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https://www.researcher-app.com/paper/130063
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math
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Modified Gravity (MOG), the speed of gravitational radiation and the event GW170817/GRB170817A.
Modified gravity (MOG) is a covariant, relativistic, alternative gravitational theory whose field equations are derived from an action that supplements the spacetime metric tensor with vector and scalar fields. Both gravitational (spin 2) and electromagnetic waves travel on null geodesics of the theory's one metric. Despite a recent claim to the contrary, MOG satisfies the weak equivalence principle and is consistent with observations of the neutron star merger and gamma ray burster event GW170817/GRB170817A.
Publisher URL: http://arxiv.org/abs/1710.11177
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CC-MAIN-2022-40
| 657 | 3 |
http://www.wyzant.com/Tutors/CA/Wilton/8119082/?g=3FI
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math
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I have tutored students in math and chemistry over the last two years. I have tutored in math starting in grades 5 and up through algebra 1 and 2 and geometry. I have tutored in chemistry and have an academic and industrial background in chemistry. I believe in learning the logic of the concepts to help students remember the formulas. In this way fewer unrelated facts need to be known and the concepts and formulas are linked together. I use illustrations and example problems with written stepwise procedures. I will go over topics covered in the student's class to add insight and memory aides through charts, graphs, and illustrations. I let the student's classroom notes and textbook be my guide on the subject matter to be taught so that time is not wasted on non relevant material.
back to top
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CC-MAIN-2014-23
| 802 | 2 |
http://www.calebpotter.blogspot.com/2008/02/wow.html
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math
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I am humbled by all of your comments-I was never seriously wanting for anyone to feel guilty about not writing in.
I realize, rather than telling, I should have asked if we should discontinue this blog--after all it belongs to you now just as well.
I am going through a really rough spell...all those tears I could not release from July 4th till now are draining through. I literally cannot stop crying. Kai, that wonderful, sweet Kai will blog for awhile to let you know of Caleb's continued progress, and I am assuming Jenny will as well. Max, as you remember, is the strong and silent type.
I will be back as soon as I can hold it together. I need this!
Thank you Beth for the reminder that things can always get worse, as they did for your family. I once remember saying that as well- and that is how I have come to know that I need to just breath deeply again and trust...I have lost perspective.
And to all the rest of you who wrote in..you are @%&*****+++%%%$$$$$$$###### ing amazing.
And to the rest of you who chose not to write in-- the same applies!
Thanky you all for being there. mumsie
PS. to Blackbird- I hope we have that talk.
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s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386163848048/warc/CC-MAIN-20131204133048-00063-ip-10-33-133-15.ec2.internal.warc.gz
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CC-MAIN-2013-48
| 1,143 | 9 |
https://www.financialpipeline.com/compound-interest/
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math
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The key to understanding “compound interest” is to distinguish it from “simple interest.”
Simple interest is the interest that is derived from only the investment in a principal amount. Say we take $100 and put it in an investment for one year at a simple interest rate of 5%. We multiply the interest rate of 5% times the principal amount of $100 to derive an interest payment of $5.
This seems fairly obvious and straightforward. But let’s think about what happens in the second year if the interest is not paid out.
In a “compound” investment, the interest is not paid out to the holder; it is built up within the investment. Consider our $100 above. If the interest was paid out after the first year, we would get our $5 in interest and have our original $100 still invested. If the interest isn’t paid out, however, we would have our original principal of $100 plus $5 in interest for a total of $105 at the end of the first year. This entire amount would bear the original interest rate of 5% for the second year. Not only would we get 5% or $5 in interest on the original principal of $100, we would also get $.25 which is 5% on the $5 interest from the first year. Our total interest received would be $5.25.
This would continue into the third year. At the end of the second year, since we have not paid out any interest, we would start with our $105 from the end of the first year plus the $5.25 we earned in the second year. This would give us a third year opening balance of $110.25, which would bear interest of 5% for the third year. Our 5% interest on $110.25 equals $5.5125. Assuming we still don’t pay out any interest, we will now have a total investment of $115.7625.
Our example shows us how we generate compound interest. We would expect that three years of simple interest at 5% should equal $15, which would combine with our principal amount of $100 to give us $115. The reason we now have an extra $0.7625 is the “compound interest” we have earned on our interest reinvested.
Mathematically, we can express compound interest as a series of multiplications. We take the percent interest expressed in hundredths and add it to one. We then multiply this together for the number of time periods and then multiply the result times the initial investment or principal amount. Using our example above, if we take 1.05 multiplied together three times (1.05×1.05×1.05) we get an answer of 1.157625. If we multiply this times our principal amount of $100 we get $115.7625, which was our total invested amount at the end of three years. The mathematical notation for this is (1.05) ³ or “raised to the power of 3″. Most calculators and common spreadsheets have a “power function” which is usually shown as “x to the nth power”, with n being the number of periods.
We now have the ability to solve for the end amount or “future value” if we have a starting principal amount or “present value,” an interest rate and the number of time periods. Inversely, we can also find the present value or how much a future sum will be in the present if we have the future value, the number of time periods and the interest rate. These concepts all revolve around the central concept of compound interest and are known as “the time value of money.”
Given a present value, or starting value, and an ending value or “future value” we can use the number of time periods to establish what the compound interest rate would be over a time period. It is in this fashion that we derive “compound investment returns” that we see featured prominently in mutual fund advertising. Mathematically, this process is the reverse of compounding into the future. We find the nth root that solves the equation that compounds the present value into the future value. Using our example, we know that we started with a present value of $100 and ended with a future value of $115.7625. We divide the future value of $115.7625 by the present value of $100 to get 1.157625, which we then take the third root of which is equal to 1.05. Subtracting the 1 that we added in, we get .05 or 5% for the compound return that makes $100 grow exactly to $115.7625.
If you’re confused by this discussion, don’t be worried. Most investment professionals need to draw a time line to figure out the more complex present value and compounding problems. Basic finance and accounting textbooks cover this issue very well. To have a firm grasp on investments, compound interest should be a subject that you spend some time on.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100583.31/warc/CC-MAIN-20231206063543-20231206093543-00307.warc.gz
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CC-MAIN-2023-50
| 4,541 | 10 |
https://bora.uib.no/bora-xmlui/handle/1956/11725
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math
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Investigating Streamless Sets
Conference object, Peer reviewed
MetadataVis full innførsel
In this paper we look at streamless sets, recently investigated by Coquand and Spiwack. A set is streamless if every stream over that set contain a duplicate. It is an open question in constructive mathematics whether the Cartesian product of two streamless sets is streamless. We look at some settings in which the Cartesian product of two streamless sets is indeed streamless; in particular, we show that this holds in Martin-Loef intentional type theory when at least one of the sets have decidable equality. We go on to show that the addition of functional extensionality give streamless sets decidable equality, and then investigate these results in a few other constructive systems.
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s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711218.21/warc/CC-MAIN-20221207185519-20221207215519-00322.warc.gz
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CC-MAIN-2022-49
| 779 | 4 |
https://pothi.com/pothi/book/isaac-todhunter-history-probability-1865-vol-1
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math
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The theory of probability and statistics emerged much later in the history of mathematics, as compared to trigonometry or algebra. Its roots lie - like a lot of science, in astronomy. However, in the modern world it has been applied to almost every area of science, economics and society. Isaac Todhunter's book is the definitive work on the subject, but some familiarity with high-school mathematics is required.
This book has been printed in two parts, so it should be purchased with Vol. 2
Binding: Paperback (Perfect Binding)
Availability: In Stock (Print on Demand)
Emerging Applications of Bayesian Statistics and Stochastic Modelling by Pramendra Singh Pundir
Poorani Dairy by Syed Sultan Ahmad Rizvi
The Science Of Mechanics ( Volume II ) by Ernst Mach
IJMAA(V3N3-D) by JS Publication
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s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039744561.78/warc/CC-MAIN-20181118180446-20181118201654-00000.warc.gz
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CC-MAIN-2018-47
| 792 | 8 |
https://orderofthepearl.org/australian-light-vrs/article.php?e0a02b=lesson-plan-system-of-linear-equations
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math
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lesson plan system of linear equations
Sciences, Culinary Arts and Personal In this algebra learning exercise, students solve linear equations and graph their answer. Graphing linear equations | Lesson Our mission is to provide a free, world-class education to anyone, anywhere. You can test out of the first two years of college and save 4. OK, so maybe graphing linear equations can be fun. Weekly Syllabus. Complete Lesson Plan # 1 move around. In this linear equations worksheet, 9th graders solve and complete 15 different problems that include writing each in slope-intercept form. In this linear systems worksheet, 9th graders solve and complete 24 different problems that include solving various linear systems. If at first you cannot graph, substitute. Solve simple cases by inspection. In this linear equation worksheet, students solve 18 problems regarding slope-intercept, linear equations, parallel lines and perpendicular lines. Khan Academy is a 501(c)(3) nonprofit organization. Well, if it has two variables, the solutions can be endless! A video lesson demonstrates how to use a linear equation to determine the depth when given a specific pressure. Hopefully you will be able to see improvement from the previous lesson plan on the Pythagorean Theorem. In this linear equation instructional activity, students solve systems of linear equations by elimination. Solve systems of equations using any strategy. In this linear equation forms lesson, students graph relevant information using slope intercepts. They graph equations using... With a graph and two linear equations, Sal explains how to graph systems of equations. By completing a set of challenge questions, they... Use the distributive property to solve equations. Armed with information about the rates different pizza places charge for their pizzas and deliveries, learners write equations to represent each restaurant. Learners then watch as the instructor shows how to use the equation to analyze the situation. This seven-page worksheet contains detailed instructions, examples, and 5 problems. Developing an intuition for the kinds and descriptions of solutions is key for success in those later courses. Students find the slope of a line given a table, graph, two points, and the intercepts. So far, all of the systems of equations we have encountered in this lesson have involved linear equations. They determine the value of each variable. Did you know… We have over 220 college courses that prepare you to earn Linear equations aren't just for graphing! It goes through graphing and solving algebraically, and then... Small but mighty, a collection of three resources features presentation slides that prepare high schoolers for the Smarter Balanced assessment. In this algebra lesson plan, students perform transformation on linear equations. Some people think that since linear equations are the simplest equations that students encounter, they are the easiest to learn. 3. Given a linear equation in standard form, scholars write the equation in slope-intercept form, identify the y-intercept, and use the slope to graph the line. They graph their lines using the slope and y-intercept. First, they graph each equation using the x- and y-intercepts of each. Learn how easy it is to use on any linear system in two variables. Integrated Algebra/Math A Regents Questions: Solving One Variable Linear Equations, Algebra I - Worksheet G12: Graphing Lines, Solving Systems of Linear Equations with Row Reductions to Echelon Form on Augmented matrices, Proportional Relationships, Lines, and Linear Equations. Variables appear in the denominator and/or in the neumerator. Using four given linear equations, scholars create a graph of a star. In this activity, pupils review The Chicken and Pigs problem and compare their solutions with the graphing... Ah, the dreaded systems of equations word problem. 2) Creating a table. The three page worksheet contains ten problems. Other Evidence: • To be decided by the teacher. In this linear equations and polynomials activity, students add, subtract, and multiply polynomials. The equations are proportions where the numerators and denominators may have more than one term. This lesson gives students a hands on approach to learning about systems of equations with something that they can all relate to...money! Students will interpret a variety of systems of linear equations including several real-world word problems. Write a system of equations to model a situation and interpret the solution in context (MP.4). The quick video uses a linear equation to determine the size of a bookshelf given a diagram of the shelves and the amount of wood to use. You can solve a system of linear equations by graphing the equations on the same coordinate plane. In this linear equations instructional activity, 9th graders solve and complete 7 different problems that include a number of linear equations. An instructional video... Water pressure decreases at a constant rate with a change in depth. 1. Linear equations are the focus of activities that ask learners to first complete a task that involves interpreting algebraic expressions and solving linear equations. TI-nspire CAS calculator required. Your graphing calculator can do lots of amazing things, even solve equations graphically. Students can also classify the pairs of lines in the system. A lesson Plan intended for the demonstration in applying a position Teacher I Solving Systems of Linear Equations in Two Variables by Substitution MethodLesson Plan For Demonstration- Junior High Lesson Plan For Demonstration-Junior High ELTON JOHN BALIGNOT EMBODO Applicant 2. In this linear equation learning exercise, students write linear equations in point-slope forms. Starting with the basics, students discover that systems of equations are a useful skill. Young scholars write and solve linear equations in one variable based on descriptions of the operations that are applied to the unknown variable in an algebra machine. They identify the input and output of each relation. First, they distribute to remove the parentheses and combine like terms. The fifth part in a unit of nine works with the different equivalent forms of linear equations. 8.EE.7.B Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equation. Incorporate geometry into the solving linear equations instructional activity. In this matrices worksheet, students write matrix notation to represent linear systems. Young scholars graph linear equations. In this systems of equations worksheet, students use the substitution method to solve systems of linear equations. This activity includes a set of linear equation cards to be cut and place face up on a table. Middle and high schoolers solve systems of equations relating to money and explore substitution and elimination strategies. Students express their answers using an augmented matrix in reduced... For this algebra worksheet, students solve one step linear equations by division. Students create six graphs of given equations. An interactive has them adjust lines on a coordinate plane to see changes in each form of... Sal teaches how to use graphing as a tool, not a hindrance, in this helpful video. 1. Ax+by=c. For this graphing lines worksheet, students solve linear equations and then graph the points on the line. For this linear equations worksheet, students solve 8 short answer problems. In this systems of linear equations worksheet, students use both the elimination and the substitution methods to solve 20 problems involving systems of two linear equations. The collection includes lessons and practice for your students... Never miss a centimeter with a set of measurement lessons and activities, focusing on area and perimeter. Illuminate the world of mathematics for your class - specifically, the process of linear equations. When this is done, one of three cases will arise: Case 1: Two Intersecting Lines . His lecture would make be a good resource for your algebra class, particularly as the focus shifts to graphing problems and linear systems. Anyone can earn credit-by-exam regardless of age or education level. Educators earn digital badges that certify knowledge, skill, and experience. Using given data, your class checks their answers with graphing calculators. What's Point-Slope Form of a Linear Equation? Pupils follow rules to combine circles and squares to add the two equations together, resulting in one... Cover it up to reveal the values of variables. In this linear equations worksheet, students solve linear equations using a given variable. Introduction to Systems of Linear Equations Lesson This is the product of more experience with lesson planning, teaching, and the realization that the two heavily influence each other. This lesson will provide students with an introduction to solving equations and inequalities numerically (using a table), graphically, and algebraically. Because the relationship between the... Ninth graders review the information that they have already learned It then shows how to transfer those values to a graph resulting... Study linear equations with this algebra lesson. In this linear equation worksheet, students write the linear equation of a line that passes through a given point. This three-page activity contains 50 problems. The workbook provides a class activity and homework for... What does it mean to solve an equation? Examples, worksheets, videos and solutions to help Grade 8 students learn the elimination method for solving a system of linear equations. In this lesson we will see how to solve a system consisting of 3 linear equations and 3 variables. Digital Citizenship | Curriculum, Lessons and Lesson Plans, Plans for a Common Core Standards Open Resource, Biology Lesson Plans: Physiology, Mitosis, Metric System Video Lessons, Insurance Underwriter: Employment Info & Career Requirements, Band Management: Employment Info & Career Requirements, Best Online Bachelor's Degrees in Public Administration, How to Be an Emergency Medical Technologist, Working Scholars® Bringing Tuition-Free College to the Community, Definition and examples; graphing systems of equations, Converting stories into equations for solving; substitution method, Solving linear system problems in three variables that have an answer, Linear system in three variables with no definite answer, Solving linear system problems in three variables that have no or infinite solutions, Real Numbers: Types & Properties Lesson Plans, Working with Linear Equations Lesson Plans, Working with Complex Numbers Lesson Plans, Working with Quadratic Functions Lesson Plans, Higher-Degree Polynomial Functions Lesson Plans, Rational Functions & Difference Quotients Lesson Plans, Rational Expressions & Function Graphs Lesson Plans, Exponential & Logarithmic Functions Lesson Plans, Using Trigonometric Functions Lesson Plans, Solving Trigonometric Equations Lesson Plans, Analytic Geometry & Conic Sections Lesson Plans, Vectors, Matrices & Determinants Lesson Plans, Polar Coordinates & Parameterizations Lesson Plans, Circular Arcs, Circles & Angles Lesson Plans. Using four different equations, the resource reinforces the steps to solving a two-step linear equation. A video lesson demonstrates how to write a linear equation to represent a specific situation. Using systems of simultaneous equations, they graph each scenario to determine the best value. Students practice more formal strategies for representing and solving systems of equations. This two-page worksheet contains approximately 36 problems. The next 10 problems have a given equation, but your... Super simple equations are yours for the solving. What's Slope-Intercept Form of a Linear Equation? If the lines intersect, the solution is that intersection point. Probably one of the best ways to connect algebra to the real-world, here is a video that goes through the steps on how to detect key words and create two equations that model the scenario. Then,... Middle schoolers solve 18 equations with a variable and an extra number operation. Then, students determine the equation that... Ninth graders develop an understanding of and the applications for linear equations and their graphical displays. In this linear equations learning exercise, students interpret graphs. How much wood would ...? In this algebra lesson, students solve linear equations by addition, subtraction, division and multiplication. Students solve linear equations using the slope and y-intercept. A comprehensive online edtech PD solution for schools and districts. Here are a variety of lessons and worksheets to hit your content. Linear systems are the most common type of system that appear in real life problems. In this linear equations worksheet, learners read and interpret graphed linear equations. Explore methods for solving linear systems with your classes and introduce learners to using matrices as a viable method. Students write linear equations from exploring graphs and tables. Save time lesson planning by exploring our library of educator reviews to over 550,000 open educational resources (OER). Pupils determine the orientation of the line and, through a... Show your class that linear equations produce graphs of lines. This one-page worksheet contains eight problems, with answers. They then take part in a card activity matching equations, situations,... Scholars first complete an assessment task on writing linear equations to model and solve a problem on a running race. Common difference is to arithmetic sequences as what is to linear equations? Explore linear equations with this video about lines with equal signs! Additionally, students rewrite 4 standard form equations in... Let the learners take the driving wheel! They use systems of linear equations to solve the problem. What does the rate of change mean when graphing a line? How Do You Graph a Linear Equation by Making a Table? Is it a linear equation? Lead a discussion on how to manipulate the sum of a geometric series to figure out a formula to find the sum at any step. In this Algebra II lesson, 11th graders solve s system of linear equations with row reduction to Echelon Form on an augmented matrix. Students solve one variable equations by isolating the variables. There are 10 questions. The sixth lesson in a 33-part series has scholars solve equations that need to be transformed into simpler equations first. One way to solve a system of linear equations is by graphing each linear equation on the same -plane. Direct instruction. For this algebra worksheet, students solve linear equations through graphing and the use of formulas. In this video, Sal teaches about linear equations in standard form after two videos on slope-intercept and slope-point forms. Solve simultaneous linear equations, otherwise known as systems of linear equations. He uses a table to pick points, completes the equations, and plots the lines on the graph. To find x, you have to get it by itself, correct? Middle schoolers problem solve and calculate the answers to eighteen linear equations involving a variety of variables. Test your knowledge of the entire course with a 50 question practice final exam. Here is a complete chapter on graphing linear equations and functions. (Answer: slope) Pupils learn how arithmetic sequences can be considered as linear patterns. A lesson has pupils determine solutions for two-variable equations using tables. This five-page worksheet cotnains 12 problems. Get in line to explore linear equations. Answers are not provided. From practice sheets to collaborative games, your high schoolers will love geometry! There are 34 questions ranging from addition and subtraction of linear equations to graphing and substitution. In this lesson, you'll learn how to take a word problem and convert it into the system of equations that will allow you to find the answer using either substitution or elimination. This one-page worksheet contains 15 problems, with answers. Students then write equations for 4 problems given the slope and y-intercepts. This example does not fit into the standard form and is not a linear equation. In this linear equations worksheet, students solve 20 multiple choice and short answer questions.
Campbell Run Apartments, Colorado Tree Identification, Utopia Definition Bible, Laughing Owl Letterpress, The Whole Is Greater Than The Part Example, Residence Inn Watertown, Ma, Highest Grossing Movies Bollywood, Best Wood For Firewood Australia,
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CC-MAIN-2021-17
| 16,570 | 3 |
https://wifusion.org/and-pdf/1754-cosine-and-sine-law-pdf-968-695.php
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math
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File Name: cosine and sine law .zip
All of our Grade 9 Through Grade 12 Math worksheets, lessons, homework, and quizzes.
- Oblique Triangles
- The Laws of Sines and Cosines
- Section 4: Sine And Cosine Rule
- Service Unavailable in EU region
In trigonometry, the law of cosines also known as the cosine formula, cosine rule, or al-Kashi's theorem relates the lengths of the sides of a triangle to the cosine of one of its angles. Titanic 1 and Titanic 2 leave port at 5pm. Test your ability to use the law of cosine formula to solve for unknown sides and angles of triangles in this quiz and worksheet combo.
Juan and Romella are standing at the seashore 10 miles apart. Interactive Demonstration of the Law of Cosines Formula. The interactive demonstration below illustrates the Law of cosines formula in action. Drag around the points in the triangle to observe who the formula works. Try clicking the "Right Triangle" checkbox to explore how this formula relates to the pythagorean theorem. Applet on its own. Law of Cosines.
These laws are used when you don't have a right triangle — they work in any triangle. You determine which law to use based on what information you have. In general, the side a lies opposite angle A, the side b is opposite angle B, and side c is opposite angle C. Worksheets are Extra practice, Find each measurement round your answers Law of Cosines. Get help with your Law of cosines homework. Access the answers to hundreds of Law of cosines questions that are explained in a way that's easy for you to understand.
The Laws of Sines and Cosines
The relationship explains the plural "s" in Law of Sines : there are 3 sines after all. Another important relationship between the side lengths and the angles of a triangle is expressed by the Law of Cosines. Why do we use the plural "s" in the Law of Cosines? The expression itself involves a single cosine , but by rotation or, as A. Einstein might have said, by symmetry similar formulas are valid for other angles:. In fact, I do not know the exact sources of the existing nomenclature.
Section 4: Sine And Cosine Rule
You will need to know at least one pair of a side with its opposite angle to use the Sine Rule. Practice Questions Work out the answer to each question then click on the button marked to see if you are correct. Finding Sides If you need to find the length of a side, you need to know the other two sides and the opposite angle. Sides b and c are the other two sides, and angle A is the angle opposite side a.
Measurement and Geometry : Module 24 Year : PDF Version of module.
Service Unavailable in EU region
Round your answers to the nearest hundredth. That role is played by atan2 in the haversine formula. May 8 from pm.
In trigonometry, the law of sines also known as the sine law, sine formula, or sine rule is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angles. According to the law,. Sometimes the law is stated using the reciprocal of this equation:. The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are knowna technique known as triangulation. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the formula gives two possible values for the enclosed angle, leading to an ambiguous case. The law of sines is one of two trigonometric equations commonly applied to find lengths and angles in a general triangle, the other being the law of cosines.
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CC-MAIN-2021-25
| 3,503 | 17 |
http://www.dailyranger.com/ranger_headlines.php?page=294&category=
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math
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Apr 5, 2013 - By Paul V.M. Flesher: The dilemma makes it difficult for religions to look to the future, because they always look to the past, and stay in line with their previous ... [more]
Apr 5, 2013 - Staff: Linda J. Shields Gotcher, of Lander, died Tuesday, March 5, 2013, at Kindred Hospital in Denver surrounded by her loving family. She was 64 years old. ... [more]
Apr 5, 2013 - Staff: The Rev. John Clark Herrington died Wednesday, April 3, 2013, at the Wyoming Medical Center in Casper after a period of declining health. He was 78 years old. ... [more]
Apr 5, 2013 - Staff: Marjorie (Marge) Ricketts Reynick died Saturday, March 30, 2013, at Shepherd of the Valley in Casper after a lengthy battle with heart disease. She was 83 ... [more]
Apr 5, 2013 - Staff: Deaths
Apr 5, 2013 - By Eric Blom, Staff Writer: Applause and whistles filled the War Bonnet room in the Best Western Inn at Lander when a speaker introduced Del McOmie as the guest of honor. The Fremont ... [more]
Apr 5, 2013 - Staff: Holdaway earns excellence award
Apr 5, 2013 - Staff: Riverton Middle School recently recognized the students who were named to the third quarter honor roll for the 2012-13 school year.
Apr 5, 2013 - By David Lauter, Tribune Washington Bureau: WASHINGTON -- A majority of Americans support legalizing marijuana, a new poll shows, with the change driven largely by a huge shift in how the baby boom ... [more]
Apr 5, 2013 - Lew Diehl, Riverton: Editor:
Apr 5, 2013 - By Steven R. Peck: April is National Poetry Month, so consider a favorite verse -- or find one
Apr 5, 2013 - The Associated Press: Mountain pine beetle declines in state
Apr 5, 2013 - By Mead Gruver, The Associated Press: CHEYENNE -- A legal settlement between ranchers and the U.S. Bureau of Land Management would reduce wild horse numbers by about half on more than 4,300 square ... [more]
Apr 5, 2013 - From staff reports: Steven Amos, 41, of Riverton, was arrested for indecent dress and exposure at about noon Thursday after reportedly leaving Riverton Memorial Hospital wearing ... [more]
Apr 5, 2013 - By Eric Blom, Staff Writer: Snowpack in the Wind and Sweetwater river basins remains below average, but Riverton has seen more precipitation this year compared with last year, and ... [more]
Apr 5, 2013 - By Alejandra Silva, Staff Writer: The Kuyasa Kids children's choir from the Kayamandi township in South Africa visited Wyoming this week with a message of hope and overcoming sorrow and loss ... [more]
Apr 5, 2013 - By Katie Roenigk, Staff Writer: The station's general manager says viewers won't notice a change.
Apr 4, 2013 - Staff: Central Wyoming College student Shanelle Anderson (back to camera) watched students, from left, Taylor Stagner, Sara Ludgate, Katie Wagner and Sarah Widdoss ... [more]
Apr 4, 2013 - By Bruce Tippets, Sports Editor: Mike Broadhead is moving ahead with his idea of having an American Legion baseball team for players from the Wind River Indian Reservation.
Apr 4, 2013 - By Bruce Tippets, Sports Editor: Events begin at 9 a.m. and should be finished by 5 p.m. at Wolverine Field.
Apr 4, 2013 - Staff: Deaths
Apr 4, 2013 - Staff: Jay Patrick Corbett, of Gunnison, Colo., died Saturday, March 23, 2013, as a result of accidental drowning while he was enjoying a vacation in Hawaii. He was ... [more]
Apr 4, 2013 - Staff: Patty "Pat" M. Petek died Tuesday, April 2, 2013, after a battle with leukemia. She was 79 years old.
Apr 4, 2013 - By The Philadelphia Inquirer: By agreeing to a toothless invasion-of-privacy settlement with Google, federal and state authorities blew a chance to take a bolder stand against the Internet ... [more]
Apr 4, 2013 - Daniel Talley, Riverton: Editor:
Apr 4, 2013 - By Betty Starks Case: Is it spring yet?
Apr 4, 2013 - By Steven R. Peck: Fremont County gets slapped again from another time zone
Apr 4, 2013 - The Associated Press: Supreme Court accepts Hill lawsuit
Apr 4, 2013 - By Bob Moen, The Associated Press: CHEYENNE -- The percentage of University of Wyoming students defaulting on federal loans is well below the national average, while five of the state's seven ... [more]
Apr 4, 2013 - By Eric Blom, Staff Writer: State statute requires counties to pay the funeral costs of people who cannot provide for themselves.
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| 4,367 | 30 |
https://optiwave.com/forums/reply/11454/
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math
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One way to look at the variables that are created in the workspace, is to first enable the parameter “Load Matlab” in the Matlab component. Then after the simulation you can write “workspace” into the Matlab Command Window which will appear.
It is possible to choose and then combine desired modes in a Matlab component by simply deleting parts of the input data structure. However, using OptiSystem’s built-in Mode Selector component may be even simpler. I have attached an example of using the Mode Selector component to capture and combine the 3, 5, 9 and 12 modes of a multimode fiber.
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| 599 | 2 |
https://fastassignmentshelp.com/1-show-how-education-can-signal-the-worker-s-innate-ability-in-the-labor-market-what-2652644/
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math
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1. Show how education can signal the worker's innate ability in the labor market. What is a pooled equilibrium? What is a perfectly separating signaling equilibrium?
2. How can we differentiate between the hypothesis that education increases productivity and the hypothesis that education is a signal for the worker's innate ability?
3. Discuss the difference between general training and specific training. Who pays for and collects the returns from each type of training?
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| 473 | 3 |
http://klcourseworkaxxd.jayfindlingjfinnindustries.us/analysis-of-graph-theory.html
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math
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Patent big data analysis using graph theory sunghae jun department of statistics, cheongju university, chungbuk 360-764, korea [email protected] abstract. Graphs provide a structural model that makes it possible to analyze and graphdata, exampledata — curated collection of theoretical and empirical graphs. We have developed a freeware matlab-based software (braph–brain analysis using graph theory) for connectivity analysis of brain.
To analyze a risk based on major threat scenarios of application of graph theory algorithms and parameters to analyze exemplary transportation system are. The theory of continuous graphs also known as graphons heavily relies on functional analysis of the lebesgue space l 1 as one has to deal. One of the main mathematical tools to perform this type of analysis is graph theory, and indeed we saw in these last few years an impressive progress also in this.
Cambridge core - discrete mathematics information theory and coding - hybrid graph theory and network analysis - by ladislav novak. Elective in robotics 2014/2015 analysis and control of multi-robot systems elements of graph theory dr paolo robuffo giordano cnrs, irisa/inria rennes. Am's initial success and later inability to generate new results were analyzed mainly by furthermore, operators specialized for the graph theory domain were. The explicit linking of graph theory and network analysis began only in 1953 and has been rediscovered many times since analysts have taken from graph.
Graphtea your buddy to teach, learn and research on graph theory features download tutorials topological-indices publications support video preview. Social network analysis lecture 2-introduction graph theory donglei du (ddu @unbca) faculty of business administration, university of new brunswick, nb. Theory what is a network • network = graph • informally a graph is a set of nodes graph theory - history wasserman & faust, social network analysis .
Graph theory analysis (gta) is a method that originated in mathematics and sociology and has since been applied in numerous different fields in neuroscience. Network topologies may be treated as a directed graph specific methods and definitions for analyzing network topology using graph theory are. Graph theory analysis of protein-protein interaction graphs through clustering method abstract: graph mining is an active progressive field of research in recent. Jordán, c gómez, ja conejero, ja an analysis of the influence of graph theory when preparing for programming contests mathematics.
Graph theory, branch of mathematics concerned with networks of points half of the 19th century became known as analysis situs—the “analysis of position. However, without adding in a level of social graph theory analysis, we found ourselves blindly reaching out to influencers with very little rol. Protein-protein interaction data available has made graph theory function using this analysis leads to building predictive models for hypoth. Graph theory definition is - a branch of mathematics concerned with the study of graphs.
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CC-MAIN-2018-43
| 3,077 | 6 |
https://slideplayer.com/slide/5242097/
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math
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Continuous Random Variables and Probability Distributions
Published byModified over 6 years ago
Presentation on theme: "Continuous Random Variables and Probability Distributions"— Presentation transcript:
1 Continuous Random Variables and Probability Distributions Chapter 6Continuous Random Variables and Probability Distributions
2 Continuous Random Variables A random variable is continuous if it can take any value in an interval.
3 Cumulative Distribution Function The cumulative distribution function, F(x), for a continuous random variable X expresses the probability that X does not exceed the value of x, as a function of x
4 Cumulative Distribution Function F(x)11Cumulative Distribution Function for a Random variable Over 0 to 1
5 Cumulative Distribution Function Let X be a continuous random variable with a cumulative distribution function F(x), and let a and b be two possible values of X, with a < b. The probability that X lies between a and b is
6 Probability Density Function Let X be a continuous random variable, and let x be any number lying in the range of values this random variable can take. The probability density function, f(x), of the random variable is a function with the following properties:f(x) > 0 for all values of xThe area under the probability density function f(x) over all values of the random variable X is equal to 1.0Suppose this density function is graphed. Let a and b be two possible values of the random variable X, with a<b. Then the probability that X lies between a and b is the area under the density function between these points.The cumulative density function F(x0) is the area under the probability density function f(x) up to x0where xm is the minimum value of the random variable x.
7 Shaded Area is the Probability That X is Between a and b abx
8 Probability Density Function for a Uniform 0 to 1 Random Variable f(x)11x
9 Areas Under Continuous Probability Density Functions Let X be a continuous random variable with the probability density function f(x) and cumulative distribution F(x). Then the following properties hold:The total area under the curve f(x) = 1.The area under the curve f(x) to the left of x0 is F(x0), where x0 is any value that the random variable can take.
10 Properties of the Probability Density Function f(x)CommentsTotal area under the uniform probability density function is 1.1x01x
11 Properties of the Probability Density Function CommentsArea under the uniform probability density function to the left of x0 is F(x0), which is equal to x0 for this uniform distribution because f(x)=1.f(x)1x01x
12 Rationale for Expectations of Continuous Random Variables Suppose that a random experiment leads to an outcome that can be represented by a continuous random variable. If N independent replications of this experiment are carried out, then the expected value of the random variable is the average of the values taken, as the number of replications becomes infinitely large. The expected value of a random variable is denoted by E(X).
13 Rationale for Expectations of Continuous Random Variables Similarly, if g(x) is any function of the random variable, X, then the expected value of this function is the average value taken by the function over repeated independent trials, as the number of trials becomes infinitely large. This expectation is denoted E[g(X)]. By using calculus we can define expected values for continuous random variables similarly to that used for discrete random variables.
14 Mean, Variance, and Standard Deviation Let X be a continuous random variable. There are two important expected values that are used routinely to define continuous probability distributions.The mean of X, denoted by X, is defined as the expected value of X.The variance of X, denoted by X2, is defined as the expectation of the squared deviation, (X - X)2, of a random variable from its meanOr an alternative expression can be derivedThe standard deviation of X, X, is the square root of the variance.
15 Linear Functions of Variables Let X be a continuous random variable with mean X and variance X2, and let a and b any constant fixed numbers. Define the random variable W asThen the mean and variance of W areandand the standard deviation of W is
16 Linear Functions of Variable An important special case of the previous results is the standardized random variablewhich has a mean 0 and variance 1.
17 Reasons for Using the Normal Distribution The normal distribution closely approximates the probability distributions of a wide range of random variables.Distributions of sample means approach a normal distribution given a “large” sample size.Computations of probabilities are direct and elegant.The normal probability distribution has led to good business decisions for a number of applications.
18 Probability Density Function for a Normal Distribution 0.40.30.20.10.0x
19 Probability Density Function of the Normal Distribution The probability density function for a normally distributed random variable X isWhere and 2 are any number such that - < < and - < 2 < and where e and are physical constants, e = and =
20 Properties of the Normal Distribution Suppose that the random variable X follows a normal distribution with parameters and 2. Then the following properties hold:The mean of the random variable is ,The variance of the random variable is 2,The shape of the probability density function is a symmetric bell-shaped curve centered on the mean as shown in Figure 6.8.By knowing the mean and variance we can define the normal distribution by using the notation
21 Effects of on the Probability Density Function of a Normal Random Variable 0.4Mean = 6Mean = 188.8.131.52.0x184.108.40.206.220.127.116.11.5
22 Effects of 2 on the Probability Density Function of a Normal Random Variable 0.4Variance =0.30.2Variance = 10.10.01.52.18.104.22.168.57.58.5x
23 Cumulative Distribution Function of the Normal Distribution Suppose that X is a normal random variable with mean and variance 2 ; that is X~N(, 2). Then the cumulative distribution function isThis is the area under the normal probability density function to the left of x0, as illustrated in Figure As for any proper density function, the total area under the curve is 1; that is F() = 1.
24 Shaded Area is the Probability that X does not Exceed x0 for a Normal Random Variable f(x)x0x
25 Range Probabilities for Normal Random Variables Let X be a normal random variable with cumulative distribution function F(x), and let a and b be two possible values of X, with a < b. ThenThe probability is the area under the corresponding probability density function between a and b.
26 Range Probabilities for Normal Random Variables f(x)abx
27 The Standard Normal Distribution Let Z be a normal random variable with mean 0 and variance 1; that isWe say that Z follows the standard normal distribution. Denote the cumulative distribution function as F(z), and a and b as two numbers with a < b, then
28 Standard Normal Distribution with Probability for z = 1.25 0.8944z1.25
29 Finding Range Probabilities for Normally Distributed Random Variables Let X be a normally distributed random variable with mean and variance 2. Then the random variable Z = (X - )/ has a standard normal distribution: Z ~ N(0, 1)It follows that if a and b are any numbers with a < b, thenwhere Z is the standard normal random variable and F(z) denotes its cumulative distribution function.
30 Computing Normal Probabilities A very large group of students obtains test scores that are normally distributed with mean 60 and standard deviation 15. What proportion of the students obtained scores between 85 and 95?That is, 3.76% of the students obtained scores in the range 85 to 95.
31 Approximating Binomial Probabilities Using the Normal Distribution Let X be the number of successes from n independent Bernoulli trials, each with probability of success . The number of successes, X, is a Binomial random variable and if n(1 - ) > 9 a good approximation isOr if 5 < n(1 - ) < 9 we can use the continuity correction factor to obtainwhere Z is a standard normal variable.
32 The Exponential Distribution The exponential random variable T (t>0) has a probability density functionWhere is the mean number of occurrences per unit time, t is the number of time units until the next occurrence, and e = Then T is said to follow an exponential probability distribution.The cumulative distribution function isThe distribution has mean 1/ and variance 1/2
33 Probability Density Function for an Exponential Distribution with = 0.2 f(x)Lambda = 0.20.20.10.0x1020
34 Joint Cumulative Distribution Functions Let X1, X2, . . .Xk be continuous random variablesTheir joint cumulative distribution function, F(x1, x2, . . .xk) defines the probability that simultaneously X1 is less than x1, X2 is less than x2, and so on; that isThe cumulative distribution functions F(x1), F(x2), . . .,F(xk) of the individual random variables are called their marginal distribution functions. For any i, F(xi) is the probability that the random variable Xi does not exceed the specific value xi.The random variables are independent if and only if
35 CovarianceLet X and Y be a pair of continuous random variables, with respective means x and y. The expected value of (x - x)(Y - y) is called the covariance between X and Y. That isAn alternative but equivalent expression can be derived asIf the random variables X and Y are independent, then the covariance between them is 0. However, the converse is not true.
36 CorrelationLet X and Y be jointly distributed random variables. The correlation between X and Y is
37 Sums of Random Variables Let X1, X2, . . .Xk be k random variables with means 1, 2,. . . k and variances 12, 22,. . ., k2. The following properties hold:The mean of their sum is the sum of their means; that isIf the covariance between every pair of these random variables is 0, then the variance of their sum is the sum of their variances; that isHowever, if the covariances between pairs of random variables are not 0, the variance of their sum is
38 Differences Between a Pair of Random Variables Let X and Y be a pair of random variables with means X and Y and variances X2 and Y2. The following properties hold:The mean of their difference is the difference of their means; that isIf the covariance between X and Y is 0, then the variance of their difference isIf the covariance between X and Y is not 0, then the variance of their difference is
39 Linear Combinations of Random Variables The linear combination of two random variables, X and Y, isWhere a and b are constant numbers.The mean for W is,The variance for W is,Or using the correlation,If both X and Y are joint normally distributed random variables then the resulting random variable, W, is also normally distributed with mean and variance derived above.
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CC-MAIN-2021-39
| 10,833 | 42 |
http://myhomeagency.com/en/realestate/207/house-300-m2-for-sale-dramalj
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math
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Apartment house suitable for holiday rental
House for sale consisting of 6 apartments with 2 on each floor.
All apartments are fully furnished, have their own parking and a garden beside the house. Has a central heating and each apartment has air conditioning.
The house was built in 2007.
- Custom ID: 277
- Bedroom Count: 6
- Bathroom Count: 6
- Room Count: 12
- Area Outside: 400,00 m2
- Year Built: 2007
- Distance from the Town Center: 300 m
- Distance from the Sea: 600 m
- Energy Class: B
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CC-MAIN-2019-22
| 522 | 13 |
http://perplexus.info/show.php?pid=3256&cid=24722
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math
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You are on a treasure hunt, wondering where the "Truth Treasure" is buried. You have no idea where it is buried, so you go Truth Town (made up of only Liars and Knights) because it is famous for holding the treasure's secret location.
When you enter the city, you see two twins and they know each other's types, but you don't. One twin says "(Exactly) one of us is a Knight..." and the other twin says "...and (exactly) one of us is a Liar." You can only ask one question and only to one person. You only care about where the treasure is kept, and obviously a liar would give you a wrong answer, so what should you do?
(In reply to re: Solution - Not convinced
From the word "possibly" in "what answer could he possibly give me?", Twin B the Liar could say w, x or y, if the location was z, so Twin A would be telling the truth if he gave any of those.
If Twin A were allowed to reply, "Twin B could say any of w, x, y or z", then he would be lying without giving away the location. So the question could be rephrased "What is one answer he could possibly give me ?"
Posted by Penny
on 2005-07-05 12:00:13
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s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823621.10/warc/CC-MAIN-20181211125831-20181211151331-00174.warc.gz
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CC-MAIN-2018-51
| 1,105 | 7 |
http://slashdot.org/~DusterBar
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math
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6% for 40 years is a bit over 10x (not 8%)
The math is (1 + %) ^ 40 == 10 which means if you put in 6% you get 1.06 ^ 40 == 10.28 thus 40 years at 6% will get you 10.28 times the number.
Now, add in the extras (anti-lock breaks, water cooled engine, airbags, air conditioning, power windows, etc) and you seem to have a much better/nicer/safer car for around the same price given your 6% number.
Note that inflation has been all over the map over the last 40 years. Since time value of money can not trivially be reduced to averages, it is not clear what the right number is, but if you look at the 3.5% number I have seen as the effective average over the last 40 years we see that you get only a 4x multiple rather than the 10x. This seems much closer to reasonable considering the significant differences between the VW bug from 1968 (a very simple device) to the one from 2008 (a rather complex and sophisticated device).
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CC-MAIN-2015-14
| 925 | 4 |
https://www.reddit.com/r/argentina/comments/svhu5/living_in_argentina_costquality_of_living/?sort=hot
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math
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here is a spin-off from another thread
I'm considering the opportunity of moving to Argentina and more precisely to the Mendoza Area; I don't dislike Rosario, Cordoba and Buenos Aires as well... but I must admit that I don't lile big towns, so I don't know if I could be ok in BA.
Anyway, I'd lile to know something about the quality and the cost of living in theese towns. Let's suppose an annual salary of 60k pesos, is it possible to live in any of these towns?
1. how much does gasoline cost in Argentina?
2. ...and elecricity?
3. What about water? Do you have a public water distribution system?
4. What about public transportation in these towns? Is it expensive? Is it a good transportation network?
5. Are these towns "cycling"? I mean: is it possible to move by bike?
6. How much is a pint of a good local beer? ;-)
7. How much does a broadband connection cost?
Thanks in advance!
EDIT: first salary hypothesis was 75k.
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CC-MAIN-2015-40
| 928 | 12 |
https://www.teacherspayteachers.com/Product/Minnesota-Adapted-Books-Level-1-and-Level-2-Minnesota-State-Symbols-4364221
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math
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Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing.
Save money, buy the US States bundle. • Minnesota state adapted books, differentiated for non-readers and readers, are a great way to introduce your students to the state of Minnesota and its symbols.
What is included with the adapted book set of 2 books?
• Level 1 - 14 page book with 4 comprehension questions (22 total Velcro answer pieces); a shorter book with vocab words on each page
• Level 2 - 14 page book with 6 comprehension questions (26 total Velcro answer pieces); a longer book in sentence format
• Paper comprehension tests (matches the questions in each book - great for keeping data)
What information is included in the Minnesota state adapted book?
• State abbreviation
• State capital
• State nickname
• State flag, bird, and flower
• Important state symbol/monument
• # of state, as entered into the union
• Region in the US
• Neighboring states
• Major Export/Crop
• List of 2 famous people from the state
Find more Adapted Books here.
How do I put these adapted books together?
You may find this blog post about adapted books helpful.
• Print on card stock and laminate.
• Cut on the dotted line on each page.
• Bind the adapted book together. OR hole punch in the upper left corner and use a 1" binder ring to hold the book together.
• Put a piece of Velcro in each empty box or circle. Put a piece of Velcro on the back of each answer piece.
Connect with me:
Don't forget about the green ★ to follow my store to get notifications of new resources and freebies!
Thanks for Looking and Happy Teaching!
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CC-MAIN-2019-22
| 1,680 | 28 |
https://math.stackexchange.com/questions/37806/extended-euclidean-algorithm-with-negative-numbers/37810
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math
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Does the Extended Euclidean Algorithm work for negative numbers?
If I strip out the sign perhaps it will return the correct
GCD, but what exactly to do if I also want $ax + by = GCD(|a|,|b|)$ to be correct (I guess it won't hold anymore as soon as I've stripped out the signs of $a$ and $b$)?
== UPDATE ==
It couldn't be that simple.
If $a$ was negative, then after stripping out signs EEA returns such $x$ and $y$ that $(-a)*x + b*y = GCD(|a|,|b|)$. In this case $a*(-x) + b*y = GCD(|a|,|b|)$ also holds.
The same for $b$.
If both $a$ and $b$ were negative, then $(-a)*x + (-b)*y = GCD(|a|,|b|)$ holds and, well $a*(-x) + b*(-y) = GCD(|a|,|b|)$ ought to hold.
Am I right? Should I just negate $x$ if I have negated $a$ and negate $y$ if I have negated $b$?
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s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141176256.21/warc/CC-MAIN-20201124111924-20201124141924-00170.warc.gz
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CC-MAIN-2020-50
| 757 | 9 |
http://bigwww.epfl.ch/teaching/projects/abstracts/vaussard/
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math
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|Adaptive Wiener filtering using polyharmonic wavelet packets|
Section Microtechnique, EPFL
|Lots of images show a fractal-like behaviour, characterized by a power spectrum density following a
decreasing exponential law. This provides a way to easily perform a Wiener filter, which reduces the problem to the estimation of only three parameters
(the Hurst exponent H, the sigma0 coefficient, and the noise level sigma). In this semester project, the goal was to perform this filtering locally on the
image. Indeed images are more likely to have a fractal-like behaviour locally than globally.|
To do this, the wavelet packet transform is used. While the Fourier transform gives only a frequency
information, the wavelet packet transform does a frequency / time decomposition allowing the localization of the estimation. The isotropic polyharmonic
B-splines wavelets with quincunx subsampling are used because of their isotropic behaviour, more adapted to the isotropic behaviour of fractals.
The result gives a good estimator for the fractal parameters and the noise level. This allows to perform
the local Wiener filtering with a small improvement compared to the global Wiener filter. But denoising using the isotropic polyharmonic B-splines is not
as efficient as using more classical transformations.
a) Original image; b) Map with Hurst estimations;
c) Noisy image (SNR = -1.59dB); d) Filtered image (SNR = 10.7dB)
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CC-MAIN-2023-14
| 1,419 | 14 |
https://math.berkeley.edu/people/faculty/luke-oeding
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math
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Ph.D. Texas A&M University, 2009.
Hyperdeterminants of polynomials, Advances in Mathematics 231 (2012).
Toward a salmon conjecture (with Dan Bates), Experimental Mathematics, (2011).
Secant varieties of P^2xP^n embedded by O(1,2) (with Dustin Cartwright and Daniel Erman), J. London Math. Soc. (2012).
Set-theoretic defining equations of the tangential variety of the Segre variety, Journal of Pure and Applied Algebra, (2011).
Set-theoretic defining equations of the variety of principal minors of symmetric matrices, Algebra and Number Theory, (2011).
Copyright © 2011–2015 Regents of the University of California
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CC-MAIN-2017-17
| 618 | 7 |
http://www.harmonydentaltx.com/promotions/
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math
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$39 Exam and X-rays
$59 Exam,X-rays and a Basic Cleaning
For New Cash Patients
Free Orthodontic Consultation and NO down payment
$799 for removal of 4 impacted or non impacted wisdom teeth with ORAL sedation Included.
$1,499 Dental Implants (Surgery,Abutment,Crown Included )
$1,399 each if more than 2 Implants
$1,299 each if more than 3 Implants
Payment Plans Available
Offer ends July 31st ,2017
$79 for Simple Extraction and $165 for Surgical Extraction.
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s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549423716.66/warc/CC-MAIN-20170721042214-20170721062214-00583.warc.gz
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CC-MAIN-2017-30
| 458 | 11 |
https://www.raspberrypi.com/news/tag/wolfram-language/
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math
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Tag: Wolfram LanguageAll news
Astronaut-made virtual co-pilot
Helping to prevent aircraft crashes
MagPi 72: AI made easy for your Raspberry Pi
Work with voice and image recognition on your Raspberry Pi
Solar Eclipses from Past to Future, Earth to Jupiter
What would a solar eclipse look like on Mars?
Where’s Wally? Find him faster with Wolfram Language
That's Waldo for those of you not in the UK.
A Big Year for Dwarf Planets
Just how big ARE dwarf planets?
A Smart Programming Language for a Smart Cities Hackathon
Digging deep into public data with the Wolfram Language
Controlling Telescopes with Raspberry Pi and Mathematica
Astrophotography and more
Mathematica 10 – now available for your Pi!
Liz: If you use Raspbian, you'll have noticed that Mathematica and the
Vernier sensors and the Wolfram Language
Here's another guest post from Allison at Wolfram Research. Today we're
Give us your best (Mathematica) one-liner
Here's another guest post from Allison Taylor at Wolfram, which is part of a
Mathematica and the Wolfram Language on Raspberry Pi
Have you been staring at the Mathematica and Wolfram Language icons on your
The Wolfram Language and Mathematica on Raspberry Pi, for free
One of the best things about working on Raspberry Pi has been the
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s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945381.91/warc/CC-MAIN-20230326013652-20230326043652-00101.warc.gz
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CC-MAIN-2023-14
| 1,265 | 25 |
https://sciencenotes.org/how-to-calculate-concentration-of-a-chemical-solution/
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math
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The concentration of a chemical solution refers to the amount of solute that is dissolved in a solvent. Although it’s common to think of a solute as a solid that is added to a solvent (e.g., adding table salt to water), the solute could exist in another phase. If the solute and the solvent are in the same phase, then the solvent is the substance presence in the largest percentage. For example, if we add a small amount of ethanol to water, then the ethanol is the solute and the water is the solvent. If we add a smaller amount of water to a larger amount of ethanol, then the water would be the solute.
Units of Concentration
Once the solute and solvent have been identified, you can determine the concentration of the solution. There are several ways to express concentration. The most common units are percent composition by mass, mole fraction, molarity, molality, or normality.
Percent Composition by Mass (%)
This is the mass of the solute divided by the mass of the solution (mass of solute plus mass of solvent), multiplied by 100.
Determine the percent composition by mass of a 100 g salt solution which contains 20 g salt.
20 g NaCl / 100 g solution x 100 = 20% NaCl solution
Mole Fraction (X)
Mole fraction is the number of moles of a compound divided by the total number of moles of all chemical species in the solution. The sum of all mole fractions in a solution must equal 1.
What are the mole fractions of the components of the solution formed when 92 g glycerol is mixed with 90 g water? (molecular weight water = 18; molecular weight of glycerol = 92)
90 g water = 90 g x 1 mol / 18 g = 5 mol water
92 g glycerol = 92 g x 1 mol / 92 g = 1 mol glycerol
total mol = 5 + 1 = 6 mol
xwater = 5 mol / 6 mol = 0.833
x glycerol = 1 mol / 6 mol = 0.167
It’s a good idea to check your math by making sure the mole fractions add up to 1:
xwater + xglycerol = .833 + 0.167 = 1.000
Molarity is probably the most commonly used unit of concentration. It is the number of moles of solute per liter of solution (not volume of solvent).
What is the molarity of a solution made when water is added to 11 g CaCl2 to make 100 mL of solution?
11 g CaCl2 / (110 g CaCl2 / mol CaCl2) = 0.10 mol CaCl2
100 mL x 1 L / 1000 mL = 0.10 L
molarity = 0.10 mol / 0.10 L
molarity = 1.0 M
Molality is the number of moles of solute per kilogram of solvent. Because the density of water at 25 °C is about 1 kilogram per liter, molality is approximately equal to molarity for dilute aqueous solutions at this temperature. This is a useful approximation, but remember that it is only an approximation and doesn’t apply when the solution is at a different temperature, isn’t dilute, or uses a solvent other than water.
What is the molality of a solution of 10 g NaOH in 500 g water?
10 g NaOH / (4 g NaOH / 1 mol NaOH) = 0.25 mol NaOH
500 g water x 1 kg / 1000 g = 0.50 kg water
molality = 0.25 mol / 0.50 kg
molality = 0.05 M / kg
molality = 0.50 m
Normality is equal to the gram equivalent weight of a solute per liter of solution. A gram equivalent weight or equivalent is a measure of the reactive capacity of a given molecule. Normality is the only concentration unit that is reaction dependent.
1 M sulfuric acid (H2SO4) is 2 N for acid-base reactions because each mole of sulfuric acid provides 2 moles of H+ ions. On the other hand, 1 M sulfuric acid is 1 N for sulfate precipitation, since 1 mole of sulfuric acid provides 1 mole of sulfate ions.
You dilute a solution whenever you add solvent to a solution. Adding solvent results in a solution of lower concentration. You can calculate the concentration of a solution following a dilution by applying this equation:
MiVi = MfVf
where M is molarity, V is volume, and the subscripts i and f refer to the initial and final values.
How many millilieters of 5.5 M NaOH are needed to prepare 300 mL of 1.2 M NaOH?
5.5 M x V1 = 1.2 M x 0.3 L
V1 = 1.2 M x 0.3 L / 5.5 M
V1 = 0.065 L
V1 = 65 mL
So, to prepare the 1.2 M NaOH solution, you pour 65 mL of 5.5 M NaOH into your container and add water to get 300 mL final volume.
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http://slideplayer.com/slide/763143/
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math
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3.4 Linear Programming 10/31/2008
Optimization: finding the solution that is either a minimum or maximum
Linear Programming Optimize an objective function subject to constraints Graph of constraints is called the Feasible Region A minimum or maximum can only occur at a vertex of the feasible region
Example 1 (ex1)C = - x +3y Objective Function Find the min/max subject to the following constraints: Step 1: Graph the system of inequalitiesGraph
Step 2: Find intersections of the boundary lines: List of Vertices: (2,0), (5,0), (2,8) and (5,2)
Step 3: Test the vertices in the objective function C= -x +3y Minimum Maximum
Example 2 (ex 2) For the objective function C= x+5y find the minimum and maximum values subject to the following constraints:
Graph System Graph constraints Find intersections points: Intersection points (0,2) and (1,4)
Test the vertices in the objective function: Minimum Maximum??? Wait this is smaller???
Closure Note: If the feasible region is unbounded (open on a side) there may not be a minimum or maximum.
Solve the system of inequalities by graphing. x ≤ – 2 y > 3
3.4 Linear Programming.
5.1 Modeling Data with Quadratic Functions. Quadratic Function: f(x) = ax 2 + bx + c a cannot = 0.
8.6 Linear Programming. Linear Program: a mathematical model representing restrictions on resources using linear inequalities combined with a function.
Find the solutions. Find the Vertex and Max (-1, 0) (5, 0) (2, 10)
Is the shape below a function? Explain. Find the domain and range.
1 LP Duality Lecture 13: Feb Min-Max Theorems In bipartite graph, Maximum matching = Minimum Vertex Cover In every graph, Maximum Flow = Minimum.
Linear Programming Problem
Linear Programming 1.6 (M3) p. 30 Test Friday !!.
Ch 2. 6 – Solving Systems of Linear Inequalities & Ch 2
Linear Programming?!?! Sec Linear Programming In management science, it is often required to maximize or minimize a linear function called an objective.
Linear Programming Unit 2, Lesson 4 10/13.
Linear Programming Special Cases Alternate Optimal Solutions No Feasible Solution Unbounded Solutions.
Objectives: Set up a Linear Programming Problem Solve a Linear Programming Problem.
Linear Programming Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Linear programming is a strategy for finding the.
Linear Programming Operations Research – Engineering and Math Management Sciences – Business Goals for this section Modeling situations in a linear environment.
Chapter 5 Linear Inequalities and Linear Programming Section R Review.
Math – Getting Information from the Graph of a Function 1.
Algebra 2 Chapter 3 Notes Systems of Linear Equalities and Inequalities Algebra 2 Chapter 3 Notes Systems of Linear Equalities and Inequalities.
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http://tajimisunma.com/?ebooks/math-1720-pre-calculus-ii-trigonometry-algevra-and-trigonometry-with-analytic-geometry-10-th
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math
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Format: PDF / Kindle / ePub
Size: 7.76 MB
Downloadable formats: PDF
He said the math standards: …will be a mandate for reform math — a method of teaching math that eschews memorization, favors group work and student-centered learning, puts the teacher in the role of “guide” rather than “teacher” and insists on students being able to explain the reasons why procedures and methods work for procedures and methods that they may not be able to perform. — http://www.educationnews.org/education-policy-and-politics/the-pedagogical-agenda-of-common-core-math-standards/ Ze’ ev Wurman has written extensively about the Common Core Standards (in Pioneer Institute publications and elsewhere).
Publisher: Brooks/Cole (2004)
Elementary treatise on plane trigonometry
A2 Core Mathematics: Workbook: Calculus and Trigonometry
Graphing calculator lessons are scattered throughout the 7-program series and help students keep up with today's math education usage of graphing calculators Plane And Spherical Trigonometry. Solving Right Triangles Using Trigonometry - … Using Trigonometry Right Triangle Trigonometry Web supplement for Solving Right Triangles Using Trigonometry Trigonometry Right Triangles Web supplement for Solving Right Triangles Using Trigonometry Triangle Trigonometry Attached Files This file will contain: (a) Solving Right Triangles Using Trigonometry Examples, Worksheet, Worksheet Key … Introduction to Trigonometry - … may find useful when preparing a trigonometry unit include: * Lewis, Geoff. "Sharing Teaching Ideas Elements of Plane and Spherical Trigonometry with Logarithmic and Other Mathematical Tables and Exam. An unusual book in format that is aimed at the serious student, but is definitely worth having: Perrot, Pierre. Oxford. 1998. 0198565569 Three books that are as elementary as can be at the calculus level are: Some more advanced texts that are still at the undergraduate level. There are books that try to explain quantum physics to the layman, i.e. without mathematics Algebra with Trigonometry for College Students. Basic Principles of Curriculum and Instruction (Paperback ed.). Chicago, IL: The University of Chicago Press. (Original work published 1949) This section will give a detailed description of each lesson with in the trigonometry unit. In total there are five lesson plans, after each lesson plan there is the included pages from Discovery Geometry (Serra, 2003) as well as some supplemented pages from Geometry Common Core (Randall et al., 2012) Tb Alg Trig Analy Geo 12e. Casio FX-115ES and Sharp EL-W516 are two calculators below $20 with this ability. Casio FX-115ES Manual This is easier to search than the printed one Math 1720- PreCalculus II, Trigonometry, Algevra and Trigonometry with Analytic Geometry, 10th Edition online. Chapter-wise tests to help you assess your preparation status after studying a chapter The Humongous Book of Trigonometry Problems. The goal of this website is to provide new education tools to teachers, parents and off course students who can benefit from it. Mathebook.net has Online Tutorials as well as Downloadable pdf tutorials which one can save on computer and use later, even it can be used by teachers for home work purpose, as this pdf is editable and kids can e-mail the homework tutorials by e-mail back to Teachers Five-Place Logarithmic and Trigonometric Tables.
This is the first course in a highly theoretical sequence in analysis, and is intended for the most able students Mandatory Package College Algebra with Trigonometry with Smart CD (MAC)
. Use track and field to apply the concepts of arc length and the power theorems for chords Logarithmisch-Trigonometrische Tafeln Mit Sechs Decimalstellen: Mit Rücksicht Auf Den Schulgebrauch
. Angles are given either in degrees (1 complete circle = 360o) or radians (1 complete circle = 2p). For pure mathematics, radians are preferred, although both measures of angles are in very common use. Generally speaking, when we are interested in sine and cosine as functions (of the angle q), we use radians. Radians are pure numbers, the same as 0, 1, 4.3874653 and p = 3.14159
, while degrees carry units (denoted with the degree sign o) A treatise on plane trigonometry: containing an account of hyperbolic functions, with numerous examples
. The Abul Wafa crater of the Moon is named in recognition of his work in astronomy Analytic Trigonometry: The Commonwealth and International Library of Science, Technology, Engineering and Liberal Studies: Mathematics Division (Commonwealth Library)
Optimized Local Trigonometric Bases
In 'elementary' right triangle trigonometry, we teach the children, er... students that trigonometry is a type of 'indirect measurement.' So you might conduct a measurement like this: A 50-foot high tower is observed having an angle of elevation of 25 degrees from the observer. How far are we from the tower? d = 50/tan 25^o = 50/0.4663 = 107.22 feet. Now suppose we are off by 0.1 degrees, a relative error of 0.4%, and the angle of elevation is 25.1 degrees College Algebra & Trigonometry [[2nd (Second) Edition]]
. The point C will have marked out a semi-circle and the angle formed AOAf is sometimes called a straight angle. Now let the rotating arm continue to rotate, in the same direction as before, until it arrives back at its original position on OA Student Solutions Manual for Larson/Hostetler/Edwards' Algebra and Trigonometry: A Graphing Approach and Precalculus: A Graphing Approach
. These papers are designed by Educationkranti experts to ensure availability of standardized tests for every student The Civil Engineer's Pocket-Book
. Get the ideal app to measure angles with several nice features such as: This is a colorful teaching tool for your kids to learn the basic of numbers. It suitable for age from 2 to 4. the numbers one by one and read it loudly. e.g. First tap 5, and then tap next 2 blocks, count the blocks from 1 to 2 while tapping. The answer will be 7. e.g. 16 - 4 = 5, First tap 16, and then tap previous 4 blocks, count the blocks from 1 to 4 while taping, A simple App that lets a kid enter a number, then reads the number aloud Review digest of plane trigonometry
. But the rest of the world knows what they asked for the first place that involved x. So I have to go back and get rid of this Final Exam Review: College Trigonometry
. Neither GlobalRPh Inc. nor any other party involved in the preparation of this program shall be liable for any special, consequential, or exemplary damages resulting in whole or part from any user's use of or reliance upon this material ALGEBRA 2 3RD EDITION PRESENTATION PRO CD-ROM 2004C
McDougal Littell Structure & Method: Tests (Blackline) Book 2
A Treatise On Spherical Trigonometry, Part One: With Applications To Spherical Geometry (1912)
Prep-Course: Trigonometry: A general review on Algebra and an overview of what is most important to retain from Trigonometry in order to be successful in future courses. (Volume 2)
Plane and Spherical Trigonometry
Instructor's guide for Trigonometry
Algebra &Trigonometry Structure &Method Book 2 - 2000 publication
Algebra & Trigonometry 5th Edition - Annotated Instructor's Edition (Algebra & Trigonometry)
Integrated college algebra and trigonometry
Student Study Guide and Solutions Manual for Larson's Trigonometry
Logarithmisch-trigonometrische Tafeln: Enthaltend Die Logarithmen Für Alle Ganze Zahlen Von 1 Bis 10.000 In Sieben Decimal-bruchstellen ...
A New Trigonometry For Schools Part I.
Trigonometry with Applications
Trigonometry for Beginners: With Numerous Examples
Today, a gnomon is the vertical rod or similar device that makes the shadow on a sundial. At midday the shadow of a stick is shortest, and the civilisations of Mesopotamia, Egypt, and China took the North - South direction from this alignment. In contrast, the Hindus used the East - West direction, the rising and setting of the sun, to orient their "fire-altars" for religious practices. To do this they constructed the "gnomon circle" whose radius was the square root of the sum of the square of the height of the gnomon and its shadow [See Note 2 below] Algebra and Trigonometry with Analytic Geometry (11th Edition with CD-ROM) (Available Titles CengageNOW)
. Included in the links will be links for the full Chapter and E-Book of the page you are on (if applicable) as well as links for the Notes, Practice Problems, Solutions to the Practice Problems and Assignment Problems PLANE AND SPHERICAL TRIGONOMETRY
. The student is expected to: (A) determine the domain and range of a linear function in mathematical problems; determine reasonable domain and range values for real-world situations, both continuous and discrete; and represent domain and range using inequalities; (B) write linear equations in two variables in various forms, including y = mx + b, Ax + By = C, and y - y1 = m(x - x1), given one point and the slope and given two points; (C) write linear equations in two variables given a table of values, a graph, and a verbal description; (D) write and solve equations involving direct variation; (E) write the equation of a line that contains a given point and is parallel to a given line; (F) write the equation of a line that contains a given point and is perpendicular to a given line; (G) write an equation of a line that is parallel or perpendicular to the X or Y axis and determine whether the slope of the line is zero or undefined; (H) write linear inequalities in two variables given a table of values, a graph, and a verbal description; and (I) write systems of two linear equations given a table of values, a graph, and a verbal description. (3) Linear functions, equations, and inequalities Teach Yourself Trigonometry
. Assume that all variables are positive, and note that I used the variable t instead of x to avoid confusion with the x’s in the triangle: Click on Submit (the arrow to the right of the problem) to solve this problem Answer To Dr. Priestley's Letters To A Philosophical Unbeliever: Part I
. If R is the radius of the circumscribed circle, then sin B = b/2R and S = abc/4R = 2R2sin A sin B sin C Precalculus (6th Edition)
. However, suggestions for further improvement, from all quarters would be greatly appreciated. Or want to know more information about Math Only Math. Use this Google Search to find what you need. "Courses like these are critical for many families who are deciding whether or not they can handle homeschooling through high school NY Algebra 2 & Trigonometry, Studentworks CD-ROM
. Three well-known ones can be mentioned here. the cylinder. the cone and the sphere Title: LARSON TRIGONOMETRY 7ED(PASADENA)CPC
. Of course he may have been thinking about the contacts he could possibly make in Partition is fairly common between different types of. Then he would have about my reaction to like a ball and. The Scandinavians are spread were some Likud higher News where hed fit major complication because you. The reason I believe the way it did after the inadvertent space vote with download Math 1720- PreCalculus II, Trigonometry, Algevra and Trigonometry with Analytic Geometry, 10th Edition pdf.
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https://www.careerstoday.in/maths/parts-of-a-circle
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math
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Parts of a Circle
Circumference of Circle
Many objects that we come across in our daily life are ‘round’ in shape. Coin, bangles, bottle caps, the Earth, wheels etc. In layman terms, the round shape is often referred to as a circle. A closed plane figure, which is formed by the set of all those points which are equidistant from a fixed point in the same plane, is known as a circle. The fixed point is called the center of the circle and the constant distance between any point on the circle and its center is called the radius. The circumference of a circle can be defined as the distance around it.
In other words, a circle can be described as the locus of a point moving in a plane, in a way that its distance from a fixed point is always constant.
Figure 1 given above represents a circle with radius ‘r’ and center ‘O’. A circle of any particular radius can be easily traced using a compass. The pointed leg of the compass is placed on the paper and the movable leg is revolved as shown. The traced figure gives us a circle.
In the figure shown above, various points are marked lying either outside or inside the circle or on the circle. Based on this any point can be defined as:
- Exterior Point: Points lying in the plane of the circle such that its distance from its center is greater than the radius of the circle are exterior points. A point X is exterior point w.r.t to circle with center ‘O’ if OX > r. In fig. 2 D, G and B are exterior points.
- Interior Points: Point lying in the plane of the circle such that its distance from its center is less than the radius of the circle is known as the interior point. A point X is interior point w.r.t to circle with center ‘O’ if OX < r. In fig. 2 C, F, and E are interior points.
- Point on the circumference of a circle: Points lying in the plane of the circle such that its distance from its center is equal to the radius of a circle. In simple words, a set of points lying on the circle are points on the circumference of a circle.
A point X is said to lie on the circumference of a circle with center ‘O’ if OX = r
In fig. 2, points P, S and R lie on the circumference of a circle and on joining these points with center, i.e. OR, OP and OS will represent the radius of the given circle.
Now that we know about a point and its relative position with respect to a circle let us discuss about a line and its relative position with respect to a circle. Given a line and a circle, it could either be touching the circle or non-touching as shown below:
In the first fig. the line AB intersects the circle at two distinct points P and Q. The line AB here is called secant of the circle. The line segment PQ is known as the chord of the circle as its endpoints lie on the circumference of the circle. A chord passing through a center of the circle is known as the diameter of the circle and it is the largest chord of the circle.
In the second figure, the line AB touches the circle exactly at one point, P. A line touching the circle at one single point is known as the tangent to the circle.
In the last figure the line does not touch the circle anywhere, therefore, it is known as a non- intersecting line.
Now let us discuss about the circular region which is cut off from the rest of the circle by a secant or a chord.
Arc of a circle
A part of a circumference of the circle is known as an arc. An arc is a continuous piece of the circle.
The arc PAQ is known as the minor arc and arc PBQ is the major arc. Part of a circle bounded by a chord and an arc is known as a segment of the circle. The figure given below depicts the major and the minor segment of the circle.
Sector of a circle is the part bounded by two radii and an arc of a circle. In the given fig AOB is a sector of a circle with O as center.
The figure given below illustrates the various terms related to circles as explained above.
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https://www.smartmathz.com/question/grade-2-4/
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math
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Frank has 23 diamonds. He gave 13 to a charity home to help poor children. How many diamonds does Frank have left?
This is a subtraction problem because we want to know how many diamonds Frank has left after he gives some to the charity. We take the total number of diamonds that Frank had before, and then we subtract the number that he gave to the Charity.
The keywords to look out for in a problem involving subtraction are: take away; how many more; how many less; how many left; greater; smaller.
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https://www.math-edu-guide.com/CLASS-2-Concept-Of-Division.html
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math
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CONCEPT OF DIVISION
DIVISION – Division is repeated subtraction of same number from big number through simple way with investment of minimum time. Sign of division is ‘÷’
Division mean equal sharing or grouping between same or different group.
The number which is to be divided is called ‘DIVIDEND’
The number which divides the ‘Dividend’ is called ‘DIVISOR’
The answer of Division is called ‘QUOTIENT’
DIVISOR ) DIVIDEND ( QUOTIENT
------------- = QUOTIENT
Example. 1- Suppose there is a number which we have to subtract 5 times and the number is 2, this number 2 will be repeatedly subtract from 12.
So, we have subtracted 2 from 12 by 6 times.
but instead of repeated subtraction of 2 for 6 times we can use short method and that is = 12 ÷ 2 = 6 (Answer).
Example.2 – Suppose a person for a special occasion he bought a big cake and cut into 20 pieces, his invited guest was 5 no.s, now the question is each of his guest how many pieces of cake can get?
In repeated subtraction we can say each of his guest can get no.s of cake pieces,
Please notice, how many times 5 have been subtracted from 20, and the answer is 4. So we can say each of his guest can get 4 pieces of cake.
But we can give the answer in simple way with investment of less time, i.e each of his guest can get no.s pieces of cake is = 20 ÷ 5 = 4 no.s cake pieces
Instead of subtraction 5 four times, we can write 20 ÷ 5 = 4. (Ans.)
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http://giuseppe-zanotti.org/category/web-resources/
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math
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Finding Assistance For Mathematics For Your Child In School.
Most high school’s students find it difficult to get high scores in math as they argue they find it hard and tricky and they can hardly get even fifty percent in it which prompts teachers and parents to find a combination of mutual efforts to try to come with amicable solution for their children. There are numerous tricks and methods of passing and excelling in mathematics that you should be aware of in your high school life and this article will guide you in knowing some of them so that you can shift from bottom to top in mathematics performance.
In high school mathematics course, you are required to have knowledge of using the logarithm calculators that have been availed by the education sector and required by the curriculum developers as they will enhance you to have insight and knowledge of solving problems related to calculations and number operations. You need to have knowledge of using and calculating log tables and these calls for logarithm help from your tutor so that you can be skilled with knowledge on how to interpret and conduct operations relating to such mathematics table as they are imperative in the success of your mathematics subject And overall grade.
When learning mathematics, it’s advisable to have with you all the necessary mathematics course books and their revision booklets that will enable you to refer to them for solutions and complicated tactics involving logarithms and other operations so that you can succeed and understand them better. Mathematics can’t be learned without the assistance and advice from mathematics counselors and teachers and they ought to be listened and followed as they know every trick and formulae of doing any type of calculation and getting solution to any operations.
It’s also important to note that group works and team works with other learners offers thought-of assistance and understanding of mathematics as they can show you
where they know and you don’t know and this equips you with advanced knowledge of basic logarithm calculations. The learning of mathematics with exquisite help can only be started from within self and this means the right attitude and dedication will make you grasp logarithm calculations and understanding of formulae concepts easily and quickly to save their time and to for solving operations and this will be the beginning for your success and achievement in mathematics.
It’s easier to succeed in mathematics and with the advancement of technology, numerous mathematics websites with requisite formulae and tricks are being opened and regularly updated with the aim of assisting mathematics learners with basic help in acquiring mathematics formulae for success. In conclusion, calculation of numbers involves interests and practice which you ought to employ in your routine for you to excel.
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https://reference.wolfram.com/language/tutorial/TroubleshootingInternetConnectivity.html
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math
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Troubleshooting Internet Connectivity Problems
If the test fails, note how long it took. If there was a minutes-long delay, skip to "Check Your Proxy Settings" for troubleshooting information.
By default, the Wolfram System will attempt to use system proxy settings, if your operating system has them, using the setting Use Proxy Settings from My System or Browser in the Proxy Settings section of the Internet Connectivity dialog (Help ▶ Internet Connectivity).
If you are experiencing connectivity problems, it is possible that your system has incorrect proxy information or that you do not use a proxy to access the internet. To test this, select the setting Direct Connection to the Internet.
On Windows, the Wolfram System will by default use the proxy settings found in Windows under Control Panel ▶ Internet Options ▶ Connections. These are the same proxy settings used by Internet Explorer.
For Windows Firefox users, proxy settings are found in Tools ▶ Options. Click Advanced at the top, and then click the Network tab that appears below. Click the Settings button in the Connections section, and a proxy settings window will appear.
If you are using Windows, try unchecking the Use same proxy server for all protocols checkbox in the Proxy Settings dialog under Control Panel ▶ Internet Options ▶ Connections and then clicking the Advanced button in the Proxy Settings section.
If the Test Network Connectivity button (described in "Test Internet Connectivity") fails after a minutes-long delay, an incorrect SOCKS proxy configuration is likely to be the problem. The Wolfram System does not require a SOCKS proxy, but will attempt to use one if the SOCKS proxy information is available.
If so, configure the firewall to allow the Java Runtime Environment in the Wolfram System layout to use the internet. The Wolfram System connects to the internet using the standard HTTP port, port 80.
For more information about the Wolfram System's connectivity capabilities, see "Internet Connectivity".
If the information in this document does not help you resolve your connectivity problems, consult the Wolfram Research Technical Support troubleshooting guide at support.wolfram.com/kb/12420.
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| 2,207 | 11 |
https://www.physics-world.com/ap-physics-1/4-circular-motion/
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math
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In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body.
Examples of circular motion include: an artificial satellite orbiting the Earth at the constant height, a stone which is tied to a rope and is being swung in circles, a car turning through a curve in a race track, an electron moving perpendicular to a uniform magnetic field, and a gear turning inside a mechanism.
Since the object’s velocity vector is constantly changing direction, the moving object is undergoing acceleration by a centripetal force in the direction of the center of rotation. Without this acceleration, the object would move in a straight line, according to Newton’s laws of motion.
|Topical Notes, Problems, Presentations, Quiz, Test, Investigations and Videos|
|Uniform Circular Motion|
|Centripetal Acceleration and Centripetal Force|
|Satellites in Circular Orbits|
|Test your Understanding: Chapter 3 MCQ Quiz 1 Here Take Chapter 3 ReQuiz MCQ Quiz 2 Here|
In physics, uniform circular motion describes the motion of a body traversing a circular path at constant speed. Since the body describes circular motion, its distance from the axis of rotation remains constant at all times. Though the body’s speed is constant, its velocity is not constant: velocity, a vector quantity, depends on both the body’s speed and its direction of travel. This changing velocity indicates the presence of an acceleration; this centripetal acceleration is of constant magnitude and directed at all times towards the axis of rotation. This acceleration is, in turn, produced by a centripetal force which is also constant in magnitude and directed towards the axis of rotation.
In the case of rotation around a fixed axis of a rigid body that is not negligibly small compared to the radius of the path, each particle of the body describes a uniform circular motion with the same angular velocity, but with velocity and acceleration varying with the position with respect to the axis.
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https://masterconceptsinchemistry.com/index.php/2019/08/14/how-to-calculate-percentage-abundance-using-atomic-and-isotopic-masses/
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math
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To calculate percentage abundance, you must recall the atomic mass of an element is calculated by using the formula:
In the above formula you see fractional abundance. How do you get that? To get fractional abundance, you usually divide the percentage abundance of each isotope by 100. And when you add all the fractional abundance values of all the isotopes, you will notice they all add up to 1. To calculate the percent abundance of each isotope in a sample of an element, chemists usually divide the number of atoms of a particular isotope by the total number of atoms of all isotopes of that element and then multiply the result by 100. Now, let’s apply our understanding to solve the following question:
Silver (Ag) has two stable isotopes: silver-107(107Ag) and silver-109 (109Ag). Silver-107 has a mass of 106.90509 amu and silver-109 has a mass of 108.90476 amu. Calculate the percentage abundance of each isotope.
To calculate percentage abundance, we must first know the fractional abundance of each isotope. But from the question, we are not given these values, which means we must think of a way of finding them. One way we can find them is to remember that the:
fractional abundance of isotope 1 plus the fractional abundance of isotope 2 = 1
We know this because the sum of the percentage abundance values always equal 100. And since the fractional abundance is obtained by dividing the percentage abundance by 100, then, it follows that the sum of the fractional abundance must equal 1. Once we understand this, we can then let X represent the fractional abundance for isotope 1 and (1-X) represent the fractional abundance for isotope 2. Once we substitute these variables into the formula above, we will generate an algebraic equation from which we can solve to find X and then (1-X). Once we get the values for fractional abundance (X and 1-X), we can then multiply each fractional abundance by 100 to get percentage abundance.
Also, notice the question gave only the isotopic masses and not the atomic mass of silver. How do we get the atomic mass of silver to plug into the above equation? We get the atomic mass of silver by reading its value from the periodic table. And If you do, you will notice the atomic mass of silver is 107.8682. After gathering all the information necessary to solve the question, here is how the setup will appear based on the above information and formula:
Next, we then apply our algebra skills to solve for X. To do this, we will expand the right side of the equation by multiplying X and 1-X by the numbers in front of them. If we do, here is what we will get:
107.8682 amu = 106.90509X amu + 108.90476 amu – 108.90476X amu
Next, we bring like terms together by moving 108.90476 amu from the right over to the left side of the equation. Since it’s a positive number on the right, it will become a negative number when it moves across the equal sign to the left. When we move it, here is how the rearranged equation will appear:
107.8682 amu – 108.90476 amu = 106.90509 X amu – 108.90476 X amu
Next, we subtract like terms
-1.03656 amu = -1.99967 X amu
Next, we divide by -1.99967 amu to isolate X.
amu/-1.99967 amu = -1.99967 amu X/ -1.99967 amu
X = 0.518
Here is a more clear representation of the division step
Notice that the negative signs and units cancel each other when we divide by -1.99967 amu, hence, fractional abundance has no units
X= 0.518 and 1 – X = 1-.518 = 0.482
Therefore, the fractional abundance of isotope 1 (Silver-107) is 0.518 and isotope 2 (Silver-109) is 0.482.
How to find percentage abundance
To get the percentage abundance, we will simply multiply each fractional abundance by 100. Recall that fractional abundance is calculated by dividing the percentage abundance by 100. Therefore, to get back percentage abundance, we multiply fractional abundance by 100. If we do, the percentage abundance for silver-107 is 0.518 x 100 = 51.8%. And percentage abundance for silver-109 is 0.482 x 100 = 48.2%
To learn how to calculate atomic mass using percentage abundance and isotopic masses click here.
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https://www.arxiv-vanity.com/papers/nucl-th/0102041/
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math
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The stability of the scalar interaction
A scalar field theory with a interaction is known to be unstable. Yet it has been used frequently without any sign of instability in standard text book examples and research articles. In order to reconcile these seemingly conflicting results, we show that the theory is stable if the Fock space of all intermediate states is limited to a finite number of loops associated with field that appears quadradically in the interaction, and that instability arises only when intermediate states include these loops to all orders.
Scalar field theories with a interaction (which we will subsequently denote simply by ) have been used frequently without any sign of instability, despite a proof in 1959 by G. Baym showing that the theory is unstable. For example, it is easy to show that, for a limited range of coupling values , the simple sum of bubble diagrams for the propagation of a single particle leads to a stable ground state, and it is shown in Ref. that a similar result also holds for the exact result in “quenched” approximation. However, if the scalar interaction is unstable, then this instability should be observed even when the coupling strength is vanishingly small , as pointed out recently by Rosenfelder and Schreiber (see also Ref. ). Both the simple bubble summation and the quenched calculations do not exhibit this behavior. Why do the simple bubble summation and the exact quenched calculations produce stable results for a finite range of coupling values?
A clue to the answer is already provided by the simplest semiclasical estimate of the ground state energy. In this approximation the gound state energy is obtained by minimizing
where is the bare mass of the matter particles, and the mass of the “exchanged” quanta, which we will refer to as the mesons. The minimum occurs at
The ground state is therefore stable (i.e. greater than zero) provided
This simple estimate suggests that the theory is stable over a limited range of couplings if the strength of the field is finite. In this letter we develop this argument more precisley and show under what conditions it holds.
We start in the Heisenberg representation, where the fields depend on time and the states are independent of time. The fields are expanded in terms of creation and annihilation operators
with . The equal-time commutation relations are
The Lagrangian for the theory is
and the hamiltonian is a normal ordered product of interacting (or dressed) fields and
This hamitonian conserves the difference between number of matter and the number of antimatter particles, which we denote by . Eigenstates of the hamiltonian will therefore be denoted by , where represents the other quantum numbers that define the state. Hence, allowing for the fact that the eigenvalue may depend on the time,
In the absence of an exact solution of (9), we may estimate it from the equation
where is the time translation operator which carries the hamiltonian from time to later time . We have also chosen to be the time at which the interaction is turned on, , and the last step simplifies the discussion by permitting us to work with a hamiltonian constructed from the free fields and . [If the interaction were turned on at some other time , we would obtain the same result by absorbing the additional phases into the creation and annhilation operators.]
At the hamiltonian in normal order reduces to
and . To evaluate the matrix element (10) we express the the eigenstates as a sum of free particle states with matter particles, pairs of particles, and mesons:
where is a normalization constant (defined below), the time dependence of the states is contained in the time dependence of the coefficients and , and
with , and
The particle masses in and have been suppressed; their values should be clear from the context. The normalization of the functions and is chosen to be
which leads to the normalization
The expansion coefficients and are vectors in infinite dimensional spaces.
In principle the scalar cubic interaction in four dimensions requires ultraviolet regularization. However the issue of regularization and the question of stabilty are qualitatively unrelated. For example, the cubic interaction is also unstable in dimensions lower than four, where there is no need for regularization. The ultraviolet regularization would have an effect on the behavior of functions , and , which are left unspecified in this discussion except for their normalization.
The matrix element (10) can now be evaluated. Assuming that and , it becomes:
where the constants , , and are
and the time dependent quantities are
Note that and are the average number of matter pairs and mesons, respectively, in the intermediate state.
The variational principle tells us that the correct mass must be equal to or larger than (19). This inequality may be simplified by using the Schwarz inequality to place an upper limit on the quantities and . Introducing the vectors
we may write
Hence, suppressing explicit reference to the time dependence of and , Eq. (19) can be written
Minimization of the ground state energy with respect to the average number of mesons occurs at
At this minimum point the ground state energy is bounded by
This result shows that the ground state is stable for couplings in the interval with
This interval is nonzero if the number of matter particles, , and the average number of pairs, , is finite. In particular, if there are no diagrams or loops in the intermediate states, then the ground state will be stable for a limited range of values of the coupling.
This result also suggests strongly that the system is unstable when , or when (implying that ). However, since Eq. (26) is only a lower bound, our argument does not provide a proof of these latter assertions.
To strengthen our understanding of the causes of instability in a theory, we turn to the Feynman-Schwinger representation (FSR). This can be used to show that the ground state is (i) stable when -diagrams are included in intermediate states, but (ii) unstable when matter loops are included.
The FSR is a path integral approach for finding the exact result for propagators in field theory. It replaces integrals over fields by integrals over all possible covariant trajectories of the particles. It has been applied to the interaction in Refs. [2, 6, 7, 8, 9, 10].
The covariant trajectory of the particle is parametrized as a function of the proper time . In theory the FSR expression for the 1-body propagator for a dressed -particle in quenched approximation in Euclidean space is given by
where the integrations are over all possible particle trajectories (discretized into segments with variables and boundary conditions , and ) and the kinetic and self energy terms are
where is the Euclidean progagator of the meson (suitably regularized), , and
(The substitution does not alter the results, but is necessary to correctly transform the original integral from Minkowsky space to Euclidean space, where it can be numerically evaluated. For a detailed discussion of this technical point, see Ref. .)
In preparation for a discussion of the effects of -diagrams and loops, we first discuss the stability of Eq. (28) when neither -diagrams nor loops are present. To make the discussion explicit, consider the one body propagator in 0+1 dimension. Since the integrals converge, we make the crude approximation that each integral is approximated by one point (since we are excluding -diagrams, the points may lie along the classical trajectory). If the boundary conditions are and the points along the classical trajectory are , and
If the interaction is zero, this has a stationary point at , giving
yielding the expected free particle mass . [Note that half of this result comes from the sum over .] The potential term (30) may be similarily evaluated; it gives a negative contribution that reduces the mass.
We now turn to a discussion of the effect of -diagrams. For the simple estimate of the kinetic energy, Eq. (32), we chose integration points uniformly spaced along a line. The classical trajectory connects these points without doubling back, so that they increase monotonically with proper time, . However, since the integration over each is independent, there also exists trajectories where does not increase monotonically with . In fact, for every choice of integration points there exist trajectories with monotonic in and trajectories with non-monotonic in . The latter double back in time, and describe -diagrams in the path integral formalism. Two such trajectories that pass through the same points are shown in Fig. 1. These two trajectories contain the same points , but ordered in different ways, and both occur in the path integral.
Now, since the total self energy is the sum of potential contributions from all pairs, irrespective of how these coordinates are ordered, it must be the same for the straight trajectory and the folded trajectory :
However, according to Eq. (29), the kinetic energy of the folded trajectory is larger than the kinetic energy of the straight trajectory
because it includes some terms with larger values of . Since the kinetic energy term is always positive, the folded trajectory (-graph) is always suppressed (has a larger exponent) compared with a corresponding unfolded trajectory (provided, of course, that ).
This argument holds only for cases where the trajectory does not double back to times before or after . An example of such a trajectory is shown in Fig. 2 (upper panel). Here we compare this folded trajectory to another folded trajectory, , with point closer to the starting point (lower panel of Fig. 2). This new folded trajectory has points spaced closer together, so that the kinetic energy is smaller and the potential energy is larger, and therefore
It is clear that the larger the folding in the trajectory, the less energetically favorable is the path, and the most favorable path is again an unfolded trajectory with no points outside of the limits .
While these arguments have been stated in 0+1 dimensions for simplicity, they are not dependent on the number of dimensions, and hold for the realistic case of 3+1 dimensions.
We conclude that a calculation in quenched approximation, where the creation of particle-antiparticle pairs can only come from -graphs, must be more stable (produce a larger mass) than a similar calculation without any Z-graphs or pairs. The quenched theory therefore is bounded by the same limits given in Eq. (27). This conclusion supports, and is supported by, the results of Refs. [2, 9, 10] which show, in the quenched approximation, that the interaction is stable for a finite range of coupling strengths.
It is now clear that the instability of theory must be due to either (i) the possibility of creating an infininte number of closed loops, or (ii) the presence of an infinite number of matter particles (as in an infinite medium). Indeed, the original proof given by Baym used the possibility of loop creation from the vacuum to prove that the vacuum was unstable. In fact, the FS representation can be used to show explicitly that the critical coupling decreases as , where is the number of closed loops, in agreement with the estimate of Eq. (27).
This work was supported in part by the US Department of Energy under grant No. DE-FG02-97ER41032. The Southeastern Universities Research Association (SURA) operates the Thomas Jefferson National Accelerator Facility under DOE contract DE-AC05-84ER40150.
- G. Baym, Phys. Rev. 117, 886 (1959).
- Ç. Şavklı, J. A. Tjon, and F. Gross, Phys. Rev. C 60 055210 (1999).
- R. Rosenfelder, A.W. Schreiber, Phys. Rev. D 53 3337, 1996; Phys. Rev. D53, 3354, 1996; hep-ph/9911484.
- B. Ding and J. Darewych, J. Phys. G, 26, 907 (2000).
- Yu. A. Simonov and J. A. Tjon, Ann. Phys. 228, 1 (1993).
- T. Nieuwenhuis and J. A. Tjon, Phys. Rev. Lett. 77, 814 (1996).
- T. Nieuwenhuis, PhD-thesis, University of Utrecht (1995), unpublished.
- Ç. Şavklı, F. Gross, J. A. Tjon, Phys Review D 62 116006
- Ç. Şavklı Oct 2000. 33pp. Lectures given at 13th Indian Summer School: Understanding the Structure of Hadrons, Prague, Czech Republic, 28 Aug - 1 Sep 2000. To be published in Czech. J. Phys, hep-ph/0011249
- Ç. Şavklı, hep-ph/9910502, ( accepted for publication in Comp. Phys. Comm.)
- Ç. Şavklı, F. Gross, and J. A. Tjon, in preparation.
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https://gladtidingsclearfield.org/and-pdf/1300-greatest-mathematicians-and-their-contributions-pdf-16-761.php
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Greatest mathematicians and their contributions pdf
Top 10 Indian Mathematicians and their ContributionsShare facts or photos of intriguing scientific phenomena. The greatest scholar of the ancient era, Archimedes made phenomenal contribution in the field of mathematics. His works include finding various computation techniques to determine volume and area of several shapes, including the conic section. Euclid, the 'father of Geometry', wrote the book ,"Euclid's Elements", that is considered to be the greatest piece of historical works in mathematics. The book is divided into 13 parts and in it, Euclid has discussed in details about geometry what is now called Euclidean geometry. His works are also well-known in the fields of spherical geometry, conic sections, and number theory.
15 Famous Greek Mathematicians and Their Contributions
He had already turned down maths' most prestigious honour, designed formulae for the volumes of surfaces of revolution and also invented a technique for expressing extremely large numbers. He also identified the spiral that bears his name, the Fields Medal in He discovered the position of nine planets and stated that these planets revolve around the sun. Alan Turing was a British mathematician who has been call the father of computer science.This was a revolutionary idea, and it led to probability theory, When he went blind in his late 50s his productivity in many areas increas? Cavalieri's works on indivisibles were reissued with his later corrections in Death Carl Ludwig Siegel breathed his last on April 4.
Here is some information about the same. Gauss was maathematicians, for whom he was always ready to do anything in his power, considered by many to be the greatest scientist of all time, astronomy and many other subjects that underlie our modern wor. Being devoted to mathema. We start our list with Sir Isaac Newton.
Contributions This article enlists the names of those mathematicians whose theorems, results and inventions paved a path for deep research in mathematics. Here is some information about the same. There is certainly no end to the series of mathematicians whose works created the platform for others to produce seminal works in mathematics. But since it is indeed not possible to list every mathematician who has contributed to this great science, I have made a humble attempt to compile a list of some really well-known mathematicians whose great works revolutionized the scientific and mathematical world. Famous Mathematicians List Mathematics has witnessed some of the most genius brains pondering over complex problems and solving them to unravel mysteries of Universe, science and life. The world salutes the great mathematicians and their contributions.
Everywhere you look it is likely mathematics has made an impact, greates the faucet in your kitchen to the satellite that beams your television programs to your home. He attended lectures on number theory by Prof. However, make him one of the great teachers of religion in the ancient Greek wor. He wrote on the rectification of the parabola and of the cycloid.
We picked five of the most brilliant mathematicians whose work continues to help shape our modern world, Turing bent his brain to the problem of breaking Nazi crypto-code and was the one to finally unravel messages protected by the infamous Enigma machine? In he published works on modular lattices in algebra. He established strong foundations in the field of physics, hydrostatics and explained the principle greatwst the lever, sometimes hundreds of years after their death. During Gfeatest War II.Log in to Reply this helped greafest in homework Log in to Reply I like aryabhata Log in to Reply nice but Shakuntla devi is left you must add her also Log in to Reply That page was good and it was much helpful for my maths project. Learn more about Scribd Membership Bestsellers. Newton is so famous for his calculus, optics and laws of motion? Carl Friedrich Gauss was born to a poor family in Germany in and quickly showed himself to be a brilliant mathematician?
He discovered Puiseux series almost two centuries before they were re-invented by Puiseux. Role in the Seminar Along with Dehn, he was known as the forerunner of geometrical knowledge and went on to contribute greatly in the field of mathematics, Hellinger and Epstein. Although little is known about Euclid's early and personal life. Great Mathematicians.
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https://connectedmentor.com/qa/quick-answer-how-much-mortgage-can-i-get-for-1000-a-month.html
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- How much house can I afford with a monthly payment of $1200?
- What’s the payment on a $300 000 mortgage?
- How much do I need to make to afford a 250k house?
- What is the 28 36 rule?
- How much is 10k on a mortgage?
- How do you estimate PMI?
- How big of a mortgage can I afford?
- What is a good down payment on a house?
- Why does it take 30 years to pay off $150000 loan even though you pay $1000 a month?
- What mortgage can I afford on 60k?
- How much would a 1000 mortgage cost?
- How much does every 10000 add to monthly mortgage?
- What mortgage can I get for 500 a month?
- What happens if I pay an extra $200 a month on my mortgage?
- How do you know how much to spend on a house?
- How much house can I afford for $900 a month?
- How much should you make to buy a 500000 house?
- How do you calculate a house payment?
How much house can I afford with a monthly payment of $1200?
How Much House Can You Afford?Monthly Pre-Tax IncomeRemaining Income After Average Monthly Debt PaymentMaximum Monthly Mortgage Payment (including Property Taxes and Insurance) with the 36% Rule$3,000$2,400$480$4,000$3,400$840$5,000$4,400$1,200$6,000$5,400$1,5604 more rows.
What’s the payment on a $300 000 mortgage?
Monthly payments on a $300,000 mortgage At a 4% fixed interest rate, your monthly mortgage payment on a 30-year mortgage might total $1,432.25 a month, while a 15-year might cost $2,219.06 a month.
How much do I need to make to afford a 250k house?
How much do you need to make to be able to afford a house that costs $250,000? To afford a house that costs $250,000 with a down payment of $50,000, you’d need to earn $43,430 per year before tax.
What is the 28 36 rule?
According to this rule, a household should spend a maximum of 28% of its gross monthly income on total housing expenses and no more than 36% on total debt service, including housing and other debt such as car loans and credit cards.
How much is 10k on a mortgage?
30 Year $10,000 Mortgage LoanLoan Amount2.50%5.50%$10,000$39.51$56.78$10,005$39.53$56.81$10,010$39.55$56.84$10,015$39.57$56.8616 more rows
How do you estimate PMI?
You can calculate PMI percentage fee with just your monthly statement. To calculate the exact percentage fee of your loan, you take the PMI required per month and multiply it by 12. Next, divide the original loan amount by the PMI required per year. The resulting amount should be between 0.30 percent and 1.15 percent.
How big of a mortgage can I afford?
To calculate ‘how much house can I afford,’ a good rule of thumb is using the 28%/36% rule, which states that you shouldn’t spend more than 28% of your gross monthly income on home-related costs and 36% on total debts, including your mortgage, credit cards and other loans like auto and student loans.
What is a good down payment on a house?
Lenders require 5% to 15% down for other types of conventional loans. When you get a conventional mortgage with a down payment of less than 20%, you have to get private mortgage insurance, or PMI. The monthly cost of PMI varies, depending on your credit score, the size of the down payment and the loan amount.
Why does it take 30 years to pay off $150000 loan even though you pay $1000 a month?
Why does it take 30 years to pay off $150,000 loan, even though you pay $1000 a month? … Even though the principal would be paid off in just over 10 years, it costs the bank a lot of money fund the loan. The rest of the loan is paid out in interest.
What mortgage can I afford on 60k?
The usual rule of thumb is that you can afford a mortgage two to 2.5 times your annual income. That’s a $120,000 to $150,000 mortgage at $60,000. … Lenders want your principal, interest, taxes and insurance – referred to as PITI – to be 28 percent or less of your gross monthly income.
How much would a 1000 mortgage cost?
30 Year $1,000 Mortgage LoanLoan Amount2.50%6.00%$1,000$3.95$6.00$1,005$3.97$6.03$1,010$3.99$6.06$1,015$4.01$6.0916 more rows
How much does every 10000 add to monthly mortgage?
THE DWELL MORTGAGE RULE OF THUMB: Every $10,000 in purchase price only adds an additional $40 to your monthly payment.
What mortgage can I get for 500 a month?
30 Year $500 Mortgage LoanLoan Amount2.50%5.00%$500$1.98$2.68$505$2.00$2.71$510$2.02$2.74$515$2.03$2.7616 more rows
What happens if I pay an extra $200 a month on my mortgage?
Adding Extra Each Month Simply paying a little more towards the principal each month will allow the borrower to pay off the mortgage early. Just paying an additional $100 per month towards the principal of the mortgage reduces the number of months of the payments.
How do you know how much to spend on a house?
To determine how much house you can afford, most financial advisers agree that people should spend no more than 28 percent of their gross monthly income on housing expenses and no more than 36 percent on total debt — that includes housing as well as things like student loans, car expenses, and credit card payments.
How much house can I afford for $900 a month?
A payment of $900 would have a mortgage balance of $191,976. If you include your monthly taxes, insurance and mortgage insurance payment of $300 a month, you now have a payment of $1,200 a month.
How much should you make to buy a 500000 house?
A generally accepted rule of thumb is that your mortgage shouldn’t be more than three times your annual income. So if you make $165,000 in household income, a $500,000 house is the very most you should get.
How do you calculate a house payment?
If you want to do the monthly mortgage payment calculation by hand, you’ll need the monthly interest rate — just divide the annual interest rate by 12 (the number of months in a year). For example, if the annual interest rate is 4%, the monthly interest rate would be 0.33% (0.04/12 = 0.0033).
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https://www.physicsforums.com/threads/geometry-question.259923/
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math
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1. Prove the property of exterior angle bisectors discussed in class. Given triangle ABC and point D on line AB such that CD bisects the exterior angle at C, then BD/AD=BC/AC. The only thing I can compute about this problem is the Property of Exterior Angle, but I don't know how to apply it to the problem. Exterior Angle is the angle between any side of a shape, and a line extended from the next side. <-- That's is all I understand. Help on where to start this problem out with. Thanks.
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https://shivanshdubey.com/lines-and-planes/
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math
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Here. are some discussions about the Lines and Planes so Read the Context given below:
Unlike a plane, a line in three dimensions does have an obvious direction, namely, the direction of any vector parallel to it. A line can be defined and uniquely identified by providing one point on the line and a vector parallel to the line (in one of two possible directions) that is, the line consists of exactly those points we can reach by starting at the point and going for some distance in the direction of the vector. Let’s see how we can translate this into more mathematical language.
Lines and planes are probably the easiest curves and places in a three-dimensional area and they will seem important as we seek to understand curves and more complex areas.
The number of lines in two dimensions is ax + by = c; it is reasonable to expect that a line in three dimensions is given an ax + by + cz = d; it makes sense, but it’s not right – it turns out that this is the plane number.
The plane does not have a clear “direction” as does the line It is possible that the plane is aligned with the direction in a very useful way, however: there are two direct directions to the aircraft. Any vector with one of these two indicators is said to be normal in a plane.
Suppose two points (v1, v2, v3) and (w1, w2, w3) are in the plane; then the vector ⟨w1 – v1, w2 – v2, w3 – v3⟩ is like a plane; especially, when this vector is placed with its tail in (v1, v2, v3) its head is in (w1, w2, w3) and lies on the plane. As a result, any vector relative to the plane corresponds to ⟨w1 – v1, w2 – v2, w3 – v3⟩. It is easy to see that an airplane contains those points (w1, w2, w3) where w1 – v1, w2 – v2, w3 – v3⟩ corresponds to the average in aircraft, as shown in Figure 12.5.1.That is, suppose we know that ⟨a, b, c⟩ is common in a plane containing a point (v1, v2, v3).Then (x, y, z) is in the plane when and when Ophelia, b, c⟩ corresponds to ⟨x – v1, y – v2, z – v3⟩. Next, we know that this is especially true when ⟨a, b, c⟩⋅⟨x – v1, y – v2, z – v3⟩ = 0. So, (x, y, z) you are on a plane if and only if ⟨A, b, c⟩⋅⟨x – v1, y – v2, z – v3⟩a (x – v1) + b (y – v2) + c (z – v3) ax + ngo + cz – av1 – bv2 −cv3ax + by + cz = 0 = 0 = 0 = av1 + bv2 + cv3. To work backwards, note that if (x, y, z) is a point that satisfies the ax + by + cz = d then ax + ngu + czax + ngu + cz – da (x – d / a) + b (y – 0) + c (z – 0) ⟨a, b, c⟩⋅⟨x – d / a, y, z ⟩ = D = 0 = 0 = 0.
That is, ⟨a, b, c⟩ corresponds to the verb with the tail in (d / a, 0,0) and the head in (x, y, z). This means that points (x, y, z) satisfy the equation ax + in + cz = d forming a plane in ngokuyaa, b, c⟩. (This does not apply if = 0, but in that case we can use b or c in the paragraph of. That is, it can be (x – 0) + b (y – d / b) + c (z −0) = 0 or a (x – 0) + b (y – 0) + c (z – d / c) = 0.)
Figure 12.5.1. Vehicle defined by vectors according to standard.
Therefore, when given the vector ⟨a, b, c⟩ we know that all planes corresponding to this vector have ax + form + by + cz = d, and any part of this form is a plane pointing to ⟨a, b, c ⟩.
In single variable calculus, we find that a unique function is intrinsically linear. In other words, if we zoom in on the map of the variance function at some point, the map looks like a tangent to the function at that point. Line functions play an important role in single variable calculus, which is useful in accessing variance functions and roots of functions (see Newton’s method) and in arriving at first-order solutions of variance equations (see Euler’s method). In Miscellaneous Calculus, we’re going to study curves in space; Curves also differ in the form of a local line. Furthermore, when we analyze the functions of the two variables, we can see that if the surface defined by the function looks like an enlarged plane (the touch plane), then such a function is analogous to a local.
Consequently, we need to understand both lines and planes in space because it also defines linear functions. (Note that if a function is a polyhedron, all its terms are less than or equal to the numerator. For example, this refers to a single variable linear function and a two-variable linear function, but not a linear one because both In, the sum of the failures of its factors.) We begin our work from scratch with familiar ideas.
The intersection of two planes
If two planes intersect, the junction is always dashed the vector equation for the line of intersection is given
r = r_0 + tvr = r0+ TV
where r_0r 0
A point on the line and vv is the result of the transverse vector product of the normal vectors of the two planes.
The parameter equations for the line of intersection are given
x = ax = a, y = by = b, and z = cz = c
The vector equations aa, bb and cc are r = a\bold i + b\bold j + c\bold kr = ai + bj + ck.
Finding the parameter equations representing the line of intersection of two planes
In Parameters, an example of the problem of how to find the fort where two planes meet
Find the parameter equation for the line of intersection of the planes.
2x + y-z = 32x + y-z = 3
x-y + z = 3x-y + z = 3
We have to find the vector equation of the line of intersection to achieve this, we first need to find the cross-product vv of the normal vectors of a given plane.
normal vectors for planes
2x + y -z = 32x + y -z = 3 For the plane, the normal vector \ langle2,1, -1 \ rangelea⟨2,1, −1⟩
x -y + z = 3x -y + z = 3 For the plane, the normal vector b \langle1, -1,1 \rangleb⟨1, −1,1⟩
The cross product of the normal vectors is we need a point at the intersection to achieve this, we will use the given plane equations as the system of linear equations. If we set z = 0z = 0 in both the equations, we get
2x + y-z = 32x + y-z = 3
2x + y-0 = 32x + y-0 = 3
2x + y = 32x + y = 3
x-y + z = 3x-y + z = 3
x-y + 0 = 3x-y + 0 = 3
X-Y = 3X-Y = 3
Intersection parameter line of two planes. Look at the line of intersection, find a point in the first line, and find the cross product of the normal vectors Now we will put the equations together.
(2x + x) + (y – y) = 3 + 3 (2x + x) + (y – y) = 3 + 3
3x + 0 = 63x + 0 = 6
x = 2x = 2
If we put x = 2x = 2 again in x-y = 3x-y = 3, we get
2-y = 32-y = 3
-y = 1 -y = 1
y = -1y = -1
Combine these values and the point on the line of intersection
(2, -1,0) (2, 1,0)
r_0=2\bold i-\bold j+0\bold kr
0= 2i – j + 0k
r_0 = \ lang 2, -1,0 \ rangler
0= 2, −1,0⟩
Now we have vv and r_0r. can mix
0 In the vector equation.
r = r_0 + tvr = r
r = (2\bold i-\bold j+0\bold k) + t (0\bold i-3\bold j-3\bold k) r = (2i-j + 0k) + t(0i-3j -3k)
r=2\bold i-\bold j+0\bold k+0\bold it-3\bold jt-3\bold ktr = 2i-j+0k+0it-3jt-3kt
r=2\bold i-\bold j-3\bold jt-3\bold ktr=2i-j-3jt-3kt
r = (2) \ bolus i + (-1-3t) \ bolus j + (-3t) \bold kr = (2) i + (-1−3t) j + (-3t) k
Using the vector equation for the hand cutting line, the parameter equations for both lines can be found. In our vector equation r = a \ turn i + b \ bold j + c \ bold kr = ai + bj + ck matches r = (2) \ turn i + (-1-3t) \bold j + (– 3t) ) \bold kr = (2) i + (- 1−3t) j + (- 3t) k, we can say that
a = 2a = 2
b = -1-3tb = -1−3t
c = -3tc = -3t
Therefore, the parameter equation for the line of intersection
x = 2x = 2
y = -1-3ty = -1−3t
z = -3tz = -3t
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http://vkportal.ba/relative-dating-and-absolute-dating/
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As follows http://vkportal.ba/ 1. They find. This is a system of material that they use 2 methods stratigraphic, that they find. What is. Other items considered to determine the age. Some scientists prefer the rocks they leave behind, while radiometric dating is older than another. E. Geologists often were the rocks they use 2 methods. Here is a sequence. A specific time order. Scientists use absolute. In contrast with relative percentage of the historical remains in years. E. To give rocks becomes one example where absolute dating: matches and more dating?
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https://www.scribd.com/document/76722526/Yen-Lin-Chen-Ergun-Akleman-Jianer-Chen-and-Qing-Xing-Designing-Biaxial-Textile-Weaving-Patterns
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Yen-Lin Chen, Ergun Akleman, Jianer Chen and Qing Xing Texas A&M University, College Station, Texas 77843 May 27, 2010 Abstract
Most textile weaving structures are biaxial, which consist of row and column strands that are called warp and weft, in which weft (filling) strands pass over and under warp strands. Widely different patterns can be obtained by playing with ups and downs in each row. In textile weaving, there mainly exist three fundamental types of weaving patterns: plain, satin and twill. We observe that all possible versions of the three fundamental types can be represented by a notation that is suitable for user interface development. To obtain this simple notation, we first view weaving structures as a set of rows with horizontal periodic patterns that are cyclically shifted and vertically stacked over each other. We then show that it is possible to generalize this notation to represent a wider variety of weaving structures. In our general notation, the initial weaving pattern in the first row is represented by a row of binary numbers and the shift operation is represented by a column of integers. This set of integers provides us with a very simple notation to represent any biaxial textile weaving structure. Based on this notation, we develop a java system that can let users design their own textile weaving structures with a simple user interface. Using the system it is easy to understand the patterns produced by different weaving structures.
Introduction and Motivation
It has recently been shown that any arbitrary twist of the edges of an extended graph rotation system induces a cyclic weaving on the corresponding surface [2, 1]. This theoretical result allows us to study generalized versions of widely used weaving structures as cyclic weaving structures on arbitrary surfaces. To generalize textile weaving to cyclic weaving on surfaces, we first need to understand and classify existing textile weaving. This paper provides a basic and general framework for analyzing, understanding and classification of existing and possible textile weaving structures. Most textile weaving structures are biaxial. Biaxial weaves are 2-way, in which the strands run in two directions at right angles to each otherwarp or vertical and weft or horizontal, and 2-fold, in which there are never more than two strands crossing each other. In other words, most textile weaves consists of row and column strands that are called warp and weft, in which weft (filling) strands pass over and under warp strands. The popularity of biaxial weaving comes from the fact that the textile weaving structures are usually manufactured using loom devices by interlacing of two sets of strands at right angles to each other. Since the strands are at right angles to each other, they form rows and columns. The names weft and warp are not arbitrary in practice. In the loom device, the weft (also called filling) strands are considered the ones that go under and over warp strands to create a fabric. The basic purpose of any loom device is to hold the warp strands under tension such that weft strands can be interwoven. Using this basic framework, it is possible to manufacture a wide variety of weaving structures by raising and lowering different warp strands (or in other words by playing with ups and downs in each row) such as the ones shown in Figure 1. The last pattern in the figure is an unnamed weaving structure in addition to three fundamental biaxial textile weaving structures. In this paper, we are interested in the mathematical representation and construction of repeating patterns that can be obtained by this manufacturing process such that we can precisely name any possible biaxial textile weaving using a simple notation. We start with a basic framework for the characterization of three
widely used textile structures: plain, twill and satin. Based on this characterization, we provide a general framework to obtain a wider variety of biaxial textile structures. We have developed a software that allows to create these patterns using a very simple interface.
(c) Another Twill
Figure 1: Examples of textile weaving patterns.
The mathematics behind such 2-way, 2-fold woven fabrics such as plain, satin and twill, are first formally investigated by Grunbaum and Shephard . They also coined the word, isonemal fabrics to describe fabrics that have a transitive symmetry group on the strands of the fabric . One of the important concepts introduced by Grunbaum and Shephard is hang-together or fall-apart. They pointed out that weaving patterns that look perfectly reasonable may not produce interlacements that hang-together. In such patterns some interlaced warp and weft threads may not be interlaced with the rest of the fabric if the fabric is woven. Such a fabric would come apart in pieces. Determining whether or not a pattern is hang-together or fall-apart cannot be identified by simple visual inspection of the pattern. Several procedures has been developed to identify the hang-together property of a weaving pattern. If a weaving pattern is hanging-together, it is then called a fabric. Grunbaum and Shephard provided images of all non-twill isonemal fabrics (i.e. hanging-together weaves) of periods up to 17 [11, 12]. Their papers were followed by several papers giving algorithms for determining whether or not a fabric hangs together [9, 3, 4, 6, 5]. Roth , Thomas [20, 19, 18] and Zelinka [23, 24] and Griswold [7, 8] theoretically and practically investigated symmetry and other properties of isonemal fabrics. In this paper, we are particularly interested in a certain isonemal fabrics that are called genus-1 by Grunbaum and Shephard . Genus-1 means that each row with length n is obtained from the row above it by a shift of s units to the right, for some fixed value of the parameter s. They call these fabrics (n, s)-fabrics where n is the period of the fabric and n and s’s are relatively prime. The genus-1 family of fabrics includes plain, satins and twills. If s = 1 the fabric is a twill and s2 = 1modn correspond to twills for the fabrics that contain exactly one black square in every row. In Grunbaum and Shephard’s notation plain weaving in Figure 1(a) is called (2, 1)-fabric, twill weaves in Figure 1(b) and (c) are called (4, 1) and (5, 4)-fabrics respectively, satin weaving in Figure 1(d) is called (8, 3)-fabric and finally the unnamed weave in Figure 1(e) now has a name and is called (13, 4)-fabric. Genus-1 fabrics not only include satins and twills but also they are the only isonamal fabrics that exist for every integer n. The other isonemal fabrics do not exist for odd n’s . Our approach for representing genus-1 fabrics is based on a notation that is suitable for the development of a simple user interface.
To analyze biaxial weaving structures we first need to have a simple mathematical notation that captures the essence of the three fundamental textile structures. We have observed that all possible versions of the three fundamental types can be classified as genus-1 and they can be viewed as a set of rows with horizontal periodical patterns that are shifted and vertically stacked over each other. Grunbaum and Shephard’s (n, s) notation ignores the initial row pattern. In this work, we first express the initial pattern by two integers a and b, where a is the number of up-crossings, and b is the number of down-crossings. An additional integer c denotes the shift introduced in adjacent rows. Any such weaving pattern can be expressed by a triple a/b/c. Note that a + b correspond to n and c corresponds to s in Grunbaum and Shephard’s (n, s) notation. The Figure 2(a) shows a basic block of a biaxial weaving structure and the role of these three integers, a > 0, b > 0 and c. The value of c is any integer, but, a/b/c and a/b/c + k(a + b) are equivalent since c = cmod(a + b). Therefore, unless stated otherwise we assume that the value of the c is between two numbers as − (a + b − 1)/2 ≤ c ≤ (a + b)/2
where x is the largest integer less than or equal to x. Figure 2(b) shows four examples of biaxial weaving structures that can be described by the a/b/c notation. Note that now we can name each weaving structure uniquely. The a/b/c notation can also be used to characterize all three fundamental weaving structures.
(a) Basic block of 6/7/4 weaving pattern.
(b) Examples of industrial weaving patterns.
Twill: 3/2/ − 1
Figure 2: Examples of weaving patterns that can be given by the a/b/c notation.
Plain and Twill Weaving
The most basic and simplest one of all biaxial weaving structures is plain weaving, which corresponds to alternating links. Plain weaving is very strong and therefore used in a wide variety of fabrics from fashion and furnishing to tapestry. In plain weaving, each weft strand crosses the warp strands by going over one, then under the next, and so on. The next weft strand goes under the warp strands that its neighbor goes over, and vice versa. Plain weaving creates a checkerboard pattern as shown in Figure 1(a). Twill weaving is widely used in fabrics such as denim or gabardine. Two structures that are classified as twill are shown in Figure 1(b) and (c). Since twill weaving uses less number of crossings, the strands in twill woven fabrics can move more freely than the strands in plain woven fabrics. This property makes twill weaving softer, more pliable, and better draped than plain weaving. Twill fabrics also recover better
from wrinkles than plain woven fabrics. Moreover, strands in twill weaving can be packed closer since there are less number of crossings. This property makes the twill woven fabrics more durable and water-resistant, which is the reason twill fabrics are often used for sturdy work clothing or durable upholstery.
(b) 2/2/ − 1
(d) 1/3/ − 1
Figure 3: Two length 4 biaxial twill examples. Note that 2/2/1 and 2/2/ − 1 are the same weaving that is rotated, so are 3/1/1 and 1/3/ − 1. To illustrate rotation, in (a) and (c), weft strands are painted yellow. On the other hand, in (b) and (d), weft strands are painted blue. In twill weaving, the weft strands pass over and under two or more warp strands to give an appearance of diagonal lines. The numbers of over-passes and under-passes create different twills. In the a/b/c notation, a is the number of over-passes and b is the number of under-passes. In twill weaving, each row is offset by c = 1 or c = −1. This offset creates a characteristic diagonal pattern that makes the weaving visually appealing. The sign of the offset defines the orientation of the twill. In other words, b/a/ − 1 is 900 rotated version of a/b/1. For instance, 2/2/1 and 2/2/ − 1 in Figure 3 define exactly the same twill structures that are 900 rotated versions of each other. Similarly, 3/1/1 and 1/3/ − 1 define exactly the same twill weaving structures. This is not unexpected since one up in weft must correspond to one down in warp and one down in weft must correspond to one up in warp. Thus, the total number of ups and the total number of downs of all wefts and warps should be equal. Note that in plain or twill weaving the offset numbers c other than 1 or −1 are not used. On the other hand, the c values other than 1 or −1 are needed to be used for satin weaving. But, before discussing satin weaving, we first discuss a/b/c for general cases.
Weave Equivalency and Simplification
The notation a/b/c does not uniquely represent a weaving structure. We have already seen that a/b/c and a/b/c + k(a + b) are equivalent for any integer k and we, therefore, chose − (a + b − 1)/2 ≤ c ≤ (a + b)/2 . Even after removing these cases, the two weaving structures given by a/b/c notation can still be equivalent. We say two weaving structures given as a1 /b1 /c1 and a2 /b2 /c2 are equivalent if a1 gcd(a1 , b1 , |c1 |) b1 gcd(a1 , b1 , |c1 |) c1 gcd(a1 , b1 , |c1 |) = = = a2 , gcd(a2 , b2 , |c2 |) b2 , gcd(a2 , b2 , |c2 |) c2 gcd(a2 , b2 , |c2 |)
where gcd(ai , bi , |ci |) is the greatest common divisor of ai , bi and |ci |. For instance, 2/2/2, 3/3/3, 4/4/4 and 5/5/5 weaving patterns shown in Figure 4 are all equivalent to plain weaving i.e. 1/1/1. Our reason behind
this equivalency comes from the fact that these weaving structures can simply be woven as plain weaving by using warp strands that are wider than weft strands. Similarly, 2/4/2 weaving structure shown in Figure 4 is equivalent to 1/2/1, 3/6/3 or 4/8/4. The structure of the weaving, therefore, can be uniquely described and represented as a b c / / gcd(a, b, |c|) gcd(a, b, |c|) gcd(a, b, |c|) Therefore, 2/4/2 or 3/6/3 can simply be represented by 1/2/1 and similarly 2/2/2, 3/3/3, 4/4/4 and 5/5/5 can be represented by 1/1/1.
2/2/2 ≡ 1/1/1
3/3/3 ≡ 1/1/1
4/4/4 ≡ 1/1/1
5/5/5 ≡ 1/1/1
2/4/2 ≡ 1/2/1
Figure 4: Examples of weaving structures that can be simplified by a simpler weaving pattern.
Figure 5: Examples of non-weaving structures for a + b = 4.
Guaranteeing Hang-Togetherness by Avoiding Falling Apart
The notation a/b/c can represent a non-fabric that can fall-apart. However, unlike a general isonemal weaving case, it is easy to avoid the fabric fall-apart. A genus-1 weave pattern, such as the one given by the notation a/b/c, is not a fabric if and only if any row or column has no alternating crossing. In other words, for genus-1 weaving there is no other case of a weaving pattern that will not hang-together. These cases are also very simple and easy to detect by simple visual inspection of the pattern. It is easy to prove this property of genus-1 weaving based on the algorithm presented in . Such non-fabrics can easily be avoided in rows by choosing the value of b non zero as we have stated earlier. If b = 0 we can get the non-fabrics shown in Figure 5(a). To avoid occurrence of non-fabric in columns, similarly, a must be non-zero, otherwise we can get the structures shown in Figure 5(b). However, these two conditions alone are not sufficient to avoid non-fabric problems in columns since for many combinations of n = a + b and c, non-weaving structures can be produced as shown in Figures 5(c) and 6(a), (c), (d), (e), (h) and (i). A common property of these non-weaving structures is that n = a + b and c are not relatively prime (i.e gcd(n, |c|) = 1). In other words, if the greatest common divider of n and c is not 1, a non-weaving structure
(a) non-weave: 1/5/0
(b) weave: 1/5/1
(c) non-weave: 1/5/2
(d) non-weave: 1/5/3
(e) non-weave: 2/4/0
(f) weave: 2/4/1
(g) weave: 2/4/2
(h) non-weave: 2/4/3
(i) non-weave: 3/3/0
(j) weave: 3/3/1
(k) weave: 3/3/2
(l) weave: 3/3/3
Figure 6: Examples of weaving and non-weaving structures for a + b = 6. The trivial cases a = 0 and b = 0 are not included. Rotated versions that correspond to c = −2 and c = −1 are also not included. The 3/3/2 is an example that result in a weaving structure although gcd(6, 2) = 2 = 1. can be produced even with a = 0. For instance, in Figure 5(c), n = 4 and c = 2, and gcd(n, |c|) = gcd(4, 2) = 2 not 1. Similarly, in Figure6(c), (d), and (h) neither gcd(6, 2) = 2 nor gcd(6, 3) = 3 is 1. Note that for 2/2/2 or 3/3/3, although gcd(n, |c|) = 1, for their representative weaving 1/1/1, gcd(n, |c|) = 1. Similarly, for 2/4/2, gcd(n, |c|) = 1, but for its representative weaving 1/2/1, gcd(n, c) = 1. Moreover, 3/3/2 shown in Figure 6(k) is also acceptable although gcd(n, |c|) = 1. This gives us the last condition: if min(a, b) ≥ gcd(n, |c|) even if gcd(n, |c|) = 1 we can have weaving. As a result, a structure defined by an a/b/c notation is a non-fabric if a = 0 or b = 0 or (gcd(n, |c|) = 1) and (min(a, b) < gcd(n, |c|)) . Note that Grunbaum and Shephard enforces the third condition by choosing g and s relatively prime. The next section introduces the third important textile weaving called satin.
Satin weaving is distinguished from plain and twill weaving by its silky appearance, which is obtained by creating long rows of parallel weft strands. To create long parallel weft strands, the goal is to minimize the number of crossings where warp is up and maximize the distances between such crossings. Minimizing the number of warp-up crossings reduces the amount of scattering that is caused by alternating crossing. As a result, satin structure allows strong light reflection by creating a silky appearance and feeling. Based on this description, we can state that in each row there should be no more than 1 crossings where warp is up. Therefore, a satin weaving must always be in the form of a/1/c. The question is to define c
in such a way that the distance between warp-up positions are maximized. Like Grunbaum and Shephard , we have identified that the best solution is c2 = ±1mod(a + 1). Figure 7 shows some examples of satin weaving.
Figure 7: Examples of satin weaving structures. Note almost hexagonal shapes appear in 7/1/3 and 23/1/5. The missing 8/1/3 and 24/1/5 are excluded since they are non-weaving structures.
Generalized Notation and Implementation
The a/b/c notation can represent all three fundamental weaving structures. However, it is possible to improve the notation to represent a larger variety of weaving structures. We observe that there are two shortcomings of a/b/c notation. In this section, we discuss how to overcome these shortcomings with an improved notation for software development. The first shortcoming is that the up and down sequence in a/b/c notation must always be in the form of a-ups and b-downs. The sequences such as 2-up, 2-down, 1-up and 1-down, which can give interesting weaving structures, cannot be expressed with this notation. It is easy to overcome this shortcoming by representing initial sequence with a simple binary number where 1 corresponds to up and 0 corresponds to down. So, a = 2 and b = 3 can turn to a binary number (11000), which is 24 in decimal system. Another way to represent the initial sequence is using sets, for instance, a sequence of 2-up, 2-down, 1-up and 1-down can be given (2, 2, 1, 1). Either notation can successfully be used in software development. The second shortcoming of a/b/c notation is that we cannot use all possible offsets for each row. It is better to use a notation that gives shifts in each row as an ordered set of integers between 0 and n − 1 where n is the length of the initial row, which corresponds to a + b in a/b/c notation. Thus, offsets can be given as (k0 , k1 , k2 , . . . , kn−1 ), where ki is the shift in row i and always k0 = 0. It is always possible to replace this notation with c. For instance if a + b = 5, in this case we only have 4 possible cases for c = 1, 2, 3 and 4. c = 1 ⇒ (0, 1, 2, 3, 4) c = 2 ⇒ (0, 2, 4, 1, 3) c = 3 ⇒ (0, 3, 1, 4, 2) c = 4 ⇒ (0, 4, 3, 2, 1)
However, the opposite is not correct. For instance, (0, 1, 3, 2, 4) does not correspond to any c value and it can correspond to some interesting weaving structures as shown in Figure 8. It is also interesting to note that shifts in rows corresponds to permutations in columns. In terms of software development shifts are more suitable since they provide an easy-to-use interface. The new notation is also useful since we can use the
same shifts in different rows. By choosing same shifts in different rows, we can obtain weaving structures such as equivalents of 1/1/1 as shown in Figure 4(a) and (b).
a. (10000)(0, 1, 3, 2, 4)
b. (10100)(0, 1, 3, 2, 4)
c. (10000)(0, 1, 2, 4, 3)
d. (11000)(0, 1, 2, 4, 3)
Figure 8: Examples of weaving structures that can be obtained with generalized notation. Note that the results are not unique: The weaving structures in a. and c. are the same We have developed a java program to construct weaving structures. All weaving structures in the paper are screen-captures from the program. To construct a weaving structure, users design an initial pattern by clicking the squares that correspond to crossings. Each click works as a toggle operation by changing an up to a down or vice versa. The shifts are also handled by clicking squares in a column. Each click will shift the corresponding row by one. The java program is available in http://www.viz.tamu.edu/faculty/ergun/research/weaving/WeavingPattern/index.html.
a. (1100)(0, 0, 2, 2)
b. (111000)(0, 0, 0, 3, 3, 3)
c. (111000)(0, 2, 4, 0, 2, 4)
Figure 9: The weaving structures in (a) and (b) are equivalent to 1/1/1. The last one is 3/3/2.
In this paper, we have introduced two mathematical notations for the representation of repeating patterns that can be obtained by the manufacturing process using loom devices. Based on the notation, we have developed a simple user interface to such genus-1 weaving patterns. We show that it is easy for users to guarantee that the pattern will hang-together with visual inspection. We are planning to extend the java software to include isonemal fabrics from other genera and universal fabrics . Hoskins and Thomas , and Windeknecht and Windeknecht , and Roth showed that by applying alternating colors to weft or warp threads, the number of possible patterns expand enormously. We are also planning to allow users to paint warps and wefts with periodic patterns such as thin or thick striping described by Roth .
E Akleman, J. Chen, J. Gross, and Q. Xing. Extended graph rotation systems as a model for cyclic weaving on orientable surfaces. Technical Report - Computer Science Department, Texas A&M University - TR09-203, 2009. E Akleman, J. Chen, Q. Xing, and J. Gross. Cyclic plain-weaving with extended graph rotation systems. ACM Transactions on Graphics; Proceedings of SIGGRAPH’2009, pages 78.1–78.8, 2009. C.R.J. Clapham. When a fabric hangs together. Bulletin of the London Mathematics Society, 12:161– 164, 1980. C.R.J. Clapham. The bipartite tournament associated with a fabric. Discrete Mathematics, 57:195–197, 1985. C. Delaney. When a fabric hangs together. Ars Combinatoria, 15:71–70, 1984. T. Enns. An efficient algorithm determining when a fabric hangs together. Geometriae Dedicata, 15:259–260, 1984. R. E. Griswold. Color complementation, part 1: Color-alternate weaves. Web Technical Report, Computer Science Department, University of Arizona, 2004. R. E. Griswold. From drawdown to draft a programmers view. Web Technical Report, Computer Science Department, University of Arizona, 2004. R. E. Griswold. When a fabric hangs together (or doesnt). Web Technical Report, Computer Science Department, University of Arizona, 2004. B. Grunbaum and G. Shephard. Satins and twills: an introduction to the geometry of fabrics. Mathematics Magazine, 53:139–161, 1980. B. Grunbaum and G. Shephard. A catalogue of isonemal fabrics. Annals of the New York Academy of Sciences, 440:279–298, 1985. B. Grunbaum and G. Shephard. An extension to the catalogue of isonemal fabrics. Discrete Mathematics, 60:155–192, 1986. B. Grunbaum and G. Shephard. Isonemal fabrics. American Mathematical Monthly, 95:5–30, 1988. J.A. Hoskinks and R.S.D. Thomas. The patterns of the isonemal two-colour two-way two-fold fabrics. Bulletin of the Australian Mathematical Society, 44:33–43, 1991. S. Oates-Williams and A. P. Street. Universal fabrics. Lecture Notes in Mathematics, Springer Berlin / Heidelberg, 884:355–359, 1981. R. L. Roth. The symmetry groups of periodic isonemal fabrics. Geometriae Dedicata, Springer Netherlands, 48(2):191–210, 1993. R. L. Roth. Perfect colorings of isonemal fabrics using two colors. Geometriae Dedicata, Springer Netherlands, 56:307–326, 1995. R.S.D. Thomas. Isonemal prefabrics with only parallel axes of symmetry. Discrete Mathematics, 309(9):2696–2711, 2009. R.S.D. Thomas. Isonemal prefabrics with perpendicular axes of symmetry. Discrete Mathematics, 309(9):2696–2711, 2009. R.S.D. Thomas. Isonemal prefabrics with no axis of symmetry. Discrete Mathematics, 310:1307–1324, 2010. M. B. Windeknecht. Color-and weave ii. Self Published, 1994.
M. B. Windeknecht and T. G. Windeknecht. Microcomputer graphics and color and wedge effect in handweaving. ACM Southeast Regional Conference archive, Proceedings of the 18th annual Southeast regional conference, 18:174–179, 1980. B. Zelinka. Isonemality and mononemality of woven fabrics. Applications of Mathematics, 3:194198, 1983. B. Zelinka. Symmetries of woven fabrics. Applications of Mathematics, 29(1):14–22, 1984.
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s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257830066.95/warc/CC-MAIN-20160723071030-00223-ip-10-185-27-174.ec2.internal.warc.gz
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CC-MAIN-2016-30
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https://paperforpay.com/describe-any-three-self-oriented-values-that-companies-can-use-to-encourage-customers-to-support-green-market/
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math
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Describe any three self oriented values that companies can use to encourage customers to support green market?
Break-Even Sales Under Present and Proposed Conditions
Battonkill Company, operating at full capacity, sold 125,900 units at a price of $117 per unit during 2014. Its income statement for 2014 is as follows:
|Cost of goods sold||5,226,000|
|Income from operations||$5,331,300|
The division of costs between fixed and variable is as follows:
|Cost of goods sold||40%||60%|
Management is considering a plant expansion program that will permit an increase of $1,287,000 in yearly sales. The expansion will increase fixed costs by $171,600, but will not affect the relationship between sales and variable costs.
1. Determine for 2014 the total fixed costs and the total variable costs.
|Total fixed costs||$|
|Total variable costs||$|
2. Determine for 2014 (a) the unit variable cost and (b) the unit contribution margin.
|Unit variable cost||$|
|Unit contribution margin||$|
3. Compute the break-even sales (units) for 2014.
4. Compute the break-even sales (units) under the proposed program.
5. Determine the amount of sales (units) that would be necessary under the proposed program to realize the $5,331,300 of income from operations that was earned in 2014.
6. Determine the maximum income from operations possible with the expanded plant.
7. If the proposal is accepted and sales remain at the 2014 level, what will the income or loss from operations be for 2015?
8. Based on the data given, would you recommend accepting the proposal?
In favor of the proposal because of the reduction in break-even point.
In favor of the proposal because of the possibility of increasing income from operations.
In favor of the proposal because of the increase in break-even point.
Reject the proposal because if future sales remain at the 2014 level, the income from operations of will increase.
Reject the proposal because the sales necessary to maintain the current income from operations would be below 2014 sales.
Choose the correct answer.
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s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335365.63/warc/CC-MAIN-20220929194230-20220929224230-00122.warc.gz
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CC-MAIN-2022-40
| 2,317 | 32 |
https://ghdhairstraighteners-inc.com/qa/how-many-steps-is-a-meter.html
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math
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- How many steps is 700 meters?
- How many steps are in 25 meters?
- How long does it take to walk 100 Metres?
- How many calories does 3000 steps Burn?
- How far is 15 meters in steps?
- How do I calculate my step distance?
- How many feet is 100 meters?
- How many paces is a meter?
- Is one step a meter?
- How do you count 100 meters?
- How many miles is 10000 steps?
- How many steps does it take to run 100 meters?
- How many steps are in 10 meters?
- How many Metres is 5000 steps?
- What is 10 meters called?
How many steps is 700 meters?
meters to steps Conversion Chart Near 690 metersmeters to steps of690 meters=905.5 (905 1/2 ) steps700 meters=918.6 (918 5/8 ) steps710 meters=931.8 (931 3/4 ) steps720 meters=944.9 (944 7/8 ) steps9 more rows.
How many steps are in 25 meters?
meters to steps Conversion Chart Near 19 metersmeters to steps of23 meters=30.18 (30 1/8 ) steps24 meters=31.5 (31 1/2 ) steps25 meters=32.81 (32 3/4 ) steps26 meters=34.12 (34 1/8 ) steps9 more rows
How long does it take to walk 100 Metres?
You are going to take about 30 seconds to brisk walk 100 meters,plus 1 or 2 seconds as you inevitably slow down a little. 100 meters in 31 seconds.
How many calories does 3000 steps Burn?
Convert Your Steps to Calories2,000 Steps per Mile (Height 6 Feet and Above) Calories Burned by Step Count and Weight1,00028 cal.602,000551203,000831804,00011024018 more rows
How far is 15 meters in steps?
meters to steps Conversion Chart Near 9 metersmeters to steps of12 meters=15.75 (15 3/4 ) steps13 meters=17.06 (17) steps14 meters=18.37 (18 3/8 ) steps15 meters=19.69 (19 5/8 ) steps9 more rows
How do I calculate my step distance?
Divide the number of feet in your measured distance by the number of steps you took from the first mark to the second. Distance in feet/number of steps = step length. For example, if it took you 16 steps to cover 20 feet, your step length would be 1.25 feet (15 inches).
How many feet is 100 meters?
328 feet1 meter is 3.28 feet, so 100 meters is 100\times 3.28=328 feet.
How many paces is a meter?
Please share if you found this tool useful:Conversions Table1 Paces to Meters = 0.76270 Paces to Meters = 53.342 Paces to Meters = 1.52480 Paces to Meters = 60.963 Paces to Meters = 2.28690 Paces to Meters = 68.584 Paces to Meters = 3.048100 Paces to Meters = 76.211 more rows
Is one step a meter?
A step was a Roman unit of length. A step is equal to 2½ Roman feet (pedes) or ½ Roman pace (passus). One step is approximately equal to 0.81 yards or 0.74 meters as per standardization under Agrippa. Metre (m) or Meter is a unit of length in the metric system.
How do you count 100 meters?
Measure walked distance and divide this by steps you took. You will get your pace of walking in units of linear measurement ie., feet or metres. Normal pace of walking is 50–65 cm divide 100 by yours worked out to get the result in numbers. Walk the calculated number of steps now to know distance as 100 metres.
How many miles is 10000 steps?
5 milesAn average person has a stride length of approximately 2.1 to 2.5 feet. That means that it takes over 2,000 steps to walk one mile and 10,000 steps would be almost 5 miles.
How many steps does it take to run 100 meters?
Most sprinters will have a step frequency between 3 and 5 during their races. Example. If a sprinter’s average stride length is exactly 2.0 meters, it will take exactly 50 steps to complete the 100m. To complete 50 steps in 10.0 seconds, the sprinter will have to average 5 steps per second.
How many steps are in 10 meters?
Please share if you found this tool useful:Conversions Table9 Meters to Steps = 11.811600 Meters to Steps = 787.401610 Meters to Steps = 13.1234800 Meters to Steps = 1049.868820 Meters to Steps = 26.2467900 Meters to Steps = 1181.102430 Meters to Steps = 39.37011,000 Meters to Steps = 1312.33611 more rows
How many Metres is 5000 steps?
3810 meterssteps to meters Conversion Chart Near 4400 stepssteps to meters of5000 steps=3810 meters5100 steps=3886 meters5200 steps=3962 meters5300 steps=4039 meters9 more rows
What is 10 meters called?
dekameterUnits larger than a meter have Greek prefixes: Deka- means 10; a dekameter is 10 meters. Hecto- means 100; a hectometer is 100 meters. Kilo- means 1,000; a kilometer is 1,000 meters.
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CC-MAIN-2021-10
| 4,285 | 45 |
https://stage.geogebra.org/m/UY5FF7qe
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math
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Angle Measure in Degrees
- Albert Navetta
Degrees vs Radians
Most readers will be familiar with angles expressed in degrees. As there are radians in a circle, there are 360 degrees. We denote 360 degrees with the shorthand . If an angle is expressed without the degree symbol (), then the angle is in radians. For example, means angle theta is equal to 20 degrees, but means angle alpha is equal to 20 radians. Note that the angle for half of the unit circle is and also rad, so we use the fractions to convert between the different units of angle measure. The technique is known as unit analysis and is useful in all sciences. EX: Convert to radians.
rad.EX: Convert rad to degrees.
Sub Units: Degrees, Minutes, Seconds
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CC-MAIN-2023-50
| 720 | 6 |
https://artofproblemsolving.com/wiki/index.php?title=2021_JMPSC_Invitationals_Problems/Problem_1&oldid=157949
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math
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2021 JMPSC Invitationals Problems/Problem 1
The equation where is some constant, has as a solution. What is the other solution?
Since must be a solution, must be true. Therefore, . We plug this back in to the original quadratic to get . We can solve this quadratic to get . We are asked to find the 2nd solution so our answer is
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CC-MAIN-2021-39
| 328 | 3 |
http://discoveryandwonder.com/
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math
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June 13, 2015 6 Comments
Mathematics is not often thought of in regards to healing. Instead, we think of meditation, absorbing the wonders of nature, relaxation, living in the moment, letting go, listening to soothing music. However, lately mathematics has been a source of healing. The following example will show you how.
Recently, my Dad’s 93 year-old brain has been on the on and off mode. Sometimes he cannot distinguish between dream and reality and acts with extreme apprehension concerning the circumstances of his dreams. The more practical side of his mind is obsessed with financial transactions, often spending hours trying to solve problems that seem impossible, but are really quite doable. It dawned on me that a simple algebra problem might gather Dad’s thoughts in a tighter and more purposeful direction. I might say that mathematics seems to me like a demanding yet supportive parent. Like a parent, there are rules that must be obeyed and an order to be preserved. However, there is also something truly soothing about logical boundaries, and a clear set of rules, which, if followed to the letter, lead to future paths of discovery. Parts of the mind are held in check, but others are ever expanding, testing, and exploring. So too, an effective parent guides the child through progressive steps toward exploration. With these thoughts in mind, I gave Dad the following problem: Solve for x: (x)(x)-7=9. Dad simplified the equation to (x)(x)=16. Then he began to test numbers such as 2, 4, 6, 8, to see which ones might work. He also talked about square roots. After about ten minutes, he came up with the answer 4. When I told him that was only half the solution, he was nonplussed. It took him awhile(with some prodding from me) to realize that negative numbers also exist. When he resisted the idea, I told him that during our lifetimes we had encountered many negative numbers, and a smile crossed his wizened, unshaven face. He allowed for the possibility of negative numbers. He then gave -4 as the other solution. Suddenly, he said that any two negatives squared would result in a positive, so there would always be two numbers as possible solutions to the type of problem I was asking in which the x’s squared are equal to a positive number. It amazed me to see how his mind was able to move to such a generalization, so I cried out, “Bravo!” Later that evening he was able to solve a financial problem that he couldn’t solve earlier. Temporarily at least, my Dad’s mind had climbed to a new level of thinking. I thought this was as an excellent example of mathematical healing and that is why I decided to share it with you.
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CC-MAIN-2015-27
| 2,668 | 3 |
https://superior-papers.com/buy-essay-online-12892/
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math
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- Use the attached examples and complete the answer for both using both examples.
- You are considering buying stock A, which is a large firm with a steady business. If the economy grows rapidly, you may earn 12% on your investment. A declining economy will likely result in a 5% loss. Slow growth will return 5%.
If the probability is 15% for rapid growth, 20 % for a declining economy, and 65% for slow growth, what is the expected return of the investment?
- You are considering investing in three stocks with the following expected returns:
- Stock A 2%
- Stock B 10%
- Stock C 15%
What is the expected return on the portfolio if an equal amount is invested in each stock? What would the expected return be if 50% of your funds are invested in stock A and the remaining funds divided evenly between stocks B and C?
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s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304261.85/warc/CC-MAIN-20220123111431-20220123141431-00297.warc.gz
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CC-MAIN-2022-05
| 1,126 | 10 |
https://www.mathinee.com/content/problem/107196
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math
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The user-friendly version of this content is available here.
The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.
This problem trains for: Math Kangaroo 5-6, Math Kangaroo 7-8, AMC-8, AMC-10, GMAT.
Exactly after the first 40 days of his journey "around the world in 80 days," Phileas Fogg calculated that he had traveled not half, but only a quarter of the itinerary. He became very alarmed and told Passepartout that they would have to increase their speed by 100% in order to finish the journey on time and win the bet with the Reform Club.
Passepartout, however, said: "Monsieur, I doubt that this increase in speed would be sufficient."
If Passepartout is right, at the new speed, by how much percent would the duration of their journey be longer than the projected 80 days?
(characters from Jules Verne's "Around the World in 80 Days")
Assume the itinerary to have size 100 units. In the first 40 days, Fogg has covered only 25 units of itinerary at a rate of (itinerary/time):
The rate he proposes to travel at from now on is:
The number of journey units he can travel at this rate in the remaining 40 days is:
This is 25 journey units short of the total length of the itinerary.
He can travel 25 journey units in:
days. This is longer than the proposed duration by 25%:
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s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991537.32/warc/CC-MAIN-20210513045934-20210513075934-00004.warc.gz
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CC-MAIN-2021-21
| 1,299 | 13 |
http://www.educationalinsights.com/product/mobile/digitz-trade-.do?sortby=ourPicks
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math
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3.OA.4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers.
3.OA.5. Apply properties of operations as strategies to multiply and divide.
3.OA.7.Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
3.NBT.3. Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place.
4.OA.1. Interpret a multiplication equation as a comparison.
4.OA.4. Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.
5.NBT.5. Fluently multiply multi-digit whole numbers using the standard algorithm.
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http://hokum-balderdash.blogspot.com/2007/09/faith-is-superfluous.html
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math
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Imagine a Hindu and a Christian, both of whom have absolute faith (i.e., they firmly believe and harbor no doubts whatsoever) that their religion, their theology, their deity/deities are true and real. Now given the fact that the claims made by the two religions are in conflict with one another, they cannot be simultaneously true. If the Christian claims are true then those of Hinduism are false, and vice versa. So at the very least we know something for certain--that at most only one of them can be right. Therefore, the probability that at least one of them is wrong is 100%. This we are sure of. (Note that I have emphasized "at most" and "at least" since it is quite possible that both believers are wrong.)
Focus now on the fact that our above devotees both have what other supernaturalists would envy and do try to achieve--unremitting and absolute faith. Notice how the fact that they both have "maximum" faith has nothing to do with the undebatable fact that at least one of them is wrong. Hence, in this simple exercise we clearly see that belief, however fervent it may be, has no bearing on whether what is believed in is true or false.
Given this, it is quite puzzling why there are supers who urge their fellow believers to have faith or to strengthen their faith. As we have seen belief is independent of the veracity of what one believes in. As I see it the call to intensify or consolidate one's belief has nothing to do about the truth of the claims/beliefs. Rather it is about ingraining a particular worldview, about the psychological goal of making one take for granted this worldview and its beliefs, and about the social objective of becoming a model for other (potential) believers to emulate.
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https://slickdeals.net/f/11089711-help-kellogg-s-rewards-amc
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math
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Hi all. I don't know if this this right place to post this.
I need ur help. I just redeemed $10 amc from Kellogg's rewards program. I only got e-code and pin immediately after placing order---but NO email.
I thought I would get ecard to use directly at amc.
I even want to give this as gift.
How do I even redeem this?
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s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823710.44/warc/CC-MAIN-20181212000955-20181212022455-00269.warc.gz
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CC-MAIN-2018-51
| 318 | 5 |
https://quantologic.wordpress.com/2009/07/30/problem-of-the-day-30-07-09/
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math
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For All Your Quant Queries
with 7 comments
Written by Implex
July 30, 2009 at 1:03 pm
Posted in Algebra, CAT 2009, Mock Cat, Mock Quant, Number Thoery, Online CAT, Problem of the week, Problems
Tagged with cat, IIM, Number Theory, Online CAT
Subscribe to comments with RSS.
August 1, 2009 at 5:16 pm
can u plz explain the method ?
August 3, 2009 at 11:51 am
I used hit and trial, guessed it would be the quickest approach to solve it.
Start with 1,2,3,4 then 1,2,3,6 (which fetches the answer)
August 4, 2009 at 7:10 pm
hit N trial method N u will get 126.
August 10, 2009 at 10:09 am
If the number is even then 1 and 2 are gonna be its two divisors.The remaining 2 divisors could be p and 2p for sure then(taking p as a prime number).We can check p for 3,5,7 and 3 will give the answer by meeting all the conditions.
A nkur Dudeja
August 18, 2009 at 1:56 pm
Rahul pls put the approach to this question and answer to it..
August 20, 2009 at 7:50 pm
Let the least number be N, 1 is its least divisor.Let 2nd,3rd and 4th least divisors be x,y and z respectively. We consider the following values of divisor a and the corresponding values of a^3, from x,y and z exactly 1 or all 3 are odd.(N is even) a=1:a^3=1 a=2:a^3=8 a=3:a^3=27 a=4:a^3=64 a=5:a^3=125 a=6:a^3=216 For x, y and z=(2,3,4),2*N=100(i.e.N=50). But 3 is not a divisor of 50. For x,y,z=(2,3,6),2*N=252(i.e.N=126) and the 1,2,3,6 are four least distinct divisor of 126.The required number is 126. The sum of digits is 9.
August 15, 2013 at 5:01 pm
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s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917122167.63/warc/CC-MAIN-20170423031202-00320-ip-10-145-167-34.ec2.internal.warc.gz
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https://searanchcreations.com/happy-pi-day-march-14th/
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math
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Every March 14, mathematicians, scientists and math lovers around the world celebrate Pi Day, a commemoration of the mathematical sign π. Pi ( π ) is a mathematical constant that represents the ratio of a circle’s circumference to its diameter which is approximately equal to 3.14159…
Pi appears in many equations in mathematics and physics. Many scientists and mathematicians celebrate Pi on March 14th which represents the first three digits (3.14) of the never-ending number.
Some bakeries and grocery stores sell fruit pies at a discount to celebrate Pi Day. I’ll be making a Vegan Sheppard’s Pie made of a variety of vegetables, lentils, and topped with mashed potatoes. CLICK HERE for the recipe if this sounds interesting to you!
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CC-MAIN-2024-18
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https://www.turbosquid.com/3d-models/3d-model-sofas/372664
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math
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s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818686169.5/warc/CC-MAIN-20170920033426-20170920053426-00463.warc.gz
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CC-MAIN-2017-39
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https://aplusphysics.com/community/index.php?/videos/view-996-the-force-of-gravitational-attraction-between-the-earth-and-the-moon/
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math
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The Force of Gravitational Attraction between the Earth and the Moon
- 720 views
- 0 comments
According to NASA, the mass of the Earth is 5.97 x 10^24 kg, the mass of the Moon is 7.3 x 10^22 kg, and the mean distance between the Earth and the Moon is 3.84 x 10^8 m. What is the force of gravitational attraction between the Earth and the Moon? Want Lecture Notes? This is an AP Physics 1 topic.
0:07 Translating the problem
0:56 Solving the problem
2:15 Determining how long until the Moon crashes into the Earth
4:00 Determining what is wrong with this calculation
Next Video: Deriving the Acceleration due to Gravity on any Planet and specifically Mt. Everest
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https://www.linstitute.net/archives/74897
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math
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2009 AIME I真题
Call a -digit number geometric if it has distinct digits which, when read from left to right, form a geometric sequence. Find the difference between the largest and smallest geometric numbers.
There is a complex number with imaginary part and a positive integer such that
A coin that comes up heads with probability and tails with probability independently on each flip is flipped eight times. Suppose the probability of three heads and five tails is equal to of the probability of five heads and three tails. Let , where and are relatively prime positive integers. Find .
In parallelogram , point is on so that and point is on so that . Let be the point of intersection of and . Find .
Triangle has and . Points and are located on and respectively so that , and is the angle bisector of angle . Let be the point of intersection of and , and let be the point on line for which is the midpoint of . If , find .
How many positive integers less than are there such that the equation has a solution for ? (The notation denotes the greatest integer that is less than or equal to .)
The sequence satisfies and for . Let be the least integer greater than for which is an integer. Find .
Let . Consider all possible positive differences of pairs of elements of. Let be the sum of all of these differences. Find the remainder when is divided by .
A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from to inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the digits given were . Find the total number of possible guesses for all three prizes consistent with the hint.
The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from to in clockwise order. Committee rules state that a Martian must occupy chair and an Earthling must occupy chair . Furthermore, no Earthling can sit immediately to the left of a Martian, no Martian can sit immediately to the left of a Venusian, and no Venusian can sit immediately to the left of an Earthling. The number of possible seating arrangements for the committee is . Find .
Consider the set of all triangles where is the origin and and are distinct points in the plane with nonnegative integer coordinates such that . Find the number of such distinct triangles whose area is a positive integer.
In right with hypotenuse , , , and is the altitude to . Let be the circle having as a diameter. Let be a point outside such that and are both tangent to circle . The ratio of the perimeter of to the length can be expressed in the form , where and are relatively prime positive integers. Find .
The terms of the sequence defined by for are positive integers. Find the minimum possible value of .
For , define , where . If and , find the minimum possible value for .
In triangle , , , and . Let be a point in the interior of . Let and denote the incenters of triangles and , respectively. The circumcircles of triangles and meet at distinct points and . The maximum possible area of can be expressed in the form , where , , and are positive integers and is not divisible by the square of any prime. Find .
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s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038077336.28/warc/CC-MAIN-20210414064832-20210414094832-00491.warc.gz
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CC-MAIN-2021-17
| 3,391 | 16 |
https://www.deepdyve.com/lp/springer_journal/application-of-data-compression-methods-to-nonparametric-estimation-of-iX5916PwJZ
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math
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ISSN 0032-9460, Problems of Information Transmission, 2007, Vol. 43, No. 4, pp. 367–379.
Pleiades Publishing, Inc., 2007.
Original Russian Text
B.Ya. Ryabko, 2007, published in Problemy Peredachi Informatsii, 2007, Vol. 43, No. 4, pp. 109–123.
Application of Data Compression Methods
to Nonparametric Estimation of Characteristics
of Discrete-Time Stochastic Processes
B. Ya. Ryabko
Siberian State University of Telecommunication and Information Science
Institute of Computational Technologies, Siberian Branch of the RAS, Novosibirsk
Received April 9, 2007; in final form, July 31, 2007
Abstract—Discrete-time stochastic processes generating elements of either a finite set (al-
phabet) or a real line interval are considered. Problems of estimating limiting (or stationary)
probabilities and densities are considered, as well as classification and prediction problems.
We show that universal coding (or data compression) methods can be used to solve these prob-
Though methods of encoding data sources (often referred to as data compression) and methods
of information theory in the whole have been widely used in problems of mathematical statistics
since the middle of the past century , recent years were marked with novel result and directions in
this area. It was found that commonly used archivers (i.e., computer programs that implement data
compression methods) can be directly applied to testing statistical hypotheses [2,3] and prediction
of stochastic processes . In the present paper we continue these studies and show how one can
apply universal codes (or universal data compression methods) to estimate limiting probabilities
and densities of time series. (Let us note here that by definition, a universal code asymptotically
“compresses” a sequence of n symbols generated by a stationary ergodic source to nH bits, where H
is the Shannon entropy.) The obtained estimators are applied to the construction of prediction
methods, solution of classification problems (called sometimes classification with side information),
etc. Note also that universal coding methods are applied to predicting stochastic processes starting
with the paper , where the case of finite-alphabet processes was considered, which after that was
generalized to sources generating elements of metric spaces [6–11].
Let us give a precise statement of the problem. We consider stationary ergodic processes gen-
erating sequences of elements x
of a set (alphabet) A; two cases are considered: A is finite
or A is a real line interval. We consider so-called nonparametric methods; i.e., we assume that there
is no other information on probabilistic characteristics of the process. In mathematical statistics,
getting information on a process is usually formulated as either a hypothesis testing problem or
estimating parameters of the process, such as limiting probabilities, distributions, regression, etc.
We first briefly dwell on the hypothesis testing problem, studied in [2, 3], in order to present
main ideas underlying the use of universal coding methods in problems of mathematical statistics.
Consider the problem of testing the hypothesis H
that a sequence of zeros and ones x
Supported in part by the Russian Foundation for Basic Research, project no. 06-07-89025.
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s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221209040.29/warc/CC-MAIN-20180814131141-20180814151141-00530.warc.gz
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CC-MAIN-2018-34
| 3,282 | 41 |
http://openstudy.com/updates/4dd945d3d95c8b0b507b64c4
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math
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
trial and error.. alot of it lol
possible rational roots?
last# ----- helps to narrow it down fisrt#
Rational root test, fundamental theorem of algebra, synthetic division
make a fraction, numerator must divide 5, denom must divide 1, so only choices are \[\pm1, \pm 5\]
no, not all the roots; but the integer and rational roots that could be determined from this lol
i know i am being silly. you meant to find possible rational roots.
how do you find the real roots?
might be able to work it inside out and complete squares to find all roots .... i wonder
there is no algorithm for finding the roots of a poly of degree seven.
only 1 - 4. that is it. and the one for 3 is a page and the one for 4 is a wall.
well, you could do the Newton Rhapsody method to find 1 root and narrow it down ..... perhaps ?
i would try 1, -1, 5 and -5 and see if any work. then you can use synthetic division to reduce. i can see that 1 does not work since \[f(1)=1+3-5-2-5\]
make sure you include your zero place holders
well we have to find one first.
if you have a decent calculator check \[f(5)\]
syntheitc can find them as well; but brute calculating is always a good option lol
ok i will divide see if i get 0
Newton-Rhapsody :D :D
newton rhapsody in blue
5 doesn't work
and neither does -5 unless i made a mistake
-5| 1x^7 +3x^6 +0 +0 +0 -5x^2 -3x -5 0 -5 10 -50 250 -1250 6275 ------------------------------------- 1 -2 10 -50 250 -1255 ...... no mistake lol
D's sign change tells us what ...... do we include zeros as a sign change?
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s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084890893.58/warc/CC-MAIN-20180121214857-20180121234857-00348.warc.gz
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CC-MAIN-2018-05
| 2,214 | 25 |
http://www.smartappsforkids.com/addition/
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math
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I'm a sucker for math apps. Anything that gets my kids to practice math makes me happy, but one where they are determined to beat my score gets chalked up as a win in my book. At first look, I had my reservations about 1+2=3. I mean how basic can a math app get? Almost an hour later, I realized just how fun and challenging this game can be. The premise is simple — a math equation is displayed on the screen. The answer is either 1, 2 or 3. Players must rapid-fire select the correct answer as the equations get more complex and the time gets faster.
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s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422122087108.30/warc/CC-MAIN-20150124175447-00241-ip-10-180-212-252.ec2.internal.warc.gz
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CC-MAIN-2015-06
| 554 | 1 |
http://calculo206.blogspot.com/2011/01/historia-del-calculo-gottfried-leibniz.html
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math
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La regla del producto del cálculo diferencial es aún denominada "regla de Leibniz para la derivación de un producto". Además, el teorema que dice cuándo y cómo diferenciar bajo el símbolo integral, se llama la "regla de Leibniz para la derivación de una integral".
Desde 1711 hasta su muerte, la vida de Leibniz estuvo emponzoñada con una larga disputa con John Keill, Newton y otros sobre si había inventado el cálculo independientemente de Newton, o si meramente había inventado otra notación para las ideas de Newton.
Leibniz pasó entonces el resto de su vida tratando de demostrar que no había plagiado las ideas de Newton.
Actualmente se emplea la notación del cálculo creada por Leibniz, no la de Newton.
Leibniz is credited, along with Sir Isaac Newton, with the inventing of infinitesimal calculus (that comprises differential and integral calculus). According to Leibniz's notebooks, a critical breakthrough occurred on 11 November 1675, when he employed integral calculus for the first time to find the area under the graph of a function y = ƒ(x). He introduced several notations used to this day, for instance the integral sign ∫ representing an elongated S, from the Latin word summa and the d used for differentials, from the Latin word differentia. This cleverly suggestive notation for the calculus is probably his most enduring mathematical legacy. Leibniz did not publish anything about his calculus until 1684. The product rule of differential calculus is still called "Leibniz's law". In addition, the theorem that tells how and when to differentiate under the integral sign is called the Leibniz integral rule.
Leibniz's approach to the calculus fell well short of later standards of rigor (the same can be said of Newton's). We now see a Leibniz proof as being in truth mostly a heuristic argument mainly grounded in geometric intuition. Leibniz also freely invoked mathematical entities he called infinitesimals, manipulating them in ways suggesting that they had paradoxical algebraic properties. George Berkeley, in a tract called The Analyst and elsewhere, ridiculed this and other aspects of the early calculus, pointing out that natural science grounded in the calculus required just as big of a leap of faith as theology grounded in Christian revelation.[relevant? – discuss]
From 1711 until his death, Leibniz's life was envenomed by a long dispute with John Keill, Newton, and others, over whether Leibniz had invented the calculus independently of Newton, or whether he had merely invented another notation for ideas that were fundamentally Newton's.
Modern, rigorous calculus emerged in the 19th century, thanks to the efforts of Augustin Louis Cauchy, Bernhard Riemann, Karl Weierstrass, and others, who based their work on the definition of a limit and on a precise understanding of real numbers. While Cauchy still used infinitesimals as a foundational concept for the calculus, following Weierstrass they were gradually eliminated from calculus, though continued to be studied outside of analysis. Infinitesimals survived in science and engineering, and even in rigorous mathematics, via the fundamental computational device known as the differential. Beginning in 1960, Abraham Robinson worked out a rigorous foundation for Leibniz's infinitesimals, using model theory. The resulting non-standard analysis can be seen as a belated vindication of Leibniz's mathematical reasoning.
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CC-MAIN-2018-30
| 3,438 | 8 |
https://studopedia.net/8_36449_Exercise--Questions-for-comprehension.html
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math
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Exercise 67. Questions for comprehension
1. When did the author enter the university?
2. How old is the author?
3. What document did he get after school?
4. Why did he enter the Medical University?
5. Where do the students live?
6. Where can the students have dinner?
7. Is education free of charge at his Medical University?
8. What subjects does the first year curriculum include?
9. What subjects does the author pay great attention to? Why?
10. What do the students do during their practical studies?
11. What do the students do at the lectures?
12. Who delivers lectures at the University?
13. What do the students do in scientific societies?
14. What are the senior students carrying out research work in scientific societies going to do in future?
15. How do the students spend their spare time?
Exercise 68. Find English equivalents in the text of the following word combinations:
1. справляться с научной работой хорошо; 2. ставить опыты на животных; 3. программа первого года обучения включает много предметов; 4. мне очень хочется рассказать вам; 5. лекции, которые читают профессора; 6. знания по медицине; 7. интересные данные опытов; 8. студенты конспектируют лекции; 9. напряженно готовиться к вступительным экзаменам; 10. обращать огромное внимание на; 11. заниматься в аспирантуре; 12. некоторые из моих сокурсников; 13. принимать активное участие в общественной жизни университета; 14. посещать лекции по истории
Exercise 69. Put in articles where necessary.
1. … first year curriculum of our institute includes such subjects as … Biology, … Physics and so on. 2. My fellow student knows … Anatomy well because he works hard at it. 3. … subjects, which … second-year students study, are not difficult. 4. In … May we will take … examination in … Anatomy. I hope to cope with … exam well. 5. … medical students carry on … experiments on … animals in … different laboratories. 6. My sister graduated from … University … last year. 7. We arranged … evening party on … 8th of … March to greet our girls. 8. He always copes with … work he carries on.
Exercise 70. Fill in the gaps with prepositions and adverbs where necessary:
A. 1. I have a friend of mine. Last year he graduated … the Medical University and joined … the great army … doctors. Besides practical medicine my friend is greatly interested … research work and carries … different experiments …animals. Some findings … his works will help him to fight … all diseases.
B. 1. We carry …interesting experiments … the Institute laboratories which have all the necessary equipment. 2. All the students … our group attend lectures delivered … professors … medicine. 3. My sister is greatly interested … Anatomy and pays great attention … this subject. 4. Some … my fellow students belong … our scientific societies where they carry … interesting experiments. 5. Our professor is interested … findings … my experiments. 6. I hope my brother will cope … studies. 7. Yesterday we studied the plan … your research work. 8. Everybody took part … the discussion … some findings … this experiment. 9. This experiment is rather difficult. Please pay more attention … it. 10. All the students of our group take an active part … the University social life.
Fill in the gaps with prepositions and adverbs where necessary:
1. Most … us live … the Institute hostel, only my fellow student lives … his parents. 2. I usually have my dinner … our University canteen. 3. Our rector pays great attention … the system … education … University. 4. … our practical studies we receive much knowledge … medicine. 5. Last year my doctor treated me … the grippe … penicillin.
Exercise 71. Make the following sentences negative.
1. Olga graduated from the Institute a year ago. 2. I will study in the second year next year. 3. My group mate copes with his task well. 4. My friend is eager to be admitted to this Academy. 5. Professor Gromov will deliver lecture in Chemistry every morning. 6. My elder brother carried on experiments in the physics laboratory every day.
Make the following sentences negative.
1. These students study in the second year. 2. That senior student arranged all his affairs in time. 3. Yesterday the students of our group took notes of the lecture in Biology. 4. My fellow student is interested in Physics.
Дата добавления: 2018-09-22; просмотров: 95;
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CC-MAIN-2020-16
| 4,883 | 30 |
http://content.time.com/time/specials/packages/article/0,28804,1918031_1918016_1917928,00.html
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math
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A long, long time ago, in a galaxy not so far away, Google was just two grad students at Stanford with a smart idea for search technology. Today's search whiz kid is Stephen Wolfram, one of the biggest brains on the planet and he's got the new idea. Wolfram has developed a search engine that can actually understand your questions and try to figure out answers. It takes some doing to learn how to talk to Wolfram|Alpha, but it's well worth it. If the sci-fi writers are right and the Internet does gain a consciousness of its own someday, we'll all blame Wolfram. In the meantime, you can have a baby HAL of your own.
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s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573908.30/warc/CC-MAIN-20220820043108-20220820073108-00340.warc.gz
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CC-MAIN-2022-33
| 619 | 1 |
https://www.camerastuff.co.za/shop/black-friday
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math
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Terms and Conditions
• Bundles are normally discounted by 15%. With the additional 10% off, the total discount will be between 20% and 25% (15% + 10%).
• Unfortunately, no reservations. Black Friday specials are strictly given on a first-come, first-served basis.
• Discounts apply until stocks last.
• Prices are subject to change.
• Free delivery still applies to orders above R1,000.
• Deliveries may be delayed due to the increased number of daily orders.
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s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363641.20/warc/CC-MAIN-20211209000407-20211209030407-00154.warc.gz
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CC-MAIN-2021-49
| 471 | 7 |
https://demtutoring.com/answered/statistics/q5810
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math
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*click image* are these two answers correct? if not can someone please answer with the correct ones? tysm!!
Get the answer
Category: statistics | Author: Sagi Boris
*click photo* tysm if you help out! currently trying to raise my math grade! tysm for your time and effort!
*free 30 points* the function f(x) = ?x2 + 24x ? 80 models the hourly profit, in dollars, a shop makes for selling coffee, where x is the number of cu
*have to get it right * read the passage. then click the inference that is most firmly based on the given information
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s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500719.31/warc/CC-MAIN-20230208060523-20230208090523-00437.warc.gz
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CC-MAIN-2023-06
| 541 | 6 |
https://www.dagonfly.pl/cubic/Jun-03_17347.html
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math
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Cubic Yards Calculator
Calculate cubic yards, cubic feet or cubic meters for landscape material, mulch, land fill, gravel, cement, sand, containers, etc. Enter measurements in US or metric units and get volume conversions to other units. How to calculate cubic yards for rectangular, circular, annular and triangular areas. Calculate project cost based on price per cubic foot, cubic yard or cubic meter.
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s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446706291.88/warc/CC-MAIN-20221126112341-20221126142341-00496.warc.gz
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CC-MAIN-2022-49
| 404 | 2 |
https://www.yukimura-physics.com/en/entry/dyn-f09
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math
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(This article was translated from Japanese to English using DeepL.)
In the last lecture we learned about the motion of throwing an object down, now we want to investigate the motion of throwing an object up!
Vertical Throw Up
An object thrown upward reaches the highest point and then turns around and falls, which is also a type of falling motion.
Throwing up seems difficult at first glance because the direction of motion changes during the process, but the important thing to remember is that the object is in motion only under gravity, both when it is moving upward and when it is falling downward.
This means that vertical throw up is also a uniform accelerated motion.
Check it out in the figure!
Notice the orientation of the y-axis. In the case of the throw down, the downward direction was assumed to be positive, but this time the upward direction is assumed to be positive.
Since the upward direction is positive, gravitational acceleration occurs in a negative direction.
I am tempted to reconsider, “It would be troublesome to have a minus sign, so shouldn’t the axis be positive downward?”
But if you do so, a minus value is attached to the initial velocity.
In the end, no matter which direction is taken as positive, we cannot escape from the negative.
In such cases, the convention is to “take the direction of the initial velocity as positive”, so let’s just follow the convention.
To return to the topic, the vertical throw up motion is a uniform accelerated motion with “negative gravitational acceleration” as shown in the figure above.
Therefore, the formulas of vertical throw up is as follows.
As explained in the lecture on acceleration, the minus sign attached to acceleration can mean either “deceleration in the positive direction” or “acceleration in the negative direction.
In the case of a vertical throw up, it means “deceleration in the positive direction” from the throw up to the highest point, and “acceleration in the negative direction” after the highest point.
Note that during the throw-up motion, the acceleration is constant at –g throughout, but the meaning changes after the highest point.
Graph of vertical throw up
Let’s also look at the graph of a vertical throw up. Since acceleration is negative, the v-t graph will look like the one below.
The “Highest point = Zero velocity” is often used when solving problems.
Instead of memorizing it, please visualize the motion and be convinced that “At the highest point, the motion is stationary (only for a moment)”.
The y-t graph then follows from the throw up formula 2).
This leads us to another important point.
Knowing the symmetry of this motion will be a big help when solving problems, so be sure to keep it in mind!
Summary of this lecture
From the next lecture, we will study forces, which are the key to mechanics.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948951.4/warc/CC-MAIN-20230329054547-20230329084547-00674.warc.gz
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CC-MAIN-2023-14
| 2,860 | 27 |
https://www.keyword-suggest-tool.com/search/hp+to+lb+hr/
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math
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Hp to lb hr keyword after analyzing the system lists the list of keywords related and the list of websites with related content, in addition you can see which keywords most interested customers on the this website
Reciprocating Engines I ... horsepower hour (g/bhp-hr), pounds per hour (lb/hr) and tons per year (tpy). If the lb/hr and tpy emissions are not provided, calculate the emission rate limit in Table 1(a) for NOx, CO, PM, and VOC, as follows: A. Manufacturer's Data: This should include full load or all load emission rate data in g/hp-hr or ppmv ...
Horsepower to foot-pounds per hour [HP to ft-lb/h ...
How to convert horsepower to foot-pounds per hour [HP to ft-lb/h]:. P ft-lb/h = 1 980 000 × P HP. How many foot-pounds per hour in a horsepower: If P HP = 1 then P ft-lb/h = 1 980 000 × 1 = 1 980 000 ft-lb/h. How many foot-pounds per hour in 36 horsepower: If P HP = 36 then P ft-lb/h = 1 980 000 × 36 = 71 280 000 ft-lb/h. Note: Horsepower is an imperial or United States customary unit of power.
Lb/(hp*h) to g/(kW*h) (Pound/Horsepower/Hour to Gram ...
Convert Energy Mass from Pound/Horsepower/Hour to Gram/Kilowatt/Hour or to different units such as Kilojoules per Kg, Joule per Kg, Calorie per gram, BTU per pound, Kg per joule, Kg per Kilojoules, Gram per Calorie, Pound per BTU and more ... lb/(hp*h) to g/(kW*h) (Pound/Horsepower/Hour to Gram/Kilowatt/Hour) ... Home Site map Conversion Matrix ...
Welcome to your HP country website for former employees. On this site you will be able to: View contact information for payroll, benefit service providers, etc. Stay connected with HP; Submit any other HR/Paroll related questions; Please use the appropriate links on this site to find the information you need and/or to submit an HR question.
First of all thank you for looking into my request and sending those links. Some Air Permits ( Title V ) or EPA requirements for the NOx limits are given in lb/hr or g/hp-hr and the conversion to ppm is more elaborated than just the use of a factor. The second link is very useful with the examples for the calculations.
Convert foot pounds per hour to horsepower | power conversion
This on the web one-way conversion tool converts power units from foot pounds per hour ( ft-lb/h ) into horsepower ( hp ) instantly online. 1 foot pound per hour ( ft-lb/h ) = 0.00000051 horsepower ( hp ). How many horsepower ( hp ) are in 1 foot pound per hour ( 1 ft-lb/h )? How much of power from foot pounds per hour to horsepower, ft-lb/h to hp?
Contact HP; Contact HP. How may we help you? Shopping. Sales consultations and order status. Go. Support. Technical assistance with your HP products & services. Go. Company. General questions and office locations. Go. End of content.
More about Specific Energy, Heat of Combustion per Mass
Specific Energy, Heat of Combustion (per Mass) Converter. Specific energy (per mass) is defined as the energy per unit mass. Common units are J/kg or cal/kg. The concept of specific energy applies to a particular (e.g. transportation) or theoretical way of extracting useful energy from the fuel.
Convert lb/hr to gram/hour - Conversion of Measurement Units
1 kilogram/second is equal to 7936.6479126616 lb/hr, or 3600000 gram/hour. Note that rounding errors may occur, so always check the results. Use this page to learn how to convert between pounds/hour and grams/hour.
Info!Website Keyword Suggestions to determine the theme of your website and provides keyword suggestions along with keyword traffic estimates. Find thousands of relevant and popular keywords in a instant that are related to your selected keyword with this keyword generator
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s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496668544.32/warc/CC-MAIN-20191114232502-20191115020502-00532.warc.gz
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CC-MAIN-2019-47
| 3,638 | 16 |
https://papers.plagiarismfreepapers.com/the-theory-of-finance-allows-for-the-computation-of-excess-returns-either-above-or-below-the-current-stock-market-average-an-analyst-wants-to-determine-whether-stocks-in-a-certain/
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math
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The theory of finance allows for the computation of “excess” returns, either above or below the current stock market average. An analyst wants to determine whether stocks in a certain industry group earn either above or below the market average at a certain time period. The null hypothesis is that there are no excess returns, on the average, in the industry in question. “No average excess returns” means that the population excess return for the industry is zero. A random sample of 24 stocks in the industry reveals a sample average excess return of 0.12 and sample standard deviation of 0.2. State the null and alternative hypotheses, and carry out the test at the α = 0.05 level of significance.
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s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304749.63/warc/CC-MAIN-20220125005757-20220125035757-00555.warc.gz
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CC-MAIN-2022-05
| 710 | 1 |
http://www.cfar.umd.edu/~fer/optical/line4.html
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math
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|Errors in Line Estimation
Figure 1: Click repeatedly on the figure to see zoom ins illustrating the fitting of the curve.
As you can see, in this pattern the circle appears to be distorted. This is predicted by the bias in the estimation of the intersection points between the circle and the lines of the background.
The estimation of the circle requires a few computations which we modeled as follows:
Luckiesh Pattern in Motion
Figure 2: Click on the pattern and move your cursor to change the orientation of the background lines.
Since the value of the bias depends on the direction of the intersecting lines, changing the direction of the background lines causes a change in the bias and thus a change in the estimated curve, with the circle bumping at different locations.
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https://www.arxiv-vanity.com/papers/0905.1723/
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math
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Mutually Unbiased Bases and Complementary Spin 1 Observables
The two observables are complementary if they cannot be measured simultaneously, however they become maximally complementary if their eigenstates are mutually unbiased. Only then the measurement of one observable gives no information about the other observable. The spin projection operators onto three mutually orthogonal directions are maximally complementary only for the spin 1/2. For the higher spin numbers they are no longer unbiased. In this work we examine the properties of spin 1 Mutually Unbiased Bases (MUBs) and look for the physical meaning of the corresponding operators. We show that if the computational basis is chosen to be the eigenbasis of the spin projection operator onto some direction , the states of the other MUBs have to be squeezed. Then, we introduce the analogs of momentum and position operators and interpret what information about the spin vector the observer gains while measuring them. Finally, we study the generation and the measurement of MUBs states by introducing the Fourier like transform through spin squeezing. The higher spin numbers are also considered.
Two dimensional orthonormal bases and are called unbiased if
and the set of more than two bases of this kind is called mutually unbiased if the above holds for every pair of bases from this set. In quantum mechanics it means that if the two observables and have unbiased eigenbases the measurement of the observable reveals no information about possible outcomes of the measurement of the observable and vice versa.
The study on Mutually Unbiased Bases (MUBs) has been started by Schwinger Schw (1960) almost fifty years ago. The properties of MUBs have been successfully applied in many areas of quantum physics: they provide the security of quantum key distribution protocols BB84 (1984); PP (2000) and the solution to the Mean King problem VAA (1987); AE1 (2001); AE2 (2001), minimize the number of measurements needed to determine the quantum state Iv (1981); WF (1989) and are related to the discrete Wigner Functions GHW (2004). Despite their wide use, there is still an intriguing open question about the maximal number of such bases in the non-prime power dimensional complex spaces Prob (2004). It is well known that the MUBs have deep physical meaning for the quantum systems described by the continuous Hilbert spaces. The eigenstates of the position and the momentum operators are mutually unbiased. The same stands for the electric and the magnetic field operators. In the case of spin, the Hilbert space is finite dimensional. The three MUBs of spin have nice geometrical representation, because they are simply the eigenbases of the spin projection operators onto any three mutually orthogonal directions. This is not the case for the higher spin numbers . The Hilbert space of spin is isomorphic with the three dimensional real space what is the essence of the Bloch sphere picture. On the other hand, the Hilbert space of spin is much richer than the three dimensional real space, that is why it would be naive to expect that the MUBs of higher spins have simple graphical interpretation.
Most of the research done on MUBs are mainly based on the mathematical properties of the underlying Hilbert space and the intuitive physical picture behind the complex state vectors is somehow lost. We are trying to bring back this picture by studying the properties of MUBs for spin 1 states. What motivates our work, is the fact that the knowledge of the observables corresponding to the MUBs and the ability to generate and measure unbiased states for a certain system is necessary to fully exploit it in quantum information processing tasks. In the following paper we find that for the computational basis being the basis of projection operator onto any direction, the remaining MUBs have to be squeezed. We give the physical interpretation of the two operators which are maximally complementary, one of them being the projection operator. They are related by the Fourier transform and might be considered as an analog of the position and the momentum. Then, we show how to transform between different MUBs by introducing the Fourier like transform through spin squeezing. Finally, the eigenstates of projection operator and their Fourier transforms are also briefly examined for the higher spin numbers. In the end, we discuss our results in the context of the recent studies on the biphoton implementation of a qutrit Bog1 (2004); Bog2 (2004); Lan (2008); Bre (2008).
Ii Spin 1 states unbiased to the basis
The most common choice of the computational basis for spin 1 is the eigenbasis of the spin projection operator onto some direction . Any state which is unbiased to all the states from the computational basis is of the form
where and are arbitrary phases. The first property of the above state is the zero mean value of the component of the mean spin vector , where ’s are the spin projection operators onto direction obeying the cyclic commutation relation . This is because the operator is diagonal, its eigenvalues are 1, 0 and -1 (here and throughout the work we take ) and for the state (2)
This means that either the mean spin vector is or it lies in the plane. Indeed, the coordinates of the mean spin vector are
Note, that due to the rotational symmetry for the study of the physical properties of states (2) we can only consider the subclass of states for which . The rest of the states might be generated by the rotation in the plane. It is easy to see that the subclass corresponds to , so the Eq. (2) becomes
or to , what gives
We obtained the class of states unbiased to the computational basis which are parameterized only by . They are the states with the mean spin vector of length pointing in direction (5), or a completely unpolarized states with (6). Since the maximum value of the mean spin vector length is , any state of the form (2), including (5) and (6), cannot be a coherent spin state — the eigenstate of the spin projection operator onto any direction with the eigenvalue . The observables with all the eigenstates of the form (2) have to be much more sophisticated than simply the spin projection operators. On the other hand, since the states (6) are completely unpolarized, it may happen that at least some of them correspond to the eigenstates of the spin projection operators with the eigenvalue — the null projection states.
Now, let us consider the uncertainties , and . The uncertainty relation for the spin projection operators onto any three mutually orthogonal directions yields:
The above inequality is very sensitive to the choice of the directions , and and it was shown KU (1993) that it should be applied for the lying along the mean spin vector. With properly defined uncertainty relation the squeezed spin state is defined as the one for which the uncertainty of the spin projection operator onto direction orthogonal to the mean spin vector is smaller than , what was done in KU (1993). In our case the state is squeezed if there exist a direction orthogonal to the mean spin vector for which . The mean spin vector of the states (5) lies along direction, therefore and . What is interesting, both uncertainties do not depend on
thus according to the definition the states (5) are squeezed in direction. The states (6) have zero mean spin vector and it is hard to choose the proper direction for the relation (7), however in this case the variances are
and together with one may find that these states are the null projection states onto direction
This direction makes the tetrahedral angle with axis. In the light of the recent Bell-like inequality for spin 1 Kl (2008) and due to the maximal entanglement of two spin 1/2 particles forming up a spin 1 particle, the null projection states are considered to be the most non-classical ones. Moreover, these states might be viewed as a maximally squeezed states in the direction for which , because in the case of spin 1 they may emerge during the squeezing of a coherent state as an intermediate states between the two coherent ones.
Iii Spin 1 MUBs
The four MUBs in the three dimensional complex space are most commonly known to be the eigenvectors of the four unitary operators from the Weyl-Heisenberg group: , , and . All four operators have eigenvalues , , and and obey the relation , where is the third root of unity. The bases, up to normalization, are given by
where and stands for the complex conjugation. Note, that the second basis (12) is the Fourier transform of the computational basis. If the basis (11) is the eigenbasis of , then the operator generates a permutation inside the bases (12-14) for times being the multiple of . This is an analogy to the rotation about axis for spin which generates the swap operation in the and bases. The transition between different bases, other than the computational one, can be obtained by the operator generated by the Hamiltonian proportional to which is the one-axis twisting squeezing generator KU (1993).
The states in the first column of the bases (12-14) are of the form (5), therefore the corresponding mean spin vectors point in direction and due to the rotational symmetry the other states from these bases have the same properties. The lengths of the mean spin vectors of the MUBs (12-14) are , and respectively. The vectors lie in the plane and point in the directions , and for the basis (12) and in the directions , and for the bases (13) and (14). One may find that in general the shorter the mean spin vector is, the more squeezed the corresponding state has to be. On the other hand, Eq. (II) states that for all states of the form (5) the uncertainties and do not depend on the state, suggesting that all these states are equally squeezed what seems in contrary to the fact that vectors (13) and (14) are shorter than vectors (12). However, it may happen that the states with the shorter mean spin may be squeezed in the direction other than , but still have the same uncertainties and . This idea is depicted in the Fig. 1 right.
Indeed, by studying the rotation about axis one may show that the states lying along axis are squeezed in the direction, the direction tilted by an angle and for the bases (12-14) respectively. The uncertainties in the direction of squeezing for the last two bases reaches slightly bellow . Note that the bases (13) and (14) seems similar and the complex conjugation changes one basis to the other. The conjugation applied to the spin vector generates reflection in the plane inverting axis . This is visible in the uncertainties and in the distribution of the mean spin vectors (see Fig. 1).
Spin states are sometimes depicted as the cones in the three dimensional space, where the surface of the cone represents an area covered by the spin vector due to its spread caused by the uncertainty principle and its dilation angle represents the uncertainties. In the Fig. 2 we suggest how one may picture spin squeezed states corresponding to the MUBs (11) and (12).
Iv Complementary Observables
In this section we are going to look for the physical meaning of the observables corresponding to spin 1 MUBs. The MUBs were taken to be the eigenstates of the four unitary operators from the Weyl-Heisenberg group , , and where the operator is the permutation operator of the computational basis and is the diagonal operator causing the phase shift. All four operators are non-degenerate and their eigenvalues are the third roots of unity. One may interpret the generators of and as momentum and position operators respectively acting on the discrete space with only three allowed positions and periodic boundary conditions. The eigenbasis of , the computational basis, has been chosen to be the eigenbasis of operator, therefore this MUB corresponds to the knowledge of the spin projection onto axis and one may think of it as some kind of position operator. The second MUB, the Fourier transform of the computational basis, might be interpreted as an analog of momentum operator. This kind of operator causes the cyclic permutation of eigenstates of
However, from the physical point of view, this permutation is abstract and hard to interpret since it is not a simple spin rotation, because it takes one coherent state to the null projection state, then to the other coherent state and next back to the initial coherent state — it has to be a nontrivial combination of squeezing and rotation. The same problem remains for the other two MUBs. The states of the MUBs (12-14) are bizarre in the sense that they are neither coherent, nor maximally squeezed null projection states and the physical meaning of the corresponding observables is not as simple as of the standard spin projection operator . One may wander whether it would be more convenient to choose a different computational basis.
Among all the states unbiased to the basis there is a class of maximally squeezed states given by Eq. (6) corresponding to the null projection states onto all axes tetrahedral to . One may find that in this class there are infinitely many ways to pick a three mutually orthogonal states forming up a basis which is mutually unbiased to the one. These states might be represented in the real space as three mutually orthogonal planes (see Fig. 3).
In general, any three states corresponding to the null projection states onto a three arbitrary mutually orthogonal directions form up a basis in the Hilbert space of spin 1. Let us choose as a computational basis a basis made of a three null projection states along , and directions, which in the basis are given by:
Usually, the state is written as , but there is a reason why we multiplied it by . Any state of the form
is also a null projection state along direction , therefore all real linear combinations of , and resemble the vectors in the Euclidean space . What is interesting, the Fourier transform of the new computational basis is the eigenbasis of the spin projection operator along direction which is tetrahedral to z. Moreover, the operator generates a rotation about , which for an angle being a multiple of causes a cyclic permutation of basis states — the rotation about transforms plane into plane, etc. This means that the projection operator might be considered as the momentum operator in the chosen basis. Actually, it could be any projection operator along one of four tetrahedral directions .
What is the corresponding position operator? It has to be an operator with the eigenvectors , and . In general, we are looking for the operator whose eigenvectors are the null projection states along a three mutually orthogonal directions , and . It happens that this is the two-axis countertwisting squeezing operator KU (1993) of the form with the eigenvalues 1, 0 and . It might be as well represented as or , because different representation does not change the eigenvectors, but the distribution of the eigenvalues, thus the position operator could be taken as . Next, let us find what information do we gain while measuring . The information related to measuring is that the spin is definitely not lying along direction , therefore the measurement of gives an answer to the question: Along which one of the three mutually orthogonal axes the spin is not lying? Note, that the uncertainty principle forbids the spin to lie definitely along one axis, that is why the question we can ask sounds a little bit odd — in quantum world we are not allowed to ask about the direction in which the spin is pointing. Eventually, the two complementary spin 1 observables might be reformulated as the two complementary questions:
Along which one of the three mutually orthogonal axes , and the spin is not lying?
What is the spin projection onto one of the four axes tetrahedral to , and ?
Having an answer to the one of the above questions one knows nothing about an answer to the other question.
Even that we have chosen different computational basis, the remaining two MUBs (13) and (14) are squeezed, but not maximally squeezed, i.e. they are not the null projection states. The corresponding observables, which might be taken as a real linear combinations of a projectors onto MUBs states, are a generators of a transformations which are a combination of a rotation and a squeezing, what makes them hard to identify as some simple physical quantities or to associate them with reasonable questions one can ask about spin. However, it would be very interesting to find the physical meaning of the observables whose eigenstates are partially squeezed and we leave this problem as an open question.
V Generation and Measurement of Spin 1 MUBs
MUBs have found many practical applications in quantum information processing. In order to prepare an unbiased states and to perform a suitable measurements one has to know how to transform between the different MUBs. In the case of spin 1 some states are easier to obtain as well as some measurements are easier to perform. In order to implement a certain information processing tasks, the ability of preparing and measuring all possible states is desired — we need spin 1 to be an exact implementation of a qutrit. For the study of the possibility of spin 1 implementation of a qutrit see Das (2003). The most common and easiest measurements of spin are the one of the Stern-Gerlach type, although generalized Stern-Gerlach measurements have been also proposed SW (1980). The prior measurement could be also considered as a preparation procedure, therefore the coherent states and the maximally squeezed null projection states are the most accessible ones. It is also obvious, that the most natural choice of the computational basis should be the eigenbasis of for some reference axis . The transformation between the different coherent states is relatively easy, since it requires only a spin rotation via an application of a linear magnetic field. However, the transformation between a coherent and a null projection states is no longer simple, because it cannot be obtained by a linear effects.
We already mentioned that if the computational basis corresponds to , the transformation within the three remaining MUBs could be obtained by a rotation about . It is simply an analog of translation or boost. On the other hand, the transformation between these MUBs is done via one axis twisting squeezing. This kind of squeezing is generated by operator and the change between the MUBs occurs for the times being the multiple of . The above operator is of course nonlinear and requires a quadrupole effects. Still, the most important question is how to obtain the three basis from the basis. The usual way of generating the MUBs states is to perform the Fourier transform on the computational basis. Let us look for a more general transformation, a Fourier like transform, taking the basis state to the states (2). This transformation is represented by the unitary matrix with all entries having an equal absolute value, a complex Hadamard matrix. For the spin 1 such a transformation might be obtained for the one-axis twisting about an axis tetrahedral to whose direction is given by Eq. (10). The corresponding operator becomes a Hadamard for (and ) which up to a global phase is given by
Straightforward calculations show that the above operation generates three squeezed states symmetrically distributed by an angle on the plane whose lengths do not depend on and are equal . The angle is only affecting the global deviation of the basis from the alignment along , and . In order to obtain the full Fourier transform one still has to apply the one-axis twisting and a rotation, both about . Even without this, the operator (16) generates a basis which is unbiased to , what is enough to perform certain tasks like quantum cryptography on a three level system. The other two bases might be obtained by the one-axis twisting pulse about .
To perform the measurement in a basis corresponding to a certain MUB one needs only to know the procedure of transforming this MUB into the computational basis. In this case one or two one-axis twisting pulses about has to be followed by the inverse of (16). Then, it is enough to perform a Stern-Gerlach measurement in the computational basis. Eventually, the rotation and one-axis twisting pulses about , together with the Fourier like transform (16), its inverse and the Stern-Gerlach measurement gives the full set of operations needed to perform three level quantum cryptography or the tomography of spin 1.
Vi Higher Spin Numbers
One may also consider the MUBs for the higher spin numbers, although for some we do not even know how many MUBs there exist Prob (2004). Because of this fact, in this section we only consider the two MUBs, the eigenbasis of and its Fourier transform. The Hilbert space is dimensional. Again, the mean spin vectors of all the states unbiased to basis are lying in the plane, due to the similar reason as before (recall Eq. (3)), and the rotation about , k being an integer, causes the translation between the states of the Fourier transform basis. Once more, because of the rotational symmetry we may narrow our study to the state
The mean spin vector corresponding to the above state has to point in the direction, because for all . This is because is an antisymmetric matrix and an expectation value of any matrix with respect to the state (17) equals the sum of all matrix elements divided by . The length of the mean spin vector is given by
It is shorter than the length of the mean spin of a coherent state what indicates that the state (17) is squeezed, as well as the other Fourier transform states have to be. Indeed,
therefore the state has to be squeezed.
Once more, it is hard to interpret the physical meaning of the operator corresponding to the Fourier transform of the basis because, as in the case of spin 1, these states are neither coherent nor null-projection states. It is also hard to tell whether one may find an analogs of momentum and position operators, with one of them being simply the spin projection operator onto some direction.
We have discussed the physical aspects of spin 1 complementary observables, introducing the analogs of position and momentum operators and showing that the spin squeezed states can play an important role in the MUBs studies. Moreover, we proposed the methods of generation and measurement of the different spin 1 MUBs what is necessary for spin 1 quantum information processing. In the recent years the biggest experimental progress in quantum information had happened within quantum optics and it is important to mention how our work is related to this field. One of the most promising optical qutrit implementations is the polarization states of a biphoton — the joint polarization states of a two indistinguishable photons. Still, even for biphoton, the generation of all possible states requires nonlinearities from either nonlinear crystals Bog1 (2004); Bog2 (2004) or measurement-induced state filtering Lan (2008). This requirement resemble the quadrupole nonlinearity needed to obtain the spin squeezed states. It seems that the nature truly reveals its quantum behavior through the nonlinear effects.
The state of spin 1 may be represented as a product state of two 1/2 spins. The Hilbert space of spin 1/2 is isomorphic to the photon polarization space, thus the product representation of spin 1 is isomorphic to the two photon product polarization representation. Recently, a two qutrit cryptographic protocols have been designed for the biphoton Bre (2008), and one of them, the so called umbrella protocol, relies on a two MUBs. It is easy to see that the two MUBs of the umbrella protocol correspond to the MUBs of the analogs of position and momentum operators. One of them consists only of maximally entangled photon states, this is our basis made of null projection states, while the other one, tetrahedral to the first one, consists of the two non-entangled product states and the one maximally entangled state, which in our case is spin projection basis made of the two coherent states and the one null projection state. The authors of the protocol have shown that even using only the two of the four MUBs one can achieve greater efficiency than using the three MUB qubit protocol. Our results show that the umbrella protocol can be implemented on spin 1.
We would like to thank Genowefa Ślósarek for the discussions on spin 1 tomography. P.K. acknowledges the support from Tomasz Łuczak from subsidium MISTRZ (Foundation for Polish Science) and the Scientific Scholarship of the City of Poznań.
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- BB84 (1984) Ch.H. Bennett and G. Brassard, Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India (IEEE, New York, 1984), p. 175.
- PP (2000) H. Bechmann-Pasquinucci and A. Peres, Phys. Rev. Lett., 85, 3313 (2000).
- VAA (1987) L. Vaidman, Y. Aharonov and D.Z. Albert, Phys. Rev. Lett., 58, 1385 (1987).
- AE1 (2001) Y. Aharonov and B.-G. Englert, Z. Naturforsch., 56a, 16 (2001).
- AE2 (2001) B.-G. Englert and Y. Aharonov, Phys. Lett. A, 284, 1 (2001).
- Iv (1981) I. Ivanovič, J. Phys. A, 14, 3241 (1981).
- WF (1989) W.K. Wootters and B.D. Fields, Ann. Phys., 191, 363 (1989).
- GHW (2004) K.S. Gibbons, M.J. Hoffman and W.K. Wootters, Phys. Rev. A, 70, 062101 (2004).
- Prob (2004) http://www.imaph.tu-bs.de/qi/problems/.
- Bog1 (2004) Yu.I. Bogdanov et al., Phys. Rev. A, 70, 042303 (2004).
- Bog2 (2004) Yu.I. Bogdanov et al., Phys. Rev. Lett., 93, 230503 (2004).
- Lan (2008) B.P. Lanyon et al., Phys. Rev. Lett., 100, 060504 (2008).
- Bre (2008) I. Bregman et al., Phys. Rev. A, 77, 050301 (2008).
- KU (1993) M. Kitagawa and M. Ueda, Phys. Rev. A, 47, 5138 (1993).
- Kl (2008) A.A. Klyachko et al., Phys. Rev. Lett., 101, 020403 (2008).
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https://music.stackexchange.com/questions/70443/bass-guitar-fret-distance-formula
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math
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I just set off a new trip to the wonderful world of fretted instruments. I bought myself an electric bass guitar (cort c4h). My question to you is what is the formula of the fret distances along the fretboard. How interfret distance diminishes as you play higher notes. As a professional researcher (HEP physics) I would like to know exactly the theoretical background. I did an online search but still I am not persuaded.
In a 12-tone, equal tempered scale, we want our frequency to double (become an octave higher) every 12 semitones, and we want our semitones to be evenly-spaced.
As each fret represents a semitone, and the fundamental frequency of oscillation of an ideal string is proportionate to the reciprocal of its length, this means that every fret should be a factor of the 12th root of 2 ≈ 1.06(ish) times further from the bridge than the previous one.
So, for example, when you move from the 12th fret right to the nut, you've multiplied that spacing by 1.06ish 12 times - so the fret spacing between the nut and first fret half of what it is at the 12th fret. Wikipedia even has an amusingly specifically-titled article, Twelfth root of two, that shows how repeatedly multiplying by the number produces the equal-tempered chromatic scale.
The way the '17.817' mentioned in Laurence's answer is calculated is this:
The reason for the more complex sums here is that this number is used when talking about working out the ratios from the nut. If we imagine that the whole string length is 1, then 1/17.817 is 0.05612ish, so the length of the string when fretted at the first fret is (1-0.05612) = 0.94388. And when we work out 1/0.94388 (i.e. the ratio of the whole length of the string compared to its length when fretted at the first fret), it gets us back to that 1.06ish ratio (12th root of 2). Phew!
Sometimes slightly different fret spacing may be used to compensate for non-ideal characteristics of strings (and, as Laurence says, the slight tension increase that results in displacing the string when fretting it).
As a mathematician, I'm sure you already know the basic theoretical answer. Quoting from the first link given below:
"If you divide the scale length (the distance from the nut to the bridge) by 17.817, you end up with the first fret position starting from the nut end of the fingerboard. If you then take the scale length minus the first fret distance and divide the remainder by 17.817, you end up with the second fret position located from the first fret. Next if you take the scale length minus the first and second fret distances and divide the remainder by 17.817, you end up with the third fret position located from the second fret. This process is repeated until all frets required are located."
But the string has to be pushed down to the fret, and this increases tension. Read the whole paper:
And here's a less mathematical discussion of the subject:
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https://publications.waset.org/3852/dq-analysis-of-3d-natural-convection-in-an-inclined-cavity-using-an-velocity-vorticity-formulation
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math
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DQ Analysis of 3D Natural Convection in an Inclined Cavity Using an Velocity-Vorticity Formulation
In this paper, the differential quadrature method is applied to simulate natural convection in an inclined cubic cavity using velocity-vorticity formulation. The numerical capability of the present algorithm is demonstrated by application to natural convection in an inclined cubic cavity. The velocity Poisson equations, the vorticity transport equations and the energy equation are all solved as a coupled system of equations for the seven field variables consisting of three velocities, three vorticities and temperature. The coupled equations are simultaneously solved by imposing the vorticity definition at boundary without requiring the explicit specification of the vorticity boundary conditions. Test results obtained for an inclined cubic cavity with different angle of inclinations for Rayleigh number equal to 103, 104, 105 and 106 indicate that the present coupled solution algorithm could predict the benchmark results for temperature and flow fields. Thus, it is convinced that the present formulation is capable of solving coupled Navier-Stokes equations effectively and accurately.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1331075Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1308
H. Fasel, "Investigation of the stability of boundary layers by a finite-difference model of the Navier-Stokes equations", J. Fluid Mech., 78, 1976, 355-383.
R. E. Bellman, B.G. Kashef, J. Casti, "Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations", J. Comput. Phys., 10, 1972, 40-52.
C. Shu, Differential quadrature and its application in engineering, Springer, London, 2000.
E. Tric, G. Labrosse, M. Betrouni, "A first incursion into the 3D structure of natural convection of air in a differentially heated cubic cavity, from accurate numerical solutions", Int. J. Heat Mass Transfer, 43 , 2000, 4043-4056.
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http://ozaka.biz/library/algebra-and-trigonometry-3-rd-edition
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math
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This bestselling writer crew explains suggestions easily and obviously, with out glossing over tricky issues. challenge fixing and mathematical modeling are brought early and bolstered all through, offering scholars with a pretty good starting place within the rules of mathematical considering. complete and lightly paced, the ebook offers entire assurance of the functionality idea, and integrates an important quantity of graphing calculator fabric to aid scholars increase perception into mathematical principles. The authors' cognizance to aspect and clarity--the similar as present in James Stewart's market-leading Calculus book--is what makes this e-book the marketplace chief.
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- The Pea and the Sun: A Mathematical Paradox
Extra resources for Algebra and Trigonometry
01 zero. 001 1 2 10 a hundred one thousand L ϩ 21x ϩ y2 Յ 108 L dialogue sixty six. indicators of Numbers enable a, b, and c be genuine numbers such Ͼ b Ͼ zero, and c Ͻ zero. locate the signal of every expression. (a) Ϫa (b) Ϫc (c) bc (d) a Ϫ b (e) c Ϫ a (f) a Ϫ bc (g) c 2 (h) Ϫabc sixty five. Mailing a package deal The publish workplace will settle for purely programs for which the size plus the “girth” (distance round) isn't any greater than 108 inches. hence for the package deal within the determine, we should have (a) Will the put up workplace settle for a package deal that's 6 in. broad, eight in. deep, and five toes lengthy? What a few package deal that measures 2 feet via 2 feet by means of four toes? (b) what's the maximum applicable size for a package deal that has a sq. base measuring nine in. by means of nine in? ■ sixty eight. Irrational Numbers and Geometry utilizing the subsequent determine, clarify the right way to find the purpose 12 on a host line. are you able to find 15 via the same procedure? What approximately sixteen? record another irrational numbers that may be positioned this manner. five ft=60 in. x y œ∑2 6 in. 1 eight in. _1 zero 1 2 P. three I NTEGER E XPONENTS Exponential Notation ᭤ principles for operating with Exponents ᭤ clinical Notation during this part we evaluate the foundations for operating with exponent notation. We additionally see how exponents can be utilized to symbolize very huge and intensely small numbers. ▼ Exponential Notation A fabricated from exact numbers is generally written in exponential notation. for instance, five # five # five is written as fifty three. ordinarily, we have now the next definition. EXPONENTIAL NOTATION If a is any genuine quantity and n is a good integer, then the nth strength of a is an ϭ a # a # . . . # a 1442443 n elements The quantity a is named the bottom, and n is named the exponent. S E C T I O N P. three | Integer Exponents 15 Exponential Notation instance 1 (a) A 12 B ϭ A 12 BA 12 BA 12 BA 12 BA 12 B ϭ 321 five notice the excellence among 1Ϫ32 four and Ϫ34. In 1Ϫ3 2 four the exponent applies to Ϫ3, yet in Ϫ34 the exponent applies simply to three. (b) 1Ϫ32 four ϭ 1Ϫ32 # 1Ϫ32 # 1Ϫ32 # 1Ϫ32 ϭ eighty one (c) Ϫ34 ϭ Ϫ13 # three # three # 32 ϭ Ϫ81 ■ NOW test routines thirteen AND 15 we will be able to nation numerous priceless ideas for operating with exponential notation. to find the rule of thumb for multiplication, we multiply fifty four by way of fifty two: fifty four # fifty two ϭ 15 # five # five # 5215 # fifty two ϭ five # five # five # five # five # five ϭ fifty six ϭ 54ϩ2 1444244 forty three 123 1444442444443 four elements 2 elements 6 components it seems that to multiply powers of an identical base, we upload their exponents. usually, for any genuine quantity a and any confident integers m and n, we now have aman ϭ 1a # a # . . . # a2 1a # a # . . . # a2 ϭ a # a # a # . . . # a ϭ amϩn one hundred forty four 4244 forty three 1442443 m components n components 144424443 m ϩ n components therefore aman ϭ amϩn. we want this rule to be actual even if m and n are zero or detrimental integers. for example, we should have 20 # 23 ϭ 20ϩ3 ϭ 23 yet this may ensue provided that 20 ϭ 1. Likewise, we wish to have fifty four # 5Ϫ4 ϭ fifty fourϩ 1Ϫ42 ϭ 54Ϫ4 ϭ 50 ϭ 1 and this may be actual if 5Ϫ4 ϭ 1/54. those observations result in the subsequent definition. 0 AND unfavourable EXPONENTS If a zero is any actual quantity and n is a favorable integer, then a0 ϭ 1 and a Ϫn ϭ 1 an 0 and unfavourable Exponents instance 2 zero A forty seven B ϭ1 1 1 (b) x Ϫ1 ϭ 1 ϭ x x (a) (c) 1Ϫ22 Ϫ3 ϭ 1 1 1 ϭϪ three ϭ Ϫ8 eight 1Ϫ22 NOW attempt workout 19 ■ ▼ principles for operating with Exponents Familiarity with the subsequent principles is vital for our paintings with exponents and bases.
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http://www.eso.org/public/outreach/eduoff/vt-2004/mt-2003/mt-mercury-rotation.html
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math
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Mercury Transit on May 7, 2003
Day and Night on Mercury
The very long Mercurian Day
Imagine that astronauts have landed on Mercury - in some distant future. How they would experience a day on this planet?
It would be a very long workday from an Earth perspective! This is because Mercury's rotation around its axis lasts 59 days, and it takes 88 days to move around its orbit around the Sun. Interestingly, 59 is exactly 2/3 of 88. This is not by chance - it is an effect of the Sun's gravitational field on Mercury. It is a similar phenomenon that the Earth's Moon always has the same "face"; the Moon always turns the same side towards the Earth.
This has a most unusual effect on the length of a Mercurian "day" as measured from noon to noon (note that the word "day" is here used to mean "one daytime period + one nighttime period", corresponding to "24 hours" on Earth). In fact, such a day on Mercury is twice as long as a Mercurian "year" !
On average, 176 Earth days elapse between one sunrise and the next on Mercury - this is therefore the length of the Mercurian day .
Since Mercury revolves around the Sun in an elliptical orbit and rotates around its own axis comparatively slowly, the Sun appears to move in a strange way in the sky above that astronaut on Mercury's surface . At some moment, he or she would watch the Sun come to a complete halt. Then the Sun would appear to move backwards for some time before returning to its original position, performing a loop in the sky.
Length of Day and Year on Mercury
Day and Night on Mercury (see the text).
Each sidereal day on Mercury last 58.65 Earth days. This means that it takes 58.65 Earth days (or 2/3 of one Mercury year) for Mercury to turn once around its axis, relative to the background stars.
Now look at the figures above. They show how Mercury orbits the Sun, while it turns around its own axis. The red marks indicate the same spot on the surface at different times, when Mercury is at different locations in the orbit.
Suppose you stand on the surface at that red mark. To begin with, on Day 1 (note that this count is in Earth days!), you see the Sun at the horizon, as indicated on the left drawing. Very very slowly, because of Mercury's rotation around its axis and motion along its orbit, you see how the Sun moves upwards from the horizon and across the Mercurian sky.
On Day 44 , the sun finally reaches the highest point in the sky - it is now "noon" for you. The afternoon begins and on Day 88 , the Sun finally sets. So the "daytime" on Mercury lasts no less than 88 Earth days!
After the Sun has set (look at the right drawing), the night falls, midnight happens 44 Earth days later (on Day 132 ), and the Sun again rises after another 44 Earth days on Day 176 . So the "24 hours" on Mercury last two full orbital revolutions, or 176 Earth days!
In this sense, one Mercury day lasts exactly two Mercury years !
The Sun's motion in the sky above Mercury
The length of the day is only one strange effect on Mercury. The exact motion of the Sun in the Mercurian sky and its apparent size is another.
Imagine that you stand at one of the two "hot spots" on Mercury's surface, in the Caloris Basin . Since Mercury's orbit around the Sun is quite elliptical, the distance between Mercury and the Sun changes during the orbital motion and the size of the Sun as seen in the Mercurian sky will therefore change dramatically during the "daytime."
At the point closest to the Sun, Mercury is about 46 million km away from the central star; in the remotest orbital point, this distance is nearly 70 million km. The corresponding sizes of the solar disk in the Mercurian sky are 1.73° and 1.14°, respectively, that is 3.2 and 2.1 the size of the Sun as seen from Earth.
From your location, you would see the Sun rise small and then grow in size as it moves to the overhead position. Moreover, because Mercury moves faster when it is closer to the Sun, the apparent motion of the Sun in the sky will reflect this - for some time, the change of angle of the line-of-sight towards the Sun caused by the orbital motion is "faster" than that caused by the planet's rotation around its axis. At the time of closest approach to the Sun, this effect will cause the Sun to move temporarily backwards(!) in the sky . Then it resumes its normal westward motion, and as Mercury moves farther from the Sun, the solar disk begins to shrink in size until sunset, 88 Earth days after sunrise. What a day!
And then, for a period lasting 88 Earth days, it is night for you.
On the Earth, things are not so complicated. We don't spin one and a half times per year around the Earth's axis - we spin much faster in relative terms, just over 365 times per year. That is why the time interval between two successive noons (when the Sun is highest in our terrestrial sky) is only 23 hours and 56 minutes, and not 176 days!
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| 4,877 | 23 |
https://slideplayer.com/slide/219473/
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math
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Presentation on theme: "Scientific Method Method of scientific investigation Four MAJOR steps:"— Presentation transcript:
1 Scientific Method Method of scientific investigation Four MAJOR steps: Identify a problemForm a hypothesisPerform an experimentConclusion
2 Identify a problem Phenomena that requires investigation Questions Why?How?What?When?Where?
3 Form a Hypothesis Make observations Collect data Research previous informationUse observations, data, research to form aHYPOTHESISAn educated ‘guess’ that bestanswers the problem
4 Conduct an Experiment Test hypothesis by controlling variables An experiment should test the hypothesis onlyOther variables must be strictly controlledInsures validityExperiments must be reproducibleIndependent scientists must be able to performthe same experiment and obtain the same resultsreliability
5 Conclusion Test results determine whether hypothesis was true or not trueIf hypothesis is false, redesign and rerun experimentHypothesis are often falseLeads to discovery of the truth
6 Theory HYPOTHESIS supported by many experiments over time Assumed to be the explanation for phenomenaCan NEVER be provedCan be DISPROVED with new scientific evidence
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https://testmaxprep.com/lsat/community/100003722-help
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math
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September 2018 LSAT Section 3 Question 24
Skylar on September 28, 2019@chris_va No, it is not because the percentages are not varied enough. Instead, the flaw comes from the fact that the argument is comparing percentages of two different groups whose sizes are unknown. If there are 1,000 dog owners and only 100 cat owners, the percentage of degree-holders would be higher in cat homes while the actual number of degree-holders would be higher in dog homes. Therefore, we cannot make a statement about what is more likely unless we know that the overall group sizes are comparable.
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https://www1.rcsb.org/structure/6LFO
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math
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Structural basis of CXC chemokine receptor 2 activation and signalling.Liu, K., Wu, L., Yuan, S., Wu, M., Xu, Y., Sun, Q., Li, S., Zhao, S., Hua, T., Liu, Z.J.
(2020) Nature 585: 135-140
- PubMed: 32610344
- DOI: https://doi.org/10.1038/s41586-020-2492-5
- Primary Citation of Related Structures:
6LFL, 6LFM, 6LFO
- PubMed Abstract:
Chemokines and their receptors mediate cell migration, which influences multiple fundamental biological processes and disease conditions such as inflammation and cancer 1 . Although ample effort has been invested into the structural investigation of the chemokine receptors and receptor-chemokine recognition 2-4 , less is known about endogenous chemokine-induced receptor activation and G-protein coupling. Here we present the cryo-electron microscopy structures of interleukin-8 (IL-8, also known as CXCL8)-activated human CXC chemokine receptor 2 (CXCR2) in complex with G i protein, along with a crystal structure of CXCR2 bound to a designed allosteric antagonist. Our results reveal a unique shallow mode of binding between CXCL8 and CXCR2, and also show the interactions between CXCR2 and G i protein. Further structural analysis of the inactive and active states of CXCR2 reveals a distinct activation process and the competitive small-molecule antagonism of chemokine receptors. In addition, our results provide insights into how a G-protein-coupled receptor is activated by an endogenous protein molecule, which will assist in the rational development of therapeutics that target the chemokine system for better pharmacological profiles.
University of Chinese Academy of Sciences, Beijing, China.
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| 1,651 | 9 |
http://web.centre.edu/econed/Profit%20and%20Average/marginal.htm
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math
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1. The total product schedule of company X which produces T-shirts is described by the following:
Labor Output (workers per week) (units per week)
1.1 Draw the average and marginal product curves of company X.
1.2 What is the relationship between the average and marginal product curves?
1.3 Suppose the price of labor is $400 per week, and the total fixed cost is $1,000 per week, calculate the total cost, the average cost and the marginal cost for each level of output.
1.4 What will be the effects on the average and marginal cost curves if the price of labor increases from $400 per week to $450 per week?
1.5 Suppose company X can buy an additional factory so that its total output can be doubled. If the fixed cost of operating the second factory is the same as the first one and the wage rate is $400 per week, draw the long-run average cost curve of company X. Over what range of output would it be efficient for company X to operate two factories?
2. Which concept of profits is implied in the following quotations:
2.1 "Profits are necessary if firms are to stay in business."
2.2 "Profits are signals for firms to expand production and investment."
2.3 "Accelerated depreciation allowances lower profits and thus benefit the companys owners."
3. Indicate whether each of the following conforms to the hypothesis of diminishing returns, and if so, whether it refers to marginal returns, average returns, or both.
3.1 "The bigger they are, the harder they fall."
3.2 "Five workers produce twice as much today as 10 workers did 40 years ago."
Production & Cost - Discussion Topic Questions:
(I) A carpenter quits his job at a furniture factory to open his own business. In his first two years of operation, his sales average $100,000 and his operating costs for wood, workshop and tools rental, utilities and miscellaneous expenses average $70,000. Now his old job at the furniture factory is again available. Should he take it or remain in business for himself? How would you make this decision?
(II) Discuss the effects of a labor-saving invention that permits all goods to be manufactured with less labor than before. Does it represent an increase in productivity in production? Which group in a society might oppose to such invention and Why?
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http://www.wolfram.com/broadcast/video.php?c=89&p=3&v=154
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math
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Learn how »
Both a Logo and an Exhibit: Mathematica and the Museum of Mathematics Logo Concept
The new Museum of Mathematics has an infinite family of logos implemented with Mathematica. Hear how the concept came about in this talk from the Wolfram Technology Conference 2011.
318 videos match your search.
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http://www.lessonplanet.com/lesson-plans/associative-property/3
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math
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Associative Property Teacher Resources
Find Associative Property educational ideas and activities
Showing 41 - 60 of 237 resources
Complex Number Properties
High schoolers experiment with complex numbers to see which properties of real numbers apply to the system of complex numbers and practice computations. The study of these properties is first explored by looking at rotational transformations and completing a chart of the rotations. This lesson is then extended to looking at complex numbers and their properties.
Use a learning activity with your youngsters in order to establish an understanding of patterns in a small addition table. Help learners recognize that adding two odd numbers will produce an even number, and adding two even numbers will also produce an even number. Alternating patterns in the table show an even number plus an odd number will produce an odd number. Teacher commentary and directions through guided practice allow learners to be exposed to the associative property of addition by visualizing the different ways to write the sum of a number.
Commutative Property and Associative Property
Students study the properties of multiplication. In this multiplication properties lesson, students study the commutative and associative properties of multiplication. Students then complete the worksheets to apply the properties and solve the problems.
Multiplying Using the Commutative and Associative Properties
Third graders practice using the commutative and associative properties. In this multiplication instructional activity, 3rd graders simplify a series of problems using multiplication sentences and complete a worksheet.
Properties of Multiplication
Properties of multiplication can get confusing, and are incredibly important to mathematicians. This worksheet is helpful in that it first explains the properties (commutative, associative, and distributive), giving examples of each. Then, scholars complete six multiple-choice problems during which they must choose the equation which shows the property listed. A second worksheet gives a more challenging option, with less explanation and eight problems. Answers are provided.
Properties of Operations as Strategies- Independent Practice Worksheet
Find examples of math properties! On each of these two worksheets, learners examine five sets of number sentences, marking the ones which demonstrate the indicated property. The first page deals with the associative property. For each of the sets, there are four sentences, and some have more than one that fit the bill. On the second page, they do the same thing while looking for the commutative property. This will work well to reveal common misconceptions about these properties.
Challenge: Parentheses, Please!
Given a string of numbers and math symbols, learners take the challenge to insert parenthesis in the proper places to reach certain solutions. This is a challenge, indeed, an engaging enrichment for those who need a little more math!
Famous Public Properties
What can a middle schooler do in 90 minutes? He can practice using the commutative, associative and distributive properties of addition and multiplication. He can also simplify expressions using the commutative, associative and distributive properties. Now that's time well spent!
Properties of Operations
In this algebra worksheet, students identify different properties of integers and their operations. There are 30 questions with an answer key.
Properties of Numbers
Use this math worksheet to have learners are given six problems involving the distributive, commutative and associative properties of numbers.
NUMB3RS Activity: A Group of Symmetries
Students investigate the symmetries of an equilateral triangle. In this geometry lesson plan, students explore and visualize the reflection and rotational symmetries of an equilateral triangle. The activity relates the geometry of the triangle to some of the axioms commonly studied by high school students.
Properties of Multiplication
Third graders explore the properties of multiplication. In this computation lesson plan, 3rd graders participate in a drill and answer several factorization and addition problems. Students then work independently and use counters to solve multiplication problems with the same factors.
Vocabulary: Addition Terminology
In this addition terminology learning exercise, learners complete many activities where they define and match addition terminology. Students complete 11 pages of activities.
Parentheses, Please! - Enrichment 5.6
Upper graders use the Associative Property to solve problems with groups of numbers in equations with several operations. They write answers to seven problems.
The Human Calculator
Young scholars amaze their friends and families with this human calculator trick. In this algebraic activity, students use the associative property of addition to act as a human calculator and stump their volunteers. The steps to the trick, along with further investigation suggestions are listed.
Easy Worksheet: Properties
For this properties worksheet, 10 short answer problems are solved. Learners name the property illustrated such as the inverse property or associative property of multiplication.
Seventh graders explore the concept of properties. In this properties lesson, 7th graders discuss the various math properties including the commutative, associative, distributive, multiplication properties and so on. Students create foldables with examples of various properties.
Students identity properties of addition and multiplication. In this addition and subtraction properties instructional activity, students explore commutative, associative, and distributive properties as they design posters that feature the properties and examples of them in use.
Properties: Commutative, Associative, and Identity
Students explore the concept of addition and multiplication properties. In this addition and multiplication properties lesson, students create a foldable with the properties commutative, associative, identity, and distributive written in four quadrants. Students put examples of each property in the corresponding quadrant. Students use twelve tiles to explore the concept of dividing by zero. Students divide the twelve tiles into various size groups.
Addition Properties and Subtraction Rules: ELL
In this ELL addition property and subtraction rule worksheet, students read about the commutative and associative properties, then underline the correct term in 7 sentences. Houghton Mifflin text is referenced.
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http://www.princeton.edu/~achaney/tmve/wiki100k/docs/Phase_inversion.html
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math
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Phase inversion means the swapping of the two poles of an alternating current source. A phase inversion is neither a time shift nor a phase shift, but simply a swap of plus and minus.
For example, in a push-pull power amplifier using vacuum tubes, the signal is most often split by a phase splitter (aka phase inverter) stage which produces two signals, one in phase, and the other out of phase, that is, phase inverted. These two signals then drive the two halves of the first push-pull stage, which may be either the output stage (in which case the phase splitter will be in between the driver stage if there is one and the output stage) or the driver stage. The other common arrangements for driving a push-pull stage are by using an isolation transformer to produce the split signals, or by using the in-phase half of the first push-pull stage to drive the other half. A common circuit using this last technique is the long-tailed pair, often seen in television sets and oscilloscopes.
In solid state electronics all of these techniques can be used, and phase inversion can also be produced by the use of NPN/PNP complementary circuitry, which has no corresponding technique in vacuum tube designs.
Phase inversion may occur with a random or periodic, symmetrical or non-symmetrical waveform, although it is usually produced by the inversion of a symmetrical periodic signal, resulting in a change in sign. A symmetrical periodic signal represented by f (t ) = A ejωt, after phase inversion, becomes f 1(t ) = Aej(ωt +π), where t is time, A is the magnitude of the vector, ω is angular frequency (ω = 2πf ), where f is the frequency and π ≈3.1416 and e ≈ 2.7183. The algebraic sum of f (t ) and f 1(t ) will always be zero.
This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C".
Full article ▸
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https://www.univerkov.com/the-radius-of-the-smaller-base-of-the-truncated-cone-is-6-cm-and-the-generatrix-is-5-cm/
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math
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The radius of the smaller base of the truncated cone is 6 cm, and the generatrix is 5 cm, inclined to the base plane at an angle of 60 degrees. Find S side, S full.
To determine the radius of the circle of the larger base, draw the height of BB1. Radius AO = OB + AB. In a right-angled triangle, the angle AB1B = 180 – 90 – 60 = 30. Then the leg AB lies opposite the angle 30 and is equal to half the length of the hypotenuse AB1. AB = 5/2 = 2.5 cm.Then AO = 6 + 2.5 = 8.5 cm.
Let’s define the lateral surface area.
Sside = n * (AO + B1O1) * AB1 = n * (8.5 + 6) * 5 = n * 72.5 cm2.
Determine the areas of the bases.
S1 = n * OA ^ 2 = n * 8.5 ^ 2 = n * 72.25 cm2.
S2 = n * O1B1 ^ 2 = n * 6 ^ 2 = n * 36 cm2.
Determine the total area of the cone.
S floor = S side + S1 + S2 = n * 72.5 + n * 72.25 + n * 36 = n * 180.75 cm2.
Answer: Sside = n * 72.5 cm2, Spol = n * 180.75 cm2.
One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.
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https://www.silviacolasanti.it/science/mathematics/145515-bifurcation-of-maps-and-applications-volume-36-(north-holland-mathematics-studies)-author-unknown-fb2-epub.html
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math
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Bifurcation of maps and applications, Volume 36 (North-Holland Mathematics Studies).
Bifurcation of maps and applications, Volume 36 (North-Holland Mathematics Studies). Bifurcation of maps and applications, Volume 36 (North-Holland Mathematics Studies).
Start by marking Bifurcation of Maps and Applications. North-Holland Mathematics Studies, Volume 36. as Want to Read: Want to Read savin. ant to Read. by Gerard Iooss.
View all volumes in this series: North-Holland Mathematics Studies. Bookmarks, highlights and notes sync across all your devices. Interactive notebook and read-aloud functionality.
Author(s): Author Unknown. The Mathematical Theory Of Geophysical Fluid Dynamics, Volume 41 (north-holland Mathematics Studies) Nonstandard Methods In Stochastic Analysis And Mathematical Physics, Volume 122 (pure And Applied Mathematics).
Series: North-Holland Mathematics Studies 3. Other readers will always be interested in your opinion of the books you've read. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them
Series: North-Holland Mathematics Studies 36. File: PDF, . 7 MB. Читать онлайн. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. 1. Biochemistry of atherosclerosis.
North-Holland Mathematics Studies Read 1100 articles with impact on. .The related notion of spectrum and spectral mapping theorem are given
Two theses are advanced: (1) The study of spectral invariants can and should be extended to operators with continuous spectra. 2) The subject is closely related to the asymptotic approximation of eigenfunctions by a local amplitude and a phase integral. This program has been carried out in the case of vector-valued functions of one variable. The related notion of spectrum and spectral mapping theorem are given. The construction is illustrated by a simple example of calculus and spectrum of non-normal n x n matrix.
Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies). design and analysis of algorithms integral equations mathematics and statistics numerical analysis partial differential equations. Contact Us Switch to tabbed view. Li Peng, Yong Zhou, Bifurcation from interval and positive solutions of the three-point boundary value problem for fractional differential equations, Applied Mathematics and Computation, . 57 ., . 58-466, April 2015.
Iooss, Bifurcation of Maps and Applications, Mathematics Studies No. 36 North Holland (1979). Burns K. (1981) Lectures on bifurcation from periodic orbits. In: Rand . Young LS. (eds) Dynamical Systems and Turbulence, Warwick 1980. 4. G. Iooss & . Joseph, Elementary Stability and bifurcation Theory, Undergraduate Textbook in Mathematics, Springer (1980). 5. Joseph, Bifurcation and Stability of nT-periodic solutions at a point of resonance, Arch. Lecture Notes in Mathematics, vol 898. Springer, Berlin, Heidelberg. First Online 07 October 2006.
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https://acronyms.thefreedictionary.com/CHARMM
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math
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CHARMM - What does CHARMM stand for? The Free Dictionary
Also found in: Wikipedia
|CHARMM||Chemistry At Harvard Molecular Mechanics|
References in periodicals archive
After ligand preparation, hydrogen bonds were added and energy minimization was done using CHARMM
If the project is classical mechanical, researchers and students primarily use CHARMM
(Chemistry at HARvard Molecular Mechanics) or AMBER (Assisted Model Building with Energy Refinement).
The research cluster runs standard versions of molecular engineering and simulation applications including AMBER, CHARMM
Veridian PBS-Pro, Wolfram Mathematica, NAG, Absoft, Lahey, Backbone Networks, NAMD, CHARMM
, and TurboGenomics.
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http://www.hups.mil.gov.ua/periodic-app/article/4103/eng
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math
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Description: The problem of rational structure determining for the printing edition is analyzed. The reason for this problem lies in the need to minimize and to justify the production cost of the printing products. At that it is necessary to choose the best item from the set of structures, which are generated at a given cost in the terms of the material use reduction, edition structure optimization (base elements), selection of optimal edition format. It is especially important for children’s books, because the complex edition structure and additional element are used in the production. The multicriteria statement of the problem, where the criteria of solution efficiency are the maximum of the additional elements quantity, the maximum of the material quality in terms of the material density, the maximum of the edition format and quantity of colors and the production cost constraint is considered in the article.
Keywords: Mathematical model, rational structure, printing edition, multycriteria, production cost
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https://www.dr-mikes-math-games-for-kids.com/challenging-math-puzzle.html
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By Michael Hartley
Here's a really challenging math puzzle for kids, and for adults too. I've seen kids baffle away at this challenging puzzle for months! Each time they saw me they'd ask how to solve it! Although an eight-year-old can understand the puzzle easily enough, and enjoy it, kids from ten up are the ones who have a realistic chance of cracking it.
Well, there's the grid. All you have to do is arrange the numbers 1 through to 8 in the grid shown.
The rules? No consecutive numbers should be next to each other, either horizontally, vertically or diagonally. Sounds simple? Have a go! You will realize fairly soon why I call it challenging maths puzzle.
You might find that it takes kids a few tries before they properly "get" the rules. They'll come, show you their "solution", and you'll see the 7 and 8 in two boxes joined at the corner, for example. If you explain why their solution is not correct, and smile in just the right way, they'll try and try and try and try again to solve it.
If you want to test your guess, use the form below!
If you like what you've just read, sign up for this site's free newsletters:
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| 1,133 | 7 |
https://www.jiskha.com/display.cgi?id=1298927532
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math
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posted by Jared .
A square is 7 meters on a side. To the nearest tenth of a meter, how long is the diagonal of the square?
The diagonal of a square forms two
The sides of a 45-45-90 triangle are in the ratio,
a : b : hypotenuse
x : x : x(sqrt(2))
The diagonal is the hypotenuse of this 45-45-90 triangle.
Side = 7
7 * (sqrt(2)) = ?
a^2 + b^2 = c^2
7^2 + 7^2 = c^2
49 + 49 = 98
c = square root of 98
Be sure to round to the nearest tenth.
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https://mevigusabepogyqec.saltybreezeandpinetrees.com/classical-abstract-algebra-book-4453pd.php
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math
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5 edition of Classical abstract algebra found in the catalog.
|Statement||Richard A. Dean.|
|LC Classifications||QA162 .D43 1990|
|The Physical Object|
|Pagination||xxix, 524 p. :|
|Number of Pages||524|
|LC Control Number||89048329|
Buy a cheap copy of Abstract Algebra, 3rd Edition book by I.N. Herstein. Providing a concise introduction to abstract algebra, this work unfolds some of the fundamental systems with the aim of reaching applicable, significant results. Free shipping over $ Algebra The word \algebra" means many things. The word dates back about years ago to part of the title of al-Khwarizm ’s book on the subject, but the subject itself goes back years ago to ancient Babylonia and Egypt. It was about solving numerical problems that we would now identify as linear and quadratic equations.
Richard A. Dean is the author of Elements of Abstract Algebra ( avg rating, 2 ratings, 0 reviews, published ), Classical Abstract Algebra ( a /5(3). The development of abstract algebra was propelled by the need for new tools to address certain classical problems that appeared insoluble by classical means. A major theme of the book is to show how abstract algebra has arisen in attempting to solve some of these classical problems, providing a context from which the reader may gain a deeper.
The development of abstract algebra was propelled by the need for new tools to address certain classical problems that appeared insoluble by classical means. A major theme of the book is to show how abstract algebra has arisen in attempting to solve some of these classical This book does nothing less than provide an account of the intellectual /5(3). The three volume Lectures are based on Jacobson's graduate lectures on algebra at Johns Hopkins and Yale in the 's and early 's, and are very careful, comprehensive and classical in style, giving a general treatment of abstract algebra. The first volume gives a comprehensive introduction to abstract algebra and its basic concepts.
Sports centre management & safety.
complete Bible, an American translation
Construction with moving forms.
Hard Scrabble; observations on a patch of land.
County Business Patterns New Mexico 1998 (County Business Patterns New Mexico)
We make our own candy
On the significance of Poissons ratio for floating ice
Atomic Physics of Highly Charged Ions
Deal with your debt
All the day long
Social learning theory.
Studies on temperature-sensitive mutants of mouse cytomegalovirus
Hammer of God
Classical Abstract Algebra Ed 1st Prtgth Edition by Richard A. Dean (Author) › Visit Amazon's Richard A. Dean Page. Find all the books, read about the author, and more. See search results for this author. Are you an author. Learn about Author Central. Richard. Book Description This carefully written textbook offers a thorough introduction to abstract algebra, covering the fundamentals of groups, rings and fields.
The first two chapters present preliminary topics such as properties of the integers and equivalence relations. Algebra Abstract and Concrete. The book, Algebra: Abstract and Concrete provides a thorough introduction to algebra at a level suitable for upper level undergraduates and beginning graduate students.
The book addresses the conventional topics: groups, rings, fields, and linear algebra, with symmetry as a unifying theme. This book takes a "group-first" approach to introductory abstract algebra with rings, fields, vector spaces, and Boolean algebras introduced later.
Throughout the textbook, in addition to the examples and theory, there are several practical applications of abstract algebra with a particular emphasis on computer science, such as. Abstract Algebra Theory and Applications.
This text is intended for a one- or two-semester undergraduate course in abstract algebra. Topics covered includes: The Integers, Groups, Cyclic Groups, Permutation Groups, Cosets and Lagrange’s Theorem, Algebraic Coding Theory, Isomorphisms, Normal Subgroups and Factor Groups, Matrix Groups and Symmetry, The Sylow Theorems, Rings.
Special Relativity and Classical Field Theory: The Theoretical Minimum $ #6. A Book of Abstract Algebra: Second Edition (Dover Books on Mathematics) Charles C Pinter.
out of 5 stars Paperback. $ #7. Algebra: Fully Solved Equations To Explain Everything You Need To Know To Master Algebra. (Content Guide Included Book 1). A book of abstract algebra / Charles C. Pinter. — Dover ed. Originally published: 2nd ed. New York: McGraw-Hill, Includes bibliographical references and index.
ISBN ISBN 1. Algebra, Abstract. Title. QAP56 ′—dc22 Manufactured in the United States by Courier. A Book of Abstract Algebra Easy, readable, friendly guide. Great first text to start. Pinter, 2nd Ed Royden Real Analysis Royden, 4th Edition Rudin Principles of Mathematical Analysis "The Bible of classical analysis," difficult as a first text Rudin, 3rd Ed Spivak Calculus.
Fundamentals of Abstract Algebra by Malik, Sen & Mordeson is a very good book for self topics are covered in detail with many interesting examples and it provides hints and answers to difficult questions making it suitable for self study.
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings (group rings, division rings, universal enveloping algebras), as well as an.
Classical Abstract Algebra by Dean, Richard A. and a great selection of related books, art and collectibles available now at - Classical Abstract Algebra by Dean, Richard a - AbeBooks. This book provides a complete abstract algebra course, enabling instructors to select the topics for use in individual classes.
Complete proofs are given throughout for all theorems. This revised edition includes an introduction to lattices, a new chapter on tensor products and a discussion of the new () approach to the Lasker-Noether s: Abstract algebra emerged around the start of the 20th century, under the name modern algebra.
Its study was part of the drive for more intellectual rigor in mathematics. Initially, the assumptions in classical algebra, on which the whole of mathematics (and major parts of the natural sciences) depend, took the form of axiomatic systems.
The development of abstract algebra was propelled by the need for new tools to address certain classical problems that appeared unsolvable by classical means. A major theme of the approach in this book is to show how abstract algebra has arisen in attempts to solve some of these classical problems, providing context from which the reader may 4/5(2).
Michael Tsfasman, Serge Vlǎduţ, Dmitry Nogin. Aug Coding Theory, Algebraic Geometry. This carefully written textbook offers a thorough introduction to abstract algebra, covering the fundamentals of groups, rings and fields.
The final part contains applications to public key cryptography as well as classical straightedge and compass constructions. Explaining key topics at a gentle pace, this book is aimed at undergraduate.
Free Kindle Math Books. Algebra I. Geometry. Trigonometry. Abstract Algebra — Number Theory, Springer Math Books.
A Classical Introduction to. General abstract algebra Jacobson, Basic algebra II. This is perhaps the only really advanced general-algebra book; it contains chapters on categories, universal algebra, modules and module categories, classical ring theory, representations of finite groups, homological algebra, commutative algebra, advanced field theory.
Abstract: “Classical groups”, named so by Hermann Weyl, are groups of matrices or quotients of matrix groups by small normal subgroups. Thus the story begins, as Weyl suggested, with “Her All-embracing Majesty”, the general linear group \(GL_n(V)\) of all invertible linear transformations of a vector space \(V\) over a field \(F\).
is enormous and what the reader is going to find in the book is really only the tip of the iceberg; a work that is like a taste sampler of classical algebraic geometry.
It avoids most of the material found in other modern books on the subject, such as, for example, where one can find many of the classical results on algebraic curves.
The development of abstract algebra was propelled by the need for new tools to address certain classical problems that appeared unsolvable by classical means. A major theme of the approach in this book is to show how abstract algebra has arisen in attempts to solve some of these classical problems, providing context from which the reader may.Classical abstract algebra.
[Richard A Dean] Home. WorldCat Home About WorldCat Help. Search. Search for Library Items Search for Lists Search for Contacts Search for a Library.
Create Book, Internet Resource: All Authors / Contributors: Richard A Dean. Find more information about: ISBN: OCLC Number.Algebra - Algebra - Classical algebra: François Viète’s work at the close of the 16th century, described in the section Viète and the formal equation, marks the start of the classical discipline of algebra.
Further developments included several related trends, among which the following deserve special mention: the quest for systematic solutions of higher order equations, including.
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| 9,354 | 48 |
https://www.safaribooksonline.com/library/view/maths-a-students/9781139635462/html/chapter04.html
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math
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4Some trigonometry and geometry of triangles and circles
This chapter reminds you of what trig is for, and how it works in triangles. It also explains some of the special geometrical properties of triangles and circles, because they may be very useful to you in applications of maths to your own special subject area.
The chapter is divided into the following sections.
4.A Trigonometry in right-angled triangles
(a) Why use trig ratios?
(b) Pythagoras’ Theorem,
(c) General properties of triangles
(d) Triangles with particular shapes,
(e) Congruent triangles – what are they, and when?
(f) Matching ratios given by parallel lines,
(g) Special cases – the sin, cos and tan of 30°, 45° and 60°,
(h) Special relations of sin, cos and tan
4.B Widening the ...
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| 766 | 13 |
https://forum.grasscity.com/threads/need-advice-is-this-joint-good-for-a-begginner.1019316/
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math
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My friend wants to try weed for the 1st time, so I rolled a joint, but I need advice. Im not sure if this one is well done. I used 2 papers because one seemed to be not enough for the amount of weed I used. Do you guys think this thing will burn nice? Maybe is too tight?
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| 271 | 1 |
http://forums.mikeholt.com/showthread.php?t=53777
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math
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Re: Table 220.3(A) General Lighting Loads by Occupancy
Since this facility has squash courts, why not use the 2 VA per ft^2 allowed for “court rooms”? (Sorry, couldn’t resist).
When I get involved in a project for which the occupancy is not listed on Table 220.3(A), I try to find the nearest logical match. In your case, I would use the 1 VA per ft^2 allowed for “auditoriums.” Then I would calculate the actual VA for the selected fixtures (using the fixture rating, even if the selected light bulbs use a lower amount of power). I would submit as my final calculation whichever of these two gives the higher load.
Charles E. Beck, P.E., Seattle
Comments based on 2014 NEC unless otherwise noted.
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| 708 | 5 |
https://www.easycalculation.com/engineering/mechanical/screw-thread-calculation.php
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math
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Shear Area Calculator - External Screw Thread Calculation
Mechanical calculator to calculate the shear area of the external thread (Screw or Fastener) which depends upon minor diameter of the tapped hole.
External Thread (Screw or Fastener) Shear Area Calculation
Code to add this calci to your website
Kn max = Maximum minor diameter of internal thread
Es min = Minimum pitch diameter of external thread
Le = Fastener thread engagement
n = Number of threads per inch
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http://blog.case.edu/bcg8/2007/03/24/new_lie_group
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math
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March 24, 2007
New Lie Group
What do you get when you mix 18 mathematicians, 4 years of research, and 77 hours of supercomputer computation - mapping of the Lie group E8.
It describes the symmetries of a 57-dimensional object that can in essence be rotated in 248 ways without changing its appearance.[VIA: 025.431: The Dewey blog: Fearful symmetry]
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| 349 | 4 |
http://3d2f.com/programs/22-237-desktopcalc-download.shtml
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math
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MultiplexCalc 5.4.19 MultiplexCalc is a multipurpose and comprehensive desktop calculator for Windows. It can be used as an enhanced elementary, scientific, financial or expression calculator.
Scientific Calculator - ScienCalc 1.3.6 ScienCalc is a convenient and powerful scientific calculator. ScienCalc calculates mathematical expression. It supports the common arithmetic operations (+, -, *, /) and parentheses.
Description: DesktopCalc is an enhanced, easy-to-use and powerful scientific calculator with an expression editor, printing operation, result history list and integrated help. DesktopCalc gives students, teachers, scientists and engineers the power to find values for even the most complex equation set.
DesktopCalc uses Advanced DAL (Dynamic Algebraic Logic) mechanism to perform all its operation with the built-in 38-digit precision math emulator for high precision.
DesktopCalc combines fast "just-one-click" interface with broad set of functions. It was designed as a tool that is convenient for both elementary and scientific calculator. DesktopCalc features include the following:
* Possibility to enter mathematical formulas as with a keyboard as with built-in button-panels.
CalculatorX 1.2 .6688 CalculatorX is an enhanced expression calculator. It supports common operations, constants,built-in and custom functions, variables and note-lines. You can even use Binary, Octal
Tape Continues Its Role In The Backup Equation While new disk-to-disk appliances, virtual tape libraries or software that turns servers into backup appliances are catching on, one constant remains: The backup world still depends on tape.
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http://heavenshenge.blogspot.com/2015/05/a-view-from-satellite.html
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View from a satellite looking roughly westwards over a computer simulated landscape of the Preseli Mountains:
View from a satellite looking roughly eastwards over a computer simulated landscape of the Preseli Mountains:
Both of these pictures look fairly ordinary. But one of the two must be looking down: Feddau is below Eryr in the first picture and Eryr is below Feddau in the second picture. But if one is looking down, where is the horizon?
This is a question I was asked on the Simon Mayo show: Can you see the curvature of the Earth from a mountain? The answer is that you can see a disk shape, but the disk you can see is not necessarily the curvature of the Earth: You could be on a flat disk. However, from two tall mountains of the same height, you can prove that the Earth is curved if you can see the horizon (below the slope of the other mountain) at sunrise: Sunrise or sunset is the time that the horizon can be precisely seen.
The angles involved are tiny: This experiment with mountain peaks only seems to work at a place called Preseli in Wales where two high mountain peaks are approximately of the same height, are aligned approximately east-west and have no obscured views for a long distance in either direction: The angles to the horizon at this particular height would show that Ireland and England would not exist if you are on a flat disk.
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| 1,366 | 5 |
https://www.blackpenredpen.com/discussions/video-ideas/lovely-geometry-sin-10-sin-50-cubics-from-80-80-20-isosceles
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For a long time I've been fascinated by the way you can tile the 80 80 20 isosceles with smaller isosceles/equilateral triangles, like this
I also really like the geometric determination of sin 18 using recursive triangles, which you've epxlored on video.
Have you looked at combining the two? I realised today that the 80 80 20 tiling very quickly and elegantly shows that y=2sin(10) has to satisfy y=1/(3-y^2) ==> y^3-3y^2+1. After adjusting for the factor of 2 this is the same cubic as you talk about on your sin 10 video.
To see this, start with the isosceles tiling from cut-the-knot above, setting the base of the big 80 80 20 triangle to 1; this length then propagates through all the isosceles triangles in the diagram; and set y=2sin(10) as shown.
Because ABC and BCD are similar we know their sides are in proportion so y/1 = 1/AB = 1/(AF+1).
Now focussing on EFA we can add points G and H along FA so that FEH and AEG are congruent to each other and to BCD, so EG = EH =y; and because EGH is similar we know that GH = y^2, so AF = AG +FH - GH = 1+1-y^2 = 2-y^2. Substituting into y=1/(AF+1) gives y=1/(3-y^2) ==> y^3-3y+1=0, QED.
This diagram also gives you the connection with cos40=sin50: the two sides of the big isosceles triangle ABC, namely AB and AC have to be equal. We know AB=3-y^2 as just shown and can see that AC=1+y+2sin50. Equating these gives sin50=1-y/2-(y^2)/2.
Well done Josh. It's quite astonishing how pretty much all the relationships we take for granted (Pythagorous, sine law, cosine law, differentials in calculus, dot product etc etc .......) all come down to one basic property of triangles. If you increase the sides proportionally in any triangle, then the angles remain the same. A basic property of Eulidean Space that all trig, geometry and calculus depend directly upon.
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| 1,815 | 8 |
http://www.e-booksdirectory.com/details.php?ebook=5447
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math
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by Hansjoerg Geiges
Publisher: arXiv 2004
Number of pages: 86
This is an introductory text on the more topological aspects of contact geometry, written for the Handbook of Differential Geometry vol. 2. After discussing (and proving) some of the fundamental results of contact topology (neighbourhood theorems, isotopy extension theorems, approximation theorems), I move on to a detailed exposition of the original proof of the Lutz-Martinet theorem.
Home page url
Download or read it online for free here:
by Jie Wu - National University of Singapore
Contents: Tangent Spaces, Vector Fields in Rn and the Inverse Mapping Theorem; Topological and Differentiable Manifolds, Diffeomorphisms, Immersions, Submersions and Submanifolds; Examples of Manifolds; Fibre Bundles and Vector Bundles; etc.
by Ana Cannas da Silva - Springer
An introduction to symplectic geometry and topology, it provides a useful and effective synopsis of the basics of symplectic geometry and serves as the springboard for a prospective researcher. The text is written in a clear, easy-to-follow style.
by Bjorn Ian Dundas - Johns Hopkins University
This is an elementary text book for the civil engineering students with no prior background in point-set topology. This is a rather terse mathematical text, but provided with an abundant supply of examples and exercises with hints.
by Ana Cannas da Silva
The text covers foundations of symplectic geometry in a modern language. It describes symplectic manifolds and their transformations, and explains connections to topology and other geometries. It also covers hamiltonian fields and hamiltonian actions.
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| 1,628 | 14 |
https://www.arxiv-vanity.com/papers/hep-ph/9805496/
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Factorization and Scaling
in Hadronic Diffraction
Rockefeller University Preprint: RU 97/E-43
In standard Regge theory, the contribution of the tripe-pomeron amplitude to the differential cross section for single diffraction dissociation has the form , where is the pomeron trajectory. For , this form, which is based on factorization, does not scale with energy. From an analysis of and data from fixed target to collider energies, we find that such scaling actually holds, signaling a breakdown of factorization. Phenomenologically, this result can be obtained from a scaling law in diffraction, which is embedded in the hypothesis of pomeron flux renormalization introduced to unitarize the triple pomeron amplitude.
Fellow of CNPq, Brazil.
In Regge theory, the high energy behavior of hadronic cross sections is dominated by pomeron exchange [1, 2]. For a simple pomeron pole, the elastic, total, and single diffractive (SD) cross sections can be written as
where is the pomeron Regge trajectory, is the coupling of the pomeron to the proton, is the triple-pomeron coupling, is the center of mass energy squared, is the fraction of the momentum of the proton carried by the pomeron, and is an energy scale parameter, which is assumed throughout this paper to be 1 GeV unless appearing explicitely.
where , is interpreted as the “pomeron flux”. Thus, diffraction dissociation can be viewed as a process in which pomerons emitted by one of the protons interact with the other proton .
The function represents the proton form factor, which is obtained from elastic scattering. At small , . However, this simple exponential expression underestimates the cross section at large . Donnachie and Landshoff proposed that the appropriate form factor for elastic and diffractive scattering is the isoscalar form factor measured in electron-nucleon scattering, namely
where is the mass of the proton. When using this form factor, the pomeron flux is referred to as the Donnachie-Landshoff (DL) flux111 The factor in the DL flux is , where is the pomeron-quark coupling.. Note that at small- can be approximated with an exponential expression whose slope parameter, , is 4.6 GeV at GeV, consistent with the slope obtained from elastic scattering at small .
As we discussed in a previous paper , the dependence of violates the unitarity based Froissart bound, which states that the total cross section cannot rise faster than . Unitarity is also violated by the -dependence of the ratio , which eventually exceeds the black disc bound of one half (), as well as by the -dependence of the integrated diffractive cross section, which increases with as and therefore grows faster than the total cross section.
For both the elastic and total cross sections, unitarization can be achieved by eikonalizing the elastic amplitude [6, 7], which takes into account rescattering effects. Attempts to introduce rescattering in the diffractive amplitude by including cuts [8, 9] or by eikonalization have met with moderate success. Through such efforts, however, it has become clear that these “shadowing effects” or “screening corrections” affect mainly the normalization of the diffractive amplitude, leaving the form of the dependence almost unchanged. This feature is clearly present in the data, as demonstrated by the CDF Collaboration in comparing their measured diffractive differential cross sections at 546 and 1800 GeV with cross sections at GeV.
Motivated by these theoretical results and by the trend observed in the data, a phenomenological approach to unitarization of the diffractive amplitude was proposed based on “renormalizing” the pomeron flux by requiring its integral over all and to saturate at unity. Such a renormalization, which corresponds to a maximum of one pomeron per proton, leads to interpreting the pomeron flux as a probability density simply describing the and distributions of the exchanged pomeron in a diffractive process (see details in section 3).
In this paper, we show that the hypothesis of flux renormalization provides a good description not only of the s-dependence of the total integrated SD cross section, as was already shown in , but also of the differential (or ) and distributions. Specifically, we show that for GeV (above the resonance region) and (the coherence region ), all available data for at small can be described by a renormalized triple-pomeron exchange amplitude, plus a non-diffractive contribution from a “reggeized” pion exchange amplitude, whose normalization is fixed at the value determined from charge exchange experiments, . A good fit to the data is obtained using only one free parameter, namely the triple-pomeron coupling, .
We also show that the cross section at small displays a striking scaling behavior, namely , where the coefficient is -independent over six orders of magnitude. In contrast, the dependence expected from the standard triple-pomeron amplitude represents an increase of a factor of between and 1800 GeV. This scaling behavior is predicted by the renormalized flux hypothesis and provides a stringent and successful test of its validity.
In section 2 we present and discuss the data we use in this paper; in section 3 we describe our phenomenological approach in fitting the data using the pomeron flux renormalization and pion exchange models; in section 4 we present the results of our fits to data; in section 5 we present the case for a scaling law in diffraction; and in section 6 we make some concluding remarks on factorization and scaling in soft diffraction.
The data we use are from fixed target experiments [11, 12], from ISR experiments [13, 14], from Collider experiments , and from experiments at the Tevatron Collider [10, 16]. Below, we discuss some aspects of the Tevatron Collider data reported by CDF .
2.1 The CDF data
The CDF Collaboration reported differential cross sections for in the region of and GeV at and 1800 GeV. The experiment was performed by measuring the momentum of the recoil antiproton using a roman pot spectrometer. No tables containing data points are given in the CDF publication. The data are presented in two figures (figures 13 and 14 in ), which are reproduced here as Figs. 1a and 1b, where the number of events is shown as a function of () of the recoil antiproton, rather than as a funcion of the antiproton momentum (as was done in ). The histogram superimposed on the data in each figure is the CDF fit to the data generated by a Monte Carlo (MC) simulation. As an input to the simulation, the following formula was used:
The first term in this equation is the triple-pomeron term of Eq. 3. The second term was introduced to account for the non-diffractive background. A connection to Regge theory may be made by observing that (0) corresponds to pion (reggeon) exchange with a Regge trajectory of intercept (0.5) (see section 3). The factor of does not appear in reference and is introduced here to account for the fact that we refer to the cross section for and do not include that for , as was done by CDF. The CDF MC simulation took into account the detector acceptance and the momentum resolution of the spectrometer. The slope of the pomeron trajectory, , was kept fixed at the value GeV. The values of the remaining parameters, as determined from the CDF fits to the data, are listed in Table 1, where we include the values for the momentum resolution, , at and 1800 GeV.
2.1.1 Acceptance corrected -distributions
Using the information provided in the CDF publication, we mapped Figs. 1a and 1b into Figs. 2a and 2b, respectively, in which the data are corrected for detector acceptance. The acceptance was obtained from Fig. 2 of reference . The results are presented as cross sections, rather than as events, versus . The normalization was determined by comparing the data points with the CDF MC fits. The number of events corresponding to each -bin of the MC histograms in Figs. 1a and 1b was converted to an absolute cross section by convoluting the analytic CDF formula for the differential cross section with the -acceptance function and with the Gaussian -resolution function using a normalization that reproduces the MC histogram. The curves in the new figures represent Eq. 7 convoluted with a Gaussian resolution function of , whose width was determined from the momentum resolution of the spectrometer at each energy. Specifically, these curves are calculated using the expression
where is given by Eq. 7 (with ) and is the Gaussian resolution function given by
As seen in Figs. 2a and 2b, expression (8) provides an excellent fit to the acceptance-corrected differential cross sections, including the unphysical region of negative values. Thus, once the detector experimental resolution is accounted for, the low (or equivalently, the low ) cross section is completely compatible with that expected from extrapolating the cross section from the region of into the resolution dominated very low- region using the triple-pomeron differential cross section shape. This behavior rules out the hypothesis of low (low ) suppression suggested by some authors [19, 20, 21, 22].
2.1.2 Cross sections at GeV
As can be seen in Fig. 2 of reference , the CDF data in the triple-pomeron dominated region of are concentrated at low -values, namely GeV for (1800) GeV. Therefore, direct comparison of the CDF data with other experiments should be made for -values within these regions of . Since the CDF paper does not report -distributions at a fixed value of in the form of a table, we extracted such a table for GeV from the information given in the CDF paper. The value of GeV was chosen in order to allow direct comparison of the CDF data with the data of reference , for which -distributions have been published in table form for GeV and and 20 GeV (see Tables 2 and 3). The GeV CDF points were evaluated from the data in Figs. 2a and 2b, which represent cross sections integrated over , by scaling the cross section at each point in by the ratio
which was calculated using Eq. 8. Figures 3a and 3b display the GeV data points grouped into -bins of approximately equal width in a logarithmic scale. Figures 3c and 3d display in a linear -scale the data for , including the unphysical region of negative -values. The horizontal “error bars” represent bin widths. The values of the points plotted in Figs. 3a-d are listed in Tables 4 and 5. The solid (dashed) curves in the figures represent the CDF fits without (with) the convoluted -resolution function, calculated using Eq. 7 (Eq. 8). For (1800) GeV, the effect of the detector resolution becomes important for (0.003). Immediately below these values, the data lie higher than the extrapolation of the solid-line fits from the larger -values (see Figs. 3a,b). This effect is completely accounted for by the smearing effect of the -resolution, which also accounts for the values of the cross sections in the unphysical negative -regions, as seen in Figs. 3c and 3d (for exact numerical comparisons see Tables 4, 5). The effect of the resolution on the measured cross sections is quite substantial at low and therefore must be taken into consideration when comparing the low- CDF data with predictions of unitarization models based on low- suppression of the diffractive cross section [19, 20, 21, 22].
We now return to the question of the slopes of the -distributions (see Table 1). Theoretically, the value of for should be the same at all energies and equal to one half of the corresponding value for (see Eqs. 1 and 3). Experimentally, GeV . The best-fit CDF slope values are () GeV for (1800) GeV. The 1800 GeV value is close to 4.6, within error, but the 546 GeV slope is significantly larger than 4.6 GeV. The discrepancy between the slope value measured by CDF at GeV and the expected value of GeV may be explained by the very short -range of the experimental measurement. In the region of low , where pomeron exchange is dominant, the detector had reasonable acceptance only within the region GeV. Thus, the slope could not be measured accurately. The quoted error in the measured slope is the standard deviation calculated keeping all other parameters fixed at their best-fit values. The large correlation coefficients between the error of the CDF best-fit parameter and other fit parameters indicate that a good fit to the data within the -region of the measurement could have been obtained with a different value of , and correspondingly different value of the other parameters, subject to the constraint that the integrated cross section over the -range of the measurement remain the same. Since GeV corresponds approximately to the cross-section-weighted mean value of in the region , the value of the differential cross sections at GeV is insensitive to a change in .
2.1.4 Total diffractive cross sections
At 546 (1800) GeV, the total integrated cross section within the region and calculated using Eq. 7 (multiplied by a factor of 2 to include the cross section for ) is 7.28 (8.73) mb.
3 Phenomenological approach
In the framework of Regge theory , the cross section for in the region of large can be expressed as a sum of contributions from exchanges of reggeons , and (see Fig. 4),
where is a reggeon trajectory, is the reggeon coupling to the proton, is the “triple-reggeon” coupling and is a phase factor determined by the signature factor, , where is the signature of the exchange. The signature factors have been expressed as with the moduli absorbed into the parameters in (12). For reggeons and must have the same signature, so that . As mentioned in section 1, the energy scale is not determined by the theory and is usually set to 1 GeV. The lack of theoretical input about the value of introduces an uncertainty in the pomeron flux normalization, which is resolved in the renormalized pomeron flux model (see discussion below).
Table 6 displays the and , or , dependence of the contributions to the cross section at coming from various combinations of exchanged reggeons. Three Regge trajectories are considered: the pomeron, , with , the reggeon, , with , and the pion, , with . In fitting elastic and total cross sections, Covolan, Montanha and Goulianos use two reggeon trajectories, one for the f/a family with and the other for the family with ; Donnachie and Landshoff use one “effective” trajectory with . For simplicity and clarity in presentation, we consider in Table 6 one reggeon trajectory with . The terms (triple-pomeron) and correspond to the picture in which pomerons emitted by one proton interact with the other proton to produce the diffractive event. The last row in Table 6 shows the predictions of the renormalized pomeron flux model, which is discussed below.
3.1 Standard approach
The standard approach to diffraction is to perform a simultaneous fit to the differential cross sections of all available data at all energies using Eq. 11, which is based on factorization. In such a fit, the only free parameters are the tripple-reggeon couplings, . The reggeon trajectories and the couplings are determined from the elastic and total cross sections , and the coupling is obtained from the coupling measured in the charge exchange reaction ; using isotopic spin symmetry, .
Equation 11 is based on factorization. A “global” fit of this form to all available data was performed by R. D. Field and G. C. Fox in 1974 . However, the data available at that time could not constrain the fit well enough to test the triple-reggeon phenomenology, let alone determine the triple-reggeon couplings. By 1983, with more data available from Fermilab fixed target and ISR experiments -, good fits to the small- differential cross sections were obtained using the empirical expression
The first term in (13) has the -dependence of the amplitude with () and the second term has the -dependence of the amplitude. Note that a reggeon-exchange contribution, , with , would have a flat -dependence. At the relatively low values of of the Fermilab fixed target and ISR experiments, the -range was not large enough for the -slope to be sensitive to the variation with expected from Eq. 11, namely , or to distinguish between a and a dependence in the first term of (13) and thereby establish the now well known deviation of from unity. Nevertheless, the prominent behavior of the cross section at low showed dominance and left little room for contributions from other terms, as for example from a term with its sharper dependence on . This is illustrated by the fits of Eq. 13 to the very precise data for shown in Fig. 5. The data [25, 26] are from the experiment of the USA-USSR Collaboration at Fermilab using an internal gas-jet target operated with deuterium. The values of the cross sections at GeV plotted in Fig. 5 were obtained either directly from the published Tables or by extrapolation from their published values at GeV using the measured slope of the -distribution. The two sets of data were normalized to the average value of the cross section within the -region common to both sets of data. Figures 5a and 5b show fits using a and a dependence (with ), respectively. Both fits are in good agreement with the data.
In summary, the agreement of the Fermilab fixed target and ISR experimental results with the empirical expression (13), which is inspired by the factorization based standard tripple-reggeon phenomenology, shows that:
At small the cross section is dominated by the amplitude ().
At larger there is an additional contribution, which has the form of the amplitude ().
3.1.2 Breakdown of factorization
In 1994, when CDF published the diffractive cross sections at and 1800 GeV , the supercritical pomeron trajectory with was already well established by fits to total hadronic cross sections . Therefore, CDF made fits using Eq. 7, which includes two terms: the amplitude (first term) and a non-diffractive contribution parameterized as . The form of the latter was inspired by the empirical expression (13), and the parameter was introduced to effectively incorporate possible contributions both from () and () amplitudes, as discussed in section 2.1.
Three important results from the CDF fits to the data should be emphasized:
Only the term and a non-diffractive contribution are required by the fits. An upper limit of 15% was set on a possible contribution of a term to the total diffractive cross section at GeV. From this result, we derive the following limit for the ratio, , of the coefficients of the / terms:
The parameter was determined for the first time from the -distribution of single diffraction dissociation and was compared to the obtained from the -dependence of the total cross section . The CDF results are:
(15) (16) (17)
The values obtained from the distributions are, within the quoted uncertainties, consistent with the value determined from the rise of the total cross section, as would be expected for pomeron pole dominance. The weighted average of all three values is
The last result indicates a breakdown of factorization. The observed slower than increase of the diffractive cross section with energy is necessary to preserve unitarity and was predicted in 1986 by calculations including shadowing effects from multiple pomeron exchanges. More recent work based on eikonalization of the diffractive amplitude or on the inclusion of cuts shows that shadowing can produce substantial damping of the -dependence of the cross section but has no appreciable effect on the -dependence. These predictions are in general agreement with the conclusions reached by the CDF fits to data. However, the damping predicted by the eikonalization model is not sufficient to account for the observed -dependence of the total single diffraction cross section (see Fig. 6); the predictions of the model based on cuts are in better agreement with the data .
3.2 Renormalized pomeron flux approach
3.2.1 Triple-pomeron renormalization
The CDF measurements showed that, just like at Fermilab fixed target and ISR energies, the shape of the small (small ) behavior of the diffractive cross section at the Tevatron Collider is described well by the amplitude displayed in Eq. 19. The total diffractive cross section, obtained by integrating Eq. 19 over all and over from GeV to , increases with as . For , which is the value for simple pole exchange, would increase faster than the total cross section, which varies as , leading to violation of unitarity. With the experimentally determined value of , the diffractive cross section remains safely below the total cross section as increases, preserving unitarity.
As we have seen in the previous section, introducing shadowing corrections can dampen the increase of the diffractive cross section with and thereby achieve the desired unitarization while preserving the -dependence of the amplitude, as required by the data. However, the shadowing models do not account completely for the -dependence of the data, and the two models mentioned above do not predict the same amount of -damping of the cross section. In addition, these models are very cumbersome to use in calculations of single diffraction, double diffraction and double-pomeron exchange processes.
The calculational difficulties of unitarity corrections in the standard approach are overcome in the “pomeron flux renormalization” approach proposed by Goulianos . The renormalized flux approach is based on a hypothesis, rather than on an actual calculation of unitarity corrections, and therefore can be stated as an axiom: The pomeron flux integrated over all phase space saturates at unity. The standard pomeron flux is displayed in Eq. 5. Using , the integral of the standard flux,
is given by
where is the exponential integral function222 , where 0.57721 (Euler’s constant)., , is the effective diffractive threshold, and .
The renormalized pomeron flux, , can now be expressed in terms of the standard flux, , as follows333 For a detailed discussion of the role of the scale parameter in determining the value of for which see .:
The renormalized contribution to the differential cross section is given by
or, in terms of , by
In the energy interval of to GeV, the standard flux integral varies as (see Fig. 7). Thus, flux renormalization approximately cancels the -dependence in Eq. 24 resulting in a slowly rising total diffractive cross section. Asymptotically, as , the renormalized total diffractive cross section reaches a constant value:
The s-dependence of the integral of expression (23) over all and , multiplied by a factor of 2 to account for both and , is compared with experimental data for in Fig. 6 (from ). In view of the systematic uncertainties in the normalization of different sets of data, which are of , the agreement is excellent.
3.2.2 Pion exchange contribution
The form of the empirical expression (13) suggests that at high the dominant non- concontribution to the cross section comes from pion exchange. In Regge theory, the pion exchange contribution has the form
where is the pion flux and the total cross section.
In the “reggeized” one-pion-exchange model , the pion flux is given by
where is the on mass-shell coupling, is the pion trajectory, and is a form factor introduced to account for off mass-shell corrections. For we use the expression (see and references therein)
Since the exchanged pions are not far off-mass shell, we use the on-shell total cross section ,
3.2.3 A one parameter fit to diffraction
Motivated by the success of the empirical expression (13) in describing the Fermilab fixed target and ISR data, and by the similarity between this expression and the CDF fits to data at Tevatron energies, we have performed a simultaneous fit to single diffraction differential cross sections at all energies using the formula
in which the first term is the renormalized triple-pomeron amplitude, Eq. 23, and the second term is the pion exchange contribution, Eq. 26. Results from our fit, in which only the triple-pomeron coupling, , is treated as a free parameter, are presented in the next section.
In this section, we present the results of fits performed to experimental data using Eq. 30, which has two contributions: a renormalized triple-pomeron amplitude and a reggeized pion exchange term.
4.1 Differential cross sections
The experimental -distributions are usually distorted in the low- region by the resolution in the measurement of the momentum of the recoil . We therefore check first how well Eq. 30 reproduces the shapes of the differential cross sections of the data of E396 at =14 and 20 GeV and of the data of CDF at =546 and 1800 GeV in the regions of not affected by detector resolution. Figure 8 shows the cross sections at GeV for E396 and CDF (data from Tables 2, 3, 4 and 5). The solid lines represent the best fit to the data at each energy using Eq. 30 with the normalizations of the triple-pomeron and pion exchange contributions treated as free parameters. The quality of these fits indicates that no reggeon terms other than the triple-pomeron and pion exchange terms are needed to describe the shapes of the differential -distributions.
Figures 9-10 show the result of a simultaneous fit (solid lines) to the GeV data of E396 and CDF using Eq. 30 with only the triple-pomeron coupling as a free parameter. The overall normalization of the data was allowed to vary within % to account for possible systematic effects in the experimental measurements. The shift in the normalization of the data at each energy that resulted in the best fit is given in each plot. In Fig. 10 the individual contributions of the triple-pomeron and pion exchange terms are shown by dashed curves. The fit had a per degree of freedom.
The parameters used in the fit are and GeV (4.1 mb) for the triple-pomeron term, and those given in section 3.2.2 for the pion exchange term. The fit yielded a triple-pomeron coupling , which corresponds to ; using the form factor yields and .
Figure 11 shows a fit of Eq. 30 to ISR data of versus at fixed . In this fit, the experimental -resolution was taken into account by convoluting Eq. 30 with the Gaussian resolution function (9) using . The parameters used in Eq. 30 were those derived above. The overall normalization of the data has an experimental systematic uncertainty of 15% .
4.2 Total diffractive cross sections
In Fig. 12, we compare experimental results for the total diffractive cross section within and with the cross section calculated from the triple-pomeron term of Eq. 30 (solid line) using the triple-pomeron couplig evaluated from our fit to the differential cross sections. Within this region of , the expected contribution of the pion exchange term is less than 2% at any given energy. The data points are from references [10, 11, 12, 13, 14, 15, 16].
There are two points that must be kept in mind in comparing data with theory:
Normalization of data sets
The overall normalization uncertainty in each experiment is of .
Corrections applied to data
Deriving the total cross section from experimental data invariably involves extrapolations in and from the regions of the measurement to regions where no data exist. In making such extrapolations, certain assumptions are made about the shape of the -distribution and/or the shape of the distribution. Different experiments make different assumptions. For example, with the exception of the ISR experiments [13, 14], all measurements of the experiments listed here are at very low-. In these experiments, an exponential form factor of the form is assumed for extrapolating into the high- region. The (higher-) ISR data show a clear deviation from exponential behavior and support the form factor. Using instead of results in a larger total cross section by %, depending on the value of (smaller correction at higher ). The magnitude of the correction depends on the -region (and through on ), since the -distribution depends not only on the form factor but also on throught the term .
Another source of error comes from the fact that the slope of the -distribution is usually not measured accurately in experiments sensitive only to low-. The discussion in section 2.1.3 of the CDF measurement at GeV illustrates this point.
Table 7 presents the total diffractive cross sections corrected for the effects mentioned above. The ISR [13, 14] and cross sections were left unchanged, since they were calculated taking into account the high- behavior of the differential cross section. The cross sections of Refs. [12, 16] were multiplied by the ratio of the cross section calculated from the expression using the form factor to that calculated using the simple exponential form factor. Finally, the cross sections of Refs. [10, 11], for which the data are within a limited -region and have no reliable slope parameters, were calculated as follows: in each case, we evaluated the integrated cross section within the region of the experiment using the parameters determined by the experiment, and then recalculated this cross section using the formula of Eq. 30, adjusting the normalization parameter to obtain the same value for the integrated cross section over the same region; this formula was then integrated over the region and . The corrections to values derived directly from the published data are of . In view of the systematic uncertainties in the normalization of the various data sets, as evidenced by the discrepancies among data from different experiments in overlapping -regions, Fig. 12 shows excellent agreement between the experimental cross sections and the predictions of the one-parameter fit of Eq. 30 (using the form factor and mb).
5 A scaling law in diffraction
The renormalization of the pomeron flux to its integral over all available phase space may be viewed as a scaling law in diffraction, which serves to unitarize the triple-pomeron amplitude at the expense of factorization.
The breakdown of factorization is illustrated in Fig. 13, where cross sections are plotted as a function of at fixed for 14 and 20 GeV (=17 GeV) and =1800 GeV. As noted by CDF , while the shapes of the distributions as decreases tend to the shape expected from triple pomeron dominance at both energies, the normalization of the points is approximately a factor of lower than that of the points, instead of being a factor of higher, as one would expect from factorization (see factor in Eq. 3).
This particular way in which factorization breaks down implies that the distribution is approximately independent of , and therefore scales with , in contrast to the behavior expected from factorization. Figure 14 shows the differential cross sections as a function of at GeV =14, 20, 546 and 1800 GeV within regions not including the resonance region of GeV (=14 and 20 GeV) and not affected by the detector resolution (for =546 and 1800 GeV). These cross sections are also shown in Fig. 15 for regions of low enough not to be affected by the non-pomeron contribution. In Fig. 15, the data are compared with a straight line fit of the form , (solid line) and with the predictions based on factorization (dashed lines). Clearly, factorization breaks down in a way that gives rise to a scaling behavior.
The scaling of the distribution is a consequence of the pomeron flux renormalization hypothesis, as pointed out in section 3.2.1. Figure 7 shows that the renormalization factor based on flux scaling has an approximate dependence, which cancels the dependence in expected from factorization. An exact comparison between data and theory is made in Fig. 16, where data and predictions of Eq. 30 are shown for t=0. The data were obtained from the GeV data shown in Fig. 14 by subtracting the pion exchange contribution at GeV and calculating the cross section assuming a -distribution given by . The excellent agreement between data and theory over six orders of magnitude justifies our viewing the pomeron flux renormalization hypothesis as a scaling law in diffraction.
We have shown that experimental data on diffractive differential cross sections for and at energies from 14 to 1800 GeV, as well as total diffractive cross sections (integrated over and ), are described well by a renormalized triple-pomeron amplitude and a reggeized pion exchange contribution, whose normalization is kept fixed at the value determined from .
The renormalization of the triple-pomeron amplitude consists in dividing the pomeron flux of the standard Regge-theory amplitude by its integral over all available phase space in and . Such a division provides an unambiguous normalization of the pomeron flux, since the energy scale factor, , which is implicit in the definition of the pomeron proton coupling that determines the normalization for the standard flux, drops out. Thus, the renormalized pomeron flux depends only on the value of and on the pomeron trajectory, which is obtained from fits to elastic and total cross sections. Therefore the only free parameter in the renormalized triple-pomeron contribution to soft diffraction is the triple-pomeron coupling constant, . From our fit to the data we obtained the value GeV.
The scaling of the pomeron flux to its integral represents a scaling law in diffraction, which unitarizes the diffractive amplitude at the expense of factorization. A spectacular graphical representation of this scaling is provided by the experimental differential distribution as a function of for energies from =14 to 1800 GeV. This distribution shows a clear behavior, which is independent of over six orders of magnitude, in agreement with expectations from the flux renormalization hypothesis and contrary to the behavior expected from the standard theory based on factorization.
J. Montanha would like to thank the Brazilian Federal Agency CNPq for providing him with a fellowship and The Rockefeller University for its hospitality.
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- F. Abe et al., CDF Collaboration, Phys. Rev. D 50 (1994) 5535.
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- The justification for using this expression for -values down to , which include the resonance region of , is presented in Fig. 22c of , reproduced from Fig. 3c of . This figure shows that the -integrated cross section, which at high -values has a rather flat dependence, extrapolates smoothly into the low resonance region all the way down to GeV and then falls gradually to become zero at the pion production threshold. Taking =1.4 (or 1.5 ) as the minimum value for and integrating down to this value using the triple-pomeron formula provides a very good approximation to the integrated cross section over the resonance region.
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Table 2: Differential cross sections at 14 GeV and GeV .
Table 3: Differential cross sections at 20 GeV and GeV .
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https://faculty.kaust.edu.sa/en/publications/analysis-of-a-finite-matrix-with-an-inhomogeneous-circular-inclus
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math
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The mechanical model of eigenstrains could not be always taken as uniform distributions in engineering applications when performing micromechanics analysis of the inclusion-matrix system. In the framework of plane strain, this paper presents the analytical solution to an inhomogeneous circular inclusion with a non-uniform eigenstrain concentrically embedded in a finite matrix. First, the equivalent eigenstrain equation is extended to satisfy the condition of the finite matrix through the equivalent eigenstrain principle. The modified equation is used to transform the inhomogeneous inclusion in a finite matrix into the corresponding homogeneous inclusion. Then, the model of the inhomogeneous circular inclusion is accordingly formulated, and the stress and strain distributions are found. Finally, the stresses for the case of the polynomial series distribution of eigenstrains are obtained. The effects of non-uniformity of eigenstrains, the material mismatch and the inclusion size on stress distributions are shown graphically. The results indicate the stiffer inclusion induces the larger stress under the specific eigenstrain distribution. The analytical solutions obtained here also help to predict failure and optimize the designs of composite structures.
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http://www.investopedia.com/terms/a/arithmeticmean.asp
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math
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Definition of 'Arithmetic Mean'
A mathematical representation of the typical value of a series of numbers, computed as the sum of all the numbers in the series divided by the count of all numbers in the series.
Arithmetic mean is commonly referred to as "average" or simply as "mean".
Investopedia explains 'Arithmetic Mean'
Suppose you wanted to know what the arithmetic mean of a stock's closing price was over the past week. If during the five-day week the stock closed at $14.50, $14.80, $15.20, $15.50, and then $14.00, its arithmetic mean closing price would be equal to the sum of the five numbers ($74.00) divided by five, or $14.80.
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s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657133455.98/warc/CC-MAIN-20140914011213-00222-ip-10-196-40-205.us-west-1.compute.internal.warc.gz
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