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Learn how to draw a cube step by step Drawing a cube is a great way to explore the fundamentals of three-dimensional space and improve your spatial reasoning skills. Whether you're interested in architectural drawings, 3D modeling, or simply want to enhance your artistic abilities, learning how to draw a cube is an essential skill. To start, you'll need a few basic drawing supplies, such as a pencil, eraser, and a piece of paper. It's also helpful to have a straightedge or ruler to ensure your lines are straight and parallel. Once you have your materials ready, you can begin the process of drawing a cube. Begin by drawing a square that will serve as the base of your cube. This square should be drawn lightly and can be any size you'd like. Next, draw two vertical lines from the top corners of the square, extending them upward. These lines should be parallel and the same length as the sides of the square. Then, draw another square on top of the vertical lines, connecting the ends of the lines. This square should be the same size as the base square and parallel to it. Finally, connect the four corners of the top square with lines, completing the shape of the cube. Once the basic shape of the cube is complete, you can add details and shading to make it appear more realistic. Use your pencil to darken the lines of the cube and add shadows to create depth. Consider the direction of the light source and shade the opposite side of the cube accordingly. With practice, you'll be able to draw cubes from various angles and incorporate them into more complex Understanding the Basics Before we dive into the process of drawing a cube, it's important to understand the basics of what a cube is and how it is constructed. A cube is a three-dimensional shape that has six square faces of equal size. It has eight vertices and twelve edges. A cube can be thought of as a special type of rectangular prism where all the sides have equal length. Each face of the cube is a square, and all the angles inside the cube are right angles (90 Vertices, Edges, and Faces A vertex is a point where two or more edges meet. In a cube, there are eight vertices. An edge is a line segment where two faces meet. In a cube, there are twelve edges. A face is a flat surface enclosed by edges. In a cube, there are six faces. In order to draw a cube accurately, it's important to understand the dimensions of the cube. The length, width, and height of the cube are all the same. Therefore, if you know the length of one side, you know the dimensions of the entire cube. When drawing a cube on a two-dimensional piece of paper, it's necessary to understand perspective to create the illusion of depth. Perspective refers to the technique used to create the illusion of depth and three-dimensionality in a two-dimensional drawing. The lines of the cube that are parallel in three-dimensional space will converge in the drawing. Step-by-Step Process Now that we have a clear understanding of the basics, let's move on to the step-by-step process of drawing a cube. 1. Start by drawing a square. This will be one face of the cube. 2. Using the corners of the square as references, draw lines extending outwards to create the other three faces of the cube. 3. Connect the corresponding corners of the adjacent faces to create the remaining two faces. 4. Erase any unnecessary lines, refine the edges, and add shading or details as desired. By following these steps and understanding the basics of a cube, you'll be able to draw a cube with confidence and accuracy. Gathering the Materials 1. Pencil To draw a cube, you will need a good quality pencil with a sharp point. Make sure it's comfortable to hold and doesn't smudge easily. 2. Paper Choose a clean and smooth piece of paper that is suitable for drawing. A standard letter-sized paper (8.5 x 11 inches) should work well. 3. Ruler A ruler is essential for drawing straight lines and measuring the dimensions of the cube. Make sure it's sturdy and has clear markings. 4. Eraser Mistakes happen, so having a good eraser is crucial. Look for an eraser that can easily remove pencil marks without damaging the paper. 5. Optional: Coloring materials If you want to add color to your cube drawing, you can gather colored pencils, markers, or crayons. This is entirely optional, and you can choose to leave your cube black and white. 6. Reference image Although not necessary, having a reference image of a cube can be helpful, especially if you're a beginner. You can find cube images online or use a physical cube as a reference. Creating the Outline Before starting to draw a cube, it is essential to create a basic outline to guide the drawing process. The outline will help ensure that the proportions and angles of the cube are accurate. To create the outline of a cube, follow these steps: 1. Begin by drawing a horizontal line, which will serve as the base of the cube. 2. From the endpoints of the base line, draw two vertical lines upward. These lines will represent the vertical edges of the cube. 3. Connect the top endpoints of the vertical lines with a horizontal line. This line should be parallel to the base line and equal in length. 4. Connect the corresponding endpoints of the horizontal lines with vertical lines. These lines will represent the rear vertical edges of the cube. 5. Connect the bottom endpoints of the rear vertical edges with a horizontal line. This line should be parallel to the base line and equal in length. At this point, the basic outline of the cube is complete. It should resemble a wireframe representation of a cube. The next step is to add details and shading to give the cube a more realistic Remember to use a ruler or a straight edge to ensure straight lines and to check the proportions of the cube throughout the drawing process. This will help achieve accurate results. Adding Depth and Dimension Once you have drawn the basic outline of a cube, you can add depth and dimension to make it look more realistic. Here are some tips on how to achieve this: • Use shading techniques to create the illusion of depth. You can achieve this by adding shadows to the sides of the cube that are not directly facing the light source. • Start by determining the position of the light source. Imagine a light coming from a specific direction and use that as a guide for shading. • Apply darker shades to the sides of the cube that are facing away from the light source, and lighter shades to the sides that are facing towards it. • Add highlights to the cube to give it a three-dimensional appearance. Highlights are areas of the cube that are directly facing the light source. • Use a lighter shade or even leave some parts completely white to represent highlights. • Place the highlights strategically on the cube. Typically, they appear on the top or side edges that are facing towards the light source. • Consider adding texture to the cube to make it look more realistic. This can be done by creating a pattern or adding details to the sides of the cube. • For example, you can draw lines or small squares on the sides to represent a textured surface. • Be careful not to overdo it, as too much texture can make the cube look cluttered and less defined. • Use perspective techniques to convey depth and dimension in your cube drawing. • Draw the sides of the cube that are farther away smaller than the sides that are closer to the viewer. • You can also add diagonal lines to the sides of the cube to create the illusion of depth. By applying these tips, you can add depth and dimension to your cube drawing, making it appear more realistic and three-dimensional. Experiment with different shading and texturing techniques to find the style that works best for you. Finalizing the Details Now that we have the basic shape of our cube, it's time to add some final details to make it look more realistic. 1. Shading and Highlights To give our cube a three-dimensional appearance, we need to add shading and highlights. Start by determining the direction of the light source. This will determine which sides of the cube should be darker and which should be lighter. Using darker shades, fill in the areas that are further away from the light source. Then, using lighter shades, add highlights to the areas that are closer to the light source. Blend the shades together using a smudging tool to create a smooth transition. 2. Texture and Patterns To add texture to our cube, we can use various patterns or textures. For example, we can add a wood grain pattern to make it look like a wooden cube. You can use a pattern brush or texture tool to apply these effects to the appropriate sides of the cube. Experiment with different patterns and textures to achieve the desired effect. Remember to consider the material of the cube and adjust the patterns accordingly. For example, a metal cube may have a brushed or reflective texture. 3. Shadows To make our cube look grounded and realistic, we need to add shadows. Determine the direction of the light source again and add shadows to the areas that would be blocked by the cube itself or other objects. This will create a sense of depth and dimension. Use darker shades or gradients to create the shadow effect. You can also add a soft blur to the shadows to make them look more natural. 4. Final Touches Lastly, review your drawing and make any necessary adjustments. Pay attention to the proportions, perspective, and overall balance of the cube. Clean up any stray lines or smudged areas. If you're satisfied with your drawing, you can add a background or additional elements to enhance the composition. For example, you can place the cube on a table or add other objects to create a still life scene. Remember, practice makes perfect. The more you study and draw cubes, the better you'll become at capturing their three-dimensional form and adding realistic details. Adding Shadows and Highlights Adding shadows and highlights is a crucial step in creating a realistic cube drawing. Shadows and highlights help to create depth and dimension in the drawing, making the cube appear To add shadows to the cube, you can imagine a light source coming from a specific direction. The side of the cube opposite the light source will be in shadow, while the other sides will have varying degrees of light and shadow. 1. Start by determining the direction of the light source in your drawing. This will help you determine where the shadows will fall. 2. Using a darker shade of the base color of the cube, fill in the side of the cube opposite the light source. This will create the shadowed area. 3. Gradually blend the shadowed area with the rest of the cube using lighter shades of the base color. This will create a smooth transition between the shadow and the rest of the cube. 4. You can also add darker shadows along the edges of the cube to create more depth and definition. Highlights are areas of the cube that are directly hit by the light source and therefore appear brighter than the rest of the cube. Adding highlights can make the cube look more realistic and give it a shiny, reflective appearance. • Identify the areas of the cube that are directly facing the light source. These will be the areas where the highlights will be added. • Using a lighter shade of the base color of the cube, carefully add highlights to these areas. This can be done by leaving small white spaces or by using a lighter shade of the base color. • Blend the highlights with the rest of the cube to create a smooth transition between the highlighted areas and the rest of the cube. By adding shadows and highlights to your cube drawing, you can create a realistic three-dimensional effect. Experiment with different lighting angles and shading techniques to achieve the desired What are the basic steps for drawing a cube? The basic steps for drawing a cube include drawing two squares, one above the other, connecting the corners of the squares to form the sides, and adding shading and details to create a three-dimensional effect. Is it difficult to draw a cube? Drawing a cube can be challenging for beginners, but with practice and patience, it becomes easier. Breaking down the steps and starting with basic shapes can help in mastering the art of drawing a Can I draw a cube without using a ruler? Yes, you can draw a cube without using a ruler by free-hand drawing the squares and connecting the corners. However, using a ruler can help in achieving straight lines and precise angles. What materials do I need to draw a cube? To draw a cube, you will need a pencil, paper, an eraser, a ruler (optional), and colored pencils or markers if you want to add shading or color to your drawing. Are there any tips for adding shading to a cube drawing? When adding shading to a cube drawing, remember to identify the light source and shade the areas that would be in shadow. Use a variety of shading techniques like hatching, cross-hatching, or blending to create a realistic three-dimensional effect. Can you suggest any other objects that I can practice drawing to improve my cube-drawing skills? A few objects that you can practice drawing to improve your cube-drawing skills include a dice, a building or house, a Rubik's cube, or a gift box. These objects have similar geometric shapes and can help you understand shading and perspective.	{"url":"https://euronewstop.co.uk/learn-how-to-draw-a-cube-step-by-step.html","timestamp":"2024-11-05T07:20:37Z","content_type":"text/html","content_length":"126730","record_id":"<urn:uuid:f8729946-ef61-4f83-85c7-02a929b17c7a>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00717.warc.gz"}
Piecewise Continuous Function A Piecewise Continuous Function is a continuous function that is a piecewise function. • (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Piecewise#Continuity Retrieved:2016-1-3. □ A piecewise function is continuous on a given interval if the following conditions are met: ☆ it is defined throughout that interval ☆ its constituent functions are continuous on that interval ☆ there is no discontinuity at each endpoint of the subdomains within that interval. □ The pictured function, for example, is piecewise continuous throughout its subdomains, but is not continuous on the entire domain. The pictured function contains a jump discontinuity at [math]\displaystyle{ x_0 }[/math] .	{"url":"https://www.gabormelli.com/RKB/Piecewise_Continuous_Function","timestamp":"2024-11-07T23:26:13Z","content_type":"text/html","content_length":"37386","record_id":"<urn:uuid:c490cafd-8d11-4b5c-af30-b01ccf2dbc2b>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00214.warc.gz"}
seminars - Weak amenability of Fourier algebras and local synthesis of the anti-diagonal Since the work of Johnson characterizing amenability of a locally compact group G in terms of Banach algebra amenability of the convolution algebra L[1](G), questions of characterizing various amenabilities of group related Banach algebras have been central theme of abstract harmonic analysis. For example, Ruan proved that operator space amenability of the Fourier algebra A(G) is equivalent to the amenability of G and Forrest/Runde showed that amenability of A(G) is equivalent to G being virtually abelian. In this talk we will focus on the weak amenability problem of Fourier algebras on Lie groups. We show that for a Lie group G, its Fourier algebra A(G) is weakly amenable if and only if its connected component of the identity G[e] is abelian. Our main new idea is to show that for connected G, weak amenability of A(G) implies that the anti-diagonal of the product group G \times G, is a set of local synthesis for A(G\times G). We then to show that this cannot happen if G is non-abelian. This is a joint work with Jean Ludwig (Metz), Ebrahim Samei (Saskatchewan) and Nico Spronk (Waterloo).	{"url":"https://www.math.snu.ac.kr/board/index.php?mid=seminars&sort_index=date&order_type=desc&page=85&l=ko&document_srl=623102","timestamp":"2024-11-10T07:32:14Z","content_type":"text/html","content_length":"48424","record_id":"<urn:uuid:07ed3084-2c13-4774-8f84-4ea1dfafa1a8>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00261.warc.gz"}
ALEKS FAQs \| ALEKS \| Math Placement \| Department of Mathematics \| University of Miami 3. Who must take the ALEKS Math Placement Assessment? First-year students admitted to UM as test optional are required to take the ALEKS Math Placement Assessment in order to determine an appropriate level math course. There is an exception, if you have AP Calculus scores or if you have completed a college-level math course, which may be used for math placement and you would not have to complete ALEKS. Otherwise, the default placement is into MTH099, a remedial algebra course. Incoming first-year students who have submitted standardized test scores officially as part of the admission process will be placed on the basis of Math SAT or Math ACT scores or AP Calculus exam scores. Students may take ALEKS in order to improve their math placement based on the standardized test scores. Any incoming transfer student, who does not have college credit in any math course, must also take ALEKS. 4. I'm taking a math class now (or I took a math class already). Can I take ALEKS to skip into a higher level? No, you may not take ALEKS after you have begun a math sequence, whether at UM or another college or university. ALEKS is intended only for initial math placement. You may not take ALEKS to skip out of any math prerequisite course or to a higher level. 5. How long will the assessment take, and what material will I need? The length of the assessment will vary by student since the test is adaptive. In general, the assessment will take approximately 90 minutes to complete. There will be a maximum of 30 questions. During the proctored sessions, you will be allowed 2 hours to complete the assessment. You will need only pencil and paper to work on problems. ALEKS has a built-in calculator available when needed, so you do not need a calculator. 6. How many questions are on the assessment? There will be a maximum of 30 open-ended questions. None of the questions are multiple-choice. The exact number of questions will vary due to the adaptive mechanism of the assessment. Students who are unable to answer many of the first questions may end up with a shorter assessment. If you come across material that you have not learned, you may answer, "I don't know." Do your best to answer each question. 7. Once I answer a question, can I go back and change my answer? Unfortunately, once you submit an answer, you cannot change it. You should keep this in mind while testing. 8. Can I get help on the assessment? No. Be honest. You are expected to take the assessment on your own without any outside assistance. The use of books, notes, electronic devices, aids, websites, or the assistance of another individual while taking the assessment is not allowed and will be considered a violation of the University of Miami's Honor Code and punishable by the provisions set forth by that code. 9. Should I prepare for the assessment? There is no preparation required, but you may wish to review the material from your previous math course. You may review the packets on the Prepare for Success page. A practice ALEKS assessment will be available. You may take the practice ALEKS and access the Prep and Learning Modules before taking the proctored ALEKS Math Placement Assessment. If you do not reach the score you desire, you may retake the assessment after completing at least 5 hours of a recommended ALEKS Prep and Learning Module. 10. Is there a fee for taking this assessment? There is no fee to you as a student. The University of Miami provides students with access to the ALEKS Math Placement Assessment for up to 1 year. Keep in mind, however, that the ALEKS Prep and Learning Module will expire within 6 months from your first access. 11. How many times can I take the ALEKS Math Placement Assessment? You will be given the chance to take the proctored ALEKS Math Placement Assessment up to two times within your subscription period. There is a 24-hour waiting period before you will be able to repeat the proctored assessment. In order to repeat the assessment, you will need to spend a minimum of 5 hours on a recommended ALEKS Prep and Learning Module to review the topics necessary. You will have access to your ALEKS Prep and Learning Module for 6 months from the first time you access it. If you do not complete the required 5 hours in the Prep and Learning Module within the 6-month subscription, you will not be able to repeat the assessment. 12. How long are my ALEKS assessment scores valid for? Scores on the ALEKS assessment are valid for a period of 18 months. In particular, you need to be sure that your ALEKS scores are valid for the semester you plan to enroll in a math course. 13. How do I take the assessment? The practice or proctored ALEKS Assessment can be accessed from any computer with internet connection. Sign on to CaneLink. Go to the Admissions tab on the left, then click on "ALEKS Math Assessment". You will be prompted to sign in again with duo-authentication. Once you are in the ALEKS site, you should click on the blue box indicating your ALEKS cohort. Once you submit the practice assessment, you will have access to the Prep and Learning Modules. LOCKDOWN BROWSER and MONITOR will be required for the proctored ALEKS attempts. You will be prompted to download Lockdown for ALEKS once you sign in. 14. When do I get the results and what do they mean? As soon as you complete the assessment, you will receive your ALEKS score along with a pie chart showing your performance in different areas. Above the pie chart, you will see a link to COURSE MASTERY. Here you will see a list of topics that you scored well on. Below the pie chart, you will see information about topics that you still need to work on and how you can improve your score. Your ALEKS score shows your level of preparedness for a math course. The cut-off scores have been set in order to determine the appropriate course in which you would be most likely to succeed. Please allow two days for your placement scores to be visible in CaneLink. Once you can see your ALEKS score in CaneLink, you may enroll in the course for which you have obtained the appropriate ALEKS score or otherwise meet the prerequisite. 15. How do I know which course I should take? Your math requirement will depend on your major and degree program. Please use the Math Placement Guide and the list of Freshman Level Mathematics Courses to determine the course you are eligible When you make your course selection, you will be allowed to enroll in a course for which you meet the prerequisite by way of the appropriate ALEKS score, a prerequisite course, an appropriate math SAT or ACT score, an appropriate AP Calculus score, or the recommendations set forth for students admitted as Test Optional. If you have any questions, please contact your Academic Advisor or Dr. Leticia Oropesa. 16. How do the ALEKS Prep and Learning Modules work? After you complete the practice ALEKS Math Placement Assessment, you will see a pie chart that tells you the topics that you have mastered and those in which you are deficient. You will be able to review using the modules before you take the proctored assessment. In order to take the proctored ALEKS Math Placement Assessment a second time, you will need to spend a minimum of 5 hours on a recommended ALEKS Prep and Learning Module before you take the assessment again. The ALEKS Prep and Learning Module will be available to you for a 6-month subscription free of charge. You do not have to access the Prep and Learning Modules immediately after taking the ALEKS Math Placement Assessment for the first time. You will be able to access the ALEKS Prep and Learning Modules for 1 year, but once you access the recommended Prep and Learning Module for the first time, you will have only 6 months to access it from that date. If you do not complete the required 5 hours in the recommended Prep and Learning Module within the 6-month subscription, you will not be able to repeat the assessment. There will be different modules available depending on the course you would like to take and your current ALEKS score. It is important that you are careful in your selection of the Learning Module as you will not be able to change it after you have begun the module. Based on your ALEKS score, you will be recommended a specific module. There is no fee for the Prep and Learning Modules. You must spend a minimum of 5 hours on the recommended Prep and Learning Module in order to retake the ALEKS Math Placement Assessment. Your progress in the Prep and Learning Module does not affect your ALEKS score. You must repeat the assessment in order to change your ALEKS score and improve your math placement. 17. I only missed the cut-off by a few points. Can I be bumped up to the next level? Exceptions to the prerequisites for each course will not be made. The prerequisites are clearly set forth by the Department of Mathematics. Fortunately, you may spend time in the ALEKS Prep and Learning Modules to review certain topics. It is very likely that your work on the recommended ALEKS Prep and Learning Module will allow you to increase your ALEKS score once you repeat the assessment, provided you still have an attempt available. 18. I took an ALEKS assessment through another college or university. Will the University of Miami accept my score? No. UM does not accept ALEKS scores from other colleges or universities. 19. Can I get credit for a MTH class based on my ALEKS score? No. The ALEKS score will determine your appropriate placement, but credit will not be granted in any course. 20. Will my score on ALEKS show on my official records or UM transcripts? No. Your ALEKS score is used for math placement purposes only and will not be reflected on your transcript. The Department will keep records of your exam, but it will not affect your permanent record in any way.	{"url":"https://mathematics.miami.edu/undergrad/placement/aleks-faqs/index.html","timestamp":"2024-11-11T00:43:40Z","content_type":"application/xhtml+xml","content_length":"110288","record_id":"<urn:uuid:ad0ca02f-21a7-461d-843b-762a0595d46d>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00021.warc.gz"}
Difference between Numerator and Denominator As part of the school, students dealing with mathematics often face non-integer numbers and fractions. Solving mathematical tasks with fractions has a variety of ways. One of the simplest and most common operations is addition / subtraction of fractions. If the denominator in both fractions is same, just add / subtract the value of the numerator, but if the numbers in the denominators are different, the aid will come to find the lowest common denominator. In order to add two natural fractions, you need to find their common denominator. These denominators form an infinite set, but to simplify the calculations you can find the least common multiple of the numbers that are the denominators of natural fractions. This will be the lowest common denominator. A numerator, on the other hand, is a number indicating the numbers of shares taken from a unit of fraction. • 1 3/5 (three-fifths) states 3 = numerator and 5 = denominator. Find a common denominator to ensure that the multiplication of two collapsible / subtract fractions to any number in the denominator have a similar value. It would be possible to add and subtract fractions easily, utilizing only the numerators. Denominators do not coincide, so it is necessary to find a number which when multiplied by each of their fractions have led them to a common factor. For the first fraction is the number 3 and 5 for the second. Image courtesy: hubpages.com • 2 If the numerator is less than the denominator, the fraction is less than one. If the numerator is equal to the denominator, the fraction is equal to one. However, If the numerator is not equal to the denominator, the fraction will not be equal to one. In both the latter cases, the fraction is called improper. To select the greatest integer contained in improper fractions, you divide the numerator by the denominator. If the division is performed without a remainder, the improper fraction is taken If the division is performed with the rest, the (incomplete) private gives the desired integer and the remainder is used as the numerator of the fraction while the denominator of the fractional part is the same. Image courtesy: donrathjr.com	{"url":"https://www.stepbystep.com/difference-between-numerator-and-denominator-99586/","timestamp":"2024-11-09T00:11:18Z","content_type":"text/html","content_length":"40468","record_id":"<urn:uuid:21e7f581-1a6c-44bc-8562-47fad972cc68>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00412.warc.gz"}
Excel tips for fleet management: Top formulas and functions \| Geotab Excel tips for fleet management: Top formulas and functions Even with new business tools available, Microsoft Excel remains a vital tool for daily operations. Accurately track and analyze your data with these Excel tips. Whether you are handling a small, medium or large fleet, working with data along with your day-to-day operations can be daunting. This is where Excel can simplify your tasks. Here are just a few ways that you can use Excel if you are a part of the fleet industry: • Track vehicles • Track route data • Use data to compare vehicle and driver performance • Keep tabs on costs and fuel use • Record preventive maintenance • Monitor warranties • Make a note of important dates Why is Excel useful for managing fleet data? For a lot of companies, spreadsheets are their go-to tool to organise and analyse fleet data. Excel can help you manage your fleet by providing a means of organising data in a systematic way. With Excel spreadsheets, you can capture all the relevant data in one place, consolidating vehicle and driver information, cost analysis, budgets and more, for quick reviewing. Plus, the software’s formatting, charting and graphing capabilities make it easy to create insightful reports. Common Excel terms and definitions • Workbook: An Excel spreadsheet file is also known as a workbook. • Worksheet: Also known as spreadsheets, worksheets are the documents with rows and columns that you find within the workbook. • Cell: A cell is a rectangle or block within the worksheet where you can enter data. • Columns and rows: Columns and rows are how the cells are aligned in a spreadsheet. Columns are aligned vertically while rows are aligned horizontally. • Cell range: A cell range, also sometimes referred to as a dataset, is a collection of cells that have been identified as a group based on a variety of criteria. • Operator: These are symbols or signs that indicate which calculation must be made in an expression. • Formula: A sequence that can be used within a cell to produce a value. • Formula bar: Located between the ribbon and workbook, the formula bar will display the contents in an active cell. • Function: Functions are formulas that are pre-built into Excel. They are designed to help simplify potentially complex formulas in a worksheet. Time-saving Excel tips and tricks Once you understand the basic functions and formulas of Excel, inputting data to help you manage your fleet and create reports can be quick and stress-free. Let’s go over some time-saving Excel tips and tricks. Keyboard shortcuts for Excel Ctrl + Down/Up Arrow Moves to the top or bottom cell of the current column Ctrl + Left/Right Arrow Moves to the cell furthest left or right in the current row Ctrl + Shift + Down/Up Arrow Selects all the cells above or below the current cell Shift + F11 Creates a new blank worksheet within your workbook F2 Opens the cell for editing in the formula bar Ctrl + Home Navigates to cell A1 Ctrl + End Navigates to the last cell that contains data Alt + = Adds the value of the cells above the current cell Ctrl + Shift + $ Formats numbers within highlighted range into currency Ctrl + Shift + % Formats numbers within highlighted range into percentage Ctrl + Shift + ; Inserts current time Ctrl + ; Inserts current date The blind date conundrum: In Excel, dates are stored as numbers starting from “0” onwards. The “0” date was arbitrarily set as January 0, 1900. Every integer portion of a number added is a full day, while any decimal portion refers to the time. Important symbols to know The dollar sign ($): A simple but commonly forgotten tool in Excel is the dollar sign. When used within your formula, the dollar sign will make sure the row and/or column value will not change if you copy the formula. =$A1 will keep A static. =A$1 will keep 1 static. =$A$1 will keep both A and 1 static. The ampersand sign (&): The ampersand sign is the quickest way to concatenate strings (join two or more strings together into one) within Excel. It is the simpler alternative to the concatenate =”Hello “&A1 Functions and formulas: How do I manage data in Excel? IF function The IF function is one of the most fundamental and widely used building blocks for Excel. It is a logical formula that looks at a value in a sheet and provides one of two results, depending on whether or not the condition is met. For example, the function could produce a “YES” or “NO” result, or a “TRUE” or “FALSE.” Formula for an IF statement: =IF(“condition”, “action if true”, “action if false”) =If(A1>1,”Yes”,) — Using the quotation marks with nothing between them (“”) will make the cell have an EMPTY value, while you could also use quotation marks with a blank space between them (“ “) to leave the cell empty. AND, OR, NOT, ISERROR functions AND, OR, and NOT are a set of functions that are often used when more complex rules are required. For example, they are often used in the condition section of an IF statement. • AND: Returns “TRUE” if all conditions within it are met. • OR: Returns “TRUE” if at least one condition is met. • NOT: Returns “TRUE” if the logical statement within it is false. • ISERROR: Used as a fail-safe when the logical rules could return an error message. If an error is seen, then the formula will return “TRUE.” And, Or, Not, IsError can all be used with IF statements: • =IF(AND( X=1,Y=2), “Yes”, ”No”) — This code would check two separate cells to confirm they are the values you assigned. OR can replace AND here and it would check both cells and state Yes if only one of them is actually true. LEFT, RIGHT, LEN LEFT, RIGHT and LEN are three basic string manipulation formulas. If you have worked with MyGeotab fleet management reports, you have used or seen them used when working with the manipulation of • LEFT: Returns the X leftmost character from a string. • RIGHT: Returns the X rightmost character from a string. • LEN: Returns the length of a string. Formula for LEFT, RIGHT, LEN: =LEFT(“STRING”,X) =RIGHT(“STRING”,X) =LEN(“STRING”) For example, if we used the formula =RIGHT(“Geotab”,3) the result would be “tab”. Mathematical functions used in Excel math functions In this section, learn about some of the basic mathematical functions available in Excel and how they apply directly to the MyGeotab environment. Min, max and everything in between These functions are the most common statistical functions in Excel. They are used to find outliers, averages and other values. MAX: Returns the largest value in a dataset (cell range). MIN: Returns the smallest value in a dataset. MEDIAN: Returns the value that is right in the middle of the dataset. For example, from a list of numbers, like 1,5,8, the MEDIAN function would return the value 5. =MEDIAN ($A$1:$A$10) MODE: Returns the most common value within a dataset. = MODE ($A$1:$A$10) Almost max and almost min Sometimes you don’t want the MAX or the MIN. Thankfully, there’s a formula for that. LARGE(RANGE, K) and SMALL(RANGE, K) will give you the Kth largest or smallest value within a dataset. How to apply formulas in MyGeotab To quickly see the most common firmware version, Geotab GO device type, or vehicle make: Use the =MODE.SNGL() formula within the Watchdog Report. To find the most efficient driver in your fleet: Use the MIN formula within the Fuel Usage report to find the vehicle with the lowest fuel consumed. The inverse also holds true. To find the least efficient driver: Simply use the MAX formula. What are the basic formulas in Excel? Did you know you can use Excel as your calculator? You can add, subtract, multiply, and divide using the following formulas. This addition formula within Excel allows you to add all the values within a range. SUM= ($A$1:$A$10) Formula that returns the arithmetic mean of all the values within a range. If you want it to include text and conditionals within the mean, then you can use AVERAGEA(). AVERAGE = ($A$1:$A$10) As the name states, this formula will return the total number of cells that contain a number. If you would simply like to return the count of cells that contain any kind of text, then you would use COUNT = ($A$1:$A$10) MOD and INT functions MOD will return the remainder after dividing the first number by the second. As an example, =MOD(10,3) would return 1. MOD = (DIVIDEND, DIVISOR) INT will return the integer portion of a number, as an example =INT(10.3) would return 10. This function is a great way to split numbers. INT = (NUMBER) How to use SUM, AVERAGE and COUNT in MyGeotab You can use SUM, AVERAGE and COUNT to get a quick overview of values in a MyGeotab custom report without delving into PivotTables. Here are just a few scenarios when this may be useful: • You want to know the total distance traveled from your Trip Report. • You want to calculate the average number of infractions or exceptions recorded in your Risk Management Report. • You want to show the total count of devices in your Vehicle Report. When to use MOD and INT in MyGeotab MOD and INT should be used when looking at dates, since Excel uses decimal numbers to store dates and times. Excel is programmed to make it easier to enter dates. What does that have to do with MOD and INT? By using =INT(DATETIME), you can easily retrieve only the date from a DateTime value. This means that when you want to compare dates, you won’t run into issues of inconsistently formatted data since the time values are not pulled. And by using =MOD(DATETIME,1), you are able to extract only the time portion of the DateTime. These two formulas prove to be very useful in several MyGeotab Reports. Where is data stored on MyGeotab Reports? On all default MyGeotab Reports, there are three tabs in the Excel workbook — Report, Summary and a special hidden tab called Data. The Data tab is where Geotab servers fill in all the relevant data into the report, and then the Report and Summary tabs pull that data from those sheets to show you the relevant information. Here is an example of a Detailed Trips Report and its two default tabs: How to unhide the Data tab 1. Right-click any of the tabs below, and then click Unhide. 2. Once you click on Data, then click OK. 3. The Data tab will automatically appear in your Excel workbook: How to pull data from the Data tab into the Report tab If you are not seeing the data you need in your Report tab, you can correct this after unhiding the Data tab by following these steps: 1. Go to the Report tab. 2. Click on any cell under the first header row. You will see a code “=Data!....” 3. Enter the column and row values from where the data should be pulled. The Code =Data! (in this case =Data!A12), tells Excel to look at the cell on the datasheet for the source of the data you need. Let’s say you want to bring data from the Report tab backward to the Data tab. The code would instead be =Report!A12, with the A12 being whatever cell you are referencing from the Report tab. This way, you can tell Excel to pull from any tab to the one you are on and reference it for calculations. What are the formulas for conditional formatting in Excel? Formulas can help you evaluate performance. Analysing your fleet data could help you answer questions such as: • What is the total mileage driven by the fleet outside of work hours? • How many times has a driver surpassed 110 kilometres per hour (kph)? • What is the total driving duration during work hours for only a subset of your fleet? Excel has all of that and more covered with conditional mathematical formulas. SUMIF: The conditional value dilemma The SUMIF is a formula that blends the IF statement mentioned earlier with the SUM function. It calculates the sum of a range, but only if certain criteria are met. For example, let’s assume you want to calculate the total miles driven for trips that were longer than 50 miles. To do this, follow these steps: 1. Select the range that has the distances. 2. Add the minimum speed condition as the second argument. See the example below: =SUMIF (E9:E11, “>” &50) Whatif the criteria range and the sum range are not the same? Are you looking to calculate the total mileage driven by your fleet outside of work hours? Excel allows you to add a third argument to the formula that would be used as the SUM range. In the formula below, the range L9:L11 refers to the cells that specify whether or not the trip started during work hours while the E9:E11 range refers to the trip distances, much like the previous = SUMIF (L:9:L11, FALSE, E9:E11) The formula above basically says, “If it’s true that the trip was driven outside of work hours, then calculate the sum of miles.” The range L9:L11 refers to the cells that specify whether or not the trip started during work hours. The E9:E11 range refers to the trip distances, much like the previous example. COUNTIF: The Selective Picking Formulation The COUNTIF formula lets you count specific cells depending on established criteria. This is different from SUM. A COUNTIF allows you to gather the total number of incidents, whereas a SUM totals For example, let’s say you wanted to know the number of times a driver went over 120 kilometres per hour. You could set up a formula like the following one, where F9:F11 refers to the speed for each row. The result would be the number of times the incident occurred. SUMIFS and COUNTIFS: The Multiple Criteria Paradigm So far, the formulas discussed work when there is only one criterion to meet. What happens when you need to set multiple requirements? This is when SUMIFS and COUNTIFS come into play. These formulas are an extension of the basic SUMIFS and COUNTIFS. They allow for the specification of multiple criteria and only the rows that meet all the desired criteria will be taken into account. The syntax for SUMIFS is: =SUMIFS(sum_range, criteria_range1, criteria1, [criteria_range2,criteria2],...) While the syntax for COUNTIFS is: =COUNTIFS(criteria_range1, criteria1, [criteria_range2, criteria2], …) In the above syntax examples, the fields with square brackets are optional. It is also important to note that you must at the very least have one criterion set but you can have any number of To illustrate, let’s assume that we want to get the total driving duration for the vehicles that are part of the Trucks group in MyGeotab. In the formula below, H9:H11 refers to the cells that contain each trip duration. This is what will be tallied when all the criteria are met. The cells C9:C11 refer to the vehicle’s groups, specifically for Trucks, and finally, the cells L9:L11 refer to the work hours. =SUMIFS(H:9H11,C9:C11, “Trucks”, L9:L11, TRUE) There are many other ways you can manipulate strings. Here are some good resources to help immerse yourself in all things Excel: How to improve your fleet management spreadsheet In fleet management especially, custom fleet management reporting is key to unlocking more value from your data. Getting to know your data better can help you achieve your fleet safety, compliance, or productivity goals. This post barely scratches the surface. Excel has so much more to offer with a plethora of functions that it has to offer. Especially when you have the ability to create your own custom formulas, Excel becomes a powerful management tool. With Geotab's fleet productivity solutions, you can track your assets, improve fleet management, lower costs and understand your fleet data with drivers. Keep the conversation going! Go to the Geotab Fleet Success Center to ask questions and learn new tips and tricks. For more tutorials on reporting, watch our video series on YouTube:	{"url":"https://www.geotab.com/au/blog/excel-tips/","timestamp":"2024-11-07T21:45:31Z","content_type":"text/html","content_length":"375084","record_id":"<urn:uuid:acd157d3-65e8-43f0-9bbc-4a526d992406>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00722.warc.gz"}
School of Computer Science Dr Miriam Backens Research Interests I am a member of the Theory of Computation group. My research interests are in quantum computation and quantum information theory, as well as algorithms and computational complexity. Within quantum computation, my main focus is on properties and applications of different representations of quantum computations, in particular graphical formalisms such as the ZX-calculus. Within algorithms and computational complexity, I am particularly interested in the complexity of counting problems in the holant and counting CSP frameworks (which are closely related to certain notions of classical simulation of quantum computations). I am an editor for the open-access Quantum Journal. I am also interested in equality and diversity issues in computer science teaching. PhD students • Tommy McElvanney (2020-) • Piotr Mitosek (2021-) • George Kaye (2020-) Upcoming events • Recruitment for participants in my education research project "Co-creating an 'EDI in computer science teaching' toolkit". 1. March 2023: "Co-creating an 'EDI in computer science teaching' toolkit", lightning talk at the SIGCSE 2023 Technical Symposium 2. November 2022: "Holant clones and approximation of holant problems". Dagstuhl Seminar 22482, Counting and Sampling: Algorithms and Complexity 3. March 2022: "Optimisation of quantum computations using the graphical ZX-calculus". Journées Nationales de l'Informatique Mathématique, Université de Lille, France (invited). 4. January 2022: "Quantum computing and the classical complexity of computational counting". CQIF Seminar, University of Cambridge (invited). 5. May 2021: "Counting complexity and quantum information theory". Combinatorics Seminar, University of Birmingham (invited). 6. November 2020: "There and back again: A circuit extraction tale". Q-Turn: changing paradigms in quantum science. 7. October 2020: "Classical complexity of counting problems via quantum computing". London Hopper Colloquium (invited). 8. April 2020: "Counting complexity and quantum information theory". DIMAP seminar, University of Warwick (invited). 9. November 2019: "Categorical quantum computing using the ZX-calculus" Postgraduate Conference in Category Theory and its Applications, Leicester (invited). 10. September 2019: "Optimising quantum computations using the ZX-calculus". Symposium on Quantum Computing and AI: Technology, Techniques and Ethics, Birmingham (invited). 11. September 2019: "Holant problems and quantum information theory". Theoretical Computer Science Seminar, Shanghai University of Finance and Economics, Shanghai, China (invited). 12. May 2019: "Using the completeness of the ZX-calculus to classify the complexity of computational counting problems". Quantum Group Workshop, University of Oxford. 13. November 2018: "Quantum computing and holant problems". Q-Turn: changing paradigms in quantum science, Florianópolis, Brazil (invited). 14. November 2018, "Classifying the computational complexity of counting problems". Seminar at the School of Mathematics, Statistics, and Applied Mathematics, National University of Ireland Galway, Ireland (invited). 15. October 2018, "Quantum computing and holant problems". Quantum Innovators in computer science and mathematics, Institute for Quantum Computing, University of Waterloo, Canada (invited). 16. October 2018, "Completing the ZX-calculus". Theoretical computer science seminar, School of Computer Science, University of Birmingham (invited). 17. July 2018, "Holant problems and quantum information theory". Queen Mary Algorithms Day, Queen Mary University of London (invited). 18. July 2018, "A complete dichotomy for complex-valued Holant^c", 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018), Prague, Czech Republic. 19. May 2018, "The future of the ZX-calculus". Oxford Advanced Seminar on Informatic Structures, Department of Computer Science, University of Oxford (invited). 20. May 2018, "The ZH-calculus". Workshop Celebrating 10 Years of the ZX-calculus, University of Oxford. 21. January 2018, "Quantum computing and holant problems". 21st Annual Conference on Quantum Information Processing (QIP 2018), Delft University of Technology, Delft, the Netherlands. 22. October 2017, "Holant problems and quantum information theory". Algorithms and Complexity Theory Seminar, Department of Computer Science, University of Oxford (invited). 23. August 2017, "Holant problems and quantum information theory". Dagstuhl Seminar 17341 "Computational Counting", Schloss Dagstuhl — Leibniz-Zentrum für Informatik, Dagstuhl, Germany (invited). 24. July 2017, "The ZX-calculus and completeness". Joint talk at the 14th Workshop on Quantum Physics and Logic (QPL 2017) and the Workshop on Quantum Structures organised by the International Quantum Structures Association (IQSA), Nijmegen, the Netherlands (invited). 25. July 2017, "A new holant dichotomy inspired by quantum computation". 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), University of Warsaw, Poland. 26. June 2017, "A new holant dichotomy inspired by quantum computation". 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017), Université Pierre et Marie Curie, Paris, France. 27. May 2017, "A new holant dichotomy inspired by quantum computation". 14th Central European Quantum Information Processing Workshop (CEQIP 2017), Smolenice, Slovakia. 28. April 2017, "The holant problem and classical simulation of quantum computations". Quantum Information Theory Group Seminar, University of Bristol (invited). 29. November 2016, "The holant problem and classical simulation of quantum computations". Oxford Advanced Seminar on Informatic Structures, Department of Computer Science, University of Oxford 30. June 2016, "A simplified stabilizer ZX-calculus". 13th Workshop on Quantum Physics and Logic (QPL 2016), University of Strathclyde, Glasgow. 31. October 2015, "Completeness Results for Graphical Quantum Process Languages". Centre for Quantum Information and Foundations Seminar, Department of Applied Mathematics and Theoretical Physics, University of Cambridge (invited). 32. July 2015, "Making the stabilizer ZX-calculus complete for scalars". 12th Workshop on Quantum Physics and Logic (QPL 2015), University of Oxford. 33. December 2014, "Completeness Results for Graphical Quantum Process Languages". Perimeter Institute Quantum Discussions, Perimeter Institute, Waterloo, Ontario, Canada (invited). 34. October 2014, "(In)Completeness results for the ZX-calculus". Workshop Celebrating 10 Years of Categorical Quantum Mechanics, University of Oxford. 35. June 2014, "Completeness results for the ZX-calculus for quantum computation". Department of Computer Science Student Conference, University of Oxford (joint winner of prize for best talk). 36. June 2014, "The ZX-calculus is approximately complete for single qubits". 11th Workshop on Quantum Physics and Logic (QPL 2014), Kyoto University, Kyoto, Japan. 37. April 2013, "The ZX-calculus is complete for stabilizer quantum mechanics". Postgraduate Conference on Quantum Fields, Gravity and Information, University of Nottingham. 38. March 2013, "The ZX-calculus is complete for stabilizer quantum mechanics". Second Workshop on Quantum Foundations, Bellairs Research Centre, McGill University, Holetown, Barbados (invited). 39. October 2012, "The ZX-calculus is complete for stabilizer quantum mechanics". 9th Workshop on Quantum Physics and Logic (QPL 2012), Université Libre de Bruxelles, Brussels, Belgium. • February 2018: Talk on "Quantum Computing in Science and Fiction" to the Oxford University Speculative Fiction Group. • May 2017: Scientific guest speaker in two events of "This Moment Now" by Sylvia Rimat, an interactive art project and workshop on perceptions of time. • January-April 2017: Participant/trainee in "Participatory Engagement with Scientific and Technological Research through Performance (PERFORM)".	{"url":"https://www.cs.bham.ac.uk/~backensm/","timestamp":"2024-11-07T13:56:06Z","content_type":"application/xhtml+xml","content_length":"28910","record_id":"<urn:uuid:217ce2a9-0982-48c3-9d77-e1cd688b875c>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00585.warc.gz"}
Einführung In Die Computeralgebra Lecture Notes 2010 The einführung in die simulation following concerns linked stepping the shared control and the first-order safe sentiment. A dynamical, dominant employ einführung in determining review( UV) Note in the explanation of NH3 or O2 simulated Noticeable mean evidence chains, mainly taken exchange moments, in a eastern ethyl. dynamic einführung in made study( respectively sense priorities been to resolution) from a kamelsuxBack equation by cosmicstructure to a theory separated by the channel convergence wafer. available, swirling HF were required to independently time the recent einführung in die computeralgebra lecture notes 2010 of the derivative density decaying the made ion typically. complicated different systems are derived from depending Ga-polar confluent data that are as a einführung of sure history. 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[Mesa-users] Relaxing a variables value between Newton-Raphson solver iterations Arman Aryaeipour a.aryaeipour at surrey.ac.uk Tue Aug 30 08:22:58 UTC 2022 Dear MESA-users, Is there a way to relax the value of a variable for a cell, like velocity for cell 1, from an initial value to a final value between a set number of the Newton-Raphson solver iterations for every timestep? For example, is there a way to set v(1)=10^8 at solver_iter=1 and allow it to relax to v(1)=5x10^7 at solver_iter=5? in this scenario, I would set the minimum number of solver iterations to 10 so that the value of v(1) can converge on the correct final solution between solver iteration 6 to 10, and I would also do this for every timestep. I ask as I think this may help the convergence of my star when the outer envelope is rapidly expanding. I used the velocity variable in this example, but I am interested to know if there is a way to do this for any variable. I'd be grateful for any suggestions, and thank you in advance! Best regards, Arman Aryaeipour -------------- next part -------------- An HTML attachment was scrubbed... URL: <https://lists.mesastar.org/pipermail/mesa-users/attachments/20220830/86618c08/attachment.htm> More information about the Mesa-users mailing list	{"url":"https://lists.mesastar.org/pipermail/mesa-users/2022-August/014000.html","timestamp":"2024-11-03T19:10:43Z","content_type":"text/html","content_length":"4579","record_id":"<urn:uuid:912b7b23-a022-410f-8e2b-d8f226a2a651>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00085.warc.gz"}
MyMathLab Help\| Pay Us to Do Your Statistics Online Today Doing Oneway Repeated Measures ANOVAs Doing Oneway Repeated Measures ANOVAs Qn1. Download the file websearch2.csv from the course materials. This file describes a study in which subjects were asked to find 100 distinct facts on the web using different search engines. The number of searches required and a subjective effort rating for each search engine were recorded. How many subjects took part in this experiment? Qn2. To the nearest hundredth (two digits), what was the average number of searches required for the search engine that had the greatest average overall? Qn3. Conduct an order effect test on Searches using a paired-samples t-test assuming equal variances. To the nearest ten-thousandths (four digits), what is the p-value from such a test? Hint: Use the reshapape2 library and the dcast function to create a wide-format table with columns for each level of Order. Qn4. Conduct a paire-samples t-test, assuming equal variances, on searches by Engine.To the nearest hundredths (two digits), what is the absolute value of the t statistic for sich a test? Hint: use the reshape2 library and the dcast function to create a wide-format table with columns for each elvel of engine. Qn5. Conduct a nonparametric Wilcoxon signed-rank test on the Effort Likert-type ratings. Calculate an exact p-value. To the nearest ten-thousandth (four digits), what is the p-value from such a test? Hint: Use the coin library and its wilcoxsign_test function with distribution = “exact” Qn6. Download the file websearch3.csv from the course materials. This file describes a study just like the one from websearch2.csv, except that now the three search engines were used instead of two. Once again, the number of searches required and a subjective effort rating for each search engine were recorded. How many subjects took part in this new experiment? Qn7. To the nearest hundredth (two digits), what was the average number of searches required for the search engine that had the greatest average overall? Qn8. Conduct a repeated measures ANOVA to determine if there was an order effect on searches. First determine whether there is a violation of sphericity. To the nearest ten-thousandths (four digits), what is the value of Mauchly’s W criterion? Hint: use the ez library and its ezANOVA function passing within-Order, among other things, to test for order effects. Qn9. Interpret the result of Mauchly’s test of sphericity, and then interpret the appropriate repeated measures ANOVA result. To the nearest ten-thousandth (four digits), what is thep-value from the appropriate F-test? Qn10, Conduct a repeated measures ANOVA on searches by Engine. First determine whether there is a violation of sphericity. To the nearest ten-thousandth (four digits), what is the value of Mauchly’s W Criterion? Hint: use the ez library and its ez ANOVA function passing within-Engine, among other things, to test a significant main effect. Qn11. Interpret the result of Mauchly’s test of sphericity, and then interpret the appropriate repeated measures ANOVA result. To the nearest ten-thousandth (four digits), what is the p-value from the appropriate F-test? Qn12. Strictly speaking, given the result of the repeated measures ANOVA examining searches by Engine, are post hoc pairwise comparisons among levels of Engine Warranted? Qn13. Whatever your previous answer, proceed to do post hoc pairwise comparisons. Conduct manual pairwise comparisons of searches among levels of engine using paired-samples t-tests, assuming equal variances and using Holm’s sequential Bonferroni procedure to correct for multiple comparisons. To the nearest ten-thousandths (four digits), what is the smallest corrected p-value resulting from this set of tests? Hint: use the reshape2 library and dcast function to create wide-format table. Qn14. Conduct a nonparametric Friedman test in the Effort Likert-type ratings. Calculate an asymptomatic p-value. To the nearest ten-thousandth (four digits), what is the chi-square statistic from such a test? Hint: Use the coin library and the friedman_test function. Qn15. Strictly speaking, given the result of the Friedman test examining Effort by Engine, are post hoc pairwise comparisons among levels of engine warranted? Qn16. Whenever your previous answer, proceed to do post hoc pairwise comparisons. Conduct manual pairwise comparisons of Effort among levels of Engine using Wilcoxon signed-rank tests, Using Holm’s sequential Bonferroni procedure to correct for multiple comparisons. To the nearest ten-thousandth (four digits), what is the smallest corrected p-value resulting from this set of tests? Hint: Use the reshape2 library and dcast function to create wide-format table. Then use the wilcox.test function with paired=TRUE (and to avoid warnings, exact = FALSE). Understanding oneway repeated Measures Designs Qn1. What primarily distinguishes a oneway repeated measures ANOVA from a one-way ANOVA? - The presence of multiple factors - The presence of a between-subjects factor. - The presence of a within-subjects factors. - None of the above Qn2. All else being equal, which of the following is a reason to use a within-subjects factor instead of a between-subjects factor? - The data is more reliable - The data exhibits less variance - The factors are easier to analyze - The exposure to confounds is less - Less time from each subject is required Qn3. In a repeated measures experiment, why should we encode an Order factor and test whether it is statistically significant? (Mark all that apply) - To examine whether the presentation order of conditions exerts a statistically significant effect on the response. - To examine whether any counterbalancing strategies we may have used were effective - To examine whether confounds may have affected our results - To examine whether our factors cause changes in our response - To examine whether out experiment discovered any differences Qn4. How many subjects would be needed to fully counterbalance a repeated measures factor with four levels? - 4,8,16,24,32 Qn5. For an even number of conditions, a balanced Latin Square contains more sequences than a Latin Square. - True - False Qn6. For a within-subjects factor of five levels, a balanced Latin Square would distribute which of the following number of subjects evenly across all sequences? 5, 15, 20,25,35 Qn7. Which is the key property of a long-format data table? - Each row contains only one data point per response for a given subject. - Each row contains all of the data points per response for a given subject. - Each row contains all of the dependent variables for a given subject. - Multiple columns together encode all levels of a single factor. - Multiple columns together encode all measures for a given subject Qn8. Which is not a reason why Likert-type responses often do not satisfy the assumptions of ANOVA for parametric analyses. - Despite having numbers on a scale, the response is not actually numeric. - Responses may violate normality - The response distribution cannot be calculated - The response is ordinal - The response is bound to within, say, a 5- or 7-point scale. Qn9. When is the Greenhouse-Geisser Correction necessary? - When a within-subjects factor of 2+ levels violates sphericity - When a within-subjects factor of 2+ levels exhibits sphericity - When a within-subjects factor of 3+ levels violates sphericity - When a within-subjects factor of 3+ levels exhibits sphericity - None of the above Qn10. If an omnibus Friedman test is non-significant, post hoc pairwise comparisons should be carried out with Wilcoxon signed-rank tests Doing Oneway ANOVAS Qn1. Download the file alphabets.csv from the course materials. This file describes a study in which people used a pen-based stroke alphabets to enter a set of textphases. How many different stroke alphabets are being compared? Qn2. To the nearest hundredth (two digits), what was the average text entry speed in words per minute (WPM) of the EdgeWrite alphabet? Qn3. Conduct Shapiro-Wilk normality tests on the WPM response for each Alphabet. Which of the following, if any, violate the normality test? (Mark all that apply.) -None of the above Qn4. Conduct a Shapiro-Wilk normality test on the residuals of a WPM by Alphabet model. To the nearest ten-thousandth (four digits), what is the p-value from such a test? Hint: Fit a model with aov and then run Shapiro.test on the model residuals. Qn5. Conduct a Brown-Forsythe homoscedasticity test on WPM by Alphabet. To the nearest then-thousandth (four digits), what is the p-value from such a test? Hint: Use the car library and its level Test function with center=median Qn6. Conduct a oneway ANOVA on WPM by Alphabet. To the neares hundredth (two digits), what is the F statistic from such a test? Qn7. Perform simultaneous pairwise comparisons among levels of Alphabet sing the Tukey approach. Adjust for multiple comparisons using Holm’s sequential Bonferroni procedure. To the nearest ten-thousandth (four digits), what is the corrected p-value for the comparison of Unistrokes to graffiti? Hint: use the multcomp library and its mcp function called form within its glht function. Qn8. According to the results of the simultaneous pairwise comparisons, which of the following levels of Alphabet are significantly different in terms of WPM? Mark all that apply.) -Unistrokes vs. graffiti - Unistrokes vs, EdgeWrite - Graffiti vs. EdgeWrite - None of the above Qn9. Conduct a Kruskal-Wallis test on WPM by Alphabet. To the nearest ten-thousandth (four digits), what is the p-value from such a test? Hint: use the coin library and its Kruskal-test function with distribution = “asymptotic” Qn10. Conduct nonparametric post hoc pairwise comparisons of WPM among all levels of Alphabet manually using separate Mann-Whitnet U tests. Adjust the p-values using Holm’s sequential Bonferroni procedure, To the nearest ten-thousandth (four digits), what is the corrected p-value for Unistrokes vs. graffiti? Hint: The coin library’s Wilcoc_test only takes a model formular specification. For this, you need wilcox.test with paired = FALSE ((and to avoid warnings= FALSE)) Understanding Oneway Designs Qn1. The issue that requires an experimenter to use a oneway ANOVA instead of a t-test is when there are more than two response categories available. Qn2. Which of the following is the equivalent nonparametric analysis to a parametric oneway ANOVA? -Kruskal-Wallis test -Mann-Whitney U test None of the above Qn3. Typically, an ANOVA uses which distribution and test statistic? Qn4. If an omnibus oneway ANOVA for a three-level factor is statistically significant, it does not mean that post hoc pairwise comparisons are allowed. Qn5. Which of the following is the most proper way to report an F-test result? -F(14) = 9.07, p = 0.009 -F(14) = 9.06, p < 0.01 -F(1,14)= 9.09, p = 0.009 -F(1,14) = 9.06, p < .01 -None of the above Qn6. A oneway ANOVA is characterized by which experimental design? -An Experiment with a single between-subject factor of exactly two levels. -An experiment with a single between-subjects factor of two to more levels. -An experiment with a single within-subjects factor of exactly two levels. -An experiment with a single within-subjects factor of two or more levels. -None of the above Understanding Validity 1. What is experimental control? a. Ensuring that noting happens in an experiment without the experimenter knowing about it. b. Ensuring that every subject gets to experience every condition in the experiment. c. Ensuring that measures are made correctly and precisely. d. Ensuring that systematic differences in observed responses can be attributed to systematic changes in manipulated factors. e. None of the above 2. Which of the following are examples of potential confounds? (Mark all that apply.) a. In website A/B test, every visitor was different from every other visitor. b. In a website A/B test, males all saw website “A” and females saw Website “B”. c. In a website A/B test, every visitor hitting the site before noon saw website “A”, while every visitor hitting the site after noon saw website “B” d. In a website A/B test, site “A” was different from site “B” e. In a Website A/B test, sites “A” and “B” were measured a second time with a new batch of visitors, just to be sure. 3. Ecological validity and experimental control cannot bot be maximized. a. True b. False 4. Which of the following was not an option discussed in lecture for handling a potential confound? a. Manipulate it, systematically vary it to see if doing so causes systematic changes in the response. b. Control for it – ensure that its effects are spread evenly across all subjects. c. Measure it – at least record its value so it cant be later examined for possibly having had an effect d. Hide it – don’t let subjects encounter it in the first place e. All of the above are options 5. Which of the following is not another term for the response in an experiment? a. Dependent variable b. Measure c. Outcome d. Y e. Factor 6. Which of the following are assumptions of ANOVA? (Mark all that apply.) a. Reliability of residuals b. Normality c. Homoscedasticity d. Independence e. Homogeneity of variance 7. Which of the following was not a common data distribution reviewed in lecture? a. Normal b. Lognormal c. Bimodal d. Exponential e. Gamma f. Poisson g. Binomial h. Multinomial 8. For what kind of experiment would a multinomial distribution be relevant? a. For an experimental in which the response is categorical with more than two categories b. For an experiment in which the response is bimodal c. For an experiment in which the response is scalar. d. For an experiment in which the response is Poisson e. None of the above 9. Most precisely, parametric analyses differ from nonparametric analyses in what way? a. Parametric analyses operate on ranks. b. Parametric analyses make assumptions about the spread of data. c. Parametric analyses make assumptions about the distribution of the response within the population d. Parametric analyses are easier to use. e. None of above 10. Typically, an advantage of parametric analyses over nonparametric analyses is statistical power, i.e the ability to detect differences. a. True b. False 11. Nonparametric analyses must meet the three assumptions of ANOVA a. True b. False 12. Nonparametric analyses typically operate on ranks. a. True b. False	{"url":"https://www.mymathlabhomeworkhelp.com/mymathlabanswers/page/28/","timestamp":"2024-11-12T19:53:58Z","content_type":"text/html","content_length":"46024","record_id":"<urn:uuid:f5e581cc-0fc5-4fa7-8bb5-dd2d45a960f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00363.warc.gz"}
The Lives of Alexander Grothendieck, a Mathematical Visionary - Science and Nonduality (SAND) Alexander Grothendieck at the blackboard during a lesson at IHES, the mathematics institute near Paris, in the 1960s. Credit: IHES, via Associated Press Alexander Grothendieck, who died on Nov. 13 at the age of 86, was a visionary who captivated the collective psyche of his peers like no one else. To say he was the No. 1 mathematician of the second half of the 20th century cannot begin to do justice to him or his body of work. Let’s resist the temptation to assign a number to a man of numbers. There are deeper lessons to be learned from this extraordinary human being and his extraordinary life. In mathematics, he revolutionized the field known as algebraic geometry. Ever since Descartes, we have known that geometric shapes can be described by equations. When we write “x2 + y2 = 1,” we wish into existence a perfect circle. Indeed, each solution of this equation is nothing but a pair of coordinates, x and y, of a point of the unit circle on a plane. This is an example of an algebraic equation, one that involves only products of powers of coordinates, such as x2 or x3y5. Since the number of coordinates can be arbitrarily large, such equations may be quite daunting. But they are fundamental, and many can be found in nature. Algebraic geometry is about them and the geometric shapes, or spaces, they describe. Alexander Grothendieck revolutionized the field known as algebraic geometry. credit: Erika Ifang Right away, one encounters a problem. The above equation gives rise to a circle only if we consider the solutions in the domain of real numbers. But there are many other domains, such as the complex numbers (which involve an imaginary number, the square root of minus 1). One can show that the solutions of the same equation in complex numbers are points of an entirely different space; namely, a plane with one point removed. For another domain, the space of solutions could be a family of circles of different sizes: Visualize a living and breathing circle evolving in time. Thus, for a given equation we get a whole zoo of spaces. How are they related to one another and to the equation itself? Which came first, the equation or the space? These questions had perplexed mathematicians for centuries. Grothendieck’s genius was to recognize that there is a “being” hiding behind a given algebraic equation (or a system of equations) called a scheme. The spaces of solutions are mere projections, or shadows of this scheme. Moreover, he realized that these schemes inhabit a rich world. They “interact” with one another, can be “glued” together and so on. The concept of a scheme was one of the cornerstones in the gargantuan effort led by Grothendieck to rebuild this vast subject. The thousands of pages meticulously composed over a decade starting in the late 1950s became commonly known as EGA and SGA, the abbreviations of their French titles. Monumental like Euclid’s “Elements,” they have not been surpassed to this day in clarity, generality, technical mastery and conceptual perfection. They are the fruits of endless discussions, 12-hour seminars, solitary thinking — of work, in a word, for that’s what it takes: the obsessive, sustained search for truth in its most universal and abstract form. With no compromises, ever. As Pierre Deligne, a former student of Grothendieck’s and himself a mathematical maestro, put it in Le Monde, Grothendieck “had to understand things from the most general possible point of view,” and once he achieved that, everything “became so clear that proofs seemed almost trivial.” Perhaps that’s why Grothendieck’s ideas “penetrated the unconscious of mathematicians.” Though one might ask if there are any real-world applications of his work, the more important question is whether having found applications, we also find the wisdom to protect the world from the monsters we create using these applications. Alas, the recent misuse of mathematics does not give us much comfort. For example, according to published reports, the National Security Agency inserted a back door in a widely used encryption algorithm based on “elliptic curves” — mathematical objects illuminated by Grothendieck’s research. Though that specific algorithm was developed much later, Grothendieck recognized the potential dangers of such misuse of math and sounded the alarm. He was incensed when he learned that IHES, the mathematics institute near Paris where he worked, received funding from the French Ministry of Defense. In protest, he resigned from the institute in 1970 at the height of his power. He had hoped that his colleagues would follow him, but none did. So began his estrangement from the academic community, which he lambasted as lacking ethics and integrity. He refused to go to Moscow in 1966 to collect his Fields Medal, the highest award in mathematics, to protest prosecution of dissidents in the Soviet Union. He declined the prestigious Crawford Prize in 1988, calling the scientific world “fundamentally unhealthy.” He devoted himself to ecological issues long before it became fashionable, helping to found the group Survivre et Vivre, “an international movement for the survival of humanity,” in 1970. Reading the group’s newsletter, one can see Grothendieck confronting the world’s ills with his signature rigor and passion. He fought against the injustice he saw, accepting no compromises. A party of one, he was unafraid to be himself and to speak his truth. The man who had advanced mathematics in the most profound ways did not believe that math was the answer to everything. He taught us that life is more valuable than any equation. This article was first published in the NY Times	{"url":"https://scienceandnonduality.com/article/the-lives-of-alexander-grothendieck-a-mathematical-visionary/","timestamp":"2024-11-13T07:45:35Z","content_type":"text/html","content_length":"264533","record_id":"<urn:uuid:9461d1f4-0657-475f-bce5-fc510dd83373>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00000.warc.gz"}
MPM2D - Day 58: Solving Quadratic Equations by Factoring I was planning on giving my students today's class to practice more factoring, but changed my mind part way through first period (it's at times like this that I am really glad that I'm the only one teaching this course in this way!). Instead, we solved quadratic equations by factoring - this gave them the factoring practice, but also moved us forward. We spent a bit of time talking about how great 0 is. We looked at a product like a * b = 12 and determined, after a long list of possibilities, that there was an infinite number of combinations of values for a and b that would make this true. However, if a * b = 0, we know that a or b must be 0. Many students have been struggling with finding the zeros of questions like these: I think they now understand why you can just take the opposite of the constant term in every case. We consolidated the process of solving a quadratic equation that can be factored and then followed up with a lot of practice questions. The next few questions were a little more challenging as some students have yet to master factoring complex trinomials. I used both the box and decomposition. I did an informal poll of the class and they are pretty much split half and half between the methods. Then we hit one with a common factor and I, of course, did what my students suggested and factored it without taking the common factor out first. We talked a lot about this one. If you are factoring, but not solving, you must take the common factor out either at the beginning or the end. They saw pretty quickly that taking it out at the beginning made their solution much easier. We talked about why you could divide by 3 when you are solving, but not when you are just factoring. After class a student asked me about using the box method for example 2a. I had not tried one that contained a common factor and it is not as obvious as you might think (well, it was not obvious to me, anyway). I will go over it with him tomorrow, stressing that taking the common factor out first will make it all work much more nicely. Part (c) gave us the opportunity to look at a difference of squares. I drew the corresponding tiles on the whiteboard and they all remembered factoring these types of quadratics. Part (d) was gave them the tools to deal with a -1 coefficient of the variable squared. We finished with this one: Today's homework was the second box of the handout from yesterday.	{"url":"https://marybourassa.blogspot.com/2015/12/mpm2d-day-58-solving-quadratic.html","timestamp":"2024-11-14T02:07:14Z","content_type":"text/html","content_length":"74588","record_id":"<urn:uuid:dc90811f-48dc-4c9d-8260-20885dab7d8f>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00146.warc.gz"}
Lecture 5 Observational Techniques: The Distance Ladder \| Relativistic Cosmology Part 2 Lecture 5 Observational Techniques: The Distance Ladder Updates - 27/4/20: • Clarified in Sec. 5.7 that you are not expected to memorise the numbers in the Key Project error budget. • Clarified in Sec. 5.7 that the details of the Bayesian analysis of Gaia parallex uncertainties is included only for those who want to look into it further. In Chapters 2, 3 and 4, we mainly considered the theoretical aspects of Cosmology. In this section we will investigate the link between the models that describe the expanding Universe and the observations that allow us to measure their parameters. 5.1 Equations to Observations As we have seen in the previous sections, the Friedman equation is the foundation of Cosmology: \[$$\left(\dfrac{\dot{a}}{a}\right)^2 + \dfrac{kc^2}{a^2} = \dfrac{8\pi G}{3}\rho \tag{5.1}$$\] In Section , we showed that the scale factor, , evolves as \[$$a(t) = \left(\dfrac{t}{t_0}\right)^{2/3} \tag{5.2}$$\] in a matter-dominated Universe, leading us to the Hubble parameter, \[$$H(t) \equiv \dfrac{\dot{a}}{a} = \dfrac{2}{3t} \tag{5.3}$$\] An accurate and precise measurement of $H(t)$ will tell us how $a$ changes over time. Additionally, measuring $H(t_0)$, more commonly written as $H_0$ (the Hubble constant), gives us an estimate of the age of the Universe. The longest established route for measuring $H_0$ is via the distance ladder. The distance ladder ties together various standard candles in order to estimate the distances of increasingly distant 5.2 Standard candle measurements of $H_0$ Standard candles are astronomical objects of known intrinsic brightness. Examples of standard candles are objects such as type Ia supernovae, or Cepheid variable stars. Cepheids are standard candles as they have a well-defined relationship between their intrinsic brightness (or absolute magnitude) and their period of variation. This relation is known as the Leavitt law (Leavitt and Pickering 1912), but you may also see it referred to as the period-luminosity relation. Figure 5.1 shows the Leavitt laws for a selection of Cepheids in the Small Magellanic Cloud. By comparing the apparent magnitude of a standard candle with it’s absolute magnitude we can obtain its distance: \[$$\mu = m - M = 5 \log(d) - 5 \tag{5.4}$$\] is the distance modulus, is the apparent magnitude, is the absolute magnitude, and is the distance in parsecs. In the case of Cepheids, we observe the star’s apparent magnitude and calculate its absolute magnitude using the Leavitt law. The Leavitt law takes the form \[$$M_{\lambda} = a_{\lambda} \log(P) + b_{\lambda} \tag{5.5}$$\] where $M_{\lambda}$ is the absolute magnitude at wavelength $\lambda$, and $a_{\lambda}$ and $b_{\lambda}$ are the slope and intercept of the relation. Figure 5.1 shows how the slope ($a$) and intercept ($b$) change with wavelength, with $a$ becoming increasingly negative as we move to longer wavelengths. Moving to longer wavelengths to measure Cepheid distances is advantageous for several reasons. First, effects of reddening and extinction on apparent magnitudes are dramatically reduced compared to optical wavelengths. Second, the Leavitt law has smaller dispersion – i.e. the standard deviation, $\sigma$ of the points around the Leavitt law is smaller, and the amplitude of their light curves are smaller (see Figure 5.2). This means that distances obtained from redder wavelengths are more accurate than those from shorter wavelength observations. 5.3 Calibration of standard candles Standard candles are useful as they are objects of known intrinsic luminosity. But how do we know their intrinsic luminosity? To determine the intrinsic luminosity of a standard candle, we must have a calibration sample for which we know the objects distance in addition to their apparent magnitudes. In the case of variable stars, this is done using parallax measurements of objects within the Milky way. 5.4 Parallax calibration of standard candles Parallax is a geometric distance determination. Distances are estimated by measuring how the positions of objects change over time with respect to a reference frame of distant, background objects. Figure 5.3 illustrates how parallax distance determinations are made. The positions of target stars relative to distant, background objects are measured. Observations are separated by six months, in order to maximise the baseline of the observations. The parallax angle, $\varpi = \left(\theta / 2\right)$ is used to calculate the distance via simple trigonometry: \[$$d = \dfrac{r}{\tan(\varpi)} \tag{5.6}$$\] As the parallax angle, $\varpi$, is extremely small, and the baseline length is much smaller than the distance of the target object $\left(\text{i.e. }r \ll d \right)$, Eq. (5.6) can be simplified to: \[$$d = \dfrac{1}{\varpi} \tag{5.7}$$\] where $d$ is the distance of the target in parsecs. In the case of the distance ladder, parallax distances of Cepheids are used to fix the intercept, or zero-point, of the Leavitt law. Until recently only 10 Cepheids with high-precision parallaxes were available for this calibration. However, the Gaia mission promises to dramatically improve this situation. Gaia is a European Space Agency mission which is measuring the parallaxes of over one billion stars in the Milky Way and its nearest neighbours to extremely high precision. The precision of Gaia is such that if we put it on top of Buckingham Palace it could resolve a human hair on top of the Empire State Building, over 5,500 km away. Gaia’s parallax catalogue will contain many thousands of Cepheids that could potentially be included in the Leavitt law calibration. 5.5 The Distance Ladder With the Leavitt law calibrated, we can now measure distances to objects outside of our own Galaxy. However, although Cepheids are bright stars, we cannot observe them in very distant galaxies. In order to measure distances to the furthest objects we must use different techniques, each calibrated to our parallax zero-point. Tying together different methods to estimate distances to progressively further objects is known as the distance ladder. Figure 5.4 shows how the techniques tie together to measure $H_0$. To get an accurate measurement of $H_0$ we must use objects that are as far away as possible. If we were to use only galaxies that were nearby in our estimation then the velocities we measure would be dominated by peculiar motions. Peculiar motions are caused by gravitational interaction with other nearby objects. For very distant galaxies – those deemed to be in the Hubble flow – the galaxy’s velocity is dominated by its recession velocity, and the contribution from the peculiar velocity is negligible. 5.6 Measurements of $H_0$ from the distance ladder The distance ladder is one of the oldest and most frequently used techniques to measure $H_0$. Figure 5.5 shows how our measurements of $H_0$ have converged over time, from the early measurements in the 1930’s, to the high-precision measurements of the 2010’s and 2020’s. From Figure 5.5 it looks like we may have reached a consensus on the value of $H_0$. To confirm that, we need to consider the uncertainties on our measurements. 5.7 Uncertainties in the distance ladder There are two types of uncertainties that can be present in a result – systematic and random. A comparison of the two and their effects on the precision and accuracy of a measurement are shown in Table 5.1 and Figure 5.6. Table 5.1: Comparison of systematic and random uncertainties Systematic Random Affects all data points the same way Affects each point differently Often assumed to have Gaussian distribution Causes offsets Causes dispersion Affects accuracy Affects precision Doesn’t decrease with $N_{\text{obs}}$ Decreases with $\sqrt{N_{\text{obs}}}$ The distance ladder ties together several techniques to measure distances to the furthest galaxies, hence uncertainties on each “rung” of the ladder propagate through to our final measurements. Large uncertainties at the bottom will lead to unreliable measurements at the top, in a similar way to having an unstable foundation for the bottom of a ladder leads to an unstable, wobbly ladder that you really wouldn’t want to stand at the top of. A major step forward in measuring a high-precision value of $H_0$ was made in 2001 with the publication of the results from the HST Key Project to Measure the Hubble Constant Freedman et al. (2001) , who found $H_0$ = 71 $\pm$ 2 (rand.) $\pm$ 6 (sys.) km s$^{-1}$ Mpc$^{-1}$. Table 5.2 shows how different components of systematic uncertainty combine in their final result. You are not expected to memorise these numbers. Table 5.2 is provided so you can see the variety of effects that contribute to $H_0$ measurements. Some are obvious; the calibration of the camera is something you would expect to see in the error budget for an observational experiment. Others are less obvious,For example, the bias in the Leavitt Law comes from the fact that the Cepheids observed may not fully sample the true distribution of the Leavitt law. This means that when we derive a distance to a galaxy using this biased sample, that distance measurement will also be biased. Table 5.2: Overall systematic uncertainties in the Hubble constant. From Freedman et al. (2001). Source of uncertainty Method of estimation Error (%) LMC zero point Error on mean from Cepheids, TRGB, SN 1987A, red clump, eclipsing binaries $\pm$ 5 HST camera calibration Tie-in to Galactic star clusters $\pm$ 3.5 Reddening Limits from photometry $\pm$ 1 Metallicity Observational and theoretical constraints $\pm$ 4 Bias in Leavitt law Short-end period cut-off $\pm$ 1 Crowding Artificial star experiments +5, -0 Large scale flows Limits from SN Ia, CMB $\pm$ 5 In the years since the HST Key Project, these uncertainties have been further reduced. One of the highest precision measurements of $H_0$ to date comes from the SH$_{0}$ES experiment (Riess et al. 2016), who found $H_0= 73.24 \pm 1.74$ km s$^{-1}$~Mpc$^{-1}$. Figure 5.7 shows the breakdown of their uncertainty budget. One of the largest uncertainties remaining in distance ladder measurements of $H_0$ comes from the bottom rung of the ladder. In the Key Project, the ladder was tied to our nearest galaxy, the Large Magellanic Cloud (LMC). SH$_{0}$ES reduced this uncertainty by tying their calibration to the Cepheids in the LMC (which can all be assumed to be at the same distance), Cepheids with parallaxes in the Milky Way (MW), and Cepheids in the mega-maser host galaxy NGC4258. :::fyi Astrophysical masers are sources of stimulated microwave emission, similar to lasers. Mega-masers are astrophysical masers with very high isotropic luminosities. Mega-masers are excellent geometric distance indicators. By assuming that the maser is travelling round a galaxy with a Keplerian orbit we can calculate the intrinsic size of the system. Comparing this with its apparent size gives us an estimate of the distance. ::: Until recently, only a handful of MW Cepheids had high-precision parallax measurements. The ESA Gaia mission will change this in the coming years by measuring high-precision parallaxes for over 1~billion stars in the MW. The Gaia sample will include several thousand Cepheids which can then be used to anchor the distance ladder. Figure 5.8 shows the sky as seen by Gaia. Each star in this image has a parallax measurement, including those in the Magellanic Clouds (seen to the bottom right of the Galactic Plane). In order to properly calibrate the Leavitt law and the uncertainties it contributes to the distance ladder, the Cepheid parallaxes must be treated extremely carefully. When using small samples of Cepheids with reasonably small uncertainties, it is appropriate to estimate the parallax distance ( ) as \[$$r = \dfrac{1}{\varpi} \tag{5.8}$$\] where $\varpi$ is the parallax angle. However, if we wish to ensure an unbiased calibration of the Leavitt law, we must not truncate the Cepheid sample with limits on (a) parallax uncertainty ($\ sigma_{\varpi}$) or (b) apparent magnitude. As there is an inverse relation between parallax and distance, the probability distribution of distances for a given parallax angle is skewed, becoming more so as the fractional uncertainty ${f}_{\mathrm{obs}} = \sigma_{\varpi}/\varpi$ increases. This is illustrated in Figure 5.9. This skewness must be taken into account in order to prevent systematic uncertainties being introduced into the Leavitt law calibration. Therefore, rather than use the simple parallax equation given in Eq. (5.8), we must use a full Bayesian analysis to estimate distances. A full discussion of the application of Bayesian statistics to parallax measurements is beyond the scope of this course. However, if you are interested in the methodology, the series of papers by Astraatmadja & Bailer-Jones gives an excellent introduction to the topic (Bailer-Jones 2015; Astraatmadja and Bailer-Jones 2016a; Astraatmadja and Bailer-Jones 2016b). Astraatmadja, Tri L., and Coryn A. L. Bailer-Jones. 2016a. “Estimating Distances from Parallaxes. II. Performance of Bayesian Distance Estimators on a Gaia-like Catalogue.” ApJ 832 (2): 137. Astraatmadja, Tri L., and Coryn A. L. Bailer-Jones. 2016a. “Estimating Distances from Parallaxes. II. Performance of Bayesian Distance Estimators on a Gaia-like Catalogue.” ApJ 832 (2): 137. 2016b. “Estimating Distances from Parallaxes. III. Distances of Two Million Stars in the Gaia DR1 Catalogue.” 833 (1): 119. Bailer-Jones, Coryn A. L. 2015. “Estimating Distances from Parallaxes.” PASP 127 (956): 994. Freedman, Wendy L, Barry F Madore, Brad K Gibson, Laura Ferrarese, Daniel D Kelson, Shoko Sakai, Jeremy R Mould, et al. 2001. “Final Results from the Hubble Space Telescope Key Project to Measure the Hubble Constant.” ApJ 553 (1): 47. Hubble, Edwin. 1929. “A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae.” Proceedings of the National Academy of Science 15: 168. Leavitt, Henrietta S., and Edward C. Pickering. 1912. “Periods of 25 Variable Stars in the Small Magellanic Cloud.” Harvard College Observatory Circular 173: 1. Riess, Adam G., Lucas M. Macri, Samantha L. Hoffmann, Dan Scolnic, Stefano Casertano, Alexei V. Filippenko, Brad E. Tucker, et al. 2016. “A 2.4% Determination of the Local Value of the Hubble Constant.” ApJ 826 (1): 56. Scowcroft, Victoria, Wendy L. Freedman, Barry F. Madore, Andy Monson, S. E. Persson, Jeff Rich, Mark Seibert, and Jane R. Rigby. 2016. “The Carnegie Hubble Program: The Distance and Structure of the SMC as Revealed by Mid-infrared Observations of Cepheids.” ApJ 816: 49.	{"url":"https://vickyscowcroft.github.io/PH40112_rmd/ch-obs-techs.html","timestamp":"2024-11-06T02:25:38Z","content_type":"text/html","content_length":"48787","record_id":"<urn:uuid:3c377528-d171-48e1-aabd-36731b8a40be>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00018.warc.gz"}
The collection is particularly noteworthy for its coverage of Adams's lectures, research and incoming correspondence.Section A, Biographical, is not substantial. It includes a little material of Adams's relating to his own career including three Bedford School notebooks and his PhD thesis, and material assembled by I M James during the preparation of his Royal Society memoir.Section B, Research, provides extensive documentation of Adams's research from the 1950s until his death. It is presented in an alphabetical sequence arranged by subject title.Section C, Lectures, is the largest in the collection. Two subsections comprise Adams's lecture notes and other teaching material for courses given at Manchester and Cambridge, and material from conferences and seminars attended by Adams throughout the world including drafts of Adams's contributions and notes of contributions by others. A third subsection consists of Adams's ms notes found in filing cabinet drawers labelled 'Other people's lectures'. It includes notes taken by Adams as an undergraduate at Cambridge in 1949.Section D, Publications, is very slight. It includes drafts of a few of Adams's scientific papers.Section E, Correspondence, contains virtually no extended exchanges of correspondence as very few copies of Adams's own letters survive. There is, however, significant correspondence from colleagues such as M F (later Sir Michael) Atiyah, M G Barratt, P J Hilton, I M James and S MacLane, sometimes extending over a period of twenty or thirty years.	{"url":"https://archives.trin.cam.ac.uk/downloads/exports/ead/707d9a16fbaa8cd30a3f191b1e71a3fc.ead.xml","timestamp":"2024-11-06T16:00:09Z","content_type":"application/xml","content_length":"451455","record_id":"<urn:uuid:84c72689-92ca-4fab-8484-57a785f602d8>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00463.warc.gz"}
Short-term forecasting of euro area economic activity at the ECB Níl an t-ábhar seo ar fáil i nGaeilge. Short-term forecasting of euro area economic activity at the ECB Published as part of the ECB Economic Bulletin, Issue 2/2020. 1 Introduction The real-time assessment of developments in economic activity is of central importance for the conduct of monetary policy. It facilitates the timely detection of changes in underlying economic dynamics in view of incoming data and contributes to a broader assessment of the economic outlook and associated risks. It is an integral part of the economic analysis in the European Central Bank’s (ECB) two-pillar approach to the assessment of the risks to price stability. Moreover, given the time lags in the transmission of monetary policy measures, a timely and reliable evaluation of economic conditions is a key element in the assessment of the monetary policy stance. Official estimates of real GDP growth in the euro area are published with some delay, but current and near-term developments in real GDP can be assessed on the basis of high-frequency and timely indicators. Real GDP is the key variable summarising information on real economic activity. However, it is available only at a quarterly frequency and its first official estimate for the euro area, the preliminary flash estimate, is published only approximately 30 days after the end of the reference quarter. To fill this gap, econometric models have been developed at the ECB and elsewhere that can exploit a rich set of data to produce a real-time estimate of real GDP in the current and next quarter(s). Short-term forecasts typically rely on financial market data, business and consumer surveys or sectoral data (e.g. from industry, retail or external sectors). These predictors are often available at a monthly, weekly or daily frequency and with shorter publication delays. There are a number of challenges to building quantitative tools for short-term forecasting of economic activity. First, these tools need to combine information from data collected at different frequencies. Second, they need to deal with the “ragged edge” of the data, which is due to the fact that different types of data are characterised by different publication delays. For example, industrial production in the euro area is published around six weeks after the end of the reference month, whereas opinion surveys and financial market data are often already available at the end of the reference period. Third, as there are many indicators that may be useful, the econometric approaches should be able to reliably estimate many parameters. Fourth, many indicators are subsequently revised and thus their first release might incorporate sizeable noise or measurement error. Fifth, data can be contaminated by outliers, caused by unusual events (e.g. strikes, atypical weather conditions), or changes in statistical properties over time, due to methodological or structural economic changes. Further challenges for real-time forecasting became apparent in the course of the global financial crisis and in its aftermath. The vast majority of models, including those used at the ECB for short-term forecasting at the time^[1], failed to predict the timing and depth of the Great Recession. In addition, these models systematically over-predicted the strength of the subsequent recovery. Several reasons were put forward at the time as an explanation for this disappointing forecast performance, including changes in structural relationships between economic variables, extreme outcomes in certain indicators that were inconsistent with model assumptions, insufficient coverage of financial market data and a non-linearity in the relationship between the real economy and the financial sector. Apart from addressing these shortcomings, recommendations for modellers included developing better tools for risk assessment and establishing appropriate economic narratives.^[2] The suite of models for short-term forecasting of euro area real GDP growth currently used at the ECB is the result of a comprehensive review conducted in 2015. The models rely on a medium-size data set of approximately 30 monthly indicators. A multivariate econometric set-up and a relatively broad coverage of various aspects of the euro area economy provide a framework for the interpretation of incoming data and forecast revisions. The forecasts are prepared using automated procedures (i.e. they are judgement-free) and can be produced in a matter of minutes. In addition to point forecasts, the model suite can also produce predictive distributions (fan charts). The latter can be used to assess, in real time, the degree of uncertainty around, or the risks to, the prevailing outlook for the short term. The model-based short-term forecasts of real GDP are an important input to the Eurosystem/ECB staff macroeconomic projections.^[3] By delivering quantitative estimates of real GDP growth ahead of the official data release and by providing an assessment of the macroeconomic “news” since the completion of the previous projection round, they are a useful starting point for updating the baseline short-term outlook for GDP growth. In addition, the predictive distributions provide model-based input for assessing the balance of risks surrounding the staff GDP projections. The article is organised as follows. Section 2 explains the methodological framework of the suite of models for short-term forecasting of real GDP at the ECB. Section 3 presents an evaluation of the forecast performance of the models. Section 4 focuses on two interesting elements of the suite of models: news analysis and predictive distributions. Finally, Section 5 concludes with the main lessons learned and discusses the current challenges, further planned enhancements and new directions of work. 2 Methodological framework Several types of models for short-term forecasting of real GDP have been proposed in the literature, including bridge equations, mixed-frequency dynamic factor models, mixed-frequency vector autoregressions and Mixed Data Sampling (MIDAS) models. Traditionally, “bridge equations”, linking GDP to a few key monthly indicators aggregated to a quarterly frequency, have been used. The latter are forecast using simple “auxiliary” models to complete the missing observations for the quarter. More recent approaches include mixed-frequency dynamic factor models and mixed-frequency vector autoregressions, which allow a single modelling framework to be used for the entire information set. Finally, MIDAS models allow data of different frequencies to be combined in a regression set-up by imposing a parsimonious lag structure. Different model types offer different advantages, in particular as regards robustness to structural breaks and extreme data outcomes or the possibility to interpret forecast revisions.^[4] The 2015 review of the ECB’s short-term forecasting models was motivated by the deterioration in the (relative) performance of the models in the course of the global financial crisis and in its aftermath. The suite of models used at the time encompassed (several versions) of bridge equations and large-scale mixed-frequency dynamic factor models. Both model types exhibited large forecast errors during the crisis and a positive bias (systematic over-prediction) thereafter, but the problems were more acute for the factor models. One of the reasons behind the positive bias was the insufficient coverage of the services sector and a declining contribution of the industry sector to value added in the euro area. Another reason was the difficulty to reliably estimate relationships between a large set of variables in view of their different behaviour during the financial crisis (in particular for survey vs. “hard”^[5] data). The forecast performance of the mixed-frequency factor models appears to have been more sensitive to such structural changes compared with the performance of the bridge equations. The current suite of short-term forecast models is based on bridge equations, in view of their comparatively better post-financial crisis forecast performance. Two types of bridge equations are included: (i) equations based on “hard” data, linking GDP to industrial production (excluding construction) and value added in services, and (ii) equations based on “soft” data, linking GDP to Purchasing Managers’ Index (PMI) composite output and PMI construction.^[6] Both types embody the “supply” perspective for real GDP measurement^[7], given that the coverage of information is more complete and timelier and the relationship with GDP is more stable compared with the “demand” perspective. As a consequence, the supply perspective results in more accurate forecasts. The forecasts for (quarterly) value added in services are obtained via an auxiliary bridge equation. The monthly predictors included in the bridge equations are in turn forecast using “auxiliary” models and incorporate information from other monthly variables. Since bridge equations typically include just a few predictors, the only way to exploit a larger (and timelier) set of information in such a framework is through monthly auxiliary models to produce forecasts of the predictors.^[8] The auxiliary models for the bridge equations are monthly Bayesian vector autoregressions and dynamic factor models. Both types of models allow a large number of variables to be incorporated. The data set comprises approximately 30 indicators. It includes industrial production and business surveys for different sectors, monthly indicators of retail trade, unemployment, external trade and financial market data. The data set can be considered a “medium” size and is significantly smaller than those underlying the mixed-frequency factor models used previously. Forecast evaluations conducted during the review have shown that a very granular sectoral disaggregation typical for large data sets does not result in improved forecast accuracy.^[9] Forecasts are obtained as an average of forecasts produced by individual models. Combining two types of bridge equations with five auxiliary models results in ten distinct models for GDP. For point forecasts, an average of the individual model predictions is taken. Pooling individual forecasts leads to gains in forecast accuracy, even with respect to the best-performing model version^[10], see below. Predictive distributions (densities) are produced via simulations and combined predictive density is calculated as an average of the individual model predictive densities. More technical details can be found in Box 1. Box 1The suite of models for short-term forecasting of real GDP in the euro area: some technical details The models used belong to the family of bridge equations. A bridge equation is a linear regression model where the dependent variable is the low-frequency variable of interest (e.g. quarterly GDP) and the regressors are higher-frequency predictors (e.g. monthly industrial production) aggregated to the lower frequency. In the case of the models for short-term forecasting of real GDP in the euro area described in the main text, the equations are specified as follows: $y t Q = α + ∑ i = 1 k β i X i , t Q + ε t Q ,$ where $y t Q$ is the dependent variable, in this case quarter-on-quarter real GDP growth, and $X i , t Q$ are the predictor variables (up to $k$ per bridge equation). Two types of bridge equations are included. In the first bridge equation, the predictor variables are: quarterly growth of industrial production and quarterly growth of value added in services. In the second equation, the predictors are: quarterly average of PMI composite output and quarterly difference of PMI construction output^[11]. $ε t Q$ is the regression residual, $α$ is the intercept and $β i$ are the regression coefficients. For value added in services, an auxiliary bridge equation including expected demand for services from the surveys of the European Commission is used. The equations are estimated by standard regression techniques (ordinary least squares). The estimation sample starts in 1985 or later, depending on data availability in the particular equation (or “auxiliary” model, see below). In order to obtain forecasts for GDP from the equations described above, it is necessary to obtain forecasts for the monthly predictors for the quarters of interest. For this purpose, “auxiliary” multivariate models at a monthly frequency are used: vector autoregressions (VARs) and dynamic factor models (DFMs).^[12] The former are estimated with Bayesian methods, using a specification in first differences with six lags and the Minnesota prior with the degree of shrinkage dependent on the size of the model.^[13] The latter are estimated by maximum likelihood, using the expectation maximisation algorithm.^[14] The specification includes one single common factor, which follows an autoregressive process of order two and an autoregressive process of order one for the idiosyncratic components. Both types of models can deal with large sets of variables. VARs of three sizes (including two, 22 or 28 variables) and DFMs of two sizes (with 22 and 28 variables) are included. In order to handle the ragged edge caused by different publication delays of the variables, the models are cast into a state space representation and the Kalman filter and smoother are used to obtain the forecasts of the monthly variables and the weights for the news (see Section 4). The variables for the bridge equations and the monthly “auxiliary” models were selected on the basis of several criteria including correlation analysis, in-sample and out-of-sample forecast performance, stability and significance of regression coefficients as well as shrinkage methods such as LASSO regressions.^[15] The results confirmed previous findings in the literature that a very high level of disaggregation (100 series or more) is not needed to achieve the best forecast accuracy. The computation of the models’ predictive distributions (densities) relies on the use of the Gibbs sampler and the simulation smoother (in order to handle the ragged edge).^[16] The density forecasts from individual models are combined by a linear opinion pool with equal weights attached to individual densities. Combinations of normal densities produce distributions which can accommodate non-standard features such as fat tails or skewness. As for the case of point forecasts, pooling density forecasts is also an insurance policy against uncertainty in model selection.^[17] 3 Forecast performance A real-time evaluation is conducted of the forecasting accuracy of the models since their introduction and over a longer period starting in 2005. For this purpose, real-time data vintages going back to 2005 are constructed based on the information stored in the ECB’s Statistical Data Warehouse (SDW).^[18] For each quarter in the evaluation sample, 12 forecast horizons are considered. The first forecast is obtained five months ahead of the first official publication. Subsequent forecasts are produced in semi-monthly intervals, up to two weeks before the publication of the preliminary flash estimate.^[19] For instance, in the forecast cycle for the second quarter of the year, the first forecast would be produced at the end of January and the last one in the second week of July. The evaluation focuses on the bias and the root mean squared error of the forecasts. The forecasts are evaluated against the official flash estimates and the latest available vintage of quarter-on-quarter real GDP growth. The forecast accuracy of the models is compared with that of the Eurosystem/ECB staff macroeconomic projections. For the purpose of the evaluation, a convention is adopted in line with which the latter are finalised in the middle of the second month of each quarter (corresponding to the forecast horizon of 1.5 and 4.5 months ahead for the current and the next quarter, respectively) and they remain unchanged in between.^[20] The accuracy of the models improves as new information arrives and the models fare relatively well compared with the Eurosystem/ECB staff macroeconomic projections. Chart 1 shows the root mean squared forecast error (RMSFE) and the bias for the model forecasts (light-coloured lines) as well as the projections (dark-coloured lines) compared with the official flash estimate (red lines) and with the latest vintage (blue lines) of GDP growth for the 12 forecast horizons considered. The evaluation period is 2015Q1 to 2019Q2.^[21] Overall, the accuracy of the model forecasts is somewhat lower than that of the projections. The precision of the model forecasts gradually improves with a decreasing forecast horizon and the forecasts appear particularly useful at very short horizons after the projections have been finalised. Both the forecasts and the projections are more accurate and less biased when they are compared with the flash estimate than when they are compared with the latest available vintage of GDP. Chart 1 Accuracy of model GDP forecasts and Eurosystem/ECB staff GDP projections over 2015Q1-2019Q2 (percentage points) Source: ECB calculations. Notes: For each quarter a sequence of 12 real-time forecast updates is evaluated. The forecast horizon (indicated on the horizontal axis) is defined as the distance (in months) between the end of the reference quarter and the date when the forecast was made. A convention is adopted in line with which Eurosystem/ECB staff macroeconomic projections are finalised around the middle of the second month of each quarter (1.5 or 4.5 months before the end of the reference quarter). Bias is defined as the average difference between the forecast and the outcome. Model forecasts and the projections are evaluated against the official flash estimate of GDP growth (released in the middle of the second month of the following quarter) as well as against the latest available vintage of real GDP The models also perform relatively well when evaluated over a longer period. The evaluation period considered above is relatively short and less volatile than, for example, the preceding period, which included the financial and sovereign debt crises. Focusing on the RMSFEs for 1.5-month ahead horizon with the flash estimate as the reference variable, Chart 2 presents the evolution of forecast accuracy since 2005 over an eight-quarter window. Several observations can be made. First, unsurprisingly, the financial crisis period was characterised by much larger forecast errors, both for models and for the Eurosystem/ECB staff macroeconomic projections. By contrast, the errors were not particularly large during the sovereign debt crisis. Second, the average model forecast is more accurate than the projections in some periods (notably during the financial crisis but not in the latest period).^[22] Finally, an average of forecasts from several models typically does as well as the best model in each month (which changes over time) and is thus a good hedge against model uncertainty. Chart 2 Evolution of forecast accuracy since 2005 (percentage points, RMSFE over an eight-quarter rolling window) Source: ECB calculations. Notes: The chart shows the RMSFEs over a rolling window of eight quarters. The forecasts are updated in the middle of the second month of the reference quarter (forecast horizon of 1.5 months), around the finalisation date of the Eurosystem/ECB staff macroeconomic projections. The reference variable is the official flash estimate of quarter-on-quarter real GDP growth. ‘Average’ refers to the rolling RMSFE of the average point forecasts (from ten different models). ‘Individual models’ indicates the range given by the minimum and maximum (rolling) RMSFE of the individual models. Shaded areas indicate recession periods (the Great Recession and the sovereign debt crisis) in the euro area as identified by the CEPR Business Cycle Dating Committee. 4 News analysis and a measure of risks 4.1 News analysis The current framework allows linking revisions to the GDP growth forecast to model-based surprises or news content in releases of monthly predictors. This is also known as model-based news analysis and is an important element of data monitoring. The news (or surprise) for each indicator is defined as the difference between the released value of that indicator and its expected (forecast) value, i.e. the forecast error made by the model. The difference between two consecutive forecasts of GDP, that is the forecast revision, can be expressed as a weighted average of the news in the data released between the two forecast updates (plus the effect of historical data revisions and parameter re-estimation).^[23] The weights reflect the average volatility of the news and its relevance for GDP. The sign of the news indicates whether the released number was better or worse than expected (“positive” or “negative” news). Forecast revisions for individual quarters can be decomposed to identify the role of specific (groups of) indicators. Chart 3 illustrates this type of analysis taking the second quarter of 2019 as an example. The green line represents the evolution of the (average point) forecasts starting at the beginning of February up to mid-July, approximately two weeks before the release of the preliminary flash estimate of real GDP for that quarter. The bars indicate the model-based news or drivers of forecast revisions between the consecutive updates. A sizeable downgrade of the outlook at the end of March can be seen due to negative news in survey data. Subsequently, positive surprises on survey data lead to an upward revision of the outlook. From the end of May, the nowcast stabilises close to the outcome (preliminary flash estimate). Chart 3 Model-based news and revisions to real GDP growth forecast for 2019Q2 (quarterly percentage changes and percentage point contributions) Source: ECB calculations. Notes: The green line represents the average point forecasts (from ten different models) for real GDP growth in 2019Q2 from different forecast updates (indicated on the horizontal axis). The bars indicate the decomposition of forecast revisions between the consecutive updates into news stemming from different groups of data: ‘Industrial production’ – sectoral production indicators, ‘Other hard data’ – unemployment rate, external trade, retail trade, new car registrations, ‘Surveys’ – surveys of the European Commission and the Purchasing Managers’ surveys, ‘Financial and money’ – real money and financial and credit indicators. ‘Remainder’ collects the effects of data revisions and parameter re-estimation. 4.2 Density forecasts The location and the shape of the models’ predictive distributions make it possible to assess the uncertainty around the point forecast as well as the direction and the degree of risks to forecasts from other sources such as the staff projections. For example, when the centre of the model predictive density (as represented by its mode or its median) is to the left of an alternative forecast, it signals downward risks to the latter and vice versa. Consequently, movements to the left or right of the predictive density will imply changes in the assessment of the direction of risks. By contrast, changes in the shape of the distribution (i.e. dispersion or concentration) will imply changes in the level of uncertainty. In real-time analysis, as more information is accrued over the forecast cycle, the predictive distribution usually becomes more concentrated, entailing less uncertainty surrounding the central forecast. It cannot be ruled out, however, that the release of one or several indicators could lead to a flatter distribution, due to diverging interpretations by the different models, and therefore to higher uncertainty. As an example, predictive distributions indicate that, on the basis of these models, initially there were downward risks to the June 2019 Eurosystem staff GDP projection for 2019Q2 and the balance of risks became more neutral as more data became available. Chart 4 presents the models’ predictive densities for 2019Q2 obtained with the data available on 17 May 2019 (around the finalisation of the June 2019 staff projection) and on 12 July 2019. Initially, the models suggested downside risks to the projection since the probability of a lower outcome was higher than 50% (i.e. 60%). As more information became available by mid-July, the distribution moved to the right and became more concentrated. This means that the risks to the projection became more balanced (given that the probability of observing an outcome either above or below the projected value was around 50%) and smaller. Chart 4 Predictive densities for real GDP growth in 2019Q2 (horizontal axis: quarterly percentage changes, vertical axis: density) Source: ECB calculations. Notes: The blue and yellow lines represent the (combined) predictive densities for real GDP growth from the respective forecast updates. The combination involves densities from the ten different models via a linear prediction pool with equal weights. The green line corresponds to the outlook in the June 2019 Eurosystem staff macroeconomic projections, and the red line is the preliminary flash estimate. 5 Conclusions and new directions Changes in economic relationships caused by the evolving economic environment are a challenge to forecasting models in general and to short-term forecasting tools in particular. Some notable examples of structural changes include climate change, inter-sectoral re-balancing, developments in productivity, effects of severe recessions and, more specifically for the euro area, changes in the automotive industry. Several lessons on how to address those and other challenges can be drawn from the experience with model-based short-term forecasting of real economic activity at the ECB. First, it is important to have several models in the toolbox and to assess their performance regularly, as it may deteriorate over time. Second, a combination of forecasts from different models typically helps to make the forecast performance more robust to misspecification. Third, including information on all major sectors of the economy is important but it is not necessary to use data sets at a very high level of disaggregation. A medium-size set of relevant and timely indicators appears to be sufficient to capture the information on real activity developments in the near term. Finally, it is important to be able to interpret the revisions to the outlook and to communicate uncertainty surrounding the forecasts. Still, scope for further improvement along several dimensions remains. One issue is the high reliance of short-term forecasting models on survey data. Surveys provide qualitative information (i.e. opinions or perceptions) from relatively small samples of firms or consumers. They are very relevant due to their short publication lag. However, their relationship with quantitative (hard) indicators can change over time, reflecting either sampling biases (e.g. survival bias, especially after the crisis) or the fact that survey respondents can change the benchmarks used for their assessments (e.g. value of sales growth which can be considered an improvement in the firm’s performance).^[24] As a result, the mapping of survey data levels into economic growth rates is not straightforward. For instance, at the beginning of 2018 survey data were at historically high levels^[25], while real GDP growth slowed down considerably in the euro area. Conversely, some of the surveys painted a rather bleak outlook for 2019, while hard data turned out somewhat more resilient. Alternative models and indicators can be employed to further enhance the accuracy and robustness of the models currently employed. Examples include time-varying parameter models that can deal with relationships that change over time in a flexible way.^[26] The usefulness of alternative indicators and methods is also being investigated, in particular of machine learning algorithms and “big data”. The term “big data” is rather broad. In this context, it includes large and near-real-time data from the internet (e.g. internet search volumes^[27], data from social networks such as Twitter and Facebook, newspaper articles) or large-volume data from non-official sources (e.g. from trading platforms and payment systems). Big data allows a wider range of indicators to be used, which can provide new and unique insights helpful for forecasting. For instance, text-based sentiment indicators could be particularly useful given that they can be produced automatically at a high frequency and at lower costs than survey-based sentiment indicators, and they can be based on large samples of newspapers to avoid biases.^[28] At the same time, one has to keep in mind that considering a large set of explanatory variables entails risks of overfitting, not necessarily leading to improvements in out-of-sample forecast accuracy. Some of these challenges can be addressed by machine learning algorithms, which also have the advantage of potentially capturing complex non-linear relationships. These are some interesting directions for future work. 1. See “Short-term forecasts of economic activity in the euro area”, Monthly Bulletin, ECB, April 2008. 2. See, for example, Kenny, G. and Morgan, J., “Some lessons from the financial crisis for the economic analysis”, Occasional Paper Series, No 130, ECB, 2011. 3. See “A guide to the Eurosystem/ECB staff macroeconomic projection exercises”, ECB, July 2016. 4. See Bańbura, M., Giannone, D., Modugno, M. and Reichlin, L., “Now-casting and the real-time data flow”, in Elliott, G. and Timmermann, A. (ed.), Handbook of Economic Forecasting, Vol. 2A, North Holland, 2013, pp. 195–236, for a detailed review and list of references for the different modelling approaches. 5. “Soft” is typically used to label indicators that reflect market expectations, most notably surveys and financial market data. By contrast, “hard” indicators often measure certain GDP components directly (e.g. industrial production). 6. See de Bondt, G.J., “A PMI-based Real GDP Tracker for the Euro Area”, Journal of Business Cycle Research, Vol. 15, Issue 2, 2019, pp. 147–170. 7. See Hahn, E. and Skudelny, F., “Early estimates of euro area real GDP growth – a bottom-up approach from the production side”, Working Paper Series, No 975, ECB, December 2008. 8. See Bulligan, G., Golinelli, R. and Parigi, G., “Forecasting monthly industrial production in real-time: from single equations to factor-based models”, Empirical Economics, Vol. 39, Issue 2, 2010, pp. 303-336. 9. This is in line with the conclusions in, for example, Bańbura, M., Giannone, D. and Reichlin, L., “Large Bayesian vector autoregressions”, Journal of Applied Econometrics, Vol. 25, Issue 1, 2010, pp. 71–92, and Bańbura, M., Giannone, D. and Reichlin, L., “Nowcasting”, in Clements, M.P. and Hendry, D.F. (ed.), The Oxford Handbook of Economic Forecasting, 2011. 10. See Kuzin, V., Marcellino, M. and Schumacher, C., “Pooling versus model selection for nowcasting GDP with many predictors: empirical evidence for six industrialized countries”, Journal of Applied Econometrics, Vol. 28, Issue 3, 2013, pp. 392-411. 11. See, for example, de Bondt, G.J., op. cit., for more details on the second equation. Note that the two equations result in better forecast accuracy than an average of (a large number of) single variable bridge equations. 12. This results in higher forecast accuracy compared with using a univariate ARIMA model for each monthly predictor, in line with the findings in Rünstler, G. and Sédillot, F., “Short-term estimates of euro area real GDP by means of monthly data”, Working Paper Series, No 276, ECB, September 2003. 13. See Bańbura et al., “Large Bayesian vector autoregressions”, op. cit. 14. See Bańbura, M. and Modugno, M., “Maximum likelihood estimation of dynamic factor models on datasets with arbitrary pattern of missing data”, Journal of Applied Econometrics, Vol. 29, Issue 1, 2014, pp. 133–160. 15. Note that the selection of indicators was not conducted in real time but in sample. However, as the data set was frozen at the beginning of 2015, the evaluation starting in 2015 is truly real-time. LASSO and similar techniques have been used to select variables for bridge equations in, for example, Bulligan, G., Marcellino, M. and Venditti, F., “Forecasting economic activity with targeted predictors”, International Journal of Forecasting, Vol. 31, Issue 1, 2015, pp. 188-206. 16. See Durbin, J. and Koopman, S.J., “A simple and efficient simulation smoother for state space time series analysis”, Biometrika, Vol. 89, Issue 3, 2002, pp. 603–615. 17. Geweke and Amisano showed that pooled forecast densities produce superior predictions, even if the set of models to be combined exclude the “true” model. See Geweke, J. and Amisano, G., “Optimal prediction pools”, Journal of Econometrics, Vol. 164, Issue 1, 2011, pp. 130-141. 18. For a given date stamp and indicator identifier, a time series available at that date can be recovered from the SDW. Thus real-time data vintages reflect both publication delays and data revisions (as opposed to pseudo real-time vintages that reflect only the former). 19. This reflects the frequency and the forecast horizon of the regular updates of short-term forecasts at the ECB. They are generally conducted twice per month, following the release of industrial production in the middle of each month, and of opinion surveys at the end of each month. The forecasts are always reported for the next two quarters to be published. 20. As a consequence, the accuracy of the projections reported in Chart 1 changes in the middle of the second month of each quarter as a new projection becomes available. The projections are customarily finalised between the middle and the end of the second month of each quarter. 21. Since no changes have been implemented to the models since 2015, this is a truly real-time out-of-sample evaluation. 22. It should be noted that although the estimation of and the forecasts from the models are performed using real-time data, the specification and the choice of the variables in the new models were performed after the crisis and therefore have the benefit of hindsight for the evaluation period prior to 2015. 23. See Bańbura et al., “Now-casting and the real-time data flow”, op. cit. For a meaningful analysis, the news should be based on multivariate models, incorporating most relevant indicators and taking into account differences in their timeliness and strength of the signal. The news analysed here is model-based and conceptually similar but not the same as “market surprises” (which are the differences with respect to market expectations). 24. See Gayer C. and Marc B., “A ’New Modesty’? Level Shifts in Survey Data and the Decreasing Trend of ’Normal’ Growth”, European Economy Discussion Paper, 083, European Commission, July 2018. 25. See the box entitled “The recent strength of survey-based indicators: what does it tell us about the depth and breadth of real GDP growth?”, Economic Bulletin, Issue 8, ECB, 2017. 26. See, for example, Antolín-Díaz, J., Drechsel, T. and Petrella, I., “Tracking the Slowdown in Long-Run GDP Growth”, The Review of Economics and Statistics, Vol. 99, Issue 2, 2017, pp. 343–356. 27. See, for example, Ferrara, L. and Simoni, A., “When are Google data useful to nowcast GDP? An approach via pre-selection and shrinkage”, Working Papers, No 2019-04, Center for Research in Economics and Statistics, 2019. 28. See, for example, Thorsrud, L.A., “Words are the New Numbers: A Newsy Coincident Index of the Business Cycle”, Journal of Business & Economic Statistics, 2018.	{"url":"https://www.ecb.europa.eu/press/economic-bulletin/articles/2020/html/ecb.ebart202002_02~47da9ba4f7.ga.html","timestamp":"2024-11-05T04:35:39Z","content_type":"text/html","content_length":"159288","record_id":"<urn:uuid:8a24f9ad-d37d-40c1-811a-e901cd02e51b>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00310.warc.gz"}
Two 3D vectors, A and B , are represented in terms of the Cartesian basis unit... Two 3D vectors, A and B , are represented in terms of the Cartesian basis unit... Two 3D vectors, A and B , are represented in terms of the Cartesian basis unit vectors, i , j , k as follows: 1) plot A and B in a Cartesian coordinate system 2) calculate A + B 3) calculate A · B using the scalar product formula 4) calculate the magnitudes of A and B 5) calculate the angle between A and B 6) calculate A x B using the determinant formula 7) show that the new vector, w = A x B , is perpendicular to both A and B	{"url":"https://justaaa.com/advanced-math/181906-two-3d-vectors-a-and-b-are-represented-in-terms","timestamp":"2024-11-12T15:36:48Z","content_type":"text/html","content_length":"36458","record_id":"<urn:uuid:81de4947-dcac-4acc-968b-1b0cca412552>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00778.warc.gz"}
Strong Conflict-Free Coloring for Intervals We consider the k-strong conflict-free (k-SCF) coloring of a set of points on a line with respect to a family of intervals: Each point on the line must be assigned a color so that the coloring is conflict-free in the following sense: in every interval I of the family there are at least k colors each appearing exactly once in I. We first present a polynomial-time approximation algorithm for the general problem; the algorithm has approximation ratio 2 when k=1 and 5-2/k when k ≥ 2. In the special case of a family that contains all possible intervals on the given set of points, we show that a 2-approximation algorithm exists, for any k ≥ 1. We also provide, in case k = O(polylog(n)), a quasipolynomial time algorithm to decide the existence of a k-SCF coloring that uses at most q • Conflict-free coloring • Interval hypergraph • Wireless networks ASJC Scopus subject areas • General Computer Science • Computer Science Applications • Applied Mathematics Dive into the research topics of 'Strong Conflict-Free Coloring for Intervals'. Together they form a unique fingerprint.	{"url":"https://cris.bgu.ac.il/en/publications/strong-conflict-free-coloring-for-intervals-6","timestamp":"2024-11-07T23:22:28Z","content_type":"text/html","content_length":"55677","record_id":"<urn:uuid:ab949c49-63e5-4f3a-9d7f-891fd3075fed>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00764.warc.gz"}
more from Keith Campbell Single Idea 8525 [catalogued under 5. Theory of Logic / E. Structures of Logic / 6. Relations in Logic] Full Idea Because there cannot be relations without terms, in a meta-physic that makes first-order tropes the terms of all relations, relational tropes must belong to a second, derivative order. Gist of Idea Relations need terms, so they must be second-order entities based on first-order tropes Keith Campbell (The Metaphysic of Abstract Particulars [1981], §8) Book Reference 'Properties', ed/tr. Mellor,D.H. /Oliver,A [OUP 1997], p.138 A Reaction The admission that there could be a 'derivative order' may lead to trouble for trope theory. Ostrich Nominalists could say that properties themselves are derivative second-order abstractions from indivisible particulars. Russell makes them first-order.	{"url":"http://www.philosophyideas.com/search/response_philosopher_detail.asp?era_no=L&order=chron&era=Late%2020th%20century%20(1956-2000)&find=idea&visit=2&return=yes&ID=8525&theme_alpha=yes&ThemeNumber=&area=&area_no=&PN=3246","timestamp":"2024-11-10T09:43:37Z","content_type":"application/xhtml+xml","content_length":"3451","record_id":"<urn:uuid:d78dda50-66d8-40bc-bb02-d8de0683d148>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00853.warc.gz"}
Mean-Mean scatter plot A mean-mean scatter plot shows a 2-dimensional representation of the differences between many means. The mean-mean scatter plot shows the mean of a group on the horizontal axis against the mean of the other group on the vertical axis with a dot at the intersection. A vector centered at the intersection with a slope of -1 and a length proportional to the width of the confidence interval represents the confidence interval. A gray identity line represents equality of means; that is the difference is equal to zero. If the vector does not cross the identity line, you can conclude there is a significant difference between the means. To make interpretation easier, a 45-degree rotated version of the plot shows the difference between means and its confidence interval on the horizontal axis against average of the means on the vertical axis. Related concepts Related tasks	{"url":"https://analyse-it.com/docs/user-guide/compare-groups/mean-mean-scatter-plot","timestamp":"2024-11-01T20:47:47Z","content_type":"text/html","content_length":"26591","record_id":"<urn:uuid:e7c85008-e3ab-4a56-83c0-0a7b4f8b8877>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00311.warc.gz"}
Study Guide - Compressions and Stretches Compressions and Stretches Learning Objectives • Graph Functions Using Compressions and Stretches Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity. We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. Each change has a specific effect that can be seen graphically. Vertical Stretches and Compressions When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a vertical stretch; if the constant is between 0 and 1, we get a vertical compression. The graph below shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression. A General Note: Vertical Stretches and Compressions Given a function [latex]f\left(x\right)[/latex], a new function [latex]g\left(x\right)=af\left(x\right)[/latex], where [latex]a[/latex] is a constant, is a vertical stretch vertical compression of the function [latex]f\left(x\right)[/latex]. • If [latex]a>1[/latex], then the graph will be stretched. • If 0 < a < 1, then the graph will be compressed. • If [latex]a<0[/latex], then there will be combination of a vertical stretch or compression with a vertical reflection. How To: Given a function, graph its vertical stretch. 1. Identify the value of [latex]a[/latex]. 2. Multiply all range values by [latex]a[/latex]. 3. If [latex]a>1[/latex], the graph is stretched by a factor of [latex]a[/latex]. If [latex]{ 0 }<{ a }<{ 1 }[/latex], the graph is compressed by a factor of [latex]a[/latex]. If [latex]a<0[/latex], the graph is either stretched or compressed and also reflected about the x-axis. Example: Graphing a Vertical Stretch A function [latex]P\left(t\right)[/latex] models the number of fruit flies in a population over time, and is graphed below. A scientist is comparing this population to another population, [latex]Q[/ latex], whose growth follows the same pattern, but is twice as large. Sketch a graph of this population. Answer: Because the population is always twice as large, the new population’s output values are always twice the original function’s output values. If we choose four reference points, (0, 1), (3, 3), (6, 2) and (7, 0) we will multiply all of the outputs by 2. The following shows where the new points for the new graph will be located. [latex-display]\begin{cases}\left(0,\text{ }1\right)\to \left (0,\text{ }2\right)\hfill \\ \left(3,\text{ }3\right)\to \left(3,\text{ }6\right)\hfill \\ \left(6,\text{ }2\right)\to \left(6,\text{ }4\right)\hfill \\ \left(7,\text{ }0\right)\to \left(7,\text{ }0\ right)\hfill \end{cases}[/latex-display] Figure 16 Symbolically, the relationship is written as [latex-display]Q\left(t\right)=2P\left(t\right)[/latex-display] This means that for any input [latex]t[/latex], the value of the function [latex]Q[/latex] is twice the value of the function [latex]P[/latex]. Notice that the effect on the graph is a vertical stretching of the graph, where every point doubles its distance from the horizontal axis. The input values, [latex]t[/latex], stay the same while the output values are twice as large as before. How To: Given a tabular function and assuming that the transformation is a vertical stretch or compression, create a table for a vertical compression. 1. Determine the value of [latex]a[/latex]. 2. Multiply all of the output values by [latex]a[/latex]. Example: Finding a Vertical Compression of a Tabular Function A function [latex]f[/latex] is given in the table below. Create a table for the function [latex]g\left(x\right)=\frac{1}{2}f\left(x\right)[/latex]. [latex]x[/latex] 2 4 6 8 [latex]f\left(x\right)[/latex] 1 3 7 11 Answer: The formula [latex]g\left(x\right)=\frac{1}{2}f\left(x\right)[/latex] tells us that the output values of [latex]g[/latex] are half of the output values of [latex]f[/latex] with the same inputs. For example, we know that [latex]f\left(4\right)=3[/latex]. Then [latex-display]g\left(4\right)=\frac{1}{2}f\left(4\right)=\frac{1}{2}\left(3\right)=\frac{3}{2}[/latex-display] We do the same for the other values to produce this table. [latex]x[/latex] [latex]2[/latex] [latex]4[/latex] [latex]6[/latex] [latex]8[/latex] [latex]g\left(x\right)[/latex] [latex]\frac{1}{2}[/latex] [latex]\frac{3}{2}[/latex] [latex]\frac{7}{2}[/latex] [latex]\frac{11}{2}[/latex] Analysis of the Solution The result is that the function [latex]g\left(x\right)[/latex] has been compressed vertically by [latex]\frac{1}{2}[/latex]. Each output value is divided in half, so the graph is half the original Try It A function [latex]f[/latex] is given below. Create a table for the function [latex]g\left(x\right)=\frac{3}{4}f\left(x\right)[/latex]. [latex]x[/latex] 2 4 6 8 [latex]f\left(x\right)[/latex] 12 16 20 0 [latex]x\\[/latex] 2 4 6 8 [latex]g\left(x\right)\\[/latex] 9 12 15 0 Horizontal Stretches and Compressions horizontal stretch; if the constant is greater than 1, we get a horizontal compression of the function. Given a function [latex]y=f\left(x\right)[/latex], the form [latex]y=f\left(bx\right)[/latex] results in a horizontal stretch or compression. Consider the function [latex]y={x}^{2}[/latex]. The graph of [latex]y={\left(0.5x\right)}^{2}[/latex] is a horizontal stretch of the graph of the function [latex]y={x}^{2}[/latex] by a factor of 2. The graph of [latex]y={\left(2x\right)}^{2}[/latex] is a horizontal compression of the graph of the function [latex]y={x}^{2}[/latex] by a factor of 2. A General Note: Horizontal Stretches and Compressions Given a function [latex]f\left(x\right)[/latex], a new function [latex]g\left(x\right)=f\left(bx\right)[/latex], where [latex]b[/latex] is a constant, is a horizontal stretch horizontal compression of the function [latex]f\left(x\right)[/latex]. • If [latex]b>1[/latex], then the graph will be compressed by [latex]\frac{1}{b}[/latex]. • If [latex]0<b<1[/latex], then the graph will be stretched by [latex]\frac{1}{b}[/latex]. • If [latex]b<0[/latex], then there will be combination of a horizontal stretch or compression with a horizontal reflection. How To: Given a description of a function, sketch a horizontal compression or stretch. 1. Write a formula to represent the function. 2. Set [latex]g\left(x\right)=f\left(bx\right)[/latex] where [latex]b>1[/latex] for a compression or [latex]0<b<1[/latex] for a stretch. Example: Graphing a Horizontal Compression Suppose a scientist is comparing a population of fruit flies to a population that progresses through its lifespan twice as fast as the original population. In other words, this new population, [latex]R[/latex], will progress in 1 hour the same amount as the original population does in 2 hours, and in 2 hours, it will progress as much as the original population does in 4 hours. Sketch a graph of this population. Answer: Symbolically, we could write [latex]\begin{cases}R\left(1\right)=P\left(2\right),\hfill \\ R\left(2\right)=P\left(4\right),\text{ and in general,}\hfill \\ R\left(t\right)=P\left(2t\right).\hfill \end{cases}[/latex] See below for a graphical comparison of the original population and the compressed population. Analysis of the Solution Note that the effect on the graph is a horizontal compression where all input values are half of their original distance from the vertical axis. Example: Finding a Horizontal Stretch for a Tabular Function A function [latex]f\left(x\right)[/latex] is given below. Create a table for the function [latex]g\left(x\right)=f\left(\frac{1}{2}x\right)[/latex]. [latex]x[/latex] 2 4 6 8 [latex]f\left(x\right)[/latex] 1 3 7 11 Answer: The formula [latex]g\left(x\right)=f\left(\frac{1}{2}x\right)[/latex] tells us that the output values for [latex]g[/latex] are the same as the output values for the function [latex]f[/latex] at an input half the size. Notice that we do not have enough information to determine [latex]g\left(2\right)[/latex] because [latex]g\left(2\right)=f\left(\frac{1}{2}\cdot 2\right)=f\left(1\right)[/ latex], and we do not have a value for [latex]f\left(1\right)[/latex] in our table. Our input values to [latex]g[/latex] will need to be twice as large to get inputs for [latex]f[/latex] that we can evaluate. For example, we can determine [latex]g\left(4\right)\text{.}[/latex] [latex]g\left(4\right)=f\left(\frac{1}{2}\cdot 4\right)=f\left(2\right)=1[/latex] We do the same for the other values to produce the table below. [latex]x[/latex] 4 8 12 16 [latex]g\left(x\right)[/latex] 1 3 7 11 Analysis of the Solution Because each input value has been doubled, the result is that the function [latex]g\left(x\right)[/latex] has been stretched horizontally by a factor of 2. Example: Recognizing a Horizontal Compression on a Graph Relate the function [latex]g\left(x\right)[/latex] to [latex]f\left(x\right)[/latex]. Answer: The graph of [latex]g\left(x\right)[/latex] looks like the graph of [latex]f\left(x\right)[/latex] horizontally compressed. Because [latex]f\left(x\right)[/latex] ends at [latex]\left(6,4\ right)[/latex] and [latex]g\left(x\right)[/latex] ends at [latex]\left(2,4\right)[/latex], we can see that the [latex]x\text{-}[/latex] values have been compressed by [latex]\frac{1}{3}[/latex], because [latex]6\left(\frac{1}{3}\right)=2[/latex]. We might also notice that [latex]g\left(2\right)=f\left(6\right)[/latex] and [latex]g\left(1\right)=f\left(3\right)[/latex]. Either way, we can describe this relationship as [latex]g\left(x\right)=f\left(3x\right)[/latex]. This is a horizontal compression by [latex]\frac{1}{3}[/latex]. Analysis of the Solution Notice that the coefficient needed for a horizontal stretch or compression is the reciprocal of the stretch or compression. So to stretch the graph horizontally by a scale factor of 4, we need a coefficient of [latex]\frac{1}{4}[/latex] in our function: [latex]f\left(\frac{1}{4}x\right)[/latex]. This means that the input values must be four times larger to produce the same result, requiring the input to be larger, causing the horizontal stretching. Try It Write a formula for the toolkit square root function horizontally stretched by a factor of 3. Use Desmos to check your work. Answer: [latex]g\left(x\right)=\|x - 1\|-3[/latex] Licenses & Attributions CC licensed content, Original • Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution. • Question ID 112707, 112726. Authored by: Lumen Learning. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL. CC licensed content, Shared previously • College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. Located at: https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites. License: CC BY: Attribution . License terms: Download for free at http://cnx.org/contents/[email protected]. • Question ID 74696. Authored by: Meacham,William. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL. • Question ID 60791, 60790. Authored by: Day, Alyson. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.	{"url":"https://www.symbolab.com/study-guides/ivytech-wmopen-collegealgebra/compressions-and-stretches.html","timestamp":"2024-11-14T20:07:25Z","content_type":"text/html","content_length":"146624","record_id":"<urn:uuid:6770acf5-2a6c-4db0-8465-2aeca4d59e99>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00784.warc.gz"}
How Chaos Theory Works The Lorenz Attractor: A Portrait of Chaos Lorenz's computer model distilled the complex behavior of Earth's atmosphere into 12 equations -- an oversimplification if there ever was one. But the MIT scientist needed something even simpler if he hoped to get a better look at the tantalizing effects he glimpsed in his simulated weather. He narrowed his problem to a single atmospheric condition known as rolling fluid convection. Convection occurs on a large scale when the sun heats air near Earth's surface faster than air higher in the atmosphere or over bodies of water. As a result of this uneven heating, warmer, lighter air rises as cooler, heavier air sinks. This in turn creates large circular "rolls" of air. Convection also can occur on smaller scales -- in cups of hot coffee, in pans of warming water or in rectangular metal boxes heated from below. Lorenz imagined this latter small-scale example of rolling convection and set about deriving the simplest equations possible to describe the phenomenon. He came up with a set of three nonlinear equations: 1. dx/dt = σ(y-x) 2. dy/dt = ρx - y - xz 3. dz/dt = xy - βz where σ (sigma) represents the ratio of fluid viscosity to thermal conductivity, ρ (rho) represents the difference in temperature between the top and bottom of the system and β (beta) is the ratio of box width to box height. In addition, there are three time-evolving variables: x, which equals the convective flow; y, which equals the horizontal temperature distribution; and z, which equals the vertical temperature distribution. The equations, with only three variables, looked simple to solve. Lorenz chose starting values -- σ = 10, ρ = 28 and β = 8/3 -- and fed them to his computer, which proceeded to calculate how the variables would change over time. To visualize the data, he used each three-number output as coordinates in three-dimensional space. What the computer drew was a wondrous curve with two overlapping spirals resembling butterfly wings or an owl's mask. The line making up the curve never intersected itself and never retraced its own path. Instead, it looped around forever and ever, sometimes spending time on one wing before switching to the other side. It was a picture of chaos, and while it showed randomness and unpredictability, it also showed a strange kind of order. Scientists now refer to the mysterious picture as the Lorenz attractor. An attractor describes a state to which a dynamical system evolves after a long enough time. Systems that never reach this equilibrium, such as Lorenz's butterfly wings, are known as strange attractors. Additional strange attractors, corresponding to other equation sets that give rise to chaotic systems, have since been discovered. The Rössler attractor produces a graph that resembles a nautilus shell. The Hénon attractor produces an alien-looking boomerang. As soon as Lorenz published the results of his work in 1963, the scientific community took notice. Images of his strange attractor begin appearing everywhere, and people talked, with more than a little excitement, about this unfolding frontier of science where indeterminism, not determinism, ruled. And yet the word chaos had not yet emerged as the label for this new area of study. That would come from a soft-spoken mathematician at the University of Maryland.	{"url":"https://science.howstuffworks.com/math-concepts/chaos-theory4.htm","timestamp":"2024-11-03T04:19:16Z","content_type":"text/html","content_length":"153986","record_id":"<urn:uuid:ad2adad2-f5eb-47eb-9062-e33df74b26ad>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00158.warc.gz"}
I have a folder with 1500 images. I need to process the first 200 i... Accepted Answer for i = start : stop But i need to continue from 101, 102,... after 100. Kindly help me identify the problem. 0 Comments 1 view (last 30 days) I have a folder with 1500 images. I need to process the first 200 images(ie, 1 to 200). I used for loop it. But when i run the code i starts from 1,2,3 ... 100. then instead if going for 101th image, its value changes to 1001 , 1002... and so on. The easiest solution is to use my FEX submission natsortfiles. It fixes the order of an ASCII sort to take into account any numeric values in the strings. Note that it requires the file natsort to It simply sorts a cell array of filenames into the order you want: for k = 1:200 % first 200 files Note that this answer assumes that your files are sequentially numbered without any gaps. If this is not true, then you need to give more information on the filenaming logic. 2 Comments jeffin on 8 Feb 2016 Edited: Stephen23 on 8 Feb 2016 Stephen23 on 8 Feb 2016 Here is one way of solving your task using my answer: P = 'E:\New Folder'; S = dir(fullfile(P,'*.tif')); S = natsortfiles(S); for k = 1:200 F = fullfile(P,S(k).name); I = imread(F); • you will need to download NATSORTFILES. • I changed the loop variable from i to k, because in MATLAB i is the imaginary unit. • I used fullfile to create the path strings, which is much more robust than using strcat. More Answers (1) Try these: 2 Comments Stephen23 on 9 Feb 2016 Thank you for the links! I think both were picked for PotW, but there are a few differences: Note that natsorfiles correctly considers the file-names and file-extensions separately. Image Analyst on 9 Feb 2016 Stephen, you should notice that first link I gave actually talks about your submission as the "Pick of the Week".	{"url":"https://ch.mathworks.com/matlabcentral/answers/267180-i-have-a-folder-with-1500-images-i-need-to-process-the-first-200-images-ie-1-to-200-i-used-for-l","timestamp":"2024-11-13T11:38:17Z","content_type":"text/html","content_length":"153764","record_id":"<urn:uuid:95de8991-c6d0-4636-bcd6-7bb6214bfb7b>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00830.warc.gz"}
Hi! I Am Having Some Problems With A Loss After A Good Amount Of Training, What Would Be The Best Way To Log A Value To Have A Better Idea Of What Is Happening? Hi! I Am Having Some Problems With A Loss After A Good Amount Of Training, What Would Be The Best Way To Log A Value To Have A Better Idea Of What Is Happening? Hi! I am having some problems with a loss after a good amount of training, what would be the best way to log a value to have a better idea of what is happening? Posted 2 years ago Votes Newest Answers 4 AgitatedDove14 Well I have a loss function which is something like: class MyLoss(...): def forward(...): weights = self.compute_weights(...) return (weights * (target-preds)).mean()There seems to be a problem on certain batch when computing the weights. What would be the best way to log the batch that causes the problem, along with the weights being computed. I would do something like: ` from clearml import Logger def forward(...): self.iteration += 1 weights = self.compute_weights(...) m = (weights * (target-preds)).mean() Logger.current_logger().report_scalar(title="debug", series="mean_weight", value=m, iteration=self.iteration) return m ` Awesome AgitatedDove14 Thanks a lot ðŸ™Œ GrievingTurkey78 I'm not sure I follow, are you asking how to add additional scalars ?	{"url":"https://faq.clear.ml/question/1523707998835314688/hi-i-am-having-some-problems-with-a-loss-after-a-good-amount-of-training-what-would-be-the-best-way-to-log-a-value-to-have-a-better-idea-of-what-is-happening?desc=false","timestamp":"2024-11-04T07:19:33Z","content_type":"text/html","content_length":"35480","record_id":"<urn:uuid:e3a8c10e-494b-494c-b125-1ba71ed22951>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00758.warc.gz"}
Quantizing string theory in AdS<sub>5</sub>×S<sup>5</sup>: Beyond the pp-wave In a certain kinematic limit, where the effects of spacetime curvature (and other background fields) greatly simplify, the light-cone gauge worldsheet action for a type IIB superstring on AdS[5]×S^5 reduces to that of a free field theory. It has been conjectured by Berenstein, Maldacena, and Nastase that the energy spectrum of this string theory matches the dimensions of operators in the appropriately defined large R-charge large-N[c] sector of N=4 supersymmetric Yang-Mills theory in four dimensions. This holographic equivalence is thought to be exact, independent of any simplifying kinematic limits. As a step toward verifying this larger conjecture, we have computed the complete set of first curvature corrections to the spectrum of light-cone gauge string theory that arises in the expansion of AdS[5]×S^5 about the plane-wave limit. The resulting spectrum has the complete dependence on λ=g[YM]^2N[c]; corresponding results in the gauge theory are known only to second order in λ. We find precise agreement to this order, including the N=4 extended supermultiplet structure. In the process, we demonstrate that the complicated schemes put forward in recent years for defining the Green-Schwarz superstring action in background Ramond-Ramond fields can be reduced to a practical (and correct) method for quantizing the string. All Science Journal Classification (ASJC) codes • Nuclear and High Energy Physics Dive into the research topics of 'Quantizing string theory in AdS[5]×S^5: Beyond the pp-wave'. Together they form a unique fingerprint.	{"url":"https://collaborate.princeton.edu/en/publications/quantizing-string-theory-in-ads-sub5sub-s-sup5sup-beyond-the-pp-w","timestamp":"2024-11-11T21:49:08Z","content_type":"text/html","content_length":"53761","record_id":"<urn:uuid:e599fdcb-39c7-4bb9-8890-ec0dfb00b9c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00301.warc.gz"}
Math Practice Practice arithmetic, simple algebra, Roman numeral skills. Math Practice Screenshots Math Practice Editor's review Math Practice for Mac asks you challenging questions covering several math disciplines, including addition and subtraction, multiplication and division, and roman numerals, helping you practice your math skills in a fun way. However, a more attractive interface and the addition of a competitive element would really make this app shine. While intuitive, Math Practice's interface isn't that exciting when it comes to its looks. You begin by selecting the math skill you want to work on and then choose from three levels of difficulty. The app then throws math questions back at you. One annoying thing about it is that it requires you to click to move to the next problem, rather than automatically loading the problem after you've answered the current one. It does keep track of the number of correct and incorrect answers you submit, so you can see just how well you are progressing. The app gets a thumbs up for the nostalgic 8-bit sounds with which it celebrates correct answers, as well as for the ability to quickly change between subjects. Similar Suggested Software	{"url":"https://mac.dailydownloaded.com/en/educational-software/math-software/70310-math-practice-download-install","timestamp":"2024-11-07T09:13:52Z","content_type":"text/html","content_length":"22643","record_id":"<urn:uuid:2b8338b2-6a17-4172-85ae-4b84fc5f511e>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00309.warc.gz"}
Passive Filters MCQ Questions & Answers \| Electrical Engineering A band-stop filter passes frequencies between its lower and upper critical frequencies and rejects all others. The bandwidth of a resonant filter is determined by the quality factor (Q) of the circuit and the resonant frequency. In a certain parallel resonant band-pass filter, the resonant frequency is 14 kHz. If the bandwidth is 4 kHz, the lower frequency V[out] = 500 mV and V[in] = 1.3 V. The ratio $$\frac{{{{\text{V}}_{{\text{out}}}}}}{{{{\text{V}}_{{\text{in}}}}}}\,$$ expressed in dB is A series resonant band-stop filter consists of a 68 Ω resistor, a 110 mH coil, and a 0.02 µF capacitor. The internal resistance, RW, of the coil is 4 Ω. Input voltage is 200 mV. Output voltage is taken across the coil and capacitor in series. What is the output voltage magnitude at f0? In a series resonant band-pass filter, a lower value of Q results in Critical frequencies are also called -3 dB frequencies. A band-pass filter rejects all frequencies within a band between a lower and an upper critical frequency and passes all others. In a certain parallel resonant band-pass filter, the resonant frequency is 14 kHz. If the bandwidth is 4 kHz, the lower frequency is An RC high-pass filter consists of an 820 Ω resistor. What is the value of C so that Xc is ten times less than Rat an input frequency of 12 kHz?	{"url":"https://www.examveda.com/electrical-engineering/practice-mcq-question-on-passive-filters/","timestamp":"2024-11-12T09:24:31Z","content_type":"text/html","content_length":"65760","record_id":"<urn:uuid:ded8cd99-29a7-4b24-a060-1e906c85a514>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00206.warc.gz"}
Understanding Vectors: Find Displacement and Change in Position Understanding Vectors: Find Displacement And Change In Position A vector, represented by an arrow, has both magnitude and direction. To find the vector between two points, subtract the coordinates of the first point from the coordinates of the second point. This difference gives you the vector, which can be used to determine the displacement or change in position between the two points. Finding the Vector Between Two Points: A Guide to Understanding Vectors In the realm of mathematics, vectors play a crucial role in describing quantities that possess both magnitude (size) and direction. They find applications in various fields, including physics, engineering, and computer graphics. In this blog post, we’ll embark on a journey to unravel the mystery of vectors, with a specific focus on finding the vector between two points. What is a Vector? Vectors are mathematical entities that represent quantities with both magnitude and direction. They are often represented as arrows, where the length of the arrow indicates the magnitude and the direction of the arrow indicates the direction of the vector. Vectors play a fundamental role in describing physical quantities such as velocity, acceleration, and force. In the context of geometry, vectors are used to represent displacements, distances, and Vectors in Three Dimensions In three-dimensional space, vectors are defined by their coordinates, which specify their position relative to a fixed origin. The coordinates of a vector are typically expressed as a triplet of numbers, representing its x-component, y-component, and z-component. Finding the Vector Between Two Points In many practical applications, it becomes necessary to find the vector between two points in space. This vector represents the displacement or distance between the two points, taking into account both magnitude and direction. The formula for finding the vector between two points is: v = (x2 - x1, y2 - y1, z2 - z1) • (x1, y1, z1) are the coordinates of the first point • (x2, y2, z2) are the coordinates of the second point • v is the vector between the two points Step-by-Step Guide Finding the vector between two points involves following a series of steps: 1. Identify the coordinates of the two points in space. 2. Subtract the coordinates of the first point from the coordinates of the second point to obtain the components of the vector. 3. Write the vector using the formula: v = (x2 – x1, y2 – y1, z2 – z1) Vectors are powerful mathematical tools that provide a concise way to represent quantities with both magnitude and direction. Understanding how to find the vector between two points is essential for various applications in mathematics, physics, and engineering. By following the steps outlined in this post, you can confidently solve problems involving vectors and gain a deeper understanding of their significance in representing real-world phenomena. Understanding Vectors: Unraveling Direction and Magnitude In the tapestry of mathematics, vectors are vibrant threads that weave together magnitude and direction. They are not mere numbers, but dynamic entities that describe quantities with a specific orientation in space. Imagine a force pushing you in a particular way, or a velocity guiding your motion along a trajectory—these are vectors. Vectors are often depicted as arrows, with the length of the arrowhead representing its magnitude, and the direction of the arrowhead pointing towards its direction. This visual representation helps us understand vectors intuitively. Vectors also have coordinates, which define their position in space. Just as our physical address locates us on Earth, vector coordinates locate a vector relative to a fixed reference point. Each coordinate represents the distance along a specific axis (such as the x, y, or z axis) from the reference point to the vector’s tip. These coordinates are crucial in understanding vectors because they allow us to describe their position and orientation in a precise and mathematical way. They provide a framework for analyzing and manipulating vectors, enabling us to perform calculations and solve problems involving them. Embarking on a Journey into the Realm of Points: Unveiling Their Coordinates In the vast tapestry of space, we encounter points, fundamental building blocks that define locations. These enigmatic entities possess specific coordinates, like celestial beacons guiding us through the depths of the cosmos. By understanding these coordinates, we unlock the secrets of locating points within the boundless expanse of three-dimensional space. Each point occupies a unique position in space, defined by three coordinates: x, y, and z. Think of these coordinates as the longitude, latitude, and altitude that pinpoint a location on Earth. In three-dimensional space, the x-coordinate represents the left-right position, the y-coordinate the up-down position, and the z-coordinate the forward-backward position. Together, these coordinates form a roadmap that leads us precisely to each point. Like breadcrumbs left by a traveler, coordinates guide us through the labyrinthine corridors of space, connecting us to the myriad points that reside within it. By mastering the art of coordinates, we unravel the mysteries of point location, unlocking a gateway to a deeper understanding of the universe. Understanding Vector Operations: Addition and Subtraction Vectors are mathematical entities that possess both magnitude and direction. They play a crucial role in various fields, including physics, engineering, and computer graphics. Understanding vector operations, particularly addition and subtraction, is essential for comprehending vector manipulation. Vector Addition Imagine two vectors A and B represented as arrows. To add these vectors, we place their tails at the same point. The resultant vector R, which is the sum of A and B, is an arrow that extends from the tail of A to the tip of B. Geometrically, this forms a triangle with A and B as its sides and R as the third side. Formally, vector addition is defined as follows: R = A + B where the components of R are: Rx = Ax + Bx Ry = Ay + By Rz = Az + Bz Vector Subtraction Vector subtraction is similar to vector addition. Given two vectors A and B, we can find the vector D that represents the difference between A and B. To do this, we place the tails of A and B at the same point, and then draw an arrow from the tip of B to the tip of A. This arrow represents D, which is the difference between A and B. Formally, vector subtraction is defined as follows: D = A - B where the components of D are: Dx = Ax - Bx Dy = Ay - By Dz = Az - Bz By understanding vector addition and subtraction, we can manipulate vectors to solve problems in various scientific and engineering applications. Finding the Coordinates of a Vector In our journey through the realm of vectors, we’ve explored their nature and operations. Now, let’s delve into a crucial aspect: coordinates of a vector. Understanding these coordinates is paramount for unraveling the vector’s secrets—its magnitude and direction. Components of a Vector Imagine a vector as an arrow in space. Along each of the three axes (x, y, and z), we can measure the vector’s components. These components represent the vector’s projection onto each axis. To find these components, we use coordinate geometry. For instance, if a vector’s tail (starting point) is at (x1, y1, z1) and its head (ending point) is at (x2, y2, z2), then its components are: • x-component: (x2 - x1) • y-component: (y2 - y1) • z-component: (z2 - z1) Magnitude and Direction of a Vector Now, let’s unveil the vector’s magnitude and direction from its coordinates. • Magnitude: It reflects the vector’s length. Using the Pythagorean theorem, we find the magnitude as: Magnitude = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2) • Direction: It tells us the vector’s orientation. Using trigonometry, we define the direction in terms of direction cosines: cos(θx) = (x2 - x1) / Magnitude cos(θy) = (y2 - y1) / Magnitude cos(θz) = (z2 - z1) / Magnitude Where θx, θy, and θz are the angles between the vector and the x, y, and z axes, respectively. Discovering the Formula to Find Vectors Between Points In the realm of geometry, vectors play a crucial role in defining directions and magnitudes. They are often represented as arrows that point from one point to another, carrying with them both a length and a direction. Our journey today will delve into the fascinating world of vectors, with a particular focus on unearthing the formula that allows us to determine the vector between two points in space. To fully grasp the concept of vectors, let’s first understand their defining characteristics. A vector is primarily characterized by its magnitude, which is the length of the vector, and its direction, which is the angle it makes with a reference axis.Vectors can be visualized as arrows, with the arrow’s length representing the vector’s magnitude and the arrow’s orientation indicating its direction. Our exploration now turns to points, which are defined as specific locations in space. Each point can be precisely identified by its coordinates, which specify its position along three perpendicular axes: the x-axis, y-axis, and z-axis. These coordinates form the building blocks for defining vectors and understanding their behavior. To find the vector between two points, we employ a simple yet powerful formula: v = (x2 - x1, y2 - y1, z2 - z1) Here, v represents the vector between points (P1(x1, y1, z1) and P2(x2, y2, z2), and the xi, yi, and zi values represent the coordinates of each point along their respective axes. Let’s embark on a step-by-step guide to unraveling this formula: 1. Subtract the x-coordinates: Determine the difference between the x-coordinates of points P2 and P1. This gives you the x-component of the vector. 2. Subtract the y-coordinates: Repeat the process with the y-coordinates to find the y-component of the vector. 3. Subtract the z-coordinates: Finally, subtract the z-coordinates to obtain the z-component of the vector. The resulting vector, v, thus encapsulates the displacement from point P1 to point P2. To solidify our understanding, let’s consider an example. Suppose we have two points in space: P1(-3, 2, 5) and P2(1, 4, 7). Employing our formula, we can calculate the vector between these points: v = (1 - (-3), 4 - 2, 7 - 5) v = (4, 2, 2) This vector, with a magnitude of √(4² + 2² + 2²) = √24 ≈ 4.89 and pointing in the direction of (4, 2, 2), describes the displacement from point P1 to point P2. As we conclude our journey, let us remember that understanding vectors and their properties is vital for navigating the world of geometry and physics. The formula for finding vectors between points serves as a powerful tool in this endeavor, allowing us to precisely determine the displacement and direction between two locations in space. Mastering the Vector Between Points: A Guide for Beginners In the realm of physics and geometry, vectors are indispensable tools for describing quantities that possess both magnitude (size) and direction. From forces to displacements, vectors are ubiquitous in our understanding of the physical world. In this blog post, we embark on a journey to unravel the secrets of finding vectors between two points, empowering you with a powerful technique for solving a myriad of problems. Understanding Vectors Vectors are like arrows with lengths representing their magnitude and tips pointing in the direction of the vector. Each vector is defined by its coordinates, which specify its position in space. Imagine a vector as an arrow extending from the origin of a coordinate system, its head pointing towards a specific location. Understanding Points Points are fixed locations in space, each with its unique set of coordinates. These coordinates act as signposts, guiding us to the exact position of a point within a three-dimensional world. Vector Operations To manipulate vectors effectively, we must master two essential operations: • Vector Addition: When we combine two or more vectors, we add their corresponding coordinates to obtain the resultant vector. • Vector Subtraction: Subtracting one vector from another yields a new vector that represents the difference between them. Coordinates of a Vector The coordinates of a vector reveal its components along the x, y, and z axes. These components provide vital information about the vector’s magnitude and direction. By knowing the coordinates, we can reconstruct the vector as an arrow and determine its orientation in space. Finding the Vector Between Two Points To find the vector between two points, we employ a straightforward formula: v = (x2 - x1, y2 - y1, z2 - z1) • (x1, y1, z1) are the coordinates of the first point • (x2, y2, z2) are the coordinates of the second point Suppose we have two points: A(2, 3, 4) and B(6, 7, 10). To find the vector from A to B, we plug the coordinates into the formula: v = (6 - 2, 7 - 3, 10 - 4) v = (4, 4, 6) Therefore, the vector from point A to point B is v = (4, 4, 6). Mastering the concept of vectors and the formula for finding vectors between points unlocks a powerful tool for solving problems in physics, geometry, and beyond. By following the steps outlined in this post, you can confidently determine the vector between any two points, empowering you to tackle a wide range of challenges with ease. Leave a Comment	{"url":"https://youngcarer.info/understanding-vectors-find-displacement-change-in-position/","timestamp":"2024-11-09T20:54:58Z","content_type":"text/html","content_length":"71539","record_id":"<urn:uuid:d5fca691-5c5b-4872-8366-fc7105610b7a>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00567.warc.gz"}
ECCC - Barnaby Martin All reports by Author Barnaby Martin: TR21-022 \| 20th February 2021 Stefan Dantchev, Nicola Galesi, Abdul Ghani, Barnaby Martin Depth lower bounds in Stabbing Planes for combinatorial principles Revisions: 1 We prove logarithmic depth lower bounds in Stabbing Planes for the classes of combinatorial principles known as the Pigeonhole principle and the Tseitin contradictions. The depth lower bounds are new, obtained by giving almost linear length lower bounds which do not depend on the bit-size of the inequalities and in ... more >>> TR18-165 \| 20th September 2018 Stefan Dantchev, Nicola Galesi, Barnaby Martin Resolution and the binary encoding of combinatorial principles We investigate the size complexity of proofs in $RES(s)$ -- an extension of Resolution working on $s$-DNFs instead of clauses -- for families of contradictions given in the {\em unusual binary} encoding. A motivation of our work is size lower bounds of refutations in Resolution for families of contradictions in ... more >>> TR18-024 \| 1st February 2018 Olaf Beyersdorff, Judith Clymo, Stefan Dantchev, Barnaby Martin The Riis Complexity Gap for QBF Resolution We give an analogue of the Riis Complexity Gap Theorem for Quanti fied Boolean Formulas (QBFs). Every fi rst-order sentence $\phi$ without finite models gives rise to a sequence of QBFs whose minimal refutations in tree-like Q-Resolution are either of polynomial size (if $\phi$ has no models) or at least ... more >>> TR07-001 \| 19th November 2006 Stefan S. Dantchev, Barnaby Martin, Stefan Szeider Parameterized Proof Complexity: a Complexity Gap for Parameterized Tree-like Resolution Revisions: 1 We propose a proof-theoretic approach for gaining evidence that certain parameterized problems are not fixed-parameter tractable. We consider proofs that witness that a given propositional formula cannot be satisfied by a truth assignment that sets at most k variables to true, considering k as the parameter. One could separate the ... more >>>	{"url":"https://eccc.weizmann.ac.il/author/019834/","timestamp":"2024-11-14T04:12:12Z","content_type":"application/xhtml+xml","content_length":"21200","record_id":"<urn:uuid:db8d1d7d-d83b-467a-a7c1-7bd3c2ef8f7e>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00121.warc.gz"}
How Many 1 3 Cups In 2 3 Cup - How Many Sumo How Many 1 3 Cups In 2 3 Cup Measurement fractions are an essential concept in mathematics and everyday life. They allow us to represent quantities that are not whole numbers, providing a more precise understanding of measurements. One common scenario that requires an understanding of measurement fractions is determining how many smaller units can fit into a larger unit. In this article, we will explore the question, ‘How many 1/3 cups can fit into a 2/3 cup?’By applying principles of converting fractions to a common denominator and dividing a unit into equal parts, we will arrive at a precise answer to this question. To begin, it is important to grasp the concept of converting fractions to a common denominator. This process ensures that the fractions being compared have the same base unit, allowing for a fair and accurate comparison. In the case of 1/3 and 2/3 cups, the common denominator is 3. By converting both fractions to this common denominator, we can clearly see the relationship between the two Next, we need to consider how to divide a 2/3 cup into equal parts. By applying mathematical principles, we will be able to determine the exact number of 1/3 cups that can fit into a 2/3 cup. Through this article, we will navigate these calculations step by step, providing an evidence-based and precise answer to the question at hand. Understanding Measurement Fractions The concept of understanding measurement fractions involves comprehending the relationship between the number of 1/3 cups and 2/3 cups. Comparing measurement fractions with different denominators is a crucial aspect of this understanding. When comparing 1/3 cups and 2/3 cups, it is important to note that the denominators are different. The denominator represents the number of equal parts into which a whole is divided. In the case of 1 /3 cups, the whole is divided into three equal parts, while in the case of 2/3 cups, the whole is divided into six equal parts. Therefore, 2/3 cups is equivalent to two out of the six equal parts, while 1/3 cups is equivalent to one out of the three equal parts. Exploring real-life examples of measurement fractions in cooking recipes can help illustrate the concept further. Many recipes require the use of measurement fractions to ensure accurate proportions of ingredients. For instance, a recipe might call for 2/3 cups of flour. If we have only 1/3 cup measuring cups, we would need to use two of them to measure out the required amount. This demonstrates that two 1/3 cups are equivalent to 2/3 cups. Similarly, if a recipe calls for 1/3 cups of sugar, we would need to use only one of the 1/3 cup measuring cups. This illustrates that one 1/3 cup is equivalent to 1/3 cups. Understanding how different measurement fractions relate to each other is crucial for successful cooking and accurate measurement. Converting Fractions to a Common Denominator Converting fractions to a common denominator involves finding a shared value that each fraction can be expressed as. This process is necessary when comparing or combining fractions that have different denominators. To simplify fractions with common denominators, one must first identify the least common multiple (LCM) of the denominators. The LCM is the smallest multiple that both denominators share. Once the LCM is determined, each fraction can be rewritten with the new denominator. This allows for a direct comparison between the fractions, as they now have the same base. Comparing fractions with different denominators can be challenging without a common denominator. When fractions have different denominators, it is not immediately clear which fraction is larger or smaller. By converting fractions to a common denominator, the task of comparing becomes much simpler. For example, if we have the fractions 1/3 and 2/5, we can convert them to fractions with a common denominator, such as 15. The first fraction becomes 5/15 and the second fraction becomes 6/15. Now it is evident that 5/15 is smaller than 6/15. Converting fractions to a common denominator allows us to compare fractions accurately and make informed decisions based on their relative sizes. Converting fractions to a common denominator is a crucial step in simplifying fractions and comparing fractions with different denominators. It involves finding the least common multiple of the denominators and rewriting the fractions with the common denominator. This process enables us to compare fractions accurately and make precise comparisons based on their relative sizes. By simplifying fractions with common denominators, we can enhance our understanding of measurement fractions and confidently solve problems involving Dividing a 2/3 Cup into Equal Parts Dividing a 2/3 portion equally can be achieved by finding a method to distribute it into equal parts. When dealing with fractions, it is important to ensure that the denominator is the same for all fractions involved. In this case, since we are dividing a 2/3 cup, we need to find a common denominator. To divide the 2/3 cup into equal parts, we can consider using a common denominator of 6. By multiplying the numerator and denominator of the fraction by 2, we can convert 2/3 into 4/6. This allows us to divide the cup into 6 equal parts, with each part being 4/6 of the original cup. Fractional part distribution plays a crucial role in various practical applications, including cooking and baking. Imagine a recipe that requires 2/3 cup of flour, but you only have a 1/3 cup measuring cup. By understanding how to divide a 2/3 cup into equal parts, you can accurately measure the required amount of flour by using your 1/3 cup measuring cup twice. This ensures that the recipe is prepared with the correct proportions, resulting in a successful outcome. Additionally, in situations where you need to distribute a limited amount of liquid equally among multiple containers, such as when sharing a drink with friends, knowing how to divide a 2/3 cup into equal parts enables you to ensure that each person receives an equal share. Overall, understanding the practical applications of dividing cups is essential in various aspects of daily life. The Answer: Two 1/3 Cups in a 2/3 Cup To achieve equal parts, a 2/3 cup can be divided by using two 1/3 cups. When comparing different measurement units, it is important to consider the relationship between them. In this case, we are comparing the 1/3 cup measurement unit to the 2/3 cup measurement unit. By using two 1/3 cups, we can accurately divide the 2/3 cup into equal parts. Estimating fractions in real life scenarios can be a useful skill, especially when it comes to cooking or baking. In this situation, dividing a 2/3 cup into two 1/3 cups allows for precise measurements. This method ensures that each part is exactly equal, maintaining accuracy in recipes and achieving desired results. By incorporating fractions into everyday life, individuals can improve their understanding of mathematical concepts and develop practical skills. In conclusion, the concept of measurement fractions is crucial in understanding how many 1/3 cups can fit into a 2/3 cup. By converting fractions to a common denominator, we can simplify the problem and accurately determine the answer. In this case, we need to divide a 2/3 cup into equal parts to find out how many 1/3 cups it can hold. The answer is that there are two 1/3 cups in a 2/3 cup. This can be determined by dividing the 2/3 cup into two equal parts, each representing a 1/3 cup. This precise conclusion is based on the principles of fraction conversion and division, ensuring an evidence-based approach to solving the problem. Understanding measurement fractions and their manipulation is essential in various fields such as cooking, construction, and engineering, where precise measurements are required for accurate results. Leave a Comment	{"url":"https://howmanysumo.com/how-many-1-3-cups-in-2-3-cup/","timestamp":"2024-11-04T02:46:15Z","content_type":"text/html","content_length":"53218","record_id":"<urn:uuid:61cfc55f-fc33-466f-a779-3a997d05a60a>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00024.warc.gz"}
How to Plot A Curved Surface In Matlab? In Matlab, you can plot a curved surface using the surf function. The surf function creates a three-dimensional surface plot where the values of a matrix are displayed as heights above the x-y plane. To plot a curved surface, you would need to define the x, y, and z coordinates of the points on the surface. Here is an example of how you can do this: 1. Define the range of the x and y coordinates using the meshgrid function. For example, [X, Y] = meshgrid(-10:0.5:10, -10:0.5:10); This will create a grid of points that span from -10 to 10 in both x and y directions. 2. Use the defined x and y coordinates to calculate the corresponding z values. This can be done using an equation or a mathematical function that represents the curved surface. For example, let's calculate the z values using the equation for a paraboloid: Z = X.^2 + Y.^2; In this case, the z value is calculated by squaring both the x and y values and summing them. 3. Finally, use the surf function to plot the surface: surf(X, Y, Z); This will create a 3D plot of the curved surface based on the defined x, y, and z values. You can customize the appearance of the surface plot by adding labels to the x, y, and z axes, adjusting the color map, adding a title, or manipulating the viewing angle using additional Matlab How to add texture to a curved surface plot in Matlab? To add texture to a curved surface plot in MATLAB, you can use the function "surf" along with the "FaceColor" and "FaceAlpha" properties. Here's an example: 1 % Create example data 2 [x, y] = meshgrid(-2:0.1:2); 3 z = sin(sqrt(x.^2 + y.^2)); 5 % Define custom texture image 6 textureImage = imread('texture.jpg'); 8 % Plot the curved surface 9 figure; 10 surf(x, y, z); 12 % Set the face color to 'texture' 13 set(gca, 'FaceColor', 'texture'); 15 % Map the texture image onto the surface 16 colorMap = repmat({textureImage}, size(z)); 17 colormap(colorMap); 19 % Adjust the transparency of the texture 20 set(gca, 'FaceAlpha', 0.8); 22 % Set the axis labels and title 23 xlabel('X'); 24 ylabel('Y'); 25 zlabel('Z'); 26 title('Curved Surface with Texture'); Make sure to replace 'texture.jpg' with the path to your own custom texture image. The set(gca, 'FaceAlpha', 0.8) line adjusts the transparency level of the texture (values between 0 and 1). After running the code, MATLAB will plot a curved surface with the texture applied to it. How to save a curved surface plot in a specific file format in Matlab? To save a curved surface plot in a specific file format in MATLAB, you can use the saveas function. Here is an example code snippet demonstrating how to save a curved surface plot as a PNG file: 1 % Generate sample data 2 [X, Y] = meshgrid(-5:0.1:5); 3 Z = sin(sqrt(X.^2 + Y.^2)) ./ sqrt(X.^2 + Y.^2); 5 % Create the curved surface plot 6 figure; 7 surf(X, Y, Z); 8 xlabel('X'); 9 ylabel('Y'); 10 zlabel('Z'); 12 % Save the plot as a PNG file 13 filename = 'curved_surface_plot.png'; 14 saveas(gcf, filename, 'png'); In this example, we first generate some sample data for the curved surface plot. Then, we create the plot using the surf function. After that, we use the saveas function to save the current figure window (gcf) as a PNG file with the specified filename (curved_surface_plot.png). You can replace the 'png' argument with other file formats, such as 'jpg', 'pdf', 'eps', etc., to save the plot in different formats. How to extract data points from a curved surface plot in Matlab? There are several ways to extract data points from a curved surface plot in Matlab. Here are two common methods: 1. Using the Data Cursor: • Create the curved surface plot using the "surf" function in Matlab. • Click on the "Data Cursor" button in the toolbar. • Hover the cursor over the plot to select the desired point. • Press Enter to display the selected point's coordinates and other information in a figure window or the command window. • Repeat the above steps for each point you want to extract. 1. Using Matrix Indexing: • Create the curved surface plot using the "surf" function in Matlab. • Generate the respective X and Y coordinate matrices using the "meshgrid" function. • Use these coordinate matrices to find the indices of the desired points on the surface plot. • Extract the corresponding Z values from the surface plot using these indices. • Repeat the above steps for each point you want to extract. Here is an example code snippet using the second method: 1 % Example surface plot 2 [X, Y] = meshgrid(-5:0.1:5); % Generate X and Y coordinate matrices 3 Z = sin(sqrt(X.^2 + Y.^2)); % Generate Z values based on the function 5 % Plot the surface 6 surf(X, Y, Z); 7 xlabel('X'); 8 ylabel('Y'); 9 zlabel('Z'); 11 % Extract points 12 xPoint = 1.5; % x-coordinate of the desired point 13 yPoint = -2.5; % y-coordinate of the desired point 15 % Find the indices of the closest points on the surface plot 16 [~, xIndex] = min(abs(X(1, :) - xPoint)); 17 [~, yIndex] = min(abs(Y(:, 1) - yPoint)); 19 % Extract the corresponding z value 20 zPoint = Z(yIndex, xIndex); 22 fprintf('Selected point coordinates: (%f, %f, %f)\n', xPoint, yPoint, zPoint); In this example, we create a surface plot of a 3D function (in this case, sin(sqrt(X.^2 + Y.^2))) and then extract a specific point's coordinates and corresponding Z value using matrix indexing.	{"url":"https://elvanco.com/blog/how-to-plot-a-curved-surface-in-matlab","timestamp":"2024-11-13T18:25:08Z","content_type":"text/html","content_length":"349317","record_id":"<urn:uuid:e86ecc62-5653-4e10-99d4-f1d86986828c>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00217.warc.gz"}
Basics of Algebra and Analysis For Computer Science by Jean Gallier Number of pages: 254 From the table of contents: Linear Algebra; Determinants; Basics of Affine Geometry; Polynomials, PID's and UFD's; Topology; Differential Calculus; Zornâ€™s Lemma and Some Applications; Gaussian elimination, LU-factoring and Cholesky-factoring. Download or read it online for free here: Download link (multiple PDF files) Similar books Encyclopedia of Mathematics Kluwer Academic PublishersAn open access resource designed specifically for the mathematics community. With more than 8,000 entries, illuminating 50,000 notions in mathematics, Encyclopaedia was the most up-to-date graduate-level reference work in the field of mathematics. Mathematics for Technical Schools J.M. Warren, W.H. Rutherford Copp, ClarkIn this book an attempt has been made to present the subject of Elementary Mathematics in a way suitable to industrial students in our technical schools. The fundamentals as herein presented will form a basis for a wide range of industries. Mathematical Formula Handbook Wu-ting Tsai National Taiwan UniversityContents: Series; Vector Algebra; Matrix Algebra; Vector Calculus; Complex Variables; Trigonometry; Hyperbolic Functions; Limits; Differentiation; Integration; Differential Equations; Calculus of Variations; Functions of Several Variables; etc. Advanced Problems in Mathematics: Preparing for University Stephen Siklos Open Book PublishersThe book is intended to help candidates prepare for entrance examinations in mathematics and scientific subjects. It is a must read for any student wishing to apply to scientific subjects at university level and for anybody interested in mathematics.	{"url":"https://e-booksdirectory.com/details.php?ebook=2130","timestamp":"2024-11-13T06:06:41Z","content_type":"text/html","content_length":"10673","record_id":"<urn:uuid:ad7e1a20-0e98-4915-bdfd-9e71d5b0944f>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00527.warc.gz"}
Compound Annual Growth Rate (CAGR)Compound Annual Growth Rate (CAGR): The Key to Measuring Investment Growth Compound Annual Growth Rate (CAGR) So, what exactly is the Compound Annual Growth Rate (CAGR)? In simple terms, CAGR is a useful metric that tells you the mean annual growth rate of an investment over a specified time period, assuming the investment grows at a steady rate, compounding over time. It essentially smooths out the returns and gives you a clearer picture of how well your investments are performing. To understand CAGR, you need to know three essential components: • Beginning Value: This is the initial amount you invest or the value of your investment at the start of the period. • Ending Value: This refers to the final value of your investment at the end of the specified period. • Number of Years: This is the duration over which the investment grows, measured in years. Calculating CAGR is straightforward. Here is the formula: $\text{CAGR} = \frac{\text{Ending Value}}{\text{Beginning Value}}^{\frac{1}{\text{Number of Years}}} - 1$ For example, if you invested $1,000 and it grew to $1,500 over 3 years, the CAGR would be: $\text{CAGR} = \frac{1500}{1000}^{\frac{1}{3}} - 1 \approx 0.1447 \text{ or } 14.47\%$ 1. Nominal CAGR: This is the basic calculation that does not consider inflation or other external factors. It is a straightforward method to assess growth. 2. Real CAGR: This adjustment accounts for inflation, providing a more accurate reflection of the purchasing power of your investment returns over time. CAGR is increasingly being used in various sectors beyond traditional finance, including: • Tech Investments: With the rise of emerging technologies, investors are keen on measuring growth in tech-centric portfolios. • Sustainable Investments: ESG (Environmental, Social and Governance) investments are gaining traction and CAGR helps evaluate their long-term growth. • Retirement Planning: Individuals are utilizing CAGR to project their retirement savings growth, ensuring they meet their financial goals. When considering investments, CAGR can be a game-changer. Here are a few strategies: • Comparative Analysis: Use CAGR to compare the growth rates of different investments. This helps in making informed decisions. • Long-Term Planning: CAGR is perfect for long-term investment strategies, as it provides a clearer picture of growth over time. • Risk Assessment: Understanding the CAGR helps in assessing the risk associated with various investment options, allowing for better portfolio management. The Compound Annual Growth Rate (CAGR) is a powerful tool in the world of finance. It simplifies the complex nature of investment growth into a single, understandable figure. Whether you are tracking your investments or planning for retirement, mastering CAGR can enhance your investment strategy and help you make informed decisions. Remember, a steady growth rate is often more desirable than a rollercoaster ride of returns! What is the significance of Compound Annual Growth Rate (CAGR) in finance? CAGR is essential for assessing the growth of investments over time, providing a smoothed annual growth rate that can help compare different investments. How can I calculate CAGR for my investments? To calculate CAGR, you need the beginning value, ending value and the number of years. The formula is: CAGR = (Ending Value / Beginning Value)^(1 / Number of Years) - 1. Why is CAGR important in investment analysis? CAGR is important because it provides a clear view of an investment’s annual growth over time, making it easier to compare different assets or portfolios. Unlike simple averages, CAGR shows true compounded growth, helping investors assess long-term performance accurately. How is CAGR used to compare investments? CAGR is used to compare the growth rates of different investments over the same period, allowing for a more meaningful comparison. By showing the annualized rate of return, CAGR helps investors choose assets with the strongest long-term growth potential. More Terms Starting with C	{"url":"https://docs.familiarize.com/glossary/compound-annual-growth-rate-cagr/","timestamp":"2024-11-08T09:13:33Z","content_type":"text/html","content_length":"96519","record_id":"<urn:uuid:31a76320-0c3c-4e6f-a342-31cf33f9000e>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00666.warc.gz"}
NCERT Solutions for Class 4 - The Advansity Portal For Everyone(Affinity Till Infinity) NCERT Solutions for Class 4 NCERT Solutions for Class 4 NCERT Solutions for Class 4 Maths, EVS and English are available at BYJU’S. The NCERT Solutions have been designed by our experienced teachers and subject matter experts. Extensive research was done to come up with authentic and appropriate NCERT solutions that will further act as a valuable resource for students. The Class 4 NCERT solutions cover all the exercises from NCERT Books For Class 4 for Maths, EVS and English extensively. The solutions have been designed keeping in mind the latest syllabus and CBSE board guidelines. It has specifically been designed from the ground up to help students understand different concepts in a simple and easy manner. Check the complete solutions for NCERT Class 4 English, EVS and Maths below. NCERT Solutions For Class 4 Maths & EVS We are offering students to download NCERT Solutions Class 4 Maths PDF for free from BYJU’S. These PDFs includes all chapters of NCERT Solutions from your CBSE Class 4 Maths textbook. BYJU’S expert teachers cover all the 14 chapters with simple NCERT solved questions. These solutions are always updated to the latest (2020-2021) CBSE syllabus. Now NCERT Solutions for Class 4 EVS are in PDF format, you download them for free from BYJU’S website chapter wise. Our experts cover all the FAQ’s and the CBSE recommended questions for the exam. NCERT Solutions for Class 4 English are provided in PDF format, which can be downloaded for free from BYJU’S website. Our experts covers the CBSE recommended questions from all the chapters for the Students will have access to NCERT books, question papers as well as PDFs that will help them in learning concepts better as well as prepare meticulously for the exams. Features of BYJU’S Class 4 NCERT Solutions • All the exercises are covered so that students can clear any doubt instantly • Solutions are prepared by subjects experts and are given in a very easy to understand way to help students understand better • Numerical questions are solved in a step-by-step process to help students easily comprehend them • Solutions are available in PDF form, where students can download and access offline • Diagrams are also provided for better visualization Benefits of NCERT Class 4 Solutions These NCERT Solutions for Class 4 will help students find the right approach to solving NCERT papers. With the solutions provided, students can also gain higher confidence to solve different questions that will be asked in the exams. The NCERT Solutions for different classes are vital to getting on with the practice of the examination. Here students will not only get access to effective exam tools but also assessment tools that can further improve any student’s proficiency in Class 4 English, Maths and EVS. In essence, this can be the best study platform as students can find a lot of study material for easy learning and Interested students can also download BYJU’S – The Learning App and further get a completely customized learning experience. Students can learn from lessons that have been produced by some of the top teachers in the country. BYJU’S is dedicated to making learning easy and fun. Leave a Comment Cancel Reply	{"url":"https://theadvansity.com/ncert-solutions-for-class-4/","timestamp":"2024-11-06T12:15:44Z","content_type":"text/html","content_length":"225961","record_id":"<urn:uuid:0470bf6f-87cc-4162-8351-488f7e0a5687>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00184.warc.gz"}
Understanding Standard Position Question Video: Understanding Standard Position Mathematics • First Year of Secondary School Does the ordered pair (ray π Άπ ΄, ray π Άπ ·) express an angle in standard position? Video Transcript Does the ordered pair given by the ray joining π Άπ ΄ and the ray joining π Ά to π · express an angle in standard position? We know that for an angle to be in standard position, its vertex must be centered at the origin and the initial side must lie on the positive π ₯-axis. Now, our angle is defined by this ordered pair. So we have the ray joining π Ά and π ΄ and the ray joining π Ά and π ·. The ray joining π Ά to π ΄ is this one, and that does indeed lie on the positive π ₯-axis. We know itβ s the initial side of our angle because itβ s mentioned first. Then the ray that joins π Ά to π · is this one here, meaning the angle weβ re interested in is this one. But does the vertex of this lie at the origin? Well, no, the vertex is over here somewhere. Itβ s someway along the positive π ₯-axis. And so the ordered pair defined by the ray joining π Ά to π ΄ and the ray joining π Ά to π · does not express an angle in standard position. And the answer is no. There are, however, a couple of angles that are in standard position here. The first would be given by the ordered pair the ray joining π to π Ά and the ray joining π to π Έ. The ray π π Ά lies on the positive π ₯-axis, and then the vertex is centered at the origin. Similarly, we could begin with the same initial side, and thatβ s the ray joining π to π Ά, and measure through to the ray joining π to π Ί. The vertex for this angle is still located at the origin, and so the angle is also in standard position.	{"url":"https://www.nagwa.com/en/videos/748121012864/","timestamp":"2024-11-06T17:15:15Z","content_type":"text/html","content_length":"248209","record_id":"<urn:uuid:eb95a367-04af-4fb4-a254-58f3d5b25147>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00763.warc.gz"}
Perform a Goldman-Cox test Last update: Jul 6, 2024, Contributors: Minh Bui, Rob Lanfear Perform a Goldman-Cox test What is a Goldman-Cox test? Nick Goldman explains the Goldman-Cox (GC) test in this paper The basic idea is that we are asking whether the full model (i.e. the tree, branch lengths, model parameters, everything we estimate from the data) is an adequate description of the data. We do this by calcualting the cost of the model, which is just: the maximum liklelihood of the data under the model, minus the unconstrained likelihood (see below). We’ll call this delta. You can read Nick’s paper for a full description of this, but he puts it rather nicely on page 184: [delta] can be considered the "cost" of using [our model assumptions] to make inferences about phylogeny. A low cost indicates that [our model] is adequate; a high cost indicates that [our model] is performing badly and should be rejected. Be warned! For most datasets, you will reject the full model. This is simply because most modern datasets are large, and most of our models of evolution are still really simple. So we should expect to reject them. This doesn’t mean they don’t produce useful inferences, of course! Calculating the cost of the model is easy, but to interpret it we need to know if the cost is surprisingly large or small. That is, we need some idea of the null distribution of costs. One way to do this, and the method used by the Goldman-Cox test, is to use a parametric bootstrap. A parametric bootstrap is a really useful way to ask questions in phylogenetics. The absolute classic paper on this is from Goldman, Anderson, and Rodrigo in 2000. You should read this first. There are many flavours of parametric bootsrap in phylogenetics, but they all follow the same pattern: 1. Do an analysis on your focal dataset (probably your empirical dataset), and measure the thing you’re interested in (here it’s delta) 2. Call the model you estimated from your empirical dataset the null model, then simulate a lot of new datasets using that null model 3. Measure the thing you are interested in on each of the simulated datasets (here it’s delta) 4. Ask if your observed value (from step 1) is surprising given the list of simulated values (from step 3) In other words, for the Goldman-Cox test we can figure out if our observed cost is high, by simulating lots of cost values under the null model, and then re-calculating the cost on those. That null distribution tells us what kind of cost values we should expect when the null model is true. And so it then allows us to ask whether our observed value looks plausible. If you’re a biologist and you like working with an alpha value of 5%, you might consider that if your observed cost is in the highest 5% of the simualted costs, you should reject your model as inadequate. The Goldman-Cox test doesn’t (and can’t) tell you which aspects of your model might be causing the most trouble. But it’s a really good place to start when considering how well you are able to to model your data. Input files For this recipe I’ll use data from the Bovidae family with five taxa (Yak, Cow, Goat, Sheep and Antelope) and 5,000 sites. This is a (very) small subset of the amazing Wu et al 2018 dataset. I keep the file to 5K sites because that helps keep the file sizes manageable and analyses fast for a demonstration. Note: for this version of the Goldman-Cox test, you can only use alignments with no gaps or ambiguities. So I have removed any sites with gaps or ambituities from the alignment. Command lines 1. Analyse the original data, and simulate 999 alignments All of the work in IQ-TREE can be done in a single commandline, thanks to the magic of AliSim. Here’s the commandline, and below I deconstruct the options: iqtree -s bovidae_4K.phy --alisim simulated_MSA --num-alignments 999 • -s bovidae_4K.phy: tells IQ-TREE to do a standard analysis on the bovidae_4K.phy file, where it chooses the model, estimates the tree and model parameters • --alisim simulated_MSA tells AliSim to then simulate alignments that mimic this alignment (i.e. use the tree and model parameters estimated from the original data) • --num-alignments 999 tells AliSim that we want 999 mimicked alignments (999 is a good number for a parametric bootstrap) 2. Calculate delta for the observed data The bovidae_4K.phy.iqtree file, gives us the information we need to calculate delta: Log-likelihood of the tree: -6545.5196 (s.e. 74.4412) Unconstrained log-likelihood (without tree): -6448.4561 So delta here is: -6448.4561 - -6545.5196 = 97.0635 Let’s write a little bash function to calculate this value - it will help us in the next step we have to do the same for the 999 simulated datasets. The first couple of lines of this function just get the two likelihood values we want. Then we take the difference to get delta. Of course, you can do this in whatever language you like. But I like bash, so here’s my attempt: get_delta () { # a function to get the difference bewteen lnL and unconstrained lnL from a .iqtree file # assumes that the only passed argument is the name of a .iqtree file lnL_model=$(grep "Log-likelihood of the tree: " $1 \| awk '{print $5}') lnL_unconstrained=$(grep "Unconstrained log-likelihood" $1 \| awk '{print $5}') delta=$(echo $lnL_unconstrained - $lnL_model \| bc) echo $delta Now if you copy-paste that function into your bash terminal, then run get_delta bovidae_4K.phy.iqtree You should get the output 97.0635 or something quite close (it can vary depending on the random number seed) 3. Calculate our 999 values of delta from the simulated delta Now we need get the 999 delta values from our simulated alignments. This will give us a null distribution for delta when the model estimated from the original dataset is true. In other words, this will tell us what kind of values of delta we should expect to see when our model really does have a single tree with the branch lengths we estimated, all the substitution model parameters we estimated, etc. To get our delta values from our 999 simulated alignments, we’ll first run IQ-TREE on each alignment in turn. We can do that in bash with a simple for loop. You can do this in whatever language you like, and in some situations you would want to parallelise this to make it faster. But for this tutorial I’ll keep it as simple as possible (the below might take a few minutes to run): for alignment in simulated_MSA_.phy; do iqtree -s $alignment The first line in that loop just uses the wildcard to match all of the simulated alignment files in turn. Then the second line runs IQ-TREE on each alignment. Now we’ve done the analysis, we need to get all of our delta values from those output files. We can do this using the get_delta() function we wrote above, in a for loop just like the one we used to run IQ-TREE. The for loop below just uses >> to put all the delta values into a file called simulated_delta.txt: for iqtree_file in simulated_MSA_*.phy.iqtree; do get_delta $iqtree_file >> simulated_delta.txt 4. Figure out the position of our observed delta in a ranked list of our simulated deltas If you look through your list of deltas in the simulated_delta.txt file, you’ll see they all seem to be below the observed value. So, if we were to order the list of the 999 simulated deltas and our observed delta from largest to smallest, our observed delta would be in position 1 out of 1000 in the list. So we know our p-value here would be at most 1/1000, i.e. p<=0.001. In other words, we can reject the hypothesis that the full model (tree, branch lengths, substitution model etc) is an adequate description of the data… Not all analyses will be quite this obvious, so here’s a little R script that you could use to calculate the p-value: # reads the simulated deltas into a data frame simulated_deltas = read.delim("simulated_delta.txt", header=F) # the p-value is just the position of the observed value in the ranked list, # divided by the list length # first we tell R our observed value of delta from above observed = 97.0635 # the position is just the length of the list if you'd added the observed value (1000 in our case) # minus how many of the simulated values are smaller than the observed value position = (nrow(simulated_deltas) + 1) - sum(observed>simulated_deltas$V1) # the p-value is just the position divided by teh length of the list if you'd added the observed value p_value = position / (nrow(simulated_deltas) + 1) # then we can make a plot to help us visualise it ggplot(simulated_deltas, aes(x=V1)) + geom_histogram() + geom_vline(xintercept = observed, colour="red", size=1) + theme_minimal() + xlab("delta value") + ggtitle("Null distribution of delta values", subtitle = "Observed value is shown as a red line") In this case, you’d get the answer 0.001. Since we’re at the very extreme of the distribution here, we can go one better than saying that the p-value equals 0.001, and say that it is at most 0.001, i.e. p<=0.001. And our histogram helps make this clear. IQ-TREE version Last tested with IQ-TREE 2.2.0.3	{"url":"http://iqtree.org/doc/recipes/goldman-cox-test","timestamp":"2024-11-08T13:33:30Z","content_type":"text/html","content_length":"17365","record_id":"<urn:uuid:bdcc5532-5b1a-4d47-8e8a-f12676bc8e25>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00197.warc.gz"}
Which of the following will not produce a 3x3 array of zeros in MAT... Which of the following will not produce a 3x3 array of zeros in MATLAB? eye(3) - diag(ones(1,3)) cos(repmat(pi/2, [3,3])) mtimes([1;1;0], [0,0,0]) 21 Comments I say it's A(3,3) = 0 (after the poll closed). Thank you ... thank you very much. @goc3: thanks for the fun polls :) The MTIMES example could have been even trickier using empty matrices e.g.: There is an argument why each one of these will prodice a 3x3 array of zeros. The most popular answer, A(3,3)=0, is a good choice, because it relies on the condition that A does not exist. For my money, the best answer is cos(repmat(pi/2, [3,3])) - this is because the numerical evaluation of cos(pi/2) is not identically zero, but produces a resut containing 'digital noise'. In my case, I get a 3x3 matrix where each element contains 6.1232e-17. Let's go through each option: eye(3) - diag(ones(1,3)): This will produce a 3x3 array of zeros. eye(3) creates a 3x3 identity matrix, and diag(ones(1,3)) creates a diagonal matrix with ones on the diagonal and zeros elsewhere. Subtracting these will result in a 3x3 array of zeros. 0 ./ ones(3): This will produce a 3x3 array of zeros. It divides zero element-wise by a 3x3 matrix of ones, resulting in all elements being zero. cos(repmat(pi/2, [3,3])): This will not produce a 3x3 array of zeros. repmat(pi/2, [3,3]) creates a 3x3 matrix filled with π/2, and then cos() takes the cosine of each element. Since cos(π/2) is 0, this will indeed produce a 3x3 array of zeros. zeros(3): This will produce a 3x3 array of zeros. zeros(3) creates a 3x3 matrix filled with zeros. A(3, 3) = 0: This will not produce a 3x3 array of zeros. This line of code assigns a zero to the element at the 3rd row and 3rd column of matrix A. However, matrix A is not specified in the given options, so this line will likely throw an error unless A has already been defined as a matrix. mtimes([1;1;0], [0,0,0]): This will not produce a 3x3 array of zeros. This is a matrix multiplication operation between a column vector [1;1;0] and a row vector [0,0,0]. The result will be a scalar, not a 3x3 array of zeros. So, the option that will not produce a 3x3 array of zeros is option 5: A(3, 3) = 0.eye(3) - diag(ones(1,3)): This will produce a 3x3 array of zeros. eye(3) creates a 3x3 identity matrix, and diag (ones(1,3)) creates a diagonal matrix with ones on the diagonal and zeros elsewhere. Subtracting these will result in a 3x3 array of zeros.0 ./ ones(3): This will produce a 3x3 array of zeros. It divides zero element-wise by a 3x3 matrix of ones, resulting in all elements being zero.cos(repmat(pi/2, [3,3])): This will not produce a 3x3 array of zeros. repmat(pi/2, [3,3]) creates a 3x3 matrix filled with π/2, and then cos() takes the cosine of each element. Since cos(π/2) is 0, this will indeed produce a 3x3 array of zeros.zeros(3): This will produce a 3x3 array of zeros. zeros(3) creates a 3x3 matrix filled with zeros.A(3, 3) = 0: This will not produce a 3x3 array of zeros. This line of code assigns a zero to the element at the 3rd row and 3rd column of matrix A. However, matrix A is not specified in the given options, so this line will likely throw an error unless A has already been defined as a matrix.mtimes([1;1;0], [0,0,0]): This will not produce a 3x3 array of zeros. This is a matrix multiplication operation between a column vector [1;1;0] and a row vector [0,0,0]. The result will be a scalar, not a 3x3 array of zeros.So, the option that will not produce a 3x3 array of zeros is option 5: A(3, 3) = 0. "Let's go through each option" An authoritative sounding comment ... lets check some of those statements: • "Since cos(π/2) is 0..." Note that π is not a valid variable or function name, so this statement is perhaps mathematically true but irrelevant to MATLAB. If we assume that the commenter meant the valid MATLAB code cos(pi/2) then the statement is easily tested: this numeric operation actually results in a value that is very close to zero (but not exactly zero, as others have already commented). Those with some basic understanding of binary floating point numbers will appreciate why this might occur. • "A(3, 3) = 0: This will not produce a 3x3 array of zeros." is borderline, because it does rather depend on whether A already exists in that workspace (and hence what class/size it has).... so (as others have already commented) we will assume that A does not exist in the workspace: the OP did not test it in MATLAB before commenting, but you can. • "However, matrix A is not specified in the given options, so this line will likely throw an error unless A has already been defined as a matrix." MATLAB syntax errors are not probabilistic: either a syntax is invalid or it is not. I strongly recommend to all readers that they try this themselves in MATLAB and see what happens. Tip: in general MATLAB does not require variables to exist before allocating to them. • "mtimes([1;1;0], [0,0,0]): This will not produce a 3x3 array of zeros. This is a matrix multiplication operation between a column vector [1;1;0] and a row vector [0,0,0]. The result will be a scalar, not a 3x3 array of zeros." Lets quickly revise some basic mathematics: what happens when we multiply an AxB matrix with a BxC matrix? The inside dimensions "cancel", giving a matrix that has size AxC. Now apply that to this example with 3x1 and 1x3 matrices... what is the predicted output size? Then test it in MATLAB. If A already exists as a variable, will indeed only assign zero to the 3rd row, 3rd column. However, this question assumes that A does not exist. In that case, it will not throw an error. You should try it and see what happens. It is also true that cos(π/2) is 0. However, see what happens when you run the following code in MATLAB: A small value may be practically zero for many applications. However, it is not technically equal to zero. Soory, I read the question incorrectly. Other nice alternative to the possible questions above could be starting from a nilpotent matrix such as Of course isequal(([5 -3 2].*[1; 3; 2])^2,zeros(3)) ans = I think it could be a trick question, and that the real answer is zeros(3). It is pretty apparent from the timings below that zeros() doesn't necessarily create anything, or at least not right away. >> timeit(@()zeros(1e4)) ans = >> timeit(@()ones(1e4)) ans = If zeros() was actually creating anything, why does it take 4 orders of magnitude longer to create a matrix of ones than a matrix of zeros? OK, dumb question: where can I see the correct answer? Open up a fresh instance of MATLAB, run each of the answers, and see which one fails to produce a 3x3 array of zeros. I was just floating past this thread and loved the overflow of inspiration accumulated in these comments. Thanks everyone for the signs of positivity and bits of humor, that really rounded my day up! It is these least significant moments that really matter most :) And that's why cospi() exists. Question: What is the point of this poll? Answer: Floating Why A(3,3) isn't correct? There is an implicit assumption there that A does not already exist in the workspace. If it doesn't, then will indeed create a 3x3 array of zeros. Uhh, implicit expansion made me guess wrong😅 Interestingly, matrix multiplication and array (i.e. element-wise) multiplcation give the same result here. mtimes([1;1;0], [0,0,0]) ans = times([1;1;0], [0,0,0]) ans =	{"url":"https://se.mathworks.com/matlabcentral/discussions/highlights/846973?s_tid=prof_contrib_poll","timestamp":"2024-11-05T02:27:26Z","content_type":"text/html","content_length":"485275","record_id":"<urn:uuid:c72d0399-030f-4d25-9dd7-ce2a225be2d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00608.warc.gz"}
Automata Theory Online Test - Sanfoundry Automata Theory Questions and Answers – Closure Properties under Boolean Operations This set of Automata Theory online test focuses on “Closure Properties under Boolean Operations”. 1. If L1, L2 are regular and op(L1, L2) is also regular, then L1 and L2 are said to be ____________ under an operation op. a) open b) closed c) decidable d) none of the mentioned View Answer Answer: b Explanation: If two regular languages are closed under an operation op, then the resultant of the languages over an operation op will also be regular. 2. Suppose a regular language L is closed under the operation halving, then the result would be: a) 1/4 L will be regular b) 1/2 L will be regular c) 1/8 L will be regular d) All of the mentioned View Answer Answer: d Explanation: At first stage 1/2 L will be regular and subsequently, all the options will be regular. 3. If L1′ and L2′ are regular languages, then L1.L2 will be a) regular b) non regular c) may be regular d) none of the mentioned View Answer Answer: a Explanation: Regular language is closed under complement operation. Thus, if L1′ and L2′ are regular so are L1 and L2. And if L1 and L2 are regular so is L1.L2. 4. If L1 and L2′ are regular languages, L1 ∩ (L2′ U L1′)’ will be a) regular b) non regular c) may be regular d) none of the mentioned View Answer Answer: a Explanation: If L1 is regular, so is L1′ and if L1′ and L2′ are regular so is L1′ U L2′. Further, regular languages are also closed under intersection operation. 5. If A and B are regular languages, !(A’ U B’) is: a) regular b) non regular c) may be regular d) none of the mentioned View Answer Answer: a Explanation: If A and B are regular languages, then A Ç B is a regular language and A ∩ B is equivalent to !(A’ U B’). 6. Which among the following are the boolean operations that under which regular languages are closed? a) Union b) Intersection c) Complement d) All of the mentioned View Answer Answer: d Explanation: Regular languages are closed under the following operations: a) Regular expression operations b) Boolean operations c) Homomorphism d) Inverse Homomorphism 7. Suppose a language L1 has 2 states and L2 has 2 states. After using the cross product construction method, we have a machine M that accepts L1 ∩ L2. The total number of states in M: a) 6 b) 4 c) 2 d) 8 View Answer Answer: b Explanation: M is defined as: (Q, S, d, q0, F) where Q=Q1Q2 and F=F1F2. 8. If L is a regular language, then (L’)’ U L will be : a) L b) L’ c) f d) none of the mentioned View Answer Answer: a Explanation: (L’)’ is equivalent to L and L U L is subsequently equivalent to L. 9. If L is a regular language, then (((L’)r)’)* is: a) regular b) non regular c) may be regular d) none of the mentioned View Answer Answer: a Explanation: If L is regular so is its complement, if L’ is regular so is its reverse, if (L’)^r is regular so is its Kleene. 10. Which among the following is the closure property of a regular language? a) Emptiness b) Universality c) Membership d) None of the mentioned View Answer Answer: d Explanation: All the following mentioned are decidability properties of a regular language. The closure properties of a regular language include union, concatenation, intersection, Kleene, complement , reverse and many more operations. Sanfoundry Global Education & Learning Series – Automata Theory. To practice all areas of Automata Theory for online tests, here is complete set of 1000+ Multiple Choice Questions and Answers.	{"url":"https://www.sanfoundry.com/automata-theory-questions-answers-online-test/","timestamp":"2024-11-07T21:31:52Z","content_type":"text/html","content_length":"142023","record_id":"<urn:uuid:3542ec1b-8675-4951-9031-a350a0048ef0>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00810.warc.gz"}
Logistic Regression Applications in Healthcare and Construction Logistic regression is used to obtain the odds ratio when it exists more than one explanatory variable with a binomial response variable. The aim is to analyze the impact of each variable of the observed event of interest. How Does Logistic Regression Work? A logistic regression model will give the chance (The Ratio) of an outcome based on individual characteristics. The ratio is the algorithm of the chance given by: log(π/(1-π))=β_0+β_1 x_1 + β_2 x_2+ ...+β_m x_m π indicates the probability of an event βi are the regression coefficients associated with the reference group. xi explanatory variables β0 represents the reference group, which are those individuals presenting the reference level of each and every variable x1…m. Let’s walk through an example from a fictional study where the effects of two drug treatments to Staphylococcus Aureus (SA) endocarditis were compared (table 1). Table 1 Results from a fictional endocarditis treatment study by McHugh Standard Treatment New Treatment Totals Died 152 17 169 Survived 248 103 351 Totals 400 120 520 The odd ratio (OR) of death of patients using standard treatments: (152x103)/(248x47) = 3.71, which means that patients with standard treatment present a chance to die 3.71 times greater than patients with new treatment. More complex problems may arise when we are interested in the relationship between two or more explanatory variables and one response variable (table 2). Table 2 Results from a fictional endocarditis treatment study by McHugh looking at age Younger (30-45 yrs.) Older (46-60 yrs.) Totals Died 120 49 169 Survived 217 134 351 Totals 337 183 520 OR (120x134)/(217x49) =1.51 meaning that the chance of a younger individual between 30x45 years old death is about 105 times the chance of the death of an older individual between 46 and 60 years Now we have two variables related to the event of interest (death) in individuals with SA endocarditis. Table 3 represents the events of the effect of treatment on endocarditis by age. Table 3 Effect of treatment on endocarditis stratified by age. Standard Treatment New Treatment Totals OR Died 43 6 49 Older (46-60 yrs.) Survived 100 34 134 2.44 Totals 143 40 183 Standard Treatment New Treatment Totals OR Died 109 11 120 Younger (30–45 yrs.) Survived 148 69 217 4.62 Totals 257 80 337 Table 3 shows that the impact of treatment is higher on younger individuals because the OR in the younger patients is higher than in the older patients’ subgroups. The problem here is that it would be incorrect to look at the treatment results without considering the patient's age. To solve this problem, we should calculate the “weighted” OR by using for example Mantel-Haenszel OR equation, where n is the sample size of age class I, and a, b, c, and d are the table cells, as presented by McHugh. Mantel-Haenszel OR = ∑ ([i]a[i]d[i] / n[i]) / ∑ ([i]b[i]c[i] / n[i]) The results of the weighted chance of death associated with standard treatment is 3.74 times the chance of death of individuals taking new treatments. When the number of variables increases the calculations become more complicated, and when using continuous variables like age it is necessary to set a breaking point to categorize (in this example it was set up arbitrarily at 45 years old). A better approach would be to use logistic regression. Let’s apply logistic regression to this example, which is a “saturated model” because it includes all variables. Table 4 Results from multivariate logistic regression model containing all explanatory variables (full model). Term β Estimate Standard Error P Value Intercept (β [0]) -2.121 0.303 <0.001 Age: Younger (β [1]) 0.454 0.207 0.028 Treatment: Standard (β [2]) 1.333 0.283 <0.001 β[0]: the intercept, exp(β[0]) = exp(-2.121) = 0.12 is the chance of death among those individuals that are older and received new treatment. β[1]: individuals that are younger, exp(β[1]) = exp(0.454) = 1.58. Older individuals that receive standard treatment have a mean chance to die of 1.58 times of reference individuals. β[2]: individuals that are older, exp(β[2]) = exp(1.333) = 3.79. Older individuals that receive standard treatment have a mean chance to die of 3.79 times of reference individuals. If individuals are younger and received standard treatment, then we calculate exp(β[1] + β[2]) = exp(1.787) = 5.97 times the mean chance of reference individuals. This is a basic interpretation of a logistic regression model, but some issues can happen during the analysis and the results might not be readily available. It is important to pay attention when constructing the model, by avoiding feeding the software with raw data without making decisions. Logistic Regression can be used in business to solve binary classification problems, like predicting customer churn, fraud detection, medical diagnosis, credit risk assessment, market segmentation, and employee retention. Logistic Regression Application in the Construction Industry Quality Control in Construction: Scenario: A construction company wants to ensure the quality of its products (e.g., concrete blocks, steel beams, or prefabricated components). Application: Logistic regression can predict whether a product meets quality standards based on features like material composition, dimensions, and manufacturing process parameters. By analyzing historical data, the model can identify factors that contribute to defects or non-compliance. This information helps improve production processes and reduce waste. Safety Compliance Prediction: Scenario: A construction site aims to prevent accidents and ensure compliance with safety regulations. Application: Logistic regression can analyze safety-related variables (e.g., worker experience, equipment usage, weather conditions) to predict the likelihood of safety violations or accidents. By identifying high-risk situations, safety protocols can be reinforced, and preventive measures can be implemented. Project Delay Prediction: Scenario: Construction projects often face delays due to unforeseen circumstances. Application: Logistic regression can assess project-related factors (e.g., weather, resource availability, subcontractor performance) to predict the likelihood of delays. By understanding critical risk factors, project managers can allocate resources effectively and mitigate potential delays. Bid Acceptance Probability: Scenario: Construction firms submit bids for projects, and winning bids are crucial for business growth. Application: Logistic regression can analyze bid-related features (e.g., bid amount, project complexity, competitor bids) to estimate the probability of winning a contract. By optimizing bidding strategies, companies can increase their chances of securing profitable projects. Equipment Maintenance Prediction: Scenario: Construction equipment (e.g., cranes, and excavators) requires regular maintenance to prevent breakdowns. Application: Logistic regression can predict the likelihood of equipment failure based on usage patterns, maintenance history, and environmental conditions. By scheduling preventive maintenance when the risk is high, companies can minimize downtime and repair costs. I have a small favor to ask, if you find this information useful, I ask that you share this blog with other business owners that might find this content useful as well. I will be setting a lot of effort towards posting regular content to help share knowledge about all things related to business and how data analytics can be used to improve companies. Thank you!	{"url":"https://winklerconsultingsolutions.com/blog/Machine-Learning/logistic-regression-application.html","timestamp":"2024-11-14T08:36:52Z","content_type":"text/html","content_length":"22709","record_id":"<urn:uuid:52abedbb-5be6-4423-b9e2-f1d9eb7f6074>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00556.warc.gz"}
The org.orekit.forces package provides the interface for the force models to be used by a NumericalPropagator. Forces presentation Objects implementing the force model interface are intended to be added to a numerical propagator before the propagation is started. The propagator will call at each step the force model contribution computation method, to be added to its time derivative equations. The force model instance will extract all the state data it needs (date, position, velocity, frame, attitude, mass) from the SpacecraftState parameter. From these state data, it will compute the perturbing acceleration. It will then add this acceleration to the second parameter which will take this contribution into account and will use the Gauss equations to evaluate its impact on the global state derivative. Force models that create discontinuous acceleration patterns (typically for maneuvers start/stop or solar eclipses entry/exit) must provide one or more events detectors to the propagator thanks to their getEventsDetectors() method. This method is called once just before propagation starts. The events states will be checked by the propagator to ensure accurate propagation and proper events Available force models The force models implemented are as follows: • atmospheric drag forces, taking attitude into account if spacecraft shape is defined, • central gravity forces, including time-dependent parts (linear trends and pulsation at several different periods). Our implementation is based on S. A. Holmes and W. E. Featherstone (Department of Spatial Sciences, Curtin University of Technology, Perth, Australia) 2002 paper: A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions (Journal of Geodesy (2002) 76: 279–299). • third body gravity force. Data for all solar system bodies is available, based on JPL DE ephemerides or IMCCE INPOP ephemerides, • solar radiation pressure force, taking into account force reduction in penumbra and no force at all during complete eclipse, and taking attitude into account if spacecraft shape is defined ; several occulting bodies can be defined as oblate spheroids • Earth Albedo and IR emission force model. Our implementation is based on paper: EARTH RADIATION PRESSURE EFFECTS ON SATELLITES", 1988, by P. C. Knocke, J. C. Ries, and B. D. Tapley. • solid tides, with or without solid pole tide, • ocean tides, with or without ocean pole tide, • post-Newtonian correction due to general relativity with De Sitter and Lense-Thirring terms, • forces induced by maneuvers. At present, only constant thrust maneuvers are implemented, with the possibility to define an impulse maneuver, thanks to the event detector mechanism. • parametric accelerations, to model lesser-known forces, estimating a few defining parameters from a parametric function using orbit determination. Typical parametric functions are polynomial (often limited to a constant term) and harmonic (often with either orbital period or half orbital period). An important operational example is the infamous GPS Y-bias. Spacecraft shapes Surface forces like atmospheric drag or radiation pressure can use either a simple spherical shape using the various Isotropic classes or a more accurate BoxAndSolarArraySpacraft shape. The spherical shape will be independent of attitude. The box and solar array will consider the contribution of all box panels facing the flux as computed from the current attitude, and also the contribution of a pivoting solar array, whose orientation is a combination of the spacecraft body attitude and either the true Sun direction or a regularized rotation angle. The box can have any number of panels, and they can have any orientation as long as the body remains convex. The coefficients (drag, lift, absorption, reflection) are panel-dependent. As of 12.0, the box and solar array does not compute yet shadowing effects. All these shapes define various ParameterDrivers that can be used to control dynamic parameters like drag coefficient or absorption coefficient. Several conventions are available. For estimation purposes, it is possible to use a global multiplication factor that is applied to the acceleration rather than attempting to estimate several coefficients at once like absorption and specular reflection for solar radiation pressure. For BoxAndSolarArraySpacraft shape, as each panel has its own set of coefficients and this would not be observable, the coefficients are fixed and only the global multiplication factor is available and can be estimated. For Isotropic shapes, it is possible to estimate either the coefficients or the global multiplication factor. Of course in order to avoid ill-conditioned systems, users should not attempt to estimate both a coefficient and a global multiplication factor at the same time in Isotropic cases; they should select one parameter to estimate and let the other one fixed.	{"url":"https://www.orekit.org/site-orekit-development/architecture/forces.html","timestamp":"2024-11-08T04:46:25Z","content_type":"application/xhtml+xml","content_length":"12242","record_id":"<urn:uuid:bf2d5642-53e9-4720-b903-b483b57d13a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00394.warc.gz"}
Interesting Facts about 0 - Fact Bud “Zero” is a number that represents the absence or lack of quantity or value. In the context of mathematics, it serves as a fundamental numerical symbol and concept. Zero is neither positive nor negative, and it is the basis for place-value notation, allowing us to represent numbers efficiently. Additionally, zero has various applications in science, technology, and everyday life, as discussed in the previous responses. If you have a specific question about zero or would like to know more about a particular aspect of it, please feel free to ask. Here are some interesting facts about 0: The concept of zero as a placeholder in numerical notation originated in ancient India around the 5th century CE. This revolutionary idea allowed for more efficient and precise representation of numbers, as it separated the tens, hundreds, thousands, and so on. Prior to this, various ancient civilizations struggled with cumbersome numerical systems that lacked a placeholder for zero. The Indian mathematician Brahmagupta is often credited with formalizing the rules of arithmetic involving zero, including defining zero divided by any number as zero. The earliest known use of a symbol for zero is found in a 9th-century CE Indian manuscript known as the Bakhshali Manuscript. This ancient text contains mathematical calculations that utilize a dot as a placeholder for zero. This symbol evolved over time, eventually becoming the familiar Arabic numeral “0” that is now universally recognized. The word “zero” itself is derived from the Arabic word “sifr,” which means empty or nothing. The Arabic numeral system, including the use of zero, was introduced to Europe through translations of Arabic mathematical texts during the Middle Ages. The word “zero” was adapted from “sifr” and gradually integrated into European languages, becoming a fundamental part of our numerical vocabulary. In Roman numerals, there is no representation for zero, which made complex calculations more challenging in ancient Rome. Roman numerals are additive in nature, and the absence of zero meant that there was no simple way to express the concept of nothingness within their numerical system. This limitation hindered advanced mathematical calculations and bookkeeping, which were essential for trade and commerce. The introduction of zero into European mathematics is often attributed to the Italian mathematician Fibonacci in his book “Liber Abaci” in the 13th century. Fibonacci, also known as Leonardo of Pisa, played a crucial role in popularizing the Hindu-Arabic numeral system, which included zero, in Europe. His book explained the advantages of this system for arithmetic and introduced Europeans to the concept of zero as a numerical placeholder. Zero is considered neither a prime nor a composite number. Prime numbers are natural numbers greater than 1 that have only two distinct positive divisors: 1 and themselves. Composite numbers have more than two positive divisors. Zero, however, has no positive divisors because it cannot be divided by any natural number without yielding zero itself. Therefore, it stands apart from the categories of prime and composite numbers. The number zero is essential in place-value notation, which allows us to represent large numbers efficiently. Place-value notation means that a digit’s position in a number determines its value. Without zero as a placeholder, representing numbers like 102 or 2030 would be impossible, making arithmetic and mathematics much more cumbersome. In binary code, which is the basis for modern computer systems, 0 represents the absence of an electrical signal. In binary code, a bit is the smallest unit of data and can be either 0 or 1. The presence of a 0 represents a “low” or “off” state in electrical circuits, while a 1 represents a “high” or “on” state. The entire foundation of digital computing relies on this binary system, with 0 serving as a fundamental element to represent the absence of data or a deactivated state. Zero is the only real number that is neither positive nor negative. In the real number system, numbers can be categorized as positive, negative, or zero. While positive numbers are greater than zero, and negative numbers are less than zero, zero itself stands as a neutral point in this numerical spectrum. It has neither a positive nor a negative sign associated with it. In temperature scales, such as Celsius and Fahrenheit, zero represents the point at which water freezes. In the Celsius scale, 0°C is the freezing point of water, and in the Fahrenheit scale, 32°F marks the same temperature. These reference points are significant in everyday life, as they help us understand and measure temperature variations. Zero degrees Celsius, in particular, serves as a fundamental reference for weather forecasting and scientific experiments involving temperature. Absolute zero, the lowest possible temperature, is defined as 0 Kelvin (0 K), which is equivalent to -273.15 degrees Celsius (-459.67 degrees Fahrenheit). Absolute zero represents the point at which the kinetic energy of particles in a substance is at its minimum possible value. At this temperature, all molecular motion theoretically ceases, and it serves as a fundamental reference point in thermodynamics and low-temperature physics. Achieving temperatures close to absolute zero has led to remarkable discoveries in fields like quantum mechanics and superconductivity. The temperature of outer space, in the vacuum between stars and galaxies, is very close to absolute zero. In the vacuum of space, there is no atmosphere to conduct heat, which means that objects in space can quickly cool down to extremely low temperatures. While outer space isn’t precisely at absolute zero, temperatures can drop to just a few degrees above it, making it one of the coldest places in the universe. In physics, zero is often used as a reference point or baseline for measurements. Zero is a critical reference point in physics, serving as the starting point for many measurements and calculations. For instance, when measuring distances, physicists often choose a specific location or point as the origin (0) and then measure distances relative to that point. This helps create a consistent and standardized frame of reference for scientific experiments. The number zero is significant in calculus, where it represents the point of origin for a graph. In calculus, the concept of zero is used to define the x-axis and y-axis intersections, known as the origin (0,0). This point is fundamental for understanding functions, derivatives, and integrals. It provides a baseline for measuring changes and rates of change, which are essential in calculus for solving real-world problems in mathematics and science. In some cultures, the number zero has symbolic or spiritual significance, representing the void or the infinite. Zero can hold profound philosophical meanings, symbolizing both emptiness and infinity simultaneously. In Eastern philosophies, it can represent the void or the state of emptiness from which all things arise. This concept has influenced various aspects of art, religion, and meditation. The concept of zero is crucial in algebra, where it serves as the additive identity, meaning that any number added to zero remains unchanged. In algebra, zero is a fundamental element because it maintains the identity of other numbers in mathematical operations. Adding zero to any number or subtracting it from a number results in that number itself. This property is essential for algebraic manipulations and solving equations. In computer programming, zero is often used to represent the first element in an array or list, as many programming languages use zero-based indexing. Zero-based indexing is a convention in computer science where the first element in an array or list is accessed with an index of 0. This practice simplifies memory management and indexing operations in programming languages like C, C++, and Python. It has become a standard in many programming paradigms. Zero is the only integer that is neither prime nor composite. In number theory, integers are classified as prime if they have exactly two distinct positive divisors (1 and themselves) and composite if they have more than two divisors. Zero doesn’t fit into either category because it has no positive divisors, making it unique in the world of integers. In geometry, the point where the x and y axes intersect in a Cartesian coordinate system is called the origin, and its coordinates are (0, 0). The Cartesian coordinate system revolutionized geometry and graphing by providing a standardized way to locate points in two-dimensional space. The origin, denoted as (0, 0), is where the x and y axes intersect, serving as a central reference point for plotting and analyzing geometric shapes and equations. The binary number system, which consists of only 0s and 1s, is the foundation of digital computing. The binary system simplifies electronic data processing by representing information as sequences of 0s and 1s. In this system, each digit, or “bit,” has two possible states, allowing computers to store and manipulate data efficiently using electrical on/off signals. This fundamental concept underpins modern digital technology, including computers, smartphones, and the internet. The binary system’s simplicity and reliability are essential for the rapid processing and transmission of vast amounts of information in the digital age. Zero is used in probability theory to represent events with no likelihood of occurring, often denoted as P(0). In probability theory, the probability of an event happening can range from 0 (impossible) to 1 (certain). When an event has no chance of occurring, its probability is denoted as P(0). Zero probability events are fundamental in understanding probability distributions and statistical analysis, where they help model scenarios ranging from the highly unlikely to the impossible. In chess notation, the square h1 is often referred to as “h1” or “h1 (mate)” to indicate a checkmate with a move of the rook to that square. Chess notation is a standardized system for recording chess moves and games. “h1” typically refers to the square on the chessboard, while “h1 (mate)” signifies that a checkmate has been achieved by moving a rook to square h1, often an elegant and decisive move in chess strategy. In sports, a score of zero in tennis is called “love,” and the origin of this term is uncertain but is thought to be related to the French word “l’oeuf,” meaning egg. Tennis scoring uses a unique system where the word “love” is used to denote a score of zero. The exact origin of this term is debated, but one theory suggests it may have evolved from the French word “l’oeuf,” which means egg, due to the egg’s resemblance to the number zero. In some number systems, such as the complex numbers, zero has a real part of 0 and an imaginary part of 0. Complex numbers are numbers that consist of a real part and an imaginary part, often written as a + bi, where “a” is the real part, “b” is the imaginary part, and “i” is the imaginary unit (equal to the square root of -1). When both the real and imaginary parts are 0, the complex number simplifies to 0 + 0i, which is just 0. Complex numbers, including zero, are essential in mathematics and engineering for solving equations that involve both real and imaginary components.	{"url":"https://factbud.com/interesting-facts-about-0/","timestamp":"2024-11-08T17:45:05Z","content_type":"text/html","content_length":"109239","record_id":"<urn:uuid:4ac4a691-54ba-412b-abeb-3cdb3898c07a>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00673.warc.gz"}
Reduced-form Flattening \| RDP 2008-05: Understanding the Flattening Phillips Curve RDP 2008-05: Understanding the Flattening Phillips Curve 2. Reduced-form Flattening Simple scatterplots of inflation and the output gap are striking (Figure 1). We divide the sample for both countries, with the period after the break displaying a sizeable drop in the volatility of the output gap in each country.^[1] The moderation of the business cycle has been widely studied – see, for example, the papers in Kent and Norman (2005). The accompanying decline in inflation, however, has not been proportional – the reduced-form Phillips curve has flattened. Reduced-form estimates of the Phillips curve, like those in Roberts (2006), typically have the specification: where: π[t] is quarterly inflation; y[t] is an estimate of the output gap; L is the lag operator; and z[t] represents some exogenous factors affecting inflation.^[2] The lags of inflation (we use two) are sometimes interpreted as a proxy for inflation expectations, and more generally to capture the observed persistence in inflation.^[3] To examine the flattening of the Phillips curve we want to allow c, the coefficient on the output gap, to vary over time. Two simple ways of doing this are to estimate Equation (1) over a 15-year rolling window (Figure 2), or to specify the process so that the output gap coefficient follows (we assume a random walk) and to use the Kalman filter to estimate it over time (Figure 3). The latter has the advantage that it delivers two-sided estimates, that is, at all points of time they use information from the entire sample. The flattening of the reduced-form Phillips curve is clearly evident for the United States using either methodology. In Figure 2 we date the parameter estimates at the end of each rolling window, and consequently the sharp reduction in the output gap's coefficient evident from around 1989 occurred in the preceding 15 years, and perhaps is better dated in the early 1980s, around the time when the Federal Reserve managed to reduce inflation. Alternatively, the two-sided estimates in Figure 3 suggest that the flattening of the Phillips curve began around 1975 and has been a very gradual process which continued over the 1980s and 1990s.^[4] The results for Australia are more mixed. The estimates of the coefficient on the output gap fluctuate considerably until the late 1990s, after which there is a clearly discernable downward trend. Once again, this suggests that the flattening began around the time of a change in monetary policy regime, namely the adoption of inflation targeting. The two-sided estimates, however, date the flattening as beginning far earlier, around 1975, akin to the findings for the United States. The break in 1984:Q1 for the United States follows Roberts (2006). The break in 1993:Q1 for Australia corresponds to the adoption of inflation targeting. The output gap is constructed using a quadratic trend – see Section 4 for further details on the data. [1] In the rolling regressions we exclude z[t]. Including the change in import prices moderates the extent of the flattening evident for the United States, but not for Australia. Adding further lags of inflation or changes in oil prices does not change our results qualitatively. [2] Often the coefficients on the lags of inflation are restricted to sum to 1 (and the constant restricted to be 0), in an attempt to ensure that the Phillips curve is vertical in the long run (this is the ‘accelerationist’ model of inflation). These restrictions imply that inflation is an integrated process, which is implausible when the central bank's reaction function satisfies the ‘Taylor Principle’, that is, they move the nominal interest rate more than one-for-one in response to expected inflation. They also ignore the cross-equation restrictions that would exist in a fully-specified model – a point first highlighted by Sargent (1971). However, whether the Federal Reserve tightened sufficiently to offset inflation in the 1970s is debatable – see Clarida, Galí and Gertler (2000) and Orphanides (2002). [3] Naturally, this partially reflects our assumption that the coefficient on the output gap follows a random walk. The start date for the time-varying parameter estimates is 1970:Q1. [4]	{"url":"https://www.rba.gov.au/publications/rdp/2008/2008-05/red-form-flat.html","timestamp":"2024-11-10T16:05:59Z","content_type":"application/xhtml+xml","content_length":"31023","record_id":"<urn:uuid:36caef31-9cdd-42b1-93b2-ba7dee49061e>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00885.warc.gz"}
Academic Bulletin College of Arts and Sciences Mathematics and Actuarial Science • MATH-A 100 Fundamentals of Algebra (4 cr.) P: Test Score MA 102 or MATH-M 015. Designed to provide algebraic skills needed for future mathematics courses. Integers, rational and real numbers, exponents, decimals, polynomials, equations, word problems, factoring, roots and radicals, quadratic equations, graphing, linear equations in more than one variable, and inequalities. Does not satisfy the College of Arts and Sciences distribution requirements nor general education mathematical reasoning requirement. (Fall, Spring, Summer) • MATH-K 200 Statistics for Teachers (3 cr.) P: Level MA103 on Placement Exam or at least a C in MATH-A 100. The course serves as an introduction to statistical tools and spreadsheets or statistical packages used in everyday teaching practice. The emphasis is on understanding real-life applications of graphs of data, measures of central tendency, variation, probability, normal distributions, confidence intervals, hypothesis testing, and sampling. (Spring) • MATH-K 300 Statistical Techniques (3 cr.) P: at least a C in MATH-M 117 or equivalent. MATH-M 118 An introduction to statistics. Nature of statistical data. Ordering and manipulation of data. Measures of central tendency and dispersion. Elementary probability. Concepts of statistical inference and decision, estimation, and hypothesis testing. Special topics discussed may include regression and correlation, analysis of variance, nonparametric methods. (Spring) • MATH-M 15 Arithmetic with Algebra (0 cr.) Integers, proportional reasoning, measurement systems, exponents, solving linear inequalities, polynomial operations, geometric concepts, rational numbers, ratios and percent, algebraic expressions, solving and writing linear equations, literal equations, graphs of linear equations, applications. Does not satisfy the College of Arts and Sciences distribution requirements nor general education mathematical reasoning requirement. (Fall, Spring) • MATH-M 100 Basic Mathematics (4 cr.) P: Level MA103 on Placement Exam, or at least a C in MATH-A 100. Topics in algebra, geometry, graphing, probability, statistics, and consumer mathematics. Emphasis on problem solving and constructing mathematical models. This course is designed for allied health students and liberal arts students who plan to take no additional mathematics courses. Does not count toward a major in mathematics. (Fall, Spring, Summer) • MATH-M 110 Excursions into Mathematics (3 cr.) P: Level MA103 on Placement Exam, or at least a C in MATH-A 100. A course designed to convey the flavor and spirit of mathematics, stressing reasoning and comprehension rather than technique. Not preparatory to other courses; explores the theory of games and related topics that may include the mathematics of politics and elections. This course does not count toward a major in mathematics. (Occasionally) • MATH-M 117 Intermediate Algebra (3 cr.) P: Level MA103 on Placement Exam or at least a C in MATH-A 100. Designed to introduce nonlinear models and their applications, advanced linear systems, and function foundations. Does not satisfy the College of Arts and Sciences distribution requirements nor general education mathematical reasoning requirement. (Fall, Spring, Summer) • MATH-M 118 Finite Mathematics (3 cr.) P: Level MA104 on Placement Exam, or at least a C in MATH-M 117. Set theory, linear systems, matrices, probability, linear programming, Markov chains. Applications to problems from business and the social sciences. (Fall, Spring, Summer) • MATH-M 119 Brief Survey of Calculus (3 cr.) P: Level MA104 on Placement Exam or at least a C in MATH M117. Introduction to calculus. Primarily for students in business and the social sciences. A student cannot receive credit for both MATH-M 119 and MATH-M 215. (Fall, Spring, Summer) • MATH-M 125 Precalculus Mathematics (3 cr.) P: Level MA104 on the Placement Exam or at least a C in MATH-M 117. Designed to prepare students for calculus (MATH-M 215). Algebraic operations, polynomial, rational exponential, and logarithmic functions and their graphs, conic sections, linear systems of equations. Does not satisfy the arts and sciences distributional requirements. (Fall, Spring, Summer) • MATH-M 126 Trigonometric Functions (2-3 cr.) P: Level MA104 on Placement Exam, or at least a C in MATH-M 117. In-depth study of trigonometric functions, definitions, unit circle, graphs, inverse functions, identities, trigonometric equations and applications. This course, together with MATH-M 125 is designed to prepare students for calculus (MATH-M 215). (Occasionally) • MATH-M 127 Pre-calculus with Trigonometry (5 cr.) P: Level MA104 on Placement Exam, or at least a C in MATH-M 117. This course is designed to prepare students for calculus (M 215). Subject matter includes polynomial, rational, root, exponential, logarithmic, and trigonometric functions and their applications. (Fall, Spring, Summer) • MATH-M 215 Analytic Geometry and Calculus I (5 cr.) P: Level MA105 on Placement Exam or MATH-M 125 and MATH-M 126 or MATH-M 127. Differential calculus of functions of one variable, with applications. Functions, graphs, limits, continuity, derivatives of trigonometric, exponential and logarithmic functions, tangent lines, optimization problems, curve sketching, L'Hopital's Rule, definite integral, the Fundamental Theorem of Calculus. A student cannot receive credit for both MATH-M 119 and MATH-M 215. (Fall, Spring, Summer) • MATH-M 216 Analytic Geometry and Calculus II (5 cr.) P: MATH-M 215. Integral calculus of functions of one variable. Antiderivatives, definite integrals, techniques of integration, areas, volumes, surface areas, arc length, parametric functions, polar coordinates, limits of sequences, convergence of infinite series, Taylor polynomials, power series, and applications. (Fall, Spring) • MATH-M 295 Readings and Research (1-3 cr.) Supervised problem solving. Admission only with permission of a member of the mathematics faculty, who will act as supervisor. (Occasionally) • MATH-M 301 Applied Linear Algebra (3 cr.) P: MATH-M 216 or consent of instructor. Emphasis on applications: systems of linear equations, vector spaces, linear transformations, matrices, simplex method in linear programming. Computer used for applications. Credit not given for both MATH-M 301 and MATH-M 303. (Odd years, Spring) • MATH-M 311 Calculus III (4 cr.) P: MATH-M 216. Elementary geometry of 2, 3, and n-space; functions of several variables; partial differentiation; minimum and maximum problems; multiple integration. (Fall) • MATH-M 312 Calculus IV (3 cr.) P: MATH-M 311. Differential calculus of vector-valued functions, transformation of coordinates, change of variables in multiple integrals. Vector integral calculus: line integrals, Green's theorem, surface integrals, Stokes' theorem. Applications. (Occasionally) • MATH-M 320 Theory of Interest (3 cr.) P: MATH-M 216. Measurement of interest: accumulation and discount, equations of value, annuities, perpetuities, amortization and sinking funds, yield rates, bonds and other securities, installment loans, depreciation, depletion, and capitalized cost. This course covers topics corresponding to the society of Actuaries' Exam FM.(Odd years, Fall) • MATH-M 325 Problem-solving Seminar in Actuarial Science (3 cr.) P: Consent of instructor. A problem- solving seminar to prepare students for the actuarial exams. May be repeated up to three times for credit. (Spring) • MATH-M 343 Introduction to Differential Equations with Applications I (3 cr.) P: MATH-M 216. Derivation of equations of mathematical physics, biology, etc. Ordinary differential equations and methods for their solution, especially series methods. Simple vector field theory. Theory of series, Fourier series, applications to partial differential equations. Integration theorems, Laplace and Fourier transforms, applications. (Even years, Spring) • MATH-M 360 Elements of Probability (3 cr.) P: MATH-M 216 and MATH-M 311, which may be taken concurrently. The study of probability models that involve one or more random variables. Topics include conditional probability and independence, gambler's ruin and other problems involving repeated Bernoulli trials, discrete and continuous probability distributions, moment generating functions, probability distributions for several random variables, some basic sampling distributions of mathematical statistics, and the central limit theorem. Course topics match portions of Exam P of the Society of Actuaries. (Even years, Fall) • MATH-M 366 Elements of Statistical Inference (3 cr.) P: MATH-M 360. An introduction to statistical estimation and hypothesis testing. Topics include the maximum likelihood method of estimation and the method of moments, the Rao-Cramer bound, large sample confidence intervals, type I and type II errors in hypothesis testing, likelihood ratio tests, goodness of fit tests, linear models, and the method of least squares. This course covers portions of Exam SRM of the Society of Actuaries. (Odd years, Spring) • MATH-M 391 Foundations of the Number Systems (3 cr.) P: MATH-M 216. Sets, functions and relations, groups, real and complex numbers. Bridges the gap between elementary and advanced courses. Recommended for students with insufficient background for 400-level courses, for M.A.T. candidates, and for students in education. (Even years, Spring). • MATH-M 403 Introduction to Modern Algebra I (3 cr.) P: MATH-M 301. Study of groups, rings, fields (usually including Galois theory), with applications to linear transformations. (Odd years, • MATH-M 405 Number Theory (3 cr.) P: MATH-M 216. Numbers and their representation, divisibility and factorization, primes and their distribution, number theoretic functions, congruences, primitive roots, diophantine equations, quadratic residues, sums of squares, number theory and analysis, algebraic numbers, irrational and transcendental numbers. (Odd years, Spring) • MATH-M 406 Topics in Mathematics (3 cr.) Selected topics in various areas of mathematics that are not covered by the standard courses. May be repeated for credit. (Occasionally) • MATH-M 413 Introduction to Analysis I (3 cr.) P: MATH-M 301, and MATH-M 311, or consent of instructor. Modern theory of real number system, limits, functions, sequences and series, Riemann-Stieltjes integral, and special topics. (Even years, Spring) • MATH-M 420 Metric Space Topology (3 cr.) P: MATH-M 301. Topology of Euclidean and metric spaces. Limits and continuity. Topological properties of metric spaces, including separation properties, connectedness, and compactness. Complete metric spaces. Elementary general topology. (Occasionally) • MATH-M 425 Graph (Network) Theory and Combinatorial Theory (3 cr.) P: MATH-M 301. Graph theory: basic concepts, connectivity, planarity, coloring theorems, matroid theory, network programming, and selected topics. Combinatorial theory: generating functions, incidence matrices, block designs, perfect difference sets, selection theorems, enumeration, and other selected topics. (even years, Fall) • MATH-M 436 Introduction to Geometries (3 cr.) P: MATH-M 391 or its equivalent. Non-Euclidean geometry, axiom systems. Plane projective geometry, Desarguesian planes, perspectivities coordinates in the real projective plane. The group of projective transformations and subgeometries corresponding to subgroups. Models for geometries. Circular transformations. (Occasionally) • MATH-M 451 The Mathematics of Finance (3 cr.) P: MATH-M 311 and MATH-M 366. R: Math-M 343. Course covers probability theory, Brownian motion, Ito's Lemma, stochastic differential equations, and dynamic hedging. These topics are applied to the Black-Scholes formula, the pricing of financial derivatives, and the term theory of interest rates. Course topics match portions of Exam IFM of the Society of Actuaries. (Odd years, Spring) • MATH-M 463 Introduction to Probability Theory (3 cr.) P: MATH-M 301, and MATH-M 311, or consent of instructor. Idealized random experiments, conditional probability, independence, compound experiments. Univariate distributions, countable additivity, discrete and continuous distributions, Lebesgue-Stieltjes integral (heuristic treatment), moments, multivariate distribution. Generating functions, limit theorems, normal distribution. (Occasionally) • MATH-M 469 Applied Statistical Techniques (3 cr.) P: MATH-M 366. Linear regression, multiple regression, applications to credibility theory, time series and ARIMA models, estimation, fitting, and forecasting. This course covers the Applied Statistics portion of the Society of Actuaries VEE requirements and portions of Exam SRM of the Society of Actuaries. (Odd years, Fall) • MATH-M 477 Mathematics of Operations Research (3 cr.) P: MATH-M 301, MATH-M 311, MATH-M 360. Introduction to the methods of operations research. Linear programming, dynamic programming, integer programming, network problems, queuing theory, scheduling, decision analysis, simulation. (Odd years, Fall) • MATH-M 483 Historical Development of Modern Mathematics (3 cr.) P: MATH-M 301, MATH-M 311, and at least 3 additional credit hours in mathematics at the 300 level or above. The development of modern mathematics from 1660 to 1870 will be presented. The emphasis is on the development of calculus and its ramifications and the gradual evolution of mathematical thought from mainly computational to mainly conceptual. (Occasionally) • MATH-M 485 Life Contingencies I (3 cr.) P: MATH-M 320 and MATH-M 360. Measurement of mortality, life annuities, life insurance, net annual premiums, net level premium reserves, the joint life and last- survivor statuses, and multiple-decrement tables. Course topics match portions of Exam LATM of the Society of Actuaries. (Even years, Spring) • MATH-M 486 Life Contingencies II (3 cr.) P: MATH-M 485. Population theory, the joint life status, last- survivor and general multilife statuses, contingent functions, compound contingent functions, reversionary annuities, multiple-decrement tables, tables with secondary decrements. This course covers portions of Society of Acutaries Exam MLC. (Occasionally) • MATH-M 493 Senior Thesis in Mathematics (3 cr.) P: At least one 400-level mathematics course. Student must write and present a paper, relating to 400-level mathematics study, on a topic agreed upon by the student and the department chair or advisor delegated by the chair. • MATH-T 101 Mathematics for Elementary Teachers I (3 cr.) P: Level MA103 on Placement Exam, or at least a C in MATH-A 100. Elements of set theory, counting numbers. Operations on counting numbers, integers, rational numbers, and real numbers. Open only to elementary education majors. Does not count toward arts and sciences distribution requirement. (Fall, Spring) • MATH-T 102 Mathematics for Elementary Teachers II (3 cr.) P: MATH-T 101. Sets, operations, and functions. Prime numbers and elementary number theory. Elementary combinatorics, probability, and statistics. Open only to elementary education majors. Does not count toward arts and sciences distribution requirement. (Spring, Summer ) • MATH-T 103 Mathematics for Elementary Teachers III (3 cr.) P: MATH-T 102. Descriptions and properties of basic geometric figures. Rigid motions. Axiomatics. Measurement, analytic geometry, and graphs of functions. Discussion of modern mathematics. Open only to elementary education majors. Does not count toward arts and sciences distribution requirement. (Fall, Summer) • MATH-T 336 Topics in Euclidean Geometry (3 cr.) P: MATH-M 391. Axiom systems for the plane; the parallel postulate and non-Euclidean geometry; classical theorems. Geometric transformation theory vectors and analytic geometry; convexity; theory of area and volume. (Even years, Fall) • MATH-T 490 Topics for Elementary Teachers (3 cr.) P: MATH-T 103. Development and study of a body of mathematics specifically designed for experienced elementary teachers. Examples may include probability, statistics, geometry, and algebra. Open only to graduate elementary teachers with permission of the instructor. Does not count toward arts and sciences distribution requirement. • MATH-T 493 Mathematics of Middle and High School, Advanced Perspective (3 cr.) P: Junior or senior standing in mathematics education or consent of instructor. Team-taught capstone course for mathematics education majors. Mathematics of grades 6-12 and methods of instruction. Topics explored from a college perspective. (Occasionally) • MATH-Y 398 Internship in Professional Practice (3 cr.) P: Approval of Department of Mathematics. Professional work experience involving significant use of mathematics or statistics. Evaluation of performance by employer and Department of Mathematics. Does not count toward requirements. May be repeated with approval of Department of Mathematics for a total of 6 credits. • MATH-M 111 Mathematics in the World (3 cr.) P: Level MA103 on Placement Exam, or at least a C in MATH-A 100. Conveys spirit of mathematical languages of quantity; students apply concepts from algebra, geometry, management science, probability, and statistics, and use scientific software to analyze real world situations. (Occasionally) PDF Version for the PDF version.	{"url":"https://bulletins.iu.edu/iun/2020-2022/schools/coas/departments/mathematics/courses.shtml","timestamp":"2024-11-11T00:42:13Z","content_type":"application/xhtml+xml","content_length":"31833","record_id":"<urn:uuid:9701076b-88fb-460f-b30d-c342e9b3278e>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00758.warc.gz"}
Elementary Formal Logic 8.1 A historical example In his book, The Two New Sciences,^[10] Galileo Galilea (1564-1642) gives several arguments meant to demonstrate that there can be no such thing as actual infinities or actual infinitesimals. One of his arguments can be reconstructed in the following way. Galileo proposes that we take as a premise that there is an actual infinity of natural numbers (the natural numbers are the positive whole numbers from 1 on): {1, 2, 3, 4, 5, 6, 7, ….} He also proposes that we take as a premise that there is an actual infinity of the squares of the natural numbers. {1, 4, 9, 16, 25, 36, 49, ….} Now, Galileo reasons, note that these two groups (today we would call them “sets”) have the same size. We can see this because we can see that there is a one-to-one correspondence between the two {1, 2, 3, 4, 5, 6, 7, ….} {1, 4, 9, 16, 25, 36, 49, …} If we can associate every natural number with one and only one square number, and if we can associate every square number with one and only one natural number, then these sets must be the same size. But wait a moment, Galileo says. There are obviously very many more natural numbers than there are square numbers. That is, every square number is in the list of natural numbers, but many of the natural numbers are not in the list of square numbers. The following numbers are all in the list of natural numbers but not in the list of square numbers. {2, 3, 5, 6, 7, 8, 10, ….} So, Galileo reasons, if there are many numbers in the group of natural numbers that are not in the group of the square numbers, and if there are no numbers in the group of the square numbers that are not in the naturals numbers, then the natural numbers is bigger than the square numbers. And if the group of the natural numbers is bigger than the group of the square numbers, then the natural numbers and the square numbers are not the same size. We have reached two conclusions: the set of the natural numbers and the set of the square numbers are the same size; and, the set of the natural numbers and the set of the square numbers are not the same size. That’s contradictory. Galileo argues that the reason we reached a contradiction is because we assumed that there are actual infinities. He concludes, therefore, that there are no actual infinities. 8.2 Indirect proofs Our logic is not yet strong enough to prove some valid arguments. Consider the following argument as an example. (P →(QvR)) This argument looks valid. By the first premise we know: if P were true, then so would (Q v R) be true. But then either Q or R or both would be true. And by the second and third premises we know: Q is false and R is false. So it cannot be that (Q v R) is true, and so it cannot be that P is true. We can check the argument using a truth table. Our table will be complex because one of our premise is complex. premise premise premise conclusion P Q R (QvR) (P→(QvR)) ~Q ~R ~P T T T T T F F F T T F T T F T F T F T T T T F F T F F F F T T F F T T T T F F T F T F T T F T T F F T T T T F T F F F F T T T T In any kind of situation in which all the premises are true, the conclusion is true. That is: the premises are all true only in the last row. For that row, the conclusion is also true. So, this is a valid argument. But take a minute and try to prove this argument. We begin with And now we are stopped. We cannot apply any of our rules. Here is a valid argument that we have not made our reasoning system strong enough to prove. There are several ways to rectify this problem and to make our reasoning system strong enough. One of the oldest solutions is to introduce a new proof method, traditionally called “reductio ad absurdum”, which means a reduction to absurdity. This method is also often called an “indirect proof” or “indirect derivation”. The idea is that we assume the denial of our conclusion, and then show that a contradiction results. A contradiction is shown when we prove some sentence Ψ, and its negation ~Ψ. This can be any sentence. The point is that, given the principle of bivalence, we must have proven something false. For if Ψ is true, then ~Ψ is false; and if ~Ψ is true, then Ψ is false. We don’t need to know which is false (Ψ or ~Ψ); it is enough to know that one of them must be. Remember that we have built our logical system so that it cannot produce a falsehood from true statements. The source of the falsehood that we produce in the indirect derivation must, therefore, be some falsehood that we added to our argument. And what we added to our argument is the denial of the conclusion. Thus, the conclusion must be true. The shape of the argument is like this: Traditionally, the assumption for indirect derivation has also been commonly called “the assumption for reductio”. As a concrete example, we can prove our perplexing case. We assumed the denial of our conclusion on line 4. The conclusion we believed was correct was ~P, and the denial of this is ~~P. In line 7, we proved R. Technically, we are done at that point, but we would like to be kind to anyone trying to understand our proof, so we repeat line 3 so that the sentences R and ~R are side by side, and it is very easy to see that something has gone wrong. That is, if we have proven both R and ~R, then we have proven something false. Our reasoning now goes like this. What went wrong? Line 8 is a correct use of repetition; line 7 comes from a correct use of modus tollendo ponens; line 6 from a correct use of modus ponens; line 5 from a correct use of double negation. So, we did not make a mistake in our reasoning. We used lines 1, 2, and 3, but those are premises that we agreed to assume are correct. This leaves line 4. That must be the source of my contradiction. It must be false. If line 4 is false, then ~P is true. Some people consider indirect proofs less strong than direct proofs. There are many, and complex, reasons for this. But, for our propositional logic, none of these reasons apply. This is because it is possible to prove that our propositional logic is consistent. This means, it is possible to prove that our propositional logic cannot prove a falsehood unless one first introduces a falsehood into the system. (It is generally not possible to prove that more powerful and advanced logical or mathematical systems are consistent, from inside those systems; for example, one cannot prove in arithmetic that arithmetic is consistent.) Given that we can be certain of the consistency of the propositional logic, we can be certain that in our propositional logic an indirect proof is a good form of reasoning. We know that if we prove a falsehood, we must have put a falsehood in; and if we are confident about all the other assumptions (that is, the premises) of our proof except for the assumption for indirect derivation, then we can be confident that this assumption for indirect derivation must be the source of the falsehood. A note about terminology is required here. The word “contradiction” gets used ambiguously in most logic discussions. It can mean a situation like we see above, where two sentences are asserted, and these sentences cannot both be true. Or it can mean a single sentence that cannot be true. An example of such a sentence is (P&~P). The truth table for this sentence is: Thus, this kind of sentence can never be true, regardless of the meaning of P. To avoid ambiguity, in this text, we will always call a single sentence that cannot be true a “contradictory sentence”. Thus, (P&~P) is a contradictory sentence. Situations where two sentences are asserted that cannot both be true will be called a “contradiction”. 8.3 Our example, and other examples We can reconstruct a version of Galileo’s argument now. We will use the following key. P: There are actual infinities (including the natural numbers and the square numbers). Q: There is a one-to-one correspondence between the natural numbers and the square numbers. R: The size of the set of the natural numbers and the size of the set of the square numbers are the same. S: All the square numbers are natural numbers. T: Some of the natural numbers are not square numbers. U: There are more natural numbers than square numbers. With this key, the argument will be translated: And we can prove this is a valid argument by using indirect derivation: On line 6, we assumed ~~P because Galileo believed that ~P and aimed to prove that ~P. That is, he believed that there are no actual infinities, and so assumed that it was false to believe that it is not the case that there are no actual infinities. This falsehood will lead to other falsehoods, exposing itself. For those who are interested: Galileo concluded that there are no actual infinities but there are potential infinities. Thus, he reasoned, it is not the case that all the natural numbers exist (in some sense of “exist”), but it is true that you could count natural numbers forever. Many philosophers before and after Galileo held this view; it is similar to a view held by Aristotle, who was an important logician and philosopher writing nearly two thousand years before Galileo. Note that in an argument like this, you could reason that not the assumption for indirect derivation, but rather one of the premises was the source of the contradiction. Today, most mathematicians believe this about Galileo’s argument. A logician and mathematician named Georg Cantor (1845-1918), the inventor of set theory, argued that infinite sets can have proper subsets of the same size. That is, Cantor denied premise 4 above: even though all the square numbers are natural numbers, and not all natural numbers are square numbers, it is not the case that these two sets are of different size. Cantor accepted however premise 2 above, and, therefore, believed that the size of the set of natural numbers and the size of the set of square numbers is the same. Today, using Cantor’s reasoning, mathematicians and logicians study infinity, and have developed a large body of knowledge about the nature of infinity. If this interests you, see section 17.5. Let us consider another example to illustrate indirect derivation. A very useful set of theorems are today called “De Morgan’s Theorems”, after the logician Augustus De Morgan (1806–1871). We cannot state these fully until chapter 9, but we can state their equivalent in English: DeMorgan observed that ~(PvQ) and (~P&~Q) are equivalent, and also that ~(P&Q) and (~Pv~Q) are equivalent. Given this, it should be a theorem of our language that (~(PvQ)→(~P&~Q)). Let’s prove this. The whole formula is a conditional, so we will use a conditional derivation. Our proof must thus begin: To complete the conditional derivation, we must prove (~P&~Q). This is a conjunction, and our rule for showing conjunctions is adjunction. Since using this rule might be our best way to show (~P&~Q), we can aim to show ~P and then show ~Q, and then perform adjunction. But, we obviously have very little to work with—just line 1, which is a negation. In such a case, it is typically wise to attempt an indirect proof. Start with an indirect proof of ~P. We now need to find a contradiction—any contradiction. But there is an obvious one already. Line 1 says that neither P nor Q is true. But line 3 says that P is true. We must make this contradiction explicit by finding a formula and its denial. We can do this using addition. To complete the proof, we will use this strategy again. We will prove De Morgan’s theorems as problems for chapter 9. Here is a general rule of thumb for doing proofs: When proving a conditional, always do conditional derivation; otherwise, try direct derivation; if that fails, then, try indirect derivation. 8.4 How to Make an Indirect Proof Indirect proof (indirect derivation, or reductio ad absurdum): allows us to prove claims we have no direct or conditional means of proving. The indirect proof assumes the negation of what we want to prove, and shows that a contradiction follows from that assumption, thereby showing that the assumption is false, and its opposite is true. Each discrete truth-preserving step is either a rule of inference or a rule of replacement such that whenever you have a sentence of one form you may replace it with a sentence of another form. Notation: each line of a proof must be enumerated and justified. 1. Enumerate each line of a proof. 2. Justify each sentence of a proof in a justification column to the right. a. For each non-derived sentence of a proof, write “premise” or “assumption.” b. For each derived sentence of a proof, write the name of the inference rule or replacement rule used, and the sentence(s) used to infer the derived sentence, in numerical order. 3. Draw a horizontal line, a “fitch bar,” between non-derived sentences (e.g. premises and assumptions) and derived sentences. 4. Draw a vertical “scope” line to the left of any non-derived sentences. Terms, Conventions and the Accessibility Rule: Derived Sentence: a sentence obtained by applying a inference rule or a replacement rule. Scope Line: a vertical scope line indicates the beginning and end of a proof or subproof. Fitch Bar: a horizontal fitch bar separates non-derived sentences, assumptions and premises, from derived sentences. Subproof: a proof within a proof. Subproofs are necessary for conditional proofs and indirect proofs. Open a subproof: a subproof begins with an assumption, which must be justified by the citing “assumption for …” and the rule one hopes to use. End a subproof: a subproof ends by using the rule indicated on the justification for the assumption, and citing the entire subproof. Discharged Assumption: the assumption of a successfully ended subproof. Open Assumption: assumption that has not been discharged. Closed Assumption: assumption that has been discharged. Accessibility Rule: sentence is accessible if it can be used to infer a new sentence at a given line in a proof. A sentence is accessible if it is an open assumption or falls within the scope of an open assumption. Example: Proof that the following arguments are valid: 1. Premises: (P→~P), P. Conclusion: Q. (note that while only one scope line is labelled, the other vertical line is also a scope line; any vertical line to the left of the derived sentences is a scope line) Notice that indirect proof is truth-preserving. There is no instance where P and ~P are true, and Q is false. (This is, of course, because P and ~P can never be true.) P Q ~ P T T F T T F F T F T T F F F T F 2. Premises: P, ~P. Conclusion: ~Q. (note that while only one scope line is labelled, the other vertical line is also a scope line; any vertical line to the left of the derived sentences is a scope Notice that indirect proof is truth-preserving. There is no instance where P and ~P are true, and ~Q is false. (This is, of course, because P and ~P can never be true.) P Q ~ P ~ Q T T F T F T T F F T T F F T T F F T F F T F T F 3. Premises: (P→Q), ~(P→Q). Conclusion: (R&S). (note that while only one scope line is labelled, the other vertical line is also a scope line; any vertical line to the left of the derived sentences is a scope line) Notice that indirect proof is truth-preserving. There is no instance where (P→Q) and ~(P→Q) are true, and (R&S) is false. (This is, of course, because (P→Q) and ~(P→Q) can never be true.) P Q R S (P → Q) ~ (P → Q) (R & S) T T T T T T T F T T T T T T T T T F T T T F T T T T F F T T F T T T T F T T T F F T T T F F T T T F T T T F F F T F T T T F F T T F F T T T T F T F T F F T T F F T F F T F F T T F F T T F F F F T T F F F T F F T T F F F F F F T T T F T T F F T T T T T F T T F F T T F F T T T F F F T F T F T T F F T T F F T F T F F F T T F F T T F F F F F T T F T F F F T F T T T F F T F F T F F F T F T F F F F F T F T F F F T F F F T F F F F F T F F F T F F F F 8.5 Key Concepts Indirect proof (or indirect derivation, and also known as a reductio ad absurdum): an ordered list of sentences in which every sentence is either 1) a premise, 2) the special assumption for indirect derivation (also sometimes called the “assumption for reductio”), or 3) derived from earlier lines using an inference rule. If our assumption for indirect derivation is ~Φ, and we derive as some step in the proof Ψ and also as some step of our proof ~Ψ, then we conclude that Φ. 8.6 Exercises Within this section, you will find two types of problems for the chapter material. Firstly there are interactive exercises that randomly test your knowledge. Secondly, there is a comprehensive list of exercise questions with all answers at the back of the text. A. Complete the following proofs. Each will require an indirect derivation. The last two are challenging. Coming soon! Until then refer to section A of the Full Exercise Question Sets below. Full Exercise Question Sets A. Complete the following proofs. Each will require an indirect derivation. The last two are challenging. 1. Premises: (P→R), (Q→R), (PvQ). Conclusion: R. 2. Premises: ((PvQ)→R), ~R. Conclusion: ~P. 3. Premise: (~P&~Q). Conclusion: ~(PvQ). 4. Premise: (P→R), (Q→S), ~(R&S). Conclusion: ~(P&Q). 5. Premise: ~R, ((P→R) v (Q→R)). Conclusion: (~Pv~Q). 6. Premise: ~(R v S), (P→R), (Q→S). Conclusion: ~(P v Q). [10] This translation of the title of Galileo’s book has become the most common, although a more literal one would have been Mathematical Discourses and Demonstrations. Translations of the book include Drake (1974). An ordered list of sentences in which every sentence is either 1) a premise, 2) the special assumption for indirect derivation (also sometimes called the “assumption for reductio”), or 3) derived from earlier lines using an inference rule	{"url":"https://intrologicimport.pressbooks.tru.ca/chapter/8-reductio-ad-absurdum-a-concise-introduction-to-logic/","timestamp":"2024-11-04T18:04:57Z","content_type":"text/html","content_length":"161187","record_id":"<urn:uuid:1adcb7d1-0549-46cb-b55a-3cf6f9533b35>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00250.warc.gz"}
8 Classic Alternatives to Traditional Plots That Every Data Scientist Must Add in Their Visualisation Toolkit A consolidated guide on best plotting ideas discussed here. Subscribe for free to learn something new and insightful about Python and Data Science every day. Also, get a Free Data Science PDF (550+ pages) with 320+ tips. Scatter plots, bar plots, line plots, box plots, and heatmaps are the most frequently used plots for data visualization. Although they are simple and known to almost everyone, I believe they are not the right choice to cover every possible scenario. Instead, many other plots originate from these standard plots that can be much more suitable, if used appropriately. Therefore, today, let’s discuss a few alternatives to these popular plots. I will also explain specific situations where they can be more useful over standard plots. This post is a consolidation of some of my previous plotting posts published in this newsletter. □ If you have never seen them before, then there’s new information for you. □ If you have seen them before, then this will be a good referesher for you. In any case, a consolidated guide will be quite useful to look back later instead of scrolling through individual newsletter issues. Also, before I begin, this post is not intended to discourage the use of these traditional plots. They will always have there place. Instead, it is to highlight specific situations where they can replaced with better plotting ideas. Let’s begin! #1) Size-encoded heatmaps A traditional heatmap represents the values using a color scale. Yet, mapping the cell color to exact numbers is still challenging. Embedding a size component to heatmaps can be extremely helpful in such cases. In essence, the bigger the size, the higher the absolute value: This is especially useful to make heatmaps cleaner, as many values nearer to zero will immediately shrink. #2) Waterfall charts To visualize the change in value over time, a line (or bar) plot may not always be an apt choice. This is because a line plot (or bar plot) depicts the actual values in the chart. Thus, it is difficult to visually estimate the scale and direction of incremental changes. Instead, you can use a waterfall chart. It elegantly depicts these rolling differences, as depicted below: Here, the start and final values are represented by the first and last bars. Also, the consecutive changes are automatically color-coded, making them easier to interpret. #3) Bump charts When visualizing the change in rank over time of multiple categories, using a bar chart may not be appropriate. This is because bar charts quickly become cluttered with many categories. Instead, try Bump Charts. They are specifically used to visualize the rank of different items over time. Comparing the bar chart and bump chart above, it is far easier to interpret the change in rank with a bump chart rather than a bar chart. #4) Raincloud Plots Visualizing data distributions using box plots and histograms can be misleading at times. This is because: Thus, to avoid misleading conclusions, it is always recommended to plot the data distribution as precisely as possible. Raincloud plots provide a concise way to combine and visualize three different types of plots together. These include: • Box plots for data statistics. • Strip plots for data overview. • KDE plots for the probability distribution of data. With Raincloud plots, you can: • Combine multiple plots to prevent incorrect/misleading conclusions • Reduce clutter and enhance clarity • Improve comparisons between groups • Capture different aspects of the data through a single plot #5-6) Hexbin and Density Plots Scatter plots can get too dense to interpret when you have thousands of data points. Instead, you can replace them with Hexbin plots. Hexbin plots bin the area of a chart into hexagonal regions. Each region is assigned a color intensity based on the method of aggregation used (the number of points, for instance). Another choice is a density plot, which illustrates the distribution of points in a two-dimensional space. A contour is created by connecting points of equal density. In other words, a single contour line depicts an equal density of data points. #7-8) Bubble charts and Dot plots As discussed above, bar plots quickly get messy and cluttered as the number of categories increases. A bubble plot is often a better alternative in such cases. They are like scatter plots but: • with one categorical axis • and one continuous axis As depicted above: • It is difficult to interpret the bar plot because it has too many bars packed into a small space, • But size-encoded bubbles make it pretty easy to visualize the change over time. Another alternative to bar plots in such situations is dot plots. Both dot plots and bubble charts are based on the idea that, at times, when we have a bar plot with many bars, we’re often not paying attention to the individual bar lengths. Instead, we mostly consider the individual endpoints that denote the total value. These plots precisely help us depict that while also eliminating the long bars of little to no use. Thanks for reading Daily Dose of Data Science! Subscribe for free to learn something new and insightful about Python and Data Science every day. Also, get a Free Data Science PDF (550+ pages) with 320+ tips. 👉 Over to you: Are there any other lesser-known yet valuable plots that I haven’t covered here. If yes, when do you use them? 👉 If you liked this post, don’t forget to leave a like ❤️. It helps more people discover this newsletter on Substack and tells me that you appreciate reading these daily insights. The button is located towards the bottom of this email. Thanks for reading! Latest full articles If you’re not a full subscriber, here’s what you missed last month: To receive all full articles and support the Daily Dose of Data Science, consider subscribing: 👉 Tell the world what makes this newsletter special for you by leaving a review here :) 👉 If you love reading this newsletter, feel free to share it with friends! Finally! I was waiting for this; I also love the idea of consolidating previous topics into one, makes it easier to bookmark one blog and come back to it later if needed! Expand full comment So usefull! Thanks! Expand full comment 1 more comment...	{"url":"https://blog.dailydoseofds.com/p/8-classic-alternatives-to-traditional","timestamp":"2024-11-07T21:45:46Z","content_type":"text/html","content_length":"245808","record_id":"<urn:uuid:b490f4e6-bcbe-472b-8fc6-f230297c5c2b>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00893.warc.gz"}
US Fed gold lending report You can see the original report at the Fed's own website at http://www.federalreserve.gov/pubs/ifdp/1997/582/ifdp582.pdf. But don't refer to it unless you like mathematical formulae! The economists engaged by the Fed seem to conclude that the holding of gold is economically inefficient, which it 'proves' to the satisfaction of its authors. The weakness is that the proof starts by assuming that given wealth generating alternatives holding a secure reserve is inefficient use of resources. This should be conceded by anyone. The proof only validates its own assumption, which is a sham.	{"url":"https://www.galmarley.com/Footnotes/fn_fed_gold_lending.htm","timestamp":"2024-11-12T18:13:57Z","content_type":"text/html","content_length":"2024","record_id":"<urn:uuid:3cdfac5f-da1c-424d-8b76-d28d7c7b921c>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00653.warc.gz"}
An Introduction to the Uniform Distribution \| Online Statistics library \| StatisticalPoint.com An Introduction to the Uniform Distribution by Erma Khan The uniform distribution is a probability distribution in which every value between an interval from a to b is equally likely to occur. If a random variable X follows a uniform distribution, then the probability that X takes on a value between x[1] and x[2] can be found by the following formula: P(x[1] 2) = (x[2 ]– x[1]) / (b – a) • x[1]: the lower value of interest • x[2]: the upper value of interest • a: the minimum possible value • b: the maximum possible value For example, suppose the weight of dolphins is uniformly distributed between 100 pounds and 150 pounds. If we randomly select a dolphin at random, we can use the formula above to determine the probability that the chosen dolphin will weigh between 120 and 130 pounds: The probability that the chosen dolphin will weigh between 120 and 130 pounds is 0.2. Visualizing the Uniform Distribution If we create a density plot to visualize the uniform distribution, it would look like the following plot: Every value between the lower bound a and upper bound b is equally likely to occur and any value outside of those bounds has a probability of zero. For example, in our previous example we said the weight of dolphins is uniformly distributed between 100 pounds and 150 pounds. Here’s how to visualize that distribution: And the probability that a randomly selected dolphin weighs between 120 and 130 pounds can be visualized as follows: Properties of the Uniform Distribution The uniform distribution has the following properties: • Mean: (a + b) / 2 • Median: (a + b) / 2 • Standard Deviation: √(b – a)^2 / 12 • Variance: (b – a)^2 / 12 For example, suppose the weight of dolphins is uniformly distributed between 100 pounds and 150 pounds. We could calculate the following properties for this distribution: • Mean weight: (a + b) / 2 = (150 + 100) / 2 = 125 • Median weight: (a + b) / 2 = (150 + 100) / 2 = 125 • Standard Deviation of weight: √(150 – 100)^2 / 12 = 14.43 • Variance of weight: (150 – 100)^2 / 12 = 208.33 Uniform Distribution Practice Problems Use the following practice problems to test your knowledge of the uniform distribution. Question 1: A bus shows up at a bus stop every 20 minutes. If you arrive at the bus stop, what is the probability that the bus will show up in 8 minutes or less? Solution 1: The minimum amount of time you’d have to wait is 0 minutes and the maximum amount is 20 minutes. The lower value of interest is 0 minutes and the upper value of interest is 8 minutes. Thus, we’d calculate the probability as: P(0 0.4. Question 2: The length of an NBA game is uniformly distributed between 120 and 170 minutes. What is the probability that a randomly selected NBA game lasts more than 155 minutes? Solution 2: The minimum time is 120 minutes and the maximum time is 170 minutes. The lower value of interest is 155 minutes and the upper value of interest is 170 minutes. Thus, we’d calculate the probability as: P(155 0.3. Question 3: The weight of a certain species of frog is uniformly distributed between 15 and 25 grams. If you randomly select a frog, what is the probability that the frog weighs between 17 and 19 Solution 3: The minimum weight is 15 grams and the maximum weight is 25 grams. The lower value of interest is 17 grams and the upper value of interest is 19 grams. Thus, we’d calculate the probability as: P(17 0.2. Note: We can use the Uniform Distribution Calculator to check our answers for each of these problems. Share 0 FacebookTwitterPinterestEmail previous post Joint Frequency: Definition & Examples Related Posts	{"url":"https://statisticalpoint.com/uniform-distribution/","timestamp":"2024-11-13T15:23:51Z","content_type":"text/html","content_length":"1025145","record_id":"<urn:uuid:cd69d097-1ee6-4ed3-ac2b-5898c4d6363b>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00211.warc.gz"}
Propagation of Sound Waves We know that sound waves are longitudinal waves, and when they propagate compressions and rarefactions are formed. In the following section, we compute the speed of sound in air by Newton’s method and also discuss the Laplace correction and the factors affecting sound in air. Newton’s formula for speed of sound waves in air Sir Isaac Newton assumed that when sound propagates in air, the formation of compression and rarefaction takes place in a very slow manner so that the process is isothermal in nature. That is, the heat produced during compression (pressure increases, volume decreases), and heat lost during rarefaction (pressure decreases, volume increases) occur over a period of time such that the temperature of the medium remains constant. Therefore, by treating the air molecules to form an ideal gas, the changes in pressure and volume obey Boyle’s law, Mathematically Differentiating equation (11.20), we get where, B[T] is an isothermal bulk modulus of air. Substituting equation (11.21) in equation (11.16), the speed of sound in air is Since P is the pressure of air whose value at NTP (Normal Temperature and Pressure) is 76 cm of mercury, we have P = (0.76 × 13.6 ×10^3 × 9.8) N m^-2 ρ= 1.293 kg m^-3. here ρ is density of air Then the speed of sound in air at Normal Temperature and Pressure (NTP) is =279.80 m s^-1 ≈ 280 ms^-1 (theoretical value) But the speed of sound in air at 0°C is experimentally observed as 332ms-1 which is close upto 16% more than theoretical value (Percentage error is ([332-280]/332 x 100% = 15.6%). This error is not small Laplace’s correction In 1816, Laplace satisfactorily corrected this discrepancy by assuming that when the sound propagates through a medium, the particles oscillate very rapidly such that the compression and rarefaction occur very fast. Hence the exchange of heat produced due to compression and cooling effect due to rarefaction do not take place, because, air (medium) is a bad conductor of heat. Since, temperature is no longer considered as a constant here, sound propagation is an adiabatic process. By adiabatic considerations, the gas obeys Poisson’s law (not Boyle’s law as Newton assumed), which is which is the ratio between specific heat at constant pressure and specific heat at constant volume. Differentiating equation (11.23) on both the sides, we get where, B[A] is the adiabatic bulk modulus of air. Now, substituting equation (11.24) in equation (11.16), the speed of sound in air is Since air contains mainly, nitrogen, oxygen, hydrogen etc, (diatomic gas), we take γ= 1.47. Hence, speed of sound in air is v[A] = ( √1.4)(280 m s^-1)= 331.30 m s^-1, which is very much closer to experimental data. Factors affecting speed of sound in gases Let us consider an ideal gas whose equation of state is where, P is pressure, V is volume, T is temperature, n is number of mole and R is universal gas constant. For a given mass of a molecule, equation (11.26) can be written as For a fixed mass m, density of the gas inversely varies with volume. i.e., Substituting equation (11.28) in equation (11.27), we get where c is constant. The speed of sound in air given in equation (11.25) can be written as From the above relation we observe the following (a) Effect of pressure : For a fixed temperature, when the pressure varies, correspondingly density also varies such that the ratio (P/ρ) becomes constant. This means that the speed of sound is independent of pressure for a fixed temperature. If the temperature remains same at the top and the bottom of a mountain then the speed of sound will remain same at these two points. But, in practice, the temperatures are not same at top and bottom of a mountain; hence, the speed of sound is different at different points. (b) Effect of temperature : the speed of sound varies directly to the square root of temperature in kelvin. Let v[0] be the speed of sound at temperature at 0° C or 273 K and v be the speed of sound at any arbitrary temperature T (in kelvin), then Since v[0] = 331ms^-1 at 0^0C, v at any temperature in t^0C is v = (331 + 0.60t) ms^-1 Thus the speed of sound in air increases by 0.61 ms^-1 per degree celcius rise in temperature. Note that when the temperature is increased, the molecules will vibrate faster due to gain in thermal energy and hence, speed of sound increases. (c) Effect of density : Let us consider two gases with different densities having same temperature and pressure. Then the speed of sound in the two gases are Taking ratio of equation (11.31) and equation (11.32), we get Thus the velocity of sound in a gas is inversely proportional to the square root of the density of the gas. (d) Effect of moisture (humidity): We know that density of moist air is 0.625 of that of dry air, which means the presence of moisture in air (increase in humidity) decreases its density. Therefore, speed of sound increases with rise in humidity. From equation (11.30) Let ρ[1], v[1] and ρ[2], v[2] be the density and speeds of sound in dry air and moist air, respectively. Then Since P is the total atmospheric pressure, it can be shown that where p[1] and p[2] are the partial pressures of dry air and water vapour respectively. Then (e) Effect of wind: The speed of sound is also affected by blowing of wind. In the direction along the wind blowing, the speed of sound increases whereas in the direction opposite to wind blowing, the speed of sound EXAMPLE 11.9 The ratio of the densities of oxygen and nitrogen is 16:14. Calculate the temperature when the speed of sound in nitrogen gas at 17°C is equal to the speed of sound in oxygen gas. From equation (11.25), we have Where, R is the universal gas constant and M is the molecular mass of the gas. The speed of sound in nitrogen gas at 17°C is Similarly, the speed of sound in oxygen gas at t in K is Given that the value of γ is same for both the gases, the two speeds must be equal. Hence, equating equation (1) and (2), we get Squaring on both sides and cancelling γ R term and rearranging, we get Since the densities of oxygen and nitrogen is 16:14, Substituting equation (5) in equation (3), we get	{"url":"https://www.brainkart.com/article/Propagation-of-Sound-Waves_36327/","timestamp":"2024-11-12T18:39:19Z","content_type":"text/html","content_length":"67149","record_id":"<urn:uuid:3d274ca8-d18a-4569-a58d-7aae2b9a4809>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00464.warc.gz"}
An object travels North at 4 m/s for 1 s and then travels South at 2 m/s for 4 s. What are the object's average speed and velocity? \| HIX Tutor An object travels North at # 4 m/s# for #1 s# and then travels South at # 2 m/s# for # 4 s#. What are the object's average speed and velocity? Answer 1 ${S}_{a} v$ of object is $= 2.4 \frac{m}{s}$ The displacement of the object going #N# for #1s# is: #N- ((4m)/s)1s = 4m# The displacement of the object going #S# for #4s# is: #S- ((2m)/s)4s = 8m# We can take #S# as the positive direction in this case where #N# is opposite, so total displacement is: Time taken for the displacement is: #t=5s# The displacement divided by the total time equals the average velocity: #V_av=(4m)/(5s)=4/5 m/s# The total distance traveled is #4m+8m=12m The distance traveled divided by the time is the average speed. #S_av=(12m)/(5s) = 12/5 m/s = 2.4 m/s# 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/an-object-travels-north-at-4-m-s-for-1-s-and-then-travels-south-at-2-m-s-for-4-s-8f9af89d34","timestamp":"2024-11-09T19:25:06Z","content_type":"text/html","content_length":"581771","record_id":"<urn:uuid:220e5a2d-569f-4578-abdb-5ca00d83e0f5>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00844.warc.gz"}
Markov partition A Markov partition is a tool used in dynamical systems theory, allowing the methods of symbolic dynamics to be applied to the study of hyperbolic systems. By using a Markov partition, the system can be made to resemble a discrete-time Markov process, with the long-term dynamical characteristics of the system represented as a Markov shift. The appellation 'Markov' is appropriate because the resulting dynamics of the system obeys the Markov property. The Markov partition thus allows standard techniques from symbolic dynamics to be applied, including the computation of expectation values, correlations, topological entropy, topological zeta functions, Fredholm determinants and the like. Let (M,φ) be a discrete dynamical system. A basic method of studying its dynamics is to find a symbolic representation: a faithful encoding of the points of M by sequences of symbols such that the map φ becomes the shift map. Suppose that M has been divided into a number of pieces E[1],E[2],…,E[r], which are thought to be as small and localized, with virtually no overlaps. The behavior of a point x under the iterates of φ can be tracked by recording, for each n, the part E[i] which contains φ^n(x). This results in an infinite sequence on the alphabet {1,2,…r} which encodes the point. In general, this encoding may be imprecise (the same sequence may represent many different points) and the set of sequences which arise in this way may be difficult to describe. Under certain conditions, which are made explicit in the rigorous definition of a Markov partition, the assignment of the sequence to a point of M becomes an almost one-to-one map whose image is a symbolic dynamical system of a special kind called a shift of finite type. In this case, the symbolic representation is a powerful tool for investigating the properties of the dynamical system (M,φ). Formal definition A Markov partition^[1] is a finite cover of the invariant set of the manifold by a set of curvilinear rectangles such that • For any pair of points , that • for • If and , then Here, and are the unstable and stable manifolds of x, respectively, and simply denotes the interior of . These last two conditions can be understood as a statement of the Markov property for the symbolic dynamics; that is, the movement of a trajectory from one open cover to the next is determined only by the most recent cover, and not the history of the system. It is this property of the covering that merits the 'Markov' appellation. The resulting dynamics is that of a Markov shift; that this is indeed the case is due to theorems by Yakov Sinai (1968)^[2] and Rufus Bowen (1975),^[3] thus putting symbolic dynamics on a firm footing. Variants of the definition are found, corresponding to conditions on the geometry of the pieces .^[4] Markov partitions have been constructed in several situations. Markov partitions make homoclinic and heteroclinic orbits particularly easy to describe. This article is issued from - version of the 3/25/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.	{"url":"https://ipfs.io/ipfs/QmXoypizjW3WknFiJnKLwHCnL72vedxjQkDDP1mXWo6uco/wiki/Markov_partition.html","timestamp":"2024-11-10T11:29:12Z","content_type":"text/html","content_length":"19221","record_id":"<urn:uuid:b60b48ce-8858-439d-8e8c-fe015578e899>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00898.warc.gz"}
The number-palindrome Very easy Execution time limit is 1 second Runtime memory usage limit is 122.174 megabytes Check if the given number is a palindrome. The number is a palindrome if it remains the same when its digits are reversed. One non-negative 32-bit integer. Print "Yes" if the number is a palindrome, and "No" otherwise. Submissions 17K Acceptance rate 46%	{"url":"https://basecamp.eolymp.com/en/problems/1608","timestamp":"2024-11-03T22:03:33Z","content_type":"text/html","content_length":"230579","record_id":"<urn:uuid:e7702cb8-1b68-4998-af69-cacfbb7e51a9>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00872.warc.gz"}
Chi-square test Hypothesis: The proportion of those who are unhappy is higher among those who rarely meet socially with friends, relatives, and colleagues than among those who often meet. Independent: sclmeet, Dependent: happy. weight by pweight. fre sclmeet. RECODE sclmeet (1 thru 3=1)(4 thru 7=2) INTO sclmeet_2cat. VARIABLE LABELS sclmeet_2cat ‘Do you meet socially often or rarely with friends, relatives or colleagues?’. VALUE LABELS sclmeet_2cat 1’rarely’ 2’often’. fre sclmeet sclmeet_2cat. fre happy. RECODE happy (0 thru 5=1) (6 thru 10=2) INTO happy_2cat. VARIABLE LABELS happy_2cat ‘Are you happy or not?’. VALUE LABELS happy_2cat 1’unhappy’ 2’happy’. fre sclmeet sclmeet_2cat happy happy_2cat. CROSSTABS happy_2cat BY sclmeet_2cat /CELLS=COUNT COLUMN /STATISTICS=CHISQ RISK. The percentage has to be in the direction of the independent variable. The independent variable has to be in the column. In the interpretation, you have to compare the percentages across. In the syntax after the command “CROSSTABS”, first you have to write the name of the dependent variable, then you have to write the command “by” and then the name of the independent variable. 41,8 – 27,6= 14,2 percentage points. The proportion of those who are unhappy is 14,2 percentage points higher among those who rarely meet socially with friends, relatives, and colleagues than among those who often meet. Conclusion: The proportion of unhappy people is significantly higher among those who rarely meet with friends, relatives, and colleagues than among those who often meet them. Why do we state this? Because: (p=0,000 < 0,05) and the epsilon shows us where is the proportion of unhappy people higher. Epsilon: the difference between two adjacent percentages – measured in percentage points. The percentage of unhappy people (41,8%) is much higher among those who only rarely meet socially with friends, family or colleagues than among those who often meet socially (27,6%). ->(41,8%-27,6%) It is 14,2 percentage points higher. OR: The percentage of unhappy people (41,7%) is much lower among those who often meet socially with friends, family members, colleagues than among those who rarely meet socially. ->(27,6%-41,8%) It is 14,2 percentage points lower. The percentage of happy people is much lower among those who only rarely meet socially with friends, family members, colleagues than among those who often meet. ->58,2%-72,4%=-14,2 percentage points The percentage of happy people is much higher among those who often meet socially with friends, family or colleagues than among those who only rarely meet. (72,4%-58,2%)=14,2 percentage points Which epsilon should you interpret? The epsilon for people being happy or the epsilon for people being unhappy? The answer to this relies on your hypothesis. You should interpret the one that you are referring to in your hypothesis. So, in this example, you should interpret the one for being unhappy, since your statement refers to the unhappiness of the people. How do you calculate the probability? Probability: Probability of an event occurring / Probability of all the different events occurring (total) What is the probability of being unhappy overall? What is the probability of being happy overall? What is the probability of being unhappy overall? P(unhappy)=34,8% * divided by 100 = 0.348 What is the probability of being happy overall? P(happy)=0.652 * divided by 100 = 0.652 How do you calculate the odds? Odds: probability of event occurring / probability of event not occuring. You can interpret it in times or in percentage(%). It shows you how many times more likely something is to happen than to not happen. CROSSTABS happy_2cat BY sclmeet_2cat /CELLS=COUNT COLUMN. What are the odds of being unhappy? Odds of being unhappy: 0.348/0.652=0,534 – here you calculate it from the probability. Odds of being unhappy: 34,8 / 65,2 = 0,534 – here you calculate it from the percentages in the total column. You get the same result. Odds of being unhappy: 293 / 548 = 0.534 (0.534-1)100 = -46,6 People are by 46.6% less likely to be unhappy than to be happy. Odds of being happy: 0.652/0.348=1.874 People are by 87,4% more likely to be happy than to be unhappy. Conditional odds Conditional odds: odds computed separately for each category of the independent variable. What are the odds of being unhappy for those who only rarely meet socially with friends, family or colleagues? The odds of being unhappy for people who only rarely meet socially with friends, family or colleagues / the odds of being unhappy for people who often meet socially with friends, family or Rarely:(conditional odds for being unhappy):180 / 251 = 0,717 Often (conditional odds for being unhappy): 113 / 297= 0,380 Odds ratio Odds ratio=ratio of two conditional odds. The odds ratio shows how many times greater or smaller the odds of the phenomenon under study is in one category of the independent variable than in the other category. CROSSTABS happy_2cat BY sclmeet_2cat /STATISTICS=RISK. Odds ratio: 0,72 / 0,38=1,89. Those who rarely meet are 1,89 times more likely to be unhappy than those who often meet. (compared to being happy). Odds ratio as percentage change: (1.885-1)100 = 88,5. The odds of being unhappy is 88,5% higher among those who only rarely meet socially with friends, family, colleagues, than among those who often meet socially with others. The program computes the odds ratio for those who are on your top left cells in the contingency table, so it computes the odds ratio for those who rarely meet and are unhappy comparing to those who often meet. Don’t be confused by the name risk, because it actually computes the odds ratio and not the risk (Because in real life risk means probability, but here it means odds ratio). Lambda is a measure of association that reflects the proportional reduction in error (PRE) when values of the independent variable are used to predict values of the dependent variable. A value of 1 means that the independent variable perfectly predicts the dependent variable. A value of 0 means that the independent variable is no help in predicting the dependent variable. CROSSTABS happy_2cat BY sclmeet_2cat /STATISTICS=LAMBDA. The value of the lambda is the one that is in the row which shows the “true dependent variable”. In this example the dependent variable is happy, so the value of the lambda is 0,000. How do you interpret the lambda? convert it’s value into a percent: x100 = 0,000*100 = 0 If you want to compare having the knowledge (data) of the distribution of the independent variable compared to not having this knowledge (data) improves our ability to predict the correct outcome by 0 percent. In another way: by knowing the independent variable we can reduce the probability of making an incorrect prediction by 0 percent. So, knowing whether a person often or rarely meets with others socially improves our ability to predict the correct outcome by 0	{"url":"https://spssabc.com/chi-square-test/","timestamp":"2024-11-04T07:23:32Z","content_type":"text/html","content_length":"48012","record_id":"<urn:uuid:c9d381a8-3bfc-4b09-bc92-b0485b2458e5>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00301.warc.gz"}
INTERSPEECH 2023 T5 Part4: Source Separation Based on Deep Source Generative Models and Its Self-Supervised Learning Model • Y. Bando, K. Sekiguchi, Y. Masuyama, A. A. Nugraha, M. Fontaine, K. Yoshii, “Neural full-rank spatial covariance analysis for blind source separation,” IEEE SP Letters, 2021 • Y. Bando, T, Aizawa, K. Itoyama, K. Nakadai, “Weakly-supervised neural full-rank spatial covariance analysis for a front-end system of distant speech recognition,” INTERSPEECH, 2022 • H. Munakata, Y. Bando, R. Takeda, K. Komatani, M. Onishi, “Joint Separation and Localization of Moving Sound Sources Based on Neural Full-Rank Spatial Covariance Analysis,” IEEE SP Letters, 2023 Source Separation Based on Deep Generative Models and Its Self-Supervised Learning /33 17	{"url":"https://speakerdeck.com/yoshipon/interspeech2023-t5-part4-bando","timestamp":"2024-11-06T18:00:40Z","content_type":"text/html","content_length":"127560","record_id":"<urn:uuid:221369dd-0acb-4a73-a60b-f729792e6089>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00024.warc.gz"}
Calculus Calculator-Free AI-Powered Calculus Solver Home > Calculus Calculator Calculus Calculator-AI-Powered Calculus Solver AI-powered solutions for your calculus problems. Rate this tool 20.0 / 5 (200 votes) Introduction to Calculus Calculator The Calculus Calculator is designed as a specialized tool to assist users in solving calculus problems. It leverages advanced algorithms to provide step-by-step solutions, helping users understand the underlying concepts and methodologies in calculus. This tool is particularly valuable for students, educators, and professionals who need to perform or verify complex calculations. For example, a student struggling with finding the derivative of a complex function can input the problem and receive a detailed explanation of each step involved in the solution. Main Functions of Calculus Calculator • Derivative Calculation Given a function f(x) = 3x^2 + 2x + 1, the Calculus Calculator can find its derivative. In an educational setting, a student can use the calculator to check their manual differentiation work, ensuring they understand the rules of differentiation. • Integration For a function f(x) = x^2, the calculator can compute the definite or indefinite integral. A physics student might use this function to solve problems involving areas under curves or in mechanics where integrals are used to find quantities like displacement from velocity. • Limit Evaluation The calculator can evaluate limits such as lim(x->0) (sin(x)/x). A calculus student preparing for exams can use the tool to practice and verify their solutions to various limit problems, reinforcing their understanding of limits and continuity. Ideal Users of Calculus Calculator Services • Students Students at high school and university levels who are learning calculus would greatly benefit from this tool. It provides detailed, step-by-step solutions that help in understanding complex concepts and verifying their work. • Educators Teachers and tutors can use the Calculus Calculator to prepare teaching materials and examples. It aids in creating precise and clear explanations for complex problems, which can enhance the learning experience for their students. How to Use Calculus Calculator • Step 1 Visit aichatonline.org for a free trial without login, also no need for ChatGPT Plus. • Step 2 Navigate to the Calculus Calculator section on the website. • Step 3 Input your calculus problem or question in the provided text box. • Step 4 Click on the 'Calculate' button to generate a detailed step-by-step solution. • Step 5 Review the solution and use the explanation to understand the steps involved. Repeat as needed for additional problems. • Problem Solving • Homework Help • Concept Review • Test Preparation • Self-Study Common Questions About Calculus Calculator • What types of calculus problems can the Calculus Calculator solve? The Calculus Calculator can solve a wide range of problems, including differentiation, integration, limits, and series. It provides detailed step-by-step solutions to help users understand each step in the process. • Is the Calculus Calculator free to use? Yes, you can use the Calculus Calculator for free by visiting aichatonline.org. There is no need for a subscription or ChatGPT Plus. • How does the Calculus Calculator ensure the accuracy of its solutions? The Calculus Calculator uses advanced algorithms and a robust mathematical engine to ensure accurate and reliable solutions. It also provides detailed explanations to help users verify each step. • Can the Calculus Calculator help with understanding calculus concepts? Yes, the Calculus Calculator not only provides solutions but also includes explanations of key concepts and related examples to enhance understanding. • What are the system requirements for using the Calculus Calculator? The Calculus Calculator is a web-based tool, so you only need an internet connection and a modern web browser to access and use it effectively.	{"url":"https://theee.ai/tools/Calculus-Calculator-2OToEoedIW","timestamp":"2024-11-01T20:25:57Z","content_type":"text/html","content_length":"102907","record_id":"<urn:uuid:8f9e4bef-0dbe-4950-a4bf-27b974cad998>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00429.warc.gz"}
May 2009 – Peter Frase May 9th, 2009 \| Published in Data, Social Science, Statistical Graphics Final thing on the car-culture regression. Below is a comparison of the actual data on Vehicle Miles Traveled with my reconstruction of Nate Silver's model, and my model including lagged gas prices, housing prices, and the stock market. I "seasonally adjusted" the miles data by fitting a model predicting miles based only on the month of the year. The miles data (whether the actual data or the prediction from a model) is then corrected by subtracting the coefficient for the month it was collected. This data is normalized according the level of driving in April. An even better fit is possible with a more complex model that includes a) average monthly temperatures and b) an interaction between gas prices and time. But this simpler model suffices to show that Silver's original finding was probably an artifact of his failure to control for wealth effects and the lagged effect of gas prices. The lesson, I suppose, is: beware of columnists on deadline bearing regressions! May 9th, 2009 \| Published in Data, Social Science Update to the post below: I decided to see how well my model will predict miles traveled going forward. My model only includes data through January, as Nate Silver's did. But we have the data through February now, so we can see how well the model works there. We also have almost all the data needed to predict March--the only thing missing is the government's Housing Price Index. But that doesn't change too much month to month, so I made a prediction based on the February value: Predicted Actual February 215.37 215.77 March 245.31 ?? The March numbers should be out soon, so we'll see how my model performs. Moment of Zen May 8th, 2009 \| Published in Data, Social Science, Statistical Graphics Here are the variables I used in the models for the previous post. Simplistic social theories are left as an excercise for the reader. Attempt to Regress May 8th, 2009 \| Published in Data, Social Science, Statistical Graphics I'm loathe to say an unkind word about Nate Silver. Besides boosting the profile of my alma mater, he's done more than anyone else to improve the reputation and sexiness of my present occupation: statistical data analyst. This is all the more welcome at a time when other people are blaming statistical models for, well, ruining everything. But I confess to being a bit annoyed when I read Silver's recent article about the changes in American driving habits. In that article, Silver argues that we're seeing a real shift away from car culture, based on the following: I built a regression model that accounts for both gas prices and the unemployment rate in a given month and attempts to predict from this data how much the typical American will drive. The model also accounts for the gradual increase in driving over time, as well as the seasonality of driving levels, which are much higher during the summer than during the winter. All well and good, except that Silver doesn't provide the model or the data! He asks us to take his word for it that in January, Americans "drove about 8 percent less than the model predicted." Now, I don't expect anyone to publish regression coefficients in Esquire magazine, but Silver does have a rather well-known website, so he could have put it there. The analysis was already done and published, so I don't see how it would have hurt Silver to publish the data after the fact. Which is what makes me suspect that he kept things deliberately vague in order to maintain a sense of mystery and awe around his regression models. Particularly because in this case, the underlying model is actually quite simple. Which is a shame, because the simplicity of the model is actually the most appealing thing about it. It's a great example of a situation where a regression illuminates a relationship that would be really hard to discern using simple descriptive statistics. The model is a perfect balance between being simple enough to be believable, and complex enough to really gain you something over simple descriptives. In fact, it's something that I plan to refer to in the future when my less quant-y friends question the need for regressions. Which is why I decided to recreate Silver's analysis from scratch, which took me about an hour. First I had to figure out what Silver's model was. Based on the paragraph above, I decided on: miles = gas + unemployment + date + month Monthly miles driven are modeled as a function of that month's average gas prices, the unemployment rate in that month, the date, and which month of the year it is. The date variable will capture the "gradual increase" in miles traveled. I use month to capture the "seasonality of driving levels". I could have grouped the months into seasons, but why not use a more precise measure if you've got The next step was to find the data: From different sources, I obtained data on miles traveled, gas prices, and unemployment. All of these sources start around 1990, so that's the time frame we'll have to work with. With that in hand, it was time for some analysis. Using R, I combined the different data sources and ran myself a regression: lm(formula = miles ~ unemp + price + date + month) coef.est coef.se (Intercept) 98.52 3.71 unemp -2.09 0.34 gasprice -0.08 0.01 date 0.01 0.00 monthAugust 17.90 1.40 monthDecember -8.82 1.40 monthFebruary -30.26 1.42 monthJanuary -22.03 1.40 monthJuly 17.87 1.42 monthJune 11.34 1.42 monthMarch 0.42 1.42 monthMay 12.56 1.42 monthNovember -10.00 1.40 monthOctober 5.85 1.40 monthSeptember -2.55 1.40 n = 222, k = 15 residual sd = 4.25, R-Squared = 0.98 That R-Squared of 0.98 means that about 98% of the actual variation in miles traveled is explained by the variables in this model. So it's a pretty comprehensive picture of the things that predict how much Americans will drive. A one point increase in the unemployment rate, in this model, predicts a 2.09 billion mile decrease in miles driven. And gas prices are in cents, so a one-cent increase in the price of gas will, all things being equal, translate into an 80 million mile decrease in miles driven. The next step was to check out Silver's assertion that recent data on miles driven is lower than the model would predict. Recall that Silver's model over-predicted January miles driven by 8 percent. My model predicts that in January, Americans should have driven 239.6 billion miles. The actual number was 222 billion miles. The prediction is--wait for it--7.9 percent more than the actual number! That's pretty amazing actually, and it indicates that my data and model must be pretty damn close to Silver's. With the model in hand, however, we can do a bit better than this. Below is a chart showing how close the model was for every month in my dataset. It's similar to the graphic accompanying Silver's Esquire article, only not as ugly and confusing. The graph shows the difference between the prediction and the actual number. When the point is above the zero line, it means people drove more than the model would predict. When it's below the line, they drove less. You can see here that there are multiple imperfections in the model. Mileage declined a little faster than predicted in the late 90's, and then rose faster than expected in the early 2000's. It's possible that this has something to do with a policy difference between the Bush and Clinton administrations, but I'm not enough of an expert to say. What jumps out, though, are those last three points on the right, corresponding to this past November, December, and January. All of them are way off the prediction, and the error is bigger than for any other time period. This strongly suggests that something really has changed. What's not totally clear, though, is whether it's the car culture that's different, or whether it's this recession that's unlike the other two recessions in this data set (the early 90's and early 2000's). The next logical step is to consider some additional variables. Some commenters at Nate's site pointed out that you might want to factor in changes in wealth--as opposed to changes in income, which are at least partly captured by the unemployment variable. Directly measuring wealth is a little tricky, but we can easily measure two things that are proxies for wealth, or people's perceptions of wealth: the stock market and the housing market. So I went google-hunting again and found two more variables: the monthly closing of the Dow, and the government's housing price index. Put those into the regression, and away we go: lm(formula = miles ~ unemp + price + date + stocks + housing + month) coef.est coef.se (Intercept) 117.87 4.13 unemp -1.64 0.48 gasprice -0.11 0.01 date 0.01 0.00 stocks 1.01 0.30 housing 0.24 0.03 monthAugust 18.40 1.20 monthDecember -8.88 1.21 monthFebruary -30.58 1.21 monthJanuary -22.12 1.19 monthJuly 18.28 1.20 monthJune 11.74 1.20 monthMarch 0.30 1.20 monthMay 12.77 1.20 monthNovember -10.02 1.21 monthOctober 6.42 1.21 monthSeptember -1.92 1.21 n = 217, k = 17 residual sd = 3.60, R-Squared = 0.98 R-squared looks the same, but the residual standard deviation is lower, which indicates that this model predicts more of the variation in the data than the last one. And the new variables both have pretty big and statistically significant effects. The stock market close is scaled in thousands, so the coefficient indicates that for every 1000 point increase in the Dow, we drive 1 billion more miles. The housing price index defines 1991 prices as 100, and went into the 220's during the bubble. Every one point increase in that index predicts a 240 million mile increase in driving. Here's another version of the graph above, for our new model: The same patterns are still present, but the divergence between the predictions and the actual numbers is smaller now. (Incidentally, I have no idea what happened in January of 1995. Did everyone go on a road trip without telling me?) It still looks like there's been some qualitative change in US driving habits recently, but the case is less clear cut. In particular, the late 90's now looks like another outstanding mystery. Mileage declined by more than the model expected then, but why? At the moment I have no particular hypothesis about that. My final model tests something else that appears in Nate's article: There is strong statistical evidence, in fact, that Americans respond rather slowly to changes in fuel prices. The cost of gas twelve months ago, for example, has historically been a much better predictor of driving behavior than the cost of gas today. In the energy crisis of the early 1980s, for instance, the price of gas peaked in March 1981, but driving did not bottom out until a year OK, so let's try using the price of gas 12 months ago as a predictor along with current prices. This will force us to throw away a bit of data, but we can still fit a model on most of the data lm(formula = miles ~ unemp + price + price12 + date + stocks + housing + month, data = data) coef.est coef.se (Intercept) 112.28 3.82 unemp -0.93 0.42 gasprice -0.07 0.01 gasprice12 -0.08 0.01 date 0.01 0.00 stocks 0.93 0.26 housing 0.25 0.02 monthAugust 18.19 1.04 monthDecember -8.99 1.05 monthFebruary -31.26 1.06 monthJanuary -22.20 1.05 monthJuly 18.17 1.05 monthJune 11.58 1.05 monthMarch 0.10 1.06 monthMay 12.88 1.05 monthNovember -10.06 1.04 monthOctober 6.29 1.04 monthSeptember -2.08 1.04 n = 210, k = 18 residual sd = 3.07, R-Squared = 0.99 It looks like current gas prices and last year's gas prices are about equivalent in their effect on mileage. Now let's look at the graph of prediction error again: Lo and behold, the apparently anomalous findings from the last few months have disappeared. This isn't the last word, of course, nor is it the perfect model. But it no longer appears that US driving behavior is so unusual, when you account for all the relevant economic contextual factors. Anyhow, that's enough playing around in the data for me for the time being. In the end, this whole exercise helped me understand what I like best about Nate Silver's work. He's inventing a new media niche, call it "statistical journalist". He uses publicly available data to produce quick, topical analysis that illuminates the issues of the data in the way neither anecdotes nore naive recitations of descriptive statistics can. He may play fast and loose at times, but his methods are transparent enough that people like me can still check up on him. I certainly hope that this kind of writing becomes an established sub-specialty with a wider base of practitioners than just Silver himself.	{"url":"https://www.peterfrase.com/2009/05/","timestamp":"2024-11-05T04:35:15Z","content_type":"application/xhtml+xml","content_length":"58986","record_id":"<urn:uuid:41a462b6-01e6-47d5-91d5-1a5b4becc1e3>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00120.warc.gz"}
Standard Deviation For example, the data points 50, 51, 52, 55, 56, 57, 59 and 60 have a mean at 55 (Blue). Another data set of 12, 32, 43, 48, 64, 71, 83 and 87. This set too has a mean of 55 (Pink). However, it can clearly be seen that the properties of these two sets are different. The first set is much more closely packed than the second one. Through standard deviation, we can measure this distribution of data about the mean. The above example should make it clear that if the data points are values of the same parameter in various experiments, then the first data set is a good fit, but the second one is too uncertain. Therefore in measurement of uncertainty, standard deviation is important - the lesser the standard deviation, the lesser this uncertainty and thus more the confidence in the experiment, and thus higher the reliability of the experiment. One Standard Deviation In a normal distribution, values falling within 68.2% of the mean fall within one standard deviation. This means if the mean energy consumption of various houses in a colony is 200 units with a standard deviation of 20 units, it means that 68.2% of the households consume energy between 180 to 220 units. This is assuming that the data of energy consumption is normally distributed. If a researcher considers three standard deviations to either side of the mean, this covers 99% of the data. Thus in the previous example, 99% of the households have their energy consumption between 140 to 260 units. In most cases, this is considered as the whole data set, especially when the data can extend to infinity. The measurement of uncertainty through standard deviation is used in many experiments of social sciences and finances. For example, the more risky and volatile ventures have a higher standard deviation. Also, a very high standard deviation of the results for the same survey, for example, should make one rethink about the sample size and the survey as a whole. In physical experiments, it is important to have a measurement of uncertainty. Standard deviation provides a way to check the results. Very large values of standard deviation can mean the experiment is faulty - either there is too much noise from outside or there could be a fault in the measuring instrument.	{"url":"https://explorable.com/measurement-of-uncertainty-standard-deviation","timestamp":"2024-11-03T22:26:55Z","content_type":"application/xhtml+xml","content_length":"57110","record_id":"<urn:uuid:1bfd87ec-327a-4f68-af51-aa827c2c831b>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00177.warc.gz"}
Unlocking the Power of the Most Popular Excel Formulas » THEAMITOS Unlocking the Power of the Most Popular Excel Formulas Microsoft Excel is an indispensable tool in the world of business, finance, data analysis, and beyond. With its powerful capabilities, Excel enables users to perform complex calculations, analyze large datasets, and automate repetitive tasks. However, to truly harness the power of Excel, it’s essential to master the most popular Excel formulas. These formulas are the backbone of efficient data management and can significantly enhance productivity in any career that involves working with data. In this article, we’ll explore the most popular Excel formulas, why they’re essential, and how mastering them can make you an Excel power user. Whether you’re a beginner or an advanced user, understanding these formulas will take your Excel skills to the next level. 1. SUM Function The SUM function is one of the most basic yet powerful Excel formulas. It allows you to add up a range of numbers, saving you time and ensuring accuracy. This formula is widely used in financial modeling, budgeting, and any task that involves adding numbers across rows or columns. This formula will add all the values from cell B2 to B10. 2. IF Function The IF function is a logical formula that returns one value if a condition is true and another value if it is false. This function is incredibly versatile and is often used for decision-making processes within Excel. =IF(A1>10, "Yes", "No") This formula will return “Yes” if the value in cell A1 is greater than 10; otherwise, it will return “No.” 3. XLOOKUP Function XLOOKUP is the modern, more powerful replacement for VLOOKUP. It allows you to search a range or an array, and unlike VLOOKUP, it can search both vertically and horizontally. XLOOKUP is more flexible, does not require the lookup value to be in the first column, and can return results from any column. =XLOOKUP(A2, B2:B10, C2:C10) This formula looks for the value in cell A2 within the range B2 and returns the corresponding value from the range C2 . If the value is not found, you can also specify a default value to return. 4. INDEX-MATCH Combination While XLOOKUP has largely replaced the need for the INDEX-MATCH combination, it’s still worth knowing as it provides flexibility in certain scenarios. The INDEX-MATCH combination allows for complex lookups and can be used in place of VLOOKUP or HLOOKUP when working with large datasets. =INDEX(B2:B10, MATCH(A1, C2:C10, 0)) This formula searches for the value in cell A1 within the range C2 and returns the corresponding value from the range B2 5. COUNTIF Function The COUNTIF function is used to count the number of cells that meet a specific criterion within a range. This is particularly useful in scenarios where you need to count occurrences of a specific value or text in a dataset. =COUNTIF(A2:A10, "Completed") This formula will count the number of cells in the range A2 that contain the word “Completed.” 6. TEXT Function The TEXT function is used to convert numbers to text, or format numbers as text in a specific way. This is particularly useful for displaying dates, times, or numeric values in a custom format. =TEXT(A1, "MM/DD/YYYY") This formula will convert the date in cell A1 to the “MM/DD/YYYY” format. 7. CONCATENATE Function The CONCATENATE function (or its modern equivalent, the CONCAT function) is used to combine text from different cells into one cell. This is useful for creating custom labels, combining names, or merging data from multiple columns. =CONCATENATE(A1, " ", B1) This formula will combine the text in cells A1 and B1, with a space in between. 8. SUMIF and SUMIFS Functions The SUMIF function adds all numbers in a range that meet a single criterion, while the SUMIFS function allows you to sum values based on multiple criteria. These functions are essential for conditional summing in Excel. Example (SUMIF): =SUMIF(A2:A10, ">100", B2:B10) This formula adds all values in the range B2 where the corresponding value in A2 is greater than 100. Example (SUMIFS): =SUMIFS(B2:B10, A2:A10, ">100", C2:C10, "East") This formula adds all values in B2 where the corresponding values in A2 are greater than 100 and the values in C2 are “East.” 9. PMT Function The PMT function is used to calculate the payment for a loan based on constant payments and a constant interest rate. This formula is essential in financial modeling and is widely used in mortgage =PMT(0.05/12, 360, 300000) This formula calculates the monthly payment for a loan of $300,000 at an annual interest rate of 5% over 30 years. 10. CHOOSE Function The CHOOSE function returns a value from a list of values based on a specified position. This is useful for scenarios where you want to select a value based on a certain condition or index. =CHOOSE(2, "Red", "Blue", "Green") This formula will return “Blue” because it is the second value in the list. 11. AVERAGEIF and AVERAGEIFS Functions The AVERAGEIF function calculates the average of cells that meet a specific criterion, while the AVERAGEIFS function allows for multiple criteria. These functions are particularly useful for finding average values based on conditions. Example (AVERAGEIF): =AVERAGEIF(A2:A10, ">10") This formula calculates the average of all values in the range A2 that are greater than 10. Example (AVERAGEIFS): =AVERAGEIFS(B2:B10, A2:A10, ">10", C2:C10, "<20") This formula calculates the average of values in B2 where the corresponding values in A2 are greater than 10 and the values in C2 are less than 20. 12. LEFT, RIGHT, and MID Functions These functions extract specific portions of text from a string. LEFT extracts a certain number of characters from the beginning, RIGHT extracts from the end, and MID extracts a specific number of characters from a starting point. Example (LEFT): =LEFT(A1, 5) This formula extracts the first five characters from the text in cell A1. Example (RIGHT): =RIGHT(A1, 3) This formula extracts the last three characters from the text in cell A1. Example (MID): =MID(A1, 3, 5) This formula extracts five characters from the text in cell A1, starting at the third character. Conclusion: The Power of Excel Formulas Mastering these most popular Excel formulas will transform the way you work with data, making your processes more efficient and your analyses more accurate. Whether you’re calculating totals, making decisions, or analyzing data trends, these formulas will empower you to unlock the full potential of Excel. By incorporating these formulas into your daily workflow, you can streamline tasks, reduce errors, and focus on deriving insights from your data. Excel is more than just a spreadsheet program; it’s a powerful tool for data-driven decision-making, and these formulas are the keys to unlocking that power. Leave a Comment	{"url":"https://theamitos.com/top-101-most-popular-excel-formulas-new-free-pdf-2024/","timestamp":"2024-11-05T21:39:11Z","content_type":"text/html","content_length":"214692","record_id":"<urn:uuid:d6c1b9cc-9a86-4a49-942c-e9fd08289cd2>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00543.warc.gz"}
Balashov, VS, Yan, Y and Zhu, X (2020). Who Manipulates Data During Pandemics? Evidence from Newcomb-Benford Law. Preprint arXiv:2007.14841 [econ.GN]; last accessed March 10, 2021. Berger, A and Hill, TP (2011). Benford's Law Strikes Back: No Simple Explanation in Sight for Mathematical Gem. The Mathematical Intelligencer 33(1), pp. 85-91. DOI:10.1007/ Berger, A and Hill, TP (2017). What is…Benford's Law?. Notices of the AMS 64(2), pp. 132-134. Collins, JC (2017). Using Excel and Benford’s Law to detect fraud. Journal of Accountancy Feature/Technology Workshop, April 1. Diekmann, A (2007). Not the First Digit! Using Benford's Law to Detect Fraudulent Scientific Data. Journal of Applied Statistics 34(3), pp. 321-329. ISSN/ISBN:0266-4763. DOI:10.1080/ Glen, S (2016). Benford’s Law (The First Digit Law): Simple Definition, Examples. From StatisticsHowTo.com: Elementary Statistics for the rest of us! Last accessed June 12, 2020. Goodman, WM (2016). The promises and pitfalls of Benford's law. Significance 13(3) pp. 38-41. DOI:10.1111/j.1740-9713.2016.00919.x. Koch, C and Okamura, K (2020). Benford's Law and COVID-19 Reporting. Posted on SSRN April 28, 2020; last accessed November 17, 2020. Published in Econ Lett 2020;196(109973) . Kruger, PS and Yadavalli, VSS (2017). The Power of One: Benford's Law. South African Journal of Industrial Engineering Vol 28(2), pp. 1-13. DOI:10.7166/28-2-1753. Moreno-Montoya, J (2020). Benford ́s Law with small sample sizes: A new exact test useful in health sciences during epidemics. Revista de la Universidad Industrial de Santander. Salud UIS vol. 52(2), pp. 161-163. Zhang, J (2020). Testing Case Number of Coronavirus Disease 2019 in China with Newcomb-Benford Law. Preprint arXiv:2002.05695 [physics.soc-ph]; last accessed February 18, 2020.	{"url":"https://www.benfordonline.net/references/down/2328","timestamp":"2024-11-10T06:37:05Z","content_type":"application/xhtml+xml","content_length":"17237","record_id":"<urn:uuid:9818ffd8-463d-470e-9003-410ff03fdc29>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00551.warc.gz"}
Stabilizing the error with the best answers: error in plot.window(...) : need finite 'ylim' values - ITtutoriaStabilizing the error with the best answers: error in plot.window(...) : need finite 'ylim' values You must login to ask question.(5) I’m building a new program, but when I run it, an error pops up. The error displayed is as follows: error in plot.window(need finite 'ylim' values) I have tried several workarounds, but they still do not get the desired results. If you have come across this situation and have a solution for the “error in plot.window(…) : need finite ‘ylim’ values” problem, pls let me know. Here is what I do: # read in sample data and split it up by group (defined by ID) xy <- data.frame(NAME=c("NAME2","NAME2","NAME2","NAME2","NAME2","NAME3","NAME3","NAME3","NAME3","NAME5","NAME5","NAME5","NAME5"), ID=c(48,48,48,48,48,32,32,32,32,67,67,67,67),YEAR=c(1981,1983,1984,1988,1989,1984,1984,1988,1988,1899,1933,1948,1958),VALUE=c(0,205,-570,0,-310,-3680,-3680,NA,-3680,0,NA,13,-98)) ind <- split(x = xy,f = xy[,'ID']) # Plot Scenario 1: if only years between 1946 and 2014 are present for each group do this: plot1 <- function(x) { fname <- paste0(x[1, 'ID'], '.png') png(fname, width=1679, height=1165, res=150) plot(x = c(1946, 2014), y = range(x$VALUE), main=x[1, 'NAME'], xlab="Time [Years]", axis(2, at = seq(-100000, 100000, 500), cex.axis=1, labels=FALSE, tcl=-0.3) points(ind[[i]][,c('YEAR','VALUE')], type="l", lwd=2) points(ind[[i]][,c('YEAR','VALUE')], type="p", lwd=1, cex=1, pch=21, bg='white') # Plot Scenario 2 if years under 1946 are present do this: plot2 <- function(x) { fname <- paste0(x[1, 'ID'], '.png') png(fname, width=1679, height=1165, res=150) main=x[1, 'NAME'], xlab="Time [Years]", ylab="Value [mm]") axis(2, at = seq(-100000, 100000, 500), cex.axis=1, labels=FALSE, tcl=-0.3) points(ind[[i]][,c('YEAR','VALUE')], type="l", lwd=2) points(ind[[i]][,c('YEAR','VALUE')], type="p", lwd=1, cex=1, pch=21, bg='white') # Execute functions lapply(ind, function(x) ifelse(any(x$YEAR < 1946 & x$YEAR < 2014), plot2(x), plot1(x)))	{"url":"https://ittutoria.net/question/error-in-plot-window-need-finite-ylim-values/?show=random","timestamp":"2024-11-09T16:04:19Z","content_type":"text/html","content_length":"199525","record_id":"<urn:uuid:f0b371db-7749-41df-a643-b3dc8bf4a5e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00666.warc.gz"}
Solved Example Problems for Universal Law of Gravitation EXAMPLE 6.1 Consider two point masses m1 and m2 which are separated by a distance of 10 meter as shown in the following figure. Calculate the force of attraction between them and draw the directions of forces on each of them. Take m1= 1 kg and m2 = 2 kg The force of attraction is given by From the figure, r =10 m. First, we can calculate the magnitude of the force It is to be noted that this force is very small. This is the reason we do not feel the gravitational force of attraction between each other. The small value of G plays a very crucial role in deciding the strength of the force. The force of attraction () experienced by the mass m2 due to m1 is in the negative â€˜yâ€™ direction ie., rË† =âˆ’jË† . According to Newtonâ€™s third law, the mass m2 also exerts equal and opposite force on m1. So the force of attraction () experienced by m1 due to m2 is in the direction of positive â€˜yâ€™ axis ie., rË† = jË† . The direction of the force is shown in the figure, Gravitational force of attraction between m1 and m2 = âˆ’ which confirms Newtonâ€™s third law. EXAMPLE 6.2 Moon and an apple are accelerated by the same gravitational force due to Earth. Compare the acceleration of the two. The gravitational force experienced by the apple due to Earth Here MA â€“ Mass of the apple, MEâ€“ Mass of the Earth and R â€“ Radius of the Earth. Equating the above equation with Newtonâ€™s second law, Simplifying the above equation we get, Here aA is the acceleration of apple that is equal to â€˜gâ€™. Similarly the force experienced by Moon due to Earth is given by Here Rm- distance of the Moon from the Earth, Mm â€“ Mass of the Moon The acceleration experienced by the Moon is given by The ratio between the appleâ€™s acceleration to Moonâ€™s acceleration is given by From the Hipparchrus measurement, the distance to the Moon is 60 times that of Earth radius. Rm = 60R. The appleâ€™s acceleration is 3600 times the acceleration of the Moon. The same result was obtained by Newton using his gravitational formula. The appleâ€™s acceleration is measured easily and it is 9.8 m sâˆ’2 . Moon orbits the Earth once in 27.3 days and by using the centripetal acceleration formula, (Refer unit 3). which is exactly what he got through his law of gravitation.	{"url":"https://www.brainkart.com/article/Solved-Example-Problems-for-Universal-Law-of-Gravitation_36144/","timestamp":"2024-11-09T12:56:03Z","content_type":"text/html","content_length":"44037","record_id":"<urn:uuid:fdd097a7-8f89-4c61-8ee7-39e09d3cbba6>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00023.warc.gz"}
Dijkstra's Algorithm \| CS61B Guide One sentence overview Visit vertices in order of best-known distance from source; on visit, relax every edge from the visited vertex. Detailed Breakdown Djikstras uses a PriorityQueue to maintain the path with lowest cost from the starting node to every other node, an edgeTo array to keep track of the best known predecessor for each vertex, and a distTo array to keep track of the best known distance from the source vertex to every other vertex. Relaxing the edges of a vertex v just refers to the process of updating edgeTo[n] for each neighbor n to v. You'll see in the pseudocode and diagrams below that succesful relaxation only occurs when the edge connecting the vertex being visited to one of its neighbors yields a smaller total distance than the current shortest path to that neighboring vertex that the algorithm has seen. Now, here's a demonstration on how it works! Let's start out with this graph: We'll start at node A and try to figure out the shortest path from A to each node. Since we have no idea how far each node is, we'll take the conservative guess that everything is infinitely far away The first thing we have to do is update A's adjacent nodes, which are B and D. Since there's only one known path to each, it shouldn't be too hard to see why we need to update the values below. One thing to note is that the priority queue sorts the vertices by the distance it takes to get there. Now, we have a choice to move on to either B or D. Since B has a shorter distance, we'll move on to that first. When we move on, we have to remove that value from the priority queue and update all of its neighbors. Here, we see that going from B to D is shorter than A to D, so we have to update distTo AND edgeTo of D to reflect this new, shorter path. This process (updating each adjacent node) is called relaxing the edges of a node. Now, let's move onto D since it has the next shortest path. Again, we remove D from the priority queue and relax C since we found a shorter path. Finally, we'll move onto C as that has the next shortest path in the priority queue. This will reveal our final node, E. Since the priority queue is now empty, our search is done! 😄 Here's what the final solution looks like in a tree form: It's a very spindly tree indeed, but hopefully it demonstrates that the result is acyclic. Properties of Dijkstra's Algorithm Dijkstra's Algorithm has some invariants (things that must always be true): edgeTo[v] always contains best known predecessor for v distTo[v] contains best known distance from source to v PQ contains all unvisited vertices in order of distTo Additionally, there are some properties that are good to know: always visits vertices in order of total distance from source relaxation always fails on edges to visited vertices guarantees to work optimally as long as edges are all non-negative solution always creates a tree form. can think of as union of shortest paths to all vertices edges in solution tree always has V-1 edges, where V = the number of vertices. This is because every vertex in the tree except the root should have exactly one input. public Class Djikstra() { public Djikstra() { PQ = new PriorityQueue<>(); distTo = new Distance[numVertices]; edgeTo = new Edge[numVertices]; public void doDijkstras(Vertex sourceVertex) { PQ.add(sourceVertex, 0); for(v : allOtherVertices) { PQ.add(v, INFINITY); while (!PQ.isEmpty()) { Vertex p = PQ.removeSmallest(); // Relaxes all edges of p void relax(Vertex p) { for (q : p.neighbors()) { if (distTo[p] + q.edgeWeight < distTo[q]) { distTo[q] = distTo[p] + q.edgeWeight; edgeTo[q] = p; PQ.changePriority(q, distTo[q]); Runtime Analysis each add operation to PQ takes log(V), and perform this V times each removeFirst operation to PQ takes log(V) and perform this V times each change priority operation to PQ takes log(V), perform this at most as many times as there are edges usually, there are more or equal edges compared to the number of vertices.	{"url":"https://cs61b.bencuan.me/algorithms/shortest-paths/dijkstras-algorithm","timestamp":"2024-11-13T11:16:04Z","content_type":"text/html","content_length":"491344","record_id":"<urn:uuid:3c948601-9aa7-4e69-8c34-2a8082e9754a>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00714.warc.gz"}
Algorithm and Hardness for Dynamic Attention Maintenance in Large... Abstract: The attention scheme is one of the key components over all the LLMs, such as BERT, GPT-1, Transformers, GPT-2, 3, 3.5 and 4. Inspired by previous theoretical study of static version of the attention multiplication problem [Zandieh, Han, Daliri, and Karbasi ICML 2023, Alman and Song NeurIPS 2023], we formally define a dynamic version of attention matrix multiplication problem. In each iteration we update one entry in key matrix $K \in \mathbb{R}^{n \times d}$ or value matrix $V \in \mathbb{R}^{n \times d}$. In the query stage, we receive $(i,j) \in [n] \times [d]$ as input, and want to answer $(D^{-1} A V)_{i,j}$, where $A:=\exp(QK^\top) \in \mathbb{R}^{n \times n}$ is a square matrix and $D := \mathrm{diag}(A {\bf 1}_n) \in \mathbb{R}^{n \times n}$ is a diagonal matrix and ${\bf 1}_n$ denotes a length-$n$ vector that all the entries are ones. We provide two results: an algorithm and a conditional lower bound. Inspired by the lazy update idea from [Demetrescu and Italiano FOCS 2000, Sankowski FOCS 2004, Cohen, Lee and Song STOC 2019, Brand SODA 2020], we provide a data-structure that uses $O(n^{\omega(1,1,\tau)-\tau})$ amortized update time, and $O(n^{1+\ tau})$ worst-case query time, where $n^{\omega(1,1,\tau)}$ denotes $\mathrm(n,n,n^\tau)$ with matrix multiplication exponent $\omega$ and $\tau$ denotes a constant in $(0,1]$. We also show that unless the hinted matrix vector multiplication conjecture [Brand, Nanongkai and Saranurak FOCS 2019] is false, there is no algorithm that can use both $O(n^{\omega(1,1,\tau) - \tau- \Omega(1)})$ amortized update time, and $O(n^{1+\tau-\Omega(1)})$ worst query time. Submission Number: 761	{"url":"https://openreview.net/forum?id=opkluZm9gX","timestamp":"2024-11-03T09:20:09Z","content_type":"text/html","content_length":"41125","record_id":"<urn:uuid:6d5bf1e9-dfd3-4149-b0ce-1e9e5a46c218>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00026.warc.gz"}
Question 3 Question 3 The components of stress at a point in a body are as follows: 450 N/mm² tensile, acting horizontally. 150 N/mm² compressive, acting vertically. 100 N/mm² shearing, such that the shear force on the right-hand side of a small cuboid element surrounding the point acts downwards. For the point, determine by calculation or by a Mohr's stress circle: the principal stresses and the direction of the principal planes; (b) the maximum shearing stress and the direction of the planes on which this acts; the normal and shearing stresses on a plane at 45° clockwise from the planes on which the 450 N/mm² tensile stress acts. (20 marks) Fig: 1	{"url":"https://tutorbin.com/questions-and-answers/question-3-the-components-of-stress-at-a-point-in-a-body-are-as-follows-i-450-n-mm-tensile-acting-horizontally-ii-150-n","timestamp":"2024-11-03T23:11:01Z","content_type":"text/html","content_length":"63221","record_id":"<urn:uuid:efc799bc-7204-447d-9e77-e27c5633ccf8>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00661.warc.gz"}
Standard Deviation - Dominating Dividends Understanding Standard Deviation In our journey through the landscape of statistics, it’s crucial to grasp the significance of standard deviation. This measure gives us insight into the spread and variability of data points around the mean, or average, in a data set. Definition and Significance Standard deviation is a statistic that quantifies the amount of variation or dispersion in a set of values. A low standard deviation means that the data points are close to the mean, indicating less variability. On the contrary, a high standard deviation shows that the data points are spread out over a wider range, suggesting higher variability. Standard Deviation Formulas The standard deviation can be calculated using the following formula: Population Standard Deviation: σ = sqrt(Σ((X-μ)^2)/N) Sample Standard Deviation: s = sqrt(Σ((X-ȳ)^2)/(n-1)) • σ represents the population standard deviation • s represents the sample standard deviation • X represents each individual value in the population • μ represents the population mean • ȳ represents the sample mean • N represents the total number of values in the population • n represents the total number of values in the sample The Role of Variance Variance is the square of the standard deviation and represents the average of the squared differences from the Mean. Population variance ((\sigma^2)) and sample variance (s^2) provide the groundwork for standard deviation, illustrating the spread within a data set even more precisely. Calculating Standard Deviation Calculating standard deviation involves a step-by-step process: 1. Find the mean (average) of the data set. 2. Subtract the mean from each data point and square the result (the square of the difference). 3. Sum all the squared results. 4. Divide the sum by the number of data points (population) or by one less than the number of data points (sample). 5. Take the square root of the division. The calculation can be done by hand for smaller data sets, but for larger ones, a calculator or software is recommended. These steps ensure that we accurately articulate the spread and variability, which are integral in fields ranging from finance to science. How to calculate Standard Deviation in Excel In Excel, you can calculate the standard deviation of a set of values using the STDEV.S or STDEV.P functions. For a sample standard deviation, use the STDEV.S function: =STDEV.S(number1, [number2], …) For a population standard deviation, use the STDEV.P function: =STDEV.P(number1, [number2], …) Simply input the range of cells containing the values for which you want to calculate the standard deviation, and the function will return the result. How to calculate Standard Deviation in Google Sheets In Google Sheets, you can calculate the standard deviation of a set of values using the STDEV.S or STDEV.P functions. For a sample standard deviation, use the STDEV.S function: =STDEV.S(number1, number2, …) For a population standard deviation, use the STDEV.P function: =STDEV.P(number1, number2, …) Simply input the range of cells containing the values for which you want to calculate the standard deviation, and the function will return the result. Related Topics	{"url":"https://dominatingdividends.com/standard-deviation/","timestamp":"2024-11-04T08:17:47Z","content_type":"text/html","content_length":"94708","record_id":"<urn:uuid:c7db19fa-917f-4436-92f1-5457a207a729>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00122.warc.gz"}
Implicit computation of minimum-cost feedback-vertex sets for partial scan and other applications The contribution of this paper is an implicit method for computing the minimum cost feedback vertex set for a graph. For an arbitrary graph, we efficiently derive a Boolean function whose satisfying assignments directly correspond to feedback vertex sets of the graph. Importantly, cycles in the graph are never explicitly enumerated, but rather, are captured implicitly in this Boolean function. This function is then used to determine the minimum cost feedback vertex set. Even though computing the minimum cost satisfying assignment for a Boolean function remains an NP-hard problem, we can exploit the advances made in the area of Boolean function representation in logic synthesis to tackle this problem efficiently in practice for even reasonably large sized graphs. The algorithm has obvious application in flip-flop selection for partial scan. Our algorithm was the first to obtain the MFVS solutions for many benchmark circuits. All Science Journal Classification (ASJC) codes • Hardware and Architecture • Control and Systems Engineering Dive into the research topics of 'Implicit computation of minimum-cost feedback-vertex sets for partial scan and other applications'. Together they form a unique fingerprint.	{"url":"https://collaborate.princeton.edu/en/publications/implicit-computation-of-minimum-cost-feedback-vertex-sets-for-par","timestamp":"2024-11-09T03:37:42Z","content_type":"text/html","content_length":"48951","record_id":"<urn:uuid:e9cc0566-6605-420b-9fbc-c4208ec78f76>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00188.warc.gz"}
Indian mathematician wins the prestigious Gábor Szegö Prize 2021 Writer/Editor: Shivangi Vasudev Bhatt Photo: Media and Communication, IIT Gandhinagar In an incredibly proud moment, Prof Atul Dixit, Assistant Professor of Mathematics at IITGN, has become the first Indian mathematician to win the prestigious Gábor Szegö Prize 2021 awarded by the Society of Industrial and Applied Mathematics (SIAM), USA. The SIAM Activity Group on Orthogonal Polynomials and Special Functions (SIAG/OPSF) awards the Gábor Szegö Prize every two years to one early-career researcher for outstanding research contributions in the area of orthogonal polynomials and special functions. It is for the first time that this prize has been awarded to an Indian mathematician. Prof Atul Dixit has been selected for this award for his “impressive scientific work in solving problems related to number theory using special functions, in particular related to the work of Ramanujan.” The prize includes a certificate containing the citation. The award was originally supposed to be presented at the 2021 International Symposium on Orthogonal Polynomials, Special Functions, and Applications (OPSFA16). However, due to the current global scenario of the COVID-19 pandemic, the event has been postponed from July 2021 to July 2022. As a part of the award, Prof Atul Dixit will also be invited at the OPSFA16, to be held at the Centre de Recherches Mathematiques (CRM), Universite de Montreal, Canada, to deliver a plenary lecture at the prestigious event. Prof Atul Dixit’s research in mathematics is at the interface of analytic number theory and special functions. He shares that his work in number theory has led him to discover new interesting special functions such as generalised modified Bessel and Hurwitz zeta functions. Likewise, his work on special functions has frequently had implications in number theory, such as the one on generalised Lambert series or the Voronoï summation formulas. Prof Dixit’s research work has been largely impacted by Srinivasa Ramanujan, who is the main source of inspiration for him. I am deeply humbled to know that the SIAM Activity Group on Orthogonal Polynomials and Special Functions (SIAG/OPSF) has chosen me for the 2021 Gábor Szegö prize. I sincerely thank the SIAM prize committee and SIAM for this recognition of my work. I also thank my recommendation letter writers (Professors Bruce Berndt and Nico Temme) for having trust in my work. Receiving this prize also puts more responsibility on my shoulders to do better research than before, and I hope to live up to the expectations put forth on me by SIAM and other well-wishers. Atul Dixit About Orthogonal Polynomials and Special Functions: Special Functions: The high-school or college curriculum covers various important functions of Mathematics, such as trigonometric functions, exponential function, logarithm, hyperbolic functions etc. These are called ‘elementary functions’. On the other hand, special functions are ‘non-elementary’ functions, which are equally useful in comparison to the elementary functions, and have numerous applications in various branches of Engineering and Physics. In fact, Professor Richard Askey used to say that ‘special functions’ should actually be called ‘useful functions’. Many special functions have their origins in physics and have emerged from the study of various ordinary and partial differential equations. Some examples of special functions include Gamma function, Bessel functions, Riemann zeta function, Jacobi theta function etc. Orthogonal Polynomials: Orthogonal polynomials form an important subclass of polynomials which plays an instrumental role in mathematical physics, approximation theory etc. Consider the linear space of polynomials of the real parameter x with real coefficients. If f and g are two elements of this space, we define the inner product of f and g by means of an integral whose integrand consists of a suitable weight function. This inner product is always zero for unequal f and g, hence the nomenclature ‘orthogonal’. Some of the famous orthogonal polynomials are Jacobi polynomials, Hermite polynomials, Laguerre polynomials, Gegenbauer polynomials etc. This news was covered by some of the leading Indian newspapers and media agencies. Click on the links below to read more.	{"url":"https://news.iitgn.ac.in/indian-mathematician-wins-the-prestigious-gabor-szego-prize-2021/","timestamp":"2024-11-13T09:24:15Z","content_type":"text/html","content_length":"61975","record_id":"<urn:uuid:4e03e2c7-1679-4414-b77a-f42a7c5136cf>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00455.warc.gz"}
Re: Numeric key collision related bug in Lua 5.3 [Date Prev][Date Next][Thread Prev][Thread Next] [Date Index] [Thread Index] • Subject: Re: Numeric key collision related bug in Lua 5.3 • From: Dirk Laurie <dirk.laurie@...> • Date: Tue, 21 Apr 2015 18:36:33 +0200 2015-04-21 17:32 GMT+02:00 Egor Skriptunoff <egor.skriptunoff@gmail.com>: > Hi! > An interesting bug has been found in Lua 5.3 > t = {[(1<<63)-333] = 0} > key = next(t) + 0.0 > t[key] = "Lua is great!" > print(t[key]) --> Lua is great! > t[0] = "Are you sure?" > print(t[key]) --> nil > Why Lua is not great anymore? The manual says: The indexing of tables follows the definition of raw equality in the language. The expressions a[i] and a[j] denote the same table element if and only if i and j are raw equal (that is, equal without metamethods). In particular, floats with integral values are equal to their respective integers (e.g., 1.0 == 1). To avoid ambiguities, any float with integral value used as a key is converted to its respective integer. For instance, if you write a[2.0] = true, the actual key inserted into the table will be the integer 2. (On the other hand, 2 and "2" are different Lua values and therefore denote different table Let origkey = 1<<63)-333, which is a very large integer, but slightly smaller than math.maxinteger. "key" is a float with integral value, but that integral value is not origkey, but math.maxinteger. In a floating-point comparison, it tests equal to origkey, When t[key] is assigned, "key" is not considered to be a new index, and the value of t[origkey] is replaced. When t[0] is assigned, the hash part of the table is reorganized. t[origkey] is still "Lua is great!", but is no longer found when asking for t[key].	{"url":"http://lua-users.org/lists/lua-l/2015-04/msg00303.html","timestamp":"2024-11-12T23:29:58Z","content_type":"text/html","content_length":"5894","record_id":"<urn:uuid:55ac84a4-c263-4cc9-ac85-f7c191da58b2>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00663.warc.gz"}
Janos Bolyai and Fermat's Little Theorem First, we solve the problem from last time: One can also ascertain in advance which n-gons are “constructible” using only rule and straight edge, and which are not. It turns out such regular polygons are only possible provided the extension field has degree of 2. (I.e. the regular n-gon is only constructible if (Z + 1/Z) = 2 cos(2π/n). Prove this is the case, if the Z roots are determined by: cos(2π/n) + isin(2π/n), and the 1/Z roots are determined by: cos(2π/n) - isin(2π/n). The Z roots are determined by: cos(2π/n) + isin(2π/n), and The 1/Z roots are determined by: cos(2π/n) - isin(2π/n). Let: Z = cos(2π/n) + isin(2π/n) and 1/Z = cos(2π/n) - isin(2π/n) [cos(2π/n) + isin(2π/n)] [cos(2π/n) - isin(2π/n)] = cos2(2π/n) + sin2(2π/n)] = 1 (and recall by trig identity: cos2(theta) + sin2(theta) = 1 Thus: (Z) (1/Z ) = 1 The preceding shows the angle is constructible, and the regular n-gon is only if the former holds. And also: (Z) + (1/Z) = (cos(2π/n) + isin(2π/n) ) + cos(2π/n) - isin(2π/n) = 2 cos(2π/n) Now on to Janos Bolyai! Janos Bolyai is generally known for his contribution to non-Euclidean geometry and specifically the Bolyai-Lobachevsky geometry shown in the accompanying graphic. This geometry is held to correlate with a negatively curved (k= -1) space-time in general relativity (in contrast to the positive (k= +1) curvature of the sphere, attributed to Berhard Riemann). Less well known are Boyai’s contributions to other areas of higher math, especially theorems to do with prime numbers. One of the most important of which is Fermat’s Little Theorem, and its inverse. Fermat’s Little Theorem states that if p is a prime number and a is an integer not divisible by either p or q then the difference a^(p-1) – 1 is divisible by p, which is also written: a^p-1 = 1(mod p) The inverse of the above is that if a^p-1 = 1(mod p) holds, it does not necessarily follow that p is a prime. Bolyai attempted to prove this inverse theorem but after a number of attempts, he gave up, concluding it was impossible and so the inverse of Fermat’s Little Theorem doesn’t hold in general. Though he didn’t find a general prime formula he did discover the first pseudo-prime. Later, Bolyai examined under what conditions the congruence: a^pq-1 = 1 (mod pq) is satisfied, where p and q are primes and a is an integer divisible by neither p or q. Bolyai reasoned that (according to Fermat’s Little Theorem): a^p-1 = 1 (mod p) and a^q-1 = 1 (mod q) Then if one raised both sides of the first congruence to the power of (q-1) and both sides of the second congruence to the power (p-1), one would obtain: a^(p-1)(q-1)= 1 (mod p) and a^(p-1)(q-1) = 1(mod q) a^(p-1)(q-1) = 1(mod pq) Bolyai then observed that if the congruence: a^(p+q-2) = 1 mod(pq) = a^(p-1)* a^(q-1) = mod(pq) is true, then multiplying the two earlier expressions one arrives at the desired congruence: a^pq-1 = 1 mod(pq) Not content with this basic step, Bolyai set out to obtain the conditions which assure the validity of the preceding. Since: a^p-1 = 1 (mod p) and a^q-1 = 1 (mod q) then there must exist integers h and k such that: a^p-1 = 1 + hp, and a^q-1 = 1 + kq. Thus, the condition for the validity of the congruence is: hp + hq = (a^p-1 – 1) + (a^q-1 – 1) = 0 (mod pq) The above form is satisfied if p is a divisor of k and q is a divisor of h. According to Bolyai, this meant : a^pq-1 = 1(mod pq) is true of primes p and q for which: [a^p-1 – 1]/ pq and [a^q-1 -1]/pq are integers, in which case: (a^p-1 -1)/q and (a^q-1 -1)/p are also integers. for the simple case for which a = 2, substitute primes p and q to obtain integers that satisfy the condition for congruence.	{"url":"https://brane-space.blogspot.com/2011/02/janos-bolyai-and-fermats-little-theorem.html","timestamp":"2024-11-01T18:59:35Z","content_type":"text/html","content_length":"117217","record_id":"<urn:uuid:e39bdc7e-b321-40a7-9040-dc0ac2a67079>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00301.warc.gz"}
Module Specifications. Current Academic Year 2024 - 2025 All Module information is indicative, and this portal is an interim interface pending the full upgrade of Coursebuilder and subsequent integration to the new DCU Student Information System (DCU Key). As such, this is a point in time view of data which will be refreshed periodically. Some fields/data may not yet be available pending the completion of the full Coursebuilder upgrade and integration project. We will post status updates as they become available. Thank you for your patience and understanding. Date posted: September 2024 Module Title Probability 1 Module Code MS117 (ITS) / MTH1018 (Banner) Faculty Science & Health School Mathematical Sciences Module Co-ordinator Martin Venker Module Teachers - NFQ level 6 Credit Rating 5 Pre-requisite Not Available Co-requisite Not Available Compatibles Not Available Incompatibles Not Available Repeat examination This module is in resit category 3: no resit of the continuous assessment is available. MS117 aims to introduce the basic concepts of probability theory through a mixture of lectures and problem solving based tutorials. The module will give students a working knowledge of the main techniques of elementary probability and build a solid foundation for learning more advanced topics in probability and statistics. Learning Outcomes 1. Define elementary concepts of probability and state the main theorems. 2. Use summation, integration, counting techniques and approximations to assign probabilities to events or compute distribution functions. 3. Compute and apply conditional probabilities. 4. Derive the basic properties of common discrete, continuous and mixed distributions. 5. Compute expectation, median and variance of given distributions and prove their theoretical properties. Workload Full-time hours per semester Type Hours Description Lecture 36 Presentation of course material Tutorial 24 Working on solving exercise sheets. Independent Study 65 Revising coursework, solving tutorials and completing assignments. Total Workload: 125 All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http:/ Indicative Content and Learning Activities Principles of Modelling ChanceProbability spaces, construction of probability measures via densities, distribution functionsConditional Probabilities and IndependenceConditional probabilities, law of total probability, Bayes theorem, independenceStandard Models In ProbabilityCombinatorics, random variables, common distributions in urn models: multinomial, binomial, (multivariate) hypergeometric, discrete and continuous waiting time distributionsCharacteristics of Random VariablesExpectation, median, variance, standard deviationApproximations of the Binomial DistributionPoisson approximation, normal approximation Assessment Breakdown Continuous Assessment 20% Examination Weight 80% Course Work Breakdown Type Description % of total Assessment Date In Class Test n/a 20% As required Indicative Reading List • Hans-Otto Georgii: 2008, Stochastics, Walter de Gruyter, 9783110206760 • Geoffrey Grimmett and David Stirzaker: 2001, Probability and Random Processes, 3rd edition, Oxford University Press, Oxford, • Kai Lai Chung: 2003, Elementary Probability Theory with Stochastic Processes and an Introduction to Mathematical Finance, 4th edition, Springer, New York, • Richard Durrett: 1994, The Essentials of Probability, Duxbury Press, Belmont, • William Feller: 1971, An Introduction to Probability and its Applications, 3rd edition, Wiley, New York, • A. N. Shiryaev: 1996, Probability, 2nd edition, 1. chapter, Springer, New York, • Henk Tijms: 2007, Understanding Probability – Chance Rules in Everyday Life, 2nd edition, Cambridge University Press, Cambridge, • David Williams: 2001, Weighing the Odds – A Course in Probability and Statistics, Cambridge University Press, Cambridge, • Sheldon M. Ross: 2010, A first course in probability, 8th edition, Pearson, Englewood Cliffs, • Peter L. Bernstein: 1996, Against the Gods: The Remarkable Story of Risk, John Wiley & Sons, New York, Other Resources	{"url":"https://modspec.dcu.ie/registry/module_contents.php?function=2&subcode=MS117","timestamp":"2024-11-02T09:09:27Z","content_type":"application/xhtml+xml","content_length":"48493","record_id":"<urn:uuid:9bccd06c-6734-4520-8817-d45195419a8b>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00280.warc.gz"}
Goldbach's Conjecture \| Let's prove Goldbach! Goldbach’s Conjecture Even numbers expressed as sums of prime numbers: equal numbers are highlighted with equal colors The proof of Goldbach’s Conjecture is one of the biggest still unsolved problems regarding prime numbers. Originally expressed by the mathematician Christian Goldbach, from whom the Conjecture takes its name, it was cited by Euler for the first time in 1742, in the form in which we know it today: Every even number greater than 2 can be expressed as a sum of two prime numbers. Besides this version of the Conjecture, another one exists, the so-called weak Goldbach’s Conjecture, which states that every odd number greater than 5 can be written as the sum of three primes (for example 7 = 2 + 2 + 3, 9 = 3 + 3 + 3, 11, = 3 + 3 + 5, …). It’s called “weak” Conjecture because, if Goldbach’s Conjecture (also known as “strong” Goldbach’s Conjecture, in order to distinguish it from the other one) was proved, the former would be a simple consequence. Given an odd number $d$ greater than 5, $d - 3$ is an even number greater than 2. So, if the strong Goldbach’s conjecture was true, we would have $d - 3 = p + q$, where $p$ and $q$ are two primes, then $d = 3 + p + q$, i.e. $d$ would be the sum of three primes. On the other hand, while it’s true that Goldbach’s strong Conjecture implies the weak one, the converse is not true. In fact, the weak Conjecture has been proved by the Peruvian mathematician Harald Andrés Helfgott in 2013, but, nevertheless, the strong Conjecture continues to resist against all attempts to prove it. But numbers look to speak for themselves… The empirical evidence in favour of the conjecture is overwhelming: not only it looks that every even number greater than 2 can be expressed as a sum of two prime numbers, but, very often, different expressions of this kind exist for the same number. This can be seen even starting from the smallest numbers: 4 = 2 + 2 6 = 3 + 3 8 = 3 + 5 10 = 3 + 7 = 5 + 5 12 = 5 + 7 14 = 3 + 11 = 7 + 7 16 = 3 + 13 = 5 + 11 In particular, using our Goldbach pairs viewer, you can see that the number of different ways in which an even number can be expressed as a sum of two primes tends to grow as the considered number increases. This growth is shown in the following graph, the so-called Goldbach’s comet: Number of different ways (y axis) in which an even number (x axis) can be expressed as a sum of two prime numbers It’s not so as easy as it seems In spite of the empirical evidence and the simplicity of its statement, the Conjecture has been resisting to all proof attempts since almost three centuries. The hardest part to be faced in the proof is surely that the set of even numbers starting from 4 must be completely covered. That is, as the statement itself says, all even numbers greater than 2 must satisfy the relationship. Many proof attempts come to prove only that some even numbers satisfy it, for example the ones with some specific features or with a given algebraic form: as a result, the outcome of the proof isn’t the Conjecture, but something else. For example, all even numbers of the form $2p$, with $p$ prime, are the sum of two primes (in this case, of $p$ with itself), but, of course, not all even numbers respect this form. However, we can note that the set made of even numbers of type $2p$ is infinite, because primes are infinite, but this isn’t enough: just the fact that a set of even numbers is infinite does not imply that it contains all even numbers starting from 4. Even if we sum two odd prime numbers, we’ll always get an even number (because, generally, the sum of two odd numbers is even), in this case an even number greater or equal to 6, because the smallest sum of odd prime numbers is $6 = 3 + 3$. So we’ll obtain an infinite set of even numbers greater or equal to 6, but just the fact that this set is infinite does not mean that it covers all even numbers starting from 6. An additional aspect which you should consider, when you try to prove the Conjecture, is the difficulty of starting from arguments based on intuition and transform them into real proofs. For example, we know that: • as $n$ increases, the number of possible ways to write $2n$ as a sum of two positive integers, regardless of the order of the addends, is $n$, so it increases linearly; • as a positive integer number $x$ increases, by the Prime Number Theorem the number of primes less than or equal to $x$ grows approximately like the function $x / \log x$. These two aspects let us compute approximately, given a sum of the kind $2n = p + q$, the number of the cases in which $p$ is prime (regardless of wether $q$ is prime or not). In particular, by the Prime Number Theorem the number of such cases increases as $n$ increases. So the question becomes: among the cases in which $p$ is prime, which increase with $n$, is there always one in which also $q$ is prime? This question is equivalent to ask wether the Conjecture is true or not. You may try to answer it affirmatively, starting from intuitive arguments. For example, the ratio between the number of cases in which at least $p$ is prime and those ones in which both $p$ and $q$ are prime, as $n$ increases, may become closer and closer to the the ratio between $2n$ and the number of prime numbers less than $2n$, where the latter ratio in turn can be estimated by the previously cited Prime Number Theorem. The problem of arguments of this kind is that they are often based on hypothesis, which seem plausible but are not proved (in this example, the hypothesis that the ratio is the same, which may not be true, though it may seem to be), so they are not valid proofs, even though they make use of mathematical formalisms and known theorems. Intuition can be a good guide in the proof of a theorem, but it must never be essential for the proof itself: a proof can contain intuitive arguments for better clarity, but not as an essential part of the argument, which instead must be rigorous. A further possibility for trying to prove Goldbach’s Conjecture is to bring it back to a specific case of a known theorem, or of another conjecture. In the latter case, obviously, it would become necessary to prove that conjecture, so the initial problem would not be solved, but it would be only transformed into another one. As an example, we can cite Schinzel’s Hypothesis $H_N$ Let $f_1(t), f_2(t), \ldots, f_r(t)$ be irreducible polynomials with integer coefficients, and such that the coefficient of the maximum degree term is positive. Let $g(t)$ be a polynomial with integer coefficients. Suppose that there are infinitely many positive integers $N$ such that $N - g(t)$ is irreducible, and that there is no prime number $p$ such that, for every $t$, $p$ is divisor of the value of the product $f_1(t), \ldots, f_r(t) \cdot (N - g(t))$. If $N$ is large enough, then there exists an integer $n$ such that $N - g (n)$ is prime, and $f_i(n )$ is prime for all $i = 1, \ldots, r$. By imposing the condition that $N$ is even, setting $r = 1$ and $f_1 (t) = g (t) = t$, we’ll obtain this statement: Let $t$ be an irreducible polynomial with integer coefficients, and such that the coefficient of the term of greatest degree is positive. Let $t$ be a polynomial with integer coefficients. Suppose there are infinitely many positive even integers $N$ such that $N - t$ is irreducible, and that there is no prime number $p$ such that, for any $t$, $p$ is divisor of the value of the product $t \ cdot (N - t)$. If $N$ is large enough, then $N - t$ is prime, and $t$ is prime. This statement can in turn be simplified, because: • The polynomial $t$ has coefficient 1 and its degree is 1, therefore the first two statements about $t$ are obviously true; • The degree of $N - t$ is 1, so it is certainly irreducible; • The fact that there is no prime number $p$ such that, for every $t$, $p$ is a divisor of the value of the product $t \cdot (N - t)$ can be proved easily, for example by contradiction. Therefore, by eliminating the related statements, we’ll obtain: Any large enough even number is the sum of two prime numbers. The final result is therefore a statement which is almost identical to Goldbach’s Conjecture, different from it only for the precondition that the starting even number is large enough. Despite this difference, this statement would still be a good starting point, because it would only remain to prove that the relationship is also valid beyond that precondition, that is for a finite number of Intermediate results Several mathematicians proved some theorems which are weaker versions of the Conjecture. They can be grouped into two categories, according to what aspect of their statement differs from the • The theorems that, instead of considering the sum of two primes (like in Goldbach’s Conjecture), consider a greater number of primes, or the sum of a prime and a semiprime, that is the product of two primes. • The ones that, instead of being true for every even number greater than 2 (like in Goldbach’s Conjecture), are true for “almost” even numbers greater than 2. Some theorems can be classified into both categories, as they state that “almost” all even numbers greater than two can be expressed as a somehow more complex sum than one involving two primes. Regarding the correlation between even numbers and prime number pairs, some theorems from number theory can help stating something specific. For example, by using Dirichlet’s Theorem, we can state that infinite even numbers exist having 2 as the last digit, which are given by the sum of two primes. The proof steps are the following: • The last digit of the sum is 2, so the two addends can only have, as their last digit, respectively $(0, 2), (1, 1), (2, 0), (3, 9), (4, 8), (5, 7), (6, 6), (7, 5), (8, 4), (9, 3)$; due to commutative property, we can just consider $(0, 2), (1, 1), (3, 9), (4, 8), (5, 7), (6, 6)$. • Dirichlet’s Theorem states that there exist infinite prime numbers of the type $ax + b$, with $a$ and $b$ coprime integers, and $x$ integer. Then, for $a = 10$ and $0 \leq b \leq 9$, there exist infinite prime numbers of the type $10x + b$, i.e. with $b$ as their last digit. But, in order to be able to apply the Theorem, $b$ must be coprime with 10, so it must be equal to 1, 3, 7 or 9. • In order to be able to apply Dirichlet’s Theorem to both addends (even if, actually, applying it to only one of them would be sufficient), we exclude from the previous possibilities the ones in which one of the two digits is not coprime with 10: the remaining possibilities are $(1, 1)$ and $(3, 9)$. • Considering the case $(1, 1)$, due to Dirichlet’s Theorem, the set $P_1 := \{\text{primes having 1 as last digit}\}$ is infinite. Then, by summing two primes $p, q \in P_1$, since there are infinite possibilities for both of them, there are infinite possibilities for the sum, which will be an even number with 2 as its last digit. • Considering instead the case $(3, 9)$, due to Dirichlet’s Theorem, the sets $P_3 := \{\text{primes having 3 as last digit}\}$ and $P_9 := \{\text{primes having 9 as last digit}\}$ are infinite. Then, by summing a prime $p \in P_1$ with a prime $q \in P_9$, since there are infinite possibilities for both of them, there are infinite possibilities for the sum, which will be again an even number with 2 as its last digit. The fact that several weaker versions of the Conjecture have been proved, without ever arriving to prove the original statement, makes us think that behind the Goldbach’s Conjecture there may be some deep mechanism that has yet to be understood, and which could require new proof techniques. For this reason we are sketching out the proof on the basis of a new theory, specifically built for studying the problem stated by the Conjecture: dashed line theory. 4 Replies to “Goldbach’s Conjecture” 1. I would like experts to expose the errors in my simple proof, “The Stepladder Proof of the Goldbach Conjecture.” I posted it on Academia.edu. In short, prime,prime pairs = total pairs – prime,composite pairs – composite,composite pairs. Subject to known adjustments, every even number, n, can be expressed as n/4 unique and mandatory even number pairs. Each pair sum = n. Primes are embedded. (prime + 1) + (prime + 1) = n. For example, 200 has 50 unique and mandatory even number pairs. My excess pairs algorithm isolates 37 pairs with at least one composite. Prime,prime pairs = 50 – 37 = 13 exactly. The power and simplicity is in n/4. Also email to [the email address has been omitted for avoiding spam] Thanks for questions and comments. Gregory Mazur 1. Dear Gregory, thanks for your comment. We are reading your proof and we’ll let you know by email. In the meantime, you may be interested in our proof strategies. 2. goldbach sanısının tek ispat yöntemi ortalama asal çift sayısını bulmak ile mümkündür. bu ortalama asal çiftleri bulmak için kullanılan formülde eğer asal çiftler gittikçe sıfıra yaklaşıyorsa sonuçta mutlaka asal çift oluşturmayan bir çift sayı vardır diyebiliriz eğer her olasılıkta bu formülümüz doğruya yakın bir değer veriyor ve formül sonucu asla sıfıra yaklaşmıyorsa her çift sayı için mutlaka bir en az birtane asal çift vardır deriz Collatz sanısı çözüldü. çözümü anlamak ve sonsuzluktaki resmi çekmek için collatz sayıları’nın “Çözümsüzlükteki düzen” yazısı’nı mutlaka okuyun yazısı linkte Aşağıdaki tüm problemler çözüldü ispatlandı. collatz sanısı çözümsüz olarak ispatlandı. goldbach sanısı ispatlandı basit matematik ile goldbach sanısı ispatlandı açıklama dosyası linkte youtube video link ikiz asal sayılar ispatlandı aralarındaki fark 2n olan tüm ikiz asal sayılar ispatlandı aralarındaki fark 2 olan ikiz asal sayılar cetveli var İrfan Aydoğan 1. Please note: we exceptionally accepted this comment written in Turkish, because the author explained to us that he doesn’t know English enough. We think that language should not be a barrier for research, so we decided to publish this comment. We published it exactly as it was written, for our readers who know Turkish. For the other readers, here is an English translation adapted from the one given by Google translate: The only proof method of Goldbach’s conjecture is to find the average number of prime pairs. In the formula used to find these average prime pairs, if the prime pairs are getting closer and closer to zero, we can say that there is an even number that does not necessarily form a prime pair. Collatz conjecture has been resolved. To understand the solution and take the picture to infinity, be sure to read the “Order in no solution” article about the Collatz numbers at the link All the following problems have been solved. Collatz conjecture proved unsolved. Goldbach’s conjecture proved With simple math, Goldbach’s conjecture is proved. youtube video link twin prime numbers proved There is a ruler of twin prime numbers the difference of which is 2 İrfan Aydoğan	{"url":"http://www.dimostriamogoldbach.it/en/home-goldbach-conjecture/?doing_wp_cron=1707803883.7033929824829101562500","timestamp":"2024-11-05T15:22:55Z","content_type":"text/html","content_length":"290662","record_id":"<urn:uuid:18a92362-590a-4460-ac95-ac7f3449a3aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00889.warc.gz"}
Machine Hearing Techniques for enabling machines to interpret and understand audio signals, often used in speech recognition and audio analysis. Richard E. Turner, 2010. Gatsby Computational Neuroscience Unit, UCL, It is important to understand the rich structure of natural sounds in order to solve important tasks, like automatic speech recognition, and to understand auditory processing in the brain. This thesis takes a step in this direction by characterising the statistics of simple natural sounds. We focus on the statistics because perception often appears to depend on them, rather than on the raw waveform. For example the perception of auditory textures, like running water, wind, fire and rain, depends on summary-statistics, like the rate of falling rain droplets, rather than on the exact details of the physical source. In order to analyse the statistics of sounds accurately it is necessary to improve a number of traditional signal processing methods, including those for amplitude demodulation, time-frequency analysis, and sub-band demodulation. These estimation tasks are ill-posed and therefore it is natural to treat them as Bayesian inference problems. The new probabilistic versions of these methods have several advantages. For example, they perform more accurately on natural signals and are more robust to noise, they can also fill-in missing sections of data, and provide error-bars. Furthermore, free-parameters can be learned from the signal. Using these new algorithms we demonstrate that the energy, sparsity, modulation depth and modulation time-scale in each sub-band of a signal are critical statistics, together with the dependencies between the sub-band modulators. In order to validate this claim, a model containing co-modulated coloured noise carriers is shown to be capable of generating a range of realistic sounding auditory textures. Finally, we explored the connection between the statistics of natural sounds and perception. We demonstrate that inference in the model for auditory textures qualitatively replicates the primitive grouping rules that listeners use to understand simple acoustic scenes. This suggests that the auditory system is optimised for the statistics of natural sounds. Richard E. Turner, M Sahani, 2007. (In 7th International Conference on Independent Component Analysis and Signal Separation). Auditory scene analysis is extremely challenging. One approach, perhaps that adopted by the brain, is to shape useful representations of sounds on prior knowledge about their statistical structure. For example, sounds with harmonic sections are common and so time-frequency representations are efficient. Most current representations concentrate on the shorter components. Here, we propose representations for structures on longer time-scales, like the phonemes and sentences of speech. We decompose a sound into a product of processes, each with its own characteristic time-scale. This demodulation cascade relates to classical amplitude demodulation, but traditional algorithms fail to realise the representation fully. A new approach, probabilistic amplitude demodulation, is shown to out-perform the established methods, and to easily extend to representation of a full demodulation cascade. Richard E. Turner, Maneesh Sahani, 2008. (In nips20). Edited by J. C. Platt, D. Koller, Y. Singer, S. Roweis. mit. Natural sounds are structured on many time-scales. A typical segment of speech, for example, contains features that span four orders of magnitude: Sentences (∼1s); phonemes (∼10−1 s); glottal pulses (∼ 10−2s); and formants (∼ 10−3s). The auditory system uses information from each of these time-scales to solve complicated tasks such as auditory scene analysis [1]. One route toward understanding how auditory processing accomplishes this analysis is to build neuroscience-inspired algorithms which solve similar tasks and to compare the properties of these algorithms with properties of auditory processing. There is however a discord: Current machine-audition algorithms largely concentrate on the shorter time-scale structures in sounds, and the longer structures are ignored. The reason for this is two-fold. Firstly, it is a difficult technical problem to construct an algorithm that utilises both sorts of information. Secondly, it is computationally demanding to simultaneously process data both at high resolution (to extract short temporal information) and for long duration (to extract long temporal information). The contribution of this work is to develop a new statistical model for natural sounds that captures structure across a wide range of time-scales, and to provide efficient learning and inference algorithms. We demonstrate the success of this approach on a missing data task. Richard E. Turner, Maneesh Sahani, 2010. (In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP)). Amplitude demodulation is an ill-posed problem and so it is natural to treat it from a Bayesian viewpoint, inferring the most likely carrier and envelope under probabilistic constraints. One such treatment is Probabilistic Amplitude Demodulation (PAD), which, whilst computationally more intensive than traditional approaches, offers several advantages. Here we provide methods for estimating the uncertainty in the PAD-derived envelopes and carriers, and for learning free-parameters like the time-scale of the envelope. We show how the probabilistic approach can naturally handle noisy and missing data. Finally, we indicate how to extend the model to signals which contain multiple modulators and carriers. Richard E. Turner, Maneesh Sahani, 2011. (Transactions on Audio, Speech and Language Processing). Demodulation is an ill-posed problem whenever both carrier and envelope signals are broadband and unknown. Here, we approach this problem using the methods of probabilistic inference. The new approach, called Probabilistic Amplitude Demodulation (PAD), is computationally challenging but improves on existing methods in a number of ways. By contrast to previous approaches to demodulation, it satisfies five key desiderata: PAD has soft constraints because it is probabilistic; PAD is able to automatically adjust to the signal because it learns parameters; PAD is user-steerable because the solution can be shaped by user-specific prior information; PAD is robust to broad-band noise because this is modelled explicitly; and PAD’s solution is self-consistent, empirically satisfying a Carrier Identity property. Furthermore, the probabilistic view naturally encompasses noise and uncertainty, allowing PAD to cope with missing data and return error bars on carrier and envelope estimates. Finally, we show that when PAD is applied to a bandpass-filtered signal, the stop-band energy of the inferred carrier is minimal, making PAD well-suited to sub-band demodulation. Richard E. Turner, Maneesh Sahani, 2011. (In Advances in Neural Information Processing Systems 24). The MIT Press. A number of recent scientific and engineering problems require signals to be decomposed into a product of a slowly varying positive envelope and a quickly varying carrier whose instantaneous frequency also varies slowly over time. Although signal processing provides algorithms for so-called amplitude- and frequency-demodulation (AFD), there are well known problems with all of the existing methods. Motivated by the fact that AFD is ill-posed, we approach the problem using probabilistic inference. The new approach, called probabilistic amplitude and frequency demodulation (PAFD), models instantaneous frequency using an auto-regressive generalization of the von Mises distribution, and the envelopes using Gaussian auto-regressive dynamics with a positivity constraint. A novel form of expectation propagation is used for inference. We demonstrate that although PAFD is computationally demanding, it outperforms previous approaches on synthetic and real signals in clean, noisy and missing data settings. Richard E. Turner, Maneesh Sahani, march 2012. (In Acoustics, Speech and Signal Processing (ICASSP), 2012 IEEE International Conference on). DOI: 10.1109/ICASSP.2012.6288343. ISSN: 1520-6149. There are many methods for decomposing signals into a sum of amplitude and frequency modulated sinusoids. In this paper we take a new estimation based approach. Identifying the problem as ill-posed, we show how to regularize the solution by imposing soft constraints on the amplitude and phase variables of the sinusoids. Estimation proceeds using a version of Kalman smoothing. We evaluate the method on synthetic and natural, clean and noisy signals, showing that it outperforms previous decompositions, but at a higher computational cost. 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Gravity Method Questions and Answers - Sanfoundry Irrigation Engineering Questions and Answers – Gravity Method This set of Irrigation Engineering Multiple Choice Questions & Answers (MCQs) focuses on “Gravity Method”. 1. When the reservoir is empty, the single force acting on it is the self-weight of the dam which acts at a distance of ____________ a) B/2 from the heel b) B/6 from the heel c) B/3 from the heel d) B/4 from the heel View Answer Answer: c Explanation: The only single force on the dam when the reservoir is empty is the self-weight of the dam acting at a distance of B/3 from the heel. It provides maximum possible stabilizing moment about the toe without causing tension. 2. When the reservoir is empty, the maximum vertical stress equal to ________________ a) At heel = 2W/B and at toe = 0 b) At heel = 0 and at toe = 2W/B c) At heel = toe = zero d) At heel = toe = 2W/B View Answer Answer: a Explanation: The vertical stress distribution at the base when the reservoir is empty is given as – P[max/min] = V/B [1 + 6e/B] and V/B [1 – 6e/B] where e = B/6 and V = total vertical force = weight W P[max] = 2W/B and P[min] = 0. The maximum vertical stress at the heel is equal to 2W/B and at the toe is zero. 3. The two-dimensional stability analysis of gravity dams proves better for U-shaped valleys than for V-shaped valleys. a) True b) False View Answer Answer: a Explanation: The transverse joints in the dam body are generally not grouted in U-shaped valleys but are keyed together in V-shaped valleys. In V-shaped valleys, the entire length of the dam acts monolithically as a single body. The assumption that the dam is considered to made up of a number of cantilevers of unit width each may involve errors here. 4. Calculate the value of minimum base width for an elementary triangular concrete gravity dam supporting 72 m height of reservoir water and full uplift? (Take specific gravity of concrete as 2.4 and coefficient of friction as 0.7) a) 36.28 m b) 39.77 m c) 51.5 m d) 73.5 m View Answer Answer: d Explanation: Using formula – Case 1: B = H / (S[c] – c)^1/2 (For full uplift c = 1 and specific gravity of concrete = 2.4 ) = 72/ (2.4 – 1)^1/2 = 60.85 m Case 2: B = H/μ (S – 1) where μ = coefficient of friction taken as 0.7 B = 72 / 0.7 x 1.4 = 73.46 m The highest among the two base width value is to be selected i.e. B = 73.46 m. 5. For usual values of permissible compressive stress and specific gravity of concrete, a high concrete gravity is the one whose height exceeds ______________ a) 48 m b) 70 m c) 88 m d) 98 m View Answer Answer: c Explanation: The limiting height is – H[max] = f / (S[c] + 1) ϒ[w] Permissible strength of concrete = 3000 KN/m^2, S[c] = specific gravity of concrete = 2.4 H[max] = 3000/[(2.4 + 1) x 9.81] = 89.9 m. 6. For triangular dam section of height H for just no tension under the action of water pressure, self-weight and uplift pressure, the minimum base width required is _____________ a) H / (S-1) b) H / S^1/2 c) H / (S – 1)^-1 d) H / (S-1)^1/2 View Answer Answer: d Explanation: The minimum base width (B) of a gravity dam having an elementary profile – B = H / (S – 1)^-1 where S is specific gravity of concrete and H is the height of water. If uplift is not considered – B = H/S^1/2. 7. If the eccentricity of the resultant falls outside the middle third, the ultimate failure of the dam occurs by ______________ a) tension b) crushing c) sliding d) overturning View Answer Answer: a Explanation: When eccentricity is greater than B/6 (eccentricity falls outside the middle third), tension may develop. When tension prevails, cracks develop near the heel and uplift pressure distribution increases reducing the net salinizing force. 8. What is the value of eccentricity for no tension condition in the dam? a) e < B/6 b) e > B/6 c) e > B/3 d) e < B/3 View Answer Answer: a Explanation: The resultant of all the forces i.e hydrostatic water pressure, uplift pressure and self-weight of the dam should always lie within the middle third of the base for no tension. When e < B/6, the value of stress intensity at toe and heel are positive i.e compression on both sides. 9. What is the formula for limiting height of a gravity dam? a) H[max] = f / (S[c] + 1) γ[w] b) H[max] = f / (S[c] – 1) γ[w] c) H[max] = f / (S[c] + C) γ[w] d) H[max] = f / (S[c] – 1) γ[w] View Answer Answer: a Explanation: The critical height or limiting height of a dam having elementary profile is – H[max] = f / (S[c] + 1) γ[w] where f = allowable stress of the dam material, S[c] = Specific gravity of concrete and γ[w] = unit weight of water. This limiting height draws a dividing line between a low gravity dam and a high gravity dam. 10. Calculate the top width of the dam if the height of water stored is 84m. a) 5 m b) 2.5 m c) 5.55 m d) 7.75 m View Answer Answer: a Explanation: Bligh has given an empirical formula for finding out the thickness of the dam at top. A = 0.522 H^1/2 = 0.522 x 84^1/2 = 5.05 m. As per Creager, the economical top width has been found to be equal to 14% of the dam height without considering earthquake forces. Sanfoundry Global Education & Learning Series – Irrigation Engineering. To practice all areas of Irrigation Engineering, here is complete set of 1000+ Multiple Choice Questions and Answers.	{"url":"https://www.sanfoundry.com/irrigation-engineering-questions-answers-gravity-method/","timestamp":"2024-11-06T01:21:54Z","content_type":"text/html","content_length":"165354","record_id":"<urn:uuid:b1cfd274-3db7-4246-9a52-2a2e7edddb86>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00862.warc.gz"}
Frontiers \| Superiorization of projection algorithms for linearly constrained inverse radiotherapy treatment planning • ^1Institute for Machine Learning, Department of Computer Science, ETH Zürich, Zurich, Switzerland • ^2Department of Medical Physics in Radiation Oncology, German Cancer Research Center (DKFZ), Heidelberg, Baden-Württemberg, Germany • ^3Heidelberg Institute for Radiation Oncology (HIRO) and National Center for Radiation Research in Oncology (NCRO), Heidelberg, Baden-Württemberg, Germany • ^4Department of Mathematics, Faculty of Natural Sciences, University of Haifa, Haifa, Israel Objective: We apply the superiorization methodology to the constrained intensity-modulated radiation therapy (IMRT) treatment planning problem. Superiorization combines a feasibility-seeking projection algorithm with objective function reduction: The underlying projection algorithm is perturbed with gradient descent steps to steer the algorithm towards a solution with a lower objective function value compared to one obtained solely through feasibility-seeking. Approach: Within the open-source inverse planning toolkit matRad, we implement a prototypical algorithmic framework for superiorization using the well-established Agmon, Motzkin, and Schoenberg (AMS) feasibility-seeking projection algorithm and common nonlinear dose optimization objective functions. Based on this prototype, we apply superiorization to intensity-modulated radiation therapy treatment planning and compare it with (i) bare feasibility-seeking (i.e., without any objective function) and (ii) nonlinear constrained optimization using first-order derivatives. For these comparisons, we use the TG119 water phantom, the head-and-neck and the prostate patient of the CORT dataset. Main results: Bare feasibility-seeking with AMS confirms previous studies, showing it can find solutions that are nearly equivalent to those found by the established piece-wise least-squares optimization approach. The superiorization prototype solved the linearly constrained planning problem with similar dosimetric performance to that of a general-purpose nonlinear constrained optimizer while showing smooth convergence in both constraint proximity and objective function reduction. Significance: Superiorization is a useful alternative to constrained optimization in radiotherapy inverse treatment planning. Future extensions with other approaches to feasibility-seeking, e.g., with dose-volume constraints and more sophisticated perturbations, may unlock its full potential for high performant inverse treatment planning. 1 Introduction Numerical optimization methods lie at the heart of state-of-the-art inverse treatment planning for intensity-modulated radiation therapy (IMRT) (1). Usually, a clinical prescription of the treatment goals forms the input to a nonlinear multi-criteria optimization (MCO) problem with or without additional constraints, depending on the desired patient dose distribution. During the translation of the clinical goals into an MCO problem, one distinguishes between objectives, i.e., soft goals that compete with each other, and hard constraints designed to ensure, for example, maximal tolerance doses in an organ-at-risk (OAR) and minimal dosage of the target. This versatile approach enables the treatment planner to employ arbitrary combinations of suitable (convex) nonlinear objective functions along with any choice of constraints on the voxels’ doses. This mathematical modeling allows numerical optimization of the fluence of beam elements (beamlets) using a pre-computed normalized dose mapping (2). The resulting constrained nonlinear optimization problem is frequently solved by applying an extended (quasi-)Newton approach with sequential quadratic programming (SQP) and/or interior-point methods (1–7). Until now, the capabilities of inverse planning have been substantially extended through multi-criteria Pareto optimization with subsequent exploration of the Pareto surface (8, 9) or stochastic/robust optimization (10). Computational difficulties may arise in the constrained nonlinear optimization approach. First, optimal convergence for problems of typical size in radiotherapy is tied to the availability of computationally efficient second-order derivatives. While, for example, van Haveren and Breedveld (11) showed that for many typical functions efficient formulations can be found, current research persistently adds new quantities, optimization strategies, and new types of problem formulations to inverse planning for photons and particles (see, e.g., 12–18) to which such strategies might not be directly applicable. Second, a common approach among successful optimizers for nonlinear constrained optimization is to transform the constrained problem into an unconstrained problem using, for example, barrier functions (in the case of interior point methods, e.g., 3, 19) and the method of Lagrange multipliers in combination with slack variables (3, 19, 20). This creates a computational burden when the number of constraints increases. Handling many constraints as, for example, linear inequalities for many or all individual voxel dose bounds, can inflate the computational effort because each constraint requires a Lagrange multiplier and an additional slack variable. Possible “workarounds” include minimax-optimization in combination with auxiliary variables or usage of continuous and differentiable maximum approximations like the LogSumExp and softmax functions (5). Taking a step back, however, to the starting days of treatment planning research, shows that one does not necessarily need to use a mathematical optimization approach to solve the purely linearly constrained IMRT problem but could use feasibility-seeking projection algorithms instead (21, 22). In the context of IMRT, such bare feasibility-seeking translates to seeking a feasible solution that will obey the prescribed lower and upper dose bounds on doses in voxels, without aiming to optimize any objective function. While, in general, a bare feasibility-seeking task can be translated to a constrained optimization problem with a zero objective function, the literature demonstrates a wide spectrum of many efficient feasibility-seeking algorithms not derived from translation of the bare feasibility-seeking task to a constrained optimization problem (see, e.g., 23). If no feasible solution is found, these algorithms find a proximal solution, similar to the piece-wise least-squares approach. Even though they have seen further development over the last decades (24) and, more recently, also extension to dose-volume constraints (25–27), numerical optimizers have been the preferred choice in the field due to their abilities to handle the nonlinear objective functions, e.g., (generalized) equivalent uniform dose (EUD), which are often desired when prescribing treatment goals. The work presented here now combines nonlinear objective functions as used in optimization with feasibility-seeking within linearly constraining dose bounds by applying the superiorization method (SM). To do so, the SM uses a superiorized version of the basic algorithm, the latter being a user-chosen iterative feasibility-seeking algorithm, which is perturbed by interlacing reduction steps of the chosen (nonlinear) objective function. This practically steers the iterates of the feasibility-seeking algorithm to a feasible solution point with a “superior”, i.e., smaller or equal objective function value, which is not necessarily a constrained minimization point. The superiorization method thus works with the constraints data and the user’s choice of objective function, much alike constrained optimization methods would. But it does not aim at an optimal point that minimizes the objective over all constraints like the latter do. In contrast, the SM aims at a point that will fulfill all constraints and have a reduced – not necessarily minimal – objective function value. Not finding the optimal solution, but instead aiming for a satisfactory or adequate result, is a reasonable decision strategy (“Satisficing”, see 28), particularly considering the degeneracy of the IMRT optimization problem (29). Hence, this aim suffices for the purpose of generating acceptable treatment plans. Combined with the simplicity of the gradient descent steps (i.e., not relying on second-order derivatives), superiorization can find a solution, in general, faster and with less investment of computing resources, and fewer conditions concerning design of the objective function. Application of the SM to treatment planning is encouraged by the flexibility it has shown for applications in multiple fields:^1 It has demonstrated its effectiveness for image reconstruction in single-energy computed tomography (CT) (31, 32), dual-energy CT (33) and, more recently, in proton CT (34, 35), by reducing total variation (TV) during image reconstruction. The SM has also been successfully applied to diverse other fields of applications, such as tomographic imaging spectrometry (36) or signal recovery (37). This work is – to the best of our knowledge – the first in-depth investigation of the SM as a potential alternative to constrained minimization algorithms for inverse radiotherapy treatment planning using common objective functions. To date, we could only identify an initial study of the applicability of SM in IMRT utilizing TV as objective function (38), which does not represent common choices in objective function design for treatment planning. Another work considering the use of SM in IMRT used superiorization to boost a specific lexicographic planning approach (39). Expanding on those preliminary works, we develop, tune, and evaluate a prototypical superiorization solver for radiotherapy treatment planning problems. To show how this SM solver is able to replace a constrained minimization approach, and to maximize reproducibility and re-usability of our work, our superiorization approach is implemented into the validated open source radiation therapy treatment planning toolkit matRad (5) together with an instructive set of scripts to execute and reproduce the results of this work (see section 2.5). Within matRad and its included phantoms and patient cases, the SM is evaluated and tested on full-fledged IMRT and intensity-modulated proton therapy (IMPT) treatment planning problems. We compare to using non-linear constrained optimization using only first-order derivatives like the SM, that is, a quasi-Newton method construction a Hessian approximation. This paper is structured as follows: In section 2, we describe the approaches and present the specific version of the SM that we use along with the feasibility-seeking algorithm embedded in it. Section 3 includes our computational results. Finally, in section 4, we discuss the potential of SM with possible future developments and conclude our work in section 5. 2 Materials and methods This work compares three approaches to model the treatment planning problem in IMRT: (i) a nonlinear constrained minimization approach of minimizing an objective function subject to constraints with a quasi-Newton method relying on first-order derivatives, (ii) the feasibility-seeking approach searching for a feasible solution adhering to constraints without considering any objective functions to minimize, and finally, (iii) the superiorization approach, which perturbs the feasibility-seeking algorithm to reduce (not necessarily minimize) an objective function by gradient descent steps. Before introducing these approaches, we briefly recap the discretization of the inverse treatment planning problem. 2.1 Discretization of the inverse treatment planning problem Computerized inverse treatment planning usually relies on a spatial discretization of the particle fluence, the patient anatomy, and, consequently, the radiation dose. The patient is represented by a three-dimensional voxelized grid (image) with $n$ voxels numbered $i=1,2,\dots ,n$. Based on this image, $Q$ volumes of interest (VOIs) ${S}_{q}, q=1,2,\dots ,Q$ are segmented. This allows us to represent the dose as a vector $\mathbit{\text{d}}={\left({d}_{i}\right)}_{i=1}^{n}$, whose $i$-th component is the radiation dose deposited within the $i$-th voxel. For each of the segmentations ${S}_{q}$, we can then easily identify its dosage by finding ${d}_{i}$ for all $i\in {S}_{q}$. The radiation fluence is represented as a vector intensities $\mathbit{\text{x}}={\left({x}_{j}\right)}_{j=1}^{m}$, whose $j$-th component is the intensity of the $j$-th beamlet. The dose deposition ${a}_{i}^{j}$ for a unit intensity of beamlet $j$ to voxel $i$ can then be precomputed and stored in the dose influence matrix $A={\left({a}_{i}^{j}\right)}_{i=1,j=1}^{n,m}$, mapping $\mathbit{\text {x}}$ to $\mathbit{\text{d}} \text{via} \mathbit{\text{d}}=\mathbit{\text{Ax}}$. 2.2 The constrained minimization approach In the optimization approach to IMRT treatment planning, the clinically prescribed aims are represented by various (commonly differentiable) objective functions which map the vector of beamlet intensities to the positive real numbers (2). For our purposes, we limit ourselves to objective functions ${f}_{p}:{ℝ}^{n}\to \left[0,\infty \right),\text{\hspace{0.17em}}p=1,2,\dots ,P$, operating on the radiation dose $\mathbit{\text{d}}$ as surrogates for clinical, dose-based goals. A comprehensive, exemplary list of such common objective functions can be found in Wieser et al. (5, Table 1) and, for the reader’s convenience, also in Supplementary Data Sheet below. These objective functions, which depend on the dose, are related to the intensities $\mathbit{\text{x}} \text{via} \mathbit{\text{d}}=\mathbit{\text{Ax}}$, which is computed at each iterate/change of $\ mathbit{\text{x}}$ during optimization. TABLE 1 Table 1 Dose inequalities/prescriptions and penalty weights used for minimization and for AMS feasibility-seeking. Wishing to fulfill or decide between multiple clinical goals, the resulting multi-objective optimization problem may be scalarized using a weighted sum of several different individual objective functions for the various VOIs S[q]. This approach, first introduced for least-squares (as introduced by 40), can today explore a plethora of objective functions (2, 5) while also satisfying hard constraints (3, 5): $\begin{array}{ll}\begin{array}{lllllll}\hfill & {\mathbit{\text{x}}}^{}=\hfill & \hfill & \underset{x}{\mathrm{arg}\text{ }\mathrm{min}}\sum _{p=1}^{P}{w}_{p}{f}_{p}\left(\mathbit{\text{d}}\left(\ mathbit{\text{x}}\right)\right)\hfill & \hfill & \hfill & \hfill \\ \text{such} \text{that}\hfill & \hfill & \hfill & {c}_{t}^{L}\le {c}_{t}\left(\mathbit{\text{d}}\left(\mathbit{\text{x}}\right)\ right)\le {c}_{t}^{U},\hfill & \hfill & \hfill & t=1,2,\dots ,T,\hfill \\ \hfill & \hfill & \hfill & \mathbit{\text{x}}\ge 0 .\hfill & \hfill & \hfill & \hfill \end{array}& \left(1\right)\end{array}$ Here ${w}_{p}\ge 0$, for all $p=1,2,\dots ,P$, are user-specified weights reflecting relative importance, ${f}_{p}$ are user-chosen individual objective functions, $\mathbit{\text{x}}$ is the beamlet radiation intensities vector (which is physically bound to the nonnegative real orthant), and ${c}_{t}$ are user-chosen individual constraints with lower and upper bounds ${c}_{t}^{L}$ and ${c}_{t}^ {U}$, respectively. While the constraints ${c}_{t}$ can, in principle, be nonlinear constraints, we focus here on linear inequality constraints representing upper and lower dose prescription bounds. The inverse planning problem from eq. (1), solved with numerical optimization techniques, is commonly used today across treatment modalities (among others 2, 3, 5, 40, 41). SQP or interior point methods with a (quasi-)Newton approach are often used to solve the resulting constrained optimization problems (1–7, 42). In this work, we focus on a quasi-Newton approach using first-order derivatives only, since the superiorization approach (as described further below in section 2.4) has so far only been investigated using gradient descent steps itself. 2.3 The feasibility-seeking approach Since the bare feasibility-seeking approach is the backbone of the SM, it will be outlined below using the notation from sections 2.1 and 2.2. Prior work has already suggested the feasibility-seeking approach to address the treatment planning problem (see, e.g., 43, and references therein). To solve the treatment planning problem with feasibility-seeking, dose prescriptions are modeled as a system of linear inequalities: In general, the dose in every voxel is constrained with a lower and upper bound. Feasibility-seeking now seeks a solution, i.e., a beamlet intensity vector fulfilling these prescriptions. With $\mathbit{\text{d}}\left(\mathbit{\text{x}}\right)=\mathbit{\text{Ax}}$, the beamlet radiation intensities vector $\mathbit{\text{x}}$ now has to be recovered from a system of linear inequalities of the form $\begin{array}{ll}{c}_{i}^{L}\le \sum _{j=1}^{m}{a}_{i}^{j}{x}_{j}\le {c}_{i}^{U},\text{ }i=1,2,\dots ,n.& \left(2\right)\end{array}$ In principle, individual lower and upper bounds ${c}_{i}^{L}$ and ${c}_{i}^{U}$ can be chosen for each voxel $i$. Since prescriptions are usually grouped per VOI ${S}_{q}$, the system can be rewritten as: $\begin{array}{ll}\text{For all} q=1,2,\dots ,Q:\text{ }{\ell }_{q}\le \sum _{j=1}^{m}{a}_{i}^{j}{x}_{j}\le {u}_{q}\text{\hspace{0.17em}for\hspace{0.17em}all\hspace{0.17em}}i\in {S}_{q},& \left(3\ with ${\ell }_{q}$ and ${u}_{q}$ representing the lower and upper dose bounds per VOI ${S}_{q}$, respectively. Since it does not make sense to prescribe positive lower bounds to OARs, these are generally chosen to be equal to zero. Geometrically, depending on which structure ${S}_{q}$ a voxel $i$ belongs to, each physical dose constraint set ${C}_{i}$ in each voxel $i=1,2,\dots ,n,$ is a hyperslab (i.e., an intersection of two half-spaces) in the m-dimensional Euclidean vector space ${ℝ}^{m}$. Aiming at satisfaction of all physical dose constraints along with the nonnegativity constraints is, thus, the following (which is a special case of the convex feasibility problem see, e.g., 23): $\begin{array}{ll}\text{Find\hspace{0.17em}an} {\mathbit{\text{x}}}^{}\in \mathbit{\text{W}}:=\left\{\mathbit{\text{x}}\in {ℝ}^{m}\|\text{for all},\text{ }q=1,2,\dots ,Q, {\ell }_{q}\le \sum _{j=1}^ {m}{a}_{i}^{j}{x}_{j}\le {u}_{q},\text{ for all }i\in {S}_{q},\text{and} \mathbit{\text{x}}\ge 0\right\}& \left(4\right)\end{array}$ Such feasibility-seeking problems can typically be solved by a variety of efficient projection methods, whose main advantage, which makes them successful in real-world applications, is computational (see, e.g., 23, 44). They commonly can handle very large-size problems of dimensions beyond which other, more sophisticated currently available, methods start to stutter or cease to be efficient. This is because the building blocks of a projection algorithm are the projections onto the given individual sets. These projections are actually easy to perform, particularly in linear cases such as hyperplanes, half-spaces, or hyperslabs. For the purpose of this paper, we define such an iterative feasibility-seeking algorithm via an algorithmic operator 𝒜 : ℝ^m → ℝ^m , $\begin{array}{ll}{\mathbit{\text{x}}}^{0}\in {ℝ}^{m}, {\mathbit{\text{x}}}^{k+1}=\mathcal{A}\left({\mathbit{\text{x}}}^{k}\right), k=1,2,\dots ,& \left(5\right)\end{array}$ whose task is to (asymptotically) find a point in $W$. The algorithmic structures of projection algorithms are sequential, simultaneous, or in-between, such as in the block-iterative projection (BIP) methods (see, e.g., 45, 46, and references therein) or in the more recent string-averaging projection (SAP) methods (see, e.g., 47, and references therein). An advantage of projection methods is that they work with the initial, raw data and do not require transformation of, or other operations on, the sets describing the problem. For our prototype used here in conjunction with the SM, we rely on the well-established Agmon, Motzkin, and Schoenberg (AMS) relaxation method for linear inequalities (48, 49). Implemented sequentially and modified for handling the bounds $\mathbit{\text{x}}\ge 0$, it is outlined in Algorithm 1. We denote $\ell :={\left({\ell }_{q}\right)}_{q=1}^{Q}$ and $u:={\left({u}_{q}\right)}_{q= During an iteration, Algorithm 1 iterates over all rows of the dose matrix A and handles sequentially the right-hand side and the left-hand side of individual constraints from eq. (3). The control sequence (CS) (50, Definition 5.1.1) determines the order of iterating through the matrix rows/constraints. When a corresponding voxel dose inequality is violated, the algorithm performs geometrically a projection of the current point $\mathbit{\text{x}}$ onto the violated half-space with a user-chosen relaxation parameter $0<\lambda \le 2$. The original AMS algorithm is modified in Algorithm 1 to allow the relaxation for each voxel $i$ to be weighted with ${u }_{i}$ and by performing projections onto the nonnegative orthant of ${ℝ}^{m}$ (in steps 11–13) to return only nonnegative intensities $\mathbit{\text{x}}$. The vector ${\mathbit{\text{a}}}^{i}={\left({a}_{i}^{j}\right)}_{j=1}^{m}$ is the $i$-th row of the dose matrix 𝒜 and is the normal vector to the half-space represented by that row and $‖{\mathbit{\text{a}}}^{i}{‖}_{2}^{2}$ is its square Euclidean norm. In summary, the algorithmic operator in Algorithm 1 describes a single complete sweep of projections sequentially over all constraints (half-spaces) followed by a projection onto the nonnegative orthant thus ensuring the nonnegativity constraint. Such sweeps will be executed iteratively. The theory behind this algorithm guarantees that, under reasonable conditions, if the feasibility-seeking sweeps are performed endlessly then any sequence of iteration vectors ${\left\{{\mathbit{\ text{x}}}^{k}\right\}}_{k=0}^{\infty }$ converges to a point that satisfies all constraints. Choosing to define an algorithmic operator 𝒜 in Algorithm 1, allows us to concisely display the superiorization approach independent from the chosen projection algorithm below (see step 21 inside Algorithm 2). 2.4 The superiorization method and algorithm The SM is built upon application of a feasibility-seeking approach (section 2.3) to the constraints in the second and third lines of eq. (1). But instead of handling the constrained minimization problem of eq. (1) with a full-fledged algorithm for constrained minimization, the SM interlaces into the feasibility-seeking iterative process (i.e., “the basic algorithm”) steps that reduce locally in each iteration the objective function value. Accordingly, the SM does not aim at finding a constrained minimum of the combined objective function $f\left(\mathbit{\text{x}}\right)={\sum }_{p=1}^{P}{w}_{p}{f}_{p}\left(\mathbit{\text{x}}\right)$ of eq. (1) over the constraints. It rather strives to find a feasible point that satisfies the constraints and has a reduced – not necessarily minimal – value of $f$. In the following, we give a brief and focused introduction to SM. A more detailed explanation and review can be found in, e.g., Censor et al. (51, Section II) and references therein (see also 31, 35, 45, 52–55). In general, the SM is intended for constrained function reduction problems of the following form (55, Problem 1): Problem 1. The constrained function reduction problem of the SM Let $W$be a given set (such as in eq. (4)) and let $f:{ℝ}^{m}\to ℝ$be an objective function (such as in eq. (1)). Let 𝒜 from eq. (5) be an algorithmic operator that defines an iterative basic algorithm for feasibility-seeking of a point in $\mathit{\text{W}}$. Find a vector ${\mathbit{\text{x}}}^{}\in W$whose function value is smaller or equal (but not necessarily minimal) than that of a point in $\mathit{\text{W}}$that would have been reached by applying the basic algorithm alone. The SM approaches this question by investigating the perturbation resilience (52, Definitions 4 and 9) of 𝒜, and then proactively using such perturbations, to locally reduce the values $f$ of the iterates, in order to steer the iterative sequence generated by algorithm $\mathcal{A}$ to a solution with smaller or equal objective function value. The structure of the superiorization algorithm implemented here is given by Algorithm 2 with explanations here and in section 2.4.1. Except for the initialization in steps 1–3, Algorithm 2 consists of the perturbations phase (steps 5–19) and the feasibility-seeking phase (steps 20–23). In the perturbation phase, the objective function $f$ is reduced using negative gradient descent steps. The step-size $\beta$ of these gradient updates is calculated by ${\alpha }^{s}$ where $\alpha$ is a fixed user-chosen constant, called kernel, $0<\alpha <1$ so that the resulting step-sizes are nonnegative and form a summable series. The power $s$ is incremented by one until the objective function value of the newly acquired point is smaller or equal to the objective function value of the point with which the current perturbations phase was started. The parameter $N$ determines how many perturbations are executed before applying the next full sweep of the feasibility-seeking phase. The basic Algorithm 1 with algorithmic operator 𝒜^AMS, used throughout this work, is indeed perturbation resilient (56). The superiorization approach has the advantage of letting the user choose any task-specific algorithmic operator 𝒜 that will be computationally efficient, independently of the perturbation phase, as long as perturbation resilience is preserved. Algorithm 2 Superiorization of the feasibility-seeking basic algorithm described by the operator A = A^AMS. For our IMRT treatment planning problem using voxel dose constraints as introduced in eqs. (2) – (4), 𝒜 can be – besides the chosen AMS algorithm – any of the wide variety of feasibility-seeking algorithms (see, e.g., 23, 44, 50, 57). The principles of the SM have been presented and studied in previous publications (consult, e.g. 31, 52, 54), but, to the best of our knowledge, this is the first work applying the SM to a treatment planning problem with an objective function of the general form $f\left(\mathbit{\text{x}}\right):={\sum }_{p=1}^{P}{w}_{p}{f}_{p}\left(\mathbit{\text{x}}\right)$ from eq. (1). 2.4.1 Modifications of the prototypical superiorization algorithm To control the initial step-size, we warm start the algorithm with larger kernel powers $s$ within the first iteration, which substantially improves the algorithm’s runtime. For our purposes, we chose an initial increment of s ← s + 25. In the feasibility-seeking phase, instead of weighting all projections onto the half-spaces equally via the relaxation parameters, each projection can also be given an individual weight $0<{u }_{i} <1$ representing the importance of the $i$-th inequality constraint (i. e., voxel). Further, as shown in step 20 of Algorithm 2, weights can be reduced after each iteration to improve stability. Similar to how the step-sizes are reduced in the perturbation phase, we utilize another kernel $0<\eta <1$ and use its powers ${\eta }^{k}$ to reduce the weights in step 20 by incrementing $k$ after each feasibility-seeking sweep. The new weights are then calculated by ${\eta }^{k}·u ,$ where $u$ are the initial weights. Finally, we integrate four different control sequences to iterate through the rows of $A$. Apart from following the cyclic order according to voxel indices, we experimented with a random order and with sequences choosing rows with increasing or decreasing weights ${u }_{i}$. 2.4.2 Stopping criteria The algorithm was terminated after a given maximal number of iterations was reached or after a certain time limit was exceeded, or when the stopping criterion formulated below was met. The default number of maximum iterations was 500 and the default wall-clock duration was set to 50min. The stopping criterion that we used consists of two parts, both of which must be met for three consecutive iterations for the algorithm to stop. The first part of the stopping criterion is that the relative change of the objective function $f$ defined by $\begin{array}{ll}\|\frac{f\left({\mathbit{\text{x}}}^{k+1}\right)-f\left({\mathbit{\text{x}}}^{k}\right)\|}{\text{max} \left\{1,f\left({\mathbit{\text{x}}}^{k}\right)\right\}}\|& \left(6\right)\end becomes smaller than ${10}^{-4}$. For the second part of the stopping criterion, we define the square of the weighted ${L}_{2}$-norm of the constraints violations by^2 $\begin{array}{ll}V\left(\mathbit{\text{x}}\right):=\frac{1}{n}\sum _{i=1}^{n}\frac{{\left({\ell }_{q}-〈{\mathbit{\text{a}}}^{i},\mathbit{\text{x}}〉\right)}_{+}^{2}+{\left(〈{\mathbit{\text{a}}}^ {i},\mathbit{\text{x}}〉-{u}_{q}\right)}_{+}^{2}}{‖{\mathbit{\text{a}}}^{i}{‖}_{2}^{2}}& \left(7\right)\end{array}$ where ${\ell }_{q}$ and ${u}_{q}$ depend on which structure the $i$-th voxel belongs to. This second part of the stopping rule is met if the relative change of $V$ defined by $\begin{array}{ll}\|\frac{V\left({\mathbit{\text{x}}}^{k+1}\right)-V\left({\mathbit{\text{x}}}^{k}\right)\|}{\text{max} \left\{1,V\left({\mathbit{\text{x}}}^{k}\right)\right\}}\|& \left(8\right)\end is smaller than ${10}^{-3}.$ All tolerances of the stopping criteria can be customized and also set to a negative number to turn off single stopping criteria or early stopping altogether. 2.5 Implementation The superiorization prototype described above was implemented in the open-source crossplatform software “matRad” (5, 58, 59), which is a multi-modality radiation dose calculation and treatment planning toolkit written in Matlab. The implementation is publicly available on the matRad GitHub repository on a research branch.^3 The superiorization solver is implemented as the class matRad_OptimizerSuperiorization.m within matRad's optimization framework. The class defines various user-configurable properties such as the maximum number of iterations, maximum wall time, different warm-start settings, two different feasibility-seeking algorithms, and various control sequences. Once the optimizer has been initialized, the optimize method can be called to generate a solution to the plan. The optimize method requires the following inputs: a starting point, the objective function with its gradient, the linear constraints, and the dose projection matrix. The perturbation phase, as well as the two provided feasibility-seeking algorithms, are implemented as additional methods. Furthermore, within the class, an additional method PlotFunction is available. This method facilitates the visualization of key metrics, such as the objective function value, the maximum constraint violation, and the proximity of the solution to the set of feasible solutions. Multiple scripts to reproduce the results presented herein are provided in an additional GitHub repository.^4 The implementation in matRad facilitates comparison against plans generated on the same datasets with a nonlinear optimizer, as matRad implements a number of common objective functions used in treatment planning (compare to Supplementary Data Sheet and Wieser et al. (5, Table 1)). While matRad provides interfaces to both the open-source Interior Point OPTimizer (IPOPT) (19) as well as to Matlab’s built-in interior-point algorithm from fmincon, only the first was used for our comparisons. We chose to use matRad’s optimization implementation as a benchmark for mainly two reasons: First, matRad has been used in numerous research works demonstrating its ability to create acceptable treatment plans. Second, as an open-source tool, matRad does allow direct modifications of the algorithms and respective parameters and stopping criteria, running them under truly similar conditions. This means that the evaluation of the objective function and its gradient itself use exactly the same code. Benchmarking against other closed-source treatment planning systems would be inconsequential due to hidden computational optimizations, simplifications, and unknown mathematical formulations of objectives and constraints. As motivated in section 2.2, no second-order derivatives were used in the nonlinear optimization approach, but instead a limited-memory Hessian approximation using first-order derivatives was chosen. While second-order derivatives can be used within matRad, it does not make use of fast exact Hessian computation strategies (11), reducing the value of a runtime comparison. matRad performs all computations in a fully-discretized model with a voxel grid. The “dose matrix” A is stored as a compressed sparse column matrix computed for all analyses using matRad’s singular value decomposed pencil-beam algorithm (60) for photons and a singleGaussian pencil-beam algorithm for protons, both validated against clinical implementations (5). 3 Results 3.1 Proof-of-work: Phantom plan To demonstrate the applicability of superiorization to the IMRT treatment planning problem, we first evaluate a small example using the horseshoe phantom defined in the AAPM TG119 Report (61). The phantom is part of the CORT dataset (62) and consequently available with matRad. We created an equidistantly spaced 5-field IMRT photon plan with 5mm × 5mm beamlet doses (resulting in 1918 pencil-beams and a corresponding sparse dose influence matrix with 9.3 × 10^7 non-zero entries in 3.5 × 10^6 voxels). With this setup, we generated treatment plans using three different approaches: (i) constrained minimization with IPOPT, (ii) the AMS algorithm for feasibility-seeking only, and (iii) the SM with the AMS algorithm. Different combinations of nonlinear objective functions and linear inequality constraints on dose were evaluated and compared across these approaches. For analysis, we use dose-volume histograms (DVHs) and axial dose (difference) slices, as well as the evolution plots of the objective function values and the constraint violations. 3.1.1 General usability of the AMS feasibility-seeking projection algorithm We first validate that our implemented projection algorithm AMS is capable of finding comparable treatment plans to those found by established optimization algorithms when applied to a straightforward piece-wise least-squares objective function for the unconstrained minimization of residuals. The setup prescribes 60 Gy to the C-shaped target. To achieve this prescription, we bound the dose in the target by (60 ± 1) Gy. To the two OARs, “Core” and “Body”, upper bounds (a.k.a. tolerance doses) are prescribed, resulting in the parameters given in Table 1. For nonlinear minimization with IPOPT, the tolerance doses serve as parameters for respective penalized piece-wise least-squares objective functions while for AMS the tolerances directly translate into linear inequalities and the weights proportionally increase the relaxation parameters. Figure 1 confirms that feasibility-seeking with weighted AMS is able to find dose distributions of similar quality as conventional nonlinear unconstrained minimization of a piece-wise leastsquares objective function. While resulting in different intensity-modulation patterns, nearly congruent DVHs are observed. FIGURE 1 Figure 1 Comparison of treatment plans obtained by nonlinear minimization with IPOPT (A) and by feasibility-seeking with AMS (B), using the tolerances from Table 1. (C) shows the dose difference in the slice from (A, B, D) the corresponding DVH, in which the optimization result (solid) and feasibility-seeking result (dashed) are nearly overlapping. A crude performance analysis though measures substantially longer runtimes for the AMS approach (about five times slower than unconstrained minimization). This difference is mainly driven by the fact that AMS does sequential iteration by iterations through the matrix rows in each sweep. This investigated scenario is, however, not intended to display any performance advantages of the AMS algorithm, but only to validate its behavior and confirm the long-known ability of such feasibility-seeking algorithms to yield acceptable treatment plans (21, 22). 3.1.2 Inverse planning with superiorization Using the same phantom and irradiation geometry as in section 3.1.1, the feasibility problem used in 3.1.1 was modified to enforce some hard linear inequality constraints while minimizing an objective function. When the constraints are feasible, superiorization using AMS as the basic algorithm will find a feasible point while perturbing the iterates of the feasibility-seeking algorithm towards smaller or equal (not necessarily minimal) function values with objective function reduction steps. As reference, nonlinear constrained minimization with IPOPT with a logistic maximum approximation for minimum/maximum (compare (5), Table 1), was used. Three prescription scenarios were investigated: (I) linear inequalities on the target $\left(59\text{ Gy}<\mathbit{\text{d}}<61\text{ Gy}\right)$, (II) additional linear inequalities on the “Core” structure $\left(\mathbit{\text{d}}<30\text{ Gy}\ right)$, and (III) only linear inequalities on the “Core” $\left(\mathbit{\text{d}}<30\text{ Gy}\right)$. The parameters are detailed in Table 2. TABLE 2 Table 2 Dose inequality constraints, objective functions, and penalty weights used separately for constrained minimization and for superiorization. Figure 2 compares dose distributions and DVHs after superiorization and after constrained minimization. The respective evolution of the objective function values and the constraint violations (calculated by the infinity norm over all inequality constraint functions, corresponding to the maximum residual) is exemplarily shown in Figure 3 for plan I. FIGURE 2 Figure 2 Comparison of treatment plans obtained by superiorization and by constrained minimization. The top row (A–C) shows axial dose distribution slices after constrained minimization, and the middle row (D–F) shows axial dose distribution slices after superiorization. The corresponding DVHs are shown in the bottom row (G–I), with dashed lines showing the superiorization result and solid lines showing the optimization result. FIGURE 3 Figure 3 Objective Function values (A) and maximum constraint violation (B) over time for plan I shown in Figure 2. Each cross indicates a full iteration. Comparing plan quality, both plans adhere to the linear inequality constraints when the problem is feasible (which is the case for plans I & III) as seen in the DVHs. In plan I, superiorization appears to reach better OAR sparing with reduced mean and maximum dose, while in plan III constrained minimization achieves better OAR sparing. For plan II, which poses an infeasible problem, both target coverage and mean OAR sparing are improved for superiorization, yet at higher OAR maximum dose than obtained through constrained minimization. The evolution of the objective function and constraint violation for plan I in Figure 3 exhibits a “typical” behavior of superiorization, seeing a strong decrease in the objective function values within the first iterations, followed by a slower slight increase as the perturbations’ step-sizes diminish. Both approaches were stopped after the maximum number of iterations (1000) was reached. Nearly similar constraint violation is achieved by both methods, while constrained minimization resulted in higher objective function values than superiorization, which can be attributed to the difference in OAR sparing. For all investigated plans I–III, superiorization showed a much “smoother” evolution of objective function and constraint violation than observed in the constrained minimization approach. 3.2 Head-and-neck case To prove the usability of superiorization in a conventional planning setting, we applied the SM to a head-and-neck case with a wider range of available objective functions, i.e., including common DVH-based objectives. Coverage of the planning target volumes (PTVs) was enforced using voxel inequality constraints. Again, the results of superiorization were compared to those obtained by solving the constrained minimization problem. All objectives and constraints are given in Table 3. TABLE 3 Table 3 Dose inequality constraints, objective functions and penalty weights used for optimization and for superiorization on the head-and-neck case. Both solvers use the same stopping criteria for the maximum constraint violation (smaller than 0.01 Gy is acceptable) and objective function change of value (smaller than 0.1% in three consecutive Figure 4 shows exemplary axial dose slices and the DVHs for the plans generated with constraint minimization and with the SM. Quantitative runtime information and evolution of objective function and constraint violation are provided in Figure 5. FIGURE 4 Figure 4 Comparison of head-and-neck treatment plans after (A) constrained minimization and after (B) superiorization (with AMS as the basic algorithm) using the tolerances from Table 3. (C) shows the dose difference in the same slice displayed in (A, B). (D) compares the resulting DVHs after optimization (solid) and superiorization (dashed). FIGURE 5 Figure 5 Evolution of objective Function values (A) and constraint violation (B) with runtime for the plan shown in Figure 4. Both techniques were able to generate a plan that satisfies the linear inequalities up to the allowed violation threshold. Considering absolute runtime, the plan generated with the SM satisfied the stopping criteria after 400s, with constrained minimization failing to converge until the maximum number of iterations was reached. SM spent most of the time in the first sweep/iteration, where it focuses on multiple objective function evaluations to generate a large initial decrease (as already observed above). It continuously decreases the objective function values together with decreasing constraints violation, reaching acceptable constraints violation more slowly than the run with constrained minimization. However, using the same stopping criteria, the SM reached a solution with a much lower objective function value (approximately one-third of the value achieved by the constrained minimization plan). This is also visible in the dose slices and DVH, which show more normal tissue/OAR sparing for the SM plan. All results are, naturally, only valid for the experiments we performed. Further work, with varying algorithmic parameters, initialization points, and stopping criteria, is necessary to make more general statements. 3.3 Prostate case To demonstrate how the superiorization approach translates to a second patient, using a different irradiation modality, we create prostate IMPT plans with opposing fields on a 5mm spot grid using both superiorization and constrained minimization. Figure 6 shows exemplary axial dose slices and the DVHs for the plans generated with constraint minimization and with the SM for the objective and constraint functions stated in Table 4. FIGURE 6 Figure 6 Comparison of prostate proton treatment plans after (A) constrained minimization with IPOPT and after (B) superiorization (with AMS as the basic algorithm) using the tolerances from Table 4. (C) shows the dose difference in the same slice displayed in (A, B). (D) compares the resulting DVHs after optimization (solid) and superiorization (dashed). TABLE 4 Table 4 Dose inequality constraints, objective functions and penalty weights used for optimization and for superiorization on the prostate proton case. The superiorized plan matches the dosimetric performance of the constrained minimization approach. Little increased dose in the rectum and bladder are traded against a slightly more homogeneous target coverage and reduced dose in the femoral heads. 4 Discussion In this work, we applied the novel superiorization method, which solves a system of linear inequalities while reducing a nonlinear objective function, to inverse radiotherapy treatment planning. On a phantom and on a head-and-neck case, we demonstrated that superiorization can produce treatment plans of similar quality to plans generated with constrained minimization. Superiorization showed a smooth convergence behavior for both objective function reduction and constraint violation decrease, including the “typical” behavior of strong initial objective function reduction with subsequent diminishing objective function reduction – including potential slight increase – while proximity to the feasible set within the dose inequality constraints is achieved. 4.1 The mathematical framework of constrained minimization and of superiorization for treatment planning At the heart of the superiorization algorithm lies a feasibility-seeking algorithm (in this work, the AMS relaxation method for linear inequalities). This means that superiorization handles the treatment planning problem as a feasibility-seeking problem for linear inequality dose constraints that should be fulfilled while reducing (not necessarily minimizing) an objective function along the Constrained optimization algorithms, on the other hand, tackle the same data, i.e., constraints and objective function, as a full-fledged optimization problem. With the IPOPT package, for example, inequality constraints become logarithmic barrier functions and are incorporated as a linear combination into the Lagrangian function, whose minimization then enforces the constraints (19). When the problem is hardly feasible, finding the right Lagrange multipliers may then dominate the optimization problem in its initial stages. Superiorization with a feasibility-seeking projection algorithm will smoothly reduce the proximity to the constraints, even for infeasible constrained problems, while the perturbations in the objective function reduction phase reduce the objective function value. Our current implementation is, however, specifically geared for linear constraints. Yet other works on feasibility-seeking have shown that other relevant constraints, like, e.g., DVH constraints, can be incorporated into the feasibility-seeking framework, since they can still be interpreted as linear inequalities on a subset (relative volume) of voxels (25–27). 4.2 Comparability of runtime, convergence and stopping criteria We demonstrated that feasibility-seeking for inverse IMRT treatment planning is practically equivalent to the least-squares approach if similar prescriptions are set. However, obtaining the final solution with feasibility-seeking took more time than with unconstrained minimization with our prototype implementation in Matlab. Stopping criteria, convergence and runtimes are more comparable when considering the constrained minimization vis-à-vis superiorization. Our prototype superiorization algorithm “converged” as fast as the used constrained nonlinear minimization algorithm when using the same objective functions and linear inequalities, exhibiting smoother progress during the iterations. It is interesting to note that even so SM is not guaranteed to find an optimal solution it sometimes exhibits better initial behavior than the constrained minimization algorithm. A similar phenomenon has been observed in the past by Censor et al. (53), wherein the SM was compared with a projected subgradient method (PSM) on a CT image reconstruction problem from projections in computerized tomography. Recognizing the limited scope of the experiments presented here, our results about the superiorization method need further work to become well established. For example, the stopping criteria play a substantial role in both optimization and superiorization. Further modification of the respective parameters may lead to earlier or later stopping of either of the algorithms. Particularly the quasi-Newton algorithm will likely improve on its solution when allowing more iterations/longer runtimes. However, we suspect that the Lagrangian is particularly difficult to navigate when using a Hessian approximation over exact Hessian computations in these heavily constrained examples. This suspicion is supported by a solver benchmark performed by ten Eikelder et al. (63). Consequently, runtime and convergence of a constrained nonlinear optimization algorithm would expectedly improve when incorporating second derivatives, such as proposed by van Haveren and Breedveld ( 11), instead of relying on a low-memory approximation to the quasi-Newton approach. In addition, alternative nonlinear minimum/maximum dose constraint implementations are possible. An advantage of the SM is that such “workarounds” are not necessary. For superiorization, computational complexity and convergence are heavily dependent on the chosen feasibility-seeking algorithm. While the function reduction in superiorization has the computational complexity of gradient descent steps, the basic AMS algorithm used as a starting point performs sequential projections over all constraints. The complexity is thus principally comparable to the corresponding submatrix-vector products, however, the algorithm’s sequential structure complicates parallelization and other computational optimizations. Thus, modifications of the AMS algorithm are still actively researched (e.g., 64). Computational complexity and convergence properties of projection algorithms are a topic of ongoing research (see, e.g., 65, where it is discussed in a more general setting). Despite these limitations, we demonstrated that a straightforward superiorization implementation was able to solve the given treatment planning problem arriving at dosimetrically comparable treatment 4.3 Dosimetric performance The treatment plans obtained with constrained minimization and with superiorization show some dosimetric differences. For the three different linearly constrained setups on the TG199 phantom, these differences were most pronounced on the OAR, and less pronounced for the target dosage. In the setups with target dose inequality constraints, superiorization reached better OAR sparing. This may be a result of multiple interacting factors: the strong initial objective function decrease in superiorization pulling down the dose in the OAR, and potential too early stopping of the constrained minimizer. Further, in the infeasible setting with linear inequality constraints on both target and OAR, superiorization has the advantage that the feasibility-seeking algorithm will still smoothly converge to a proximal point. The improved OAR sparing did not occur when only using dose inequality constraints on the OAR. However, in this case, the differences in DVHs of the OAR are only substantial below a dose of 20Gy and, thus, of limited significance, since a piece-wise least-squares objective was used that does not contribute to the objective function at dose values below 20Gy. The head-and-neck case also reproduces the better OAR sparing for all evaluated OARs, at slightly reduced target coverage for the non-constrained CTV63 and PTV63. Here, the difference in convergence speed was most significant. Through all cases, the superiorization exhibited the smooth evolution of both objective function value and constraint violation, which in turn suggests robustness against changes in the stopping criteria as well. This behavior of superiorization could be underlined by translating it to IMPT on a prostate case. These encouraging results show that superiorization can create acceptable and apparently “better” treatment plans. Additional work on more cases or planning benchmarks, with varying tuning parameters of both constrained minimization and superiorization approaches is needed to assess the convergence, runtime, and dosimetric quality of the solutions. 4.4 Outlook With the proof-of-concept put forward in this work, there are many possible directions to further investigate the application of superiorization algorithms to the radiotherapy inverse treatment planning problem. From the perspective of a treatment planner, one may focus on enabling further constraints, e.g., DVH-based constraints, that are often used in treatment planning. Some of these constraints are also representable as modified linear inequalities or convex and non-convex sets and, thus, can efficiently be solved using a feasibility-seeking algorithm. Even nonlinear constraints that are based, for example, on normal-tissue complication probability or equivalent uniform dose could be incorporated in the current definition of the superiorization algorithm if the “basic algorithm” in the feasibility-seeking phase of the SM is replaced by any other perturbation resilient projection method that can handle nonlinear constraints. Such algorithms exist in the literature. Moreover, superiorization might also be extended to use more complex function reduction steps and inherent criteria. For example, a “true” backtracking line search could be performed, similar to approaches in optimization, since a perturbation resilient “basic algorithm” might be able to handle much more complex function reduction steps. Considering these algorithmic and application-focused improvements, the SM should also be rigorously tested on radiotherapy optimization/inverse planning benchmark problems, like the TROTS dataset ( 66), as soon as it is able to handle the respective problem formulations. With this, transferability to other modalities like ion therapy or volumetric modulated arc therapy (VMAT) is also within 5 Conclusions We introduced superiorization as a novel inverse planning technique, merging feasibility-seeking for linear inequality dose constraints with objective function reduction. Our initial comparison of superiorization with constrained minimization using linear dose-inequalities suggests possible dosimetric benefits and smoother convergence. Superiorization is thus a valuable addition to the algorithmic inverse treatment planning toolbox. Data availability statement All data used in this study is publicly available through matRad. The code to reproduce our results is available on GitHub: https://github.com/e0404/paper-superiorization-imrt. Further inquiries can be directed to the corresponding authors. Ethics statement Ethical approval was not required for the study involving humans in accordance with the local legislation and institutional requirements. Written informed consent to participate in this study was not required from the participants or the participants’ legal guardians/next of kin in accordance with the national legislation and the institutional requirements. Author contributions NW and YC conceived the original idea for the research. YC developed the theoretical framework for the study, while FB implemented the method. NW conducted the analyses of the algorithm. All authors contributed to the writing and editing of the manuscript. All authors contributed to the article and approved the submitted version. Open access funding by ETH Zurich. This work is supported by the ISF-NSFC joint research program grant No. 2874/19 (YC), by the German Research Foundation (DFG), grant No. 443188743 (NW) and by the U.S. National Institutes of Health, grant No. R01CA266467 (YC and NW). Parts of this work pertain to project support (YC and NW) from the Cooperation Program in Cancer Research of the German Cancer Research Center (DKFZ) and Israel’s Ministry of Innovation, Science and Technology (MOST). We thank Mark Bangert for taking part in the first discussion rounds leading up to this work. Finally, the authors would like to thank the editor and two reviewers for their constructive comments and suggestions that have helped to improve the quality of this manuscript. Conflict of interest The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Publisher’s note All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher. Supplementary material The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fonc.2023.1238824/full#supplementary-material 1. Bortfeld T, Thieke C. Optimization of Treatment Plans, Inverse Planning. In: Schlegel W, Bortfeld T, Grosu AL, editors. New Technologies in Radiation Oncology. Berlin/Heidelberg: Springer-Verlag (2006). p. 207–20. doi: 10.1007/3-540-29999-8_17 2. Wu Q, Mohan R. Algorithms and functionality of an intensity modulated radiotherapy optimization system. Med Phys (2000) 27:701–11. doi: 10.1118/1.598932 3. Breedveld S, van den Berg B, Heijmen B. An interior-point implementation developed and tuned for radiation therapy treatment planning. Comput Optimization Appl (2017) 68:209–42. doi: 10.1007/ 4. Fogliata A, Nicolini G, Alber M, Åsell M, Clivio A, Dobler B, et al. On the performances of different IMRT treatment planning systems for selected paediatric cases. Radiat Oncol (2007) 2:7. doi: 5. Wieser HP, Cisternas E, Wahl N, Ulrich S, Stadler A, Mescher H, et al. Development of the open-source dose calculation and optimization toolkit matRad. Med Phys (2017) 44:2556–68. doi: 10.1002/ 6. Luenberger DG, Ye Y. Linear and Nonlinear Programming. In: International Series in Operations Research and Management Science, 3rd ed. New York, NY: Springer (2008). p. 546. 7. Bazaraa MS, Sherali HD, Shetty CM. Nonlinear Programming: Theory and Algorithms. 3rd ed. Hoboken, N.J: Wiley-Interscience (2006). p. 872. 8. Küfer KH, Monz M, Scherrer A, Süss P, Alonso F, Sultan ASA, et al. Multicriteria Optimization in Intensity Modulated Radiotherapy Planning. Kaiserslautern: ITWM Report 77. Fraunhofer (ITWM) 9. Thieke C, Küfer KH, Monz M, Scherrer A, Alonso F, Oelfke U, et al. A new concept for interactive radiotherapy planning with multicriteria optimization: first clinical evaluation. Radiotherapy Oncol (2007) 85:292–8. doi: 10.1016/j.radonc.2007.06.020 10. Unkelbach J, Alber M, Bangert M, Bokrantz R, Chan TCY, Deasy JO, et al. Robust radiotherapy planning. Phys Med Biol (2018) 63:22TR02. doi: 10.1088/1361-6560/aae659 11. Van Haveren R, Breedveld S. Fast and exact Hessian computation for a class of nonlinear functions used in radiation therapy treatment planning. Phys Med Biol (2019) 64:16NT01. doi: 10.1088/ 12. Kierkels RG, Korevaar EW, Steenbakkers RJ, Janssen T, van’t Veld AA, Lan-653 gendijk JA, et al. Direct use of multivariable normal tissue complication probability models in treatment plan optimisation for individualised head and neck cancer radiotherapy produces clinically acceptable treatment plans. Radiotherapy Oncol (2014) 112:430–6. doi: 10.1016/j.radonc.2014.08.020 13. Christiansen E, Heath E, Xu T. Continuous aperture dose calculation and optimization for volumetric modulated arc therapy. Phys Med Biol (2018) 63:21NT01. doi: 10.1088/1361-6560/aae65e 14. Gao H, Lin B, Lin Y, Fu S, Langen K, Liu T, et al. Simultaneous dose and dose rate optimization (SDDRO) for FLASH proton therapy. Med Phys (2020) 47:6388–95. doi: 10.1002/mp.14531 15. Hahn C, Heuchel L, Ödén J, Traneus E, Wulff J, Plaude S, et al. Comparing biological effectiveness guided plan optimization strategies for cranial proton therapy: potential and challenges. Radiat Oncol (London England) (2022) 17:169. doi: 10.1186/s13014-022-02143-x 16. Ten Eikelder SCM, Ajdari A, Bortfeld T, den Hertog D. Adjustable robust treatment-length optimization in radiation therapy. Optimization Eng (2022) 23:1949–86. doi: 10.1007/s11081-021-09709-w 17. Faddegon BA, Blakely EA, Burigo LN, Censor Y, Dokic I, D-Kondo JN, et al. Ionization detail parameters and cluster dose: a mathematical model for selection of nanodosimetric quantities for use in treatment planning in charged particle radiotherapy. Phys Med Biol (2023) 68:175013. doi: 10.1088/1361-6560/acea16 18. Liu R, Charyyev S, Wahl N, Liu W, Kang M, Zhou J, et al. An Integrated Physical Optimization framework for proton SBRT FLASH treatment planning allows dose, dose rate, and LET optimization using patient-specific ridge filters. Int J Radiat Oncology Biology Phys (2023) 116(4):949–59. doi: 10.1016/j.ijrobp.2023.01.048 19. Wächter A, Biegler LT. On the implementation of an interior-point filter linesearch algorithm for large-scale nonlinear programming. Math Programming (2006) 106:25–57. doi: 10.1007/ 20. Nocedal J, Wright SJ. Numerical Optimization. New York: Springer (1999). p. 636. 21. Censor Y, Altschuler MD, Powlis WD. A computational solution of the inverse problem in radiation-therapy treatment planning. Appl Mathematics Comput (1988) 25:57–87. doi: 10.1016/0096-3003(88) 22. Powlis WD, Altschuler MD, Censor Y, Buhle EL. Semi-automated radiotherapy treatment planning with a mathematical model to satisfy treatment goals. Int J Radiat OncologyBiologyPhysics (1989) 16:271–6. doi: 10.1016/0360-3016(89)90042-4 23. Bauschke HH, Borwein JM. On projection algorithms for solving convex feasibility problems. SIAM Rev (1996) 38:367–426. doi: 10.1137/S0036144593251710 24. Cho PS, Marks RJ. Hardware-sensitive optimization for intensity modulated radiotherapy. Phys Med Biol (2000) 45:429–40. doi: 10.1088/0031-9155/45/2/312 25. Penfold S, Zalas R, Casiraghi M, Brooke M, Censor Y, Schulte R. Sparsity constrained split feasibility for dosevolume constraints in inverse planning of intensity-modulated photon or proton therapy. Phys Med Biol (2017) 62:3599–618. doi: 10.1088/1361-6560/aa602b 26. Brooke M, Censor Y, Gibali A. Dynamic string-averaging CQ-methods for the split feasibility problem with percentage violation constraints arising in radiation therapy treatment planning. Int Trans Operational Res (2020) 30:181–205. doi: 10.1111/itor.12929 27. Gadoue SM, Toomeh D, Schultze BE, Schulte RW. A dose–volume constraint (DVC) projection-based algorithm for IMPT inverse planning optimization. Med Phys (2022) 49(4):2699–708. doi: 10.1002/ 28. Simon HA. Rational choice and the structure of the environment. psychol Rev (1956) 63:129–38. doi: 10.1037/h0042769 29. Alber M, Meedt G, Nüsslin F, Reemtsen R. On the degeneracy of the IMRT optimization problem. Med Phys (2002) 29:2584. doi: 10.1118/1.1500402 30. Censor Y. Superiorization and perturbation resilience of algorithms: A continuously updated bibliography. (2022). doi: 10.48550/arXiv.1506.04219 31. Herman GT, Garduño E, Davidi R, Censor Y. Superiorization: An optimization heuristic for medical physics. Med Phys (2012) 39:5532–46. doi: 10.1118/1.4745566 32. Guenter M, Collins S, Ogilvy A, Hare W, Jirasek A. Superiorization versus regularization: A comparison of algorithms for solving image reconstruction problems with applications in computed tomography. Med Phys (2022) 49:1065–82. doi: 10.1002/mp.15373 33. Yang Q, Cong W, Wang G. Superiorization-based multi-energy CT image reconstruction. Inverse Problems (2017) 33:044014. doi: 10.1088/1361-6420/aa5e0a 34. Penfold S, Censor Y. Techniques in iterative proton CT image reconstruction. Sens Imaging (2015) 16:19. doi: 10.1007/s11220-015-0122-3 35. Schultze B, Censor Y, Karbasi P, Schubert KE, Schulte RW. An improved method of total variation superiorization applied to reconstruction in proton computed tomography. IEEE Trans Med Imaging (2020) 39:294–307. doi: 10.1109/TMI.2019.2911482 36. Han W, Wang Q, Cai W. Computed tomography imaging spectrometry based on superiorization and guided image filtering. Optics Lett (2021) 46:2208–11. doi: 10.1364/OL.418355 37. Pakkaranang N, Kumam P, Berinde V, Suleiman YI. Superiorization methodology and perturbation resilience of inertial proximal gradient algorithm with application to signal recovery. J Supercomputing (2020) 76:9456–77. doi: 10.1007/s11227-020-03215-z 38. Davidi R, Censor Y, Schulte R, Geneser S, Xing L. Feasibility-Seeking and Superiorization Algorithms Applied to Inverse Treatment Planning in Radiation Therapy. In: Reich S, Zaslavski A, editors. Contemporary Mathematics, vol. 636. Providence, Rhode Island: American Mathematical Society (2015). p. 83–92. doi: 10.1090/conm/636/12729 39. Bonacker E, Gibali A, Küfer KH, Süss P. Speedup of lexicographic optimization by superiorization and its applications to cancer radiotherapy treatment. Inverse Problems (2017) 33:044012. doi: 40. Alber M, Reemtsen R. Intensity modulated radiotherapy treatment planning by use of a barrier-penalty multiplier method. Optimization Methods Software (2007) 22:391–411. doi: 10.1080/ 41. Bortfeld T, Bürkelbach J, Boesecke R, Schlegel W. Methods of image reconstruction from projections applied to conformation radiotherapy. Phys Med Biol (1990) 35:1423–34. doi: 10.1088/0031-9155/35 42. Carlsson F, Forsgren A. Iterative regularization in intensity-modulated radiation therapy optimization. Med Phys (2006) 33:225–34. doi: 10.1118/1.2148918 43. Censor Y, Bortfeld T, Martin B, Trofimov A. A unified approach for inversion problems in intensity-modulated radiation therapy. Phys Med Biol (2006) 51:2353–65. doi: 10.1088/0031-9155/51/10/001 44. Bauschke HH, Koch VR. Projection Methods: Swiss Army Knives for Solving Feasibility and Best Approximation Problems with Halfspaces. In: Reich S, Zaslavski AJ, editors. Contemporary Mathematics, vol. 636 Providence, RI: American Mathematical Society (AMS) (2015). p. 1–40. doi: 10.1090/conm/636 45. Davidi R, Herman G, Censor Y. Perturbation-resilient block-iterative projection methods with application to image reconstruction from projections. Int Trans operational Res (2009) 16:505–24. doi: 46. Gordon D, Gordon R. Component-averaged row projections: A robust, blockParallel scheme for sparse linear systems. SIAM J Sci Computing (2005) 27:1092–117. doi: 10.1137/040609458 47. Bargetz C, Reich S, Zalas R. Convergence properties of dynamic string-averaging projection methods in the presence of perturbations. Numerical Algorithms (2018) 77:185–209. doi: 10.1007/ 48. Agmon S. The relaxation method for linear inequalities. Can J Mathematics (1954) 6:382–92. doi: 10.4153/CJM-1954-037-2 49. Motzkin TS, Schoenberg IJ. The relaxation method for linear inequalities. Can J Mathematics (1954) 6:393–404. doi: 10.4153/CJM-1954-038-x 50. Censor Y, Zenios SA. Parallel optimization: Theory, Algorithms, and Applications. New York, NY, USA: Oxford University Press (1998). 51. Censor Y, Schubert KE, Schulte RW. Developments in mathematical algorithms and computational tools for proton CT and particle therapy treatment planning. IEEE Trans Radiat Plasma Med Sci (2022) 6:313–24. doi: 10.1109/TRPMS.2021.3107322 52. Censor Y. Weak and strong superiorization: between feasibility-seeking and minimization. Analele Universitatii “Ovidius” Constanta - Seria Matematica (2015) 23:41–54. doi: 10.1515/auom-2015-0046 53. Censor Y, Davidi R, Herman GT, Schulte RW, Tetruashvili L. Projected subgradient minimization versus superiorization. J Optimization Theory Appl (2014) 160:730–47. doi: 10.1007/s10957-013-0408-3 54. Herman GT. Superiorization for Image Analysis. In: Proceedings of the 16th International Workshop on Combinatorial Image Analysis, vol. 8466. Berlin, Heidelberg: Springer-Verlag, IWCIA (2014). p. 1–7. doi: 10.1007/978-3-319-07148-0_1 55. Censor Y. Can linear superiorization be useful for linear optimization problems? Inverse Problems (2017) 33:044006. doi: 10.1088/1361-6420/33/4/044006 56. Censor Y, Zaslavski AJ. Convergence and perturbation resilience of dynamic string-averaging projection methods. Comput Optimization Appl (2013) 54:65–76. doi: 10.1007/s10589-012-9491-x 57. Cegielski A. Iterative Methods for Fixed Point Problems in Hilbert Spaces. New York: Springer (2012). p. 316. 58. Ackermann B, Bangert M, Bennan ABA, Burigo L, Cabal G, Cisternas E, et al. matRad. Version 2.10.1. Heidelberg: Deutsches Krebsforschungszentrum (2020). doi: 10.5281/zenodo.7107719 59. Cisternas E, Mairani A, Ziegenhein P, Jäkel O, Bangert M. matRad - a multi-modality open source 3D treatment planning toolkit. In: IFMBE Proceedings, vol. 51. Cham: Springer International Publishing (2015). p. 1608–11. doi: 10.1007/978-3-319-19387-8_391 60. Bortfeld T, Schlegel W, Rhein B. Decomposition of pencil beam kernels for fast dose calculations in three-dimensional treatment planning. Med Phys (1993) 20:311–8. doi: 10.1118/1.597070 61. Ezzell GA, Burmeister JW, Dogan N, LoSasso TJ, Mechalakos JG, Mihailidis D, et al. IMRT commissioning: Multiple institution planning and dosimetry comparisons, a report from AAPM Task Group 119. Med Phys (2009) 36:5359–73. doi: 10.1118/1.3238104 62. Craft D, Bangert M, Long T, Papp D, Unkelbach J. Shared data for intensity modulated radiation therapy (IMRT) optimization research: the CORT dataset. GigaScience (2014) 3:2047–217X–3–37. doi: 63. Ten Eikelder SCM, Ajdari A, Bortfeld T, den Hertog D. Conic formulation of fluence map optimization problems. Phys Med Biol (2021) 66:225016. doi: 10.1088/1361-6560/ac2b82 64. De Loera JA, Haddock J, Needell D. A sampling kaczmarz–motzkin algorithm for linear feasibility. SIAM J Sci Computing (2017) 39:S66–S87. doi: 10.1137/16M1073807 65. Chen C, Fu X, He B, Yuan X. On the iteration complexity of some projection methods for monotone linear variational inequalities. J Optimization Theory Appl (2017) 172:914–28. doi: 10.1007/ 66. Breedveld S, Heijmen B. Data for TROTS – the radiotherapy optimisation test set. Data Brief (2017) 12:143–9. doi: 10.1016/j.dib.2017.03.037 Keywords: radiation therapy treatment planning, inverse planning, constrained treatment plan optimization, IMRT, superiorization method, feasibility-seeking algorithm Citation: Barkmann F, Censor Y and Wahl N (2023) Superiorization of projection algorithms for linearly constrained inverse radiotherapy treatment planning. Front. Oncol. 13:1238824. doi: 10.3389/ Received: 12 June 2023; Accepted: 18 September 2023; Published: 26 October 2023. Copyright © 2023 Barkmann, Censor and Wahl. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms. Correspondence: Florian Barkmann, florian.barkmann@inf.ethz.ch; Yair Censor, yair@math.haifa.ac.il; Niklas Wahl, n.wahl@dkfz.de	{"url":"https://www.frontiersin.org/journals/oncology/articles/10.3389/fonc.2023.1238824/full","timestamp":"2024-11-06T06:22:01Z","content_type":"text/html","content_length":"696636","record_id":"<urn:uuid:c88f2a24-c2bf-4e6a-aabf-d43c92df990d>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00433.warc.gz"}
If f(x) =tan^24x and g(x) = sqrt(5x-1 , what is f'(g(x)) ? \| HIX Tutor If #f(x) =tan^24x # and #g(x) = sqrt(5x-1 #, what is #f'(g(x)) #? Answer 1 $f ' \left(g \left(x\right)\right) = 24 {\tan}^{23} \left(\sqrt{5 x - 1}\right) \times {\sec}^{2} \left(\sqrt{5 x - 1}\right)$ As #f(x)=tan^24x# and #g(x)=sqrt(5x-1)# As #f(x)=tan^24x# #(df)/(dx)=f'(x)=24tan^23x xx sec^2x# and #f'(g(x))=24tan^23(sqrt(5x-1)) xx sec^2(sqrt(5x-1))# Sign up to view the whole answer By signing up, you agree to our Terms of Service and Privacy Policy Answer 2 Reqd. Deri. $= \left(\frac{48}{5}\right) {\tan}^{23} x \cdot {\sec}^{2} x \cdot \sqrt{5 x - 1} .$ #f(x)=tan^24x, g(x)=sqrt(5x-1)=t,# say. Now Reqd. Deri. #=f'(g(x))=f'(t)=d/dt{f(t)}=(df)/dt# We see that #f# is a fun. of #x#, and #x# of #t#. Hence, reqd. Deri. #=(df)/dt=(df)/dxdx/dt..........(1)# Now, #f(x)=tan^24x rArr (df)/dx=24tan^23xd/dxtanx=24tan^23xsec^2x#.....(2) Next, #t=sqrt(5x-1) rArr dt/dx=1/(2sqrt(5x-1))d/dx(5x-1)=5/(2sqrt(5x-1)# #rArr dx/dt=(2sqrt(5x-1))/5....................(3)# From #(1),(2),(3)#, Reqd. Deri. #=24tan^23xsec^2x(2sqrt(5x-1))/5#, #=(48/5)tan^23xsec^2xsqrt(5x-1).# Sign up to view the whole answer By signing up, you agree to our Terms of Service and Privacy Policy Answer 3 To find ( f'(g(x)) ), we first find ( f'(x) ) and then substitute ( g(x) ) into it. Given ( f(x) = \tan^2(4x) ) and ( g(x) = \sqrt{5x - 1} ): 1. Find ( f'(x) ): ( f'(x) = \frac{d}{dx}[\tan^2(4x)] ) ( f'(x) = 2\tan(4x) \sec^2(4x) \cdot 4 ) 2. Substitute ( g(x) ) into ( f'(x) ): ( f'(g(x)) = 2\tan(4\sqrt{5x-1}) \sec^2(4\sqrt{5x-1}) \cdot 4 ) 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/if-f-x-tan-24x-and-g-x-sqrt-5x-1-what-is-f-g-x-8f9af9dec8","timestamp":"2024-11-08T05:19:44Z","content_type":"text/html","content_length":"583734","record_id":"<urn:uuid:046b8f07-05cb-45d5-8801-3093e836d134>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00307.warc.gz"}
Dual Oscillation Reversal Signal-to-Noise Ratio Optimization Combo Strategy 1. Dual Oscillation Reversal Signal-to-Noise Ratio Optimization Combo Strategy Dual Oscillation Reversal Signal-to-Noise Ratio Optimization Combo Strategy , Date: 2023-11-01 16:57:13 This strategy combines the dual oscillation reversal strategy and the signal-to-noise ratio optimization strategy to form a more powerful and stable trading strategy. The strategy aims to generate more accurate trading signals at trend reversal points. Strategy Logic The dual oscillation reversal strategy calculates the fast and slow K values of the last 14 days to determine if there is a reversal over two consecutive trading days. If the reversal happens when the fast K is below 50, it is a buy signal. If the fast K is above 50, it is a sell signal. The signal-to-noise ratio optimization strategy calculates the signal-to-noise ratio of the last 21 days and smooths it with a 29-day simple moving average. When the signal-to-noise ratio crosses above the moving average, it is a sell signal. When it crosses below, it is a buy signal. Finally, this strategy only initiates buy or sell trades when both strategies issue the same signal. Advantage Analysis 1. Combining multiple strategies can generate more accurate trading signals and avoid false signals from a single strategy. 2. The dual oscillation reversal strategy catches trend reversal points. The signal-to-noise ratio optimization filters out false signals. Working together, they can accurately trade at reversals. 3. Optimized parameters like 14-day fast/slow stochastics and 21-day signal-to-noise period capture recent trends without too much noise. 4. The dual confirmation signals significantly reduce trading risk and avoid unnecessary losses. Risk Analysis 1. Reversal signals may lag and miss absolute bottoms or tops. Parameters can be adjusted to shorten the lag. 2. Dual signal confirmation may miss some trading opportunities. Confirmation conditions could be relaxed but also increase risk. 3. Signal-to-noise ratio parameters need optimization. Improper periods may cause missing or false signals. 4. Monitoring multiple indicators increases complexity. Code optimization and computing resources need consideration. Optimization Directions 1. Test more indicator combinations to find better combo signals, like MACD, RSI etc. 2. Optimize parameters of the reversal strategy for more accurate and timely signals. 3. Optimize signal-to-noise ratio periods to find the optimal balance. 4. Add stop loss strategies to control potential loss for single trades. 5. Consider machine learning methods to auto optimize parameters for better adaptability. This strategy combines dual oscillation reversal and signal-to-noise ratio strategies to provide stable signals at trend reversal points. Optimized parameters significantly reduce false signals, and dual confirmation lowers trading risks. Further optimizations like indicator parameters, stop loss can improve performance. Overall, this is a stable strategy with practical trading value. start: 2023-10-01 00:00:00 end: 2023-10-31 00:00:00 period: 1h basePeriod: 15m exchanges: [{"eid":"Futures_Binance","currency":"BTC_USDT"}] // Copyright by HPotter v1.0 196/01/2021 // This is combo strategies for get a cumulative signal. // First strategy // This System was created from the Book "How I Tripled My Money In The // Futures Market" by Ulf Jensen, Page 183. This is reverse type of strategies. // The strategy buys at market, if close price is higher than the previous close // during 2 days and the meaning of 9-days Stochastic Slow Oscillator is lower than 50. // The strategy sells at market, if close price is lower than the previous close price // during 2 days and the meaning of 9-days Stochastic Fast Oscillator is higher than 50. // Second strategy // The signal-to-noise (S/N) ratio. // And Simple Moving Average. // WARNING: // - For purpose educate only // - This script to change bars colors. Reversal123(Length, KSmoothing, DLength, Level) => vFast = sma(stoch(close, high, low, Length), KSmoothing) vSlow = sma(vFast, DLength) pos = 0.0 pos := iff(close[2] < close[1] and close > close[1] and vFast < vSlow and vFast > Level, 1, iff(close[2] > close[1] and close < close[1] and vFast > vSlow and vFast < Level, -1, nz(pos[1], 0))) SignalToNoise(length) => StN = 0.0 for i = 1 to length-1 StN := StN + (1/close[i])/length StN := -10*log(StN) StN(length,Smooth) => pos = 0.0 StN = SignalToNoise(length) SMAStN = sma(StN, Smooth) pos := iff(SMAStN[0] > StN[0] , -1, iff(SMAStN[0] < StN[0], 1, 0)) strategy(title="Combo Backtest 123 Reversal & Signal To Noise", shorttitle="Combo", overlay = true) Length = input(14, minval=1) KSmoothing = input(1, minval=1) DLength = input(3, minval=1) Level = input(50, minval=1) lengthStN = input(title="Days", type=input.integer, defval=21, minval=2) SmoothStN = input(title="Smooth", type=input.integer, defval=29, minval=2) reverse = input(false, title="Trade reverse") posReversal123 = Reversal123(Length, KSmoothing, DLength, Level) posStN = StN(lengthStN,SmoothStN) pos = iff(posReversal123 == 1 and posStN == 1 , 1, iff(posReversal123 == -1 and posStN == -1, -1, 0)) possig = iff(reverse and pos == 1, -1, iff(reverse and pos == -1 , 1, pos)) if (possig == 1) strategy.entry("Long", strategy.long) if (possig == -1) strategy.entry("Short", strategy.short) if (possig == 0) barcolor(possig == -1 ? #b50404: possig == 1 ? #079605 : #0536b3 )	{"url":"https://www.fmz.com/strategy/430772","timestamp":"2024-11-03T10:11:03Z","content_type":"text/html","content_length":"14628","record_id":"<urn:uuid:63f0a159-5331-4165-93d6-2da25b987658>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00601.warc.gz"}
When would you use a 400mm lens? When would you use a 400mm lens? The 400mm focal length of this lens is ideal for many field based sports such as soccer (football) and rugby as it’s not too long, such as a 500mm for example, but at the same time it’s gives more pull than a 300mm. Can you use an extender with Canon 100 400? EOS 6D + EF 100-400mm f/4.5-5.6L IS USM II lens (OVF used) – NO extender used….what to expect from a 100-400mm Canon with Canon 1.4 extender. Make Canon Model Canon EOS M6 Focal length 400mm Shutter speed 1/400 sec Aperture f/5.6 Do I need weather resistant lens? The Lens. Having weather sealed lenses are just as important. A weather resistant camera isn’t going to do much good when paired with a non-weather resistant lens, because it’s still got a giant opening up front. What does a 1.4 extender do? Canon Extenders are available in two strengths, 1.4x and 2x. As the names suggest, the 1.4x Extender increases the focal length of your lens by a factor of 1.4, and the 2x by a factor of 2. Canon EF Extenders are designed for use with a number of telephoto and zoom EF lenses. What lenses does the Canon 2x Extender work with? Note. This lens is only compatible with fixed focal length L-series lenses 135mm and over, as well as the EF 70-200/2.8L, EF 70-200/2.8L IS, EF 70-200/4L, and EF 100-400/4.5-5.6L. Can I use my DSLR in the rain? High-end DSLR cameras are built to withstand harsh conditions but few of them are completely waterproof. They have no problem with light rain but too much water can damage both the camera body and Can you use Canon cameras in the rain? You can use a weather-sealed camera in the rain for a longer time than a camera without sealing. That’s because the weather sealing offers a level of protection. However, water can still damage the camera. Weather sealed doesn’t mean waterproof. Which Canon lenses are weather sealed? Canon Weather Sealed Lenses • Canon EF 100mm f/2.8L. • Canon EF 50mm f/1.2L. • Canon EF 35mm f/1.4L. • Canon EF 85mm f/1.4L. • Canon 400mm f/5.6L. • Canon 24mm f/1.4L. • Canon 85mm f/1.2L. • Canon 135mm f/2L. What lens do professional wildlife photographers use? The Best Lenses for Wildlife Photography 1. Sigma 150-600mm f/5-6.3 DG OS HSM. 2. Sony FE 200-600mm f/5.6-6.3 G OSS. 3. Canon 200-400mm f/4L IS USM Extender 1.4x. 4. Nikon 200-400mm f/4 VR II. 5. Canon 100-400mm f/4.5-5.6L IS II. 6. Nikon 300mm f/2.8 VR II. 7. Fujifilm XF 100-400mm f/4.5-5.6 R LM OIS WR. 8. Nikon 400mm f/2.8E. Are Canon extenders worth it? The Canon Extender 2x II works great, but it only can do so much. It really does give a much longer focal length, but with two stops less speed, while remaining pretty sharp and with very good AF performance. AF is slower, especially if it has to rack a long way in or out. Are lens extenders worth it? While teleconverters give you decent image quality, they still cause the photos to lose some of it. However, they are still much better than cropping the image, and they preserve way more quality than cropping. This probably goes without saying, bit when you’re using a longer lens, there’s more camera shake. What is the difference between an extender and a teleconverter? Quick contrast. Teleconverters act as a magnifying glass vs. an extension tube pushes the lens element closer to the subject. Teleconverters can focus to infinity, an extension tube reduces the maximum focal distance.	{"url":"https://liverpoololympia.com/when-would-you-use-a-400mm-lens/","timestamp":"2024-11-04T12:16:21Z","content_type":"text/html","content_length":"75161","record_id":"<urn:uuid:d6553eb0-f43c-4a6b-88f3-70d76dbba24f>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00260.warc.gz"}
How to Make Conditional Statements In Tensorflow? In TensorFlow, you can create conditional statements using the tf.cond function. This function takes three arguments: a condition, a true_fn function, and a false_fn function. The condition is a boolean tensor that determines whether to execute the true_fn or false_fn function. For example, if you want to implement a simple if-else statement in TensorFlow, you can do so using the tf.cond function. Here's an example code snippet: import tensorflow as tf x = tf.constant(5) y = tf.constant(10) def true_fn(): return tf.add(x, y) def false_fn(): return tf.subtract(x, y) result = tf.cond(x > y, true_fn, false_fn) with tf.Session() as sess: output = sess.run(result) print(output) In this example, the condition x > y is evaluated to be False, so the false_fn function, which subtracts x from y, is executed. The output will be -5. Overall, using the tf.cond function allows you to implement conditional statements in TensorFlow, similar to how you would in a traditional programming language. What are the limitations of using conditional statements in TensorFlow? 1. Limited expressiveness: Conditional statements in TensorFlow are limited in their ability to handle complex logical conditions and operations. They are primarily designed for simple branching based on scalar values and cannot easily handle more elaborate data-dependent branching. 2. Computational inefficiency: Conditional statements can lead to computational inefficiencies in TensorFlow, as they introduce additional control flow operations that can slow down the execution of the computational graph. 3. Graph construction issues: When using conditional statements in TensorFlow, care must be taken to ensure that the computational graph is constructed correctly. Incorrect placement or use of conditional statements can lead to errors in the graph construction process. 4. Debugging difficulties: Conditional statements can make debugging TensorFlow code more challenging, as they introduce additional complexity and potential sources of errors. Debugging conditional statements may require careful inspection of the computational graph and extensive testing to identify and fix any issues. 5. Limited support for dynamic shapes: Conditional statements may not support dynamic shapes in TensorFlow, which can limit their usefulness in scenarios where the input data or model architecture varies in size or dimensions. This can make it difficult to apply conditional statements in more flexible and adaptable settings. What is the role of control dependencies in conditional statements in TensorFlow? In TensorFlow, control dependencies play a crucial role in conditional statements by ensuring that certain operations are only executed after certain conditions have been met. When a conditional statement is executed in a TensorFlow graph, control dependencies can be used to specify that certain operations should only be executed after the condition has been evaluated. This ensures that the correct order of operations is maintained and that the graph behaves as expected. For example, consider a simple conditional statement in TensorFlow: 1 if condition: 2 operation1 = ... 3 else: 4 operation2 = ... If there are operations that depend on the results of either operation1 or operation2, control dependencies can be used to ensure that these operations are only executed after the conditional statement has been evaluated: 1 with tf.control_dependencies([operation1, operation2]): 2 dependent_operation = ... By using control dependencies in conditional statements, you can control the order of execution in your TensorFlow graph and ensure that operations are only executed when their dependencies have been satisfied. This can help to prevent issues such as race conditions and ensure that your graph behaves as expected. How to pass variables as arguments in conditional statements in TensorFlow? In TensorFlow, you can pass variables as arguments in conditional statements using the tf.cond() function. This function takes three arguments: predicate, true_fn, and false_fn. The first argument, predicate, is a boolean tensor that determines whether to execute true_fn or false_fn. The true_fn and false_fn arguments are lambda functions that specify the operations to be executed based on the value of the predicate. Here's an example of how to pass variables as arguments in conditional statements in TensorFlow: 1 import tensorflow as tf 3 # Define the variables 4 x = tf.Variable(10) 5 y = tf.Variable(20) 7 # Define the predicate 8 predicate = tf.less(x, y) 10 # Define the true_fn and false_fn lambda functions 11 true_fn = lambda: tf.add(x, y) 12 false_fn = lambda: tf.subtract(x, y) 14 # Conditionally execute the true_fn or false_fn based on the predicate 15 result = tf.cond(predicate, true_fn, false_fn) 17 with tf.Session() as sess: 18 sess.run(tf.global_variables_initializer()) 19 output = sess.run(result) 20 print(output) In this example, the tf.less() function is used to create the predicate based on whether x is less than y. The true_fn lambda function adds x and y, while the false_fn lambda function subtracts x from y. The tf.cond() function then executes either true_fn or false_fn based on the value of the predicate. You can customize the true_fn and false_fn lambda functions to include any operations you want to perform based on the condition. What is the difference between tf.cond and tf.where in TensorFlow? In TensorFlow, tf.cond and tf.where are used for conditional operations, but they are used in different contexts. 1. tf.cond: • tf.cond is used to conditionally execute operations based on a specific condition. • It takes in a predicate (condition) and two functions as arguments. The first function is called if the condition is true, and the second function is called if the condition is false. • tf.cond is typically used for controlling the flow of operations within a TensorFlow graph. 1. tf.where: • tf.where is used to select elements from two tensors based on a condition. • It takes in a boolean mask (condition) and two tensors as arguments. It returns a tensor containing elements from the first tensor where the condition is True, and elements from the second tensor where the condition is False. • tf.where is typically used for element-wise conditional selection or masking operations. In summary, tf.cond is used for controlling the flow of operations, while tf.where is used for selecting elements based on a condition. How to debug conditional statements in TensorFlow? To debug conditional statements in TensorFlow, you can use print statements along with TensorFlow's eager execution mode. Here are some steps to help you debug conditional statements in TensorFlow: 1. Enable eager execution mode: Eager execution enables you to evaluate operations immediately without building a computational graph. This makes it easier to debug your code as you can see the results of operations immediately. You can enable eager execution by adding the following code at the beginning of your script: 1 import tensorflow as tf 2 tf.enable_eager_execution() 1. Use print statements: Insert print statements within your conditional statements to see the values of tensors and variables at different stages of execution. For example: 1 x = tf.constant(5) 2 y = tf.constant(10) 4 if x > y: 5 print("x is greater than y") 6 else: 7 print("y is greater than x") 1. Use tf.print: TensorFlow provides a tf.print operation that allows you to print the values of tensors during execution. You can use tf.print within your conditional statements to debug your code. For example: 1 x = tf.constant(5) 2 y = tf.constant(10) 4 if x > y: 5 tf.print("x is greater than y") 6 else: 7 tf.print("y is greater than x") By using print statements and tf.print within your conditional statements, you can easily debug and troubleshoot issues in your TensorFlow code.	{"url":"https://article-blog.kdits.ca/blog/how-to-make-conditional-statements-in-tensorflow","timestamp":"2024-11-01T19:33:37Z","content_type":"text/html","content_length":"179027","record_id":"<urn:uuid:f91f5f82-d932-4f8a-a097-b220b3228dfc>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00391.warc.gz"}
Advanced Machine Learning and Signal Processing Quiz Answers All Weeks Advanced Machine Learning and Signal Processing Quiz Answers Advanced Machine Learning and Signal Processing Week 01 Quiz Answers Quiz : Machine Learning Q1. Please order the following terms on their dimensionality • scalar, vector, matrix, tensor • scalar, matrix, vector, tensor • vector, scalar, matrix, tensor Q2. A line can seperate two point clouds in 2D space. How would you call a line of separation in 3D space? Q3. A line can seperate two point clouds in 2D space. How would you call a line of separation in 4D space? Q4. How do you call the process of predicting a continuous value? • Classification • Regression • Clustering Q5. How do you call the process of predicting a discrete (categorical) value? • Classification • Clustering • Regression Q6. How do you call the process of finding data points which belong together? • Clustering • Classification • Regression Quiz : ML Pipelines Q1. What are Machine Learning Pipelines? Please choose all correct answers • A way to do feature engineering within the pipeline framework • A way of expressing your complete end-2-end machine learning flow within a single framework with a homogeneous API • A way of making ML algorithms run faster • A way of speeding up ML development Q2. How is the class called which transforms a string class label to a class index in SparkML? • Bucketizer • OneHotEncoder • StringIndexer • VectorAssembler Q3. What is the class called which transforms a set of columns in a data frame to a single DenseVector representation in SparkML?1 point • VectorAssembler • OneHotEncoder • Bucketizer • StringIndexer Advanced Machine Learning and Signal Processing Week 02 Quiz Answers Quiz : Linear Regression Q1. Consider the following linear regression model. y = b + w1x1 + w2x2 + w3x3 + w4x4 What answers are true? • x1 – x4 are used to predict y • y is used to predict w1 – w4 • y is used to predict x1 – x4 • w1 – w4 are used to predict y Q2. Consider the following linear regression model. y = b + w1x1 + w2x2 + w3x3 + w4x4 What is the dimensionality of the training data set? Q3. Consider the following linear regression model. y = b + w1x1 + w2x2 + w3x3 + w4x4 Now consider that the influence of x3 to predict y is very low. On other words, independently of the value of x3, y doesn’t change a lot, therefore x3 and y are weakly correlated. Please choose a value for w3 to reflect this behaviour Quiz : Splitting and Overfitting Q1. When do we have an over-fitting problem? • If we perform well on the validation set and poorly on the training set • If we perform well on the validation set and well on the training set • If we perform poorly on the validation set and poorly on the training set • If we perform well on the training set and poorly on the validation set Quiz : Evaluation Measures Q1. What’s the accuracy given the true labels t and the predicted labels p Note: This is a classification problem, so the value needs to be between 0 and 1 t = (1,3,2,4,3,2,1,4,3,2,3,4) p = (1,2,2,4,4,2,1,4,1,2,3,4) Format: 0.XX Quiz : Logistic Regression Q1. Which statement is correct? • Logistic Regression is a supervised machine learning model to predict a discrete (categorical) value • Logistic Regression is a supervised machine learning model to predict a continuous value • Logistic Regression is a unsupervised machine learning model to predict a continuous value • Logistic Regression is a unsupervised machine learning model to predict a discrete (categorical) value Quiz : Naive Bayes Q1. Bayes’ theorem is used for reversing the order of joint probabilities. Q2. A box contains 3 lemons and 6 apples. We draw two fruits at random. What’s the probability of getting 2 lemons? Please round your answer to 3 decimal places. Q3. You roll a six sided die. What’s the probability of rolling a 3 or a 6? Please round your answer to 3 decimal places. Q4. Bayesian inference is a method of inference where the probability of a _________________ is updated as new evidence becomes available. • prior • hypothesis • posterior • distribution Q5. Why is the Gaussian distribution often used in machine learning? • The process of sampling any random distribution with finite variance and adding the numbers together produces a Gaussian distribution (Central Limit Theorem) • Occurs naturally in many situations (age, height of people, blood pressure readings etc.) • It is easily described (you only need a mean and a variance) • All of the above Q6. The process of Bayesian inference involves the following steps: 1. Collect data 2. Calculate the likelihood 3. Obtain a posterior 4. Obtain a prior What is the correct order of executing the above steps? • 4, 1, 2, 3 • 1, 4, 2, 3 • 3, 1, 2, 4 • 4, 2, 1, 3 Q7. In Bayesian statistics, MAP stands for • Mean accuracy projection • Maximum a posteriori probability • Manifold associated probability • Measurement adoption process Q8. Naive Bayes is considered “naive” because • it is an outdated technique and better methods exist nowadays. • the input features are considered to be independent. • it can be used only with a Gaussian distribution. • it can’t handle multiple input features. Quiz : Support Vector Machines Q1. Why are Support Vector Machines also called “maximum margin classifier”? • Because margins from the distances are maximized when computing the boundaries of separation • Because distances from the margin are maximized when computing the boundaries of separation • Because distances from the decision boundaries are maximized when computing the hyper-plane of separation Quiz : Testing, X-Validation, GridSearch Q1. What is the purpose of a test set in contrast of a train and validation set? • A test set is used to assess over-fitting hyper-parameters • A test set is used to improve model performance • A test set is used to prevent over-fitting hyper-parameters Q2. When adding pipeline or model hyper-parameters to the search grid – what is the relation between number of tune-able hyper-parameters and the growth in computational complexity? • linear • exponential • logarithmic • cubic Quiz : Enselble Learning Q1. How are Random Forest different in re-sampling from Gradient Boosted Trees? • Re-sampling doesn’t differ in those models • Sampling is done using Bootstrapping in RandomForests wheres Gradient Boosted Trees use Boosting • Sampling is done using Boosting in RandomForests wheres Gradient Boosted Trees use Bootstrapping Q2. Which model is mostly prune to overfitting? • Random Forest • Gradient Boosted Trees • Decision Trees Quiz : Regularization Q1. Which regularization technique is penalizing large model parameters most? • L1 Regularization • L2 Regularization Q2. When is it appropriate to use Regularization • To prevent underfitting • To prevent overfitting Advanced Machine Learning and Signal Processing Week 03 Quiz Answers Quiz : Clustering Q1. Which of the following algorithms needs you to pre-specify the expexted number of clusters? • kmeans • Distribution-based clustering • Density-based clustering • Hierarchical clustering Q2. Which algorithm let’s you visually determine a good number of clusters based on it’s output? • kmeans • Density-based clustering • Distribution-based clustering • Hierarchical clustering Quiz : PCA Q1. What are the implications of using highly dimensional data? • Data becomes sparse as we add dimensions • Adding more dimensions reduces the size of the data set • Distances loose meaning in high dimensions • Adding more dimensions reduces the collinearity in the data Q2. The process of reducing the number of random variables by obtaining a smaller set of artificial features is known as • feature reduction • feature selection Q3. What are some linear methods for dimensionality reduction? • Principal Component Analysis (PCA) • Linear Discriminant Analysis (LDA) • Self-organising Maps (SOM) • Autoencoders Which line gives the direction of greatest variance in the data set plotted above? Q5. We use _______________ to measure how a group of random variables vary together. • correlation • direction of variance • covariance • Kullback–Leibler divergence Q6. Using PCA to reduce the dimensionality of a data set involves the following steps: 1. Centre the data 2. Find the eigenvalues and eigenvectors of Sigma 3. Compute the covariance matrix 4. Select new dimensions and project the data Put the steps in the correct order. • 1, 4, 3, 2 • 1, 2, 4, 3 • 1, 3, 2, 4 • 4, 1, 3, 2 Q7. In PCA, the second principal component is ______________ to the first principal component. • perpendicular • parallel • opposite • identical Q8. SystemML provides an out-of-the-box implementation of PCA. Advanced Machine Learning and Signal Processing Week 04 Quiz Answers Quiz : Fourier Transform Q1. The Fourier transform is an invertible transformation between the time and frequency domain representations of a signal. The figure above shows two signals A and B, that have the same frequency and phase shift, but different amplitudes. What would the sum (A+B) of these two signals look like? • The sum of A and B would be • The sum of A and B would be • The sum of A and B would be • The sum of A and B would be The plot above shows a signal in the • time domain • frequency domain Q4. The reduction of a continuous time signal to a discrete time signal is known as • anti-aliasing • sampling • Z-transforming • low-pass filtering Q5. You have the following continuous signal but when sampled its plot looks like this: What is the most likely explanation for this effect? • The ADC is not functioning correctly • The sampling rate is too high • The sampling rate is too low • The ADC resolution is too low to handle this frequency Q6. You have the following frequency domain plot of a signal that’s been generated by adding two separate signals (A and B) together. What can the plot tell you about the components of the original signal? Assume that the axes follow the same convention that’s been used so far in this module. • The frequencies of A and B are 3.0 and 5.0 Hz. • The frequencies of A and B are 2.0 and 3.0 Hz. • The amplitudes of A and B are 2.0 and 3.0 • The amplitudes of A and B are 3.0 and 5.0 Q7. A known limitation of FT/DFT is that it requires an infinite series of sinusoids to represent a signal, so it cannot be used efficiently in a discrete setting. Q8. Discrete Fourier Transform is slower compared to Fast Fourier Transform. • Correct. The computational complexity of DFT is O(n^2) in contrast to O(nlog(n)) for FFT. • This is incorrect. Please, review the lecture on FFT. Quiz : Wavelet Transform Q1. A signal that does not change in time is said to be generated by • a stationary process • a non-stationary process Q2. Which of the following signals are generated by a stationary process • white noise (a signal containing many frequencies with equal intensities) • electrocardiogram (ECG) • a sum of multiple sine waves, each having a fixed frequency and amplitude • the sound of a fireworks display Q3. A key limitation of Fourier transform is that it cannot provide information on when specific frequencies occur in the signal. Q4. The visual representation of a wavelet transform is called • a histogram • a spectrogram • a scaleogram • a sonograph Q5. The wavelet defined by the function ψ(t) and used in the scaling and translation process is called • base wavelet • initial wavelet • mother wavelet • pseudo-wavelet Q6. The x and y axes of a 2D scaleogram represent= • time • amplitude • scale • frequency Q7. Passing the signal through a series of low pass and high pass filters is a step in the calculation of= • Discrete Wavelet Transform (DWT) • Continuous Wavelet Transform (CWT) • Fast Fourier Transform (FFT) • Fourier Transform (FT) The signal shown on the plot above has been generated by • a stationary process • a non-stationary process Get All Course Quiz Answers of Advanced Data Science with IBM Specialization Fundamentals of Scalable Data Science Coursera Quiz Answers Advanced Machine Learning and Signal Processing Quiz Answers Applied AI with DeepLearning Coursera Quiz Answers Advanced Data Science Capstone Coursera Quiz Answers	{"url":"https://networkingfunda.com/advanced-machine-learning-and-signal-processing-quiz-answers/","timestamp":"2024-11-13T05:31:54Z","content_type":"text/html","content_length":"166769","record_id":"<urn:uuid:a6bce4ac-1842-4137-be8b-575e1b81e30c>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00289.warc.gz"}
propane metric tons to barrels For each of the cryogenic fluids, a second table of properties is provided for the vapor at a pressure of one standard atmosphere. tons CO 2 for natural gas + 0.20 metric tons CO 2 for 1 cubic meter of Propane weighs 493 kilograms [kg] 1 cubic foot of Propane weighs 30.77698 pounds [lbs] Propane weighs 523.56 m3. We would like to show you a description here but the site wont allow us. /MWh Fuentes: MIBGAS, OMIE, Enags GTS. This constant represents the approximate amount of carbon dioxide (CO 2) that is produced when you burn a gallon of propane.. Notes "The US Energy Information up to 35,000 barrels per day of propane and produce up to 1.65 billion pounds per year of polymer grade propylene (PGP). Metric Conversion; Metric Converter; Site Map; Contact; This site is owned and maintained by Wight Hat Ltd. 2003-2020. Conversion Tables. To convert US barrels (oil) to metric tons (or tonnes), multiply the value in US barrels (oil) by 0.15644376609749119367. Residual Fuel Oil - 6.287 million Btus per barrel. 10,000 Tons Of Oil Equivalent to Barrels Of Oil Equivalent = 68411.7647. 0.07. metric teaspoon. Oil producers in Europe and Asia tend to measure in metric tons. Platts Conversion Base Rates guide identifies the base rate values used energy, mass and volume conversions used throughout Platts assessment processes and published content. = 2.20462 lb. Sth. Full weight: 1,333 pounds. Since the 13-year low in February 2016, nickel prices have been showing strong cyclical growth. To convert metric tons (or tonnes) to US barrels (oil), multiply the value in metric tons (or tonnes) by 7.1475121. 1 ton of crude oil = 1 metric ton of crude oil = appr. Convert Metric Tons to Tons. Full BTU content is 18,300,000 BTUs; thats enough to provide 10,000 BTU/h for 76 days And 6 hours. Abbreviation Description; MW: megawatt: Propane: 1.0 Cubic A metric ton (2205) divided into 333 equals 6.62. As of July 1, 2022, we are continuing to restore our systems. Multiply For $/MT 1 ton reg = 17.8107606679 bbl (oil) Example: convert 15 bbl (oil) to ton reg: 15 bbl (oil) = 15 0.0561458333 ton reg = 0.8421875 ton reg. Search: 454 To 350 Swap. gallon to barrel and pint to quart Click for free download Click for free download. oxygen 1.427 metric tons. 1 m3 219.969 imperial gallons 264.172 US gallons. 12 barrels oil per day to tonnes (water mass) per year = 696.36 ton/yr. Abbreviation Description; MW: megawatt: kW.h: kilowatt hour: 6.2898 Barrels (bbl) 1.0 Cubic metres (m) 1,000 Litres (l) 1.0 Cubic metres (m) Propane: 1.0 Cubic metres (m) 25.53 Gigajoules (GJ) Butane: 1.0 Cubic metres (m) About Propane. assuming its 8 in this case, that is 250 lbs per barrel. of crude oil with 7.333 bbl. The amount of petroleum in a reservoir is measured in barrels or tons. It can hold a maximum of 200 gallons (80% rule) Empty weight: 483 pounds. Multiply For $/Bbl. 2.0301312457499E-8 Mt LNG. Conversion base : 1 MMBtu = 2.0301312457499E-8 Mt LNG. The symbol is " m ". You can A gallon is 3.7854 L, so around 10.4 million gallons. Platts Conversion Base Rates guide identifies the base rate values used energy, mass and volume conversions used throughout Platts assessment processes and Kilotons to Barrel of Oil Equivalent (US)s Conversion. The monthly data releases, including the Petroleum Supply Monthly, Natural Gas Monthly, and Electric Power Monthly, will be published next week.We will continue to post regular updates regarding the status of other data products. 1 metric ton motor gasoline = 8.53 barrels 1 metric ton LP-gas (liquefied petroleum gas) (propane) = 11.6 barrels 1 metric ton natural gas = 10 barrels 1 metric ton NGL (natural gas barrels per day: bbl: equivalent to metric ton: MMt: Million Metric tons: MMt/y: Million Metric tons per year: Electricity. = 50.802 kg. Propane increased 0.18 USD/GAL or 17.15% since the beginning of 2022, according to trading on a contract for difference (CFD) that tracks the benchmark market for this commodity. COAL TAR PRODUCTS Coal tar and materials traditionally derived from it such as benzene, toluene, and xylenes are sometimes measured in thousands of U.S. This is a revision to the 2001 edition of the ASHRAE Handbook. Metric Tons. Bookmark File PDF Integration Propane Dehydrogenation Pdhgrowing demand across core end-markets. 1 Metric Tonne LPG (Propane) = 12.7 barrels. There are Conversion rate between Barrel and mt volume units and converted BARREL/MT rate with the answer. Revised tables have been prepared for R23, R32, R124, R125, R152a, R245fa, R404A, R407C, R410A, R507A, propane, butane, isobutane, propylene, nitrogen. 0.2. Metric Tons per Year. 11 barrels oil per hour to tonnes (water mass) per hour = 1.75 ton/hr. 7.3 barrels of crude oil (assuming a specific gravity of 33 API) = 6.6-8.0 bbl. kton stands for kilotons and BOE stands for barrel of oil equivalent (us)s. The formula used in kilotons to barrel of oil equivalent (us)s conversion is 1 Kiloton = 683.898914248348 Barrel of Oil Equivalent (US). Calculate. novembro 21, 2021 Por Por 10 barrels oil per hour to tonnes (water mass) per hour = 1.59 ton/hr. Lower heating value considers vapor not to be useful and does not count its energy. O 2 / H 2O d The metric measures are a system of measurement used in all other countries except the United States Occupies a volume of the stone, m, using an electronic balance, and! Crude Oil Futures (Light, Sweet Crude & Brent Crude) The value of a $ /bbl price move on this contract = Calculate $ NYMEX E-mini Crude Oil Energy Futures. 283, 302, 307, 350, 396, 400 and 454 CID Engines 1967 - 1974, 327 and 427 CID Engines 1967-1969 (Except Conversion $12 Our Shop Trucks Our shop truck has been running on our propane kit for the last 14 years on the same propane kit! of crude 1 Barrel LPG 1 Cubic meter = 6.28981077 Oil barrels. Page 33/52. BARREL To MT Conversion 1 Megatonne or metric megaton (unit of volume) is tonne (metric) of TNT / ton of TNT / kilogram of TNT Liquefied Petrolium Gas (LPG or LP gas) is often reffered The conversion factors are used consistently in the datafiles and in this website 1 Barrel LPG (Propane) = 0.0787 metric tonnes 1 Metric Tonne LPG (Propane) = 12.7 barrels 1 Barrel LPG (Butane) = 0.090 metric tonnes 12-31-2010, 11:48 PM. Easy tons to t conversion. India has the world's 5th largest proven coal reserves with nearly 177 billion metric tons as on 1 April 2021. In India, coal is the bulk primary energy contributor with 56.90% share equivalent to 452.2 Mtoe in 2018. A ton, also referred to as a short ton, is a unit of weight equal to 2,000 pounds. Project Management 9 M DOE grant to design and build a sensor system to help farmers make changes in how they grow and harvest crops that ultimately helps reduce carbon emissions 75 million tonnes/year and crude oil/ condensate distillation capacity of 280,000 barrels per day 5 metric tons annually 5 metric tons annually. See all; Folding Ramp & Tilt Deck Trailers; 12 barrels oil per hour to tonnes (water mass) per hour = 1.91 ton/hr. Barrels of oil consumed. Search: Pttgc Project Engineer. Oil companies registered on the New York Stock Exchange Search: Pttgc Project Engineer. A metric unit of volume, commonly used in expressing concentrations of a chemical in a volume of air. Enter the number of tons to convert into metric tons. Barrel Of Oil Equivalent (BOE): A barrel of oil equivalent (BOE) is a term used to summarize the amount of energy that is equivalent to the (28 June 2021 ) Nickel prices were up 44% year-over-year in May 2021 , from $12,179 to $17,577 per metric ton . 13 barrels oil per day to tonnes (water mass) per year = 754.39 ton/yr. Metric Tons <-> Barrels. PTTGC wants to finalise their investment decision in 2016 or 2017 Schweitzer Engineering Laboratories-3: Silicon Power-3: SilverLining-3: Skytran Inc-3: SNC-Lavalin: Atkins North America: 3: SPIE International Society for Optics & Photonics-3: State University System of Florida-3: Sunnova Energy-3: SunZia Southwest Transmission Project-3: The answer is 7.3997773769789. iPhone & Android app Volume US Barrels (Oil) See also US liquid barrels, US federal barrels, US dry barrels and UK barrels. If we assume is it 6 300 Million Barrels Of Oil Equivalent to Tons Of Oil Equivalent = 43836795.6435. Calculate $/Barrel $/mmBtu. 1 Metric Tonne LPG (Propane) = 12.7 barrels 1 Barrel LPG (Butane) = 0.090 metric tonnes 1 Metric Tonne LPG (Butane) = 11.1 barrels 1 Barrel LPG (average) = 0.086 metric tonnes 6 Million Barrels Of Oil Equivalent to Tons Of Oil Equivalent = 876735.9129. When fully charged, you will pay about 4.6 per mile for the first 42 miles you drive ($1.94 for the electricity and $.00 for the gasoline). Units: tonne (metric) of TNT / ton of TNT / kilogram of TNT / barrel of oil equivalent (BOE) Liquefied Petrolium Gas (LPG or LP gas) is often reffered as simply propane or butane. This vehicle did not use any gasoline for the first 42 miles in EPA tests. In 2020, the total capacity of global refineries for crude oil was about 101.2 million barrels per day. Indias coal production has only fallen once in the last 30 years when the figure fell from 319 mt in 1997 to 316 mt in 1998. 1 metric tonne = 2204.62 lb = 1.1023 short tons 1 kilolitre = 6.2898 barrels 1 kilolitre = 1 cubic metre 1 kilocalorie (kcal) = 4.1868 kJ = 3.968 Btu 1 kilojoule (kJ) = 0.239 kcal = 0.948 Btu 1 This gives an answer of approximately 172.9 barrels (7,263 / 42). A metric ton, or tonne, is a unit of weight equal to 1,000 kilograms. One cubic meter also equals A metric ton, or tonne, is a unit of weight equal to 1,000 kilograms. Propane Bookmark File PDF Convert emissions or energy data into concrete terms you can understand such as the annual CO 2 emissions of cars, households, and power plants.. 6-8 oil barrels equal a ton. Higher heating value includes the energy of water vaporization. Metric Tons per Year. assuming its 8 in this case, that is 250 lbs per barrel. This calculator allows you to calculate the amount of each fuel necessary to provide the same energy as 1 kg of hydrogen, 1 million cubic feet natural gas, 1 barrel of crude oil, or 1 gallon of One cubic meter equals 35.3 cubic feet or 1.3 cubic yards. Barrels per Day. = 6.2898 American barrels 264.17 American gallons = 28.35 grams 0.453592 kilograms = 0.009 cwt. Sergio Russo/CC-BY-2.0. If Mont Belvieu propane costs 64cts/gal, you would multiply by 5.21 to convert the price to dollars per metric ton (also known as a tonne). Calculate. Change the API gravity first and then input a number in any blank and press the "Calculate" button. 13 barrels oil per hour to tonnes (water mass) per hour = 2.07 ton/hr. convert gross barrels to net barrels. Liquid petroleum gas converts at a rate of 11.6 barrels per metric ton, the highest ratio of all the refined oil products. Annual Fuel Cost* $1,950: Cost to Drive 25 Miles: $3.29: Cost to Fill the Tank: $67: Tank Size: 13.7 gallons Based on 45% highway, 55% city driving, 15,000 annual miles and current fuel prices. 1 metric ton = 2205 pounds. Sources. Did you ever wonder what reducing carbon dioxide (CO 2) emissions by 1 million metric tons means in everyday terms?The greenhouse gas (GHG) equivalencies calculator can help you understand just that, translating abstract measurements and emissions data into concrete terms, such as the annual emissions from cars or households.There are two options for 7 Million Barrels Of Oil Equivalent to Tons Of Oil Equivalent = 1022858.565. Carry Deck Crane 11 - 15 Ton; Cary Deck Crane 4 - 10 Ton Propane Convection & Radiant Heaters 22k - 200k Bt; Propane/ natural Gas Direct-fired Heaters 30k-2.5m; Traffic Cones & Barrels; Barricades & Warning Lights; Traffic Control Signs; Trailers. 1 US gallon of propane is equivalent 0.00221 tonne. How much is 0.00221 tonne of propane in US gallons? 0.00221 tonne of propane equals 1 US gallon. In the U.S., there are 7.33 barrels in a metric ton. There are 0.15644376609749 metric tons (or tonnes) in a US barrel (oil). 1 US barrel (oil) is equal to 0.15644376609749 metric tons (or tonnes). 1 million metric tons LNG = 1.23 million metric tons oil equivalent 1 million metric tons LNG = 52 trillion Btus 1 million metric tons LNG = 8.68 million barrels oil equivalent 1 million metric tons 15 barrels oil per day to tonnes (water mass) per year = 870.46 ton/yr. It is commonly used in the United States. 2016 Present Director/ Assistant Managing Director Project,BJC Heavy Industries Pcl (GCP), a joint venture between PTTGC and Japanese partners Sanyo Chemical Industries (SCI) and Toyota Tsusho Corporation (TTC) Industry peers and experts, from both owner operators and E&C's, will discuss how to make sharp and The value of a $ /bbl price move on this contract = Calculate $ CME Group is the worlds leading derivatives marketplace. 30.77698 pounds [lbs] of Propane fit into 1 cubic foot. To determine the number of metric tons of CO 2 equivalent emitted per litre of gasoline combusted, natural gas, heating oil, wood, and any other fuel used for heating such as coal and propane. That equals ALMOST 9 barrels of oil (8.82). What is a Cubic Meter? In a big flip-flop, propane has been the preferred feedstock for petrochemical plants on the Gulf Coast for a couple of weeks now (it had been ethane for the most part of the last 3+ years). 1 barrel of residual fuel oil = 6,287,000 Btu; 1 cubic foot of natural gas = 1,039 Btu; 1 gallon of propane = 91,452 Btu; 1 short ton (2,000 pounds) of coal (consumed by the electric Finally, convert gallons of oil to barrels by dividing the volume by the conversion of 42 gallons per barrel. Heres an example of what we wanted: 250 gallon residential propane tank is 7 feet 8 inches long, with a diameter of 30 inches. propane (liquid) = 25.53 GJ/m3 propylene = 25.53 GJ/m3 still gas = 41.727(1) MJ/L sulfur = 9.337 GJ/tonne = 42.433 GJ/tonne = 42.987 GJ/tonne HEATING VALUES ammonia 0.77 asphaltic additional >100,000 metric tons of propylene to meet Page 22/52. To The answer is 0.1351392006888. of oil equals 7,263 gallons of oil (55,125/7.59). Propane cylinders vary with respect to size; for the purpose of this equivalency calculation, a typical cylinder for home use was assumed to contain 18 pounds of propane. Thus a gallon of ethane weighs 2.972 pounds [2205 / 742 = 2.972.) Cubic meter (metre) is a metric system volume unit. Background; You can repeat = 0.01 quintal = 112 lb. How to Convert Ton Register to Barrel (US) 1 ton reg = 23.7476808905 bbl (US) 1 bbl (US) = 0.042109375 ton reg. The conversion rate for crude oil is an approximation because it is based on the worldwide average gravity of crude oil. Cubic meters of propane to tonnes; 0.1 cubic meter of propane = 0.0583 tonne: 1 / 5 cubic meter of propane = 0.117 tonne: 0.3 cubic meter of propane = 0.175 tonne: 0.4 cubic meter of CONVERSION TABLE. 50 Tons Of Oil Equivalent to Barrels Of Oil Equivalent = An oil barrel is about 42 gallons. It may use gasoline depending on how you drive. 0.2. 14 barrels oil per day to tonnes (water mass) per year = 812.43 ton/yr. Naphtha Conversion factors. edward jones rates of return. There are 742 gallons per metric ton of ethane. Unit Divide For Cts/Gal Divide For $/Bbl. You can use the formula : US barrels (oil) = metric tons (or tonnes) 7.1475121 Barrels per Day. The actual energy may vary up to 10%. Our Shop Trucks Our shop truck has been running on our propane kit for the last 14 years on the same propane kit!. taken as average = 1.16 kl. 1 Cubic meter = 8.38641436 Fluid barrels. 1 Barrel LPG (Propane) = 0.0787 metric tonnes. Trailers. 1 tonne of propane [gas] at 1.013 bar and 15 C takes up a volume of: volume = 1000 kg 1.91 kg/m3. mass = d v vcfmcf, where mcf is the conversion factor to convert from tonne to kilogram (table near the end of this page) and vcf equals 1 because the volume is already in cubic meters. LPG (propane) Gas Unit Conversions: Gas in kg, Litres, MJ, kWh & m Also Propane Gas Unit Conversion in Pounds, Gallons, BTU, Therms & ft Where LPG is propane, 1kg of LPG A metric Ton is 2205 pounds. There are 42 U.S. gallons in a barrel. There are 226.77 centimoles in 100 grams of Propane. barrels of crude oil: x: 42 a = gallons of crude oil, US: barrels of crude oil: x: 158.987 3 = liters of crude oil: barrels of crude oil: x: 0.136 = metric tons of crude oil: barrels of This constant represents the approximate amount of carbon dioxide (CO 2) that is produced when you burn a gallon of propane.. Notes "The US Energy Information Administration (EIA) estimated1 that U.S. gasoline and diesel fuel consumption for transportation in 2013 generated 1,095 million metric tons of CO 2 from gasoline and 427 million metric tons of CO2 How to convert US barrels (oil) to metric tons (or tonnes)? A standard ton is 2000 pounds. PRODUCTOS SPOT Y PROMPT - PVB PRODUCTOS FUTUROS - PVB PRODUCTOS SPOT TVB Y AVB PRECIOS DEL GAS NATURAL (/MWh) El producto Day Ahead se corresponde con el producto de entrega en el siguiente da de gas a su negociacin.. "/> Conversion of units between 1 Ton (Displacement) and Barrel (Petroleum) (1 and bl; bbl) is the conversion between different units of measurement, in this case it's 1 Ton (Displacement) and Barrel (Petroleum), for the same quantity, typically through multiplicative conversion factors ( 400 Million Barrels Of Oil Equivalent to Tons Of Oil Equivalent = 58449060.8579. 6-8 oil barrels equal a ton. 493 kilograms [kg] of Propane fit into 1 cubic meter. A metric Ton is 2205 pounds. Unregistered Share Tweet #17. A ton, also referred to as a short ton, is a unit of weight equal to 2,000 pounds. Leading international agencies have made the following nickel price predictions:The World Bank, in its commodity forecast report, estimated that nickel US teaspoon. How many barrels of diesel are in a metric ton? Barrels of oil consumed. 5.80 mmbtu/barrel 20.31 kg C/mmbtu 44 kg CO 2 /12 kg C 1 metric ton/1,000 kg = 0.43 metric tons CO 2 /barrel. Your price is now just over $333/mt. One barrel [US, petroleum] is equal to how many diesel [metric ton]? equivalent to metric ton: MMt: Million Metric tons: MMt/y: Million Metric tons per year: Electricity. The conversion factors are used consistently in the datafiles and in this website. This measurement is usually used by oil producers in the United States. Energy value of a LPG depends on a particular mixture of propane and butane. Conversion Tables. We provide conversion for a typical energy value. If we assume is it 6 barrels per ton, than 2000 divided into 6 is 333.33. m5 high profile handguard. Elaboracin propia. One metric ton is equal to 2,240 pounds, so:588,000,000 pounds / 2,240 = about 262,500 metric tons in 2 million barrels. Search: 454 To 350 Swap. ton to picojoule ton to megaton ton to cheval vapeur heure ton to hundred cubic foot of natural gas ton to gigawatt-hour ton to erg ton to celsius heat unit ton to Q unit ton to nanojoule ton to About Propane. Most recently, I picked up a GM 1982 Light Duty Truck Factory Service Manual, which includes a full wiring diagram m&p shield ez pistol important safety recall notice for pistols manufactured between march 1st, 2020 and october 31st, 2020 learn more Number of ZR350 vehicles = 1 3l gets a set of 2011 doors Now is the time to upgrade Now is the time to Conversely, residual fuel oil converts at the I could tell it was sitting higher I don't have the frame mounts for the 454 030 OVER: BORED Use all of the parts from the 454 truck in the K10 Messages: 9 I got a 454 engine from a crashed, what I believed was a, 1976 Chev or GMC 1 ton truck 2WD I got a 454 engine from a crashed, what I believed was a, 1976 Chev or GMC 1 ton truck 2WD. Mont Conversion base : 1 Mt LNG = 49257899.069014 MMBtu. How many tons are in a metric ton? Brent 7.52 barrels per metric tonne Dubai 7.20 barrels per metric tonne Conversion factors for petroleum product categories LPG 11.00-11.80 barrels per metric tonne Motor It is commonly used in most countries, except the United States, where the short ton is used instead. 5.80 mmbtu/barrel 20.31 kg C/mmbtu 44 kg CO 2 /12 kg C 1 metric ton/1,000 kg = 0.43 metric tons CO 2 /barrel. Easy t to tons conversion. Oil refineries are Liquified gas storage vessels store propane and similar gaseous fuels at pressure sufficient to maintain them in liquid form. Barrels. Most other countries use the metric ton, or "tonne". = 0.98421 long tonor English = 1.10231 Please This is a conversion chart for barrel of oil equivalent (Oil Energy Equivalent). 40 Tons Of Oil Equivalent to Barrels Of Oil Equivalent = 273.6471. Example: convert 15 ton reg to bbl (US): 15 ton reg = 15 Use the search box to find your required metric converter. Propane cylinders vary with respect to size; for The Greenhouse Gas Equivalencies calculator allows you to convert emissions or energy data to the equivalent amount of carbon dioxide (CO 2) emissions from using that amount.The calculator helps you translate Refined oil products have a variety of conversion factors due to density variations. Calculate. Metric Tons per Year. More information about this unit: diesel [metric ton] / 2.89 metric tons CO 2 equivalent/ton of waste recycled instead of That equals ALMOST 9 barrels of oil (8.82). million tonnes liquefied natural gas. The density of propane is 0.51 kg/L, and you have 20 million kilograms, thus about 39.2 million liters.	{"url":"http://www.stickycompany.com/2008/street/92027538c43512265f1aa697ee80d-propane-metric-tons-to-barrels","timestamp":"2024-11-14T01:15:17Z","content_type":"text/html","content_length":"28782","record_id":"<urn:uuid:8fc1ec4e-ac5b-433c-86a7-95b62459c6e7>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00737.warc.gz"}
Use MAX to find latest date in a list [Quick tip] » Chandoo.org - Learn Excel, Power BI & Charting Online Here is a quick tip that I learned while conducting training classes in Australia. If you have several dates in a range and you want to find out what the latest date is, just use MAX, like: =MAX(A1:A10) would give you the latest date. A Question…, Assuming you have some dates (not necessarily sorted) in column A, which formula finds the last date (not latest)? Bonus question: What if there are some gaps (cells with no value)? How would you find the last date? Go ahead and post your answers in comments. Or share your favorite formula to find latest date in a range. PS: My Australian trip is over now. On a train from Melbourne to Sydney now and will be leaving to Vizag via Bangkok (and Hyderabad) early tomorrow morning. I am very happy how the whole thing went. More on this later next week. 44 Responses to “Use MAX to find latest date in a list [Quick tip]” 1. =INDEX(A:A,IF(ISERROR(MATCH(9.999999E+306,A:A)),MATCH("",A:A,-1),IF(ISERROR(MATCH("",A:A,-1)),MATCH(9.999999E+306,A:A),MAX(MATCH(9.999999E+306,A:A),MATCH("",A:A,-1))))) This seems to do it. You can use TEXT() function to format the date to your liking. 2. I just checked a simple small or min works - even with the gaps in between but ofcourse it has to be in excel date format. □ Hi Raj ur right just max or min should work, it need not be in date format..reason is date is ultimately number and min and max should give results 3. Last date with or without gaps: □ Awsome! □ What does the 9E99 do? ☆ It's simply a very large number. Stands for 910^99 I'm using it so that the MATCH function will look for the number closest to 9E99, aka the highest number aka the latest date. If we were looking for text, I'd use "zzz" to follow a similar pattern. 4. A date in Excel is just the number of days since 1 Jaunary 1900. If you can use an Excel function on a number, you can also use it on a date. To see what is going on, just change the cell format from "date" to "number". Hours, minutes and seconds are just decimal fractions of days. Excel does not display dates before 1/1/1900 (I guess this is in order to avoid any complications with the various calendar reforms that have occurred throughout history). 5. {=MAX((MyRange<(TODAY()+1))MyRange)} Where MyRange is the the data. The curly braces indicate that you've CTRL-ALT-ENTERed the formula. It looks to me that you've got to have a contiguous range for it to work, but it handles spaces. If you want the last day before today, delete the "+1". 6. I assumed that you were asking for the last date listed, with no regard to if is is the most or least current date. So this is really just a way of finding the last populated cell in a column. From here you could bolt on an =Address() and a =indirect() if you waned to get the location or value. Thanks for these challenges, Chandoo. 7. Select the range including empty cells and do a conditional formatting using top 10 and choose 1 in the dialog box making it only the highest number (latest date). It will highlight the latest date in the selected range. 8. THANK YOU!!! 9. Behold, a less than elegant solution: {=INDIRECT("A" & MAX((A:A>1)ROW(A:A)))} Alternatively, you can use "Index(a:a..." but I wanted to spice things up a bit □ how can you find the min? ☆ @Matricia If the range has blanks but no 0's use: =MIN(C1:C10) If the range has 0's use: =MIN(IF(C1:C10>0,C1:C10)) Ctrl+Shift+Enter 10. On the topic of Max dates, I use this particular array function quite frequently. It finds the latest date for a particular item. Almost like a 'MaxIf' Suppose you have a table of prices for 3 fruit which also contains historical prices: Date Fruit Price 01/01/12 Apple $1.00 01/01/12 Banana $3.00 01/01/12 Cantaloupe $4.00 03/10/12 Apple $1.35 04/11/12 Banana $2.80 06/12/12 Cantaloupe $4.05 The following formula will return the latest date for which the apple price was entered (04/11/12) As it's an array formula, Ctl+Shift+Enter is required to get the curly braces. Formula Names assume the data is formatted in a Excel 2010 table. It works fine with standard references too. Further witchcraft can get the formula to return the latest price if required. □ I think you are mixing your apples and bananas with your description of the last date for apples, bit of a fruit salad kicking in? ☆ Right you are! Although the health benefits of fruit salad are undisputed, the above should have read "The following formula will return the latest date for which the apple price was entered (03/10/12) ○ Hi Adam, I am trying to use this but not having success. Do you have to have your data in an Excel table? Or can you just have it in the spreadsheet like normal? Using your example above, I am trying to have the raw data on one sheet (so your table above on one sheet) and then on a second sheet have a table where each fruit appears only once and the formula returns the last date that the price was changed for each fruit. So I would want the formula to reference the fruit name and then look for it on the other sheet and return the last date in the cell. Like a LOOKUP function... Does that make sense? Thanks for your help. □ stumped, do you think you can assist with the price formula? 11. Use the function LASTROW() from Morefunc. It displays the last value in a column. 12. Hi For me the formula =INDEX(A:A,MATCH(9E99,A:A)) gave me the last day in a series not the latest date in calendar □ That is what Chandoo asked in the topic, to find the last day entered in the series, not the latest and that formula will provide the required solution. 13. We can also use =LOOKUP(1000000000,B:B) to get the last value in entered in column B □ Nice! I see now that we could use for latest number, and for latest text. Cool tip! 14. Hi Jordan Goldmeier, Thanks for the formula but when I tried it was giving an #Name error. However, when I tried {=INDIRECT(ADDRESS(MAX((A2:A22>1)ROW(A2:A22)),1))} it worked. □ Remember that when you copy and paste from WordPress, the quotation marks come in as fancy, slanted-quotes, which Excel treats as characters and not quotes. If you replace the quotes in the Excel formula bar, then press CTRL+SHIFT+ENTER, it should work. 15. =OFFSET("first date in colum",SUM(COUNTA("date range"),COUNTBLANK("date range"),0)) □ *EDIT Got a little too hasty typing my response. If you know the range (A2:A100) this will give you the last entry. If you don't know the last cell in the range...well you will need a more clever solution. =OFFSET(A2, SUM(COUNTA(A2:A100), COUNTBLANK(A2:A100))-1,0) 16. Array enter Will handle Mixed data types ( Text/Numbers/both ) □ The Question mark should be the Greek Letter Omega. This is significantly faster than ☆ Interesting formula Sam. Any specific reason for omega? I think even ~ should do (~ is ASCII Character 126) ○ There is an outside chance of having ~ in the data. Omega has less chances. Infact to be very safe you can use double OmegaOmega 17. I looked at all this 9E99 stuff and could not get it to work, but knew what it was I really wanted to do. I needed to find the row that had the last actual item in it from the column in question. Then I had to return the data that was in that particular row. If I used a helper column (b) on the first row, this would have contained the formula: =(A1<>"") giving me a column that was filled with true or false statements. As these are numerical, I multiplied them in another column, by the row number that I was on, ie first row formula: =ROW(A1)(A1<>"") I now have a column of numbers that are either '0' or the 'row number'. I can use the 'MAX' command to extract the row with the last data. Taking all this into an array formula gives my solution to the problem and actually dispenses with the extra columns of data by virtualising them within the array formula. Hence: {=INDEX(A:A,MAX(ROW(A:A)(A:A<>"")))} 18. Hi guys. What if I say that I have a range o dates and I wanna know the OLDEST date by a given month? I couldn't answer it. Just could find the NEWEST with this formula: {=MAX(--(MONTH(SHEET1!$D:$D)=5)( SHEET1!$D:$D ))} The MAX formula will answer it correctly because the Zeros given to the non-match cases won't interfere, but in the MIN formula, they will, giving "01/00/00" as response. Can anybody give me a hand on it? I guess it's a very nice challenge. □ Sorry, the formula is {=MAX(- -(MONTH(SHEET1!$D:$D)=5)(SHEET1!$D:$D))} 19. Assuming: 1)my dates are in range A1:A100 2)no blank lines in between 3) dates not sorted I use the following formula: And it gives me la last date in the range as recuested by Chandoo 20. how to get the latest and oldest date if I have date format like below: 2013-10-21 00:00:00 2013-11-27 00:00:00 2014-02-01 00:00:00 2014-02-21 00:00:00 2014-03-21 00:00:00 2014-06-02 00:00:00 □ Earliest date =min(range) Latest date =max(range) 21. I have a similar question to Adam's above, but I'm not able to get the same results. Let's say my daughter wears a different shirt everyday and I want to know the last date she wore her green Date Shirt 8/21/14 Green shirt 8/22/14 Blue shirt 8/23/14 Pink shirt 8/24/14 Red shirt 8/25/14 Green shirt I'm trying this formula, but I'm getting the #REF! error: =INDEX(MAX(A:A),MATCH("Green shirt",B:B,0)) 22. I'm not at my computer to test but did you make it an array formula by hitting ctrl+shift+enter? 23. How do we get the highest value enter in one cell (not in a column)? For e.g. particular cell say A1 captures the date. First time it is updated it would have values of 9/8/15. If again next it gets updated it will now have values of 9/8/15, 10/8/15 so on... How can be the highest date captured? 24. Here's the format on Sheet1 of Excel ? suppose Columns are in A, B, C, D respectively. S. No. Box Name Arrival Date No. of Pakgs 1 Box 1 19-Oct-15 4 2 Box 2 19-Oct-15 4 3 Box 3 20-Oct-15 5 4 Box 4 07-Nov-15 5 5 Box 5 08-Nov-15 4 I want to every package delivery date in below format, Let suppose in column F, G, H and I respectively. Last Delivery Date 3rd Delivery 2nd Delivery 1st Delivery The Master data is in Sheet2, having column box name, arrival date, delivery date, goods type etc Please nelp me, how to get desire result. 25. HI, I would like t find out nearest date from today(before and after today)	{"url":"https://chandoo.org/wp/use-max-to-find-latest-date-in-a-list/","timestamp":"2024-11-13T17:52:32Z","content_type":"text/html","content_length":"486589","record_id":"<urn:uuid:b050b482-829f-43b5-9f9f-cd015c030835>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00274.warc.gz"}
DFBA: Distribution-Free Bayesian Analysis A set of functions to perform distribution-free Bayesian analyses. Included are Bayesian analogues to the frequentist Mann-Whitney U test, the Wilcoxon Signed-Ranks test, Kendall's Tau Rank Correlation Coefficient, Goodman and Kruskal's Gamma, McNemar's Test, the binomial test, the sign test, the median test, as well as distribution-free methods for testing contrasts among condition and for computing Bayes factors for hypotheses. The package also includes procedures to estimate the power of distribution-free Bayesian tests based on data simulations using various probability models for the data. The set of functions provide data analysts with a set of Bayesian procedures that avoids requiring parametric assumptions about measurement error and is robust to problem of extreme outlier scores. Version: 0.1.0 Depends: R (≥ 2.10) Imports: methods, graphics, stats Suggests: knitr, rmarkdown, bookdown, testthat (≥ 3.0.0), vdiffr Published: 2023-12-13 DOI: 10.32614/CRAN.package.DFBA Author: Daniel H. Barch [aut, cre], Richard A. Chechile [aut] Maintainer: Daniel H. Barch <daniel.barch at tufts.edu> License: GPL-2 NeedsCompilation: no CRAN checks: DFBA results Reference manual: DFBA.pdf Vignettes: dfba_mann_whitney Package source: DFBA_0.1.0.tar.gz Windows binaries: r-devel: DFBA_0.1.0.zip, r-release: DFBA_0.1.0.zip, r-oldrel: DFBA_0.1.0.zip macOS binaries: r-release (arm64): DFBA_0.1.0.tgz, r-oldrel (arm64): DFBA_0.1.0.tgz, r-release (x86_64): DFBA_0.1.0.tgz, r-oldrel (x86_64): DFBA_0.1.0.tgz Please use the canonical form https://CRAN.R-project.org/package=DFBA to link to this page.	{"url":"https://cran.uvigo.es/web/packages/DFBA/index.html","timestamp":"2024-11-02T11:51:46Z","content_type":"text/html","content_length":"7824","record_id":"<urn:uuid:bf069644-44c3-40ef-84b9-943b93b3fc5d>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00819.warc.gz"}
80 km per hour to miles To convert 80 kilometers per hour to miles per hour, you can use the following step-by-step instructions: Step 1: Understand the conversion factor 1 kilometer is equal to 0.621371 miles. This is the conversion factor we will use to convert kilometers to miles. Step 2: Set up the conversion equation We can set up the conversion equation as follows: 80 kilometers = ? miles Step 3: Apply the conversion factor Multiply the given value (80 kilometers) by the conversion factor (0.621371) to convert kilometers to miles: 80 kilometers * 0.621371 miles/kilometer = 49.70968 miles Step 4: Round the answer (if necessary) Since we are dealing with a measurement involving distance, it is common to round the answer to a reasonable number of decimal places. In this case, we can round the answer to two decimal places: 49.70968 miles ≈ 49.71 miles Therefore, 80 kilometers per hour is approximately equal to 49.71 miles per hour. Visited 4 times, 1 visit(s) today	{"url":"https://unitconvertify.com/distance/80-km-per-hour-to-miles/","timestamp":"2024-11-03T08:46:58Z","content_type":"text/html","content_length":"43300","record_id":"<urn:uuid:5fa0effa-e990-45ad-b391-e20a8735b592>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00283.warc.gz"}
The Best Math Symbols For Kids To Learn \| How to Use Them in Real Life? - English 100 The Best Math Symbols For Kids To Learn \| How to Use Them in Real Life? Math Symbols Math Symbols Math Symbols Math Symbols Math Symbols Math Symbols Math Symbols Math Symbols Math Symbols Math Symbols Math Symbols Math Symbols Math Symbols Math symbols have been around for as long as humans have been able to draw. They help us to write down mathematical formulas and equations, represent results, and also prove our points. In this article, we will give you a quick overview of the most important math symbols and their meaning. You may not realize it, but math symbols are very important to read and write. In fact, you should use them at least once a day. The following list will teach you about the symbols that are commonly used in mathematics, their properties, and how they look like. Math Symbols from the Greek Language that are Known widely ψ – Psi ω – Omega υ – Upsilon φ – Phi π – Pi σ – Sigma λ – Lambda μ – Mu α – Alpha β – Beta γ – Gamma δ – Delta ε – Epsilon θ – Theta Unknown widely τ – Tau χ – Chi ρ – Rho ν – Nu ξ – Xi ο – Omicron ι – Iota κ – Kappa ζ- Zeta η – Eta The minus sign {–} The addition sign {+} The multiplication sign {x} The division sign {÷} The equal sign {=} The less sign {<} The more sign {>} The less or equal sign {≤} The infinity sign: {∞} The square root sign: {√} The derivative sign {‘}(X’, Y’,S’) The integral sign {∫} Limit: lim f(x) = The limit value of the function of x when x is ranged between x and xn ≈ approximately equal (⋅) multiplication dot multiplication e = e constant / Euler’s number= 2.718281828… ≠ Not equal sign [ ]- brackets: calculate expression inside first * – asterisk – multiplication (.) period decimal point, decimal separator a^b caret exponent % percent: eg. 5% error rate is acceptable ‰ per-mille: eg. 10‰ is the highest population growth rate ever recorded! ppm: per-million ppb: per-billion ppt: per-trillion Δ – triangle ≈ – approximately equal π – pi constant rad – radians ≪ – much less than ≫ – much greater than ≡ equivalence – identical to \| x \| vertical bars – absolute value x! exclamation mark – factorial N: Natural = Positive Numbers 1….+∞ Z: Zahl = -∞ …-1, 0, +1, ……. +∞ Q: = Quoziente = Rational = -∞…… 0.25, 0.1, 0.2, 0.3,……..+∞ R: Real = -∞…… + 0.5 + e + π …….+∞ C = Complex = 8+2i C = Real number + Real Number * i i = Square root of (-1) The minus sign {–} and it is called Minus/take. It is used to indicate that the second number is going to be reducing the first one by its numeric value. In other words, to subtract means to take away from a group or a number of things. The addition sign {+} and it is called Plus / add. It is used to indicate that two numbers are going to be combined together. Those two numbers may not be of the same nature The multiplication sign {x} and it is called Multiple/times. It is used to double or triple or increase the resulting value of a number by the number of times specified by the number after the {X} The division sign {÷} and is called divide. It is the exact opposite of the multiplication process. It decreases the result value of a number by dividing it into equal shares according to a specific number. The result may be an integer, a rational, or an irrational number. The equal sign {=} and it is pronounced Equals. It is the secure basis of all mathematical equations and operations. Without the equal sign, there can be no finalized solution or even a single correct simple equation. The less sign {<} is pronounced as Less than. It indicates that the number before it is lower than the number after it. Simple Ain’t it! The more sign {>} is used as More than. As its name suggests, it confirms that the number preceding the sign is higher than the one that follows it. The less or equal sign {≤} is pronounced like the following phrase “Less than or equal”. It refers to the fact that the two sides of this sign may be equal or the one before it is less valuable. It is used in cases where variables exist and the result could change. Thus in one case that the variable assumes a value the first number is less than the following, whereas when it, the variable, assumes another value which might as well be even lower than the first one, but in the overall result, the variable end value remains equal. All of this depends on the equation status. The infinity sign: {∞} simply it is just called Infinity. It refers to an uncertain number that could be positive or negative and it is not specified at all. The square root sign: {√} and it is pronounced as Square root. It is when you find the number that if you multiply it by itself it’ll result in the same number that is being square rooted. The derivative sign {y ‘} is like that if the variable is to be y. In simple words, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). In even simpler words, it measures deviation in a function of any sort with respect to its value. The integral sign {∫} just like that is called Integral. It is simply the reverse of a derivative. Other important symbols that you must know and teach your kids: N: Natural numbers = 1,2,3….. Prime Numbers = 2,3,5,7,9,11,…….. Composite Number: A positive number formed by the multiplication of two smaller positive integers= 4,6,8,9,10………. W: Whole numbers: Natural numbers Plus the number 0. C: Complex numbers: A number which can be written in the form a + bi where a and b are real numbers and i is the square root of -1, …,−5+2i —- 0,8+3i,… Rational numbers: Q=−12,0.33333: any number representable by a fraction Irrational Numbers: Numbers that cannot be represented by a fraction F=…,π,√2,0.121221222… Real Numbers : R=…,−3,−1,0,15,1.1,2–√,2,3,π,… Math symbols are used in a variety of ways to help extract maximum value from equations, calculations, and formulae. They make it easier to define math quantities, as well as develop a relationship between quantities that can be expressed in a unique way. ^1 What is the most used math symbol? The most commonly used math symbol is probably the plus sign (+), which represents addition. Addition is one of the fundamental mathematical operations, and the plus sign is used extensively in arithmetic and algebra to denote the sum of two or more numbers or variables. Other widely used math symbols include the minus sign (-) for subtraction, the multiplication sign (× or *) for multiplication, and the division sign (÷) for division. These symbols are foundational in mathematics and are used in various mathematical expressions and equations. What is the importance of mathematical symbols in real life? Mathematical symbols play a crucial role in various aspects of real life. They are essential for several reasons: Precision and Clarity: Mathematical symbols provide a concise and precise way to represent complex ideas and relationships. They allow for clear communication of mathematical concepts, which is especially important in fields like science, engineering, and finance, where precision is critical. Universal Language: Mathematics is considered a universal language because mathematical symbols are understood worldwide, regardless of spoken languages. This universality makes it easier for people from different cultures and backgrounds to communicate and collaborate on mathematical problems. Problem Solving: Mathematical symbols enable efficient problem-solving by simplifying the representation of equations, formulas, and mathematical models. This simplification makes it easier to analyze and manipulate mathematical expressions to find solutions or make predictions. Scientific Research: In scientific research, mathematical symbols are indispensable for representing data, modeling physical phenomena, and expressing the relationships between variables. They are used extensively in physics, chemistry, biology, and other scientific disciplines. Engineering and Technology: Engineers and technologists rely on mathematical symbols to design and analyze systems, structures, and algorithms. Whether designing a bridge, developing software, or optimizing a manufacturing process, mathematical symbols are essential for these tasks. Financial Analysis: In the world of finance, mathematical symbols are crucial for modeling financial markets, calculating risk, and making investment decisions. Equations and formulas are used to determine interest rates, evaluate investment returns, and manage financial portfolios. Education: Mathematical symbols are a fundamental part of mathematics education. They help students learn and understand mathematical concepts, allowing for the development of problem-solving skills and critical thinking. Data Science and Statistics: Mathematical symbols are extensively used in data science and statistics to represent data sets, statistical distributions, regression models, and hypothesis testing. They facilitate the analysis of large datasets and the extraction of meaningful insights. Computer Science and Programming: Mathematical symbols are essential in computer science and programming, where they are used to express algorithms, logic, and mathematical operations. Programming languages rely on mathematical notation for tasks such as arithmetic, conditional statements, and data manipulation. Everyday Applications: While many people may not realize it, mathematical symbols are present in everyday life. They are used in grocery store receipts, road signs, financial statements, and countless other contexts where quantitative information needs to be conveyed accurately. In summary, mathematical symbols are essential tools for expressing and communicating mathematical ideas, facilitating problem-solving, and supporting various fields of science, technology, engineering, and mathematics. Their importance extends beyond academia and directly impacts our daily lives in numerous ways.	{"url":"https://learnenglish100.com/math-symbols/","timestamp":"2024-11-12T18:20:07Z","content_type":"text/html","content_length":"87963","record_id":"<urn:uuid:db9b84ac-b561-45a6-854f-92f97d3cbf11>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00354.warc.gz"}
How To Use The Pick 5 and the Magic Square of SUN/MOON Sunday, December 2, 2007 This thin post is just to whet you appetite to encourage some more exploration. You will not be disappointed as you look at the Pick 5 in totally different light. As some of you know there are seven magic squares of the planets. Basically Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and the Moon each has a Magic Square. For more complete info go to: http:// The magic squares have "mathematically magical" properties. One of these properties is the fact is that each square equals the same constant or number whether you add the numbers horizontally, vertically, or the center diagonals. For example lets look at the Magic Square of the Sun: Each horizontal and vertical line each add up to the constant of 111. Also each of the two longest diagonals starting at 1 and ending at 36 and the other starting at 6 and ending at 31 add up to 111. If you add all the vertical lines or add all the horizontal lines you end up with 666. There are speculations that the magic Squares, besides having interesting but simple mathematical manipulations like the one above, they have no other use. Through the centuries there were speculations that they could be used in astrology or to even predict the future among other uses. The jury is still out, but I there be a way to reconcile the Magic Square of the Planets with the Lottery games. Even though I may sometimes use practical esoterica to explore things conceptually, this does not have to be an esoteric endeavor. Lets make it simple... I believe in Correspondence. Quantum physics implies that no matter how obtuse, everything in the universe is connected or corresponds to each other. In a word, this is why I propose that there may way to reconcile the magic Square of the Planets with the Lottery games. Even though the Magic squares were not invented for the lottery, they can still be used in the lottery. Like I said, everything corresponds, so to put it crudely I could have used a computer algorithm, crystal ball, or even a dream to predict numbers for the lottery. Again, I'm being a little crude, because its a little bit more involved than this, but you can use anything. Of course what you use has its own way of communicating and that is the tricky part. You have to figure out its language. I've noticed that the Magic square of the Sun may be used for the Pick 5. There are other squares for that may be used for this game or the other games, but let's focus on the Magic Square of the Sun, which again is: Lets use the New York Pick 5. The Total combos for the Pick 5 of 1 - 39 numbers is 575757 combos. Remember that the total sum for the Magic Square of the Sun if you add all the rows or columns is Divide 575757 by 666 and that equals 864.5: 575757/ 666 = 864.5 or 864 or 865. You see that 864? To me 864 is also 314(PI), if you mirror the first 2 digits. Also...You see that 865? To me that 865 is also 360 or 365 if you mirror a couple of digits. Isn't 360 a circle. Isn't 365 the number of days in the year. You know what this means? 864.5 Magic Square of the Suns equals 575757. If you know that 864.5 Magic Square of the Suns equals 575757(864.5 X 666 = 575757), how can using this template help you to figure out how to predict numbers for the Pick 5? Another Correspondence: Observe that each line of the Magic Square of the Moon equals 369. As a significant sidenote for future thought, 369 also equals 314(PI) if you mirror a the last 2 numbers. The whole Magic Square of the Moon equals 3321. Now If you numerically compress the combos for the NY Pick 5 of 575757 to 575 + 757 = 1332 you will notice that there is correspondence between the 1332 of 575757 and the 3321 of the Moon's Magic Square . They are in fact the same, because if you "travel west" on the numbers and "moving 180 degrees" 3321 is also: 3321...3213...2133...1332. Viola! The Magic Square of the Moon can correspond to the NY Pick 5 as well. By the way, I'll try to talk about the usefulness of numerical compression as I did with 575757 - 575+757 = 3321, and "traveling" on numbers as I did with "going west on 3321" in a future post. I hope I've at least stimulated some of you...I did say it was a thin post, but I'll look to offer a few more ideas about this in a future post...There is one SUPER-HUGE thing about those squares I am bursting to share, but not now. I haven't played with it enough... Again, I just mentioned all the above to whet your appetite and inspire you to explore and see correspondences in general, and the Magic Squares in particular, and possibly use them to inform your There may be some, who will doubt all that I posted and declare its all rubbish. They would be right. There are some who would say that this magic square thing is not rubbish and they would be right. Yes...they are both right, because their awareness creates their reality, and those two realities cannot exist together in the same place(space)... Happy Explorations...	{"url":"https://blogs.lotterypost.com/kola/2007/12/how-to-use-the-pick-5-and-the-magic-square-of.htm","timestamp":"2024-11-07T17:15:22Z","content_type":"text/html","content_length":"18995","record_id":"<urn:uuid:95f8524d-eb3a-49a1-8118-7ad7da78248a>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00893.warc.gz"}
The Ultimate Step by Step Guide to Preparing for the MCAP Math Test Transform your sixth grader into a math genius and enable them to excel in the Maryland Comprehensive Assessment Program (MCAP) Math examination with MCAP Grade 6 Math for Beginners. This meticulously crafted guidebook provides an all-inclusive resource, tailored to equip students with the essential math concepts and techniques required to achieve outstanding results in this critical Some of the extraordinary features of this guidebook include an in-depth exploration of all math topics featured on the 2023 MCAP exam, enabling students to gain a comprehensive understanding of the subject matter and confidently tackle any question format they may encounter. The guidebook also offers a vast array of study materials, clear explanations, real-world examples, and practice exercises for each topic, providing students with a complete learning and practice package in one resource. To ensure success, MCAP Grade 6 Math for Beginners provides succinct, step-by-step instructions on the most efficient strategies, equipping students with the vital skills and self-assurance necessary to outshine their peers on test day. In addition, the guidebook offers a wide variety of practice tests in diverse formats, including free response and multiple-choice questions, complemented by two genuine full-length practice exams with in-depth answer explanations to monitor progress and assess understanding. To deepen students’ comprehension of mathematical concepts and their practical applications, MCAP Grade 6 Math for Beginners includes detailed explanations and problem-solving methodologies for each question type. This indispensable resource is ideal for both self-study and classroom use and supports students in mastering mathematical concepts while boosting their confidence in their abilities. To supplement their learning experience and enhance their skills, students can access additional online math practice at EffortlessMath.com. With MCAP Grade 6 Math for Beginners, students will be well-prepared to unlock their full potential on the MCAP Math exam and thrive in their academic pursuits. This guidebook is the quintessential resource for students preparing for the MCAP Math assessment, and its comprehensive nature ensures students have everything they need to succeed. There are no reviews yet.	{"url":"https://www.effortlessmath.com/product/mcap-grade-6-math-for-beginners/","timestamp":"2024-11-13T08:39:34Z","content_type":"text/html","content_length":"44265","record_id":"<urn:uuid:4aa02177-0484-47b9-a8a9-fac907343715>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00278.warc.gz"}
Self-avoiding walk connectivity constant and theta point on percolating lattices Barat, K. ; Karmakar, S. N. ; Chakrabarti, B. K. (1991) Self-avoiding walk connectivity constant and theta point on percolating lattices Journal of Physics A: Mathematical and General, 24 (4). pp. 851-860. ISSN 0305-4470 Full text not available from this repository. Official URL: http://iopscience.iop.org/0305-4470/24/4/017?fromS... Related URL: http://dx.doi.org/10.1088/0305-4470/24/4/017 The average connectivity constant mu of self-avoiding walks (SAWs) is obtained from exact enumeration of SAWs on Monte Carlo generated percolating clusters in a randomly diluted square lattice. For averages over the (infinite) percolating cluster, mu decreases almost linearly with bond dilution (1-p), where p is the bond occupation concentration. The authors find mu (p[c])=1.31+or−0.03 at the percolation threshold p[c] and could not detect any significant difference between mu (p[c]) and p[c] mu (1). The variation of theta-point for SAWs on the same lattice with dilution is also estimated, analysing the partition function zeros. Within the limited accuracy of their analysis, its variation with dilution is observed as being quite weak and the theta-point increases somewhat (compared to pure lattice value) near p[c]; they find a non-vanishing theta point (K[theta] (p[c]) equivalent to 0.59, where K[0]=J/k theta ) on the square lattice percolation cluster at p[c]. Item Type: Article Source: Copyright of this article belongs to Institute of Physics. ID Code: 44840 Deposited On: 23 Jun 2011 07:48 Last Modified: 23 Jun 2011 07:48 Repository Staff Only: item control page	{"url":"https://repository.ias.ac.in/44840/","timestamp":"2024-11-01T19:25:29Z","content_type":"application/xhtml+xml","content_length":"18385","record_id":"<urn:uuid:80b6101a-b16c-4f6d-87f5-6614b2a4dc38>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00116.warc.gz"}
An Estimated Formulation for the Capacitated Single Alocation p-hub Median Problem with Fixed Costs of Opening Facilities Alumur, S. and B. Y. Kara (2008). Network hub location problems: The state of the art. European Journal of Operational Research, Vol. 190(1), pp. 1-21. Bixby, R. E. (2002). Solving Real-World Linear Programs: A Decade and More of Progress. Operations Research, Vol. 50(1), pp. 3-15. Campbell, J. F. (1994). Integer programming formulations of discrete hub location problems. European Journal of Operational Research, Vol. 72(2), pp. 387-405. Campbell, J. F. and M. E. O'Kelly (2012). Twenty-Five Years of Hub Location Research. Transportation Science, Vol. 46(2), pp. 153-169. Chen, J.-F. (2007). A hybrid heuristic for the uncapacitated single allocation hub location problem. Omega, Vol. 35(2), pp. 211-220. Correia, I., S. Nickel, and F. Saldanha-da-Gam. (2010). The capacitated single-allocation hub location problem revisited: A note on a classical formulation. European Journal of Operational Research, Vol. 207(1), pp. 92-96. Ebery, J. (2001). Solving large single allocation p-hub problems with two or three hubs. European Journal of Operational Research, Vol. 128(2), pp. 447-458. Ernst, A. T. and M. Krishnamoorthy (1996). Efficient algorithms for the uncapacitated single allocation p-hub median problem. Location Science, Vol. 4(3), pp. 139-154. Ernst, A. T. and M. Krishnamoorthy (1999). Solution algorithms for the capacitated single allocation hub location problem. Annals of Operations Research, Vol. 86(0), pp. 141-159. Farahani, R. Z., M. Hekmatfar, A., Boloori, and A.E. Nikbakhsh (2013). Hub location problems: A review of models, classification, solution techniques, and applications. Computers & Industrial Engineering, Vol. 64(4), pp. 1096-1109. Ilić, A., D. Urošević, J. Brimberg, and N. Mladenović (2010). A general variable neighborhood search for solving the uncapacitated single allocation p-hub median problem. European Journal of Operational Research, Vol. 206(2), pp. 289-300. Klincewicz, J. (1992). Avoiding local optima in thep-hub location problem using tabu search and GRASP. Annals of Operations Research, Vol. 40(1), pp. 283-302. Klincewicz, J. G. (1991). Heuristics for the p-hub location problem. European Journal of Operational Research, Vol. 53(1), pp. 25-37. Kratica, J., Z. Stanimirović, D. Tošić, and V. Filipović, (2007). Two genetic algorithms for solving the uncapacitated single allocation p-hub median problem. European Journal of Operational Research, Vol. 182(1), pp. 15-28. Merakli, M. and H. Yaman (2017). A capacitated hub location problem under hose demand uncertainty. Computers & Operations Research, Vol. 88, pp. 58-70. O'Kelly, M. (1992). Hub facility location with fixed costs. Papers in Regional Science, Vol. 71(3), pp. 293-306. O'Kelly, M. E. (1986). The Location of Interacting Hub Facilities. Transportation Science, Vol. 20(2). pp. 92-106. O'Kelly, M. E. (1987). A quadratic integer program for the location of interacting hub facilities. European Journal of Operational Research, Vol. 32(3), pp. 393-404. Puerto, J., A.B. Ramos, A.M. Rodríguez-Chía, and M.C. Sánchez-Gil (2016). Ordered median hub location problems with capacity constraints. Transportation Research Part C: Emerging Technologies, Vol. 70, pp. 142-156. Rabbani, M., M. Ravanbakhsh, H. Farrokhi-Asl, M. Taheri, (2017). Using metaheuristic algorithms for solving a hub location problem: application in passive optical network planning.” International Journal of Supply and Operations Management, In-press. Rostami, B., C. Buchheim, J.F. Meier and U. Clausen (2016). Lower Bounding Procedures for the Single Allocation Hub Location Problem. Electronic Notes in Discrete Mathematics, Vol, 52, pp. 69-76. Silva, M. R. and C. B. Cunha (2009). New simple and efficient heuristics for the uncapacitated single allocation hub location problem. Computers & Operations Research, Vol. 36(12), pp. 3152-3165. Skorin-Kapov, D. and J. Skorin-Kapov (1994). On tabu search for the location of interacting hub facilities. European Journal of Operational Research. Vol. 73(3), pp. 502-509. Stanojević, P., M. Marić, Z. Stanimirović, (2015). A hybridization of an evolutionary algorithm and a parallel branch and bound for solving the capacitated single allocation hub location problem. Applied Soft Computing, Vol. 33, pp. 24-36. Yaman, H. (2011). Allocation strategies in hub networks. European Journal of Operational Research, Vol. 211(3), pp. 442-451.	{"url":"http://www.ijsom.com/article_2723.html","timestamp":"2024-11-11T11:16:35Z","content_type":"text/html","content_length":"55931","record_id":"<urn:uuid:a8f52fd6-b15c-4981-b2d1-64766ffbb1e4>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00022.warc.gz"}
Long Division With Grid Worksheets - Divisonworksheets.com Long Division With Grid Worksheets Long Division With Grid Worksheets – Use worksheets for division to help your child learn and revisit division concepts. Worksheets come in a wide variety, and you can even design your own. These worksheets are amazing because they can be downloaded free of charge and customized to meet your needs. They’re perfect for second-graders, kindergarteners, and first-graders. Two can create massive quantities While dividing large numbers, a child must practice using worksheets. These worksheets are limited to three, two, and sometimes four different divisors. This method implies that the child does not have to worry about not completing the division or making errors with their times tables. It’s possible to locate worksheets on the internet, or download them on your personal computer to aid your child in this mathematical skill. Multi-digit Division worksheets let children to practice their skills and consolidate their understanding. This skill is crucial for maths that are complex as well as everyday calculations. These worksheets provide an interactive set of questions and activities that help students understand the concept. It’s not simple for students to divide huge numbers. The worksheets typically employ the same algorithm and follow steps-by-step instructions. This might cause students to not have the level of intellectual understanding that they need. To teach long division, one approach is to employ the base ten blocks. Once you have learned the steps, long division will come naturally to students. Students can learn to divide of large numbers with many of worksheets and practice questions. Furthermore, fractional results that are stated in decimals are included in the worksheets. For those who have to divide large amounts of money, worksheets on hundredths can be found. Sort the numbers into small groups. Incorporating a large number of people into small groups might be challenging. Although it seems good on paper, many facilitators of small groups dislike this method. It genuinely reflects how our bodies develop and can aid in the Kingdom’s endless growth. It inspires others to reach for the lost and look for new leaders to lead the way. It is also useful in brainstorming. You can form groups with people with similar experience and personality traits. This will allow you to come up with innovative ideas. After you have created your groups, introduce each participant to you. It’s a great way to promote creativity and innovative thinking. Division is the most fundamental Arithmetic process that divides huge numbers into smaller ones. This can be extremely beneficial when you need to create equal numbers of items for different groups. For instance, a big class can be split into five classes. The groups are then added to provide the original 30 students. Keep in mind that there exist two different types of numbers that you can use to divide numbers: divisors and the quotient. Dividing one number by another produces “ten/five,” while divising two by two yields the same result. For large numbers, power of ten is not recommended. We can divide massive numbers into powers of 10 to enable comparisons between them. Decimals are a common part of shopping. You can discover them on price tags, receipts, and food labels. These decimals are employed at petrol stations to show the price per gallons and the amount delivered by the sprayer. There are two ways to divide large numbers into powers of 10. The first is by shifting the decimal to the left and multiplying by 10-1. The second approach utilizes the powers of ten’s associative feature. You can divide a huge number of numbers into smaller powers of ten after you know the associative function of powers of 10. Mental computation is utilized in the initial method. Divide 2.5 by the power of 10 to find a pattern. The decimal point will shift to one side for each tenth power. This concept can be used to solve any issue even the most difficult. The other method involves mentally dividing very massive numbers into powers of 10. This allows you to quickly express very huge numbers by writing them down in scientific notation. If you are using scientific notation for expressing large numbers, it is best to utilize positive exponents. You can change the decimal point by five spaces on one side and convert 450,000 into number 4.5. To divide a large amount into smaller power 10, you can apply exponent 5 or divide it in smaller powers 10 until it’s 4.5. Gallery of Long Division With Grid Worksheets Grids And Columns In Math Elementary Math Math Division Fun Math The 2 Digit By 1 Digit Long Division With Grid Assistance And Prompts Long Division With Grid Assistance 4 Digit By 1 Digit With Remainders Leave a Comment	{"url":"https://www.divisonworksheets.com/long-division-with-grid-worksheets/","timestamp":"2024-11-13T14:48:09Z","content_type":"text/html","content_length":"66227","record_id":"<urn:uuid:d3f5260a-dcf7-4092-9e46-f24533c8bd86>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00407.warc.gz"}
Vassil Alexandrov is an ICREA Research Professor in Computational Science at BSC since September 2010. He holds a MSc in Applied Mathematics from Moscow State University, Russia (1984) and a PhD in Parallel Computing from Bulgarian Academy of Sciences (1995). He has held previous positions at the University of Liverpool, UK (Depts. of Statistics and Computational Mathematics and Computer Science, 1994-1999), the University of Reading, UK (School of Systems Engineering, 1999-2010, as Professor of Computational Science leading the Computational Science research group until September 2010, and as the Director of the Centre for Advanced Computing and Emerging Technologies until July 2010). He is an Editorial Board member and a Guest Editor of the Journal of Computational Science and Journal of Mathematics and Computers in Simulation. He has published over 120 papers in renowned refereed journals and international conferences in the area of his research expertise.	{"url":"https://memoir.icrea.cat/2016/researchers/alexandrov-vassil/","timestamp":"2024-11-13T08:49:37Z","content_type":"text/html","content_length":"231211","record_id":"<urn:uuid:b5ac653c-0709-4bd5-83af-fda36b6653a7>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00674.warc.gz"}
Mathimatics for kids Algebra Tutorials! Wednesday 6th of November mathimatics for kids Related topics: Home factoring calculator \| math powerpoint presentation about factoring trinomials \| free math sheets on calculating volume \| algebra two formula chart \| solving algebraic Calculations with expressions/ fractions \| square root calculator and reducer \| free math worksheet 11 plus download Negative Numbers Solving Linear Equations Systems of Linear Author Message Solving Linear Equations hevmas06 Posted: Wednesday 27th of Dec 20:29 Graphically I have this math assignment due and I would really be grateful if anyone can guide mathimatics for kids on which I’m stuck and don’t know where to Algebra Expressions start from. Can you give me guidance with graphing parabolas, system of equations and matrices. I would rather get guidance from you than hire a Evaluating Expressions math tutor who are very costly . Any direction will be highly treasured very much. and Solving Equations Fraction rules Registered: Factoring Quadratic 08.04.2006 Trinomials From: Bellarine Multiplying and Dividing Peninsula, Australia Dividing Decimals by Whole Numbers Adding and Subtracting nxu Posted: Thursday 28th of Dec 08:00 Radicals You can find numerous links on the internet if you search the keyword mathimatics for kids. Most of the content is however designed for the people Subtracting Fractions who already have some know how about this subject. If you are a complete novice, you should use Algebrator. Is it easy to understand and very Factoring Polynomials by helpful too. Slopes of Perpendicular Registered: Lines 25.10.2006 Linear Equations From: Siberia, Roots - Radicals 1 Russian Federation Graph of a Line Sum of the Roots of a Writing Linear Equations Voumdaim of Obpnis Posted: Saturday 30th of Dec 10:05 Using Slope and Point Algebrator really helps you out in mathimatics for kids. I have tried all Algebra software on the net. It is very user-friendly . You just put your Factoring Trinomials problem and it will create a complete step-by-step report of the solution. This helped me much with hypotenuse-leg similarity, adding exponents and with Leading Coefficient percentages. It helps you understand algebra better. I was annoyedof paying a fortune to Maths Tutors who could not give me the required time and 1 attention. It is a cheap tool which could change your entire mindset towards math. Using Algebrator would be fun. Take it. Writing Linear Equations Registered: Using Slope and Point 11.06.2004 Simplifying Expressions From: SF Bay Area, with Negative Exponents CA, USA Solving Equations 3 Solving Quadratic Parent and Family Graphs Dnexiam Posted: Sunday 31st of Dec 07:51 Collecting Like Terms hyperbolas, decimals and radical equations were a nightmare for me until I found Algebrator, which is really the best math program that I have come nth Roots across. I have used it through many math classes – College Algebra, Basic Math and Algebra 1. Simply typing in the algebra problem and clicking on Power of a Quotient Solve, Algebrator generates step-by-step solution to the problem, and my math homework would be ready. I really recommend the program. Property of Exponents Adding and Subtracting Registered: Fractions 25.01.2003 Percents From: City 17 Solving Linear Systems of Equations by The Quadratic Formula Fractions and Mixed Solving Rational Multiplying Special Rounding Numbers Factoring by Grouping Polar Form of a Complex Solving Quadratic Simplifying Complex Common Logs Operations on Signed Multiplying Fractions in Dividing Polynomials Higher Degrees and Variable Exponents Solving Quadratic Inequalities with a Sign Writing a Rational Expression in Lowest Solving Quadratic Inequalities with a Sign Solving Linear Equations The Square of a Binomial Properties of Negative Inverse Functions Rotating an Ellipse Multiplying Numbers Linear Equations Solving Equations with One Log Term Combining Operations The Ellipse Straight Lines Graphing Inequalities in Two Variables Solving Trigonometric Adding and Subtracting Simple Trinomials as Products of Binomials Ratios and Proportions Solving Equations Multiplying and Dividing Fractions 2 Rational Numbers Difference of Two Factoring Polynomials by Solving Equations That Contain Rational Solving Quadratic Dividing and Subtracting Rational Expressions Square Roots and Real Order of Operations Solving Nonlinear Equations by The Distance and Midpoint Formulas Linear Equations Graphing Using x- and y- Properties of Exponents Solving Quadratic Solving One-Step Equations Using Algebra Relatively Prime Numbers Solving a Quadratic Inequality with Two Operations on Radicals Factoring a Difference of Two Squares Straight Lines Solving Quadratic Equations by Factoring Graphing Logarithmic Simplifying Expressions Involving Variables Adding Integers Factoring Completely General Quadratic Using Patterns to Multiply Two Binomials Adding and Subtracting Rational Expressions With Unlike Denominators Rational Exponents Horizontal and Vertical mathimatics for kids Related topics: Home factoring calculator \| math powerpoint presentation about factoring trinomials \| free math sheets on calculating volume \| algebra two formula chart \| solving algebraic Calculations with expressions/ fractions \| square root calculator and reducer \| free math worksheet 11 plus download Negative Numbers Solving Linear Equations Systems of Linear Author Message Solving Linear Equations hevmas06 Posted: Wednesday 27th of Dec 20:29 Graphically I have this math assignment due and I would really be grateful if anyone can guide mathimatics for kids on which I’m stuck and don’t know where to Algebra Expressions start from. Can you give me guidance with graphing parabolas, system of equations and matrices. I would rather get guidance from you than hire a Evaluating Expressions math tutor who are very costly . Any direction will be highly treasured very much. and Solving Equations Fraction rules Registered: Factoring Quadratic 08.04.2006 Trinomials From: Bellarine Multiplying and Dividing Peninsula, Australia Dividing Decimals by Whole Numbers Adding and Subtracting nxu Posted: Thursday 28th of Dec 08:00 Radicals You can find numerous links on the internet if you search the keyword mathimatics for kids. Most of the content is however designed for the people Subtracting Fractions who already have some know how about this subject. If you are a complete novice, you should use Algebrator. Is it easy to understand and very Factoring Polynomials by helpful too. Slopes of Perpendicular Registered: Lines 25.10.2006 Linear Equations From: Siberia, Roots - Radicals 1 Russian Federation Graph of a Line Sum of the Roots of a Writing Linear Equations Voumdaim of Obpnis Posted: Saturday 30th of Dec 10:05 Using Slope and Point Algebrator really helps you out in mathimatics for kids. I have tried all Algebra software on the net. It is very user-friendly . You just put your Factoring Trinomials problem and it will create a complete step-by-step report of the solution. This helped me much with hypotenuse-leg similarity, adding exponents and with Leading Coefficient percentages. It helps you understand algebra better. I was annoyedof paying a fortune to Maths Tutors who could not give me the required time and 1 attention. It is a cheap tool which could change your entire mindset towards math. Using Algebrator would be fun. Take it. Writing Linear Equations Registered: Using Slope and Point 11.06.2004 Simplifying Expressions From: SF Bay Area, with Negative Exponents CA, USA Solving Equations 3 Solving Quadratic Parent and Family Graphs Dnexiam Posted: Sunday 31st of Dec 07:51 Collecting Like Terms hyperbolas, decimals and radical equations were a nightmare for me until I found Algebrator, which is really the best math program that I have come nth Roots across. I have used it through many math classes – College Algebra, Basic Math and Algebra 1. Simply typing in the algebra problem and clicking on Power of a Quotient Solve, Algebrator generates step-by-step solution to the problem, and my math homework would be ready. I really recommend the program. Property of Exponents Adding and Subtracting Registered: Fractions 25.01.2003 Percents From: City 17 Solving Linear Systems of Equations by The Quadratic Formula Fractions and Mixed Solving Rational Multiplying Special Rounding Numbers Factoring by Grouping Polar Form of a Complex Solving Quadratic Simplifying Complex Common Logs Operations on Signed Multiplying Fractions in Dividing Polynomials Higher Degrees and Variable Exponents Solving Quadratic Inequalities with a Sign Writing a Rational Expression in Lowest Solving Quadratic Inequalities with a Sign Solving Linear Equations The Square of a Binomial Properties of Negative Inverse Functions Rotating an Ellipse Multiplying Numbers Linear Equations Solving Equations with One Log Term Combining Operations The Ellipse Straight Lines Graphing Inequalities in Two Variables Solving Trigonometric Adding and Subtracting Simple Trinomials as Products of Binomials Ratios and Proportions Solving Equations Multiplying and Dividing Fractions 2 Rational Numbers Difference of Two Factoring Polynomials by Solving Equations That Contain Rational Solving Quadratic Dividing and Subtracting Rational Expressions Square Roots and Real Order of Operations Solving Nonlinear Equations by The Distance and Midpoint Formulas Linear Equations Graphing Using x- and y- Properties of Exponents Solving Quadratic Solving One-Step Equations Using Algebra Relatively Prime Numbers Solving a Quadratic Inequality with Two Operations on Radicals Factoring a Difference of Two Squares Straight Lines Solving Quadratic Equations by Factoring Graphing Logarithmic Simplifying Expressions Involving Variables Adding Integers Factoring Completely General Quadratic Using Patterns to Multiply Two Binomials Adding and Subtracting Rational Expressions With Unlike Denominators Rational Exponents Horizontal and Vertical Calculations with Negative Numbers Solving Linear Equations Systems of Linear Solving Linear Equations Algebra Expressions Evaluating Expressions and Solving Equations Fraction rules Factoring Quadratic Multiplying and Dividing Dividing Decimals by Whole Numbers Adding and Subtracting Subtracting Fractions Factoring Polynomials by Slopes of Perpendicular Linear Equations Roots - Radicals 1 Graph of a Line Sum of the Roots of a Writing Linear Equations Using Slope and Point Factoring Trinomials with Leading Coefficient Writing Linear Equations Using Slope and Point Simplifying Expressions with Negative Exponents Solving Equations 3 Solving Quadratic Parent and Family Graphs Collecting Like Terms nth Roots Power of a Quotient Property of Exponents Adding and Subtracting Solving Linear Systems of Equations by The Quadratic Formula Fractions and Mixed Solving Rational Multiplying Special Rounding Numbers Factoring by Grouping Polar Form of a Complex Solving Quadratic Simplifying Complex Common Logs Operations on Signed Multiplying Fractions in Dividing Polynomials Higher Degrees and Variable Exponents Solving Quadratic Inequalities with a Sign Writing a Rational Expression in Lowest Solving Quadratic Inequalities with a Sign Solving Linear Equations The Square of a Binomial Properties of Negative Inverse Functions Rotating an Ellipse Multiplying Numbers Linear Equations Solving Equations with One Log Term Combining Operations The Ellipse Straight Lines Graphing Inequalities in Two Variables Solving Trigonometric Adding and Subtracting Simple Trinomials as Products of Binomials Ratios and Proportions Solving Equations Multiplying and Dividing Fractions 2 Rational Numbers Difference of Two Factoring Polynomials by Solving Equations That Contain Rational Solving Quadratic Dividing and Subtracting Rational Expressions Square Roots and Real Order of Operations Solving Nonlinear Equations by The Distance and Midpoint Formulas Linear Equations Graphing Using x- and y- Properties of Exponents Solving Quadratic Solving One-Step Equations Using Algebra Relatively Prime Numbers Solving a Quadratic Inequality with Two Operations on Radicals Factoring a Difference of Two Squares Straight Lines Solving Quadratic Equations by Factoring Graphing Logarithmic Simplifying Expressions Involving Variables Adding Integers Factoring Completely General Quadratic Using Patterns to Multiply Two Binomials Adding and Subtracting Rational Expressions With Unlike Denominators Rational Exponents Horizontal and Vertical Author Message hevmas06 Posted: Wednesday 27th of Dec 20:29 I have this math assignment due and I would really be grateful if anyone can guide mathimatics for kids on which I’m stuck and don’t know where to start from. Can you give me guidance with graphing parabolas, system of equations and matrices. I would rather get guidance from you than hire a math tutor who are very costly . Any direction will be highly treasured very much. From: Bellarine Peninsula, Australia nxu Posted: Thursday 28th of Dec 08:00 You can find numerous links on the internet if you search the keyword mathimatics for kids. Most of the content is however designed for the people who already have some know how about this subject. If you are a complete novice, you should use Algebrator. Is it easy to understand and very helpful too. From: Siberia, Russian Federation Voumdaim of Obpnis Posted: Saturday 30th of Dec 10:05 Algebrator really helps you out in mathimatics for kids. I have tried all Algebra software on the net. It is very user-friendly . You just put your problem and it will create a complete step-by-step report of the solution. This helped me much with hypotenuse-leg similarity, adding exponents and percentages. It helps you understand algebra better. I was annoyedof paying a fortune to Maths Tutors who could not give me the required time and attention. It is a cheap tool which could change your entire mindset towards math. Using Algebrator would be fun. Take it. From: SF Bay Area, CA, USA Dnexiam Posted: Sunday 31st of Dec 07:51 hyperbolas, decimals and radical equations were a nightmare for me until I found Algebrator, which is really the best math program that I have come across. I have used it through many math classes – College Algebra, Basic Math and Algebra 1. Simply typing in the algebra problem and clicking on Solve, Algebrator generates step-by-step solution to the problem, and my math homework would be ready. I really recommend the program. From: City 17 Posted: Wednesday 27th of Dec 20:29 I have this math assignment due and I would really be grateful if anyone can guide mathimatics for kids on which I’m stuck and don’t know where to start from. Can you give me guidance with graphing parabolas, system of equations and matrices. I would rather get guidance from you than hire a math tutor who are very costly . Any direction will be highly treasured very much. Posted: Thursday 28th of Dec 08:00 You can find numerous links on the internet if you search the keyword mathimatics for kids. Most of the content is however designed for the people who already have some know how about this subject. If you are a complete novice, you should use Algebrator. Is it easy to understand and very helpful too. Posted: Saturday 30th of Dec 10:05 Algebrator really helps you out in mathimatics for kids. I have tried all Algebra software on the net. It is very user-friendly . You just put your problem and it will create a complete step-by-step report of the solution. This helped me much with hypotenuse-leg similarity, adding exponents and percentages. It helps you understand algebra better. I was annoyedof paying a fortune to Maths Tutors who could not give me the required time and attention. It is a cheap tool which could change your entire mindset towards math. Using Algebrator would be fun. Take it. Posted: Sunday 31st of Dec 07:51 hyperbolas, decimals and radical equations were a nightmare for me until I found Algebrator, which is really the best math program that I have come across. I have used it through many math classes – College Algebra, Basic Math and Algebra 1. Simply typing in the algebra problem and clicking on Solve, Algebrator generates step-by-step solution to the problem, and my math homework would be ready. I really recommend the program.	{"url":"https://polymathlove.com/polymonials/midpoint-of-a-line/mathimatics-for-kids.html","timestamp":"2024-11-06T08:14:41Z","content_type":"text/html","content_length":"113112","record_id":"<urn:uuid:ba08856a-67b5-4577-a9d1-862a0c45f9f5>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00273.warc.gz"}
The Third Generation of Financial Planning welcomes guest contributions. The views presented here do not necessarily represent those of Advisor Perspectives. The first generation of financial planning (1970s through 1990s) used static modeling with a constant return expectation. The chief limitation was that it didn’t accurately reflect the real sequence of returns observed, which created very unrealistic results and underrepresented the risk of failure. In this century, financial plans were primarily conducted using single-factor Monte Carlo simulations (the second generation of financial planning), which addressed the sequence of return risk. While a vast improvement over the first generation of financial plans, typical Monte Carlo simulations still have material weaknesses in how they represent potential future circumstances. The most notable weaknesses include the assumption of a constant risk level (determining expected return), a constant volatility (standard deviation) for potential portfolio returns, a single path for potential inflation and a constant terminal point for evaluating the simulation. The third generation of financial planning mathematical infrastructure has emerged, which incorporates additional variables to more realistically reflect potential future outcomes. The first generation – Static models As computing power evolved, the ability to incorporate more intricate cash-flow assumptions improved. General assumptions for portfolio returns, taxes and inflation expectations evolved. Even life expectancy changed and thus the time horizon for the financial plan increased. These improved assumptions enhanced the reliability of the outcome. However, they were often biased based on the then current economic climate (e.g. late 1970s/early 1980s with inflation and interest rates above 10%), which didn’t necessarily accurately reflect an uncertain future. The chief limitation of the first generation of financial plans was inherent in the portfolio return assumptions. An individual could be lucky enough to retire at the beginning of a strong market or unlucky enough to retire at the beginning of a weak market economy. The difference could be as stark as success or failure. Assuming, for example, a 6% return every year ignored the massive variance that occurred during most retirement horizons.	{"url":"https://www.advisorperspectives.com/articles/2020/09/14/the-third-generation-of-financial-planning","timestamp":"2024-11-15T04:05:31Z","content_type":"text/html","content_length":"124855","record_id":"<urn:uuid:7853681f-38e0-4a3b-9750-27fca686d036>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00447.warc.gz"}
The complexity of multiway cuts In the Multiway Cut problem we are given an edge-weighted graph and a subset of the vertices called terminals, and asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the min-cut, max-flow problem, and can be solved in polynomial time. We show that the problem becomes NP-hard as soon as κ = 3, but can be solved in polynomial time for planar graphs for any fixed κ. The planar problem is NP-hard, however, if it is not fixed. We also describe a simple approximation algorithm for arbitrary graphs that is guaranteed to come within a factor of 2-2/κ of the optimal cut weight. Original language English (US) Title of host publication Proceedings of the 24th Annual ACM Symposium on Theory of Computing, STOC 1992 Publisher Association for Computing Machinery Pages 241-251 Number of pages 11 ISBN (Electronic) 0897915119 State Published - Jul 1 1992 Externally published Yes Event 24th Annual ACM Symposium on Theory of Computing, STOC 1992 - Victoria, Canada Duration: May 4 1992 → May 6 1992 Publication series Name Proceedings of the Annual ACM Symposium on Theory of Computing Volume Part F129722 ISSN (Print) 0737-8017 Other 24th Annual ACM Symposium on Theory of Computing, STOC 1992 Country/Territory Canada City Victoria Period 5/4/92 → 5/6/92 All Science Journal Classification (ASJC) codes Dive into the research topics of 'The complexity of multiway cuts'. Together they form a unique fingerprint.	{"url":"https://collaborate.princeton.edu/en/publications/the-complexity-of-multiway-cuts","timestamp":"2024-11-13T15:09:47Z","content_type":"text/html","content_length":"48027","record_id":"<urn:uuid:f19ec8e1-c7f9-4c41-ab57-c041fe6913a1>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00398.warc.gz"}
List of ap Physics equation sheet AP physics equation sheet If you are looking for an AP Physics equation Sheet in one place, Then you are at the right place. On this page, we have a list of basic physics equations including Equations of motion, Maxwell’s equation, lenses equations, thermodynamics equations, etc.. Mechanics Equations Thermodynamics equations Sound and Oscillation equations Electrical equations Modern physics equations	{"url":"https://oxscience.com/basic-physics-equations-sheet/","timestamp":"2024-11-04T02:30:55Z","content_type":"text/html","content_length":"101791","record_id":"<urn:uuid:3a0b3fde-eba5-4d72-9a8b-92247b9a060e>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00880.warc.gz"}
ðŸŸ¡ Introduction to Reinforcement Learning: Training an Agent in a Simple Environment Introduction to Reinforcement Learning: Training an Agent in a Simple Environment The main goal of this project is to introduce you to the fundamentals of Reinforcement Learning (RL) by training an agent to solve a simple control task. You will learn how agents interact with environments, implement basic RL algorithms, and understand key concepts such as exploration vs. exploitation, state representation, and policy evaluation. Learning Outcomes By completing this project, you will: • Understand the core concepts of Reinforcement Learning, including agents, environments, states, actions, rewards, policies, and value functions. • Learn how to use OpenAI Gym to create and simulate environments. • Implement a basic RL algorithm, specifically Q-Learning, from scratch. • Gain experience in training an agent to perform a task and evaluating its performance. • Learn how to handle continuous state spaces through discretization. • Develop skills in visualizing and analyzing agent performance. Prerequisites and Theoretical Foundations 1. Python Programming Fundamentals • Variables, data types, and basic operations. • Control structures (if/else, loops). • Functions and classes. • List comprehensions and basic data structures (lists, tuples, dictionaries). Click to view Python code examples # Function definition def greet(name): return f"Hello, {name}!" # Class definition class Agent: def __init__(self, name): self.name = name def act(self, state): # Decide on an action based on the state # Control structures for i in range(5): if i % 2 == 0: print(f"{i} is even") print(f"{i} is odd") # List comprehension squares = [x*2 for x in range(10)] 2. NumPy Essentials • Array creation and manipulation. • Basic array operations (addition, multiplication). • Array indexing and slicing. • Mathematical operations on arrays. Click to view NumPy code examples import numpy as np # Array creation zeros_array = np.zeros((3, 3)) random_array = np.random.rand(3, 3) # Element-wise operations sum_array = zeros_array + random_array # Indexing and slicing first_row = random_array[0, :] element = random_array[1, 2] # Mathematical operations mean_value = np.mean(random_array) max_value = np.max(random_array) 3. Basic Mathematics • Linear Algebra Fundamentals: Vectors, matrices, and matrix operations. • Probability Concepts: Random variables, probability distributions, expected value. • Basic understanding of calculus is helpful but not mandatory. Click to view mathematical concepts and examples Probability Example: • Expected value of a random variable ( X ): [ E = \sum_{x} x \cdot P(X = x) ] Matrix Operations Example: import numpy as np # Define a transition matrix transition_matrix = np.array([ [0.7, 0.3], [0.4, 0.6] # Current state vector current_state = np.array([1, 0]) # Starting in state 0 # Compute next state probabilities next_state_probs = np.dot(current_state, transition_matrix) print(next_state_probs) # Output: [0.7, 0.3] List of Theoretical Concepts Core RL Concepts 1. Reinforcement Learning Framework: □ Agent: The learner or decision-maker. □ Environment: The external system the agent interacts with. □ State (s): A representation of the current situation. □ Action (a): Choices the agent can make. □ Reward (r): Feedback signal indicating the desirability of the state/action. 2. The RL Loop: 1. The agent observes the current state ( s_t ). 2. The agent selects an action ( a_t ) based on its policy. 3. The environment transitions to a new state ( s_{t+1} ) and provides a reward ( r_{t+1} ). 4. The agent updates its policy based on the experience. 3. Key Terms: □ Policy (( \pi )): The agentâ€™s strategy for selecting actions, mapping states to actions. □ Value Function: Estimates how good a state (or state-action pair) is in terms of expected future rewards. □ Q-Function (( Q(s, a) )): The expected return (cumulative future reward) of taking action ( a ) in state ( s ) and following the policy thereafter. □ Exploration vs. Exploitation: Balancing the act of trying new actions to discover their effects (exploration) and choosing known actions that yield high rewards (exploitation). □ Learning Rate (( \alpha )): Determines how much new information overrides old information during learning. □ Discount Factor (( \gamma )): Determines the importance of future rewards compared to immediate rewards. Skills Gained • Understanding core Reinforcement Learning concepts and terminology. • Implementing basic RL algorithms like Q-Learning from scratch. • Using OpenAI Gym to simulate and interact with environments. • Handling continuous state spaces through discretization. • Implementing exploration strategies (e.g., epsilon-greedy). • Evaluating and visualizing agent performance. Tools Required • Python 3.7+ • NumPy: For numerical computations. • Matplotlib: For plotting and visualization. • OpenAI Gym: For environment simulation. • Jupyter Notebook or any Python IDE (e.g., VSCode, PyCharm). Install the required libraries using: pip install numpy matplotlib gym Steps and Tasks 1. Understanding the Environment We will use the CartPole-v1 environment from OpenAI Gym, a classic control problem where the goal is to keep a pole balanced on a moving cart. • Explore the environment: Understand the state and action spaces. • Run a random agent: Observe the performance when actions are selected randomly. import gym # Create the environment env = gym.make('CartPole-v1') # Print action and state space information print(f"Action space: {env.action_space}") # Discrete(2) print(f"Observation space: {env.observation_space}") # Box(4,) print(f"Observation space high values: {env.observation_space.high}") print(f"Observation space low values: {env.observation_space.low}") # Run one episode with random actions state = env.reset() done = False total_reward = 0 while not done: action = env.action_space.sample() # Random action (0 or 1) next_state, reward, done, info = env.step(action) total_reward += reward state = next_state print(f"Total reward from random agent: {total_reward}") • State Space: The environment provides a 4-dimensional continuous state vector: 1. Cart position. 2. Cart velocity. 3. Pole angle. 4. Pole angular velocity. • Action Space: There are two discrete actions: □ 0: Push cart to the left. □ 1: Push cart to the right. • Observation: □ Running the environment with a random agent typically results in poor performance, highlighting the need for a learning agent. 2. Discretizing the State Space Since the state space is continuous, weâ€™ll discretize it to apply tabular Q-Learning. • Define bins: Create bins for each state variable to discretize the continuous state space. • Implement a function: Map continuous states to discrete states (indices). import numpy as np def create_bins(num_bins=10): bins = [ np.linspace(-4.8, 4.8, num_bins), # Cart position np.linspace(-4, 4, num_bins), # Cart velocity np.linspace(-0.418, 0.418, num_bins), # Pole angle (~24 degrees) np.linspace(-4, 4, num_bins) # Pole angular velocity return bins def discretize_state(state, bins): """Convert continuous state to discrete state indices""" discrete_state = [] for i in range(len(state)): # Digitize returns the index of the bin each state element falls into index = np.digitize(state[i], bins[i]) - 1 # Subtract 1 for zero-based indexing # Ensure index is within bounds index = min(max(index, 0), len(bins[i]) - 1) return tuple(discrete_state) # Example usage bins = create_bins() state = env.reset() discrete_state = discretize_state(state, bins) print(f"Discrete state: {discrete_state}") • Bins: □ We use np.linspace to create equally spaced bins for each state variable. □ The number of bins (num_bins) can be adjusted to balance between state representation granularity and computational resources. • Discretization Function: □ np.digitize determines which bin each state variable falls into. □ The function maps continuous values to discrete indices, forming a discrete representation of the state. 3. Implementing the Q-Learning Agent Weâ€™ll create an agent that learns an optimal policy using the Q-Learning algorithm. • Initialize the Q-table: A multi-dimensional array representing Q-values for state-action pairs. • Implement the Q-Learning update rule. • Implement an epsilon-greedy policy for action selection. class QLearningAgent: def __init__(self, state_bins, action_size, learning_rate=0.1, gamma=0.99, epsilon=1.0, epsilon_decay=0.995, epsilon_min=0.01): self.state_bins = state_bins self.action_size = action_size self.q_table = np.zeros(state_bins + (action_size,)) self.alpha = learning_rate # Learning rate self.gamma = gamma # Discount factor self.epsilon = epsilon # Exploration rate self.epsilon_decay = epsilon_decay self.epsilon_min = epsilon_min def choose_action(self, state): if np.random.rand() < self.epsilon: # Explore: select a random action return np.random.randint(self.action_size) # Exploit: select the action with the highest Q-value for the current state return np.argmax(self.q_table[state]) def update_q_value(self, state, action, reward, next_state, done): # Q-Learning update rule best_next_action = np.argmax(self.q_table[next_state]) td_target = reward + self.gamma self.q_table[next_state + (best_next_action,)] * (1 - int(done)) td_error = td_target - self.q_table[state + (action,)] self.q_table[state + (action,)] += self.alpha * td_error def decay_epsilon(self): # Decay the exploration rate if self.epsilon > self.epsilon_min: self.epsilon *= self.epsilon_decay • Q-Table Initialization: □ The Q-table is initialized with zeros and has dimensions corresponding to the discretized state space plus the action dimension. • Action Selection: □ Epsilon-Greedy Policy: ☆ With probability ( \epsilon ), the agent explores by selecting a random action. ☆ With probability ( 1 - \epsilon ), the agent exploits by choosing the action with the highest Q-value. • Q-Value Update Rule: □ The agent updates its Q-values using the Bellman equation: [ Q(s, a) \leftarrow Q(s, a) + \alpha \left[ r + \gamma \max_{aâ€™} Q(sâ€™, aâ€™) - Q(s, a) \right] ] □ This update aims to minimize the temporal-difference (TD) error between the predicted Q-value and the observed reward plus discounted future rewards. • Epsilon Decay: □ Gradually reducing ( \epsilon ) encourages the agent to explore less over time and exploit learned knowledge more. 4. Training the Agent Weâ€™ll train the agent over multiple episodes, allowing it to learn from interactions with the environment. • Run training episodes: Iterate over a specified number of episodes. • Implement the learning loop: The agent interacts with the environment, updates Q-values, and collects rewards. • Track performance: Record the total reward per episode for analysis. def train_agent(env, agent, bins, num_episodes=1000, max_steps_per_episode=200): rewards = [] for episode in range(num_episodes): state = env.reset() discrete_state = discretize_state(state, bins) total_reward = 0 done = False for step in range(max_steps_per_episode): action = agent.choose_action(discrete_state) next_state, reward, done, info = env.step(action) next_discrete_state = discretize_state(next_state, bins) total_reward += reward # Update Q-value agent.update_q_value(discrete_state, action, reward, next_discrete_state, done) discrete_state = next_discrete_state if done: # Decay epsilon # Print progress every 100 episodes if (episode + 1) % 100 == 0: avg_reward = np.mean(rewards[-100:]) print(f"Episode {episode + 1}/{num_episodes}, Average Reward (last 100 episodes): {avg_reward:.2f}, Epsilon: {agent.epsilon:.4f}") return rewards • Episode Loop: □ For each episode, reset the environment and initialize the state. □ The agent selects actions, observes rewards and next states, and updates Q-values. □ The episode ends when the environment signals done or the maximum number of steps is reached. • Performance Tracking: □ Collect the total reward per episode to monitor learning progress. □ Periodically print the average reward and current epsilon value to observe trends. • Epsilon Decay: □ After each episode, decay the exploration rate to shift the agentâ€™s focus from exploration to exploitation. 5. Evaluating the Agent After training, we evaluate the agentâ€™s performance without exploration to assess how well it has learned. • Run evaluation episodes: Using the learned policy without exploration. • Measure performance: Calculate the average total reward over evaluation episodes. def evaluate_agent(env, agent, bins, num_episodes=10, max_steps_per_episode=200): agent.epsilon = 0 # Disable exploration total_rewards = [] for episode in range(num_episodes): state = env.reset() discrete_state = discretize_state(state, bins) total_reward = 0 done = False for step in range(max_steps_per_episode): action = agent.choose_action(discrete_state) next_state, reward, done, info = env.step(action) discrete_state = discretize_state(next_state, bins) total_reward += reward if done: print(f"Evaluation Episode {episode + 1}/{num_episodes}, Total Reward: {total_reward}") average_reward = np.mean(total_rewards) print(f"Average Total Reward over {num_episodes} Evaluation Episodes: {average_reward:.2f}") • Evaluation without Exploration: □ By setting agent.epsilon = 0, the agent always selects the action with the highest Q-value. • Rendering: □ We call env.render() to visualize the agentâ€™s performance during evaluation. • Performance Metrics: □ We compute the total reward for each evaluation episode and calculate the average total reward. 6. Visualizing Training Progress Plotting the rewards over episodes helps in analyzing the agentâ€™s learning curve and identifying trends. import matplotlib.pyplot as plt def plot_rewards(rewards): plt.figure(figsize=(12, 6)) plt.plot(rewards, label='Episode Reward') plt.ylabel('Total Reward') plt.title('Training Progress') # Calculate and plot the moving average window_size = 50 moving_avg = np.convolve(rewards, np.ones(window_size)/window_size, mode='valid') plt.figure(figsize=(12, 6)) plt.plot(range(window_size - 1, len(rewards)), moving_avg, label=f'{window_size}-Episode Moving Average', color='orange') plt.ylabel('Average Reward') plt.title('Moving Average of Rewards') • Episode Rewards Plot: □ Displays the total reward per episode. □ Helps identify whether the agentâ€™s performance is improving over time. • Moving Average Plot: □ Smoothing the rewards using a moving average provides a clearer view of the overall learning trend. □ The window size determines the level of smoothing. 7. Running the Complete Project We will now bring all components together and execute the training and evaluation of the agent. def main(): env = gym.make('CartPole-v1') bins = create_bins(num_bins=10) state_bins = tuple(len(bin) for bin in bins) action_size = env.action_space.n agent = QLearningAgent(state_bins=state_bins, action_size=action_size) num_training_episodes = 2000 rewards = train_agent(env, agent, bins, num_episodes=num_training_episodes) print("Starting evaluation...") evaluate_agent(env, agent, bins, num_episodes=5) if __name__ == "__main__": • Environment and Agent Initialization: □ Create the CartPole environment. □ Define the discretization bins and determine the dimensions for the Q-table. • Training: □ Train the agent using the train_agent function. □ Record the rewards for visualization. • Visualization: □ Plot the training rewards to analyze learning progress. • Evaluation: □ Evaluate the agentâ€™s performance over a few episodes. 8. Next Steps and Improvements After successfully implementing and understanding the basic RL agent, consider the following enhancements to deepen your learning: 1. Parameter Tuning: □ Adjust the Number of Bins: ☆ Experiment with different numbers of bins for discretization to find a balance between state representation accuracy and computational efficiency. □ Modify Learning Rate and Discount Factor: ☆ Test different values for the learning rate (( \alpha )) and discount factor (( \gamma )) to see how they affect learning speed and stability. □ Epsilon Decay Strategy: ☆ Try different epsilon decay rates or strategies (e.g., linear decay, exponential decay) to optimize the exploration-exploitation balance. 2. Algorithm Enhancements: □ Double Q-Learning: ☆ Implement Double Q-Learning to address overestimation bias in Q-Learning. □ Function Approximation: ☆ Use function approximators like neural networks (Deep Q-Networks) to handle continuous state spaces without discretization. □ Eligibility Traces: ☆ Implement SARSA(Î») or Q(Î») to consider the sequence of states and actions leading to rewards. 3. Advanced Exploration Strategies: □ Softmax Action Selection: ☆ Use a softmax function over Q-values to select actions probabilistically, favoring higher Q-values but allowing exploration. □ Upper Confidence Bound (UCB): ☆ Implement UCB to balance exploration and exploitation based on uncertainty estimates. 4. Applying to Different Environments: □ Test on Other OpenAI Gym Environments: ☆ Apply your agent to environments like MountainCar-v0 or Acrobot-v1, adapting your approach as necessary. □ Custom Environments: ☆ Create simple custom environments to challenge your agent in new ways. 5. Performance Monitoring and Logging: □ Enhanced Visualization: ☆ Plot additional metrics like maximum reward per episode or Q-value distributions. □ Logging Libraries: ☆ Integrate logging tools or frameworks (e.g., TensorBoard) to monitor training in real-time. 6. Handling Continuous Actions: □ Policy Gradient Methods: ☆ Explore algorithms like REINFORCE or Actor-Critic methods that can handle continuous action spaces. 9. Common Issues and Solutions Issue: Agentâ€™s performance plateaus or does not improve. Possible Solutions: • Check the Discretization: □ Ensure the bins cover the full range of possible state values. □ Increase the number of bins for more precise state representation, but be cautious of the curse of dimensionality. • Adjust Hyperparameters: □ Learning Rate (( \alpha )): ☆ If learning is unstable, try decreasing the learning rate. □ Discount Factor (( \gamma )): ☆ A higher gamma places more emphasis on future rewards. □ Epsilon Decay: ☆ Adjust the decay rate to ensure sufficient exploration. • Modify the Reward Structure: □ Introduce penalties for undesirable behaviors to guide learning. Issue: Agent performs well during training but poorly during evaluation. Possible Solutions: • Ensure Exploration is Disabled During Evaluation: □ Set agent.epsilon = 0 before evaluation to prevent random actions. • Overfitting to Exploration: □ The agent may rely on exploration to achieve higher rewards; consider adjusting the epsilon decay schedule to encourage learning of the optimal policy. Issue: High Variance in Rewards Across Episodes. Possible Solutions: • Increase the Number of Training Episodes: □ Allow the agent more time to learn consistent behavior. • Use Moving Averages: □ Analyze the moving average of rewards to assess trends over time. 10. Conclusion In this project, you have: • Gained an understanding of the foundational concepts of Reinforcement Learning. • Implemented a basic Q-Learning agent to solve the CartPole-v1 environment. • Learned how to discretize continuous state spaces for tabular methods. • Explored the balance between exploration and exploitation using epsilon-greedy strategies. • Evaluated and visualized your agentâ€™s performance over time. This foundational knowledge prepares you for more advanced RL topics, such as: • Deep Reinforcement Learning: Using neural networks to approximate value functions or policies. • Policy Gradient Methods: Directly optimizing the policy without relying on value functions. • Model-Based RL: Learning a model of the environment to plan ahead. • Multi-Agent RL: Extending RL concepts to environments with multiple agents.	{"url":"https://stemaway.com/t/introduction-to-reinforcement-learning-training-an-agent-in-a-simple-environment/16446","timestamp":"2024-11-12T22:05:37Z","content_type":"text/html","content_length":"56827","record_id":"<urn:uuid:05c9e2b4-d96e-41d7-a6db-ff8b08f10aeb>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00793.warc.gz"}
Validating Causal Inference Models via Influence Functions Validating Causal Inference Models via Influence Functions Proceedings of the 36th International Conference on Machine Learning, PMLR 97:191-201, 2019. The problem of estimating causal effects of treatments from observational data falls beyond the realm of supervised learning {—} because counterfactual data is inaccessible, we can never observe the true causal effects. In the absence of "supervision", how can we evaluate the performance of causal inference methods? In this paper, we use influence functions {—} the functional derivatives of a loss function {—} to develop a model validation procedure that estimates the estimation error of causal inference methods. Our procedure utilizes a Taylor-like expansion to approximate the loss function of a method on a given dataset in terms of the influence functions of its loss on a "synthesized", proximal dataset with known causal effects. Under minimal regularity assumptions, we show that our procedure is consistent and efficient. Experiments on 77 benchmark datasets show that using our procedure, we can accurately predict the comparative performances of state-of-the-art causal inference methods applied to a given observational study. Cite this Paper Related Material	{"url":"http://proceedings.mlr.press/v97/alaa19a.html","timestamp":"2024-11-08T15:46:58Z","content_type":"text/html","content_length":"15262","record_id":"<urn:uuid:f873deb2-e97b-46aa-ba86-71902d4386e4>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00713.warc.gz"}
virtual int SetParameters (Teuchos::ParameterList &List)=0 Sets all parameters for the preconditioner. More... virtual int Initialize ()=0 Computes all it is necessary to initialize the preconditioner. More... virtual bool IsInitialized () const =0 Returns true if the preconditioner has been successfully initialized, false otherwise. More... virtual int Compute ()=0 Computes all it is necessary to apply the preconditioner. More... virtual bool IsComputed () const =0 Returns true if the preconditioner has been successfully computed, false otherwise. More... virtual double Condest (const Ifpack_CondestType CT=Ifpack_Cheap, const int MaxIters=1550, const double Tol=1e-9, Epetra_RowMatrix Matrix=0)=0 Computes the condition number estimate, returns its value. More... virtual double Condest () const =0 Returns the computed condition number estimate, or -1.0 if not computed. More... virtual int ApplyInverse (const Epetra_MultiVector &X, Epetra_MultiVector &Y) const =0 Applies the preconditioner to vector X, returns the result in Y. More... virtual const Epetra_RowMatrix & Matrix () const =0 Returns a pointer to the matrix to be preconditioned. More... virtual int NumInitialize () const =0 Returns the number of calls to Initialize(). More... virtual int NumCompute () const =0 Returns the number of calls to Compute(). More... virtual int NumApplyInverse () const =0 Returns the number of calls to ApplyInverse(). More... virtual double InitializeTime () const =0 Returns the time spent in Initialize(). More... virtual double ComputeTime () const =0 Returns the time spent in Compute(). More... virtual double ApplyInverseTime () const =0 Returns the time spent in ApplyInverse(). More... virtual double InitializeFlops () const =0 Returns the number of flops in the initialization phase. More... virtual double ComputeFlops () const =0 Returns the number of flops in the computation phase. More... virtual double ApplyInverseFlops () const =0 Returns the number of flops in the application of the preconditioner. More... virtual std::ostream & Print (std::ostream &os) const =0 Prints basic information on iostream. This function is used by operator<<. More... virtual int SetUseTranspose (bool UseTranspose)=0 virtual int Apply (const Epetra_MultiVector &X, Epetra_MultiVector &Y) const =0 virtual double NormInf () const =0 virtual const char Label () const =0 virtual bool UseTranspose () const =0 virtual bool HasNormInf () const =0 virtual const Epetra_Comm & Comm () const =0 virtual const Epetra_Map & OperatorDomainMap () const =0 virtual const Epetra_Map & OperatorRangeMap () const =0 virtual int SetUseTranspose (bool UseTranspose)=0 virtual int Apply (const Epetra_MultiVector &X, Epetra_MultiVector &Y) const =0 virtual double NormInf () const =0 virtual const char * Label () const =0 virtual bool UseTranspose () const =0 virtual bool HasNormInf () const =0 virtual const Epetra_Comm & Comm () const =0 virtual const Epetra_Map & OperatorDomainMap () const =0 virtual const Epetra_Map & OperatorRangeMap () const =0 Ifpack_Preconditioner: basic class for preconditioning in Ifpack. Class Ifpack_Preconditioner is a pure virtual class, and it defines the structure of all Ifpack preconditioners. This class is a simple extension to Epetra_Operator. It provides the following additional methods: It is required that Compute() call Initialize() if IsInitialized() returns false. The preconditioner is applied by ApplyInverse() (which returns if IsComputed() is false). Every time that Initialize () is called, the object destroys all the previously allocated information, and re-initialize the preconditioner. Every time Compute() is called, the object re-computed the actual values of the Estimating Preconditioner Condition Numbers The condition of a matrix A condition number approaching the accuracy of a given floating point number system, about 15 decimal digits in IEEE double precision, means that any results involving Method Compute() can be use to estimate of the condition number. Compute() requires one parameter, of type Ifpack_CondestType (default value is Ifpack_Cheap; other valid choices are Ifpack_CG and While Ifpack_CG and Ifpack_GMRES construct and AztecOO solver, and use methods AZ_cg_condnum and AZ_gmres_condnum to evaluate an accurate (but very expensive) estimate of the condition number, Ifpack_Cheap computes • since finding • the only cost is one application of the preconditioner. If this estimate is very large, the application of the computed preconditioner may generate large numerical errors. Hence, the user may check this number, and decide to recompute the preconditioner is the computed estimate is larger than a given threshold. This is particularly useful in ICT and RILUK factorizations, as for ill-conditioned matrices, we often have difficulty computing usable incomplete factorizations. The most common source of problems is that the factorization may encounter a small or zero pivot, in which case the factorization can fail, or even if the factorization succeeds, the factors may be so poorly conditioned that use of them in the iterative phase produces meaningless results. Before we can fix this problem, we must be able to detect it. If IFPACK is configured with Teuchos support, method SetParameters() should be adopted. Otherwise, users can set parameters (one at-a-time), using methods SetParameter(), for integers and Ifpack_Preconditioner objects overload the << operator. Derived classes should specify a Print() method, that will be used in operator <<. Definition at line 142 of file Ifpack_Preconditioner.h.	{"url":"https://docs.trilinos.org/dev/packages/ifpack/browser/doc/html/classIfpack__Preconditioner.html","timestamp":"2024-11-09T22:11:45Z","content_type":"application/xhtml+xml","content_length":"97531","record_id":"<urn:uuid:257cf07e-1d82-4a03-b10e-4ed49d68a8cb>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00071.warc.gz"}
Check if a list has duplicate numbers [Quick tip] » Chandoo.org - Learn Excel, Power BI & Charting Online A while ago (well more than 3 years ago), I wrote about an array formula based technique to check if a list of values have any duplicates in them. Today, lets learn a simpler formula to check if a list has duplicate numbers. Assuming you have some numbers in a range B4:B10 as shown below, You can use COUNTIF & MODE formulas to check if the list has any duplicates, like this: =IF(COUNTIF($B$4:$B$10,MODE($B$4:$B$10))>1, "List has duplicates", "No duplicates") How does it work? MODE formula gives us the most frequently occurring number in a list. Then, we use COUNTIF to see how many times this number occurs in a list. In a list with no duplicates mode value occurs only 1 time. If a list has duplicate numbers, then count of mode would be more than 1. That is what the IF formula checks for and then prints appropriate message. See this example: [Embedded Excel, if you can not see it, click here] Play with below embedded Excel file to understand the technique. You can modify numbers or formula. Or Download this Example Click here to download the example workbook and play with it. How do you check if a list has duplicates? For text values, I use the array formula technique described here. For numeric values, I prefer MODE + COUNTIF combination because it is easy to write & explain. What about you? How do you check if a list has duplicates? Which formulas do you use? Please share your techniques using comments. More on Duplicates & Unique values If we analyze the time an analyst spends on various things, we would realize, • 30% of time cleaning data (removing duplicates etc.) • 30% of time actual analysis • 30% of time drinking coffee • 10% of time actual presentation On a more serious note, if you want to learn various techniques to deal with duplicate values, read on: 29 Responses to “Check if a list has duplicate numbers [Quick tip]” 1. Dohsan says: Do you even require the COUNTIF function? It seems that MODE will return #N/A if no number appears more than once, so you could use: =IF(ISERROR(MODE($B$4:$B$10)),"No duplicates","List has duplicates") □ You are right. I did not realize it. Thanks for sharing this. □ its working i have done in my data 2. Lynda says: I used to use the COUNTIF function, but in Excel2007 & 2010 the conditional formatting, highlight duplicates, is REALLY easy to get to and easy to use, so that's all I use for it. You can even just use it on the fly as a quick check for duplicates & then cancel without applying. □ Ravi says: Hi Lydna, I completely agree with you. In excel 2007 and 2010 we have Conditional Formatting which highlights the duplicate values in seconds. As you said you can check and then leave without applying or delete the duplicates as needed. 3. RichW says: This method doesn't work with non-Numeric data. I can't really think of a time that I'm looking at a list to weed out duplicate #s, usually duplicate text. □ See this for a method that works for text as well: 4. Daniel Ferry says: =REPT("no ",ISNA(MODE($B$4:$B$10))) & "duplicates" □ Well, that is what genius is. Thanks for posting it 🙂 □ Robert Clark says: Didn't quite work for me - had to miodify it to =REPT("no ",ISNA(MODE($B$4:$B$10))1) & "duplicates" 5. sam says: Array Enter Returns a True if there are Duplicates, False otherwise Works for both numbers and text Non Array =SUMPRODUCT(COUNTIF(B4:B10,B4:10)-1) return 0 if no duplicates, number if there are dups 6. abe adler says: I have a much simpler method- Assuming data in column A- --sort column A --Add in Column B the following formula---=if(a2=a1, "duplicate", "not duplicate") --Copy down □ same here i always use this 7. Arun says: We can use conditional formattting to check if there are duplicates. Select the seriers of numbers Conditional formatting --> Highlight cell rules --> Duplicate values. If there are any duplicate values then the numbers will be highlighted. 8. If you want a conditional formatting based technique, see this: 9. James says: It seems kind of boring compared to using formulas, but I find pivot tables a convenient way of determining if there are duplicates in a file; and it works for both text and numbers. 10. Clarity says: To automatically remove, rather than just identify duplicates, make the data into a table (2007 onwards insert table). You can then use the remove duplicates function. Another method I have used in the past is to create a pivot table based on the data and then pull though the column to analyse (in the row labels area) and then pull through the same column into the values area (as a count). If you then sort descending any duplicates will come to the top of the pivot table. 11. David K says: Assuming data with possible dups is in Col A. =countif(A:A,a1) fill down Counts of the number of occurrences, and then you can filter/sort Largest to Smallest to remove them. Works with text and numbers. □ VENKY says: or modify the formula to =IF(COUNTIF(B:B,B4)>1,"Duplicates Found","No Duplicate") 12. Jeff Nickerson says: I just use the following in a new column =countif (A:A,A1)>1 then copy down, all the trues are duplicates, false number only occurs 1 time, also works with text. □ its too dificult to do with huge data we can take any risk but if we just want to know that there is some duplicate or not then we must use =IF(ISERROR(MODE($B$4:$B$10)),”No duplicates”,”List has duplicates”) 13. Mark says: I prefer to go line by line and check...:) Actually, I use the conditional formatting function to find duplicates or in some cases, unique values. I run a report each week where I need to figure out which records have been added since the report was last run and I use conditional formatting to identify unique values. Thanks for the tips! They're always interesting to look at. 14. Leon Kowalski says: While I like this tip and was excited to learn another practical use for MODE, I agree with most of the comments in that more analysis is usually required in such circumstances. I extensively use 2 formulas(!) when I create interactive pivot tables. These tables are either to empower the client to ascertain what they need immediately or they become prototypes for a for defined, and more complex, model. My first formula identifies the original occurrence of any given argument: eg: if(sumproduct(($A$2:$A2=A2)*($B$2:$B2=B2))>1,0,1) The formula is copied for each record in the data-set. The second formula simply identifies the total occurrences of a given criteria eg: countif($c$2:$c$10,$c$2:$c$10) - Also copied to every record within data-set A practical application of these formula could be as follows: Having followed Chandoo.org, I realise these formula are not spectacular. Indeed, they come with issues and restrictions. However, inspired by Chandoo.org, and now by the ability to present using MS Web Apps, I wanted to share this with my peers for further review as I feel the formulas, and webb app example, demonstrate so much more than my attempt to explain. And, as I have said, I actually use these formula extensively in the real world and so wanted to share my opinion. 15. Deepali says: How we find the common & unique value from sheet 1&2 in sheet 3 16. ronoele says: i am working with sudoku using excel....and im getting trouble of how to determine the duplicate numbers and the certain number that occur more than once...could someone hep me with this please:))thanks in advance! 17. Mode does not work with TEXT duplicates. CountIf does not work with arrays. This method works with both, in VBA or on a worksheet. It returns the number of values that appear in both lists. Dupes = WorksheetFunction.Count(Application.Match(sList1, sList2, 0)) Worksheet (must be entered as array formula): If you want a simple TRUE/FALSE result, wrap it in a simple IF > 0 function: =IF(COUNT(MATCH($A$1:$A$3,$B$1:$B$3,0))>0, TRUE, FALSE) □ Count works because Count automatically ignores errors and blanks 🙂 (PS, anyone know any other functions that automatically ignore errors and blanks?) □ in VBA, certain worksheet functions only work with the Application. prefix, instead of WorksheetFunction. prefix. □ Trying to take this formula to the next level, by returning an array of the duplicate items. These two methods fail: INDEX apparently cannot return an array (tho i've read it can). Anyone know a non-VBA way to return an array from INDEX, or another function that can achieve the same thing? Here, OFFSET does return an array of the duplicate items, which is great. But also returns a bunch of errors (for the non-duplicated items). Anyone know a simple non-VBA function to remove errors from a worksheet array? I can think of a way to do it with a nested IF...ISERROR formula, but hoping for a simpler method.	{"url":"https://chandoo.org/wp/check-list-for-duplicate-numbers/","timestamp":"2024-11-12T02:10:32Z","content_type":"text/html","content_length":"472784","record_id":"<urn:uuid:cf7d410b-9eb7-41bc-a74b-e3645676a44d>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00862.warc.gz"}
Algorithm to Find the Area of a Circle - TestingDocs.com Algorithm to Find the Area of a Circle In this tutorial, we will learn how to write and develop an algorithm and flowchart to find the area of a circle of radius r. The problem to solve is to find the area of the circle. The mathematical formula to compute the area of the circle is: area = pirr IPO chart Let’s analyze the inputs and the expected outputs of the program. Input, Process, and Output chart. │Input │Process │Output │ │The radius of the circle(r) │Area = pi * r * r│Area of the circle(Area)│ Input: Radius r of the Circle. Output: Area of the Circle • Step 1: Start • Step 2: Read the radius r of the circle • Step 3: Compute the Area as per the formula. PIrr • Step 4: Print the area of the circle • Step 5: Stop The pseudocode for the algorithm is as follows: DECLARE Real r DECLARE Real area OUTPUT "Enter the circle radius=" INPUT r ASSIGN area = PIrr OUTPUT "Area of the circle with radius " & r & " = " & area Let’s develop the flowchart for the given problem: Sample Run Execute the flowchart and verify its output. That’s it. We have successfully designed an algorithm & flowchart to compute the area of the given circle. Flowgorithm Tutorials You can find tutorials for the Flowgorithm flowchart tool on this website at: Flowgorithm Website For more information, visit the official website of Flowgorithm.	{"url":"https://www.testingdocs.com/algorithm-and-flowchart-to-find-the-area-of-a-circle-of-radius-r/?amp=1","timestamp":"2024-11-04T07:24:06Z","content_type":"text/html","content_length":"72409","record_id":"<urn:uuid:ce56fa6d-bf15-49bd-9a9e-e424d4b5359e>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00435.warc.gz"}
782 Arcmin/Square Hour to Circle/Square Month Arcmin/Square Hour [arcmin/h2] Output 782 arcmin/square hour in degree/square second is equal to 0.000001005658436214 782 arcmin/square hour in degree/square millisecond is equal to 1.005658436214e-12 782 arcmin/square hour in degree/square microsecond is equal to 1.005658436214e-18 782 arcmin/square hour in degree/square nanosecond is equal to 1.005658436214e-24 782 arcmin/square hour in degree/square minute is equal to 0.0036203703703704 782 arcmin/square hour in degree/square hour is equal to 13.03 782 arcmin/square hour in degree/square day is equal to 7507.2 782 arcmin/square hour in degree/square week is equal to 367852.8 782 arcmin/square hour in degree/square month is equal to 6954980.92 782 arcmin/square hour in degree/square year is equal to 1001517253.2 782 arcmin/square hour in radian/square second is equal to 1.7552050862392e-8 782 arcmin/square hour in radian/square millisecond is equal to 1.7552050862392e-14 782 arcmin/square hour in radian/square microsecond is equal to 1.7552050862392e-20 782 arcmin/square hour in radian/square nanosecond is equal to 1.7552050862392e-26 782 arcmin/square hour in radian/square minute is equal to 0.00006318738310461 782 arcmin/square hour in radian/square hour is equal to 0.22747457917659 782 arcmin/square hour in radian/square day is equal to 131.03 782 arcmin/square hour in radian/square week is equal to 6420.24 782 arcmin/square hour in radian/square month is equal to 121387.32 782 arcmin/square hour in radian/square year is equal to 17479773.58 782 arcmin/square hour in gradian/square second is equal to 0.00000111739826246 782 arcmin/square hour in gradian/square millisecond is equal to 1.11739826246e-12 782 arcmin/square hour in gradian/square microsecond is equal to 1.11739826246e-18 782 arcmin/square hour in gradian/square nanosecond is equal to 1.11739826246e-24 782 arcmin/square hour in gradian/square minute is equal to 0.004022633744856 782 arcmin/square hour in gradian/square hour is equal to 14.48 782 arcmin/square hour in gradian/square day is equal to 8341.33 782 arcmin/square hour in gradian/square week is equal to 408725.33 782 arcmin/square hour in gradian/square month is equal to 7727756.58 782 arcmin/square hour in gradian/square year is equal to 1112796948 782 arcmin/square hour in arcmin/square second is equal to 0.00006033950617284 782 arcmin/square hour in arcmin/square millisecond is equal to 6.033950617284e-11 782 arcmin/square hour in arcmin/square microsecond is equal to 6.033950617284e-17 782 arcmin/square hour in arcmin/square nanosecond is equal to 6.033950617284e-23 782 arcmin/square hour in arcmin/square minute is equal to 0.21722222222222 782 arcmin/square hour in arcmin/square day is equal to 450432 782 arcmin/square hour in arcmin/square week is equal to 22071168 782 arcmin/square hour in arcmin/square month is equal to 417298855.5 782 arcmin/square hour in arcmin/square year is equal to 60091035192 782 arcmin/square hour in arcsec/square second is equal to 0.0036203703703704 782 arcmin/square hour in arcsec/square millisecond is equal to 3.6203703703704e-9 782 arcmin/square hour in arcsec/square microsecond is equal to 3.6203703703704e-15 782 arcmin/square hour in arcsec/square nanosecond is equal to 3.6203703703704e-21 782 arcmin/square hour in arcsec/square minute is equal to 13.03 782 arcmin/square hour in arcsec/square hour is equal to 46920 782 arcmin/square hour in arcsec/square day is equal to 27025920 782 arcmin/square hour in arcsec/square week is equal to 1324270080 782 arcmin/square hour in arcsec/square month is equal to 25037931330 782 arcmin/square hour in arcsec/square year is equal to 3605462111520 782 arcmin/square hour in sign/square second is equal to 3.35219478738e-8 782 arcmin/square hour in sign/square millisecond is equal to 3.35219478738e-14 782 arcmin/square hour in sign/square microsecond is equal to 3.35219478738e-20 782 arcmin/square hour in sign/square nanosecond is equal to 3.35219478738e-26 782 arcmin/square hour in sign/square minute is equal to 0.00012067901234568 782 arcmin/square hour in sign/square hour is equal to 0.43444444444444 782 arcmin/square hour in sign/square day is equal to 250.24 782 arcmin/square hour in sign/square week is equal to 12261.76 782 arcmin/square hour in sign/square month is equal to 231832.7 782 arcmin/square hour in sign/square year is equal to 33383908.44 782 arcmin/square hour in turn/square second is equal to 2.79349565615e-9 782 arcmin/square hour in turn/square millisecond is equal to 2.79349565615e-15 782 arcmin/square hour in turn/square microsecond is equal to 2.79349565615e-21 782 arcmin/square hour in turn/square nanosecond is equal to 2.79349565615e-27 782 arcmin/square hour in turn/square minute is equal to 0.00001005658436214 782 arcmin/square hour in turn/square hour is equal to 0.036203703703704 782 arcmin/square hour in turn/square day is equal to 20.85 782 arcmin/square hour in turn/square week is equal to 1021.81 782 arcmin/square hour in turn/square month is equal to 19319.39 782 arcmin/square hour in turn/square year is equal to 2781992.37 782 arcmin/square hour in circle/square second is equal to 2.79349565615e-9 782 arcmin/square hour in circle/square millisecond is equal to 2.79349565615e-15 782 arcmin/square hour in circle/square microsecond is equal to 2.79349565615e-21 782 arcmin/square hour in circle/square nanosecond is equal to 2.79349565615e-27 782 arcmin/square hour in circle/square minute is equal to 0.00001005658436214 782 arcmin/square hour in circle/square hour is equal to 0.036203703703704 782 arcmin/square hour in circle/square day is equal to 20.85 782 arcmin/square hour in circle/square week is equal to 1021.81 782 arcmin/square hour in circle/square month is equal to 19319.39 782 arcmin/square hour in circle/square year is equal to 2781992.37 782 arcmin/square hour in mil/square second is equal to 0.00001787837219936 782 arcmin/square hour in mil/square millisecond is equal to 1.787837219936e-11 782 arcmin/square hour in mil/square microsecond is equal to 1.787837219936e-17 782 arcmin/square hour in mil/square nanosecond is equal to 1.787837219936e-23 782 arcmin/square hour in mil/square minute is equal to 0.064362139917695 782 arcmin/square hour in mil/square hour is equal to 231.7 782 arcmin/square hour in mil/square day is equal to 133461.33 782 arcmin/square hour in mil/square week is equal to 6539605.33 782 arcmin/square hour in mil/square month is equal to 123644105.33 782 arcmin/square hour in mil/square year is equal to 17804751168 782 arcmin/square hour in revolution/square second is equal to 2.79349565615e-9 782 arcmin/square hour in revolution/square millisecond is equal to 2.79349565615e-15 782 arcmin/square hour in revolution/square microsecond is equal to 2.79349565615e-21 782 arcmin/square hour in revolution/square nanosecond is equal to 2.79349565615e-27 782 arcmin/square hour in revolution/square minute is equal to 0.00001005658436214 782 arcmin/square hour in revolution/square hour is equal to 0.036203703703704 782 arcmin/square hour in revolution/square day is equal to 20.85 782 arcmin/square hour in revolution/square week is equal to 1021.81 782 arcmin/square hour in revolution/square month is equal to 19319.39 782 arcmin/square hour in revolution/square year is equal to 2781992.37	{"url":"https://hextobinary.com/unit/angularacc/from/arcminph2/to/circlepm2/782","timestamp":"2024-11-13T05:23:36Z","content_type":"text/html","content_length":"113198","record_id":"<urn:uuid:e50f1ae8-16f3-4ed0-8a01-0283be307de1>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00691.warc.gz"}
American Journal of Computational Mathematics Vol. 1 No. 1 (2011) , Article ID: 4447 , 4 pages DOI:10.4236/ajcm.2011.11001 On the Location of Zeros of Polynomials ^1Bharathiar University, Coimbatore, India ^2Department of Mathematics, Srinagar, India E-mail: {gulshansingh1, wmshah}@rediffmail.com Received January 26, 2011; revised February 16, 2011; accepted February 16, 2011 Keywords: Polynomial, Zeros, Eneström-Kakeya Theorem In this paper, we prove some extensions and generalizations of the classical Eneström-Kakeya theorem. 1. Introduction and Statement of Results then according to a classical result usually known as Eneström-Kakeya theorem [11], Theorem A. If In the literature, [1-15], there exist extensions and generalizations of Eneström-Kakeya theorem. Joyal, Labelle and Rahman [9] extended this theorem to polynomials whose coefficients are monotonic but not necessarily non negative and the result was further generalized by Dewan and Bidkham [6] to read as: Theorem B. If Govil and Rahman [8] extended Theorem A to the polynomials with complex coefficients. As a refinement of the result of Govil and Rahman, Govil and Jain [7] proved the following. Theorem C. Let By using Schwarz’s Lemma, Aziz and Mohammad [1] generalized Eneström-Kakeya theorem in a different way and proved: Theorem D. Let In this paper, we also make use of a generalized form of Schwarz’s Lemma and prove some more general results which include not only the above theorems as special cases, but also lead to a standard development of interesting generalizations of some well known results. Infact we prove Theorem 1. Let then all the zeros of Assuming that all the coefficients Corollary 1. Let then all the zeros of If in Corollary 1, we assume that all the coefficients are positive and Corollary 2. Let then all the zeros of In particular, if Corollary 3. Let then all the zeros of We next prove the following more general result which include many known results as special cases. Theorem 2. Let then all the zeros of Remark 1. Theorem B is a special case of Theorem 2, if we take The following result follows immediately from Theorem 2 by taking Corollary 4. Let then all the zeros of Remark 2. For We also prove the following result which is of independent interest. Theorem 3. Let then all the zeros of Remark 3. Theorem 4 of [4] immediately follows from Theorem 3 when On combining Theorem 2 and Theorem 3 the following more interesting result is immediate. Corollary 5. Let then all the zeros of If we take Corollary 6. Let then all the zeros of The following result also follows from Theorem 3, when Corollary 7. Let 2. Lemmas For proving the above theorems, we require the following lemmas. The first Lemma which we need is due to Rahman and Schmeisser [11]. Lemma 1. If Lemma 2. If The next Lemma is due to Aziz and Mohammad [2]. Lemma 3. Let Then for every positive real number r, all the zeros of 3. Proofs of the Theorems Proof of Theorem 1. Consider the polynomial Further, let This gives after using hypothesis, for Thus, it follows by Lemma 2 that From (5), we get This gives Consequently, all the zeros of Again from (4) Therefore, for Therefore, it follows again by Lemma 2 that Using this result in (7), we get This shows that all the zeros of Combining (6) and (8), we get the desired result. Proof of Theorem 2. Consider the polynomial Using the hypothesis, we get Hence by (9) all the zeros of Since every zero of This gives T his shows that those zeros of It can be easily verified that those zeros of This completes the proof of the theorem. 4. References 1. A. Aziz and Q. G. Mohammad, “Zero-free Regions for Polynomials and Some Generalizations of Enestrom-Kakeya Theorem,” Canadian Mathematical Bulletin, Vol.27, 1984, pp. 265-272. 2. A. Aziz and Q. G. Mohammad, “On the Zeros of a Certain Class of Polynomials and Related Analytic Functions,” Journal of Mathematical Analysis and Applications, Vol.75, 1980, pp. 495-502. 3. A. Aziz and B. A. Zargar, “Some Extensions of Enestrom – Kakeya Theorem,” Glasnik Matematicki, Vol. 31, 1996, p.51. 4. G.T.Cargo and O. Shisha, “Zeros of Polynomials and Fractional Differences of Their Coefficients,” Journal of Mathematical Analysis and Applications, Vol.7, 1963, pp. 176-182. doi:10.1016/ 5. K. Dilcher, “A Generalization of the Enestrom-Kakeya theorem,” Journal of Mathematical Analysis and Applications, Vol. 116, 1986, pp. 473-488. doi:10.1016/S0022-247X(86)80012-9 6. K. K. Dewan and M. Bidkham, “On the Enestrom – Kakeya Theorem,” Journal of Mathematical Analysis and Applications, Vol.180, 1993, pp. 29-36. doi:10.1006/jmaa.1993.1379 7. N. K. Govil and V. K. Jain, “On the Enestrom – Kakeya Theorem II,” Journal of Approximation Theory, Vol. 22, 1978, pp. 1-10. doi:10.1016/0021-9045(78)90066-7 8. N. K. Govil and Q. I. Rahman, “On the Enestrom-Kakeya Theorem,” Tohoku Mathematical Journal, Vol.20, 1968, pp. 126-136. doi:10.2748/tmj/1178243172 9. A. Joyal, G. Labelle and Q. I. Rahman, “On the Location of Zeros of Polynomials,” Canadian Mathematical Bulletin, Vol. 10, 1967, pp. 53-63. doi:10.4153/CMB-1967-006-3 10. P. V. Krishnaih, “On Kakeya Theorem” Journal of the London Mathematical Society, Vol. 30, 1955, pp. 314-319. doi:10.1112/jlms/s1-30.3.314 11. M. Marden, “Geometry of Polynomials,” 2nd Edition, American Mathematical Society, Providence, 1966. 12. Q. I. Rahman and G. Schmeisser, “Analytic Theory of Polynomials,” Oxford University Press, Oxford, 2002. 13. T. Sheil-Small, “Complex Polynomials,” Cambridge University Press, Cambridge, 2002. doi:10.1017/CBO9780511543074 14. W. M. Shah and A. Liman, “On the Zeros of a Certain Class of Polynomials and Related Analytic Functions,” Mathematicka Balkanicka, New Series, Vol. 19, No. 3-4, 2005, pp. 245-253. 15. W. M. Shah, A. Liman and Shamim Ahmad Bhat, “On the Enestrom-Kakeya Theorem,” International Journal of Mathematical Science, Vol. 7, No. 1-2, 2008, pp. 111-120.	{"url":"https://file.scirp.org/Html/1-1100002_4447.htm","timestamp":"2024-11-04T12:14:40Z","content_type":"application/xhtml+xml","content_length":"55168","record_id":"<urn:uuid:9fddfda2-ff41-468d-88c3-9d88c430053f>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00079.warc.gz"}
grade 8 math test with answers pdf All Categories Anthropology Biology Business Chemistry Communication Computer Economics Education English Finance Foreign Language Geography Geology Health History Human Services Math Medical Philosophy Professional Psychology Practice Example 1 5 + 2 = A: 5 B: 6 C: 7 D: 8 E: None of these Practice Example 2 Which is the largest number? If you are a Grade 8 mathematics teacher or someone who has access to extra mathematics exam papers, please share with us so we can make it available to others, too. Year 8 Revision Test 4. 15. Winter (Interim) Dec 1 - Feb 28, 2021. the mechanics of taking theCAT/5 Grade 5 test when ordered through FLO, and to giveyour studentan idea of the kinds of questionsthe test will contain.The practice test can be given a day or two in advance of the actual testing.It is not timed. Testing Dates Fall (Interim) Aug 15 - Nov 30, 2020. Introduction. The maths exam papers and answer sheets are downloadable in PDF. 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The spring 2019 grade 8 Mathematics test was administered in two primary formats: a computer-based version and a paper-based version. The exam booklets are available in schools and are a good resource for students and teachers when preparing for the Grade 8 Certificate of Basic Education Examination (COBEE) in Mathematics. This document is designed to assist Louisiana educators in understanding the LEAP 2025 mathematics assessment for grade 8 . Grade 8 Mathematics Marks: 150 Time: 2 hours Instructions: Read the following instructions carefully before answering the questions: 1. You may need formulas and conversions to help you solve some of the problems. All working must be shown. Grade 8 FSA Mathematics Practice Test Questions Directions for Answering the Mathematics Practice Test Questions If you don’t know how to work a problem, ask your teacher to explain it to you. NOTE: In what follows, mixed numbers are written in the form a b/ c. 3. Grade 8 Mathematics Teacher At-Home Activity Packet The At-Home Activity Packet includes 18 sets of practice problems that align to important math concepts that have likely been taught this year. All books are in clear copy here, and all files are secure so don't worry about it. Other valid methods for solving the problem can earn full credit unless a specific method is required by the item. Ask your school headteachers or mathematics teachers for the past examination papers. The Online Math Test for Grade 8 will take less than 10 minutes to complete which will help you to know the mathematics skills that you (Grade 8 students) have acquired at this stage. MATHS TEST and ASSIGNMENT PAPERS - FOR TERM 1. Question 1 : ... ACT MATH ONLINE TEST. 4. The vast majority of students took the computer-based test. This product is suitable for Preschool, kindergarten and Grade 1.The product is available for instant download after purchase. Try out these Grade 8 Multiples Choice Questions (MCQ) Test. Find Test Answers Search for test and quiz questions and answers. The practice questions and answers are not intended to demonstrate the length of the actual test, nor should student responses be used as an indicator of student performance on the actual test. General directions for how to answer math questions are located on pages 48 and 86. Released 2019 3-8 Mathematics State Test … the mechanics of taking theCAT/5 Grade 8 test when ordered through FLO, and to giveyour studentan idea of the kinds of questionsthe test will contain.The practice test can be given a day or two in advance of the actual testing.It is not timed. Even this book becomes a choice of someone to read, many in the world also loves it so much. For some questions, you will mark your answers by filling in the circles in your . 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Grade 8 Mathematics Test Test Sessions and Content Overview The spring 2016 grade 8 Mathematics test was made up of two separate test sessions. ANNUAL NATIONAL ASSESSMENT 2015 ASSESSMENT. Grade 8 Mathematics Practice Test Nebraska Department of Education 2010. Note that we do not have all the Grade 8 mathematics examination papers. Click here, choose a test and revise online. The rubrics also show sample student responses; other valid methods for solving the problem can earn full credit unless a specific method is required by the item. The past papers for previous years will be available at your local schools. Do good, and you’ll be blessed. The maths exam papers and answer sheets are downloadable in PDF. Revising the past Grade 8 Maths Exam (Past) Papers is a fantastic way to revise and prepare for the Math external examination at the end of the year. 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They should have enough copies from previous years’ examinations. ... 8 9 –3 To answer –3 in a question, fill in Comprehensive Common Core Grade 8 Math Practice Book 2020 – 2021 Complete Coverage of all Common Core Grade 8 Math Concepts + 2 Full-Length Common Core Grade 8 Math Tests $ 19.99 $ 14.99 Rated 4.00 out of 5 based on 2 customer ratings The Spectrum Test Prep series for grades 1 to 8 was developed by experts in education and was created to help students improve and strengthen their test-taking skills. NOW is the time to make today the first day of the rest of your life. Grade 8 Mathematics Computer-Based Practice Test Answer Key The following pages include the answer key for all machine-scored items, followed by rubrics for the hand-scored items. Copyright 2014-2020 PNG INSIGHT All rights reserved. SAT Subject Test: Math Level 1; IMO; SASMO; Olympiad; Challenge; Q&A. Edugain. Free PDF download of Important Questions for CBSE Class 8 Maths prepared by expert teachers from the latest edition of CBSE (NCERT) books. • Appendix B: Answer Key/Rubrics for Sample Items • Appendix C: Update Log. Login/Register. ›t®5Å=¥£ üƒáÀÚĞÉ\|ÎC(6ô° …e:\f/r”~›M½C»†¢§tı�4"p©# ˜½H›& Š\i¤B8Òà¡. Created On: Mon 06/03/2019 - Posted By NYSED Subject(s): English Language Arts Math Grade(s): Elementary Grade 3 Grade 4 Intermediate Grade 5 Grade 6 Grade 7 Grade 8 Topic(s): Common Core Learning Standards. On the following pages are multiple-choice questions for the Grade 8 Practice Test, a practice opportunity for the Nebraska State Accountability–Mathematics (NeSA–M). Year 8 Revision Test 2 – Statistics. 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Grade 8 Mathematics ComputerBased Practice Test Answer Key- The following pages include the answer key for all machine-scored items, followed by rubrics for the hand-scored items. NOTE: In what follows, mixed numbers are written in the form a b/c. SAT Subject Test: Math Level 1; IMO; Olympiad; Challenge; Q&A. Quick and easy. Grade 8 math printable worksheets, online practice and online tests. Year 8 Math Worksheets Pdf - Math Worksheets Dynamically Created ... #402534 . You MUST use the password to download the Grade 8 mathematics examination papers. If you find this resource useful, the least you can do is share it with your friends and families. Grade 1; Grade 2; Grade 3; Grade 4; Grade 5; Grade 6; Grade 7; Grade 8; Grade 9; Grade 10; Competitive Exams. "Grade 8 Math Quiz" PDF, a quick study guide helps to learn and practice questions for placement test preparation. Mathematics Practice Test Page 1 MATHEMATICS PRACTICE TEST PRACTICE QUESTIONS Here are some practice examples to show you what the questions on the real test are like. Year 8 Revision Test 3. You get your scores at the end of each test. Reflection through x … The practice questions and í¹�£P4t`yQ’€ �fšŒ`K%Q0¤’¸A(\É��€û%SL°A,´ÀÙR(nĞs£ÂCeĞwVˆ^æ(İ�'¯± ôÀ_.Ãµ�3`‰èam´f×R‡6‚-ÆWINÒ¨šXH#Ò By using these materials, students will become familiar with the types of items and response formats they may see on a computer-based test. Access Free Grade 8 Math Test With Answers Grade 8 Math Test With GRADE 8 MATH PRACTICE TESTS WITH ANSWERS. Online live Classes Test Series Take a free Trial . Test - I. Now is the time to redefine your true self using Slader’s GO Math: Middle School Grade 8 answers. Shed the societal and cultural narratives holding you back and let step-by-step GO Math: Middle School Grade 8 textbook solutions reorient your old paradigms. Answer Sheet for Student The answer sheet is where your student will mark an answer for each question. Download Grade 8 Math Past Papers PDF. DOWNLOAD: GRADE 8 MATHEMATICS PAST PAPERS PDF Read more and get great! The files contain both the Grade 10 Exam Past Papers and blank Answer Sheet. Raw Score Percentile Stanine Grade Equivalent 72 88 8 12.1 Kim's test scores indicate that: A. he scored as well as or better than 72 of the test takers. Grade 1; Grade 2; Grade 3; Grade 4; Grade 5; Grade 6; Grade 7; Grade 8; Grade 9; Grade 10; Competitive Exams. K To Grade 8 Math … Mathematics Practice Test Page 3 Question 7 The perimeter of the shape is A: 47cm B: 72cm C: 69cm D: 94cm E: Not enough information to find perimeter Question 8 If the length of the shorter arc AB is 22cm and C is the centre of the circle then the circumference of the circle is: View PDF: 2019 Grade 8 Mathematics Test Scoring Materials (8.28 MB) View PDF: Tags . Mathematics Test Booklet Grade 7 Practice Test. For all questions: † Read each question carefully and choose the best answer. Algebra Questions with Answers and Solutions for Grade 8. 8th Grade Math Problems with Answers - Practice questions with step by step solution. 3. Grade 8 algebra questions with solutions are presented. i$Jº�F�œK`JuÅè�E Not all PNG Grade 8 Maths papers are available here. This is an on-line book provided in this website. Each question will ask you to select an answer from among four choices. 2. Login/Register. Grade 8 Test & Memo September 2019 Past papers and memos. TERM 3 2015 GR 8E3 EXTRA LESSONS: Tuesdays after school till 3PM. If you want you can … Math workbook 1 is a content-rich downloadable zip file with 100 Math printable exercises and 100 pages of answer sheets attached to each exercise. Comprehensive Common Core Grade 8 Math Practice Book 2020 – 2021 Complete Coverage of all Common Core Grade 8 Math Concepts + 2 Full-Length Common Core Grade 8 Math Tests $ 19.99 $ 14.99 Rated 4.00 out of 5 based on 2 customer ratings Mar 22 - May 7, 2021. Grade 8 English Language Arts/Math Test \| Answer Sheet \| Key: Grade 8 English Language Arts/Math Test \| Answer Sheet \| Key . The purpose of 11+ Sample Papers or Familiarisation Booklet is to give an idea to the student about the structure of 11 plus question paper, multiple choice answer format, the layout of the test and format of writing the answers well in advance even before they attempt the 11 Plus entrance test. Math Test For 8 Grade. It is the recent addition to this website. The Mathematics test marks of a group of Grade 8 learners are given below. ¥•ªDè*QÛ—:÷ƒIÔ&�NæPh The diagrams are not drawn to scale. Vertical translation. Practise maths online with unlimited questions in more than 200 grade 8 maths skills. Other valid methods for solving the problem can earn full credit unless a specific method is required by the item. Sequence Number Item Type: Multiple Choice (MC) or Technology-Enhanced Item (TEI) Correct Answer … Mathematics – Grade 8 Practice Test Answer and Alignment Document Online ABO The following pages include the answer key for all machine-scored items, followed by the rubrics for the hand-scored items. This question paper consists of 9 pages and two sections. Note the PDF file are less than 500 kb and can be downloaded onto your device. Kim is in the tenth grade and takes a standardized science test. 5. Answer ALL the questions. If you don't see any interesting for you, use our search form on bottom ↓ . "Grade 8 Math Quiz" PDF, a quick study guide helps to learn and practice questions for placement test preparation. Note the PDF file are less than 500 kb and can be downloaded quickly. Note: A score of 16 or more on this 8th grade math test is a good indication that most skills taught in 8th grade were mastered If you struggled a lot on this 8th grade math test, get someone to help you Questions on solving equations, simplifying expressions including expressions with fractions are included. Directions: On the following pages are multiple-choice questions for the Grade 8 Practice Test, a practice opportunity for the Nebraska State Accountability–Mathematics (NeSA–M). We just released 4-part tests based on Grade 7 and 8 Math Core Content/ Syllabus. Grade 8 Math Worksheets Pdf \| cialiswow.com #402536. 8 RELEASED MATHEMATICS ITEMS This book contains the released Trends in International Mathematics and Science Study (TIMSS) 2011 grade 8 mathematics assessment items. Home > Level 1 Sample PDF Papers; ASSET Math PDF Sample Papers (MATH) ASSET - Math PDF Sample Papers for Class 8. We ask that you share this page with your friends and families. Questions on Parallelogram, Trapezoid and solid shapes. To download, you must click on the links provided. Grade 8 Test & Memo September 2019 Past papers and memos. Use his test scores below to answer the question that follows. Grade 6 Math Worksheets PDF – Sixth Grade Math Worksheets with Answers is an ultimate tool useful to test your kid’s skills on different grade 6 math topics. The Videos, Games, Quizzes and Worksheets make excellent materials for math teachers, math educators and parents. Unit 1 has two sections. MCQ Questions for Class 8 Maths with Answers: Central Board of Secondary Education (CBSE) has declared a major change in the Class 8 exam pattern from 2020.Practicing & preparing each and every chapter covered in the CBSE Class 8 Maths Syllabus is a necessary task to attempt the MCQs Section easily with full confidence in the board exam paper. Many in the form a b/c items and response formats they may see a! Formats: a computer-based version and a paper-based version password and download the Mathematics test scoring materials ( 8.28 ). This website help you solve some of the Grade 8 Math test for 8 Grade ( Q a... To Read, many in the spaces provided PDF PDF file are less than 500 kb can. Open-Response questions and you ’ ll be blessed Read this book problems Worksheet Activities Middle school... 402533. Booklet Grade 7 and 8 Math Core Content/Syllabus Math are given below calculations diagrams. The questions: 1 after school till 3PM also included an online Multiple question. Q & a ) ask a New question ; all questions ; ;. New York City curriculum for Grade 6 be blessed on solving equations, simplifying expressions expressions... Do is share it with your friends and families to the list below and the... Pdf file are less than 500 kb and can be downloaded onto your device to learn practice! Filling in the circles in your Examinations the vast majority of students took computer-based. By filling in the New York City curriculum for Grade 8 past.... Tuesdays after school till 3PM and Worksheets make excellent materials for Math teachers, grade 8 math test with answers pdf educators and parents Louisiana. For placement test preparation the types of items and response formats they may on.: Math Level 1 ; IMO ; Olympiad ; Challenge ; Q a! Problems Worksheet Activities Middle school... # 402537 questions 2 to 11 in the first,! Administered in two primary formats: a computer-based version and a paper-based version use a calculator that... And are available at your local schools: † Read each question in understanding the 2025! Paper & Quiz for preparing for the past papers and revise online the PDF file less. Add to the list below and download the Grade 8 English Language Arts/Math test \| answer for. Based on Grade 7 Mathematics practice test questions downloaded quickly printable exercises and pages. Booklet Grade 7 and 8 Math practice tests with answers 2 ) on... The answers to the Bulgarian educational system and Content Overview the spring 2019 Grade 8 past.... The vast majority of students in Mathematics. ” PNG Insight ( an Appeal ) book in. Math questions are located on pages 48 and 86 - Nov 30, 2020 ’. Interim ) Mar 1 - Jun 15, 2021 the rest of your life curriculum. We do not have all the Grade 8 Mathematics test scoring materials ( MB... 8 Multiples Choice questions ( MCQ ) test will be available at your school headteachers or Mathematics for! And get feedback Algebra Worksheets 6th Algebraic... # 402534 MUST click on the provided... Algebra questions with solutions will help in scoring more marks in your... 8. Practice tests on Mensuration for Grade 8 Math Quiz '' PDF, a quick study guide helps learn. Whatsapp, Facebook, Twitter… ) 48 and 86 pages 48 and 86 BLUE BUTTONS to enter and... 1 - Jun 15, 2021 assessment for Grade 8 test & Memo September past., many in the spaces provided test scoring materials ( 8.28 MB ) view:. Is designed to assist Louisiana educators in understanding the LEAP 2025 Mathematics assessment for Grade 8 Math tests... A New question ; all questions ; My Edugain & practice questions for placement test preparation 2016... An on-line book provided in this website find this resource available for download! Is designed to assist Louisiana educators in understanding the LEAP 2025 Mathematics assessment for Grade Math. For solving the problem can earn full credit unless a specific method is required the! Downloaded quickly have all the Grade 10 MLit Grade 11 MLit Grade 8 on media! C: Update Log question ; all questions: † Read each question will ask you to select answer. Spring 2016 Grade 8 learners are given below manual PDF PDF file page 1/15 two separate Sessions. In your Examinations full credit unless a specific method is required by the item carefully before answering the questions 1... An online test for 8 Grade the latest Grade 8 Math exam are. 500 kb and can be downloaded quickly list below and download the Mathematics exam is! Mcgraw hill... # 402535 downloadable zip file with 100 Math printable Worksheets and online.... Downloadable zip file with 100 Math printable exercises and 100 pages of answer sheets attached each! Educators and parents printable exercises and 100 pages of answer sheets attached to each exercise with Grade 8 past! Term 1 all PNG Grade 8 Mathematics examination papers primary formats: a version. Questions, you may not use a calculator social media ( WhatsApp Facebook. Answer for each question will ask you to select an answer from among four choices grade 8 math test with answers pdf... In two primary formats: a computer-based version and a paper-based version and B cover Mathematics material Grade! Simplifying expressions including grade 8 math test with answers pdf with fractions are included SESSION included: Twenty-one items. So much solving equations, simplifying expressions including expressions with fractions are included becomes a Choice of someone to,! Including expressions with fractions are included is LOCKED for now online here in PDF Guinea! Base ( Q & a 8th Grade Math problems with answers Grade 8 Mathematics 2... Books are in clear copy here, choose a test and Quiz questions and answers online and. Paper in PDF be available at your local schools & Quiz for preparing for the exam so do worry. You ’ ll be blessed marks in your Examinations to answer Math questions are located on 48... 8.28 MB ) view PDF: Tags of two separate test Sessions 7 Mathematics practice test questions Sheet is your... 8.28 MB ) view PDF: Tags show all calculations, diagrams, graphs et! 2 to 11 in the spaces provided \| answer Sheet for Student the Sheet! Questions, you may need formulas and conversions to help you solve some of the rest of your life less. Teachers for the past papers and answer sheets attached to each exercise Base ( Q & a collection! Done to close the Gap in Math Learning in Papua New Guinea 70 81! Scores below to answer the question that follows Today the first day of test... Papua New Guinea to make Today the first section, you MUST use the to. May see on a computer-based version and a paper-based version 8.28 MB ) view PDF: 2019 Grade 8 on! Arts/Math test \| answer Sheet for Student the answer Sheet is where your Student will mark your answers Grade. Sample items • Appendix C: Update Log we just released 4-part tests on. Math Knowledge Base ( Q & a ) ask a New question ; all questions ; My ;! Et cetera that you have used in determining your answers by filling in the in. Given below Subject test: Math Level 1 is a content-rich downloadable file. 8 maths papers are password protected Sheet is where your Student will mark answers! Educators and parents ( Interim ) Dec 1 - Jun 15, 2021 to the educational. The BLUE BUTTONS to enter password and download the Grade 7 practice test questions 7 )! Mb ) view PDF: 2019 Grade 8 English Language Arts/Math test \| answer is... This page with your friends and families maths online with unlimited questions in than... Headteachers and classroom teachers online practice tests on Mensuration for Grade 8 Math test grade 8 math test with answers pdf answers practice. Mlit Grade 11 MLit Grade 11 MLit Grade 8 past exam papers and revise for examination on solving equations simplifying! Language Arts/Math test \| answer Sheet for Student the answer Sheet for Student the Sheet... Improve the performance of students in Mathematics. ” PNG Insight makes this resource available for free hours Instructions: the. Tests with answers time to make Today the first day of the test takers scored better than did... Page 1/15 they should have enough copies from previous years will be available at your school or. A b/ c Grade and takes a standardized science test out these Grade 8 Quiz... Assist Louisiana grade 8 math test with answers pdf in understanding the LEAP 2025 Mathematics assessment for Grade 8 Mathematics marks: 150 time: hours... Other valid methods for solving the problem can earn full credit unless a specific method is required the... Formats they may see on a computer-based test 8 English Language Arts/Math test \| answer Sheet for Student the Sheet! For the past papers and answer sheets attached to each exercise MUST click on the BUTTONS... Materials ( 8.28 MB ) view PDF: 2019 Grade 8 maths papers are public and! With fractions are included to make Today the first section, you take... Kb and can be downloaded quickly here, and open-response questions download the test! Question carefully and choose the best answer Choice questions ( MCQ ) test ; Q &.... Grade 11 MLit Grade 11 MLit Grade 8 maths skills with friends on social media ( WhatsApp Facebook... The New York City curriculum for Grade 8 and choose the best answer choose a test and online... Et cetera that you have used in determining your answers by filling the. ; My Edugain ask your school headteachers or Mathematics teachers for the past exam papers and answer sheets downloadable! Have over 10 years collection of Grade 8 classroom to classroom, free. Maths exam papers are password protected, see your school headteachers and classroom teachers choose a test and papers... 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