trust is built over time and should be defined in terms of the temporal process • Isolating the algorithm as an instantiation misses the socio-technical factors in trust. Categories: ai, culture, liveblog, philosophy dw	{"url":"https://www.hyperorg.com/blogger/2017/12/05/liveblog-conclusion-of-workshop-on-trustworthy-algorithmic-decision-making/","timestamp":"2024-11-06T20:48:33Z","content_type":"application/xhtml+xml","content_length":"79840","record_id":"<urn:uuid:0b4e2a25-baa0-4379-bf81-ab3e74a70b38>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00683.warc.gz"}
a library Cuba – a library for multidimensional numerical integration The Cuba library offers a choice of four independent routines for multidimensional numerical integration: Vegas, Suave, Divonne, and Cuhre. They work by very different methods, summarized in the following table: Routine Basic integration method Algorithm type Variance reduction Sobol quasi-random sample Monte Carlo Vegas or Mersenne Twister pseudo-random sample Monte Carlo importance sampling or Ranlux pseudo-random sample Monte Carlo Sobol quasi-random sample Monte Carlo globally adaptive subdivision Suave or Mersenne Twister pseudo-random sample Monte Carlo + importance sampling or Ranlux pseudo-random sample Monte Carlo Korobov quasi-random sample Monte Carlo or Sobol quasi-random sample Monte Carlo stratified sampling, Divonne or Mersenne Twister pseudo-random sample Monte Carlo aided by methods from or Ranlux pseudo-random sample Monte Carlo numerical optimization or cubature rules deterministic Cuhre cubature rules deterministic globally adaptive subdivision All four have a C/C++, Fortran, and Mathematica interface and can integrate vector integrands. Their invocation is very similar, so it is easy to substitute one method by another for cross-checking. For further safeguarding, the output is supplemented by a χ²-probability which quantifies the reliability of the error estimate. The source code compiles with gcc, the GNU C compiler. The C functions can be called from Fortran directly, so there is no need for adapter code. Similarly, linking Fortran code with the library is straightforward and requires no extra tools. In Fortran and C/C++ the Cuba library can (and usually does) automatically parallelize the sampling of the integrand. Downloads (hover over download link for MD5): Windows users: Cuba 3 and up uses fork(2) to parallelize the execution threads. This POSIX function is not part of the Windows API, however, and is furthermore used in an essential way such that it cannot be worked around simply with CreateProcess etc. The only feasible emulation seems to be available through Cygwin. Ready-made MathLink executables (Version 4.2, statically linked as far as possible): Linux x86-64: MacOS x86-64: Instructions: download and gunzip the executable, then make it executable with "chmod 755 file". Vegas is the simplest of the four. It uses importance sampling for variance reduction, but is only in some cases competitive in terms of the number of samples needed to reach a prescribed accuracy. Nevertheless, it has a few improvements over the original algorithm and comes in handy for cross-checking the results of other methods. Suave is a new algorithm which combines the advantages of two popular methods: importance sampling as done by Vegas and subregion sampling in a manner similar to Miser. By dividing into subregions, Suave manages to a certain extent to get around Vegas' difficulty to adapt its weight function to structures not aligned with the coordinate axes. Divonne is a further development of the CERNLIB routine D151. Divonne works by stratified sampling, where the partitioning of the integration region is aided by methods from numerical optimization. A number of improvements have been added to this algorithm, the most significant being the possibility to supply knowledge about the integrand. Narrow peaks in particular are difficult to find without sampling very many points, especially in high dimensions. Often the exact or approximate location of such peaks is known from analytic considerations, however, and with such hints the desired accuracy can be reached with far fewer points. Cuhre employs a cubature rule for subregion estimation in a globally adaptive subdivision scheme. It is hence a deterministic, not a Monte Carlo method. In each iteration, the subregion with the largest error is halved along the axis where the integrand has the largest fourth difference. Cuhre is quite powerful in moderate dimensions, and is usually the only viable method to obtain high precision, say relative accuracies much below 1e–3. Upward of 75% of all questions regarding Cuba have to do with how to choose bounds different from the unit hypercube in Fortran, C, and C++. The solution is not to choose bounds but to scale the integrand. For the mathematically challenged, the explicit transformation is (in one dimension) ∫ [a] ^b dx f(x) → ∫ [0] ^1 dy f(a + (b - a) y) (b - a) , where the final (b - a) is the one-dimensional version of the Jacobian. This generalizes straightforwardly to more than one dimension. For constant integration bounds, this transformation might be implemented in Fortran as integer function ScaledIntegrand(ndim, x, ncomp, result) implicit none integer ndim, ncomp double precision x(ndim), result(ncomp) integer maxdim parameter (maxdim = 16) integer Integrand external Integrand double precision upper(maxdim) common /ubound/ upper double precision lower(maxdim) common /lbound/ lower integer dim, comp double precision range, jacobian, scaledx(maxdim) jacobian = 1 do dim = 1, ndim range = upper(dim) - lower(dim) jacobian = jacobianrange scaledx(dim) = lower(dim) + x(dim)range ScaledIntegrand = Integrand(ndim, scaledx, ncomp, result) do comp = 1, ncomp result(comp) = result(comp)*jacobian This site and the programs offered here are not commercial. Cuba is an open-source package and free of charge. If you want to use Cuba in a commercial application, make sure you understand the GNU Lesser General Public License under which Cuba is distributed. Cuba is being developed at the Max Planck Institute for Physics in Munich. Data protection statement and Imprint Last update: 28 Feb 22 Thomas Hahn	{"url":"https://feynarts.de/cuba/","timestamp":"2024-11-09T06:26:49Z","content_type":"text/html","content_length":"14076","record_id":"<urn:uuid:9e3e963e-9f10-4ac3-b7b9-f201c3235898>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00566.warc.gz"}
Analytical solution for dynamic response of segment lining subjected to explosive loads The existence of various types of joints, one of the typical characteristics of prefabricated lining structures, makes the mechanical performance of shield tunnel linings quite different from that of monolithic linings. A simplified calculation method for the dynamic elastic-plastic analysis of segment lining subjected to explosive loads is proposed. The lining is composed of a number of rigid arch segments that are interconnected by elastic-plastic hinges. The dynamic interaction between the segments and the bolts, and the interaction between tunnel lining segment and soil-structure can be properly simulated with the method. As an example, the calculation of the shield section of Nanjing metro subjected to blast loading was discussed. The time-history curves of displacement and speed of some key points of section lining were obtained. Furthermore, the influences of rock grade and joint stiffness on dynamic response of tunnel lining were taken into account. The result indicates that the simplified method of blasting response analysis can reflect the response of structure subjected to blast loading accurately. The results will be a reference for antiknock analysis and design of tunnel lining. 1. Introduction The shield tunnel is becoming more and more popular because of its virtues such as saving underground space, making the planning of the underground space more flexible and so on. Static data shows that for more than 60 percent of the terrorists’ attacks on the subway belong to bombings attacks, therefore to carry out anti-terrorist bombings research and evaluation work of shield tunnel linings has very important practical significance. Designers are often concerned with the dynamic response of shield tunnel linings under external impact loads. In many cases, this problem is commonly simplified and non-joint-considered of segments lining [1-6]. The errors will be large for the exact analysis of dynamic response of segments lining structure itself if the segments lining is regarded as a whole tube body with non-joint-considered, so the joints should be considered for segments lining structure analysis. Yankelevsky et al. [7] analyzed the large dynamic elastic-plastic deflections of a multi-segmented spring-supported prefabricated circular lining, and provider a numerical solution for both the uniform and the non-uniform base. The stress and deformation state of the segment structure and the surrounding medium under initial stress field is essential in solving response to the damage of tunnel segment structure, directly affect carrying capacity of tunnel segment structure to the explosion. Yuetang Zhao et al. [8] investigated the initial stress and strain state of subway segmented tunnel by LS-DYNA, and compared the initial states at different buried depths. Considering the reinforcement of bolt in the surrounding media, Wang Yong et al. [9] studied the dynamic response of section subway tunnel under different yields of surface explosion for evaluation the blast-induced effects by using LS-DYNA. The numerical results indicated that the top part and the center at the bottom of the section subway tunnel were more damaged zones, and the section subway tunnel was safe when 100 kg TNT detonates at the height of 1.5 m. In this paper, the deformability of the segments is disregarded and the segments are assumed to be rigid body. Considering the initial segment of static state of the structure, an analytical solution for dynamic response of shield tunnel segment under explosive loads is presented. Based on a certain tunnel project, the numerical simulation method is used to verify the reliability and effectively of the analytical solution. 2. The model The design method proposed is based on lumped mass model of soil-structure dynamic interaction system, by including several additional features. It considers soil and segment interaction and focuses on each joint’s forces. The proposed model is assumed to be composed of rigid arch segments interconnected by elastic-plastic joints. The surrounded medium is simulated by visco-elastic base. The problem is reduced to a system of nonlinear algebraic and differential equations which are solved by the direct step-by-step differentiation method with a stepwise check of the state of every joint. Fig. 1Sketch of simulated condition Prediction of particle displacement, velocity, acceleration, pressure and other parameters in the earth media resulting from an explosive detonation is a complicated and difficult task. Primary guidance on this topic is from the United States Department of the Army Technical Manual “Fundamental of Protective Design for Conventional Weapons”, TM5-855-1 [10]. A circular lining under top explosion (Fig. 1) is considered. The peak values of free-field stress is given as: $\left\{\begin{array}{l}{p}_{0}=\beta f\left(\rho c\right){\left(\frac{R}{\sqrt[3]{W}}\right)}^{-n},\\ {t}_{0}=\frac{R}{c},\\ {t}_{r}=\frac{0.1R}{c},\end{array}\right\$ where ${p}_{0}$ is the peak pressure; $f$ is ground coupling factor; $\beta$ is a constant; $\rho c$ is acoustic impedance; $R$ is the distance to the explosion; $W$ is charge weight; $n$ is attenuation coefficient; ${t}_{a}$ is the elapsed time from the instant of detonation to the time at which the ground shock arrives at a given location; ${t}_{r}$ is equivalent blast duration. The outer surface of shield tunnel lining at any point, the load time of arrival time of the peak pressure and pressure are different, the peak stress distribution along the lining surface expression [11] is: ${p}_{m}={p}_{0}{K}_{e}{K}_{OTP}{K}_{\sigma },$ where ${K}_{e}$ is the attenuation coefficient; ${K}_{\sigma }$ is the lateral pressure coefficient; ${K}_{OTP}$ is the general reflection coefficient and defined as follows: ${K}_{OTP}=\frac{{K}_{OTP}^{}{\mathrm{s}\mathrm{i}\mathrm{n}}^{2}\left(\gamma +\phi \right)}{{K}_{\sigma }}+{\mathrm{c}\mathrm{o}\mathrm{s}}^{2}\left(\gamma +\phi \right),$ where ${K}_{OTP}^{}$ is the positive reflection coefficient; the incident angle $\gamma$ and $\phi$ are shown in Fig. 1. The pressure acting on the segment lining at any point can be expressed as: ${p}_{i}\left(t\right)=\left\{\begin{array}{ll}\frac{t-{t}_{0}}{{t}_{ri}}{p}_{m},& {t}_{0}\le t<{t}_{0}+{t}_{ri},\\ \frac{{\tau }_{i}-t+{t}_{0}}{{\tau }_{i}-{t}_{ri}}{p}_{m},& {t}_{0}+{t}_{ri}\le t <{\tau }_{i},\end{array}\right\$ where ${\tau }_{i}$ is the pulse duration. 3. Equations of motion 3.1. Initial static analysis The stress and deformation state of the segment structure and the surrounding medium under the initial stress field is essential in solving response to the damage of tunnel segment structure, which directly affects carrying capacity of tunnel segment structure to the explosion. In many cases, the deformation of the lining has already appeared before the external load acted on the structure or the ground has already subjected to relatively high initial stresses. So the initial stresses of considerable magnitude have to be considered as a starting point in structural engineering. Fig. 2The segment static loading system For prefabricated linings, the strain of the joints interconnecting the different segments is usually considerably lower than the limited state. In such cases, the skin friction of circumferential bolts may be disregarded. The initial static loading system acting on a single segment is shown in Fig. 2. According to the geometrical relationship, corresponding constants under the case can be derived as: $\left\{\begin{array}{l}-\left({F}_{0}+{N}_{0i}\right)\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{si}+\left({F}_{0}+{N}_{0i+1}\right)\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_{si}+{\theta }_{ci}\ right)+{f}_{0i}\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{si}+{f}_{0i+1}\mathrm{c}\mathrm{o}\mathrm{s}\left({\theta }_{si}+{\theta }_{ci}\right)-{e}_{ix}=0,\\ \left({F}_{0}+{N}_{0i}\right)\mathrm{c}\ mathrm{o}\mathrm{s}{\theta }_{si}-\left({F}_{0}+{N}_{0i+1}\right)\mathrm{c}\mathrm{o}\mathrm{s}\left({\theta }_{si}+{\theta }_{ci}\right)+{f}_{0i}\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{si}+{f}_ {0i+1}\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_{si}+{\theta }_{ci}\right)-{G}_{i}-{e}_{iy}=0,\\ {M}_{0i+1}-{M}_{0i}-{e}_{ix}{R}_{0}\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_{si}+{\theta }_ {ci}/2\right)+{N}_{0i+1}{R}_{0}-{N}_{0i}{R}_{0}+\left({e}_{iy}+{G}_{i}\right)R\mathrm{c}\mathrm{o}\mathrm{s}\left({\theta }_{si}+{\theta }_{ci}/2\right)=0,\end{array}\right\$ where ${\theta }_{ci}$ is the central angle; ${\theta }_{si}$ is the polar coordinate of the segment point with respect to the horizontal direction; ${R}_{0}$ is the centre radius; ${N}_{0i}$ is the circumferential pressure; ${F}_{0}$ is the pre-load modeling on bolts; ${G}_{i}$ is the self-weight of the tunnel lining; ${e}_{ix}$, ${e}_{iy}$ are $x$ and $y$ projections of the overburden soil pressure; ${f}_{0i}$ is the static friction, which relies on ${N}_{0i}$ as: where ${k}_{1}$ is the static friction coefficient; which depends on the soil properties and represents its shear stiffness. Fig. 3Bilinear stiffness model The bending moment in the hinge depends on the relative angular rotation of the connected segments. The analysis model in this study is in a manner similar to the bilinear stiffness model by Zhu Wei [12]. The bending moment can be defined as (Fig. 3): ${M}_{0i}=\left\{\begin{array}{l}{k}_{\theta 1}\theta +{M}_{0},\\ {k}_{\theta 2}\left(\theta -{\theta }_{1}\right)+{M}_{1},\end{array}\right\$ where ${k}_{\theta }$ is the segment joint bending stiffness, which is defined by the result of the experiment; ${M}_{0}$ is the ultimate moment at the joint; ${M}_{1}$ is the turning point moment. These terms depend on the axial force $N$ as follows: $\left\{\begin{array}{l}{M}_{0}={a}_{0}N+{b}_{0},\\ {M}_{1}={a}_{1}N,\\ {\theta }_{1}=\frac{{M}_{1}-{M}_{0}}{{k}_{\theta 1}},\end{array}\right\$ where ${a}_{0}$, ${a}_{1}$, ${b}_{1}$ are the unknown coefficients, that depend on the test data of load cushion, water-proof plastic and pretightening force of bolts. 3.2. Equations under explosive load The loading system acting on a single segment is shown in Fig. 4. Here $p\left(t\right)$ is the blast load; ${F}_{ki}$ is the interaction force between longitudinal segments and is defined as where ${r}_{0}=\frac{{N}_{0i}}{{k}_{n}}$ is the axial displacement under static equilibrium; ${k}_{n}$ is the shear stiffness; $r\left(t\right)$ is the axial displacement under dynamic load which can be defined as $r\left(t\right)=-y\left(t\right)\mathrm{s}\mathrm{i}\mathrm{n}\frac{{\theta }_{ci}}{2}$. The forces of interaction between the segments and bolts are defined as: ${Q}_{i}=\tau A=GA\gamma ,$ where $G=\frac{E}{2\left(1+\mu \right)}$ is the shear modulus of the bolt, $A$ is the cross-section area, $\gamma$ is the shear deformation. Fig. 4The segment loading system The sliding friction force on the lining simulating the normal and tangential reactions is determined by formula: where ${k}_{2}$ is the dynamic friction coefficient. According to the geometrical relationship in Fig. 4, the resultant of the external force with respect to lining actions per unit area of square action, as follows: ${p}_{xi}\left(t\right)$, ${p}_{yi} \left(t\right)$ are $x$ and $y$ projections of the resultant forces; ${M}_{pi}\left(t\right)$ is the resultant moment with respect to the lining center, are given in the form: $\left\{\begin{array}{l}{p}_{xi}\left(t\right)=\underset{{\theta }_{si}}{\overset{{\theta }_{si}+{\theta }_{ci}}{\int }}{p}_{i}\left(t\right)\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{i}{R}_{2}d{\theta }_{i},\\ {p}_{yi}\left(t\right)=\underset{{\theta }_{si}}{\overset{{\theta }_{si}+{\theta }_{ci}}{\int }}{p}_{i}\left(t\right)\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{i}{R}_{2}d{\theta }_{i},\\ {M}_ {pi}\left(t\right)=\underset{{\theta }_{si}}{\overset{{\theta }_{si}+{\theta }_{ci}}{\int }}{p}_{i}\left(t\right){d}_{i}{R}_{2}^{2}d{\theta }_{i},\end{array}\right\$ where ${d}_{i}={R}_{2}\mathrm{c}\mathrm{o}\mathrm{s}\left({\theta }_{si}+0.5{\theta }_{ci}-{\theta }_{i}\right)$. The final system of equations of motion is obtained in the following form: $\left\{\begin{array}{l}{m}_{i}\frac{{d}^{2}}{d{t}^{2}}{x}_{i}\left(t\right)=\begin{array}{l}{p}_{xi}\left(t\right)-\left({F}_{ki}+{F}_{0}\right)\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{si}+\left({F} _{ki+1}+{F}_{0}\right)\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_{si}+{\theta }_{ci}\right)\\ +\left({f}_{i}+{Q}_{i}\right)\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{si}+\left({f}_{i+1}+{Q}_{i+1}\ right)\mathrm{c}\mathrm{o}\mathrm{s}\left({\theta }_{si}+{\theta }_{ci}\right),\end{array}\\ {m}_{i}\frac{{d}^{2}}{d{t}^{2}}{y}_{i}\left(t\right)=\begin{array}{l}{p}_{yi}\left(t\right)+\left({F}_{ki} +{F}_{0}\right)\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{si}-\left({F}_{ki+1}+{F}_{0}\right)\mathrm{c}\mathrm{o}\mathrm{s}\left({\theta }_{si}+{\theta }_{ci}\right)\\ +\left({f}_{i}+{Q}_{i}\right)\ mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{si}+\left({f}_{i+1}+{Q}_{i+1}\right)\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_{si}+{\theta }_{ci}\right),\end{array}\\ {I}_{i}\frac{{d}^{2}}{d{t}^{2}}{\ theta }_{i}\left(t\right)={M}_{pi}\left(t\right)-{M}_{i}+{M}_{i+1}-\left({F}_{ki}-{F}_{ki+1}\right){R}_{0},\end{array}\right\$ where ${I}_{i}$ is the mass moment of inertia of the segment with respect to its instantaneous center; ${x}_{i}\left(t\right)$, ${y}_{i}\left(t\right)$ are the displacements of the segment with respect to a Eulerian Cartesian coordinate system $x0y$, and ${\theta }_{i}\left(t\right)$ is the segment’s angular rotation with respect to its instantaneous center. 3.3. Unloading stage Segment will cause deformation under load, and rebound deformation will occur when unloading. The segment will bears active soil pressure, and the friction between segments will decrease. The sliding friction force on the lining can be expressed as: The total pressure at unloading stage is similar to that of initial stage. The complete system of equation for the solution of the problem includes the equations of motion (13) and the initial conditions (5) that are dependent on the static solution. 4. Comparison of analytical solution and numerical modeling The finite element method is a well-recognized numerical tool to analyze geotechnical works because its ability to take into account the non-behavior of soils and the complex geometry of the works. To verify the proposed analytical solution, analysis has been performed using LS-DYNA. In the model, each segmental lining is modeled by a mesh of three-dimensional shell elements, and the segmental joint element is used to simulate the joints between the segmental linings. The contacts between the segments and bolts, and the contacts between tunnel lining segment and soil, reinforced concrete coupling, transmitting boundary setting are also considered. An application example of the tunnel linings for the first stage of Nanjing subway project is presented. Fig. 5 shows the layout of segment joints which was constructed in the clay deposit. The inside diameter and width of the tunnel linings are 5.5 m and 1.2 m, respectively. The linings were constructed by C50 concrete. The Young’s modulus of concrete is 3.45×10^4 N/mm^2 (see Table 1). The blast wave is generated by an explosion of an explosive-line charge of 5 kg TNT dynamite. Fig. 5Layout of segment joints Table 1Model parameters in numerical simulation Lining Soil Segment joint $E$ 3.45×104 MPa $\gamma$ 17.4 kN/m^3 ${k}_{\theta }$ 3.45×10^4 N·m/rad $u$ 0.2 $c$ 9 kpa $t$ 0.35 m $\phi$ 29.1° $D$ 6.2 m $k$ 10000 kN/m^3 ${k}_{n}$ 3.93×10^10 N/m $B$ 1.2 m ${K}_{0}$ 0.45 Fig. 6(a) and (b) shows the results of the time history of the cover velocity and displacement from analytical method in comparison with those obtained from numerical solution. As can be seen in the figure, the prediction of the current model agrees well with the numerical result. On the whole, compared with analytical solution result, the maximum of velocity conducted by the numerical modeling is small, while the maximum of displacement is large, the major cause of this deviation is that the two computational models, the deformability of the segments is disregarded and the segment is assumed to be rigid body in the deduction of analytical solution. Fig. 6The comparison by two kinds of solving methods b) Relative displacement of segment joint 5. Parametric studies 5.1. Influence of rock grade The conceptions of impedance ration $\kappa$ and flexibility ratio $F$ (A detailed description is given in Peck [13]) are presented to analyze the influence of surrounding rock mass character, they are defined as follows: $\kappa =\frac{\sqrt{E\rho }}{\sqrt{{E}_{1}{\rho }_{1}}},$ $F=2\left(\frac{E}{{E}_{1}}\right)\left(\frac{1-{\upsilon }_{1}^{2}}{1+{\upsilon }^{2}}\right){\left(\frac{{R}_{0}}{t}\right)}^{3},$ where $E$, $\upsilon$, $\rho$ are the Young’s modulus, Poisson’s ratio and mass density of soil; ${E}_{1}$, $\mathrm{}{\upsilon }_{1}$, ${\rho }_{1}$ are the Young’s modulus, Poisson’s ratio and mass density of the lining structure; ${R}_{0}$ is the lining’s mean radius; $t$ is the lining’s thickness. Physical properties of soil layers are shown in Table 2. Table 2Physical properties of soil layers Soil $\rho$ kg/m^3 $E$ MPa $\upsilon$ $v$ m/s $F$ $\kappa$ I 1700 63.7 0.3 120 1.59 0.035 II 1800 292.5 0.3 250 7.31 0.078 III 1900 605.2 0.3 350 15.12 0.115 IV 1900 1000.4 0.3 450 25 0.148 I: soft ground, II: medium soft soil, III: medium hard soil 1, IV: medium hard soil 2 The corresponding response of acoustic impedance and flexibility ratio is given in Figs. 7. These figures show that, the peak values of liner displacement and bending moment decrease as the acoustic impedance and flexibility ratio increase, while the ascending time decreases, the loading frequency increased and the vibration characteristics is enhanced. The peak values of displacement of bending moment change significantly when the value of flexibility ratio range between 1.59 and 15.12, acoustic impedance increases from 0.035 to 0.115, while when the flexibility ratio increases to 25, the change becomes gently. This conclusion has a good agreement with the qualitative result from the valuating criteria of relative flexibility founded by Peck. This implies that the strengthening of ground stiffness will result in reduction of the additional deformation and bending moment in cross section of tunnel. Its effect is especially significant for low flexibility ratio. The condition of surrounding rock is the important factor affecting the internal force of tunnel liner caused by explosion seismic wave. The worse surrounding rock conditions are, the more remarkable internal force caused by seismic wave produce and the worse explosive resistance performance represent. Engineering design and construction should avoid the impact of adverse geological and select characteristics of a good formation. Fig. 7Influence of rock grade a) Displacement of liner top b) Bending moment of liner top 5.2. Influence of joint stiffness Segment joint stiffness is an important parameter for the design of shield tunnel segment. Fig. 8 shows the influence with different joint stiffness of 2000, 4000 and 6000 N∙m/rad. It can be observed that as the joint stiffness increases, the liner top displacement amplitudes decrease gradually, but the bending moment amplitudes increase. It can also be found that the influence of joint stiffness on the displacement is minor while that on the bending moment is great and cannot be ignored. This result agrees with the investigative result proposed by Okamoto. Fig. 8Influence of joint stiffness a) Displacement of liner top b) Bending moment of liner top 6. Conclusions Deformation and damage of tunnel lining under the load of terrorist bombing is one of major security issues. In this study, dynamic response of shield tunnel segment subjected to explosive loads is analyzed. The main results obtained from the study are as follows. (1) An analytical solution to the dynamic response of shield tunnel segment lining has been developed. It is relatively simple yet comprehensive enough to consider several parameters and physical mechanisms that are pertinent to the current problem. The contacts between the segments and bolts, as well as the contacts between tunnel lining segment and soil are considered. (2) In order to verify the reliability of the proposed method, comparison with the numerical calculation is performed, and the results show good correspondence. (3) In combination with project practice, the influence of the rock grade and joint stiffness of the segment are analyzed. The results show that the higher the surrounding rock grade, the better the capacity to resist deformation of liner structure and the better the blast-resistant characteristics of the tunnel will be; the joint stiffness has obviously impact to bend of structure and has lightly influence to displacement. For this reason, it has distinct influence to safety factor too. • Kong De-Sen, Meng Qing-Hui, Zhang Wei-Wei, et al. Shock responses of a metro tunnel subjected to explosive loads. Journal of Vibration and Shock, Vol. 12, Issue 31, 2012, p. 68-72. • Liu Qi-Jian, Zhao Yue-Yu Dynamic stability of circular tunnel linings subjected to radial harmonic excitation. Journal of Hunan University, Natural Science, Vol. 38, Issue 9, 2011, p. 22-26. • Liu Mu-Yu, Lu Zhi-Fang Analysis of dynamic response of Yangtze river tunnel subjected to contact explosion loading. Journal of Wuhan University of Technology, Vol. 29, Issue 1, 2007, p. 113-117. • Ma Li-Qiu, Zhang Jian-Min, Zhang Ga, et al. Research of blasting centrifugal modeling system and basic experiment. Rock and Soil Mechanics, Vol. 32, Issue 3, 2011, p. 946-950. • Ma Liqiu, Zhang Jian-Min, Hu Yun, et al. Centrifugal model tests for responses of shallow-buried underground structures under surface blasting. Chinese Journal of Rock Mechanics and Engineering, Vol. 29, Issue 2, 2010, p. 3672-3678. • Liu Gan-Bin, Zheng Rong-Yue, Zhou Ye Numerical model for dynamic response of tunnel in clay. Journal of Ningbo University, Vol. 22, Issue 2, 2009, p. 263-267. • Karinski Y. S., Yankelevsky D. Z. Dynamic analysis of an elastic-plastic multisegment lining buried in soil. Engineering Structures, Vol. 29, Issue 1, 2007, p. 317-328. • Luo Kun-Sheng, Zhao Yue-Tang, Luo Zhong-Xing, et al. Numerical simulation on initial stress and strain state of subway segmented tunnel. China Civil Engineering Journal, Vol. 46, Issue 4, 2013, p. 78-84. • Luo Kun-Sheng, Wang Yong, Zhao Yue-Tang, etal. Numerical simulation of section subway tunnel under surface explosion. Journal of PLA University of Science and Technology, Vol. 8, Issue 6, 2007, p. 674-679. • TM5-855-1. Fundamentals of protective design for conventional weapons. Vicksburg, US army engineers waterways experimental station, 1986. • Jin Feng-Nian, Yuan Xiao-Jun, Zhou Jian-Nan, et al. Distribution law of blast loads on large-span compound structure. Journal of PLA University of Science and Technology, Vol. 12, Issue 6, 2011, p. 635-642 • Zhu Wei, Zhong Xiao-Chun, Qin Jian-She Mechanical analysis of segment joint of shield tunnel and research on bilinear joint stiffness model. Rock and Soil Mechanics, Vol. 27, Issue 12, 2006, p. • Peck R. B., Hendron A. J., Mohraz B. State of the art of soft ground tunneling. Proceedings of the Rapid Excavation and Tunneling Conference, Chicago, 1972. About this article 24 December 2013 explosive load shield tunnel blast-resistance analysis simplified calculation This work is substantially supported by National Basic Research Program of China (973 Program: 2010CB732003, 2013CB036005), National Natural Science Foundation of China (No. 51308542, 50878208). Copyright © 2014 JVE International Ltd. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.	{"url":"https://www.extrica.com/article/14803","timestamp":"2024-11-08T02:58:20Z","content_type":"text/html","content_length":"148385","record_id":"<urn:uuid:418ce61a-f933-4d35-8379-fbdd6aefe4ad>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00866.warc.gz"}
Molar Heat of Vaporization Return to the Time-Temperature Graph file Return to Thermochemistry Menu Here is the definition of the molar heat of vaporization: the amount of heat necessary to boil (or condense) 1.00 mole of a substance at its boiling point Note the two important factors: 1) It's 1.00 mole of a substance 2) there is no temperature change Keep in mind the fact that this is a very specific value. It is only for one mole of substance boiling. The molar heat of vaporization is an important part of energy calculations since it tells you how much energy is needed to boil each mole of substance on hand. (Or, if we were cooling off a substance, how much energy per mole to remove from a substance as it condenses.) Every substance has its own molar heat of vaporization. The units for the molar heat of vaporization are kilojoules per mole (kJ/mol). Sometimes the unit J/g is used. In that case, it is referred to as the heat of vaporization, the term 'molar' being The molar heat of vaporization for water is 40.7 kJ/mol. To get the heat of vaporization, you simply divide the molar heat by 18.015 g/mol. See Example #3 below. Molar heat values can be looked up in reference books. The molar heat of vaporization equation looks like this: q = (ΔH[vap]) (mass/molar mass) The meanings are as follows: 1) q is the total amount of heat involved 2) ΔH[vap] is the symbol for the molar heat of vaporization. This value is a constant for a given substance. 3) (mass/molar mass) is the division to get the number of moles of substance Example #1 49.5 g of H[2]O is being boiled at its boiling point of 100 °C. How many kJ is required? plug the appropriate values into the molar heat equation shown above q = (40.7 kJ / mol) (49.5 g / 18.0 g/mol) Example #2: 80.1 g of H[2]O exists as a gas at 100 °C. How many kJ must be removed to turn the water into liquid at 100 °C note that the water is being condensed. The molar heat of vaporization value is used at the solid-liquid phase change, REGARDLESS of the direction (boiling or condensing). q = (40.7 kJ/mol) (80.1 g / 18.0 g/mol) Example #3: Calculate the heat of vaporization for water in J/g divide the molar heat of vaporization (expressed in Joules) by the mass of one mole of water. (40700 J/mol) / (18.015 g/mol) = 2259 J/g You might see a value of 2257 J/g used. This results from using 40.66 kJ/mol rather than 40.7 kJ/mol. The value used by an author is often the one they used as a student. Just be aware that none of the values are wrong, they arise from different choices of values available. Example #4: Using the heat of vaporization for water in J/g, calculate the energy needed to boil 50.0 g of water at its boiling point of 100 °C. multiply the heat of vaporization (expressed in J/g) by the mass of the water involved. (2259 J/g) (50.0 g) = 112950 J = 113 kJ Example #5: By what factor is the energy requirement to evaporate 75 g of water at 100 °C greater than the energy required to melt 75 g of ice at 0 °C? Notice how the amounts of water are the same. This is deliberate. The equality is important, not the amount. Change the 75 g to one mole and solve: 40.7 kJ / 6.02 kJ = 6.76 Change the amount to 1 gram of water and solve: 2259.23 J / 334.166 J = 6.76 If you insisted that you must do it for 75 g, then we have this: (75 g * 2259.23 J/g) / (75 g * 334.166 J/g) = ??? You can see that the 75 cancels out, leaving 6.76 for the answer. Return to the Time-Temperature Graph file	{"url":"https://web.chemteam.info/Thermochem/Molar-Heat-Vaporization.html","timestamp":"2024-11-07T22:08:37Z","content_type":"text/html","content_length":"4775","record_id":"<urn:uuid:1ce49d1a-31ea-458a-a9b8-b91159e03679>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00546.warc.gz"}
ggpointless is an extension of the ggplot2 library providing additional layers. You can install ggpointless from CRAN with: To install the development version from GitHub use: Once you have installed the package, attach it by calling: What will you get • geom_pointless() – emphasizes some observations with points • geom_lexis() – draws a Lexis diagram • geom_chaikin() – applies Chaikin’s corner cutting algorithm • geom_catenary() – draws a catenary curve See vignette("ggpointless") for details and examples. geom_pointless() let’s you highlight the first, or last observations, sample minimum and sample maximum to provide additional context. Or just some visual sugar. geom_pointless() behaves similar to geom_point() except that it has a location argument. You can set it to "first", "last" (default), "minimum", "maximum", and "all", where "all" is just shorthand to select "first", "last", "minimum" and "maximum". geom_lexis() is a combination of a segment and a point layer. Given a start value and an end value, this function draws a 45° line which indicates the duration of an event. Required are x and xend aesthetics, y and yend coordinates will be calculated. See also the LexisPlotR package. Chaikin’s corner cutting algorithm let’s you turn a ragged path or polygon into a smoothed one. Credit to Farbfetzen / corner_cutting. See also the smoothr package. Draws a flexible curve that simulates a chain or rope hanging loosely between two fixed points. By default, a chain length twice the Euclidean distance between each x/y combination is used. See vignette("ggpointless") for details. Credit to: dulnan/catenary-curve In addition to the geoms & stats, the following data sets are contained in ggpointless: 1. co2_ml : CO[2] records taken at Mauna Loa 2. covid_vac : COVID-19 Cases and Deaths by Vaccination Status 3. female_leaders : Elected and appointed female heads of state and government For more examples call vignette("examples"). Code of Conduct Please note that this project is released with a Contributor Code of Conduct. By participating in this project you agree to abide by its terms.	{"url":"https://cran.ms.unimelb.edu.au/web/packages/ggpointless/readme/README.html","timestamp":"2024-11-02T13:51:05Z","content_type":"application/xhtml+xml","content_length":"20544","record_id":"<urn:uuid:a5e4ffdc-852c-4497-b385-204011225d34>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00606.warc.gz"}
Exam Praxis : Science Behind Learning... NCERT Summary 1. Thermodynamic Equilibrium • Temperature of a body is related to its average internal energy, not to kinetic energy of motion of centre of mass. • Equilibrium in thermodynamics refer to situation when macroscopic variables defining thermodynamic state of system don’t’ depend on time. 2. Zeroth Law of Thermodynamics • Two system in thermal equilibrium with third system separately are in thermal equilibrium with each other. • If T[A] = T[C] and T[B] = T[C ]then T[A ]= T[B] • Thermodynamic variable whose value is equal for two systems in thermal equilibrium is called temperature. 3. Heat,Internal Energy and Work • Heat is energy transfer arising due to temperature difference between system and surroundings. • Internal energy is simply the sum of kinetic energies and potential energies of the molecules in the frame of reference to which centre of mass of system is at rest. • Internal energy depends on state of the system, not how the state was achieved. • There are two ways to change internal energy of a thermodynamic system. a. To do work on system b. Supply heat to system So heat and work are two modes of altering the state of a thermodynamic system and changing internal energy. • Heat and work in thermodynamics are not state variables. • U is a state variable. $\Delta $V depends only on initial and final states and not on path taken by gas to go from one to another. • $\Delta $Q and $\Delta $W will depend on path taken to go from initial to final state. • Work done during thermodynamic process $\Delta W=\int\limits_{{{v}_{1}}}^{{{v}_{2}}}{PdV}$ • Area under the P – V diagram with the volume axis gives the work done in thermodynamic process. 4. Specific Heat Capacity • Molar specific heat at constant volume. ${{c}_{V}}={{\left( \frac{\Delta Q}{\Delta T} \right)}_{v}}=\left( \frac{\Delta V}{\Delta T} \right)$ • Molar specific heat at consant pressure. ${{C}_{P}}={{\left( \frac{\Delta Q}{\Delta T} \right)}_{P}}={{\left( \frac{\Delta U}{\Delta T} \right)}_{P}}+{{\left( \frac{\Delta V}{\Delta T} \right)}_{P}}$ $PV=RT \, \, \, \, \therefore P{{\left( \frac{\Delta V}{\Delta T} \right)}_{P}}=R$ • ${C_P} = {C_V} + R$ (MAYER’S Equation) • $\gamma =\cfrac{{{C}_{P}}}{{{C}_{v}}}=\cfrac{{{C}_{v}}+R}{{{C}_{v}}}=1+\cfrac{R}{{{C}_{v}}}$ ${{C}_{P}}=\gamma \times {{C}_{V}}$ 5. First Law of Thermodynamics • $\Delta Q=\Delta U+\Delta W$ (Energy conservation law) $\Delta Q$=heat supplied to system by the surrounding $\Delta W$=work done by the system on the surrounding $\Delta U$=Change in internal energy of a the system • Heat supplied to system goes in partly to increase internal energy and rest in work on environment. • This is simply the general law of conservation of energy applied to any system in which energy transfer is taken into account. • $\Delta W=P\Delta V$ $\therefore \Delta Q=\Delta U+P\Delta V$ 6. Thermodynamic State Variables • Thermodynamic state variables describe equilibrium state of system. These state variables are not necessarily independent. • The connection among state variables is called equation of state. • Equilibrium state of thermodynamic system is described by state variables. The value of state variable depends on particular state not by the path used to arrive that state. Pressure, volume, temperature and mass are state variable. Heat and work are not state variables. • For an ideal gas, equation of state is $PV = {\mu}RT$ • Thermodynamic state variables are of two types a. Extensive b. Intensive • Extensive variables indicates size of system. • Internal energy, volume and mass are extensive variables, but pressure, temperature and density are intensive variables. 7. Reversible and Irreversible Process • Spontaneous processes in nature are irreversible. • A process is reversible if the process can be turned back such that both the system and surrounding return to their original states with no any other change anywhere else in universe. • A quasi-static isothermal expansion of an ideal gas in a cylinder fitted with a frictionless movable piston is a reversible process. • A quasi-static process is an infinitely slow process such that system remains in thermal and mechanical equilibrium with surroundings throughout. In this process pressure and temperature of the environment can differ from those of system only infinitesimally. No accelerated motion of piston 8. Thermodynamic Processes • A thermodynamic process is an activity where a thermodynamic system is taken from one equilibrium state to another. • Reversible process • Irreversible process • Cyclic process 9. Isothermal Process • For isothermal processes Temperature during the process should be constant PV = constant • So pressure of given mass of a gas varies inversely as its volume. • Work done in isothermal process. If a system of ideal gas at temperature T goes from (P[1], V[1]) to (P[2], V[2]) equilibrium state, then work done $W=\mu RT\ln \left( \frac{{{V}_{2}}}{{{V}_{1}}} \right)=\mu RT\ln \left( \frac{{{P}_{1}}}{{{P}_{2}}} \right)$ • Here $\Delta T=0 \, \, \, \, \, \, \, \, \therefore \Delta U=0$ $\Delta Q=\Delta W\,\,=\mu RT\ln \left( \frac{{{V}_{2}}}{{{V}_{1}}} \right)$ 10. Adiabatic Process • In adiabatic process heat interaction between system and surrounding is zero i.e. $\Delta Q=0$ • PV^x = constant Where $\gamma $ = ratio of molar specific heats at constant pressure and at constant volume. • System is insulated from surroundings and heat absorbed or releases is zero. • Work done by gas results in decrease in its internal energy • If system change from (P[1], V[1], T[1]) to (P[2], V[2], T[2]) $\Delta W\,\,=\frac{\mu R({{T}_{1}}-{{T}_{2}})}{\gamma -1}=\frac{({{P}_{1}}{{V}_{1}}-{{P}_{2}}{{V}_{2}})}{\left( \gamma -1 \right)}\,\,where\,\,\gamma ={{C}_{p}}/{{C}_{v}}$ • If work done by gas ( W > 0), then T[2] < T[1]. 11. Isobaric Process • For isobaric process pressure during the process should be constant • Work done in isobaric process $W=P\left( {{V}_{2}}-{{V}_{1}} \right)=\mu R\left( {{T}_{2}}-{{T}_{1}} \right)$ • Heat partly to do absorbed goes partly to increase internal energy and mechanical work. $\Delta Q=\Delta U+\Delta W$ $\Delta U=\mu {{C}_{v}}\Delta T,\Delta Q=\mu {{C}_{v}}\Delta T\,\,and\,\,\Delta W=\mu R\Delta T$ $\cfrac{\Delta W}{\Delta Q}=\cfrac{R}{{{C}_{P}}}=\cfrac{\gamma -1}{\gamma }\,\,and\,\,\cfrac{\Delta U}{\Delta Q}=\cfrac{{{C}_{v}}}{{{C}_{P}}}=\cfrac{1}{\gamma }$ 12. Isochoric Process • For isochoric process volume during the process should be constant $\frac{P}{T}= \text{constant}$ • Work done in isochoric process, $\Delta W=P\Delta V=0$ • $\Delta Q=\Delta U=\Delta W$ $\Delta Q=\Delta U$ • Heat absorbed by gas goes entirely to change its internal energy and its temperature. • Change in internal energy is determined by specific heat at constant volume and temperature change 13. Cyclic Process • In any cyclic process system returns to initial state, $\Delta U=0$ • Hence total heat absorbed equals the work done by the system, $\Delta Q=\Delta W$ 14. Heat Engine • Heat engine is a device in which a system undergoes a cyclic process resulting in conversion of heat in to the sink. • Efficiency of the engine is $\eta \frac{W}{{{Q}_{1}}}=\frac{{{Q}_{1}}-{{Q}_{2}}}{{{Q}_{1}}}=1-\frac{{{Q}_{2}}}{{{Q}_{1}}}$ Q[1] = heat absorbed from source Q[2] = heat released to sink $\eta $ = efficiency of heat engine • Heat engine based on idealized reversible processes achieve the highest possible efficiency. 15. Refrigerator • A refrigerator is the reverse of a heat engine. Working substance extracts heat from cold reservoir, some external work is done on system and heat is released to reservoir at high temperature. • Coefficient of performance of refrigerator $\frac{heat\,\,extracted}{work\,\,input}$ $\beta =\frac{{{Q}_{2}}}{W}=\frac{{{Q}_{2}}}{{{Q}_{1}}-{{Q}_{2}}}=\frac{{{T}_{2}}}{\left( {{T}_{1}}-{{T}_{2}} \right)}=\frac{1-\eta }{\eta }$ • Coefficient of performance for heat pump is $\beta =\frac{{{Q}_{1}}}{W}=\frac{{{T}_{1}}}{{{T}_{1}}-{{T}_{2}}}=\frac{1}{\eta }$ 16. Second Law of Thermodynamics • Kelvin-Planck statement: No process is possible whose sole result is absorption of heat from a reservoir and complete conversion of heat into work • Clausius statement: No process is possible whose sole result is transfer of heat from cold reservoir to hotter object. • Two statements are completely equivalent. • It shows that efficiency of a heat engine can never be unity so heat released to cold reservoir can never be made zero. • Kelvin Planck and Clausius deny the prefect heat engine and refrigerator. 17. Carnot Engine • Carnot engine is a reversible engine operating between two temperature T[1] and T[2]. Cannot cycle consists of two isothermal and two adiabatic processes. Its efficiency is • Engine efficiency of Carnot engine does not depend on nature of working substance. $\eta =1-\frac{{{T}_{1}}}{{{T}_{2}}}$ • Carnot Theorem: Any other engine working between temperature T[1] and T[2] cannot have efficiency more than that of Carnot engine. The Carnot engine’s efficiency is independent of nature of working substance. In Carnot cycle $\frac{{{Q}_{1}}}{{{Q}_{2}}}=\frac{{{T}_{1}}}{{{T}_{2}}}$ is universal relation and this relation can be used to design universal thermodynamic scale.	{"url":"https://exampraxis.com/blogs/thermodynamics-basics","timestamp":"2024-11-09T00:37:51Z","content_type":"application/xhtml+xml","content_length":"83957","record_id":"<urn:uuid:01c4bdca-97fb-4497-9fd9-6ecc5619b5e8>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00391.warc.gz"}
Experience of formalized mathematics Experience of formalized mathematics There has already been considerable experience of formalized mathematics in widely differing theorem proving systems. It's important to absorb the lessons this yields about the process in general, and the strengths and weaknesses of particular formal systems and proof methodologies. Most of the following is based on personal experience, but we feel that much of it will be echoed by other Formalizing mathematics is sometimes quite difficult. It's not uncommon to make a definition, set about proving some consequences of it, fail, and then realize that the definition was faulty. Usually this is because some trivial degenerate case was ignored, sometimes because of a more substantive misunderstanding. This can be tedious, but at other times it is clarifying, and forces one to think It's an interesting question whether modest changes to normal mathematical style can render mathematics completely formal, or whether this will always be an unbearably tedious load (some say that 'rigour means rigor mortis'). It certainly doesn't seem too much trouble always to say explicitly that p is a prime, that n is a natural number, that when we talk about the poset X we are talking about the partial order [X], or to distinguish between a function and its value. But it may be that the accumulation of such details renders mathematics too inelegant. Only experience can tell. Formal proofs can be long and tedious, and tend to look quite unreadable compared with their textbook models. However we believe several existing systems such as Mizar show that one can still provide fairly readable proof texts, and the main difficulty in practice is automating just the right parts of the reasoning. In principle, there seems no reason why proofs cannot be brought down to something close to the length of textbook ones; even less perhaps. Of course, for a few exceptionally explicit and precise textbooks, this may already be the case. In any case (though this comment might seem to smack of desperation) there are advantages in the fact that proofs are more difficult. One tends to think more carefully beforehand about just what is the best and most elegant proof. One sees common patterns that can be exploited. All this is valuable. Compare the greater care one would take when writing a book or essay in the days before word In practice formalized mathematics is quite addictive (at least, for a certain kind of personality). This may be similar to the usual forms of computer addiction, but we believe there is another important factor at work. Since one can usually see one's way intuitively to the end of the proof, one has a continual mirage before one's eyes, an impression of being almost there! This must account for the times people stay up till the small hours of the morning trying to finish 'just one last proof'. Moreover, the continual computer feedback provides a stimulus, and the feeling of complete confidence that each step is correct helps dispel any feelings that the undertaking is futile since mistakes will have been made anyway. Many of the problems are not connected with the profound and interesting difficulties of formalizing and proving things, but with the prosaic matter of organizing the resulting database of theorems. Since one will often need to refer back to previous theorems, possibly proved by other people, it's useful to have a good lookup technique. In mathematics texts, readers can only remember a few 'big names' like the Heine-Borel theorem, the Nullstellensatz, and the rather ironic relic of 'Hilbert's Theorem 90' which in retrospect needed a name! For run of the mill theorems, it's better to number them: although the numeration scheme may be without significance, it does at least facilitate relatively easy lookup (though some books unhelpfully adopt distinct numbering streams for theorems, lemmas, corollaries and definitions). Mizar numbers its theorems, though inside textual units which are named. But this policy seems to us less suitable for a computer theorem prover because it's more common to want to pass from a (fancied) theorem to its identifier, rather than vice versa. Admittedly in the long term we want people to read formal proofs, but at present the main emphasis on constructing them. Still, we want to bear in mind the convenience of a later reader or user as well as the writer. With a computer system, textual lookup is at least as easy as numeric lookup, and names can have a mnemonic value, so using names throughout seems a better choice. It also makes it easier to insert or rearrange theorems without destroying any logic in the naming scheme. HOL names all theorems, but some theories do not have a consistent naming convention, notably natural number arithmetic which used to include the aptly-named TOTALLY_AD_HOC_LEMMA. A computer can offer the new possibility of looking up a theorem according to its structure, e.g. finding a theorem that will match a given term, or contains a certain combination of constants. HOL and IMPS have such facilities. Then there are the social problems: when different people work at formalizing mathematics, they will choose slightly different conventions, and perhaps reinvent the same concepts with different names. For example, one may take < as the basic notion of ordering, one may take And in practice, a given arrangement is not set in stone; people may want to add more theorems to existing libraries, perhaps slightly modify the theorems on which other libraries depend. It seems that making minor changes systematically should become easier with computer assistance, just as making textual changes becomes easier with word processing software. But sometimes it does prove to be very difficult --- the problem has been studied by [curzon-changes] with respect to large hardware verification proofs. How much worse if we want to change the foundational system more radically (e.g. constructivists might like to excise the use of nonconstructive principles in proofs). It seems that a suitably 'declarative' proof style is the key, and one should try to write proofs which avoid too many unnecessary parochial assumptions (e.g. the assumption that there are von Neumann ordinals floating around). This is comparable to avoiding machine dependencies in programming. John Harrison 96/8/13; HTML by 96/8/16	{"url":"https://www.rbjones.com/rbjpub/logic/jrh0120.htm","timestamp":"2024-11-02T20:10:16Z","content_type":"text/html","content_length":"9382","record_id":"<urn:uuid:10b4b8d8-13bd-4379-b3a7-095e4e5acb24>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00305.warc.gz"}
Real root counting for parametric polynomial systems and applications. View/Download File Persistent link to this item Real root counting for parametric polynomial systems and applications. Thesis or Dissertation Polynomial systems appear in many different fields of study. Many important problems can be reduced to solving systems of polynomial equations and usually the coefficients involve parameters. This thesis is devoted to finding practical ways to solve such problems from two fields, the studies of central configurations from the Newtonian N-body problem and Maxwell's conjecture about the electric potential created by point charges. Central configurations play an important role in the study of celestial mechanics. They determine some special solutions of the Newton's laws of motion and lead to explicit expression of the solutions. After some changes of the coordinates, we can describe the central configurations as zeros of a system of polynomials, where the coefficients of each polynomial are polynomials in the masses. Therefore, the problem of counting central configurations becomes counting the positive zeros of parametric polynomial systems. A problem studied by James C. Maxwell back in the 19th century is about finding an upper bound of the number of non-degenerate equilibrium points of the electric potential created by point charges. In the case of 3 point charges, he conjectured that there are at most 4 such equilibrium points. After given proper coordinates, the problem also becomes to count positive zeros of a parametric polynomial system. In Chapter 1, we will introduce these two problems and derive some parametric polynomial systems for which we will count positive zeros in Chapter 4. Some open questions from these two fields of studies will be given in Chapter 5. Our methods of counting positive zeros are based on classic tools such as resultants, subresultant sequences, and Hermite quadratic forms. Recently developed tools like Groebner bases make it possible to let computers perform symbolic computations of polynomials and count zeros by applying classic results. A computer algebra system (CAS), for example Mathematica, is the software to do such computations. In Chapter 2, we present those tools and demonstrate how to count zeros for polynomial systems with real or complex coefficients in a CAS. When it comes to counting zeros of parametric polynomial systems, we want to count zeros of all the real polynomial systems obtained by substituting real numbers for parameters. For example, when there is one parameter, we may want to know the numbers of positive zeros for real polynomial systems obtained by substituting parameters in an open interval (a, b). When there are two parameters, we may want to count positive zeros for all real polynomial systems obtained by substituting parameters with real pairs in an open region in R2. Our main contributions in this thesis are finding methods to achieve that goal based on standard computer algebra tools and applying these methods to some enumeration problems of central configurations and some special cases of Maxwell's conjecture. We will outline our methods and develop sufficient tools in Chapter 3. University of Minnesota Ph.D. dissertation. April 2011. Major: Mathematics. Advisor: Richard Moeckel. 1 computer file (PDF); vii, 86 pages, appendices A-B. Previously Published Citation Suggested citation Tsai, Ya-lun. (2011). Real root counting for parametric polynomial systems and applications.. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/104794. Content distributed via the University Digital Conservancy may be subject to additional license and use restrictions applied by the depositor. By using these files, users agree to the Terms of Use. Materials in the UDC may contain content that is disturbing and/or harmful. For more information, please see our statement on harmful content in digital repositories.	{"url":"https://conservancy.umn.edu/items/04dffeb9-c9c0-4269-9612-ddbd5118decd","timestamp":"2024-11-13T03:03:38Z","content_type":"text/html","content_length":"433455","record_id":"<urn:uuid:c7c974bc-d60f-4007-8536-f8d157088810>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00234.warc.gz"}