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http://zunal.com/newpage.php?w=240394&n=14144
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math
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Use the websites below to answer these questions. Be sure to write the answers to the questions in your own words. You will not receive any credit for answers that are copied directly from a website. A clear understandable answer written in complete sentences is required.
Subtraction and theRest of the Integers
Online Etymology Dictionary
Negative Numbers atMathPages
Properties of RealNumbers: Negative Numbers
History of NegativeNumbers
Math is Good for You
Questions to be researched:
1. In what century were negative integers finally accepted?
2. Which cultureswere the first to use negative integers and what contributions did they make?
3. When was the word integer introduced?
4. What does the word integer mean in Latin?
5. What is the symbol for integers, where does it come from and what does it mean?
The Public URL for this WebQuest:
WebQuest Hits: 100,293
Save WebQuest as PDF
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s3://commoncrawl/crawl-data/CC-MAIN-2023-23/segments/1685224651815.80/warc/CC-MAIN-20230605085657-20230605115657-00212.warc.gz
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CC-MAIN-2023-23
| 891 | 16 |
https://www.projecteuclid.org/journals/electronic-communications-in-probability/volume-13/issue-none/On-the-rate-of-growth-of-L%C3%A9vy-processes-with-no/10.1214/ECP.v13-1414.full
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math
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In this note, we study the asymptotic behaviour of Lévy processes with no positive jumps conditioned to stay positive and some related processes. In particular, we establish an integral test for the lower envelope at $0$ and at $+\infty$ and an analogue of Khintchin's law of the iterated logarithm at 0 and at $+\infty$, for the upper envelope of the reflected process at its future infimum.
"On the rate of growth of Lévy processes with no positive jumps conditioned to stay positive." Electron. Commun. Probab. 13 494 - 506, 2008. https://doi.org/10.1214/ECP.v13-1414
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s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100529.8/warc/CC-MAIN-20231204115419-20231204145419-00460.warc.gz
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https://reggianiautonoleggio.it/exterior-angle-of-a-triangle-is-equal-to.html
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math
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Exterior angle of a triangle is equal to
- Sep 22, 2008 · Exterior angle. = 180° - interior angle adjacent to it. = sum of two remote interior angles. aravindma1990. 1 decade ago. consider the three angles as A,B,C. A+B+C=180. now extend one of the sides...
- Exterior angle: An exterior angle of a polygon is an angle outside the polygon formed by one of its sides and the extension of an adjacent side.
- 2.Cut out each exterior angle and label them 1-6. 3.Fit the six angles together by putting their vertices together. What happens? The angles all fit around a point, meaning that the exterior angles of a hexagon add up to 360˚, just like a triangle. We can say this is true for all polygons. • Exterior Angle Sum Theorem: The sum of the ...
- An exterior angle of a triangle is equal to the sum of the two opposite interior angles. The sum of exterior angle and interior angle is equal to 180 degrees. ⇒ c + d = 180° ⇒ a + f = 180°
- If one angle of a triangle is equal to one angle of the other triangle and the sides including these angle are proportional, then the two triangle are similar.prove that asked Aug 17, 2019 in Class X Maths by Sachin Tiwari ( 15 points)
- Acute triangle: a triangle with three acute angles Obtuse triangle: a triangle with one obtuse angle Right triangle: a triangle with one right angle Exercises True or False: Give a reason or counterexample to justify your response. 1. An equilateral triangle is always acute. 2. An obtuse triangle can also be isosceles. 3. The acute angles of a ...
- - Interior angles between parallel lines adds up to 180 degrees. 3) The sum of the 3 angles of a triangle will always be 180 degrees. 4) The exterior angle sum of any polygon will always be 360 degrees. Powered by Create your own unique website with customizable templates.
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- An exterior angle must form a linear pair with an interior angle. This means that the exterior angle must be adjacent to an interior angle (right next to it - they must share a side) and the interior and exterior angles form a straight line (180 degrees). If you extend one of the sides of the triangle, it will form an exterior angle.
- Angles 3 and 6 are also interior angles but are not congruent. Angles 2 and 7 are alternate exterior angles and are congruent. Angles 1 and 8 are also alternate exterior angles but are not congruent. Triangles: The three angles of a triangle always total 180 o. An equilateral triangle is a triangle with 3 equal sides and all 3 angles are 60 o.
- of exterior ZA, 10. Calculate the angles of a triangle ABC having 34B = 4ZC and the interior ZA = caculate angles of triangle abc having 3 times angle b= 4 times angle c and interior angle a =4/7 of its exterior angle The two angles of a triangle are equal and third measures 70degree .
- • Each exterior angle of a triangle is the sum of the opposite interior angles. (*) (e = a + b). Abbreviation: ext. ∠ of ∆. ea b. Section 4.1 unknown angle proofs • 75. • Base angles of an isosceles triangle are equal. (*) (If AC = BC then a = b.) Abbreviation: base ∠s of isos. ∆. C Aa bB.
- To triangulate something means either to divide into triangles or affix an object's location via triangles. Fundamentally they are based on the The measure of an exterior triangle angle is equal to the sum of the measures of the interior angles at the other two vertices (Exterior Angle Theorem).
- An isosceles triangle has two angles equal in size. In this problem A is greater than B therefore angles B and C are equal in size. Since angle A is 30 greater than angle B then A = B + 30 o. The sum of all angles in a triangle is equal to 180 o. (B+30) + B + B = 180. Solve the above equation for B. B = 50...
- An exterior angle of a triangle is formed when one side of a triangle is extended. Theorem 26: An exterior angle of a triangle is equal to the sum of the two remote (nonadjacent) interior angles. Example 1: In Figure 1, if m ∠1 = 30° and m ∠2 = 100°, find m ∠4.
- Theorem 06 - The Exterior Angle of a Triangle. Author: SeamusMcCabe. This demonstrates that an exterior angle in a triangle is always equal to the sum of the two interior opposite angles. Drag the points to change the shape of the triangle. Although the angles change size, the exterior angle is always equal to the sum of the two interior opposite angles.
- Use the angle sum of a triangle to prove that the angle sum of a quadrilateral is 360°. (a + b + c) + (d + e + f) = 180° + 180° = 360° Explain a proof that the exterior angle of a triangle equals the sum of the two interior opposite angles, using this or another construction. Given sufficient information, calculate: • interior and ...
- Angles In A Triangle. A triangle is the simplest possible polygon. It is a two-dimensional (flat) shape with three straight sides forming an interior, closed Angles In A Triangle. How To Find The Angle of a Triangle. By the Alternate Interior Angles Theorem, we know that. ∠a. is congruent (equal) to.
Google minesweeper full screenExterior angles of a triangle supplementary angles (equal 180°). and, The sum of the three interior angles in a triangle is always 180°. x + 24 + 34 = 180 x + 58 = 180 x = 122 then, 180 - 122 = 58 Also, the exterior angle is equal to the sum of the two remote interior angles. 24 + 34 = 58 Exterior Angle Theorem The measure of an exterior angle (our w) of a triangle equals to the sum of the measures of the two remote interior angles (our x and y) of the triangle. Let's try two example problems.
- (b) the exterior angle of a triangle is equal to the sum of the interior opposite angles. Take any triangle ABC. Construct a line through C, parallel to AB. angle p = angle b (corresponding angles) angle s = angle a (alternate angles) Therefore angle p + angle s = angle a + angle b. but angle r = angle a + angle b. Therefore angle p + angle s = angle r. Properties of quadrilaterals . Sum of angles in a quadrilateral
- Angles opposite to equal side lengths are equal. 3) A scalene triangle is a triangle with all three side of unequal length. 4) An acute triangle is a triangle in which all three angles are . 5) A right triangle is a triangle with a angle. 6) An obtuse triangle is a triangle with one of the angles . Basic Properties of Angles. 1) When two ...
- Triangle Angle-Sum Theorem 3. Exterior Angle Theorem 4. Exterior Angle Inequality G. Two-dimensional figures. 1. Polygon vocabulary 2. Interior angles of polygons ...
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Industry based considerations for diversification strategy includes all of the following exceptSentry safe handle stuck
The acute angles of a righttriangle are complementary. 7. The measure of an exterior angleof a triangle is equal to the sum of the measures of its remote interior angles. Find the measure of each angle.
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FACTS: • Every triangle has 6 exterior angles, two at each vertex. • Angles 1 through 6 are exterior angles. • Notice that the "outside" angles that are "vertical" to the angles inside the triangle are NOT called exterior angles of a triangle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.
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Aug 26, 2015 · An exterior angle is equal to 180° less the size of the interior angle. So an exterior angle plus an interior angle is equal to 180°. For a triangle, there are three angles, so the sum of all the interior and exterior angles is 180° x 3 = 540°.
A more perfect union worksheet answer keyQcow2 images
Exterior angles of a triangle and finding missing angles, sum of exterior angles, in video lessons with examples and step-by-step solutions. The Exterior Angle Theorem states that An exterior angle of a triangle is equal to the sum of the two opposite interior angles.
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3. There are six exterior angles of a triangle. 4. Interior angle + corresponding exterior angle = 180. 3. If any one side is produced, Exterior angle = Remote interior angle 4. If one pair of opposite sides are equal, diagonals are equal 5. The line joining the points of intersection of the bisectors of...
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s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039563095.86/warc/CC-MAIN-20210422221531-20210423011531-00307.warc.gz
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CC-MAIN-2021-17
| 8,323 | 34 |
https://www.khanacademy.org/math/precalculus/imaginary_complex_precalc/exponential-form-complex-numbers
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math
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If you're seeing this message, it means we're having trouble loading external resources for Khan Academy.
If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.
Exponential form of complex numbers
Multiply and divide complex numbers in polar form, and plot the answer by specifying the radius and angle.
Use Euler's formula to determine the value of a complex number raised to an nth power.
Using exponential form to find complex roots
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s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443736673081.9/warc/CC-MAIN-20151001215753-00073-ip-10-137-6-227.ec2.internal.warc.gz
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CC-MAIN-2015-40
| 495 | 6 |
https://www.lessonplanet.com/teachers/multiply-with-mixed-numbers-problem-solving
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math
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Multiply with Mixed Numbers: Problem-Solving
In this multiplying mixed numbers learning exercise, students read the word problems and solve the problems by multiplying the mixed numbers.
5th - 6th Math 7 Views 7 Downloads
Math Stars: A Problem-Solving Newsletter Grade 3
Make sure your class stays up with the times using this series of math newsletters. Including word problems that cover topics ranging from consumer math and symmetry to graphing and probability, these worksheets are a fun way to...
3rd - 5th Math CCSS: Adaptable
Unit Conversions and Problem Solving with Metric Measurement
Convert young mathematicians' knowledge of place value into an understanding of the metric system with this five-lesson unit. Through a series of problem solving activities and practice exercises, students learn about the relationship...
3rd - 5th Math CCSS: Designed
Deciphering Word Problems in Order to Write Equations
Help young mathematicians crack the code of word problems with this three-lesson series on problem solving. Walking students step-by-step through the process of identifying key information, creating algebraic equations, and finally...
5th - 8th Math CCSS: Adaptable
Solving Algebraic Equations and Inequalities
Solve algebraic equations and inequalities. Work in small groups to understand and solve a number of simple algebraic equations and inequalities. Then create a PowerPoint presentation that includes the problems solved and the method used...
6th - 8th Math CCSS: Adaptable
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s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988720238.63/warc/CC-MAIN-20161020183840-00502-ip-10-171-6-4.ec2.internal.warc.gz
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CC-MAIN-2016-44
| 1,499 | 15 |
https://demonstrations.wolfram.com/SuccessiveDerivativesOfAPolynomial/
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math
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Successive Derivatives of a Polynomial
Initializing live version
Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
This shows the graphs of a cubic polynomial and its first, second, and third derivatives. Only the top slider affects the third derivative, and the bottom slider doesn't affect the second derivative. Can you explain why?
Contributed by: Joe Bolte (March 2011)
Original version by: Stephen Wolfram
Open content licensed under CC BY-NC-SA
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s3://commoncrawl/crawl-data/CC-MAIN-2023-23/segments/1685224653930.47/warc/CC-MAIN-20230607143116-20230607173116-00670.warc.gz
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CC-MAIN-2023-23
| 539 | 8 |
https://www.homehighschoolhelp.com/blogs/homeschool-curriculum-how-to-fail-in-high-school-math
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math
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>>>>Dorette asked: Why did you use Saxon Advanced Math when it looks like you used another program for Algebra 1, Algebra 2, and Geometry?<<<<
A wise homeschool Mom once said, "If it works, use it. If it doesn't work - Change!"
We used Addison-Wesley math books through elementary school. When we started Algebra, the next book in the series was Algebra 1 by Paul Foerster. It was highly rated by homeschool suppliers, so it seemed like the "right" thing to do. I chose that book because it had a solution manual. Really, that was my whole reason - it had a solution manual!
My son Kevin is very mathematically minded, but he really struggled with this book! It just didn't seem to say things in a way that Kevin could understand it. By the end of the year, Kevin scored about 75% on the final. I really wanted Kevin to succeed before he moved on, so I searched for another math book so that he could do Algebra 1 over again. We chose Jacobs Algebra. I was planning to use that in the fall, and just have Kevin repeat the course. Instead, Kevin studied it by himself, took the final exam in the Jacobs Algebra text, and got a fabulous score, so in the fall he was able to move on to geometry. We used Jacobs geometry for that course. When it was time to do Algebra 2, I bought Paul Foersters Algebra 2 and Trigonometry. Don't ask me why! This author hadn't worked for us the first time, so I don't know why I thought it would work a second time through! After a month, I realized that we simply couldn't go forward with this book. We had to make a change, and Jacobs didn't have an Algebra 2 book.
Following a friend's advice, I gave Kevin the choice, and he chose his own math book. I was SHOCKED that he would choose Saxon! Saxon had no appeal to me, because I like books with color, and photos. But Kevin loved math, and he liked the look of Saxon because it was mostly numbers - problem after problem! That's what appealed to him! (Who knew?) So we switched to Saxon Math at that point. Since Kevin had already completed Algebra 1 and Geometry using different programs, he took the Saxon placement test, and we started him with Saxon Advanced Math.
I think almost everyone "loses it" at some point with high school math. It was during Algebra 2 that I simply couldn't DO the teaching anymore. That's when I began to use DIVE CDs. You can find more information at this website: www.diveintomath.com.
So now you know the story of our math curriculum choices! Our choices don't matter to your family, of course. What really matters is the underlying philosophy: If it works, use it. If it doesn't work - CHANGE!
I notice that a lot of homeschooling families work like this. They expect their student to really master a subject before moving on. I think that's why homeschoolers have higher standardized test scores than other students! We simply want them to know it before they move on! That gives them a better foundation for more advanced learning, and ultimately makes them more successful. My son Kevin is currently a sophomore in college, studying engineering and computer science. He takes upper level math classes "for fun" with his free credits. I'm really glad we encouraged mastery in math. And I'm really REALLY glad I let him choose his own math book!
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CC-MAIN-2020-40
| 3,265 | 8 |
https://www.geogebra.org/m/rMzCC7hR
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math
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Circle Angle Investigation
Investigation of Central Angles in a Circle Use the sliders to investigate how many ways three angles can be combined to fill a circle.
You will need to find eight different combinations of angles. Make sure to name each angle and describe what kind it is - acute, obtuse, right, straight or reflex. Record each combination on your worksheet.
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s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376825512.37/warc/CC-MAIN-20181214092734-20181214114234-00020.warc.gz
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CC-MAIN-2018-51
| 369 | 3 |
https://www.teacherspayteachers.com/Product/Adding-Integers-Task-Cards-with-Video-Explanations-1856422
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math
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Includes both adding & subtracting integers
24 task cards + 2 challenge cards + 2 explanation cards
•Teaching cards: The QR codes on the explanation cards link to teaching videos that explains how to add and “subtract” integers. Give these cards to students who need extra guidance or review.
•Video explanations: Each cards has a QR code that links to a video explanation of how to solve the problem.
•Easy Differentiation: The stars at the bottom of each card indicate the difficulty level.
•Challenge cards: Challenge cards can be added to any of the other 24 cards to encourage higher-level thinking.
•Blank answer forms: Print a copy for each student and have them write their answers on the form.
•Printable answer key: The answer key contains picture explanations for each problem.
•Easy-to-Understand: The way I teach & explain adding signed numbers is a bit unconventional, but I’ve found it to be extremely easy for students to understand.
•The way I teach & explain adding signed numbers is a bit unconventional, but I’ve found it to be extremely easy for students to understand.
•When students are first learning to add signed numbers, require them to draw a picture each time. This helps them really grasp the concept and avoid confusion and mistakes.
•I tell my students that we are no longer going to think of the – sign as subtraction but instead as a negative. That’s why I’ve titled these cards “adding signed numbers” not “adding and subtracting…” – because it’s better to think of it as adding a negative.
Task cards are versatile & can be used for a variety of activities including:
• Math Centers
• Remedial Help: Because these cards include video explanations, you can simply give these cards to a student who needs additional help, and they should be able to learn somewhat independently
Any questions? Feel free to ask!
Note: I have tagged these with the corresponding Common Core standards, and they can supplement any common core classroom. However, I explain the problems in the way I believe students best understand, not necessarily the way that best matches the recommended reasoning of the Common Core standards.
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s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864364.38/warc/CC-MAIN-20180622065204-20180622085204-00135.warc.gz
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CC-MAIN-2018-26
| 2,196 | 17 |
https://community.filemaker.com/thread/178304
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math
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I need to get the Initials from First and Last fields. Doesn't seem to much on the forum relating to this. I have something (attached) that was already in my database but this doesn't work.
if( not IsEmpty(First); Left (First;1) ) & if( not IsEmpty(Last); Left (Last;1)
Upper( Left (First;1) ) & Left (Last;1))
The reason that calc does not work is that it is set up as a Case ( ) function. The way that works is the FM runs down the list of queries (e.g. not IsEmpty (First), not IsEmpty (Last) … ) in order from top to bottom until it finds one that returns True, which it then applies as per the expression for that argument. Importantly, FM then ignores all remaining queries, so it will never deliver an answer to the second query if the first returns true.
Try the expressions suggested by Nicolai instead.
Keep in mind that names have a wide variation. Consider:
Mary Ellen Burns
Mary von Beethoven
Retrieving data ...
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s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202688.89/warc/CC-MAIN-20190322180106-20190322202106-00273.warc.gz
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CC-MAIN-2019-13
| 927 | 9 |
http://openstudy.com/updates/55a8011ee4b071e6530bf47a
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math
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Part A:The area of a square is (4x2 − 12x + 9) square units. Determine the length of each side of the square by factoring the area expression completely. Show your work. (5 points)
Part B: The area of a rectangle is (16x2 − 9y2) square units. Determine the dimensions of the rectangle by factoring the area expression completely. Show your work. (5 points)
Stacey Warren - Expert brainly.com
Hey! We 've verified this expert answer for you, click below to unlock the details :)
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I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
4x^2 -12x +9
4 x 9 = 36
first take out the LCM of 36
it will be
2 x 2 x 3 x 3
4x^2 -6x -6x +9
it is showing that
2x-3 is the length for part A
for part B
what you will do
16x^2 - 9y^2
you will use this formula here
x^2 -y^2 = (x-y) (x+y)
16x^2 -9y^2 = (4x -3y) (4x +3y)
this is the answer for part B
Not the answer you are looking for? Search for more explanations.
I hope you get what i explain here
I do thank you so much can you help me with one more problem
An expression is shown below:
f(x) = –16x2 + 22x + 3
Part A: What are the x-intercepts of the graph of the f(x)? Show your work. (2 points)
Part B: Is the vertex of the graph of f(x) going to be a maximum or minimum? What are the coordinates of the vertex? Justify your answers and show your work. (3 points)
Part C: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part A and Part B to draw the graph. (5 points)
sorry idk about this one
x intercept is when you set y=0
okay but I am still confused on how you find part b and part c
well first find the x intercepts you will get something like this
like say for example your x intercepts or "roots" or sometimes called zeros are
-3 and 4
the distance of that is 4 - (-3)=7 and half of 7 is 3.5
to get to get to the point between -3 and 4 just add 3.5 to -3 which is 1.5
and for c.
TO sketch it you'd plot the x intercepts, draw a vertical line between them and use symmetry.
Oh in part b if you found the mid point or MP of the vertex then the y coordinant of the vertex is
Now connect the points with a smooth line
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CC-MAIN-2018-05
| 2,853 | 43 |
https://wowlazymacros.com/t/need-some-help-with-macro/35287
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math
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So I decided to finally make my own macro, but I am having some problems with it. Maybe someone can help me out with it. I am trying to add Execute spam to my current macros for procs and for Execute phases.
My problem is this. I have added, in Macro Variables section “/cast Execute”. Also, have tried “/cast [exists] Execute”, but the problem I am having is that the macro gets stuck and I just start auto-attacking. No matter how many times I hit the key, just auto-attack and the macro doesn’t change, just stays at Execute icon. I have tried using the Variable both before and after a “/cast…” command
Can anyone tell me what im doing wrong?
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s3://commoncrawl/crawl-data/CC-MAIN-2023-23/segments/1685224655027.51/warc/CC-MAIN-20230608135911-20230608165911-00098.warc.gz
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CC-MAIN-2023-23
| 662 | 3 |
https://books.google.com/books/about/CRC_standard_mathematical_tables_and_for.html?id=yoo_AQAAIAAJ&hl=en
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math
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CRC Standard Mathematical Tables and Formulae, 31st Edition
Taylor & Francis, Dec 26, 1995 - Mathematics - 832 pages
The 30th edition still provides you with all of the usual tools. As always, the handbook covers numbers, geometry, trigonometry, calculus, special functions, numerical methods, probability, and statistics; along with many new topics, such as control theory and design theory. This is an excellent reference handbook for modern mathematics, filled with tables, formulae, equations, and descriptions.
What people are saying - Write a review
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algebraic algorithm angle ax dx called Cartesian Cartesian tensor Christoffel symbols CN CN coefficients column complex numbers components confidence interval continued fraction converges coordinate system cosh critical value curvature curve defined definite denoted Determine the critical differential equations edge eigenvalues eigenvector element example f dx F-distribution factor finite formula Fourier function given graph G homogeneous coordinates integer integral inverse Laplace transforms linear matrix Mersenne prime method metric tensor multiplication n x n node nonzero norm normal distribution orthogonal parameter permutations plane polynomial positive integer prime numbers quadric radius random variable real numbers regular parametric representation reject the hypothesis ri ri ri root rotation sample scalar sequence sin2 sinh sinx solution square symmetric symmetric matrix tensor theorem transformation triangle vector vertex vertices zero
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s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125946256.50/warc/CC-MAIN-20180423223408-20180424003408-00082.warc.gz
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CC-MAIN-2018-17
| 1,566 | 6 |
https://www.explainxkcd.com/wiki/index.php?title=Talk:216:_Romantic_Drama_Equation&oldid=55744
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math
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Talk:216: Romantic Drama Equation
This can't be right, even at 50/50, the number of gay pairings far outnumbers the number of straight pairings.126.96.36.199 20:10, 28 February 2013 (UTC) Moved from article page
- Not quite. Consider a cast of 4 with 2 male (A, B) and 2 female (C, D). Possible gay pairings - 2 (A-B and C-D). Possible straight pairings - 4 (A-C, A-D, B-C, B-D) 188.8.131.52 (talk) (please sign your comments with ~~~~)
- He says for large casts. For 2000 cast members, with 1000 of each gender, the gay couplings comes out at 999,000 and straight at 1,000,000. Presumably this is the small cross over the diagram alludes to. If you substitute x = n/2 into the equations, then you get (n^2-2n)/4 for the gay combinations and n^2/4 for the straight combinations, so for gender balanced cast size of n, the straight combinations outnumber the gay by n/2 184.108.40.206 (talk) (please sign your comments with ~~~~)
There is a typo in his formula for gay casts. The + should be a -.
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s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363189.92/warc/CC-MAIN-20211205130619-20211205160619-00267.warc.gz
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CC-MAIN-2021-49
| 995 | 5 |
https://worldbuilding.stackexchange.com/questions/102378/strategic-value-of-a-line-of-symmetry
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math
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Well. There are a few options here, and almost all of them result in:
Avoid this place like the plague
The first option is to treat the world as an infinite, flat, static plane. Your use of cartesian co-ordinates and the assumption of a 'ground' level at Z=0 implies this is true. In this case your line of symmetry must effectively be the centre of the world: Everything happens around a 'pole' of symmetry that points directly out of the ground. This is not a nice place to be, because any air that approaches this point must be moving in a radially symmetric way, effectively setting up a perfect, permanent cyclone. Whatever quirk of physics is causing this symmetry will also cause some very severe weather conditions.
The second option is similar to the first, but with plate tectonics as well. In this scenario the line of symmetry will become an area of severe, constant vulcanism as any minor plate movement must be immediately and perfectly opposed by it's counterpart, leading to immense friction and, correspondingly, heat. Couple this with the wind and you get a singularly unpleasant place to be.
The third option is to assume your world is a sphere. In this case the line of symmetry must go through the axis of rotation of the planet, or your little world
will tear itself apart in very, very short order simply cannot exist. Wind and plate tectonics are still issues. You can switch to a polar co-ordinate system and define Z as a plane passing through the equator if you want to make thinking about this easier.
I'm still working through whether or not this sort of symmetry represents a mathematical singularity or not, but if it does then you have an even more severe problem on your hands: to whit: The laws of physics break down when you reach it.
So. While pondering the awfulness that is the singularity of symmetry I realised that electromagnetism has some unfortunate things to say about this situation. I present: Amperes right hand grip rule.
This may not seem like an issue at first, however: Your world has radial symmetry. This means any magnetic field it may have must be symmetric around the pole. Now, if we assume a flat world with no meaningful magnetic field we're OK, until we get to electromagnetic interactions of molecules nearing the singularity. Things like air, or people's hands, or the ground.
In this situation any magnetic field, no matter how small, must form a curl (The magnetic field at x,y must be moving in the opposite direction to the field at -x,-y). This in turn must form a current pointing in a direction dictated by the grip rule. That must interact with the matter around it. I believe the magnetic flux becomes infinite as you approach the line of symmetry, which means the current does too. I'm genuinely not sure of my own maths/physics here (long day), so please correct me if I'm wrong.
In short, unless I've missed my maths: Nothing can reach the line of symmetry, not because it repels itself, but because the electromagnetism responsible for holding molecules together will tear them apart as they approach the mathematical monstrosity that is at the centre of your world. You have a constant line of plasma at the centre of a volcano surrounded by the fiercest winds imaginable.
Seriously. Don't go here.
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CC-MAIN-2021-25
| 3,275 | 12 |
http://www.chegg.com/homework-help/questions-and-answers/particles-fixed-x-axis-particle-1-charge-400-10-7-c-x-600-cm-andparticle-2-charge-400-10-7-q793608
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math
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fundamental of physics 8th edition jearl walker ch22#6
0 pts endedThis question is
closed. No points were awarded.
Two particles are fixed to an x axis: particle 1 of charge-4.00 10-7 C at
x = 6.00 cm, andparticle 2 of charge +4.00 10-7 C at x = 18.0
cm. Midway between the particles, what is theirnet electric field?
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s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386164972407/warc/CC-MAIN-20131204134932-00076-ip-10-33-133-15.ec2.internal.warc.gz
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CC-MAIN-2013-48
| 317 | 6 |
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1117619
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math
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19 Pages Posted: 9 Apr 2008
Date Written: March 25, 2008
The one-input cubic production function displays a range of increasing returns to a factor, followed by a range of decreasing returns, followed by a range of negative returns. If this situation holds with respect to two factors, it generates a natural intuition that the total product function in 3-dimensional space is represented by a sort of hill or protuberance with a well defined peak at the maximum value of /TP/ with respect to the two factors, with the consequence that the production isoquants are closed oblongs. However, the explicit mathematical function typically shown in textbooks as the two-input cubic production function (designated herein the "usual" function) does not have these properties. Rather the 3-dimensional total product function rises to a ridge curve before descending. The isoquants on either side of the ridge curve isoquant are not closed oblongs but rather convex downward-sloping configurations that converge in the limit but do not join. Two alternatives to the usual cubic production function are proposed: the additive and the multiplicative cubic production functions. Both of these have the expected properties: the 3-dimensional surface is depicted as a hill with a unique peak, and the isoquants are closed oblongs.
JEL Classification: D24
Suggested Citation: Suggested Citation
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CC-MAIN-2017-43
| 1,380 | 5 |
https://serc.carleton.edu/quantskills/activities/ls_vectors.html
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math
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Vectors and slope stability
This homework or in-class activity is designed to teach students with little or no experience with vectors, free-body diagrams, or the like to apply vectors to slope stability problems.
Students first learn the basic properties of vectors, vector addition, resolving forces, etc. They then apply these techniques to establish in a simplified way how geologists can determine if a slope is stable, how much loading a slope can withstand to become unstable, and other principles of slope stability determination.
This activity is used in an introductory geohazards course with no pre-requisites. It can be completed in one to two hours.
- graphically add vectors
- resolve vectors into perpendicular components
- represent forces as vectors
- determine if a slope is stable or unstable based on the forces acting on it
- articulate why a steeper slope is less stable than a shallow slope
- determine, from the mass on a slope and the angle of the slope, how large the downslope force is
- determine how much weight can be added to a slope before it becomes unstable.
Context for Use
It can be done in an evening but often students require some additional assistance from a tutor or someone who has seen vectors before.
Description and Teaching Materials
This assignment has two parts. The first is a general introduction to vectors, graphical manipulation of vectors and vector addition. The second part of the homework asks students to look at vector diagrams to see if a house is stable on slopes of differing angles. A final question asks students to find the maximum weight for a house on a given slope with known friction.
Teaching Notes and Tips
Note that pounds are used as the force units so that students do not need to convert between units of weight and force, as would have to be done if the metric system was used. I found this reduced student errors.
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CC-MAIN-2024-18
| 1,890 | 17 |
https://app-wiringdiagram.herokuapp.com/post/solving-exponential-and-logarithms-word-problem
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math
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SOLVING EXPONENTIAL AND LOGARITHMS WORD PROBLEM
Including results for solving exponential and logarithmic word problem.Do you want results only for solving exponential and logarithms word problem?
Exponential model word problems (practice) | Khan Academy
Solve word problems about exponential situations. If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *atic and *ndbox are unblocked.Medication Dissolve · Bacteria Growth
Logarithmic Word Problems - Purplemath | Home
Logarithmic Word Problems (page 1 of 3) Sections: Log-based word problems, exponential-based word problems Logarithmic word problems, in my experience, generally involve evaluating a given logarithmic equation at a given point, and solving for a given variable; they're pretty straightforward.
SOLVING WORD PROBLEMS - sosmath
SOLVING WORD PROBLEMS APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS. To solve an exponential or logarithmic word problems, convert the narrative to an equation and solve the equation. We are going to discuss several types of word problems. Click on the one that you want to
Videos of solving exponential and logarithms word problem
Click to view on YouTube4:23Solving Logarithmic Equations, Word Problems13K viewsYouTube · 6 years agoClick to view on YouTube13:38Compound Interest & Population Growth Word Problems - Logarithms9 viewsYouTube · 1 year agoClick to view on YouTube7:22Exponential growth and decay word problems | Algebra II | Khan Academy704K viewsYouTube · 9 years agoSee more videos of solving exponential and logarithms word problem
APPLICATIONS OF EXPONENTIAL AND LOGARITHMIC - S.O.S.
LOGARITHMIC FUNCTIONS (Interest Rate Word Problems) 1. To solve an exponential or logarithmic word problems, convert the narrative to an equation and solve the equation. Example 1: A $1,000 deposit is made at a bank that pays 12% compounded annually. How much will you have in your account at the end of 10 years?
Algebra - Solving Exponential Equations (Practice Problems)
Here is a set of practice problems to accompany the Solving Exponential Equations section of the Exponential and Logarithm Functions chapter of the notes for Paul Dawkins Algebra course at Lamar University.[PDF]
Sample Exponential and Logarithm Problems 1 Exponential
Sample Exponential and Logarithm Problems 1 Exponential Problems Example 1.1 Solve 1 6 3x 2 = 36x+1. Solution: Note that 1 6 = 6 1 and 36 = 62. Therefore the equation can be written (6 1) 3x 2 = (62)x+1 Using the power of a power property of exponential functions, we can multiply the exponents: 63x+2 = 62x+2 But we know the exponential function
Exponential growth & decay word problems - Khan Academy
Let's do a couple of word problems dealing with exponential growth and decay. So this first problem, suppose a radioactive substance decays at a rate of 3.5% per hour. What percent of the substance is left after 6 hours? So let's make a little table here, to just imagine what's going on. And then we
Logarithmic Functions – She Loves Math
Solving Exponential Equations using Logs. Now we can use all these tools to solve log equations! Remember again that math is just using tools that you have to learn to solve problems. Remember to always check your answer to make sure the argument of logs (what’s directly following the log) is positive! Let’s just jump in and trying somePeople also askWhat is the log function?What is the log function?LOG function. The LOG function computes the logarithm of an expression. base. The base by which to compute the logarithm. When you do not specify a value,the function computes the natural logarithm of the expression by using e for the base where e is equal to 281828459.LOG function - docsleSee all results for this questionAre logarithms and exponents the same?Are logarithms and exponents the same?Since exponentsand logarithmsare two versions of the samemathematical concept, exponentscan be converted to logarithms, or logs. An exponentis a superscript number attached to a value, indicating how many times the value is multiplied by itself. The log is based on exponentialpowers, and is just a rearrangement of terms.How to Convert Exponents to Logs | SciencingSee all results for this questionWhat are some examples of exponential equations?What are some examples of exponential equations?Exponential equations are equations in which variables occur as exponents. For example,exponential equations are in the form a x = b y. To solve exponential equations with same base,use the property of equality of exponential functions . If b is a positive number other than 1 ,then b x = b y if and only if x = y .Solving Exponential Equations - Varsity TutorsSee all results for this questionWhat is an exponential formula?What is an exponential formula?Exponential Functions. For example,y = 2 x would be an exponential function. Here's what that looks like. The formula for an exponential function is y = abx,where a and b are constants. You can see that this conforms to the basic pattern of a function,where you plug in some value of x and get out some value of y.What Is an Exponential Function? - Video & LessonSee all results for this question
Related searches for solving exponential and logarithmic word
solving exponential equations word problemsexponential and logarithmic functions problemsword problem logarithmic equationsexponential equation word problemsexponential and logarithmic equations solverexponential word problems with solutionslogarithmic word problems pdflogarithmic word problems practiceIncluding results for solving exponential and logarithmic word problem.Do you want results only for solving exponential and logarithms word problem?
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s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526408.59/warc/CC-MAIN-20190720024812-20190720050812-00373.warc.gz
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CC-MAIN-2019-30
| 5,786 | 22 |
https://brainmass.com/statistics/regression-analysis/general-statistics-238983
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math
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1. In 2002 The Cinergy Corporation sold 35,615 million cubic feet of gas to residential customers and a increase of 1.1% over the previous year. Does this information represent descriptive statistics or inferential statistics? Why?
2. What is the difference between a qualitative variable and a quantitative variable? When would each be appropriate?
3. The Acme School of Locksmiths has been accredited for the past 15 years. Discuss how this information might be interpreted as a
a. qualitative variable
b. quantitative variable
4. The Snowbird Ski Lodge attracts skiers from several New England States. For each of the following scales of Measurements, provide one example of information that might be relevant to the lodge's business
1) Essentially, descriptive are used to 'describe' raw data. For example, when you have a bunch of numbers (raw data), this does not provide the researcher with any useful information. We would calculate descriptive stats on the data to better understand what the number mean. As well, we can take perhaps hundreds of data points and calculate descriptive stats and have one overall number that might be meaningful to the researcher.
What are some examples of stats that can be conducted?
So in summary, it is esentially taking data and describe it such that it is easy for a researcher to read.
Inferential stats on the other hand, you are looking at the descriptive stats and taking it one step further. You are trying to analyze the data or to infer conclusions from the data. We make judgement on the data. We can compare two sets of data to each other.
One example would be a t-test ...
This solution answers questions on general statistics.
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CC-MAIN-2020-45
| 1,683 | 12 |
http://www.solutioninn.com/on-august-31-there-are-five-wildcard-terms-in-the
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math
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Question: On August 31 there are five wild card terms in the
On August 31 there are five wild-card terms in the American League that can make it to the play-offs, and only two will win spots. Draw a tree diagram which shows the various possible play-off wild-card teams.
Answer to relevant QuestionsThere are four routes, A, B, C, and D, between a person’s home and the place where he works, but route B is one-way, so he cannot take it on the way to work, and route C is one-way, so he cannot take it on the way ...The price of a European tour includes four stopovers to be selected from among 10 cities. In how many different ways can one plan such a tour (a) If the order of the stopovers matters; (b) If the order of the stopovers does ...How many different bridge hands are possible containing five spades, three diamonds, three clubs, and two hearts? It is known from experience that in a certain industry 60 percent of all labor– management disputes are over wages, 15 percent are over working conditions, And 25 percent are over fringe issues. Also, 45 percent of the ...Use the Venn diagram of Figure 2.7 And the method by which we proved Theorem 2.7 to prove Theorem 2.8. Figure 2.7
Post your question
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s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886104172.67/warc/CC-MAIN-20170817225858-20170818005858-00408.warc.gz
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CC-MAIN-2017-34
| 1,215 | 4 |
http://www.rcgroups.com/forums/showpost.php?p=22952292&postcount=6
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math
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Ok, using this handy tool; Right triangle calculator
, I did some estimating.
As built, the wing (measured with opposite panel flat and from root to the bottom of the last rib) has only about 1". The length of the panel from root to last rib is 16.75". Plugging these into the handy-dandy calculator yeilds 3.42deg which, of course has to be divided by two since the opposite panel is flat, yeilds only 1.71 deg. Way too small.
Looking at some other plans that I have, the dihedral angle looks to be between 4 and 6 degrees (where did I leave my protractor). So, given that range, it looks as though the measurement, performed as above, at the tip rib would be between 2.34" and 3.52". Splitting the difference results in 2.93. Erring on the side of caution, rounding up would seem to make sense, so 3" is the final number.
Now reversing the process to check the angle results in 5.14 degrees of dihedral. So now the question is whether 5 degrees sounds reasonable.
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s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645176794.50/warc/CC-MAIN-20150827031256-00182-ip-10-171-96-226.ec2.internal.warc.gz
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CC-MAIN-2015-35
| 965 | 5 |
https://www.automaticwasher.org/cgi-bin/TD/TD-VIEWTHREAD.cgi?84853_11
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math
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Thread Number: 84853
/ Tag: Small Appliances
how to clean baked on black gunk?
|[Down to Last]|
|Post# 1093204 10/14/2020 at 21:25 by gizmo (Great Ocean Road, Victoria, Au) || |
I have recently scrounged a Philips Air Fryer XXL. It is not very old, still more or less a current model and a fancy one, too.
It has no faults but was really filthy when I picked it up. I have pulled it to bits, cleaned it and checked it, it works fine and is a good gadget. I will keep it. I suspect it was thrown out because of smoke when cooking - the area around the fan was full of vile brown goo.
The area around the fan and heating element have responded to elbow grease and the selection of chemicals I threw at it. They are now clean.
This model has a basket made of sheet metal sides, covered in some nonstick stuff. The base of the basket is separate, made of stainless steel mesh. That was almost blocked with black residue but is now clean. Being stainless, I could get stuck into it with scourers and sharp implements, without damaging it.
The nonstick basket is different. It is still covered with really hard baked black stuff. Some of it has flaked off but most is stuck hard to the nonstick surface.
I'm usually good at this sort of thing but I'm stumped.
So far I have tried BBQ cleaner from Aldi - useless.
Some sort of spray on kitchen degreaser - useless.
Simple Green - little to no effect.
Wet with cloudy ammonia and leave in a plastic bag overnight - removed the easy stuff.
Citro-Clean orange oil based spray - removed the softer brown stuff but little to no effect on the hard black stuff. It is usually magic - I love the stuff.
Baking soda paste - probably the most effective. It seemed to soften some of the muck but I have done 2 or 3 applications, each left overnight. The remaining residue won't shift.
I later tried a spray on oven cleaner, the kind of product I usually try to avoid. I had imagined it would dissolve the black layer and I'd wipe it off next day... no such luck. It had no effect at all, after 2 applications, each left overnight.
I have had a little luck using a bamboo chopstick to scrape at the residue - some of it flakes off but it is slow work and I don't want to damage the nonstick any more than i have already.
If you could mention generic types of product you suggest, or specific ingredients, rather than brand names, would be helpful as we probably have different product brands in Australia.
my final option is to replace the basket, link below. but I'd rather clean this one if I can.
CLICK HERE TO GO TO gizmo's LINK
|Post# 1093207 , Reply# 1 10/14/2020 at 22:02 by CircleW (NE Cincinnati OH area) || |
If you have a restaurant supply place near you, they should have a carbonized grease remover, such as Carbon-Off.
|Post# 1093209 , Reply# 2 10/14/2020 at 22:14 by gizmo (Great Ocean Road, Victoria, Au) || |
Thanks, sounds like what I want.
I used to be a professional cook but kept my stuff clean and never had to resort to such products...
|Post# 1094334 , Reply# 3 10/24/2020 at 10:54 by Helicaldrive (St. Louis) || |
Soak in very hot sudsy water with as much ammonia as you dare, but not enough to create uncomfortable fumes, then use a Mr Clean Magic Eraser pad.
Thatís what Iíd do....
(And set it outside to soak, or cover the sink with a big towel while soaking, if you have any pets, especially birds.)
And with Thanksgiving coming up, might as well mention that thatís my method for getting my stainless steel roaster brand new, gleaming clean. 30 minutes is usually long enough to soak. And now I usually no longer need to use Bon Ami scouring powder. Just enough hot sudsy water to cover, and it doesnít take much ammonia, just a glug.
(OMG one time I poured straight ammonia all over the shower in my elderly folksí house to remove stubborn soap scum. Clearly they hadnít cleaned it in years after they got too old. What a mistake! And I knew better. I had to run out of the room, and in intervals as long as I could hold my breath, throw open windows and use a fan to exhaust the fumes, and rinse the excess away with buckets of hot water. It got the shower clean though, when nothing else would! If they would have had a water softener, no soap scum would have ever built up after they got too old to clean. For those who insist that soap does not rinse off in hard water, the shower that never needs to be cleaned again except for drying it out with a towel after showering to prevent mold and mildew disproves that belief. Even if I didnít like soft water, Iíd still have a water softener if for no reason other than to save myself the weekly chore of cleaning the shower with that nasty bathroom spray foam cleaner.)
|Post# 1094337 , Reply# 4 10/24/2020 at 11:03 by Helicaldrive (St. Louis) || |
For those who insist that soap does not rinse off in softened water, the shower disproves that belief.
To each his own, though. Most of my friends truly detest softened water, so whatever floats your boat.
|Post# 1094353 , Reply# 5 10/24/2020 at 13:01 by fan-of-fans (Florida) || |
Maybe place it in a pot of boiling water with dish soap in it, for a few minutes, then let sit?
That's what I use to clean my (regular) electric deep fryer frypot and basket.
|Post# 1094358 , Reply# 6 10/24/2020 at 13:33 by joeekaitis (Rialto, California, USA) || |
|Post# 1094424 , Reply# 7 10/24/2020 at 22:15 by gizmo (Great Ocean Road, Victoria, Au) || |
I've tried ammonia soaking overnight. That's an old family favourite. It did nothing.
Easy-Off fume free oven cleaner did absolutely nothing. Might as well have been water. Tried 2 overnight applications.
This stuff is really rock hard. Nothing seems to soften it.
Thanks for all your suggestions.
If I can't track down some carbon removing stuff, I will probably just replace the basket.
I have already spent enough on cleaning potions that didn't work to have bought a basket any way...
|Post# 1094427 , Reply# 8 10/24/2020 at 22:21 by Brisnat81 (Brisbane Australia) || |
Selleys make a gel oven cleaner that Iíve always found to be effective.
The Gel generally only seems to be available at IGA, whereas the spray is available at Bunnings, but I havenít tried it.
See how you go
|Post# 1094430 , Reply# 9 10/24/2020 at 22:50 by gizmo (Great Ocean Road, Victoria, Au) || |
does the Selleys gel work on REALLY hard black stuff?
I have seen it mentioned online and was tempted but a jar is about $10 and a new basket is about $36...
The stuff I am trying to get off is seriously hardened. All the other products I have tried have removed anything easier. This is the toughest of tough stuff. I was having some luck chipping at the edges of the black residue with a bamboo chopstick, but what is left is wearing away the chopstick...
|Post# 1094448 , Reply# 10 10/25/2020 at 05:55 by ozzie908 (Lincoln UK) || |
If you know anyone in catering or who has a grill cleaning service they will get it clean.
Its a very strong alkaline substance that works best at a temp of 45c, I had one at the restaurant it was like a large tank with a basket you put anything that needed cleaning in left overnight and in the morning rinse in cold water and its all like new...!
There is a chemical you can buy that does a similar thing but is not so efficient.
|Post# 1094510 , Reply# 11 10/25/2020 at 18:33 by Stevet (West Melbourne, FL) || |
If you have a self cleaning oven, you may want to give that a try. Use the shortest cleaning time and see if the crud gets burned off. I have an aunt who did that to all her carbon encrusted frying pans and they all came out like nearly new. Just be prepared for some more smoke than you may usually get when the cycle is going through its paces.
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| 7,756 | 61 |
https://web2.0calc.com/questions/how-many-trees-can-the-farmer-plant
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math
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A farmer has half a hectare to plant fruit trees in. Each fruit tree must have 25 metres squared to grow. How many trees can the farmer plant?
1/2 x 10000m^2 / 25 m^2/tree = 200 trees
well done thats correct
25m -> 1 tree
5000m^2-> x trees
<=>x=5000 div 25<=>
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s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589029.26/warc/CC-MAIN-20180716002413-20180716022413-00490.warc.gz
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| 259 | 6 |
https://www.hackmath.net/en/math-problem/21763
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math
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The Earth's surface is 510,000,000 km2. Calculates the radius, equator length, and volume of the Earth, assuming the Earth has the shape of a sphere.
We will be pleased if You send us any improvements to this math problem. Thank you!
Thank you for submitting an example text correction or rephasing. We will review the example in a short time and work on the publish it.
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Related math problems and questions:
The Earth's surface is approximately 510,000,000 km2 and is 7/10 covered by oceans. Of which 1/2 covers the Pacific Ocean, the Atlantic Ocean 1/4, the Indian Ocean 1/5 and the Arctic Ocean 1/20. What parts of the Earth's surface cover each ocean?
- Volume of sphere
How many times does the volume of a sphere increase if its radius increases 2 ×?
- Sphere VS
Find the surface and volume of a sphere that has a radius of 2 dm.
- Earth rotation
How fast is the place on the Earth's equator moving if the Earth's radius is 6378 km?
- The cube
The cube has a surface area of 216 dm2. Calculate: a) the content of one wall, b) edge length, c) cube volume.
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Earth's radius is 6370 km long. Calculate the length parallel of latitude 50°.
The cube has an area of base 256 mm2. Calculate the edge length, volume, and area of its surface.
- Earth's diameter
The Earth's diameter on the equator is approximately 12750 km. How long does the Gripen flyover the Earth above the equator at 10 km if it is at an average speed of 1500 km / h?
Determine the radius of the circle, if its perimeter and area is the same number.
- Area to perimeter
Calculate circle circumference if its area is 254.34cm2
- The equator
Calculate the length of the equator. Calculate a radius of 6378 km. Round to the nearest thousand. - Procedure and calculation
- The ball
The ball has a radius of 2m. What percentage of the surface and volume is another sphere whose radius is 20% larger?
- Perimeter of the circle
Calculate the perimeter of the circle in dm, whose radius equals the side of the square containing 0.49 dm2?
- The spacecraft
The spacecraft spotted a radar device at altitude angle alpha = 34 degrees 37 minutes and had a distance of u = 615km from Earth's observation point. Calculate the distance d of the spacecraft from Earth at the moment of observation. Earth is considered a
- Twice of radius
How many times does the surface of a sphere decrease if we reduce its radius twice?
Suppose that tourist went on foot over the Globe equator. How many meters more track made his hat on his head as the shoes on your feet? The radius of the earth is 6378 km, and the height of the tourist is 1.7 m.
- Area to volume
If the surface area of a cube is 486, find its volume.
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CC-MAIN-2021-21
| 2,865 | 37 |
https://nrich.maths.org/public/leg.php?code=-68&cl=3&cldcmpid=657
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math
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What would be the smallest number of moves needed to move a Knight
from a chess set from one corner to the opposite corner of a 99 by
99 square board?
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9,
12, 15... other squares? 8, 11, 14... other squares?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
Imagine a large cube made from small red cubes being dropped into a
pot of yellow paint. How many of the small cubes will have yellow
paint on their faces?
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable.
Decide which of these diagrams are traversable.
Show that all pentagonal numbers are one third of a triangular number.
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Can you find a rule which connects consecutive triangular numbers?
Draw a pentagon with all the diagonals. This is called a pentagram.
How many diagonals are there? How many diagonals are there in a
hexagram, heptagram, ... Does any pattern occur when looking at. . . .
Show that among the interior angles of a convex polygon there
cannot be more than three acute angles.
A huge wheel is rolling past your window. What do you see?
Mathematics is the study of patterns. Studying pattern is an
opportunity to observe, hypothesise, experiment, discover and
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
A tilted square is a square with no horizontal sides. Can you
devise a general instruction for the construction of a square when
you are given just one of its sides?
Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . .
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
Imagine starting with one yellow cube and covering it all over with
a single layer of red cubes, and then covering that cube with a
layer of blue cubes. How many red and blue cubes would you need?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Draw a square. A second square of the same size slides around the
first always maintaining contact and keeping the same orientation.
How far does the dot travel?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
Can you find a way of representing these arrangements of balls?
The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?
What is the shape of wrapping paper that you would need to completely wrap this model?
When dice land edge-up, we usually roll again. But what if we
Jo made a cube from some smaller cubes, painted some of the faces
of the large cube, and then took it apart again. 45 small cubes had
no paint on them at all. How many small cubes did Jo use?
Use the interactivity to listen to the bells ringing a pattern. Now
it's your turn! Play one of the bells yourself. How do you know
when it is your turn to ring?
Some puzzles requiring no knowledge of knot theory, just a careful
inspection of the patterns. A glimpse of the classification of
knots and a little about prime knots, crossing numbers and. . . .
Imagine you are suspending a cube from one vertex (corner) and
allowing it to hang freely. Now imagine you are lowering it into
water until it is exactly half submerged. What shape does the
surface. . . .
Can you describe this route to infinity? Where will the arrows take you next?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
How many different ways can I lay 10 paving slabs, each 2 foot by 1
foot, to make a path 2 foot wide and 10 foot long from my back door
into my garden, without cutting any of the paving slabs?
Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?
These are pictures of the sea defences at New Brighton. Can you
work out what a basic shape might be in both images of the sea wall
and work out a way they might fit together?
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
Here is a solitaire type environment for you to experiment with. Which targets can you reach?
A game for 2 players. Can be played online. One player has 1 red
counter, the other has 4 blue. The red counter needs to reach the
other side, and the blue needs to trap the red.
Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .
How many moves does it take to swap over some red and blue frogs? Do you have a method?
A package contains a set of resources designed to develop pupils'
mathematical thinking. This package places a particular emphasis on
“visualising” and is designed to meet the needs. . . .
A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .
Bilbo goes on an adventure, before arriving back home. Using the
information given about his journey, can you work out where Bilbo
Can you mark 4 points on a flat surface so that there are only two
different distances between them?
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s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375095557.73/warc/CC-MAIN-20150627031815-00296-ip-10-179-60-89.ec2.internal.warc.gz
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CC-MAIN-2015-27
| 7,175 | 91 |
https://www.physicsforums.com/threads/questions-about-atoms.313095/
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math
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How did we know that the more common isotope of iron has 26 protons and 30 neutrons? I'm asking a general question so the iron example is just a special case. How did we know that oxygen has 8 protons? (and 8 neutrons), and also that the 8 neutrons is the most stable isotope and that there exists other isotopes in smaller quantities. How did we determine the quantities of the isotopes?
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CC-MAIN-2018-22
| 388 | 1 |
http://nrich.maths.org/public/leg.php?code=-420&cl=3&cldcmpid=684
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math
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Is there an efficient way to work out how many factors a large number has?
What is the largest number which, when divided into 1905, 2587,
3951, 7020 and 8725 in turn, leaves the same remainder each time?
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
The clues for this Sudoku are the product of the numbers in adjacent squares.
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
A car's milometer reads 4631 miles and the trip meter has 173.3 on
it. How many more miles must the car travel before the two numbers
contain the same digits in the same order?
What does this number mean ? Which order of 1, 2, 3 and 4 makes the
highest value ? Which makes the lowest ?
Ben passed a third of his counters to Jack, Jack passed a quarter
of his counters to Emma and Emma passed a fifth of her counters to
Ben. After this they all had the same number of counters.
Take any four digit number. Move the first digit to the 'back of
the queue' and move the rest along. Now add your two numbers. What
properties do your answers always have?
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.
Find a cuboid (with edges of integer values) that has a surface
area of exactly 100 square units. Is there more than one? Can you
find them all?
Use the differences to find the solution to this Sudoku.
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...
Show that if you add 1 to the product of four consecutive numbers
the answer is ALWAYS a perfect square.
My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?
In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G =
F and A-H represent the numbers from 0 to 7 Find the values of A,
B, C, D, E, F and H.
Five children went into the sweet shop after school. There were
choco bars, chews, mini eggs and lollypops, all costing under 50p.
Suggest a way in which Nathan could spend all his money.
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
A 2-Digit number is squared. When this 2-digit number is reversed
and squared, the difference between the squares is also a square.
What is the 2-digit number?
Show that is it impossible to have a tetrahedron whose six edges
have lengths 10, 20, 30, 40, 50 and 60 units...
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
How many different symmetrical shapes can you make by shading triangles or squares?
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
This shape comprises four semi-circles. What is the relationship
between the area of the shaded region and the area of the circle on
AB as diameter?
Chris and Jo put two red and four blue ribbons in a box. They each
pick a ribbon from the box without looking. Jo wins if the two
ribbons are the same colour. Is the game fair?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
A game for 2 or more people, based on the traditional card game
Rummy. Players aim to make two `tricks', where each trick has to
consist of a picture of a shape, a name that describes that shape,
and. . . .
All CD Heaven stores were given the same number of a popular CD to
sell for £24. In their two week sale each store reduces the
price of the CD by 25% ... How many CDs did the store sell at. . . .
There are lots of different methods to find out what the shapes are worth - how many can you find?
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
Can you explain the surprising results Jo found when she calculated
the difference between square numbers?
Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?
Can you arrange these numbers into 7 subsets, each of three
numbers, so that when the numbers in each are added together, they
make seven consecutive numbers?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
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s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698540409.8/warc/CC-MAIN-20161202170900-00327-ip-10-31-129-80.ec2.internal.warc.gz
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CC-MAIN-2016-50
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http://www.jiskha.com/display.cgi?id=1338340703
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math
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Posted by Vladimyr on .
Two cars are moving. The first car has twice the mass of the second car, but only half as much kinetic energy. Then, both cars increase their speed by 8.00 m/s. They then have the same kinetic energy. The original speeds of the two cars are, respectively,
Physics - Elena, Wednesday, May 30, 2012 at 7:36am
m1•v1²/2 =m2•v2²/2 = m1•v2²/4.
m1• (v1+8)²/2 = m2• (v2+8)²/2 =
v1 =sqrt(32)= 5.66 m/s.
v2 =11.3 m/s
Physics - Breyonna Fawling, Friday, December 5, 2014 at 2:10pm
What if the instead of 8.00m/s it was 6.52m/s?
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CC-MAIN-2017-13
| 555 | 9 |
http://www.chegg.com/courses/yc/AVT/104
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math
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There are no reviews for this course.
Be the first to write one.
There are no students in this course.
It will cost $6,000 to acquire an ice cream cart. Cart sales are expected to be $3,600 a year for three years. After the three years, the cart is
expected to be worthless as the expected life of the refrigeration unit is only three years. What is the payback period?
a. 1.67 years
b. 2.00 years
c. 1.82 years
d. 1.48 years
I need Explination please•
We can't find any books for this course.
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s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1387345760007/warc/CC-MAIN-20131218054920-00007-ip-10-33-133-15.ec2.internal.warc.gz
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CC-MAIN-2013-48
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http://mathhelpforum.com/discrete-math/71556-solved-greatest-common-divisor-problems.html
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math
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Hello, I guess these problems are not supposed to be very hard, but I am confused.
1. Let . Prove that if and only if there exists such that .
2. Prove that if with and , then .
Thanks in advance.
Follow Math Help Forum on Facebook and Google+
Let be a prime divisor of . Since has to divise or (because is irreducible).
But doesn't divide Hence and that is true for any prime divisor of
Therefore (Be more precise than me to show that )
View Tag Cloud
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CC-MAIN-2017-17
| 452 | 9 |
https://education.ti.com/en-au/activity/detail?id=759DBE175ED04B3EB5EFDB6FD3105179
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math
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The activity shows how to use TI 89 Titanium calculator to perform simple operations on vectors such as: finding magnitudes, unit vectors, dot product, vector resolutes and angle between two vectors.
During the Activity
Students work through some vectors problems using the CAS calculator. They also learn how to se he Vectors program.
Vectors.89P is a simple vector problem which finds the angle between wo vectors as one of its functions.
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s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125946564.73/warc/CC-MAIN-20180424041828-20180424061828-00008.warc.gz
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CC-MAIN-2018-17
| 440 | 4 |
https://www.proofwiki.org/wiki/Principle_of_Mathematical_Induction/Informal_Analogy
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math
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Principle of Mathematical Induction/Informal Analogy
Jump to navigation Jump to search
The Principle of Mathematical Induction is often likened to a domino cascade.
A line of dominoes is set up to be balanced on their ends so that if one of the dominoes is knocked over, it knocks over the next one in the line.
When the first domino is knocked over, the entire line topples, one after the other.
It follows that if either:
- $(1) \quad$ no domino is knocked over to start with (that is, the basis for the induction does not hold)
- $(2) \quad$ the gap between two dominoes is too large for one domino to knocked over the next one (that is, the induction step does not hold)
then the domino cascade does not work and the entire line does not fall (that is, the proposition is not proved for all $\N \ge n_0$).
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 3$: Natural Numbers: $\S 3.7$: Principle of induction
- 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction
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CC-MAIN-2023-14
| 1,132 | 11 |
http://newbreedgrace.org/amortization-computation-formula-the-treatments/
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math
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Very, why don’t we begin by explaining amortization, essentially, because process of decreasing the worth of a secured asset or the balances of that loan by a periodic quantity . Each time you making a payment on that loan you have to pay some interest alongside a part of the primary. The principal will be the original amount borrowed, or the stability that you must pay. By creating normal periodic payments, the principal gradually lowers, so when it hits zero, you have completely paid down your financial troubles.
Usually, whether you can afford that loan depends upon whether you can afford the periodic cost (generally a payment cycle). Thus, the most crucial amortization formula will be the computation for the cost quantity per duration.
Calculating the Fees Quantity per Stage
The formula for determining the payment quantity is revealed below.
Instance: what can the payment be on a 5-year, $20,000 car finance with a moderate 7.5% yearly rate of interest? We’ll assume that the first rates had been $21,000 and you’ve generated a $1,000 advance payment.
You should use the amortization calculator below to find out the fees Amount (A) are $400.76 every month.
P = $20,000 roentgen = 7.5percent every year / year = 0.625per cent per duration (this is exactly registered as 0.00625 inside the calculator) letter = five years * 12 months = 60 total menstruation
Amortization Repayment Calculator
Determining the Payment in Shine
Microsoft succeed features a number of integral features for amortization formulas. The big event corresponding on the formula above is the PMT work. In shine, you might estimate the payment per month utilizing the after formula:
Calculating the interest rate A Stage
When the few compounding intervals fits the amount of repayment durations, the interest rate per period (r) will be easy to calculate. Like the preceding example, it is simply the nominal annual speed broken down from the intervals annually. However, where do you turn when you yourself have a Canadian mortage while the compounding stage try semi-annual, but you make monthly payments? In that case, you can utilize these formula, derived from the composite interest formula.
Example: If the nominal annual rate of interest is i = 7.5per cent, and the interest are combined semi-annually ( letter = 2 ), and repayments are formulated monthly ( p = 12 ), then the speed per stage is r = 0.6155per cent.
Crucial: If the chemical duration is actually reduced compared to payment cycle, employing this formula causes unfavorable amortization (paying interest on interest). See my personal post, “negative amortization” to learn more.
If you find yourself wanting to solve for your yearly interest rate, only a little algebra gives:
Sample: utilizing the RATE() formula in shine, the pace per cycle (r) for a Canadian financial (compounded semi-annually) of $100,000 with a payment per month of $584.45 amortized over twenty five years try 0.41647percent calculated utilizing r=RATE(25*12,-584.45,100000) . The yearly speed is computed as 5.05percent making use of the formula i=2*((0.0041647+1)^(12/2)-1) .
Calculations in an Amortization Plan
Whenever you understand the installment levels, it’s very hassle free to produce an amortization routine. The sample below reveals the very first 3 and final 3 costs for all the preceding example. Each line shows the sum total fees levels along with exactly how much interest and major you will be spending. Determine just how much even more interest you only pay at first than at the conclusion of the borrowed funds!
The Interest part of the cost was computed since the rate (roentgen) period the last balance, and it is generally rounded towards closest cent. The Principal part of the installment are determined as levels – Interest payday advance Kenner Kenner. This new Balance try determined by subtracting the primary through the earlier balance. The final installment levels must feel adjusted (as in the desk above) to make up the rounding.
An amortization timetable generally will show you simply how much interest and main you’re paying each course, and often an amortization calculator might determine the sum total interest compensated within the longevity of the mortgage. Besides taking into consideration the payment, you should think of the definition of from the loan (the quantity of age needed to repay it if one makes typical repayments). The lengthier you stretch out the mortgage, the greater number of interest you are going to end having to pay in the long run. Normally you need to make a trade-off within payment therefore the overall level of interest.
To quickly make your very own amortization plan to discover the interest rate, repayment period, and amount of the loan affect the amount of interest which you pay, see a few of the amortization hand calculators down the page.
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CC-MAIN-2023-14
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https://uk.answers.yahoo.com/question/index?qid=20200920002100AAJlHNo
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math
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A skier is skiing down a frictionless 30 degree slope. She starts at rest. Her mass is 50 kg.?
a. What will her acceleration be along the slope?
b. What will be the normal force that the slope exerts on her?
c. If the coefficient of static friction is 0.20, will she slide down the hill?
- oubaasLv 71 month agoFavourite answer
acceleration a = g*sin 30° = 4.903 m/sec^2
N = m*g*cos 30° = 50*9.806*0.866 = 425 N
motive force MF = m*g*sin 30°
friction force FF = m*g*cos 30°*μ
accelerating force AF = MF-FF = m*g*(sin 30°-cos 30°*μ )
acceleration a' = AF/m = g*(sin 30°-cos 30°*μ ) = 9.806*(0.5-0.173) = 3.2 m/sec^2
..she will slide down the hill with an acceleration a' = 3.2 m/sec^2
- billrussell42Lv 71 month ago
if she stopped, her velocity and acceleration are zero
weight = 9.8x50 = 49 N
component perpendicular to slope = 49cos30 = 49(√3/2) N = 42.4 N
component parallel to slope = 49sin30 = 24.5 N
force of friction = 0.2•49(√3/2) = 8.49 N
Yes, force of friction is less than 24.5 N
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https://www.analyzemath.com/calculus/Differential_Equations/simple.html
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math
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Solve Simple Differential Equations
This is a tutorial on solving simple first order differential equations of the form
y ' = f(x)
A set of examples with detailed solutions is presented and a set of exercises is presented after the tutorials. Depending on f(x), these equations may be solved analytically by integration. In what follows C is a constant of integration and can take any constant value.
Example 1: Solve and find a general solution to the differential equation.
y ' = 2x + 1
Solution to Example 1:
Integrate both sides of the equation.
ò y ' dx = ò (2x + 1) dx
y = x 2 + x + C.
As a practice, verify that the solution obtained satisfy the differential equation given above.
Example 2: Solve and find a general solution to the differential equation.
2 y ' = sin(2x)
Solution to Example 2:
Write the differential equation of the form y ' = f(x).
y ' = (1/2) sin(2x)
Integrate both sides
ò y ' dx = ò (1/2) sin(2x) dx
Let u = 2x so that du = 2 dx, the right side becomes
y = ò (1/4) sin(u) du
y = (-1/4) cos(u) = (-1/4) cos (2x)
Example 3: Solve and find a general solution to the differential equation.
y 'e -x + e 2x = 0
Solution to Example 3:
Multiply all terms of the equation by e x and write the differential equation of the form y ' = f(x).
y ' = - e 3x
Integrate both sides of the equation
ò y ' dx = ò - e 3x
Let u = 3x so that du = 3 dx, write the right side in terms of u
y = ò (-1/3) e u du
y = (-1/3) e u = (-1/3) e 3x
Exercises: Solve the following differential equations.
a) 2y ' = 6x
b) y ' cos x = sin(2x)
c) y ' e x = e 3x
Solutions to the above exercises
a) y = (3/2) x 2 + C
b) y = -2 cos x + C
c) y =(1 / 2) e 2x + C
More references on
Differential Equations - Runge Kutta Method
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| 1,719 | 41 |
https://www.hothothoops.com/2015/4/16/8425409/miami-heat-draft-pick-lottery-odds-10-seed-nba
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math
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The Miami Heat missed the playoffs for the first time since 2008, and they will be on the fence about their draft pick for another month. The NBA Draft Lottery selection will take place on May 19. At that point, the Heat will know what their first round pick is, and whether or not they get to keep it.
Here is the break down of everything you need to know about that pick.
The Miami Heat traded away it's right to their 2015 first round pick to the Cleveland Cavaliers in exchange for LeBron James. The Cavaliers traded the pick to Minnesota in part of the package that included Kevin Love. Minnesota then sent the pick the 76ers in return for Thaddeous Young.
The pick is not to be exchanged if it is a top 10 pick. Right now, the Heat are seeded as the 10th pick, but that is before the NBA Draft Lottery.
How the Lottery works
There's always some confusion about this. There are 14 teams that don't make the NBA playoffs, they are the lottery picks. Each team is given a percentage chance to land the 1st, 2nd, or 3rd pick. The lottery is for those picks. After that, the rest of the order is by seeding.
So, each team has a chance at landing in the top 3, if not, they will return to their seeding, unless a team behind them in seeding lands in the top 3.
It looks a little like this:
1. 250 combinations, 25.0% chance of receiving the #1 pick
2. 199 combinations, 19.9% chance
3. 156 combinations, 15.6% chance
4. 119 combinations, 11.9% chance
5. 88 combinations, 8.8% chance
6. 63 combinations, 6.3% chance
7. 43 combinations, 4.3% chance
8. 28 combinations, 2.8% chance
9. 17 combinations, 1.7% chance
10. 11 combinations, 1.1% chance
11. 8 combinations, 0.8% chance
12. 7 combinations, 0.7% chance
13. 6 combinations, 0.6% chance
14. 5 combinations, 0.5% chance
And here are the chances for each team and their possible landing in the lottery.
The Heat's chances
As you can see, the Heat have a 1.1% chance to land the #1 overall pick. They have a 1.3% chance for the 2nd pick and a 1.6% chance for the 3rd pick. Miami has a 91% chance to keep their spot as a 10th pick, and a 9% chance that it falls past 10th, which would then return the pick to the 76ers.
The most likely scenario has the Heat keeping their pick and selecting at #10.
But nobody here at HHH is going to mad if the Heat land in the top 3, am I right?
Recent Miami Heat lottery picks: Michael Beasley (2008), Dwyane Wade (2003), Caron Butler (2002)
The Odds History
Twice in the last seven years has a team win a 1.7% chance to win has won the lottery (2008 Bulls and 2013 Cavs). The 1993 Magic are the lowest seeded team to win at 1.52% (11th worst chance). But it hasn't been since the 2004 Magic that the team with the worst record won the lottery.
I wouldn't be holding your breathe for a top 3 pick, what you should hope for is that the Utah Jazz, Phoenix Suns, Oklahoma City Thunder and Indiana Pacers -- who are all seeded behind the Miami Heat -- don't move up in the lottery. That would result in the Heat being bumped to 11th and not having the pick.
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s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571056.58/warc/CC-MAIN-20220809155137-20220809185137-00660.warc.gz
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CC-MAIN-2022-33
| 3,038 | 31 |
https://www.rand.org/pubs/reports/R3712.html
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math
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A New Approach to Three-Dimensional Free-Surface Flow Modeling
This report describes an efficient model for the computation of three-dimensional free-surface flows. For the time integration, two different approximations are used in succession. The advection term approximations depend on the flow direction and introduce stability into the computation. The gradients of the advection terms are generally computed directly, whereas the coefficients of the nonlinear terms must be computed by iteration. Experiments show that only two iterations are required. The finite difference approximations used in the computation are of the second order, and no time filtering or introduction of viscosity is required for stability. The timestep is limited by the accuracy desired in the results. In the formulation, it is assumed that the pressures are hydrostatic. A stability analysis that uses linearized coefficients in the nonlinear terms indicates that the computation method is unconditionally stable. To investigate the effect of different timesteps on model results, the author made several experiments that confirmed the analysis of the behavior of the computation method showing that the growth of disturbances with small wavelength and periods is inhibited. A model of a lake (IJsselmeer) showed that, with limited computer resources, effective three-dimensional flow and transport computations can be made.
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s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363418.83/warc/CC-MAIN-20211207201422-20211207231422-00030.warc.gz
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CC-MAIN-2021-49
| 1,409 | 2 |
https://goyalassociatesltd.com/digital-currency/at-what-nominal-rate-compounded-continuously-must-money-be-invested-to-triple-in-7-years.html
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math
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At what nominal rate compounded continuously must money be invested to double in years?
So we want to use we call the rule of 72. And essentially what this rule states is that a investment at one compounded continuously will double in 72 years.
At what nominal rate compounded continuously must money be invested to in 8 years?
Doubling Rate: At what nominal rate compounded continuously must money be invested to double in 8 years? In order for the initial investment to double in 8 years, the money must be invested in an account with a nominal rate of approximately 8.7% compounded continuously.
What is nominal rate compounded?
The nominal interest rate, also known as an Annualised Percentage Rate or APR, is the periodic interest rate multiplied by the number of periods per year. For example, a nominal annual interest rate of 12% based on monthly compounding means a 1% interest rate per month (compounded).
How long will it take for an investment to triple in value if it earns compounded continuously Round your answer to three decimals?
hence to the nearest year, it will it take 18 years for an investment to triple, if it is continuously compounded at 6% per year.
How many years will it take an investment to triple if the interest rate is 8 compounded continuously?
Answer and Explanation: The answer to the question is 14.3 years.
How do you find nominal interest rate compounded continuously?
Using our formula from our Effective Annual Interest Rate Calculator, where i = e^r – 1 becomes e^r = i + 1. And, by definition ln(e^r) = r , we can solve for r to get the formula: r = ln(i + 1).
How do you find the nominal rate?
How to Calculate the Nominal Rate of Return
- Subtract the original investment amount (or principal amount invested) from the current market value of the investment (or at the end of the investment period).
- Take the result from the numerator and divide it by the original investment amount.
How is nominal interest rate determined?
It states that the nominal interest rate is approximately equal to the real interest rate plus the inflation rate (i = R + h). For example, a bond investor is expecting a real interest rate of 5%, when the market shows an expected inflation rate of 3%.
What rate of interest compounded annually is required to triple an investment in 15 years?
At a rate of interest of 3.86% compounded annually investment will be tripled.
What interest rate is needed to double an investment?
Alternatively you can calculate what interest rate you need to double your investment within a certain time period. For example if you wanted to double an investment in 5 years, divide 72 by 5 to learn that you’ll need to earn 14.4% interest annually on your investment for 5 years: 14.4 × 5 = 72.
How long will an account invested at 11.5% compounded yearly be tripled?
The answer is the number of years to triple an amount at 11.5% compounded annually is 10.0925 years.
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s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335514.65/warc/CC-MAIN-20221001003954-20221001033954-00311.warc.gz
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CC-MAIN-2022-40
| 2,927 | 24 |
http://wilcoxsway.blogspot.com/2017/07/area-model-in-middle-school-classroom.html
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math
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1. Distributive Property--We all know that this is an important concept moving forward, but it can sometimes be tricky for students to wrap their minds around. I use lots of different strategies to help kids understand the distributive property, but the area model is definitely one of them.
The representation below can be seen as two rectangles, a 5 x 8 with an area of 40 and a 5 x 12 with an area of 60. Or you can see this as one rectangle, a 5 x 20 with an area of 100. This is a concept that is understandable for students, and it is a good way to reinforce our abstract ways of showing this concept.
Learning abstract representations of math can be one of the major challenges as students transition from elementary to secondary math, so connections like these can be helpful.
2. Factoring--This is the natural extension of using area model to teach distributive property. By simply leaving out the shared side length, we encourage students to factor, and help them see the connection between factoring and the distributive property.
It's good to start with an example that only shares one common factor, like this one.
Students can see that the side length has to be the same number. Next, we want them to make the connection between the same side length and a common factor of 35 and 56. Student thinking might be like this:
What 5 times what equals 35? 8 times what equals 56?
Finally, we want to students to make connections between the picture and the to the abstract work:
35 + 56 = 7(5 + 8).
Now, you can move to examples that have more than one common factor that could be factored out.
40 + 60 = 5(8 + 12).
Connecting the drawing back to the work is important....where do you find the 40, the 5, the 8, and so on in the picture?
3. Battling Common Misconceptions--If you give your students the problems (8)(4.5), would you be surprised to have some students give the answer of 32.5? Me neither! But the area model can again help us out.
If students have been using area model to show the distributive property, this representation should be familiar. This shows that the area is 36 and gives a visual illustration of why we can't multiply 8 x 4 to get 32 and simply add 0.5.
4. Reinforce proportional thinking--If I had to pick one topic that was the most important thing we do in middle school, it would be proportional reasoning. Every chance I get, every way I get, I want to reinforce proportional reasoning with my students. I want to give them different ways to see it. So what about this?
Since the side of 3 is the same for both rectangles, if you double the 4 to get 8, it also doubles the area from 12 to 24.
5. Move towards algebraic thinking--Ultimately, our middle school students need to be ready for the demands of algebraic thinking. The area model can also give us another way to get students using variables in middle school. Consider the progression of the examples shown below.
If students are consistently using the area model as a representation in our middle school classrooms, hopefully the jump to the last two representations will be easier.
So we've looked at multiplication charts and area model...what other elementary models and tools can continue to be helpful in middle school?
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| 3,226 | 19 |
https://dommaxim.ru/how-to-solve-a-pythagorean-theorem-problem-253.html
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math
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If you're behind a web filter, please make sure that the domains *.and *.are unblocked.The Pythagorean theorem is a way of relating the leg lengths of a right triangle to the length of the hypotenuse, which is the side opposite the right angle.
If you're behind a web filter, please make sure that the domains *.and *.are unblocked.The Pythagorean theorem is a way of relating the leg lengths of a right triangle to the length of the hypotenuse, which is the side opposite the right angle.Tags: Fun Cause And Effect Essay TopicsBusiness Plan Pro Premier V 12A Trip To The New York City EssayMiranda V Arizona EssayBig Words To Use In An EssayStrategic Planning EssayCritical Thinking By Example
But, the length of any side of a triangle can never be negative and therefore we only consider the positive square root.
In other situations, you will be trying to find the length of one of the legs of a right triangle.
Write an expression that shows the value of \(y\) in terms of \(x\).
Since no figure was given, your first step should be to draw one.
If the distance between the two building is 12 m, find the distance between their tops.
Solution: The vertical buildings AB and CD are 34 m and 29 m respectively.You can still use the Pythagorean theorem in these types of problems, but you will need to be careful about the order you use the values in the formula. The side opposite the right angle has a length of 12.Therefore, we will write: \(8^2 y^2 = 12^2\) This is the same as: \(64 y^2 = 144\) Subtracting 64 from both sides: \(y^2 = 80\) Therefore: \(\beginy &= \sqrt \ &= \sqrt \ &= \bbox[border: 1px solid black; padding: 2px]\end\) In this last example, we left the answer in exact form instead of finding a decimal approximation.Even though it is written in these terms, it can be used to find any of the side as long as you know the lengths of the other two sides.In this lesson, we will look at several different types of examples of applying this theorem.Due south and due west form a right angle, and the shortest distance between any two points is a straight line.Therefore, we can apply the Pythagorean theorem and write: \(3.1^2 2.8^2 = x^2\) Here, you will need to use a calculator to simplify the left-hand side: \(17.45 = x^2\) Now use your calculator to take the square root. \(\beginx &= \sqrt \ &\approx 4.18 \text\end\) As you can see, it will be up to you to determine that a right angle is part of the situation given in the word problem.The order of the legs isn’t important, but remember that the hypotenuse is opposite the right angle.Now you can apply the Pythagorean theorem to write: \(x^2 y^2 = (2x)^2\) Squaring the right-hand side: \(x^2 y^2 = 4x^2\) When the problem says “the value of \(y\)”, it means you must solve for \(y\).Therefore, we will write: \(y^2 = 4x^2 – x^2\) Combining like terms: \(y^2 = 3x^2\) Now, use the square root to write: \(y = \sqrt\) Finally, this simplifies to give us the expression we are looking for: \(y = \bbox[border: 1px solid black; padding: 2px]\) The Pythagorean theorem allows you to find the length of any of the three sides of a right triangle.It is one of those things that you should memorize, as it comes up in all areas of math, and therefore in many different math courses you will probably take.
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s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046155925.8/warc/CC-MAIN-20210805130514-20210805160514-00324.warc.gz
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CC-MAIN-2021-31
| 3,288 | 8 |
https://forum.godotengine.org/t/how-to-convert-a-torque-impulse-value-to-global-coordinate-system-as-camera-moves-around/4034
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math
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|Topic was automatically imported from the old Question2Answer platform.
Dear all, recently I’m using Godot to create a ball game. To simplify the question, assuming in the physics world there’s only one RigidBody called ball, a Camera which can be rotated around with ball.
Now I wanna give the ball a torque by using
apply_torque_impulse method, for example this torque value is
Vector3(0, 10, 0) which means (for now) rotate the ball along Z axis.
Then, I rotate the Camera around the ball for 90 degrees for example (the UP is Y axis always), and here the problem comes:
Before the camera move, the impulse value
Vector3(0, 10, 0) will let it rotate backwards (Z axis), but after moving, the same value cannot be used anymore, because the current correct axis should be X axis as I still want it to roll backwards!
So my question is how to convert/transform that value to the correct one as Camera moves to different angle degrees.
Thanks a lot.
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s3://commoncrawl/crawl-data/CC-MAIN-2024-10/segments/1707947474697.2/warc/CC-MAIN-20240228044414-20240228074414-00116.warc.gz
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CC-MAIN-2024-10
| 953 | 10 |
http://cwhomeworkbpiy.komedo.info/phd-thesis-in-math-education.html
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math
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Free essay help me phd thesis in mathematics education dissertation help glasgow teachers in the library homework help. Phd thesis on mathematics education phd thesis on mathematics education essay on my favourite game football phd thesis on mathematics education creating my resume. Phd in mathematics education at the university of massachusetts dartmouth the phd in mathematics education aims to build on doctoral work and the dissertation. Considering grad school faq: most mathematics graduate programs demand a reading knowledge of one or a master's degree may be earned without writing a thesis.
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Phd mathematics, education option up to one preliminary examination requirement can be filled by completion of a master's thesis in mathematics at virginia tech. Mathematics phd theses a selection of mathematics phd thesis titles is listed below, some of which are available on line: 2016 2015 2014 2013 2012 2011 2010 2009. Riemann's hypothesis and tests for primality probably comes close, although it's in cs/applied math, and the typesetting is a bit denser than the ones mentio. This is a list of all dissertations that have been submitted in partial satisfaction for the degree of doctorate of philosophy (phd) in mathematics at ucsd. Help with opening meningitis research paper thesis for phd in mathematics write design essay help can you write my thesis for me.
Università degli studi di milano department of mathematics phd thesis in mathematics on the p-laplace operator on riemannian manifolds date: 27th january, 2014. Dissertations at the harvard mathematics department harvard mathematics department harvard department of mathematics phd dissertations archival listing. The phd in mathematics education is designed for those with a strong background in both mathematics and mathematics education who wish to be leading researchers in. A phd student must complete a dissertation the dissertation must represent an original contribution to applied mathematics, pure mathematics, or statistics, and must. Home mathematics dissertations, theses phd candidates: you are welcome and encouraged to deposit your dissertation here.
2017 liwei zhang thesis defense: thursday, april 20th, 2017 thesis title: application of statistics inside channel information leakage analysis-modeling, metric. How to write persuasive speech phd thesis dissertation mathematics uk custom essay site essay on philosophy. The doctorate in mathematics is a degree that certifies both a high level the dissertation satisfactory progress towards the phd degree is defined as. Phd in mathematics at this point, the student chooses a thesis topic, finds a doctoral committee and presents a talk on his or her proposed research topic. How to write a psychology dissertation phd thesis on mathematics education dissertation service uk numbering doctor education course work.
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CC-MAIN-2018-43
| 3,304 | 5 |
https://pure.hud.ac.uk/en/publications/an-efficient-method-for-antenna-design-based-on-a-self-adaptive-b
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math
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Gaussian process (GP) is a very popular machine learning method for online surrogate-model-assisted antenna design optimization. Despite many successes, two improvements are important for the GP-based antenna global optimization methods, including: 1) the convergence speed (i.e., the number of necessary electromagnetic (EM) simulations to obtain a high-performance design) and 2) the GP model training cost when there are several tens of design variables and/or specifications. In both aspects, the state-of-the-art GP-based methods show practical but not desirable performance. Therefore, a new method, called the self-adaptive Bayesian neural network surrogate-model-assisted differential evolution (DE) for antenna optimization (SB-SADEA), is presented in this article. The key innovations include: 1) the introduction of the Bayesian neural network (BNN)-based antenna surrogate modeling method into this research area, replacing GP modeling, and 2) a bespoke self-adaptive lower confidence bound (LCB) method for antenna design landscape making use of the BNN-based antenna surrogate model. The performance of SB-SADEA is demonstrated by two challenging design cases, showing considerable improvement in terms of both convergence speed and machine learning cost compared with the state-of-the-art GP-based antenna global optimization methods.
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s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100531.77/warc/CC-MAIN-20231204151108-20231204181108-00137.warc.gz
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CC-MAIN-2023-50
| 1,349 | 1 |
https://lixowebssa.web.app/1317.html
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math
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Interval mathematics foundations, algebraic structures, and applications by hend dawood a thesis submitted to department of mathematics, faculty of science cairo university in partial fulllment of the requirements for the degree of master of science in computer science march 2012. The book does not presume much background in either mathematics or music. On this page you will find printable math worksheets and puzzles arranged according to levels. Almost thirty years ago i wrote a book about my experiences living in the us, and since then ive had many opportunities to work with literary professionals and to write essays apart from my regular occupation.
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Each chapter consists of text plus worked examples. Studying from text books is really important in alevels. Interval notation and linear inequalities math 0, fundamentals of mathematics the university of houston chapter 1. The files in this area contain worked examples in many of the topics studied in the various 100level mathematics courses. Aug 31, 2012 i get all my students to stick this into their maths exercise books at the beginning of the year. Mathematics learning centre, university of sydney 2 this leads us to another general rule. What are some textbooks i can use to prepare for a level cie. The skills are organized into categories, and you can click on any skill name to start practicing. Free math exercise workbooks and booklets for primary. Retypeset in 2014 for the mathematics learning centre by leon poladian. Stp mathematics 1a teachers notes and answers 3 exercise 1i p.571 732 728 1621 752 1204 322 1238 1383 1141 252 150 207 1130 1634 259 418 1582 924 757 923 752 795 178 514 623 335 935 485 836 1599 91 1128 368 1062 269 472 119 870 1085
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CC-MAIN-2024-10
| 10,102 | 16 |
http://goaskgrandpa.com/pages/sample02.html
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math
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(To Main Page)
Dear Grandpa -
reading a book that says quantum physicists believe there are at least 7
newly-discovered dimensions in addition to length, height, width, and time.
What are your thoughts? If
this is true, what in the world are these dimensions???
( Question is from Ronnie Sawyer - Age 18 )
word "dimension" has more than one meaning. When physicists talk of the four dimensions, those four have
some important characteristics.
Dimensions all apply to every physical thing that exists.
Dimensions can never be negative - only zero or positive.
Dimensions must be "mutually orthogonal."
a way to see these characteristics:
a book. Let's say it's one inch
thick. Let's say it's 6 inches
tall. Let's say it's 5 inches wide.
Let's say it was published in 1940 - and this particular copy was burned up in
- the book existed for 55 years with its thickness, width and height.
now let's consider the same book - only with one of these dimensions zero.
Say the thickness is zero. Well
- the book does not exist. There is
say the height is zero - Again, there is no book.
Or the width - No book.
say the thickness, width and height are there, but the book exists for zero time
- Still - No book. It never
you reasonably imagine a book that is minus 1 inch thick? No. Thickness is
about minus 5 inches wide? No -
width can be positive only.
6 inches tall? No - height can be
really small baby human may be only one foot long. A basketball player may be more than seven feet tall.
But nowhere is there a human that is minus seven feet tall - or minus any
is a better word for this than height. We
may turn the basketball player upside down and declare that his height measured
from his feet to his head is "minus seven feet," but he's still seven
feet long, and it's plus seven feet.
we might easily imagine a book that exists for zero time. It's
just a book that never was. But we
cannot reasonably imagine a book that exists for minus two years.
orthogonal" means that when one dimension is changed, the others are not
affected. They are independent.
Often we think of these as all at right angles to each other.
Up&Down is at a right angle to Left&Right and also at a right
angle to Forward&Back. Any one
of these three is at a right angle to the other two.
It's harder to see time this way - at right angles to the other three.
But one can see that if the book survives for two years or ten years, the
other three dimensions are unaffected by that difference.
is a "dimension" in this special way.
If we could go "backwards" in time, then we could publish a
book in 2001 - and take it with us to 1991, and we could say, "Here's a
book that is minus ten years old. Excuse
Grandpa, but "Poppycock."
We cannot move backwards in time, because having a negative time is just
like having a negative thickness or length. Something cannot cease to exist before it comes into
existence. A person cannot die before they are born. We can slow down - or speed up - but only zero to positive c
(The speed we have experimentally determined is the maximum.)
This constant, c, is not the speed of light, as often taught.
Light can travel at any speed up to c.
That's how lenses work; they slow the light down in an organized way. Responsible physicists keep their minds open, but we are
pretty sure that no actual thing can go faster than this.
mathematicians, we can define any number of other "dimensions."
We could plot the time it takes to boil water in a particular container
on a particular stove. We can call
the temperature one dimension - and time the other.
But temperature is not a dimension in the same way the "four"
are. Something could be reasonably
imagined to have zero temperature and still exist.
When temperature is raised, three of the dimensions increase with it, so
temperature is not "mutually orthogonal" to the others.
Temperature does meet the one requirement of never being negative, but
one requirement is not enough to be a "dimension."
(We talk about minus temperatures, but this is just a scale we use. The "absolute" temperature scale (Kelvin) tells us
how much heat is present, and we can't have a negative amount of heat.
On that scale, zero degrees - no heat - is roughly equivalent to -273
degrees centigrade.) Temperature is not a "dimension," in that we can't
use it to measure the "size" of some part of space or of an object in
could also invent something with ten dimensions, calling color one, and
temperature one, and smoothness one, etc. These
are dimensions, but not by the responsible physicist's definition of a
"dimension of nature." There
were only three we recognized for a long time - and with Albert Einstein, we
came to recognize that time has the same qualities and acts the same way as the
is also a popular myth that someone or something can "come from another
dimension." We can't come from
length or width, etc. Again, this
is a completely different definition of dimension and has nothing to do with
anyone discovers that there is a fifth dimension - He or she will surely win a
Nobel prize. It will not be the
subject of the voodoo metaphysicist, but of real scientists.
was once involved in a research project wherein we contacted a man who was
arguing for a 9 dimensional universe. We,
the taxpayers paid this quack more than a hundred thousand dollars.
He was making claims with absolutely nothing to back them up.
He was pretending to be a physicist with an interest in his country, but
he was actually just a thief.
sometimes thinks that it is unfortunate that a less sophisticated thief, who
takes an object, goes to jail, while these more sophisticated thieves always go
free, even while we are fully aware that they have taken much more.
Another student asked if "spirit" is a dimension.
You may figure out from the above that spirit may be regarded as a
dimension, but it is not a dimension in the same sense that length is - or the
others of the four.
Remember - no "thing" can exist unless it has measurements of
non-zero for all four of the dimensions. This
refers to physical things. We may
regard Love or Enthusiasm or Trust to exist, and these do not have widths,
lengths, thicknesses. These are not
"things" in the physical sense. They
are more like "ideas," or concepts of the mind.
So what about the dimensions themselves? Are they "things", or are they "ideas"?
Think about it. Does length have width and thickness? No. Length is
the name of one of the dimensions, but it does not have dimensions itself.
Length is an idea, not a thing. A
thing must have length, but length is not a thing itself.
That makes it clear that someone cannot "come from another
dimension." One cannot come
from a place that does not exist as a physical place.
People can't "move" from one dimension to another. Dimensions are not places.
People and other things are not in some dimension; they have non-zero
positive values for each dimension. Dimensions
do not weigh anything. They are not
(To Main Page)
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CC-MAIN-2019-13
| 6,955 | 122 |
https://justmyguitar.com/guitar-accessories-trivia-quiz/
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math
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Guitar Accessories Trivia Quiz
Answer each question to the best of your ability. Results at the end!
#1. Plectrum is another name for which guitar accessory
#2. What is the name of the second string of a standard guitar?
#3. What is the name of the fourth string of a standard guitar?
#4. Screw capos work using an adjustable mechanism that allows you to fine-tune its pressure against the?
#5. In 1964, Fender introduced dot inlays made of what material?
#6. Who founded Mosrite?
#7. The Steve Vai Signature Ibanez JEM model includes which unique feature?
Handle in the head
Handle in the body
Handle in the hand
#8. Prior to the foundation of Kramer Guitars, Gary Kramer worked with which other guitar brand
#9. The Gibson ES-330 uses which type of pickup?
#10. In addition to the word "Gibson" a J-45 from 1943 would also include what words on the headstock?
Gibson is not good enough
Only a Gibson is good enough
None of the above
All of the above
#11. The first batch of Alembic modified basses with active electronics were built out of?
Guild Starfire Bassess
Both a and b
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s3://commoncrawl/crawl-data/CC-MAIN-2024-10/segments/1707947474948.91/warc/CC-MAIN-20240301030138-20240301060138-00242.warc.gz
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| 1,078 | 22 |
https://studysoup.com/tsg/11917/university-physics-13-edition-chapter-2-problem-29e
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math
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At launch a rocket ship weighs 4.5 million pounds. When it is launched from rest, it takes 8.00 s to reach 161 km/h; at the end of the first 1.00 min, its speed is 1610 km/h. (a) What is the average acceleration (in m/s2) of the rocket (i) during the first 8.00 s and (ii) between 8.00 s and the end of the first 1.00 min? (b) Assuming the acceleration is constant during each time interval (but not necessarily the same in both intervals), what distance does the rocket travel (i) during the first 8.00 s and (ii) during the interval from 8.00 s to 1.00 min?
Solution 29E Step 1: Given data Mass of the rocket (m) = 4.5 million pounds When time t = 8 s , speed v = 161 km/h t = 1 minute or 60 s , v = 1610 km/h To find average acceleration in m/s 2 Step 2: (i) t= 8s Acceleration (a) = velocity/time = v/t = 161/8 Acceleration (a) = 20.125 m/s 2 Step 3: (ii) Acceleration between time interval 8s and 1 minute t1 = 8s t2 = 1 min or 60s Speed increases from 161 km/h to 1610 km/h Since the value of acceleration takes unit m/s , units must be changed from km/h to m/s 1000 v 1 = 161 km/p = 161* 60*60 = 44.7222 m/s v 2 = 1610 km/p = 1610* 1000 = 447.222 m/s 60*60
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CC-MAIN-2020-16
| 1,163 | 2 |
http://wlkt.ustc.edu.cn/zt/detail_1_1.htm
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math
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主讲教师：许宝刚 人气：891 更新时间: 2015年10月20日
吴文俊数学重点实验室组合图论系列讲座之六十一【许宝刚】 摘要： We show that if a graph $G$ has neither triangles nor quadrilaterals, and has no odd holes of lengthat least 7, then $chi(G)le 4$ and $chi(G)le 3$ if $G$ has radius at most $3$, and for each vertex$u$ of $G$, the set of vertices of the same distance to $u$ induces a bipartite subgraph. This answers some questions of Plummer and Zha. Joint work with Gexin Yu and Xiaoya Zha.
主讲教师：张继平 人气：756 更新时间: 2016年05月25日
摘要：Adequacy of subgroups is very important in generalizations of Taylor-Wiles method for proving the automorphy of Galois representations. I will talk about some new progress on linear groups and the application related to adequacy of subgroups.
主讲教师：郁星星 人气：688 更新时间: 2016年05月25日
A subdivision of a graph G is denoted T G. Mader conjectured that every C4-free graph with average degree d contains TK_l with l = ?(d). Koml′os and Szemer′edi reduced this problem to expanders and proved Mader’s conjecture for n-vertex expanders with average degree d < exp(log1/8 n). We show that Mader's conjecture is true for n-vertex expanders with average degree d < n3/10, which improves Koml′os and Szemer′edi’s bound to a polynomial bound. As a consequence, we show that every C4-free graph with average degree d contains a TK_l with l = ?(d/(log d) c ) for any c > 3/2. This is joint work with H. Huang and Y. Wang
主讲教师：傅翔 人气：2633 更新时间: 2015年03月22日
Dominance is a useful partial ordering defined on the root systems of Coxeter groups. In the literature it has non-trivial applications in the group-theoretic, combinatorial and computational aspects of Coxeter groups. In this talk we outline a number of recent results on dominance. In particular, we discuss how dominance can be used to better understand the asymptotic behaviours of the so-called normalized roots of infinite Coxeter groups. Furthermore, we outline the natural interpretation of dominance in terms of the Cayley graphs of Coxeter groups. Time permitting, we may outline the connection between the visual boundaries of the Davis complexes of Coxeter groups and the limits of the so-called normalized roots of Coxeter groups. This talk is based on some recent joint work with Dr. Lawrence Reeves.
主讲教师：Aleksandr Makhnev 人气：985 更新时间: 2014年12月07日
Let Γ be a distance regular cover of (k +1)-clique with λ = μ. If local subgraphs are strongly regular with parameters (v’, k’, λ’, μ’), then v’= k, k’= λ, and b1= k ? λ ? 1 = (r ? 1)μ, so k’= λ devides v’? 1. It is determined parameters of strongly regular graphs with at most 1000 vertices, which may be local subgraphs in distance regular cover of clique with λ = μ. It is classified arc-transitive distance regular cover of clique with λ = μ with local subgraphs, whose parametrs belongs to our list.
主讲教师：Naoto Yotsutani 人气：1400 更新时间: 2015年05月18日
吴文俊数学重点实验室微分几何与分析系列讲座八十三【四谷直仁】 题目: DOUBLING CONSTRUCTION OF COMPACT MANIFOLDS WITH SPECIAL HOLONOMY The aim of this talk is to give differential-geometric constructions of Calabi-Yau 3-folds, 4- folds and compact Spin(7)-manifolds. Ingredients in our constructions are (orbifold) admissible pairs (with antiholomorphic involution), which were dealt with by Kovalev in a series of his papers (2003, 2011, 2013). For our convenience, we shall mainly discuss the case of CY 3-folds. If time permitted, we also give a concrete example with this construction. This talk is based on a series of joint works with Mamoru Doi.
主讲教师：Anyue Chen 人气：3188 更新时间: 2015年03月22日
In this talk, I'll address some important topics, particularly the extinction properties and their asymptotic behaviour of the Interacting Branching Collision Process (IBCP), which consists of two strongly interacting components: an ordinary Markov branching process (MBP) and a collision branching process (CBP). After establishing some elegant necessary and sufficient conditions for existence and uniqueness of IBCP, the explicit expressions are then obtained for the extinction probabilities for the regular, critical-explosive and super-explosive cases. The subtle sub-explosive case will also be addressed. The associated expected hitting times are also investigated and revealed. Moreover, considering some of these exact explicit expressions are very complicated and thus we turn to investigate the asymptotic behaviour of these extremely complicated and divergent expressions for extinction probabilities. Surprisingly, we are able to show that the asymptotic behaviour follows an elegant and homogenous power law with an index quantity. We further show that this interesting and important index quantity takes a very simple and uniform form which could be interpreted as the "spectrum", ranging from minus infinity to positive infinity, of the interaction between the two components of the IBCP.
主讲教师：吴可 人气：1419 更新时间: 2015年05月21日
吴文俊实验室数学物理系列报告之四十六 【吴可】 报告题目：Heisenberg代数和Fermi代数的范畴化报告人：首都师范大学数学学院 吴可教授时间：2015.5.19 （星期二）下午4:30 地点：管理科研楼 1418 摘要：TBA
主讲教师：彭岳建 人气：582 更新时间: 2016年05月25日
摘要：A number α ∈ [0,1) is jump for r-uniform graphs if there exists a constant c > 0 such that for any family F of r-uniform graphs, if the Turán density of F is greater than α, then the Tur´ an density of F is at least α + c. A fundamental result in extremal graph theory due to Erd¨ os and Stone implies that every number in [0,1) is a jump for r = 2. Erd¨ os also showed that every number in [0,r!r r ) is a jump for r ≥ 3. Furthermore, Frankl and R¨ odl showed the existence of non-jumps for r ≥ 3. But there are still a lot of unknowns regarding jumps or non-jumps for hypergraphs. We give a survey on the known result.
主讲教师：王诗宬 人气：687 更新时间: 2016年12月02日
11-30吴文俊讲座【王诗宬 院士】 摘要: I will give a survey talk on the recent results (joint with Derbez, Liu and Sun) around the title
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| 6,467 | 20 |
http://www.bookrenter.com/rockswold-algebra-and-trigonometry-with-model-and-visual/search--p5
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math
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Your cart is empty!
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s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386163041301/warc/CC-MAIN-20131204131721-00074-ip-10-33-133-15.ec2.internal.warc.gz
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| 987 | 6 |
https://www.manvila.com/2022/07/29/solvedpine-street-inc-makes-unfinished-bookcases-that-it-sells-for-58-51-production-costs-are-37-56/
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math
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Pine Street Inc. makes unfinished bookcases that it sells for $58.51. Production costs are $37.56 variable and $9.67 fixed. Because it has unused capacity, Pine Street is considering finishing the bookcases and selling them for $70.45. Variable finishing costs are expected to be $8.41 per unit with no increase in fixed costs.
Prepare an analysis on a per unit basis showing whether Pine Street should sell unfinished or finished bookcases.
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s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337322.29/warc/CC-MAIN-20221002115028-20221002145028-00586.warc.gz
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CC-MAIN-2022-40
| 441 | 2 |
https://www.bilkent.edu/www/math-semineri-powers-of-edge-ideals-and-matchings-in-graphs-nursel-erey-gebze-1500-5-kasim-2020-en/
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math
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You are cordially invited to the Algebra Seminar organized by the Department of Mathematics.
Speaker: Nursel Erey (Gebze)
“Powers of edge ideals and matchings in graphs”
Abstract: Let G be a finite simple graph. The edge ideal of G, denoted by I (G), is generated by the quadratic monomials corresponding to the edges of G. Computation of Castelnuovo-Mumford regularity of powers of I(G) is a challenging problem. I will explain how this problem connects to matchings in graphs. I will also talk about recent work on squarefree powers of edge ideals.
Date: 5 November, 2020
This is an online seminar. To obtain the event link, please send a message to [email protected]
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CC-MAIN-2024-18
| 686 | 6 |
https://www.calc-tools.com/
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math
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Calc-tools.com is a fast, comprehensive, convenient, and free online calculator in the field of math calculator and scientific computing. Currently, we offer both a simple calculator and a scientific calculator experience, and we are developing more. Our goal is to be the one-stop, go-to site for anyone who needs fast calculator.In addition, we believe that the calc tools provided by the Internet should be free, convenient and fast. Therefore, all our calculator tools are completely free, no registration required and you can use it anytime, anywhere.
Online calculators help people to solve everyday calculation problems, such as: calculation of math problems, square or power of numbers, trigonometric functions sin, cos, tan, cot calculation, or as a salary calculator, mortgage calculator, etc. When you have a powerful online calculator, it is equivalent to having an almighty physical calculator. We code and develop each math calculator individually and put each calculator through rigorous, comprehensive testing and testing. However, if you find even the slightest error, please let us know, your comments are invaluable to us. Calc-tools' calculatortools.com is designed to serve people all over the world, with calculators available worldwide in over 50 languages.
Frequently Asked Questions
Q: What are the functions of the online calculator?
The online calculator is a virtual web calculator. calc-tools.com provides web versions of basic calculators and advanced calculators. Our goal is to meet all kinds of math calculator needs in people's daily life and work and study.
It provides users with basic mathematical calculations such as addition, subtraction, multiplication, and division, and can also temporarily store calculation results. and call it out through the [MR] key when needed, and then perform operations with other calculation results. The advanced calculator has added calculation functions such as pi, sine, cosine, tangent, logarithm, and natural numbers, and strives to meet the needs of the majority of users. The basic calculator and scientific calculator work together on the PC side, and on the mobile phone side, click [Basic] and [Scientific] to switch between the basic calculator and the advanced calculator.
Q: How should I use a scientific calculator?
The scientific computing function is an important part of the math calculator.The scientific calculator supports the display of 24-digit numbers. The functions include: mathematical calculation, base conversion, standard mathematical functions, percentage calculator, math equation solver, online math solver, square root calculation, logarithm, power, memory and other functions.
D(DEG) R(RAD) G(GRAD) represent the angle system, the radian system, and the degree system respectively. sin cos tan is a trigonometric function, and sin-1 cos-1 tan-1 corresponds to an inverse trigonometric function, which is equivalent to arcsin, arccos, arctan. ln is the natural logarithm ln a=loge a, and log is the logarithmic function. MC: clear the stored data, MR: Read the value in the memory and display it, M-: subtract the currently displayed value from the stored value, and then save the result, M+: add the current displayed value from the stored value, and then save the result. Ans English word is answer, the meaning of the answer.
Q: How does online calculator work?
Q: Do I need to pay for the calculator?
A: No, the calculator functionality provided by calc-tools.com is completely free, and our free calculator will work until people no longer need it.
Q: What is the calculation logic of a scientific calculator?
A: Scientific calculators on Calc-tools.com are developed based on well-known formulas or theorems in textbooks, and have been verified by internal testing and actual use by thousands of users.
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A: Thinking back to your work, what are some scenarios where you need to use a calculator? For example: salary calculator is needed to calculate worker salary, GPA calculation, trade transaction fee need to use calculator, expense and cost calculator, income calculator, tax when paying taxes, tax rate calculator. Workers need to use an hour calculator for their weekly work hours. In addition to this, income tax calculators, retired workers pension calculators, performance percentage calculators, etc. are required. Here is just a list of scenarios that almost all companies use, and some more professional use scenarios are not listed one by one.
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A: Yes, please feel free to link to any page on calc-tools.com. We would also be happy if our calculator tool was helpful to you.
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s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296817222.1/warc/CC-MAIN-20240418160034-20240418190034-00746.warc.gz
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CC-MAIN-2024-18
| 5,597 | 19 |
https://www.coursehero.com/tutors-problems/Finance/8098986-Please-help-with-steps-to-solve-parts-b-and-c-Barans-Company-c/
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math
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Please help with steps to solve parts (b) and (c):
Barans Company currently has an average collection period of 55 days and annual sales of $1 billion. Assume a 365-day year.
a. What is the firm's average accounts receivable balance?
b. If the variable cost of each product is 65 percent of sales, what is the average investment in accounts receivable?c. If the equal-risk opportunity cost of the investment in accounts receivable is 12 percent, what is the total annual cost of the resources invested in accounts receivable?
Dear Student Please find... View the full answer
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s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376829542.89/warc/CC-MAIN-20181218164121-20181218190121-00213.warc.gz
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CC-MAIN-2018-51
| 574 | 5 |
https://www.physicsforums.com/threads/evaluating-an-exponential-function-that-models-a-real-world-situation.709716/
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math
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Suppose that the velocity v(t) (in m/s) of a sky diver falling near the Earth's surface is given by the following exponential function, where time is measured in seconds.
v(t) = 55 (1-e-0.18(t))
Find the initial velocity of the sky diver and the velocity after 6 seconds.
Round your answers to the nearest whole number as necessary.
The Attempt at a Solution
I don't know what e is, so I can't even attempt this. And it's an online course with no book, just an online program we have to use for everything. It doesn't tell me what e is either.
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s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103639050.36/warc/CC-MAIN-20220629115352-20220629145352-00534.warc.gz
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CC-MAIN-2022-27
| 543 | 6 |
http://mathhelpforum.com/trigonometry/218385-law-sines.html
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math
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Using Law of Sines for the ambiguous case: SSA :
Given the triangle with the following sides: A = 25 degrees, a = 8 cm, and b = 15cm , we begin by drawing a "vague triangular diagram":
Step 1: Find the value of h ( this determines how many triangles you will have) :
Step 2: Draw in the side, a, to make an acute angle at B and darken in this triangle.
Step 3: Using Law of Sines, find the missing sides and angles of this triangle :
Step 4: Now you will draw a second triangle below, making an obtuse angle at B. Now find the values of the missing side and angles of this triangle:
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s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917120092.26/warc/CC-MAIN-20170423031200-00436-ip-10-145-167-34.ec2.internal.warc.gz
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CC-MAIN-2017-17
| 582 | 6 |
http://song4u.info/math-worksheet-for-5th-grade/
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math
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Please forward this error screen to 184. Worksheet on Speed, please forward this error screen math worksheet for 5th grade 184. Learn how to read, learn how we use Roman, in 5th grade math problems you will get all types of examples on different topics along with the solutions.
Learn how to add the numbers step, representation of Integers on a Number Line. Learn how to subtract the numbers step, addition of Integers on a Number Line. Multiplication of Fractional Number by a Whole Number.
Learn how to find the estimated sum and difference to the nearest tens, multiplication of a Fraction by Fraction. Learn how to multiply a number by a 2, properties of Multiplication of Fractional Numbers. How to estimate the product by rounding the numbers to the nearest 10, worksheet on Multiplication on Fraction.
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s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267867095.70/warc/CC-MAIN-20180624215228-20180624235228-00234.warc.gz
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CC-MAIN-2018-26
| 809 | 3 |
http://www.phys.ttu.edu/~batcam/Courses/semester%201/Labs/UNIT%2015%20ROTATION%20KINEMATICS.htm
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math
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This video shows a wheel chair in motion.
a. Have one person in your group extend their arms (as in the picture below) and rotate slowly so that their fingertips move at a constant speed. Record the time it takes their arm to sweep through a known angle, q.
Rotating arms featuring elbows and hands.
b. Calculate the speed of your elbow and the speed of your fingertips. Are your elbow and your fingertips moving at the same speed?
c. Is the speed of your elbow and the speed of your fingertips related to the distance of your elbow and your fingertips from the axis of rotation (the center of your body)?
d. The angle, q, through which you rotated was measured from a reference axis (as in the picture above). At any given time during the rotation, is the angle between your elbow and the reference axis different than the angle between your fingers and the reference axis? Explain.
e. You were asked in part a to rotate at constant speed, but you found two different speeds for your elbow and for your fingertips. What is meant by the term constant speed? How is this speed different from the speed of your fingertips and the speed of your elbow you calculated in part b? Explain.
Explain your reasoning to an instructor.
1.2 An understanding that angles can be measured in units of radians or units of degrees and the relationship between angles measured in radians and angles measured in degrees is critical in the study of rotational motion.
We first need to define an arc length. An arc length is the distance, s, subtended by an angle, q, along the circumference of a circle, as shown in the picture below.
1. A degree is defined as 1/360th of a rotation in a complete circle.
2. A radian is defined as the angle for which the arc along the circle is equal to its radius. An angle measured in radians is the ratio of the arc length to the radius
a. How many radians subtend an arc length of
· 2 p r, the circumference of a circle?
· p r, half the distance around a circle?
· 5m along the circumference of a circle of radius 6.5m?
b. How many radians equal
a. Using the data in part 3.1.a, and using the reference axis as q = 0, calculate the change in q divided by the change in time
for the elbow and for the fingertips. Would this quantity be the same for a point on your arm halfway between your elbow and your fingertips? For any point on your arm?
The quantity the change in q divided by the change in time is called the angular velocity, , where the angular speed is given by
and the direction is clockwise or counterclockwise. The units of are radians per second (rad/s).
b. Does every point on a rotating object have the same angular speed? Explain.
c. In part 1.1.b, you calculated the speed of the fingertips and the elbow of a person rotating at constant angular speed. The speed you calculated is the magnitude of the velocity of the fingertips or the elbow. The velocity is tangent to the circle through which that point is moving; it is called the tangential velocity. The tangential velocity vector is shown in the diagram below. The speed you calculated is called the tangential speed.
Do the fingertips and elbow have the same tangential speed? Explain.
d. Relate the tangential speed, v (the magnitude of the tangential velocity), of the fingertips of the person in part 3.1.a to the angular speed of the person. Write an equation relating the speed, v, to the angular speed, .
Discuss your answer with an instructor.
a. Consider a rotating object that is slowing down or speeding up. Is the angular speed changing? Explain.
b. Is the tangential velocity at a point changing? Is the point accelerating? Explain.
If the tangential velocity of a point on an object is changing, the point is accelerating. Since the point is moving in a circle, there is an acceleration toward the center of the circle, as shown in the picture below. This is called the radial acceleration, aR. If the tangential speed is changing, there is also a component of the acceleration tangent to the circle through which the point is moving, as shown in the picture below. This is called the tangential acceleration, aT.
c. If the angular velocity of the rotating object is changing, we can define an angular acceleration, . The magnitude of the angular acceleration is the change in angular velocity divided by the change in time
and the direction is clockwise or counterclockwise. The units of angular acceleration are rad/s2. Is the angular acceleration the same for each point on a rotating object? Explain.
It can be shown that the magnitude of the angular acceleration is related to the magnitude of the tangential acceleration
d. Since the angular acceleration, the angular velocity and the angle, , is the same for all points on a rotating object, it is often useful to work with these quantities when studying rotational motion.
You might notice that the definitions of and are similar the definitions of a and v for motion in a straight line in one dimension that we studied in Unit 1:
It can also be shown that
From these three equations, four equations for rotational kinematics can be derived, as they were for motion in a straight line. The four equations are shown below, and also the analogous equations for motion in a straight line.
You should understand the concept of angular speed. You should understand the concept of angular acceleration. You should understand and be able to use kinematics equations to describe the motion of objects moving with constant angular acceleration.
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CC-MAIN-2018-51
| 5,501 | 37 |
https://www.wisdom.weizmann.ac.il/~naor/PUZZLES/visual.html
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math
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Suppose that 6 thieves have deposited their loot in a numbered Swiss bank account. Being thieves, they do not trust each other not to withdraw the money and take the next plane to Brazil. However, being sentimental, they do not assume a conspiracy of two or more thieves that will try to take out the money without "authorization" and they want any two of them to able to withdraw the money. They therefore want to divide the secret number into shares so that from each share you cannot learn anything about the secret number, but from any two shares you can reconstruct the secret number.
This problem is the by now classical secret sharing problem (see Doug Stinson's bibliography on secret sharing schemes ). The twist here is that the thieves's representatives will not have a computer with them when they come to withdraw the money, so they want to be to it visually: each thief gets a transparency. The transparency should yield no information about the secret number (even implicitly). However, by taking any two transparencies, stacking them together and aligning them, the secret number should "pop out". The transparencies themselves will be prepared by a (trusted) computer. How can this be done?
For starters, can you think of a method that will let two thieves split a secret between them? Solution
Back to the Puzzler page
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s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103941562.52/warc/CC-MAIN-20220701125452-20220701155452-00463.warc.gz
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CC-MAIN-2022-27
| 1,336 | 4 |
https://www.bloomcollaborative.org/drawing-tips/quick-answer-how-to-draw-bending-moment-diagram.html
|
math
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What is bending stress formula?
The bending stress is computed for the rail by the equation Sb = Mc/I, where Sb is the bending stress in pounds per square inch, M is the maximum bending moment in pound-inches, I is the moment of inertia of the rail in (inches)4, and c is the distance in inches from the base of rail to its neutral axis.
What is the formula of bending moment?
Calculate BM: M = Fr (Perpendicular to the force) Bending moment is a torque applied to each side of the beam if it was cut in two – anywhere along its length.
What is SFD and BMD?
SFD and BMD: The shear force diagram ( SFD ) and bending moment. diagram( BMD ) of a beam shows the variation of shear. force and bending moment along the length of the beam.
What is SF and BM diagram?
SF & BM DIAGRAM In Shear force diagram and bending moment diagram, the positions of important points of beam are plotted along X axis and respective values of shear forces and bending moments at that points are plotted along Y axis. A line is then drawn joining these points of Y co-ordinates to get the diagram.
How do you make a SFD in BMD?
Shear Force & Bending Moment Diagram of Simply Supported Beam
- Draw shear force and bending moment diagram of simply supported beam carrying point load.
- Both of the reactions will be equal.
- R1 = R2 = W/2 = 1000 kg.
- When simply supported beam is carrying point loads.
- Shear force between (B – C) = S.F (B -C) = 1000 – 2000.
What is the unit of bending moment?
Explanation: Moment is a product of force and perpendicular distance and the bending moment is the algebraic sum of moments taken away from the left or the right of the section hence the SI units of bending moment is same as the moment i.e kNm.
What is deflection formula?
Generally, deflection can be calculated by taking the double integral of the Bending Moment Equation, M(x) divided by EI (Young’s Modulus x Moment of Inertia).
How do you calculate bend?
K = K-factor, which is t / T. T = material thickness. t = distance from inside face to the neutral line. A = bend angle in degrees (the angle through which the material is bent ) Bend allowance.
What is a real life example of bending?
For example, a closet rod sagging under the weight of clothes on clothes hangers is an example of a beam experiencing bending.
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s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587908.20/warc/CC-MAIN-20211026134839-20211026164839-00084.warc.gz
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CC-MAIN-2021-43
| 2,300 | 23 |
https://efirescotland.org/file-ready/jsc-math-question-2014-rajshai
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math
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Jsc math question suggestion 2014 jsc math question suggestion 2014 jsc math question suggestion 2014 k wefvmcvwumwyz 1 ssc exam . Rajshahi board ssc math question 2014 and dhaka board chittagong board barisal board sylhet board mymensingh board dinajpur board comilla board khulna board . Jsc math question suggestion jsc mathematics question final suggestion 2019 last year jsc exam held from 01 november 2018 jsc exam routine 2019 here. Jsc bord question jsc model test suggestion syllabus short suggation jsc all subject mcq suggestion questi 202 like 2014 02 17 08 49 entertainment 08 49 . Math question 2014 jsc 2003 2014 kymco 300 250 quad service manual 2007 a level math solutions paper 2 2006 marcy mathworks a answers 2003 national mathcounts sprint
How it works:
1. Register a Free 1 month Trial Account.
2. Download as many books as you like ( Personal use )
3. No Commitment. Cancel anytime.
4. Join Over 100.000 Happy Readers.
5. That's it. What you waiting for? Sign Up and Get Your Books.
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s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578582584.59/warc/CC-MAIN-20190422195208-20190422221208-00432.warc.gz
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CC-MAIN-2019-18
| 1,005 | 7 |
http://oxfordindex.oup.com/view/10.1093/oi/authority.20110803095720970
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math
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A source of information whose output has an alphabet of distinct letters or, in the case of a physical source, whose output is a signal that is discrete in time and amplitude (see discrete and continuous systems). The size of the alphabet, or the number of amplitude levels, is usually finite, although for mathematical analysis it may conveniently be regarded as potentially infinite.
The discrete memoryless source (DMS) has the property that its output at a certain time does not depend on its output at any earlier time.
The discrete memoryless source (DMS) has the property that its output at a certain time may depend on its outputs at a number of earlier times: if this number is finite, the source is said to be of finite order, otherwise it is of infinite order. DSMs are usually modeled by means of Markov chains; they are then called Markov sources.
An ergodic source has the property that its output at any time has the same statistical properties as its output at any other time. Memoryless sources are, trivially, always ergodic; a source with memory is ergodic only if it is modeled by an ergodic Markov chain.
See also information theory, Shannon's model, source coding theorem.
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s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221217901.91/warc/CC-MAIN-20180820234831-20180821014831-00430.warc.gz
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CC-MAIN-2018-34
| 1,194 | 5 |
http://maths.york.ac.uk/www/biblio/tg/M?sort=title&order=asc
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math
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Filters: First Letter Of Title is M [Clear All Filters]
M.A. Lavrentyev and Hydrodynamics. J. Appl. Mech. Tech. Phys. 6:1-12.. 1990.
Magic squares and matrix models of Lie algebras. Adv. in Math.. 180:596-647.. 2003.
Magnetohydrodynamic drift equations: from Langmuir circulations to magnetohydrodynamic dynamo? Journal of Fluid Mechanics. 698:51-61.. 2012.
Making noise: Emergent stochasticity in collective motion. Journal of Theoretical Biology. 267(3):292-299.. 2010.
Management of Traffic using Traffic Flow Control and other measures. Pres. at 1998 World Congress, Antwerp.. 1998.
Marine reserves and the evolutionary effects of fishing on size at maturation. ICES Journal of Marine Science. 67(3):412-425.. 2010.
A Markov Based Method for Generating Long-Range Dependence. Physical Reviews E. 72:026118.. 2005.
Martingale differences and the metric theory of continued fractions. Illinois J. Math.. 52(1). 2008.
Mass Dependence of Quantum Energy Inequality Bounds. J. Math. Phys.. 48:093506:1-13.. 2007.
Mass scale and curvature in metabolic scaling. Journal of Theoretical Biology.. 2011.
Mass scale and curvature in metabolic scaling. Journal of Theoretical Biology. 280(1):194-196.. 2011.
Mass structured systems with boundary delay: oscillations and the effect of selective predation. Journal of Nonlinear Science. 22. 2012.
A Mass Transference Principle and the Duffin-Schaeffer conjecture for Hausdorff measures. Annals of Math.. 164, no. 3:971-992.. 2006.
Massless flows between minimal W models. Phys. Lett. B. 537:297-305.. 2002.
Master equation approach to the assembly of viral capsids. J. Theoretical Biology. 242:713-721.. 2006.
Material symmetry and invariants for a 2D fiber-reinforced network. 9th SIMAI Conference. 82:401--412.. 2009.
Mathematical Analysis of Some Methods for Amicable Numbers. Int J of Computer Mathematics. 57:1-14.. 1995.
A Mathematical Approach to Crosslinked Structures in Viral Capsids. DNA12, LNCS. 4287:239-249.. 2006.
Mathematical models for tubular structures in the family of Papovaviridae. Bull. Math. Biol.. 67:973-987.. 2005.
Mathematical models of the Battle of Britain. Mathematics in Defence.. 2009.
A Mathematical Physicist's Approach to the Structure and Assembly of Viruses. Phil. Trans. A. 364:3357-3374.. 2006.
Mathematical Virology: A mathematical physicist's approach to the protein stoichiometry and bonding structure of viral capsids. Proceedings IX. International Conference on Symmetry Methods in Physics, Mexico, 2004.. 2005.
Department of Mathematics, University of York, Heslington, York, UK. YO10 5DD
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s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368701910820/warc/CC-MAIN-20130516105830-00023-ip-10-60-113-184.ec2.internal.warc.gz
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CC-MAIN-2013-20
| 2,573 | 24 |
https://laytonforstatesenate.com/united-states-3/
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math
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Each contract controls 100,IEEE. You are receiving 418,IEEE in Euro revenues on September 9th. Assume that currently, the September 9th forward is $1. 2560. 5. How would you properly hedge the revenues using futures contracts? Discuss why, how much you will be over or under-hedged, and the risks. A. Buy 3 August Euro-futures contracts b. Sell 3 August Euro-futures contracts c. Sell 4 September Euro-futures contracts d. Buy 4 September Euro-futures contracts e. Sell 3 September Euro-futures contracts 6. If we properly hedge the above revenue, are we over-hedged or under- hedged, and what will cause the hedge to lose money?
Explain why. A. Over- hedged, an appreciation of the Euro relative to the forward b. Over-hedged, a depreciation of the Euro relative to the forward c. Under-hedged, an appreciation of the Euro relative to the forward d. Under-hedged, a depreciation of the Euro relative to the forward e. Nothing, the hedge will work perfectly 7. If on September 9th, the Bid and Ask are $1. 2490/E and $1. 2540/E, and the futures price is for the September 20 contract is $1. 2550, did your hedge work perfectly, or did you make or lose money on the hedge? A. Worked Perfectly (no gain or loss greater than $100) b. Profit < Sl 000 C. LOSS < 31000 .
Profit > $1000 e. Loss >$1000 8. Given the above results would we have been better off over-hedging or under-hedging? Retrospectively, (using the bid-ask and futures prices in problem 7) how many futures contracts should we have bought/sold to minimize gains and losses on the hedge (minimize hedge variance)? A. Under- hedging sell 3 contracts b. Under-hedging; buy 3 contracts c. Over-hedging; buy 5 contracts d. Over-hedging sell 5 contracts e. Over-hedging; sell 6 contracts 9. Suppose instead that the resulting bid and ask are $1. 2590/E and $1. 2610/ E, and the futures price is for the September 20 contract is $1. 20, (and you hedged using the proper number Of contracts-?problem 5) did your hedge work perfectly, or did you make or lose money on the hedge? A. Worked Perfectly (no gain or loss greater than SSL 00) b. Profit < $500 c. Loss $500 d. Profit > 5500 e. LOSS >$500 10. Retrospectively, how could you have minimized your gains or losses on the hedged position in problem g: under-hedging, or over-hedging? And if you wished to minimize hedge variance (the gains/losses) how many contracts should you have used? A. Under-hedging sell 3 contracts d. Over-hedging; sell 5 contracts 11. What is the purpose of hedging with futures? Cancel deviations in the spot on the closing date from the present spot b. Cancel deviations in the spot on the closing date from the forward for the closing date c. Minimize variance of the hedged position d. Two of the above e. All of the above 12. In theory, futures prices move perfectly with: a. Deviations in the spot rate from the current spot b. Deviations in the spot rate from the forward rate on contract maturity c. Deviations of the bombard rate for the maturity date of the contract d. Two of the above 13. How is hedging different when using futures vs.. Forwards? A.
Hedging with uterus, you take delivery of the underlying currency, therefore locking in the price of the currency b. Hedging with futures, you hold the contracts to expiration c. Hedging with futures, you offset gains or losses on the spot price for the currency with gains or losses on the futures position d. Two of the above 14. What is the Yield to Maturity of a five-year, Semi-Annual Euro denominated bond with a 2. 5% annual coupon, selling at 93% of par value? Show work. A. Appear 1% b. Appear 2% c. Appear 3 % d. Appear e. Appear 5% 15. If a similar US dollar denominated bond yielded 6. 0%, which bond has the higher yield after inflation?
Is the difference less than . 2%? Assume the current spot is $1. 2200/E and one-year forward is $1. 2450/E. Show work. A. US Bond, no b. US Bond, yes c. Euro Bond, no d. Euro Bond, yes e. Can’t tell, we can’t compute depreciation 16. Suppose annual inflation rates in the U. S. And Mexico are expected to be 6% and 20%, respectively, over the next several years.
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s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400244231.61/warc/CC-MAIN-20200926134026-20200926164026-00060.warc.gz
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CC-MAIN-2020-40
| 4,113 | 5 |
https://scholarworks.umt.edu/tme/vol8/iss1/2/
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math
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Mathematics educators, including some mathematicians, have, in various ways, urged that the school curriculum provide opportunities for learners to have some authentic experience of doing mathematics, opportunities to experience and develop the practices, dispositions, sensibilities, habits of mind characteristic of the generation of new mathematical knowledge and understanding – questioning, exploring, representing, conjecturing, consulting the literature, making connections, seeking proofs, proving, making aesthetic judgments, etc. (Polya 1954, Cuoco et al 2005, NCTM 2000 - Standard on Reasoning and Proof). While this inclination in curricular design has a certain appeal and merit, its curricular and instructional expressions are often contrived, or superficial, or no more than caricatures of what they are meant to emulate. One likely source of the difficulty is that most mathematics educators have little or no direct experience of doing a substantial piece of original mathematics, in part because the technical demands are often too far beyond the school curriculum. Studying the history and evolution of important mathematical developments can be helpful, but provides a less immediate and direct experience.
"Vignette of Doing Mathematics: A Meta-cognitive Tour of the Production of Some Elementary Mathematics,"
The Mathematics Enthusiast: Vol. 8
, Article 2.
Available at: https://scholarworks.umt.edu/tme/vol8/iss1/2
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s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178366969.45/warc/CC-MAIN-20210303134756-20210303164756-00586.warc.gz
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CC-MAIN-2021-10
| 1,441 | 5 |
https://dharmata.id/vajrayana/miphams-beacon-of-certainty-illuminating-the-view-of-dzogchen-the-great-perfection/
|
math
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MIPHAM’S BEACON OF CERTAINTY
(Illuminating the View of Dzogchen, the Great Perfection)
L1: Introduction [the need to induce certainty by the two types of valid cognition – about the Two Truths]
L1: Topic 1 [The problem of negation: Emptiness is also empty; it is not the real non-dual nature beyond all conceptualization]
L1: Topic 2 [about the limited realizations of the arhats in the Hinayana: they do not realize the full extent of the emptiness of phenomena; they still reify elementary dharmas]
L1: Topic 3 [about habit pattern – modal apprehension – of rational cognition]
L1: Topic 4
L1: Topic 5
L1: Topic 6
L1: Topic 7
L1: Introduction [the need to induce certainty by the two types of valid cognition – about the Two Truths]
\ “Trapped in doubt’s net, one’s mind
\ Is released by the lamp of Manjuvajra,
\ Which enters one’s heart as profound certainty.
\ Indeed, I have faith in the eyes that see the excellent path!
\ Alas! Precious certainty,
\ You connect us with the profound nature of things;
\ Without you, we are tangled and confused
\ In this web of samsaric illusion.
\ The development of confidence through certainty
\ In the phenomena of the basis, path, and result,
\ And being roused to faith by studying them
\ Are like the authentic path and its reflection.
\ The fame of the Moon of the Amazing Dharma
\ Arises along with the light of elegant speech
\ In the vast sky of the Buddha’s teaching,
\ Vanquishing the heavy darkness of doubt.
\ The valid cognition that examines conventionalities
\ Is unerring with respect to engaging and avoiding.
\ Specifically, the textual corpus on valid cognition
\ Is the only way to acquire confidence
\ In the teacher and the teaching, and
\ The Madhyamaka of the Supreme Vehicle
\ Elucidates the stainless valid cognition
\ Of ultimate reasoning, which determines the nature of things.
\ [The two valid cognitions emphasized in] these two [systems]
\ Are the wisdom eyes of a well-trained intellect.
\ Praise to such enlightened beings who
\ Abide on the path taught by the teacher
\ Without taking detours!”
\ As the sage reflected thus,
\ A mendicant who happened along
\ Asked these seven questions
\ In order to critically examine his intellect:
\ “What’s the point of being a scholar
\ If you only repeat the words of others?
\ Give us a quick answer to these questions
\ According to your own understanding.
\ Then your philosophical acumen will be obvious.
\ Though they stretch out the elephant’s trunk of their learning,
\ Like well water, the deep water of Dharma is not tasted;
\ Yet they hope still to become famous scholars
\ Like low-caste men lusting for a queen.
\ 0.1.1.2.2.1.4 [The seven topics:]
\ 1. According to which of the two negations
\ do you explain the view?
\ 2. Do arhats realize both types of selflessness?
\ 3. Does meditation involve modal apprehension?
\ 4. Does one meditate analytically or transically?
\ 5. Which of the two realities is most important?
\ 6. What is the common object of disparate perceptions?
\ 7. Does Madhyamaka have a position or not?
\ Thus, starting with the topic of emptiness,
\ Give an answer established by reasoning,
\ Without contradicting scripture,
\ For these seven profound questions!
\ Even though pressed with the barbed lances
\ Of a hundred thousand sophisticated arguments,
\ These issues have not been penetrated before.
\ Like lightning, let your long philosopher’s tongue strike
\ These difficult points, which have confounded the great!”
\ Thus incited by intellect,
\ The speech-wind wavered somewhat,
\ And that shook the sage’s heart
\ Like a mountain in the winds at the end of time.
\ After maintaining a moment of disciplined engagement,
\ he said:
\ “Alas! If by undergoing hundreds of difficult tests,
\ And analyzing again and again,
\ The fires of great intellects blazed ever greater
\ Yet were still not refined to a flawless state,
\ How can a low person like myself possibly explain this,
\ Whose innate brilliance is weak
\ And who has not undertaken lengthy study?”
\ Then, as he cried these words of lament to Manjughosa,
\ By what seemed to be His mystic power
\ A light dawned in the mind of the sage.
\ At that moment, as he acquired a little self-confidence,
\ He reasoned analytically according to eloquent scriptures, and spoke.
L1: Topic 1 [The problem of negation: Emptiness is also empty; it is not the real non-dual nature beyond all conceptualization]
\ [“Question 1: The basis as the coalescence of appearance and emptiness”
\ – The first topic concerns the definition of the Nyingma philosophical view]
\ The dGe ldan pas say the view is an absolute negation;
\ Others say it is an implicative negation.
\ What is our own Early Translation tradition?
\ In the state of great gnosis of coalescence,
\ After making a negative judgement of “non-existence,”
\ What other thing such as an exclusive emptiness,
\ Or something that is not [that which is negated],
\ Could be implied in its place?
\ Both are just intellectually designated, and,
\ In the ultimate sense, neither is accepted.
\ This is the original reality beyond intellect,
\ Which is free of both negation and proof.
\ But if you should ask about the way in which emptiness
\ is established,
\ Then it is just an absolute negation.
\ In India the glorious Chandrakirti
\ And in Tibet Rong zom Chos bzang both
\ With one voice and one intention
\ Established the great emptiness of primordial purity.
\ Because these dharmas are primordially pure,
\ Or because they are originally without intrinsic reality,
\ They are not born in either of the two realities;
\ So why fret about the expression “non-existent”?
\ In the place of a pillar, primordially pure,
\ There is nothing non-empty whatsoever.
\ If you don’t negate it by saying, “There is no pillar,”
\ What does it mean to say, “The pillar does not exist?”
\ The emptiness that is the negation of the pillar
\ And a left-over appearance
\ Are not fit, as “empty” and “non-empty,” to coalesce;
\ It is like twisting black and white threads together.
\ To say, “a pillar is not empty of being a pillar”
\ Or “dharmata is empty of being a pillar”
\ Is to posit the basis of emptiness and something
\ of which it’s empty.
\ These are verbal and ontological extrinsic emptinesses.
\ Woe! If this is not empty of this itself,
\ The empty basis is not empty and is left over.
\ This contradicts both scripture and reasoning—
\ “Form is empty of form!”
\ Consider a pillar and the true existence of a pillar:
\ If they are one, then refuting one the other is refuted;
\ If they are different, by refuting a true existence
\ That is not the pillar, the pillar
\ That is not empty of itself would be immune to analysis.
\ “Because true existence is not found to exist,
\ There is no need to debate sameness and difference”—
\ Even though true existence does not exist,
\ Individuals still apprehend vases as truly existent.
\ So aside from a non-empty vase
\ What is there to establish as truly existent?
\ And you think you’ve determined the appearance of
\ the negandum!
\ To teach emptiness by applying some qualifier
\ Such as “true existence” to the negandum
\ Is of course well known in Svatantrika texts.
\ But in the context of analyzing ultimate reality,
\ What is the point of applying it?
\ Thinking that if it’s empty, then even deceptively
\ A pillar will be non-existent,
\ You try to avoid misinterpretation of the word
\ But this is itself a great contradiction!
\ You are not satisfied to say simply,
\ “A pillar is deceptively existent.”
\ Why must you say, “It is not empty of itself?
\ You may say, “They are the same in meaning,”
\ But it is not so; “A pillar exists” and
\ “There is a pillar in a pillar” are different statements.
\ The latter means “Something depends on something”—
\ This in fact is what you end up claiming.
\ If ultimately a pillar is not perceived,
\ Then how can a pillar not be empty of pillar?
\ In saying “Deceptively a pillar [is not empty of being a]
\ You are confused, using the same word twice.
\ If something is not empty of itself,
\ Then while it exists itself, it must be empty of
\ something else.
\ If the negandum is not something else,
\ This contradicts the claim that it is not empty of itself.
\ Generally speaking, extrinsic emptiness
\ Does not necessarily qualify as emptiness.
\ Although a cow does not exist in a horse,
\ How could one thereby establish the horse’s emptiness?
\ By seeing that horse, what harm or good
\ Will it do to the cow?
\ Therefore a non-empty nirvana and
\ An apparent samsara are unfit to be dharma and dharmata.
\ Here there is no coalescence of appearance and emptiness
\ Or equality of cyclic existence and peace.
\ “The moon in the water is not the moon in the sky”—
\ If you think the emptiness of being the moon in the sky
\ And the appearance of the moon in water
\ Are the coalescence of form and emptiness,
\ Then the realization of coalescence would be easy
\ for anyone.
\ Everyone knows a cow is not a horse;
\ They directly see the appearance of a cow.
\ How could the Mahatma have said,
\ “To realize this is amazing”?
\ Therefore, in our own system,
\ If one examines a moon in the water, that moon
\ Is not found at all, and does not exist as such;
\ When the moon in the water manifestly appears,
\ It is negated, but appears nonetheless.
\ Emptiness and existence are contradictory
\ In the mind of an ordinary person. But here, this manifest
\ Coalescence is said to be wonderful;
\ The learned praise it with words of amazement.
\ If one examines from the side of emptiness,
\ Because nothing at all is non-empty,
\ One can say simply that everything is “non-existent.”
\ But that non-existence is not self-sufficient,
\ For it arises unobstructedly as appearance.
\ That appearance is not self-sufficient,
\ For it abides in baseless great emptiness.
\ There, distinctions such as “This is empty of that,”
\ Or “That is empty of this,”
\ Or “This is emptiness and that is appearance,”
\ Are never to be found;
\ When one develops inner confidence in this,
\ The one who searches won’t be frustrated
\ By pointless analysis,
\ But will attain peace of mind—amazing!
L1: Topic 2 [about the limited realizations of the arhats in the Hinayana: they do not realize the full extent of the emptiness of phenomena; they still reify elementary dharmas]
\ Some say that sravaka and pratyekabuddha arhats
\ Do not realize phenomenal selflessness.
\ As long as the self that is the apprehension
\ Of the aggregates as the mere “I” is not eliminated,
\ By the power of that, emotional disturbances are not
\ That self is a designation made
\ With respect to the aggregates; it is the object
\ Of innate I-apprehension. That, and vases, etc.
\ Aside from being different, bases of emptiness
\ Are no different in their modes of emptiness;
\ For phenomena and persons are both
\ Empty of intrinsic establishment.
\ Thus, this is proven by scripture and reasoning.
\ To go beyond this and state that
\ “Sravakas and pratyekabuddhas do not realize emptiness”
\ Is just a claim.
\ At this point, some draw unwarranted conclusions and
\ claim that
\ The paths of vision of the three vehicles are the same
\ And that there are no distinctions of levels of realization.
\ They interpret the Prajnaparamita and mantra, all of sutra
\ and tantra,
\ As texts of provisional meaning.
\ There, when those who have already traveled lower paths
\ Achieve the the Mahayana path of vision and so forth
\ There would be such faults as not having anything
\ to abandon;
\ By reasoning, harm would befall them irrevocably.
\ Moreover, though having realized what must be realized,
\ They say that in abandoning what must be abandoned,
\ [One must] ally [one’s practice with the accumulations]—
\ [But this means] nonrealization, which contradicts the
\ claim of realization.
\ To claim that the rising sun must rely on something else
\ In order to vanquish the darkness—quite strange!
\ Some say that sravakas and pratyekabuddhas realize
\ the emptiness
\ Of the five aggregates of their own continua of experience,
\ But do not realize selflessness of other phenomena.
\ If one realizes the five aggregates to be empty,
\ Then, aside from noncomposite phenomena [like space and
\ What other dharma would be left unrealized?
\ So what is our own tradition?
\ Glorious Chandrakirti’s Autocommentary
\ Says that, in order to abandon obscurations, the Buddhas
\ Teach sravakas and pratyekabuddhas personal selflessness,
\ And in order to abandon cognitive obscurations, they teach
\ Bodhisattvas how to realize phenomenal selflessness.
\ “Well then, what does it mean to say
\ That both sravakas and pratyekabuddhas
\ Have realization of emptiness?”
\ In order to abandon just the emotional afflictions
\ Sravakas and pratyekabuddhas meditate on personal
\ But “They do not meditate on the entirety
\ Of phenomenal selflessness”—thus teaches [our tradition].
\ Klong chen rab ‘byams said of yore
\ That although earlier masters all disputed
\ Whether they did or did not [realize both forms of
\ Our own position is that whatever types of sravakas and
\ Appeared of yore and reached arhatship
\ Did not become liberated without
\ Realizing the emptiness of the self
\ That is the apprehension of the aggregates;
\ But just having that realization does not mean
\ That they realized selflessness entirely.
\ Just like the space inside a sesame seed
\ That is eaten out by a worm,
\ [Their realization] is said to be a lesser selflessness.
\ Thus, with words that refute the lesser [of possible
\ It is said that “They do not realize emptiness.”
\ This is a most excellent eloquent explanation;
\ There is nothing else like it.
\ For example, if one drinks a single gulp
\ Of the water of the great ocean,
\ One cannot say that one has not drunk the ocean.
\ Because they see the selflessness of the mere “I,”
\ Which is one phenomenon among others, it is held that
\ [Sravakas and pratyekabuddhas] see emptiness.
\ Just as by drinking a single gulp one cannot say
\ That one has drunk the entire ocean’s water;
\ Because they do not realize the nature of all knowables
\ To be emptiness, it is held that they do not see selflessness
\ If one sees the emptiness of a single thing,
\ Why wouldn’t one see the emptiness of everything?
\ If, with scripture, reasoning, and pith instructions,
\ They were to examine things, of course they would see it.
\ But, for the most part, those who are destined
\ To be sravakas and pratyekabuddhas
\ Are attached to the selflessness of persons,
\ So it is hard for them to realize the latter extremes
\ [of the catuskoti],
\ Just as those who analyze a vase
\ Might assert its particles to exist substantially.
\ If the mind that realizes [selflessness]
\ After analyzing a vase also were to
\ Analyze particles, it would be reasonable to realize
\ [their emptiness];
\ But usually, they do not realize [their emptiness].
\ Though coarse bases and partless atoms appear
\ Since [sravakas and pratyekabuddhas] are mostly bereft
\ Of those scriptures, modes of reasoning, and pith
\ They practice systems that do not contradict [the possibility
\ of personal liberation].
\ Likewise, followers of the Cittamatra system
\ Do not accept the existence of external objects,
\ So why wouldn’t they also accept the non-existence of
\ the subject?
\ Why wouldn’t Svatantrikas use the reasoning that establishes
\ Ultimate truthlessness to understand the conventional
\ Nonestablishment of intrinsic characteristics (rang mtshan)?
\ So, for you everyone would become a Prasangika!
\ How would it be possible for sravakas and pratyekabuddhas
\ To denigrate the Mahayana [if they were Prasangikas] ?
\ Thus, although the nature of one thing
\ Is also the nature of everything,
\ As long as the collection of external and internal causes
\ and conditions
\ Is not complete, realization will come slowly.
\ Generally speaking, those with sharp minds become realized
\ Under their own power, while dullards
\ Do not necessarily reach realization immediately.
\ At some point, realization is inevitable;
\ At the end of ten thousand aeons, it is said,
\ The arhat wakes up from the state of cessation,
\ And enters the Mahayana path.
\ To properly abide on the Mahayana path,
\ One must cultivate oneself for a countless aeon.
\ So why shouldn’t it be impossible for
\ Sravakas and pratyekabuddhas, who strive for their own
\ Not to realize all forms of selflessness
\ During those [ten] thousand aeons [they spend
\ in cessation]?
\ Don’t those who have attained the bhumis
\ Gradually clarify and perfect their realization?
\ With the help of the accumulations,
\ Infinite modes of reasoning, bodhicitta,
\ The conduct [that follows from it], and perfect dedication—
\ When these conditions are complete, it is certain
\ That one will achieve realization,
\ Just as complete knowledge of skillful means is a condition
\ For swift realization on the mantra path.
\ Even if one has abandoned notions of permanent self,
\ Instinctive apprehension of “I” occurs in relation to
\ the aggregates.
\ Therefore [it is said], “[As long as] there is apprehension
\ of the aggregates,
\ There is apprehension of T”—this statement [from
\ the Ratnavali]
\ Means that, as long as there is a basis of designation in
\ the aggregates
\ And a mind that apprehends them,
\ The causes for designating a self are complete,
\ And as a result, apprehension of self will not cease.
\ Thus, even if the permanent self were abandoned,
\ Since the object, in relation to which the designated self
\ Is instinctively designated, would not be eliminated,
\ There would be nothing to oppose the occurrence of
\ Thus, in abandoning emotional disturbances,
\ The assertion “One must realize the aggregates and so forth
\ to be empty”
\ Is not the meaning of the passage [in the Ratnavali].
\ That meaning was explained in this way by Chandrakirti:
\ If one recognizes the designated mere “I,”
\ That is enough to stop the apprehension of “I.”
\ Though one does not know a rope to be non-existent,
\ By seeing the lack of snake, the apprehension of snake is
\ Finally, one will definitely realize both kinds of selflessness.
\ The suchness of all phenomena is unique,
\ And the way of seeing suchness is the same,
\ So Nagarjuna and his son [Chandrakirti] have expounded
\ A line of reasoning that establishes the finality of a single
\ If, as in your system, sravakas and pratyekabuddhas
\ Had already seen reality, what would that line of reasoning
\ Do to establish a single vehicle?
\ It is just an assertion.
\ Here, the primordial wisdom of coalescence
\ That sees the ultimate
\ Is precisely identical with suchness;
\ All sublime beings head toward it, and enter it.
\ Therefore, if one understands this system well,
\ The systems of Nagarjuna and Asanga are like
\ Molasses and honey combined;
\ A hungry person will easily digest them.
\ Otherwise, as with inappropriate food,
\ One feels the discomfort of cancer within.
\ Poked and jabbed with a hundred sharp lances
\ Of scripture and reasoning, one is afraid.
L1: Topic 3 [about habit pattern – modal apprehension – of rational cognition]
\ When pursuing the main practice of the view,
\ Some say one should not apprehend anything.
\ The meaning of “not apprehending anything”
\ Can be understood well or wrongly.
\ The first [way of understanding]
\ Is free of the elaborations of the four extremes.
\ For the gnosis of sublime beings,
\ Nothing is seen to remain,
\ So modal apprehension automatically subsides;
\ It is like looking at the empty, luminous sky.
\ The second is the mindless system of Hashang:
\ Letting the mind rest blankly without analysis and
\ Without the clarity aspect of penetrating insight,
\ One remains ordinary, like a rock in the ocean depths.
\ For example, though both say “There is nothing at all,”
\ The Madhyamika sees there really is nothing,
\ And the other one just imagines the absence of form;
\ Likewise here, though the words are the same,
\ The meaning is different like earth and sky.
\ Therefore, if in the absence of elaboration of the four
\ One does not apprehend the four extremes anywhere,
\ One is beyond the four extremes, and modal apprehension
\ Because it no longer exists, we say there is no modal
\ If some idiots think “Since there is no modal apprehension,
\ From the very beginning one should relax and not grasp
\ Then because all beings are quite relaxed in their ordinary
\ Always wandering in the three worlds of samsara,
\ There is no reason to encourage or remind them!
\ Some might say, “We have recognized the nature of mind,”
\ Without really understanding it; in recognizing the ultimate,
\ One must definitely realize the absence of true existence.
\ That “Deluded appearances are one thing, and I am
\ Is obvious and requires no meditation.
\ You might say, “When examining the color, form, origin,
\ And so forth, of the mind nothing is seen;
\ That is realization of emptiness.”
\ This system of teaching is extremely profound,
\ And there are also great mistakes one can make;
\ Because mind does not have a form,
\ It is impossible for anyone to see its color, etc.
\ However, it is a very great mistake to think that merely
\ not seeing them
\ Is the same as being introduced to emptiness.
\ Though you examine your head a hundred times,
\ A ruminant’s horns cannot be found.
\ To say that not seeing something is to realize its
\ Wouldn’t that be easy for anybody?
\ Therefore, if by this rational analysis
\ One sees the nature of things precisely,
\ One will profoundly realize the essential unreality
\ Of the illusion mind, which is like an illusion.
\ Then, just like looking directly into space,
\ One will derive profound certainty in the nature of
\ one’s mind,
\ Which though moving is empty.
\ You ask, “Well then, this mind of yours—
\ Is it non-existent, like space,
\ Or does it have disparate awarenesses?”
\ Because the vibrant mind that we all possess
\ Doesn’t rest for a moment, surely everyone would say
\ There is some sort of awareness.
\ Thus, you say that mind,
\ Which is neither existent nor non-existent,
\ Is the luminous dharmakaya.
\ Although he hasn’t done much study,
\ Such a person who claims to introduce the nature of mind
\ Thinks this is a teaching such that
\ “Knowing one liberates all.”
\ The teaching of “neither existent nor non-existent”
\ in the Great Perfection
\ Is the freedom from the four extremes of elaboration.
\ If you examine this mind carefully,
\ You cannot say it exists,
\ Nor can you say it does not exist.
\ But in fact, your mind does not go beyond either
\ The extreme of both existence and non-existence
\ Nor the extreme of neither existence nor non-existence.
\ You are just thinking about the mind on the basis
\ Of “neither existence nor non-existence.”
\ Aside from a difference in name, the mind,
\ Spoken of in that way, is no different than
\ The “inconceivable self” of the apostate.
\ The mind and dharmas other than it
\ Are determined to be unreal, and on that basis
\ Appearances arise as relativity,
\ Which is beyond thought and expressions of “existence”
\ and “non-existence.”
\ This is the crucial point of freedom from elaborations of the
\ four extremes,
\ Which is without a focal point and all-pervasive.
\ [But] just saying “This is free of both existence and
\ Is to place a target in front of your mind.
\ Depending on this apprehension of self and others as real
\ One enters the river of samsara continuously.
\ The antidote that ends all of this
\ Is the modal apprehension of selflessness.
\ If one does not know the manner of absence,
\ To imagine non-existence does not help;
\ If you mistake a rope for a snake,
\ It doesn’t help to think “There’s no snake;”
\ But if you see how it does not exist, it disappears.
\ Thus, having realized emptiness through analysis,
\ You should not rest content with analysis.
\ Since the habit of clinging to real entities is beginningless,
\ You should meditate again and again with modal
\ By meditating on selflessness the view of self
\ Is uprooted, so it’s been called necessary
\ By many seers of truth who practiced intensely.
\ If this is the fail-safe entry way for beginners,
\ To say that modal apprehension should be abandoned
\ From the very beginning is a rumor spread by Mara.
\ When you acquire outstanding certainty in truthlessness
\ Induced by that modal apprehension,
\ The mere apprehension of non-existence
\ Is not the final nature of things,
\ So meditate on the great emptiness free of elaboration,
\ Free of conceptual ambivalence.
\ When you’ve really understood truthlessness,
\ Emptiness arises as relativity,
\ Without apprehension of either form or emptiness.
\ This is worthy of confidence just like
\ Gold refined by fire.
\ Though this extremely profound matter
\ Has been realized with long-standing effort
\ By the great scholar-yogis of India and Tibet,
\ Woe to those idiots who say it can be realized
\ In a moment—they are plagued with doubts!
\ In the main practice of absorption,
\ Actual and potential phenomena, samsara and nirvana,
\ Are beyond existence and non existence. If in the nature
\ of things
\ Existence and non-existence are nowhere established,
\ Biased apprehension is [nothing but] conceptual
\ Therefore, when analyzing rationally,
\ One does not see anything established anywhere;
\ So how can apprehension come about?
\ However, if you analyze the nature of
\ Freedom from the four extremes of elaboration,
\ certainty is gained.
\ By this the penetrating insight of self-arisen
\ Luminous wisdom becomes clear like a lamp.
\ Its opposite—the dark night of the
\ Four extremes of inferior intellects—
\ Is uprooted by this very antidote;
\ So when you meditate upon it, certainty should arise.
\ The fundamental space beyond intellect where
\ The elaborations of the four extremes are eliminated
\ Is difficult to see all at once
\ At the level of an ordinary person.
\ The system of study and reflection
\ Is for eliminating the elaborations of the four extremes
\ in stages.
\ To the extent that one grows accustomed to it,
\ Certainty grows ever greater;
\ One’s intellect, which causes mistaken reification to subside,
\ Improves like the waxing moon.
\ The unsound view that doesn’t apprehend anything
\ Cannot produce the confidence that
\ No real entities are established anywhere;
\ Therefore, it cannot remove obscurations.
\ Therefore, just like inferring fire by smoke,
\ The difference between these meditations
\ Is known from the dividend of abandoned defilement and
\ acquired realization.
\ The ordinary idiot’s meditation
\ Is not a cause for abandoning defilements or realization.
\ Because it is an obstacle to producing good qualities,
\ It is like pouring tea through a strainer—
\ Scriptural learning and realization slip away,
\ While emotional disturbances accumulate.
\ In particular, one has little confidence in cause and effect.
\ If one has the eyes of the authentic view,
\ Scriptural learning, experience, and realization blaze up.
\ By virtue of seeing emptiness,
\ Confidence in the infallible relativity of cause and effect
\ Will increase, and emotional disturbance will lessen.
\ With the samadhi that abides one-pointedly
\ In the state of certainty induced by analysis,
\ The ultimate meaning is seen by nonseeing.
\ One does not succumb to any particular object of seeing
\ And of course does not apprehend anything.
\ Like a mute’s taste of molasses,
\ Confidence grows in a yogi who cultivates it,
\ But it cannot be produced by analysis alone.
L1: Topic 4
\ In meditating the view of the supreme vehicle,
\ Which is right—to analyze or focus the mind?
\ Some say, “Don’t analyze, but meditate transically.
\ Analysis obscures the nature of things,
\ So without analyzing, sit like a bump on a log.”
\ Some say, “Only do analysis.
\ Meditation without analysis
\ Is like going to sleep and doesn’t help,
\ So one should always analyze.”
\ To adhere exclusively to analysis or transic
\ Meditation is not appropriate.
\ Most transic meditations without analysis
\ Can become a mere calm abiding,
\ But meditating thus will not produce certainty.
\ If certainty, the unique eye of the path of liberation,
\ Is abandoned, obscurations cannot be dispelled.
\ If you do not know the nature of dharmas,
\ However much you meditate, you are still
\ Meditating on ordinary concepts. What’s the use?
\ It’s like travelling on a path with your eyes closed.
\ The habits of beginningless delusion
\ Produce clinging to mistaken notions about the nature
\ of things.
\ Without endeavoring to investigate
\ With a hundred methods of reasoning, it is difficult
\ To achieve realization.
\ Insofar as clinging to mistaken appearances
\ And seeing the authentic meaning are mutually exclusive,
\ Here, in the darkness of existence to which
\ Sentient beings are well habituated,
\ It is difficult to obtain a glimpse of reality.
\ Through the ripening of the karma of previous practice
\ And the master’s blessing,
\ By just examining the origin, abiding, and cessation of
\ the mind,
\ It is possible to determine truthlessness.
\ But this is extremely rare;
\ Not everyone can achieve realization this way.
\ In cutting through to primordial purity,
\ One needs to perfect the Prasangika view.
\ As for the aspect of non-elaboration,
\ Those two are said to be no different.
\ In order to prevent clinging to blank emptiness,
\ The Mantrayana teaches great bliss.
\ This causes an experience of
\ The expanse of non-dual bliss and emptiness,
\ Free of subject and object.
\ Appearance, clarity, and awareness
\ Are synonyms of that bliss.
\ Here the appearance aspect is the formal Buddha bodies,
\ Which protect all beings and bring them to happiness
\ As long as samsara exists;
\ It has the nature of ultimate compassion.
\ Therefore great gnosis by its very nature
\ Does not abide in either existence or peace.
\ Because it abides in the basis,
\ By practicing the path Evam of bliss and emptiness
\ In this very life, one will manifest
\ The fruitional coalescence.
\ In fact the basis, path, and result
\ Are not divided; the path of the fourth empowerment,
\ Which is the culmination of the Vajrayana,
\ Is the self-arisen gnosis of awareness and emptiness.
\ This is exclusively emphasized
\ In the path of the vajra pinnacle of luminosity,
\ Which is the final point where all vehicles converge.
\ As long as certainty has not been born,
\ One should induce it with skillful means and analysis.
\ If certainty is born, one should meditate
\ In that state without separating from that certainty.
\ The lamp-like continuity of certainty
\ Causes false conceptuality to subside.
\ One should always cultivate it.
\ If it is lost, then induce it again through analysis.
\ At first, analysis is important;
\ If you don’t start out with analysis,
\ How can you induce an excellent certainty?
\ If an excellent certainty is not born,
\ How can miserable projections cease?
\ If miserable projections do not cease,
\ How can the foul wind of karma be stopped?
\ If the foul wind of karma is not stopped,
\ How can this awful samsara be abandoned?
\ If this awful samsara is not abandoned,
\ What can be done about this dismal suffering?
\ In reality, there is no good or evil
\ In samsara and nirvana;
\ To realize the equanimity of neither good nor evil
\ Is the nature of excellent certainty.
\ With excellent certainty, nirvana is not attained
\ By abandoning samsara.
\ The mere words may seem contradictory,
\ But in fact they are not.
\ This is the most important point of the path,
\ A crucial secret instruction on the view and activity—
\ You should examine and savor its meaning!
\ Next, you should alternate analysis and trance.
\ If you analyze, certainty will be born;
\ When you don’t analyze, and cling to the ordinary,
\ Analyze again and again, inducing certainty.
\ When certainty is born, rest in that state
\ Without distraction and meditate one-pointedly.
\ Certainty and the projecting mind
\ Are mutually exclusive;
\ So by the analysis that roots out projection,
\ You should increase certainty more and more.
\ Finally, if even without analysis
\ Certainty arises naturally, rest in that very state;
\ Since it has already been established through analysis,
\ There is no need to accomplish it again.
\ If you understand that a rope is not a snake,
\ That very certainty blocks the perception of a snake.
\ To say “Still you must go on analyzing
\ The absence of a snake” is silly, isn’t it?
\ When realization of the sublime paths occurs,
\ You will not meditate with analysis;
\ What need is there to apply
\ Inferential analysis to direct realization?
\ If you think that “When you leave off analysis
\ There is no realization of the ultimate,”
\ Then for you the gnosis of Buddhas and sublime beings,
\ And the undistorted perceptions of worldly beings,
\ Would all be mistaken.
\ Because they have already been perceived,
\ They are not subject to analysis.
\ Therefore, in the context of extraordinary certainty
\ Free of elaborations of the four extremes,
\ There is no occasion for analyzing or focusing on
\ Thoughts of “this” and “that.”
\ When the analytical apprehension of characteristics
\ Binds the thinker like a silkworm in its silk,
\ The authentic nature will not be seen as it is.
\ When this extraordinary certainty
\ Dispels the darkness that obscures reality,
\ One realizes the actual fundamental luminosity
\ And the flawless vision of thatness,
\ Which is the individually cognized gnosis.
\ How could this be analytical wisdom, a form of
\ The object of analytical wisdom is “this” or “that,”
\ Which is differentiated and conceptualized,
\ Whereas this gnosis of equanimity
\ Does not reify subject, object,
\ Appearance, or emptiness in any way;
\ It does not abide in the characteristics
\ Of mind or mentation.
\ Therefore, the stainless analytical wisdom
\ Of equipoise in supreme certainty
\ Induced by analysis is the cause by which
\ One attains the resultant gnosis of coalescence.
\ The ascertainment of the view
\ And the establishment of philosophical systems
\ Determined [by that view]
\ Is the stainless valid cognition of analytical wisdom
\ That differentiates and cognizes individually.
\ The gnosis of sublime equipoise
\ That has reached the nature of things
\ By the certainty induced by that valid cognition
\ Is the main practice of the Great Vehicle.
\ If you have it, in this very life
\ The result of coalescence is bestowed;
\ So it is both a “vehicle” and “great.”
\ According to the system of four tantric classes,
\ This path of the word empowerment in anuttarayogatantra
\ Is of course the ultimate gnosis,
\ But it is not designated as a separate vehicle.
\ However, in the explanation of
\ The glorious Kalacakratantra.,
\ The body of the gnosis of equanimity
\ Is emphasized, so it is held as the ultimate tantra.
\ Among the classes of anuttarayogatantra,
\ The gnosis of the path of the fourth empowerment
\ That is emphasized and explained here [in the Great
\ Is the basic intent of all tantric classes.
\ Just as gold smelted sixteen times
\ Is extremely pure, so too here
\ The analysis of other vehicles’ philosophical systems
\ Reveals their progressive purity, which culminates here.
\ Thus the way this is established
\ Through the valid cognition of stainless wisdom
\ Is found in all the interpretive commentaries and tantras
\ And in the analysis of Dharmabhadra.
\ If you think about it, it is beyond the realm of Mara,
\ And causes inalienable wisdom to mature.
\ However, to teach the main practice of the view
\ As an object of mind and mentation, such as
\ Adhering one-sidedly to appearance or emptiness,
\ Is to make the inexpressible into an object of expression;
\ So it contradicts the intention of the learned.
\ Since atiyoga is the inconceivable gnosis
\ Of form and emptiness inseparable,
\ It is simply beyond impure mind.
\ Here the view of cutting through—which ascertains
\ The emptiness aspect of primal purity—and
\ The view of the luminous all-surpassing realization—
\ Which determines the nature
\ Of spontaneously present Buddha bodies and gnosis
\ In the inner luminosity of the youthful vase body—
\ Are inseparable;
\ They are just the coalescence of
\ Primal purity and spontaneous presence.
\ Here in the Great Perfection the so-called “indestructible
\ Tilaka of gnosis” of other tantric systems
\ Is very clearly taught as a synonym for this.
\ Each of the pith instructions of the mental class of the Great
\ Is found in the practice of learned and accomplished
\ The Mahamudra, Path and Result, Pacification,
\ Great Madhyamaka of Coalescence, and so on,
\ Are known as its synonyms;
\ Because in fact they are all the gnosis,
\ Beyond mind, they are all the same.
\ The Buddhas’ and siddhas’ intention is the same—
\ The learned affirm this univocally.
\ Some people say, “Our system of the Great Perfection
\ Is better than other systems like Mahamudra.”
\ They have no realization and
\ No understanding of the conventions of the path.
\ If they understood, they would see that this unique
\ Cannot be divided through reasoning.
\ Likewise, all the gnoses of the fourth empowerment
\ In the anuttarayogatantras
\ Are indivisible in the Great Perfection.
\ However, the source of all of those
\ Is the gnosis of the Great Perfection, whose tantric classes
\ Are divided into “mental,” “space,” and “instructional,”
\ According to their profound, extensive, extraordinary
\ There are many instructions here that are not known
\ In other systems, which use just a fragment of them,
\ So it goes without saying that this is an “extraordinary
\ There, the ultimate Great Perfection
\ Is profound, peaceful, luminous, and unfabricated—
\ The gnosis of the Buddhas.
\ But here in the context of the paths,
\ One practices the exemplary and actual coalescences,
\ Which are like a drawing of the moon,
\ The moon in water and the moon in the sky,
\ Homologous to that gnosis.
\ Each one gradually leads to the next,
\ As one cultivates the self-arisen stainless gnosis
\ According to one’s own capacity.
\ Therefore it is like meditating homologously
\ In order to reach sublime gnosis.
\ If one directly ascertains
\ The great gnosis of the coalescence of dharmata,
\ All views that are apprehensions of mental analysis
\ Will definitely subside, and one will see non-elaboration.
\ Therefore, without citing the context,
\ Saying one-sidedly that modal apprehension
\ Should be used or not has both faults and good points,
\ Like the waxing and waning of the moon.
\ This is established through reasoning,
\ According to scriptures of definitive meaning.
L1: Topic 5
\ Which of the two truths is more important?
\ Some claim the ultimate is most important.
\ “Deceptive reality is a deluded perception,” they say,
\ Understanding it as something to be abandoned.
\ “Ultimate reality is not deluded, so that ultimate
\ Is the perfectly pure view,” they say.
\ If deceptive reality were not erroneous, were indeed true,
\ Ultimate reality could not be emptiness, so
\ They are expressed differently in this way.
\ However, no ultimate can be established
\ Over and against the deceptive;
\ The two of them are method and methodical result.
\ Without depending on an entity for examination,
\ Its nonsubstantiality cannot be established—
\ Therefore both substance and nonsubstance
\ Are the same in being mere relativity.
\ If that clinging to emptiness
\ Were to fully exclude appearance,
\ It would mess up Nagarjuna’s fine system.
\ If by cultivating the path by that seeing of emptiness,
\ One were only to realize the expanse of emptiness,
\ Then one would have to accept that the
\ Sublime equipoise on emptiness
\ Would be a cause for the destruction of substantial entities.
\ Therefore, though things are empty from the beginning,
\ Appearance and emptiness are not separate things;
\ Adhering to the statement “Only emptiness is important”
\ Is an unskilled approach to the final meaning.
\ Some people put aside the ultimate
\ And from the perspective of mere conventionality,
\ Differentiate the levels of the view in the tantric classes.
\ Viewing oneself as a deity conventionally
\ Without complementing the view with the ultimate reality
\ of emptiness
\ And thus differentiating “higher” and “lower” teachings,
\ is incorrect.
\ Without having confidence in ultimate reality,
\ Just meditating on deceptive reality as divinity
\ Is mere wishful thinking, not a view;
\ Just as some heretical awareness mantras
\ Involve visualizing oneself differently during recitation.
\ Some say deceptive reality is more important;
\ They say you must integrate the two truths,
\ But then they heap praise on deceptive reality.
\ At the time of maintaining the view of coalescence,
\ They desert coalescence and grasp a blank emptiness.
\ Thus the toddler of practice cannot keep up
\ With the mother of good explanations.
\ Therefore, here in our early translation tradition,
\ Our Dharma terminology for the basis, path, and result
\ Does not fall into extremes or bias with respect to
\ Permanence, impermanence, the two truths, and so forth;
\ We maintain only the philosophical position of
\ If deceptive and ultimate reality are separated,
\ One cannot posit the basis, path, or result on the basis of
\ Basis, path, and result are all
\ Without the distinction of abandoning one thing
\ or accepting another.
\ For if one abandons deceptive reality,
\ There is no ultimate; there is no deceptive
\ Reality apart from the ultimate.
\ Whatever appears is pervaded by emptiness,
\ And whatever is empty is pervaded by appearance.
\ If something appears, it cannot be non-empty,
\ And that emptiness cannot be established as not appearing.
\ Since both entities and nonentities should both
\ Be taken as bases for establishing emptiness,
\ All appearances are just designations,
\ And emptiness too is just a mental designation.
\ For the certainty of rational analysis,
\ These two are method and methodical result;
\ If there is one, it is impossible not to have the other,
\ As they are inseparable.
\ Therefore appearance and emptiness
\ Can each be conceived separately,
\ But in fact they are never different.
\ Therefore, they are called “coalescent,”
\ Since the confidence of seeing the nature of things
\ Does not fall to any extreme.
\ In the perspective of the wisdom of authentic analysis
\ Appearance and emptiness are considered to be
\ A single essence with different aspects, for
\ If one exists, the other exists, and if
\ One does not exist, the other does not exist.
\ Nonetheless, for beginners
\ They appear as negation and negandum;
\ At that time they are not combined as one.
\ When the nature of emptiness
\ Arises as appearance, one attains confidence.
\ Thus, everything is primordially empty,
\ And these appearances are empty;
\ Though empty, they appear; though apparent,
\ They are seen as empty—this is the birth of certainty.
\ This is the root of the profound paths
\ Of sutra, tantra, and pith instructions.
\ This is the meaning of cutting off misconceptions
\ Through study and reflection;
\ It is the unmistaken, authentic view.
\ By realizing that crucial point more and more profoundly,
\ Clinging to the characteristics of appearances of
\ Deceptive reality will gradually be abandoned.
\ The stages of the vehicles of the various tantric classes
\ Appear in that way.
\ Intellectual wishful thinking and
\ The view of certainty that finds confidence in the
\ Divine appearance of animate and inanimate phenomena
\ Cannot possibly be the same.
\ The determination that phenomena are truthless
\ By Madhyamika reasoning is a view.
\ But when a Brahmin recites a mantra over a sick person,
\ His imagining a lack of illness is not the view.
\ By realizing the abiding nature of ultimate reality,
\ One grows confident in the divine appearance of deceptive
\ Otherwise, if one dwells on the manner of deceptive
\ How can divinity be established?
\ Aside from this deluded appearance of subject and object,
\ There is no such thing as samsara;
\ The divisions of the path that abandons it
\ Are not only made from the perspective of ultimate reality,
\ Because ultimate reality has a unitary character.
\ With respect to the mental ability gained
\ Through seeing and cultivating all phenomena
\ Of apparent deceptive reality, the subject (of qualities),
\ With respect to ultimate reality, the action tantra,
\ Performance tantra, yoga tantra, and unexcelled yoga tantra
\ are taught.
\ Therefore, the tantric classes are not differentiated as higher
\ Or lower with respect to either of the two truths
\ According to one’s attainment of confidence
\ In the coalescence of the two truths,
\ The practice of [each of the tantric classes naturally] follows.
\ Therefore, if one properly practices without mistakes
\ The peerless Vajra Vehicle,
\ The path that bestows liberation in a single life,
\ Then, just like the example of water seen
\ By several different types of sentient beings,
\ With respect to pure vision
\ It will be impossible for anyone not to see
\ Actual and potential phenomena as a manifested mandala.
\ If you don’t know things that way,
\ Meditating on deities while holding
\ The nature of samsara to be impure
\ Is like spraying a vomit-filled vase with perfume.
\ Alas! That sort of meditation on the Vajra Vehicle of
\ Is just like a drawing of a butter lamp.
\ The way things appear is impure,
\ But that is the system of delusion.
\ We say that authentically seeing the nature of things
\ Is the meaning of the undivided Vajrayana system.
\ Seeing the animate and inanimate universe
\ As lacking the nature of pure support and supported,
\ But meditating while imagining that they do—
\ This path evinces an obvious contradiction,
\ And is just a reflection of the Vajrayana path.
\ Coal cannot be whitened by washing;
\ Likewise, a fabricated meditation that thinks
\ “It is not, but it is”
\ Attaining some kind of result
\ Would be like the heretical sun worshippers (nyi ma pa)—
\ Who have no confidence in the emptiness of true
\ Abandoning emotional afflictions through meditating
\ On an emptiness devoid of appearance, etc.
\ What if the action, performance, and unexcelled tantric
\ Did not have different levels of view?
\ If you have confidence in the view that realizes
\ The pure equality of actual and potential phenomena,
\ But fail to take advantage of the correct view,
\ Seeing yourself and the deity as superior and inferior
\ And discriminating things as pure and impure,
\ You will only harm yourself.
\ And, if you are still attached to what is accepted and aban-
\ doned in the lower tantras
\ But practice the equality of what is accepted and abandoned
\ in the unexcelled tantras,
\ Such as “union and liberation,” eating meat, drinking
\ alcohol, etc.,
\ This is known as the “reckless behavior of
\ Isn’t that despicable?
\ The view is defined according to one’s certainty
\ In the vision of the nature of things;
\ According to one’s confidence acquired by the view,
\ One maintains the practice of meditation and conduct.
\ “Because the vehicles are differentiated
\ By different levels of view, they are not necessarily nine in
\ From the lowest of the Buddhist philosophical systems
\ Up to the ultimate vajra pinnacle of atiyoga,
\ There is a specific reason for positing
\ The enumeration of nine classes.
\ Of course there are many levels of vehicle,
\ But they are posited by necessity, as is the three-vehicle
\ Thus, according to the relative strength
\ Of inner gnosis, the animate and inanimate
\ Worlds are seen as pure or impure.
\ Therefore, the basis of inseparable appearance and emptiness
\ Is realized as the inseparability of the two realities;
\ As you cultivate the path in that way,
\ You will see the gnosis,
\ The coalescence of the two Buddha bodies.
L1: Topic 6
\ When a single instance of water appears
\ As different substances to various sentient beings,
\ Some say there is a single object of perception
\ And that all perceptions of it are valid.
\ If water had some kind of essence,
\ Valid and invalid cognitions would be impossible [here].
\ If the various objects that appear were distinct,
\ It would not be possible for [different minds]
\ To perceive the same pillars, vases [etc.].
\ 6.2.1. 2.1.
\ Some say [that in the case of water] there is just wetness;
\ But if [different appearances] are not different aspects [of the
\ same substance,
\ But merely perceptions belonging to different perceivers],
\ Different perceptions [of the same thing] would be
\ If what one [being sees as] water, pus, and so forth,
\ Is not present to other [beings],
\ What would be the basis of [those perceptions of] water,
\ pus, etc.?
\ Moreover, what would happen to the wetness basis
\ In the case of beings of the realm of infinite space?
\ If wetness were the same as water,
\ It could not appear as pus and so on;
\ If it were different from water and so on,
\ Liquidity would not be perceived anywhere.
\ It is not possible for there to be a common object
\ Of each distinct perception,
\ Because it is not possible for a suitable common substance
\ To appear in different ways.
\ If one accepts an analytically [determined] basis
\ Other than a dependently designated one,
\ One must establish its existence in reality—
\ However you look at it, it’s unreasonable.
\ If the common object were non-existent,
\ There would be no object as in Cittamatra,
\ And one would have to accept that consciousness itself
\ is the object;
\ That is unreasonable.
\ The subjective apprehension of a non-existent object
\ Would also be non-existent in fact.
\ Both subject and object are equally apparent
\ In relative truth, so considering whatever appears
\ It is not reasonable to differentiate
\ Subject and object as existent and non-existent.
\ Although an object appears, it is false.
\ Likewise apprehension of an object appears but is
\ not established.
\ The common perceptual object is a mere appearance
\ That is established as the basis of similar and dissimilar
\ Because otherwise it would be unreasonable, as in seeing
\ a dance.
\ Aside from this mere existence [of an appearance],
\ It is not possible for it to come from some other existent;
\ Without this, all appearances
\ Would be nonapparent, like space.
\ On the basis of outer and inner conditions,
\ One does not see the thing itself as it is,
\ But in the manner of seeing horses and cattle
\ In the place of wood blessed by illusion mantras.
\ Therefore the common object of perception
\ Cannot be specified as “this” or “that.”
\ So in our system appearance and emptiness
\ Are not differentiated in the basis itself,
\ Which is not established anywhere.
\ Because it is the same in everything that appears,
\ A single substance appears as various things.
\ For whomever appearance and emptiness are possible,
\ Everything is possible;
\ For whomever appearance and emptiness are impossible,
\ Nothing is possible.
\ “Well then, the distinction of valid and invalid cognitions
\ Would be invalid.”
\ Whatever appears does not appear otherwise,
\ So it is not the case that the perception of its being thus
\ Does not establish it as a cognandum.
\ For all things naturally abide in their own essences,
\ Because they are established by valid cognitions
\ That determine their sameness and difference.
\ Thus, things by their very nature are
\ That in dependence upon which valid cognitions are
\ But they are not themselves established by valid cognition;
\ If they were, they would be reality itself.
\ An instance of water that is established
\ By the valid cognition of one’s own apprehension
\ Is not independently established under its own power.
\ It is not established by ultimate reasoning,
\ Nor is it [established] for a hungry ghost.
\ If one determines the objects of one’s own perception
\ By means of direct perception and inference,
\ One is not deceived with regard to engaging and avoiding
\ The objects of those [valid cognitions];
\ So valid cognition is not pointless.
\ Thus, when we say “a single instance of water,”
\ We refer to the visual perception of human beings.
\ In the divine context,
\ A single instance of nectar is understood as the basis
\ of perception.
\ When water is seen as pus, water, and nectar,
\ The three are not mixed together.
\ If one of those three were not valid,
\ Then it could not be established as validly cognized
\ By being cognized as a different substance, and
\ All three objects of visual perception would be non-existent.
\ If this instance of water perceived by a human being
\ Were not water, it would not be viable as water for another,
\ And “water” would be completely non-existent.
\ In such a system, a system of valid cognition
\ Would also be untenable.
\ Thus, the object of a sense faculty
\ That is undistorted by accidental conditions
\ Should be established as validly cognized,
\ As in the appearance of water and mirages.
\ Thus, in the context of hungry ghosts
\ Karmic obscurations cause clean water
\ To appear as pus, but if the fault [of such obscuration]
\ Is dispelled, it then appears as water.
\ For this reason, what is seen by human beings
\ Is posited contextually as validly cognized,
\ Because the other is distorted by perceptual fault.
\ For now water is established by a valid cognition.
\ But if one analyzes with ultimate reasoning,
\ Everything is the appearance of karmic propensity.
\ Since [for sublime beings] water appears
\ As the pure realms and kayas,
\ The human perception cannot itself
\ Be established one-sidedly as the [only] valid cognition.
\ Thus, by progressively purifying the causes of obscuration,
\ It is reasonable to posit higher forms of seeing
\ In relation to lower forms of seeing.
\ Since the final nature of things is unique,
\ The valid cognition that sees only it
\ Is likewise unique; a second type is impossible.
\ Reality is a unique truth, coalescence,
\ And valid cognition is self-arisen gnosis.
\ Since there is nothing to abandon except unawareness,
\ It is simply a case of awareness and unawareness.
\ Thus, this system of valid cognition
\ Establishes the nature of all appearances as deities.
\ This is the unique tradition of the early translations,
\ The lion’s roar of the elegant works
\ Of the omniscient Rong zom Pandita.
\ Other [systems] do not explain [this] point correctly;
\ In this respect whatever other systems say is contradictory.
\ The claim that the common object of perception
\ Is either appearance or emptiness is untenable.
\ If it were only emptiness,
\ It would be possible for any sentient being
\ To perceive space as vases,
\ And vases would disappear like space.
\ If emptiness without appearance
\ Were viable as an object of perception,
\ What would not appear?
\ Things would either be permanently existent,
\ Or become entirely non-existent, being causeless;
\ Either way, it is the same.
\ In the context of emptiness there is no appearance,
\ Because they are contradictory;
\ If there were something non-empty,
\ It would contradict the position
\ That mere emptiness is the basis of appearance.
\ “Well, didn’t you say earlier
\ That appearance and emptiness are not contradictory?”
\ Here, the object of visual perception is understood
\ In the context of conventional valid cognition,
\ For which existence and non-existence are contradictory;
\ On the basis of a single thing the two truths
\ Are noncontradictory only for gnosis.
\ If a mere appearance bereft of emptiness
\ Were not viable as the basis of appearance,
\ That appearance could appear any which way;
\ For there is no appearance that is not
\ Distinguished in one way or another.
\ [A non-empty appearance] is not established as the basis
\ of appearance,
\ It is not perceived by a valid cognition that causes one
\ to know it;
\ To say that it exists is only a claim.
\ If whatever appeared were entirely separate,
\ Nothing other than it could appear;
\ Because it would be a non-empty appearance,
\ It would be immune to an ultimate analysis.
\ Whether one understands the basis as water, pus,
\ Nectar, or whatever, there is contradiction.
\ If that water were pus,
\ How could it appear as water?
\ If it were water and not pus,
\ How would it appear otherwise as pus, etc.?
\ If you say that the object that appears to hungry ghosts
\ Is water, then you would have to accept that the pus
\ That appears is non-existent.
\ For aside from whatever appears to oneself,
\ There is no separate basis of appearance,
\ Because if there were it would be something different,
\ Like pillars and vases, having a single basis but being
\ Therefore the coalescence of appearance and emptiness,
\ Or the absence of true existence and mere appearance,
\ [Is equivalent to] the original pure equality of all
\ In the great equal taste of the coalescence
\ That is free of partiality and extremes.
\ In that way, when one determines the essence of accomplishment
\ In the Great Perfection of equality,
\ In the context of the path where one cultivates
\ [that essence],
\ In dependence upon the vision of purity,
\ Impure appearances self-liberate.
\ Hence one attains confidence in the meaning
\ Of the statement from the vajra scriptures,
\ “Dharmakaya, which is the purity of all appearances.”
\ So, in the Magical Net Tantra, it is taught that
\ The continuous appearance of the five aggregates
\ Is the “pure divine body of thatness”;
\ This is confidence in the intended meaning [of that
\ Similarly, when the apprehension of pus is removed,
\ It is realized to be delusion, and by cultivating that
\ Water appears in its place.
\ A great bodhisattva [on the] pure [stages]
\ Sees countless Buddha fields in each drop of water,
\ And water itself manifests as Mamaki.
\ On the bhumi where the two obscurations are finally
\ One sees the great equal taste of coalescence.
\ As for pure vision,
\ If in order to abandon all obscurations
\ The unerring reality of things is seen
\ By it and it alone,
\ It is taken to be the final valid cognition
\ And is established for those with the eyes of reason
\ Who abide on the pinnacle of the establishment of
\ the statement
\ “Everything abides originally in the purity of dharmakaya.”
\ Moreover, this vehicle has thousands
\ Of wonderful rays of light.
\ The low-minded, like spirit birds,
\ Are as if blind to it.
\ Although it cannot be incontrovertibly proven
\ That the final space of equality
\ Only appears as divinity,
\ To the extent that the expanse of original natural purity
\ And its apparent aspect, the wisdom body,
\ Are inseparable, the apparent aspect is
\ Originally pure divinity,
\ And cannot be harmed by ultimate reasoning,
\ For the expanse of coalescent form and emptiness,
\ Which is free of the two obscurations,
\ Is the final suchness of things.
\ Aside from this, whatever else one analyzes
\ Is not the final meaning;
\ For if the two obscurations are not completely abandoned,
\ Abiding and apparent natures are always discordant.
\ Contextual appearances in the practice of the path
\ Are like healing a cataract;
\ By purifying defilements of the subject,
\ The object is likewise seen in its purity,
\ Because for a pure subject
\ There are no impure objects.
\ Thus, when an ordinary person becomes a Buddha,
\ [There is no impurity], but impurity still appears to others,
\ Because they obscure themselves with their own
\ Thus, although object and subject
\ Are originally pure,
\ They are obscured by adventitious defilements,
\ So one should strive to purify them.
\ Because there is nothing impure with respect to
\ The purity of one’s own nature,
\ There is the equality of natural luminosity.
\ Not realizing it, one apprehends
\ Various appearances individually.
\ A childish person whose mind is attached
\ Is an ignorant child whose ignorance enslaves him;
\ But everyone who realizes this will seize
\ The citadel of fruition in the state of equality,
\ And become victorious in self-arisen gnosis
\ In the fundamental expanse of the three times and
\ of timelessness.
\ This system, which accepts the principle
\ Of great pure equality, is well established.
\ Because appearance and emptiness are not established,
\ Whatever can appear appears anywhere and everywhere.
\ However else you look at it,
\ Nothing can appear anywhere.
\ The way to gain confidence in this system
\ Is the path of emptiness and dependent origination.
\ If one gains certainty in appearance and emptiness,
\ In the self-arisen changeless mandala,
\ Profound tolerance will be born within oneself
\ For the inconceivable dharmata
\ And for the emptying and non-emptying [of the limits of
\ In the width of an atom
\ One sees as many Buddha fields as are atoms,
\ And in a single instant an aeon appears.
\ With certainty in the absence of true existence
\ Which is like an illusion,
\ One can enter the range of Buddhahood.
\ One may have disciplined oneself and thought for a
\ hundred years
\ About the meaning of the words of different philosophical
\ systems, such as
\ The undifferentiability of one’s own appearances [and
\ their basis],
\ The absence of partiality and extremes,
\ The inconceivability of the fundamental expanse,
\ The dharmata that is not established anywhere,
\ The coalescence of form and emptiness, etc.,
\ Yet if one lacks the cause of prior familiarity,
\ Then, even if one’s intellect and training are not
\ One will not get it.
\ Thus the hundred rivers of elegant explanations
\ In which flow the quintessences
\ Of all philosophical systems
\ Pour into this great ocean, which is amazing.
\ Other modes of appearance
\ That appear in the process of transformation are indefinite;
\ The consummate gnosis of coalescence
\ Sees the infallible meaning and is changeless.
L1: Topic 7
\ When analyzing whether or not there is a position
\ In the Great Madhyamaka of non-elaboration,
\ Earlier scholars univocally stated
\ That our own Madhyamika system has no position,
\ Because existence, non-existence, being, and nonbeing
\ Do not exist anywhere.
\ In our texts, all the philosophical explanations
\ Of path and result and relativity
\ Are accepted as our own position, so
\ To say that all conventions are only set forth
\ From other people’s perspective
\ Is to contradict both the words and the meaning.
\ According to Klong chen rab ‘byams,
\ Earlier scholars veered to the extremes of
\ Asserting that Madhyamaka has or does not have a position;
\ Each of those positions has defects and qualities.
\ Thus, when approaching the nature of reality,
\ Nothing is established in the original state;
\ What then is there to accept as a position?
\ Therefore, because a philosophical system
\ Is a position about the nature of things, at the time of
\ debate, etc.,
\ No position is taken, in accordance with the original state.
\ In meditative aftermath, the systems of path and result—
\ Whatever and however they are posited—
\ Are expounded according to their respective positions,
\ Without confusing them.
\ Klong chen pa said, “From now on, if someone knows
\ how to
\ Expound this, it is because of my elegant explanation.”
\ In that respect, some Tibetan scholars
\ Established and overestablished the fact that
\ Their own systems had a position.
\ But if one does not differentiate the context,
\ Because the meaning of the original state
\ Is not established anywhere, it is difficult
\ To assert one-sidedly that one has a position.
\ If you say “Madhyamaka is our system,”
\ It should refer to the way that the Madhyamika system
\ Approaches the ultimate meaning.
\ Anything else is not our own system,
\ Because when other systems are approached
\ By a Madhyamika, they cannot be established.
\ Thus, if the Madhyamika accepts [deceptive reality],
\ Then he accepts it as established by its own power,
\ Because it is established by the force of reasoning.
\ That position would be established ultimately
\ And thus be immune to analysis.
\ If our own system had no position,
\ This would contradict the statement,
\ “We do have a position
\ [That accords with worldly renown].”
\ We would have two positions according to
\ Whether or not there is analysis.
\ If both of them were definitely true,
\ Would “our system” be each of them separately,
\ Or would it be both of them together?
\ If it were each of them separately, then
\ Each would contradict the other.
\ If we do not accept “existence”
\ But do accept “non-existence,”
\ The position of “existence” would not
\ Even be conventionally acceptable,
\ Because of only accepting non-existence.
\ If we accepted both of them together,
\ Having removed that which is susceptible to analysis,
\ We would posit something not harmed by reasoning.
\ Thus, both existence and non-existence
\ Would be immune to analysis.
\ Accordingly, both existence and non-existence
\ Cannot be mixed together;
\ For if they were, then even though one
\ Could realize [coalescence] through analysis,
\ When not analyzing, existence would be engaged.
\ So what good would analysis do
\ For eliminating clinging to deceptive realities?
\ For deceptive reality to be established
\ Through analysis is irrational.
\ If there were no reality beyond the mere
\ Exclusion of a negandum, an absolute negation,
\ That modal apprehension could not have
\ An apparent aspect; so why would this be any different
\ Than the position of someone who thinks
\ That view, meditation, and action are simply non-existent?
\ For there would never be any need to meditate
\ In accordance with the nature of things.
\ Therefore, according to the statement
\ of the Omniscient One,
\ Our system should be understood as follows:
\ If ours is to be a definitive Madhyamika system,
\ It must be the Great Madhyamaka of coalescence,
\ Or the nonelaborated Madhyamaka.
\ Because, by defining it according to
\ The gnosis of sublime equipoise,
\ All extremes of existence, non-existence, and so forth,
\ Are completely pacified.
\ That path that objectifies emptiness alone
\ Succumbs to each of the two realities one-sidedly;
\ That trifling point of view
\ Is neither coalescent nor unelaborated.
\ Coalescence means the equality of
\ Existence and non-existence, or of form and emptiness;
\ Whereas that view is just the subjective aspect
\ Of the expanse of ultimate emptiness.
\ Among all types of reification, such as
\ The elaborations of existence and non-existence,
\ This is nothing but an elaboration of non-existence,
\ Because it reifies [emptiness].
\ Therefore, from the perspective of Great Madhyamaka
\ There is no position whatsoever.
\ In order to realize the equality of appearance and emptiness,
\ It is free of all proof and negation such as
\ Reality, unreality, existence, and non-existence.
\ According to the sense of [ultimate] reality, all things
\ Cannot be asserted through rational proof;
\ Therefore, there is nothing to have a position about.
\ Thus, although the ultimate meaning of reality
\ Has no position, in the way things appear
\ There is a position on the conventions of each of the
\ two realities;
\ With respect to how the two realities abide inseparably,
\ They are both simply ways of appearing.
\ With respect to the gnosis that
\ Sees that they are inseparable, both valid cognitions
\ Are fragmentary, because with only one of them
\ Both realities cannot be apprehended.
\ Therefore, if the wisdom of ultimate and
\ Conventional valid cognition
\ Both engage a vase, etc.,
\ Two essences are found.
\ But when one is engaged, the other is not, for
\ In the mind of an ordinary person the two realities
\ Can only appear in succession.
\ Thus, the positions based on each type of engagement
\ Are established in fact.
\ “Well, don’t the faults of having or not having a position,
\ And the internal contradiction of the two realities
\ That you have ascribed to others above
\ Apply just as well to you?”
\ By making subtle distinctions,
\ I have differentiated the path Madhyamaka and
\ The equipoise Madhyamaka that is the main practice.
\ Since my explanation distinguishes great and little
\ With respect to coarseness and subtlety,
\ Cause and effect, consciousness and gnosis,
\ How can that defect apply to me?
\ Thus, the Great Madhyamaka
\ With no position is our ultimate system.
\ In the context of meditative aftermath,
\ When the two realities appear separately,
\ All the proofs and negations engaged by
\ The validating cognitions of each of the two realities
\ Are for negating various misconceptions;
\ But in the original state, there is
\ No position of refutation or proof.
\ Therefore, in the original state
\ The two realities are not divided,
\ Because neither of their positions
\ Is established in truth.
\ If [a position] is posited [conventionally about either] of
\ the two [truths],
\ It is only with respect to the way things appear.
\ For the time being, each is established as true
\ In its own context, so there is no contradiction,
\ And the fault of immunity to analysis, etc. does not apply.
\ Real entities are not immune to analysis;
\ Nor are unreal entities immune to analysis.
\ In the final analysis, they are the same;
\ They are just designated contextually.
\ Something that exists by consent, without investigation,
\ Is a mode of appearance, not the way things are;
\ Whatever is seen by the rational knowledge
\ That analyzes truthlessness is considered
\ As the way things really are.
\ This is an ultimate reality in relation to
\ Deceptive reality, but in the final analysis
\ It is just a conceptual ultimate.
\ If the way things appear and the way things are
\ Are mutually exclusive,
\ The four faults of the two realities being different are
\ If the two realities are mutually inclusive,
\ The four faults of the two realities being identical are
\ In this way, Buddhas and sentient beings
\ Are just the way things are and the way things appear;
\ The claim that they are cause and effect
\ Should be known as the Hinayana system.
\ Because the way things are and the way they appear
\ Are not posited as either the same or different,
\ There is absolutely no logical fault, such as
\ Sentient beings appearing as Buddhas,
\ The path and practice being pointless,
\ The cause residing in the effect.
\ However things may be in reality,
\ They are obscured by obscurations,
\ And do not appear as such.
\ Everyone accepts the need to practice the path.
\ Because the two truths are not contradictory,
\ Though the two views of “existence” and “non-existence”
\ Are posited, how could they be contradictory?
\ Because they are not mutually inclusive,
\ The two positions are formulated.
\ For this reason, as long as the two realities
\ Are engaged by minds for which
\ They appear separately,
\ Both realities are quite equivalent in force,
\ And there is no one-sided position about either of them.
\ The determination of the emptiness of truth as
\ And the determination of appearance as “existence”
\ Are the objects found or seen alternately by each
\ Of the two valid cognitions at the time of their engagement,
\ And are said to be the two truths.
\ Because those two are neither the same nor different,
\ It is not possible to one-sidedly discard one
\ And accept the other.
\ The wisdom that analyzes these two
\ Differentiates their respective positions.
\ For example, when the dharmakaya is finally attained,
\ All minds and mental events without exception
\ Cease, conventionally speaking;
\ But ultimately there is no cessation.
\ In all the texts of all sutras and treatises,
\ Among the various kinds of proof and negation
\ Some posit ultimate reality,
\ And some are stated with respect to deceptive reality.
\ With respect to ultimate reality alone,
\ The path, Buddhas, sentient beings, and so forth,
\ Are rightly said to be “non-existent.”
\ It is not the case, however, that
\ Without relying on conventions, they are simply
\ Though they do not exist, all appearances of samsara and
\ Appear, and are established through direct perception.
\ Therefore, with respect to conventional valid cognition,
\ The path, Buddhas, sentient beings, and so forth,
\ Are rightly said to be “existent.”
\ But this doesn’t mean that they are really existent
\ Without reference to ultimate reality.
\ They exist, but are not established as such,
\ Because they can be determined by
\ An analytical cognition of ultimate reality.
\ Thus, those two can never exist
\ One without the other.
\ “When both are true with equal force,
\ Will existent things be non-empty?”
\ Both are not established by their intrinsic nature,
\ Nor are they, as objects, really different;
\ Whatever appears is empty, so what can be non-empty?
\ Both are equally apparent,
\ So they are established as empty;
\ If they were not apparent, how would emptiness be known?
\ Thus, both appear together as cause and effect,
\ Without contradiction.
\ If one is certain that one exists, the other does too:
\ They are always inseparable.
\ There is no case where one does not
\ Encompass the other; therefore,
\ Whichever one investigates, it is correct.
\ By knowing appearance as emptiness,
\ One realizes appearance as realitylessness;
\ And by knowing emptiness as appearance,
\ One will not conceive emptiness as real.
\ Therefore, when they are seen as inseparable,
\ One will not revert to seeing them as real.
\ The abiding character of whatever appears
\ Is emptiness, so they are inseparable.
\ If one rejects appearance,
\ Emptiness cannot be established independently.
\ Therefore, one cultivates the wisdom
\ Of meditating on the two realities alternately.
\ In the context of this samsara of dualistic perception,
\ Gnosis does not appear,
\ So the two stainless analytical wisdoms
\ Should be upheld without ambivalence.
\ When one of these is incomplete,
\ The coalescence of gnosis
\ That arises from them will definitely not arise,
\ Just as fire will not occur without
\ Two pieces of wood rubbed together.
\ Therefore, a path where method and emptiness
\ Are separated is inauthentic
\ According to all the Buddhas and vidyadharas.
\ Therefore, if one abandons these two causes,
\ There is no other way for the great gnosis to arise.
\ The essence of gnosis
\ Is beyond thought and expression.
\ Therefore, aside from symbolic means and mere words,
\ It cannot actually be indicated.
\ Thus, the teaching of the word empowerment in the
\ In the tantras of the vajra essence, and so forth,
\ It is taught by words and methods.
\ The supramundane gnosis
\ Cannot be understood without relying on
\ Some kind of verbal expression,
\ So the path of the Madhyamaka of the two realities is taught.
\ The result of analyzing in the manner of two realities
\ Can be established as coalescence itself.
\ Therefore, when the two realities are ascertained,
\ Appearance and emptiness are taught alternately
\ As negation and negandum.
\ Their result, the gnosis of coalescence,
\ Is taught by many synonyms in tantra.
\ Thus, all Madhyamika systems
\ Are established by way of the two realities;
\ Without relying on the two realities,
\ Coalescence will not be understood.
\ Whatever the Buddhas have taught
\ Has relied entirely on the two realities;
\ Therefore, the Madhyamaka that contains
\ The positions of each of the two truths
\ Is the little Madhyamaka of alternation,
\ Which gives the result’s name to the cause.
\ The emptiness of the analyzed five aggregates
\ Is the mere absolute negation exclusive of the negandum;
\ In that respect there is the position of “non-existence.”
\ Whatever the causal or path Madhyamaka
\ Posits as the two truths,
\ Both are our own system.
\ It makes no sense to posit the ultimate as our system,
\ And say that conventional reality
\ Is only from other people’s perspective.
\ If that were so, then our own system of the ultimate
\ Would be a blank nothingness,
\ And we would wind up totally denigrating
\ All appearances of the basis, path, and result
\ As “delusions to be abandoned.”
\ Then a mere expanse of emptiness without obscuration
\ Would be left over, while the two types of omniscience
\ Would be negated. This would be similar to the sravaka
\ Which asserts a remainderless nirvana,
\ Just like the blowing out of a candle.
\ Thus, the Buddha said that these
\ Spaced-out people who denigrate
\ The expanse of coalescence as mere nothingness
\ Are thieves who destroy the Sakya Dharma.
\ With reasoning, one can see how
\ That system denigrates the existent as non-existent,
\ And one is able to destroy the mountain of bad views
\ With the vajra-fire of certainty.
\ Thus, in all Madhyamika texts,
\ Without establishing the causal Madhyamaka
\ Of analytical wisdom through rational analysis,
\ The fruitional coalescence is not established.
\ Therefore, even if one has rationally determined
\ The character of the two realities,
\ The fruition is the establishment of the inseparability
\ Of the two realities. This is the quintessence of all vehicles.
\ Therefore, gnosis
\ Does not abide alternately in the two extremes,
\ And is beyond intellect;
\ Thus it is Madhyamaka, and also great.
\ As long as one has not reached gnosis
\ By means of alternation, this is not
\ The ultimate Madhyamaka that is
\ The heart of all Buddhas’ realization (dgongs pa).
\ Like fire stirred up by a fire-stick,
\ The fire of coalescent gnosis induced
\ By the stainless analytical wisdom of the two realities
\ Pacifies all elaborations of the four extremes
\ Such as existence, non-existence, both, and neither.
\ This is the gnosis of sublime equipoise,
\ And is considered the fruitional Madhyamaka of
\ Not falling into the extremes of the two realities—
\ For the analytical wisdom of meditative aftermath
\ This may be considered the “coalescence of
\ Appearance and emptiness,”
\ But for the great gnosis of equipoise,
\ Appearance, emptiness, and coalescence
\ Are not reified as having some essence.
\ Appearance is the object of conventional valid cognition,
\ Emptiness is the object of ultimate analysis,
\ And coalescence combines these two components.
\ Since these are objects of words and concepts,
\ The equipoise that transcends them
\ Is merely designated as “gnosis known for oneself.”
\ [In the context of sublime equipoise,]
\ “Apparent,” “nonapparent,” and so forth,
\ Are not established by authentic reasoning.
\ Thus, as long as one meditates on the two realities
\ Alternately, this is analytical wisdom,
\ And when there is no such alternation,
\ One attains the coalescent gnosis.
\ Then one transcends the bare emptiness
\ That is the absolute negation that
\ Is the analytical exclusion of the aggregates.
\ Negation and negandum no longer appear separately.
\ The great nonelaborated emptiness that
\ Is consummately endowed with the aspect
\ Of appearance as method,
\ Mahamudra of coemergence, and so forth,
\ Have many synonyms.
\ Because these are all the gnosis that transcends mind,
\ They are inconceivable by any other concepts.
\ Because this gnosis is not the object of words and concepts,
\ It is not differentiated by
\ Implicative and absolute negations,
\ Nor as different, nondifferent, apparent, or empty, etc.
\ Because it does not fall into any extreme or partiality,
\ It is beyond having and not having a position,
\ And appears as the nonabiding self-arisen gnosis of
\ The coalescent Evam.
\ Thus, the ultimate meaning, free of reification and negation,
\ That is beyond all positions,
\ The state of awareness and the expanse inseparable,
\ Is held to be without any expression or indication of “this”
\ or “that.”
\ However, unlike the “thoughtless agent,”
\ It is not something that cannot be known by anyone,
\ Because the Dharma lamp of certainty
\ Is the consummate gnosis attained subsequent
\ To the individually cognized gnosis induced
\ By the analysis of stainless reasoning,
\ What appears directly to those [yogis] who
\ Are free of the darkness of doubt.
\ In the sutra path, both method and wisdom
\ Are considered in light of each other,
\ But here both method and wisdom
\ Are realized and cultivated inseparably.
\ Both the Great Madhyamaka of coalescence and
\ The Great Perfection of luminosity
\ Have the same meaning, and their names are synonymous.
\ There is no view higher than that,
\ For anything other than the absence of the elaborations
\ Of the four extremes—which is the nonapprehension
\ Of appearance and emptiness alternately—
\ Is nothing but some sort of elaboration.
\ However, the meaning of coalescence in the sutra system
\ Is ascertained through analysis;
\ In mantra, it is established through directly experiencing
\ The expanse of intrinsic awareness.
\ Therefore, “Madhyamaka” refers to the
\ Path Madhyamaka of analytical wisdom that
\ Investigates each of the two realities,
\ And the single savor of the two realities induced by it,
\ Which is the Result Madhyamaka of coalescence.
\ With respect to the causal and resultant views of sutra
\ and mantra,
\ The former is the aspect of analytical wisdom,
\ And the latter is just gnosis.
\ Therefore, this latter is praised
\ With the word “great.”
\ As for the “the way things are”:
\ There is the way things are as the emptiness of entities,
\ And the way things are as the inseparability of the two truths.
\ The term is the same in both cases, but in fact
\ The difference is like the earth and sky.
\ Accordingly, the terms “nature of things,” “expanse
\ of reality,”
\ “Emptiness,” “non-elaboration,” “limit of cessation,”
\ “Ultimate,” and so forth, function similarly in different
\ But their difference—in terms of final or partial significance—
\ Is great, so one must explain them in context,
\ Like the word sendhapa.
\ Thus, when the seven profound questions
\ Were explained with profound, vast, meaningful words,
\ The questioner said, with great respect:
\ “Alas! Like a frog at the bottom of a well,
\ Having not seen the depths
\ Of the Dharma ocean of other textual traditions,
\ And having tasted only the flavor of the well
\ Of our own arrogant view, our pride is crushed
\ By these words of yours!
\ In the great ocean of sublime spirituality,
\ The ecstatic dance of Manjusri,
\ Known as “Rong zom” and “Klong chen pa,”
\ Is an ocean of the sublime enlightened mind,
\ Which possesses many and sundry bejeweled Dharma
\ Those who abandon them and hanker after
\ The trinkets of other systems are surely deceived!
\ Those who have the discerning intellect
\ Born of the analysis of the excellent Dharma (chos bzang)
\ Are never obstructed by demons.
\ As this great lion’s roar of the path of reasoning
\ Is proclaimed, will they not find confidence in
\ This outstanding tradition of the Lake-born’s teaching?
\ Please grant us the opportunity to firmly grasp
\ The handle of wisdom’s sword, which cannot be stolen away
\ By the refutations of arrogant extremism!
\ The profound meaning that is found in the
\ Nectar ocean of Dharma learning
\ Is like a jewel that should be taken, wherever it
\ happens to be;
\ One should not just follow the external behavior of
\ another person.
\ It’s not enough to receive a lot of teachings and talk
\ about them,
\ For though one seems talented and well trained, one’s
\ cannot get this profound point, like a buried treasure.
\ But whoever does get it should be known as a spiritual
\ As if it were a jewel-encrusted vessel
\ For a hundred thousand spiritual treasures,
\ My mind realized that it was time
\ To accept the beneficence of instructions
\ Accomplished in the great ocean of profundity and vastness,
\ And I joyfully drank the ocean of the glorious
\ King of Nagas.
\ Having definitely realized the vast extent of the
\ analytical mind
\ By the river of eloquent explanations that descend from him,
\ One should realize that the source of these explanations
\ Is the oral tradition of the vidyadhara lineage,
\ Which is like the Lord of Nagas himself.
\ Please brighten the lamp of the amazing Dharma,
\ Which causes the mind to acquire great strength
\ By receiving the springtime nectar that benefits the heart,
\ The quintessence that is imbibed
\ All at once from the limits of space!”
\ When he had shown his respect with these words,
\ The sage advised him again,
\ Condensing the meaning of what he said before,
\ Which converts a shallow mind to a deep one:
\ “The lion’s milk of the supreme Dharma
\ Is only contained by the vessel of a sound mind.
\ Though others may try, it won’t stay in place.
\ A vessel that can hold it is like this:
\ A is the door of unborn dharmas;
\ Ra is the door free of particles;
\ Pa is the door of the appearance of the ultimate;
\ Tsa is the absence of death, transmigration, and birth;
\ Na is the absence of names;
\ Dhih is the door to profound intelligence.
\ If one focuses on all of these six doors
\ In the manner of the two truths
\ And accomplishes the samadhi of illusion,
\ With one gulp, one will be able to stomach
\ The water of the great infinite ocean of phenomena,
\ And in the stainless gem of one’s heart,
\ The dharani of spiritual brilliance will blaze with glory.
\ By the path of certainty that eliminates
\ The elaborations of four extremes,
\ May we abide in the expanse of fundamental luminosity
\ Beyond mind that reaches the original state,
\ The state of the Great Perfection Manjusri.
\ Having seen the real meaning of remaining in the
\ equanimity of
\ The vast expanse of the regal view without extremes,
\ All the darkness of the crude mind of the four extremes
\ Will naturally disappear as the sun of luminosity rises.”
\ Thus, the questions asked by that wanderer
\ Were explained in the number corresponding to
\ The [seven] accoutrements of royalty.
\ Thus, a feeble-minded intellectual like myself
\ Has received this extremely profound and abstruse meaning
\ From the heart of sublime great-minded beings
\ And presented it here.
\ This elegant explanation like a shower of Dharma
\ Is the path trodden by millions of bodhisattvas;
\ By listening joyfully, hoping to attain the great goal,
\ And by inquiring, the joyous opportunity for blessing has
\ Therefore, I have considered these profound
\ And vast subjects again and again,
\ And just as they arose in the face of the mind’s mirror,
\ The Dhih-named one arranged them playfully.
\ The profound way of the Buddhadharma, like the limit
\ of space,
\ Cannot be put into words entirely,
\ But if you rely on this Beacon of Certainty,
\ You can discover the amazing path of the supreme vehicle.
|
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http://csdt.rpi.edu/na/pnwb/symmetry2.html
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math
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Symmetry in Pacific Northwest Baskets
What is symmetry?
Many animals have bodies in which the right half is the mirror image of the left half. That is called "reflection symmetry." Some things have reflection symmetry between top and bottom. Footballs and soup cans, for example, have one shape that is the same for top and bottom, and a different shape that is the same for left and right.
Many Pacific Northwest basket designs display various forms of symmetry. The decorative design on the basket on the right displays reflection symmetry.
(Image scanned from Paul, Frances. 1944. Spruce Root Basketry of the Alaska Tlingit. Lawrence, Kansas: Haskell Press.)back | continue
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http://swassignmentdrse.fieldbee.us/mathmetical-economics.html
|
math
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Mathematical methods in economics by schofield, norman and a great selection of similar used, new and collectible books available now at abebookscom. The mathematical economics major is intended for students with a strong intellectual interest in both mathematics and economics and, in particular, for students who. Applied mathematical economics major the major consists of 42 credit hours divided according to the following: 18 credit hours in economics courses and 15 credit. R introduction to mathematical economics homework for econ 4808/5808 summer 1993, frank hsiao exercise 32 p40 1, 3, 4, 5 33 p45 la, 3ad, 5a. Bachelor of arts in economics - option in mathematical economics and economic theory (120 units) requirements lower division: complete each of the following courses. Math preparation for graduate school although economics graduate programs have varying admissions requirements, graduate training in economics is highly mathematical. The most formal and abstract of the economic models are the purely mathematical models.
I thank hans maks and peter senn for their helpful suggestions that i have incorporated into this paper – “there are more things in heaven and earth, horatio. 2 econ 4808 introduction to mathematical economics edward morey fall 2003 (august 22, 2003) course description: econ 4808 is a course that will improve your. To receive a bachelor of science degree in joint mathematics/economics, no more than three upper-division courses taken externally from uc san diego can be counted. Graduate-level text provides complete and rigorous expositions of economic models analyzed primarily from the point of view of their mathematical properties, followed.
Positivism is an interpretation of the methodology of physics ballooned into a general theory of knowledge for all fields such a course in economics would be likely. Must-have math tools for graduate study in economics 131 mathematical expectation talking about economics using the shorthand of mathematics. Undergraduate program majors majoring in economics at mit the undergraduate curriculum in economics is designed to: - provide a firm grounding in modern economic. Learn more about the department of economics, including course offerings, major and minor requirements, and requirements for honors.
I’ve been getting some comments from people who think my magazine piece was an attack on the use of mathematics in economics it wasn’t math in. Displaying 1 of 1 buying options we’re committed to providing you with high-value course solutions backed by great service and a team that cares about your success. Much of the study of economics requires an understanding of mathematical and statistical methods, so what exactly is mathematical economics discover it here. Academiaedu is a platform for academics to share research papers.
Read the latest articles of journal of mathematical economics at sciencedirectcom, elsevier’s leading platform of peer-reviewed scholarly literature.
Chiang's fourth edition provides readers with the mathematical concepts and knowledge necessary to succeed in upper-level and graduate economics courses. Mathematical economics also available in co-op sharpen your mathematics skills and solve global economic problems mathematical economics is pivotal to. Math underpins much of our current economic theory, and most branches of economics use math, statistics, and computational concepts extensively. This text offers a comprehensive presentation of the mathematics required to tackle problems in economic analyses to give a better understanding of the mathematical. I have to disagree with vilnis krumins while mathematical economics is on some level just a mathematical toolbox for economists, there are a couple good reasons why. Mathematical economics mathematical economics is the application of mathematical methods to represent theories and analyze theoretical problems in economics by. Conference on econometrics and mathematical economics (ceme) since 1970, the conference on econometrics and mathematical economics (ceme) has received support from.
This major helps students understand the mathematical basis of economics the program prepares students for careers as economic analysts and for rigorous graduate work. The economics and mathematics major is intended for students with a strong interest in both mathematics and economics and for students who may pursue a graduate. Mathematical economics is a form of economics that relies on quantitative methods to describe economic phenomena, it has come to be known as econometrics. Where can mathematical economics take me a compilation of competence: the mathematical economics degree the mathematical economics major offers students a degree.
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http://fitness.stackexchange.com/questions/tagged/ketogenic-diet+muscle-gain
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math
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Physical Fitness Meta
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https://www.hindawi.com/journals/abb/2020/8894399/
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math
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Analysis of Stiffness and Energy Consumption of Nonlinear Elastic Joint Legged Robot
In order to reduce the energy consumption of the legged robot in walking, this paper designs a kind of nonlinear elastic joint from the flexible variable-stiffness joint based on the mammal walking on the limb and optimizes the leg structure of the legged robot. The motor is rigidly connected to the articulated lever. When the lever is accelerated or decelerated, the elastic unit is introduced. The system can be considered as a special variable-rate elastic system. This paper will study it from theory and simulation experiments. Based on the dynamic analysis, a functional relationship between the output torque and the torsion spring stiffness and between the energy consumption and the torsion spring stiffness was established. By finding the extremum, the two optimum torsional spring stiffness that can minimize the required output average torque and the energy consumed during one cycle of motion were deduced. The results show that using this design in a reasonable position can effectively reduce the energy consumption of the system and can achieve up to a 50% reduction in energy consumption.
Compared with the wheeled robot, the foot robot has great advantages in adaptability to the external complex environment and has great research value . With the increase of the number of feet, the structure of the foot robot is more redundant, but it improves the fault-tolerant ability of the robot: the damage of one leg will not have a great impact on the robot, it can move smoothly with the cooperation of other legs. However, the foot robot needs to coordinate its limbs, and its motion efficiency is relatively low, and its energy loss is relatively large. In the process of movement, the friction loss between the body and the ground as well as between the mechanisms will also be dissipated by the heat generated by the driving winding due to the need to maintain a fixed joint angle in the process of standing still. Therefore, for the foot robot, how to reduce energy consumption to improve energy utilization is particularly important, which will bring great performance improvement to the foot robot.
In recent years, the research on energy-saving optimization of foot robot can be classified into the following four categories: (1) study the energy conversion of bionic joint by analyzing the physiological structure of leg joint of foot animal ; (2) optimize the end trajectory curve under different conditions ; (3) reasonable gait planning and optimal gait generation parameters ; and (4) how to reasonably distribute the foot force . A large number of experiments have shown that human beings and some creatures use their own physiological structure to complete energy transfer and transformation when they are in motion, which is a highly coupled function brought by evolution and the body . For example, in the movement system composed of human bones, muscles, and tendons, the flexible components of muscles and tendons produce the effect similar to that of springs, which are regularly shaped in motion. Variable can reduce energy consumption and overall impact of the system, which can be approximately considered as a load spring inverted pendulum slip model [7, 8]. Inspired by this idea, this paper introduces the spring design idea to the joint motion of the foot robot for energy conversion. Nowadays, many scholars have studied the elastic joint of robot and discussed the structure and control method of the elastic joint, but most of them focus on the performance research of damping and reducing the impact force ; this paper will study the energy consumption brought by it.
However, the flexibility of the robot joint has a significant impact on the positioning accuracy of the legs and the accuracy of the end of the trajectory when the robot moves , and the degree of freedom of its control system will increase by one level, resulting in its control more complex and easy to lose stability. Therefore, a large number of scholars have studied the impact on the control model after introducing the flexibility of the joint. Nowadays, there are many significant achievements in system modeling and dynamic modeling of robot with elastic joints. However, for the foot robot with high redundancy, such as the six-legged robot, the lack of accuracy of the foot end trajectory of one leg is not enough to cause the stability imbalance of the whole robot, but the resulting energy consumption is reduced, which can greatly improve the endurance of the foot robot. Secondly, most of the current research uses linear elastic elements and will participate in the whole movement. In this paper, a joint motor is used to drive the joint directly, and a torsion spring is installed at the joint which is in contact with the bar. By improving some elastic joints, a nonlinear elastic system is formed. Only when the joint motor decelerates to the stop and the initial speed is zero to the acceleration, the elastic joint will start to play an obvious role. In this way, it can help the joint motor decelerate in the deceleration process and accelerate in the acceleration process, so as to achieve an energy conversion function and reduce energy loss. In the following chapters, we will start with a simplified model of nonlinear elastic connections. Through dynamic analysis, the relationship between motor output torque and spring stiffness coefficient is found. Then, through the analysis of energy consumption, the value of joint stiffness coefficient is found. After that, we will design some simulation experiments to verify the effect of nonlinear elastic connection on the system. Finally, the relevant conclusions are summarized.
2. Dynamic Modeling and Analysis
The single-leg structure of the legged robot model and the prototype are shown in Figure 1. The robot can switch arbitrarily between a hexapod robot and a spherical robot. Each leg has three degrees of freedom and three joint motors. This paper focuses on the second joint, that is to say, only the leg joints are subjected to a simulation experiment with nonlinear elastic joints. For each revolute joint, a similar dynamic model will be generated when the torsion spring of the elastic joint is assembled in a suitable position. They have the same properties and only differ in numerical values.
The single-leg simplified model with nonlinear elastic joints is shown in Figure 2. Since this paper only does theoretical analysis and simulation, in order to simplify the model, the ideal position is taken during assembly without actual assembly. Some elastic joints are horizontal and symmetrical with the axis of the revolute joint.
Through dynamic modeling, the relationship between joint torque and joint motion parameters of a nonlinear elastic joint robot can be deduced. At the same time, the relationship between energy consumption and joint movement can also be deduced. This paper uses the Lagrangian method to derive the dynamic equations of the elastic joint model. The elastic part of the torsion spring has no moment of inertia and is elastically deformed only in one direction, ignoring the damping of the elastic unit.
Assume that the initial momentum of the system is zero. In the theoretical derivation and simulation experiments, this paper concludes that the trajectory is accurately tracked, and there is no error such as offset and phase difference. Figure 3 shows the simplified nonlinear elastic joint mechanism. The parameters of the mechanism diagram are shown in Table 1; the torsion spring that mainly acts in the elastic joint is defined as the main torsion spring , and its stiffness is far greater than that of the secondary torsion spring that only reduces the impact, that is, is much larger than and , and the barycentric coordinate of the lever is set as .
The joint space is divided into two parts: an elastic joint with a very small stiffness and an elastic joint with a large stiffness. In order to simplify the equation, it is necessary to make the following assumption. In the collision process, the collision is stable and the time is not counted. The collision is surface contact. The position and direction of the lever and the contact surface do not change during the collision. The inertia of each component does not change, and the deformation does not occur during the collision. Lagrangian equation is still established during collision .
As shown in Figure 3, we separately model the dynamics of different stiffness. In an ideal state, assuming that the rigid joint driving torque is , the torque of the torsion spring and the joint rotation angle in the elastic system are, respectively, the functions of the relationship and . When the lever has not yet reached the torsion spring with the large stiffness, the required torque for the joint can be given by
The required torque after the lever comes into contact with the large stiffness torsion spring can be given by
After the trajectory curve is determined, the joint output torque curve in each period is also determined, that is, the relationship between and time is a determined curve. The sum of the output torque of the joint and the torque generated by the elastic joint is a certain value at a certain point of time. At this time, the greater the torque generated by the elastic joint, the smaller the required output torque of the joint.
However, in order to make the dynamic model more accurate, the two parts of the joint space are now integrated into a dynamic analysis. The D-H method is used to establish the coordinate system of the joint. Only the second joint is taken as the research object.
The kinetic equation of the robot is represented as follows: where can be given by
Equation (4) can be rewritten as
Gravity energy can be given by
There is almost no elastic potential energy in the system before the torsion spring with a large stiffness is touched, and the stiffness coefficient can be regarded as linear after the main torsion spring is contacted. So it can be considered as a special nonlinear variable stiffness elastic model in one cycle. The elastic potential energy can be given by where is the angle conversion factor when the torsion spring and the rotation center do not coincide with each other, and keeps the value of the rotation angle of the torsion spring and the rotation angle of the joint united. is the rotation angle of the torsion spring with respect to the joint and can be approximated as the joint rotation angle , because the lever and the motor are rigidly connected. is a function of the nonlinear torsion spring for the joint rotation, which can be expressed in this paper by a piecewise function:
The total potential energy can be given by
The Lagrangian operator can be given by
The Lagrangian kinetic equation can be given by
According to the Lagrange equation, the joint torque can be obtained as
Assume that the joint output rotation function is , then the output torque after contact with the elastic joint can be given by
Equation (14) has the same conclusion as equations (1) and (2). Equation (14) shows that before the end of the stance phase and the swing phase, the robot needs a joint motor to decelerate. With the participation of the elastic joint, due to the compression of the torsion spring, the torsion spring will produce an increasingly larger countertorque. It can effectively reduce the output torque of the joint motor under the same combined torque. At the beginning of a new swing phase or stance phase, the joint motor output torque is required to provide sufficient acceleration to the joint. In the model with elastic joints, the torsion spring is compressed when the previous movement decelerates. So the torsion spring and the joint motor collectively output torque in the acceleration phase. When the required torque is the same, the output torque of the joint motor can be greatly reduced. Therefore, theoretically, the design of the elastic joint can reduce the output peak torque of the joint motor, which can effectively reduce the energy consumption of the robot during walking.
In order to obtain the best stiffness function of the torsion spring with the purpose of minimizing the output torque, differentiate equation (14) of time and let its value be zero:
can be obtained from equation (15). If , then (/rad) is the function of the torsion spring stiffness that minimizes the output torque under this input function.
For the robot’s trajectory design, many scholars have done a lot of researches [15, 16], such as composite cycloids, modified elliptical trajectories, and classic multi-item fitting curves. For the joint tip trajectory planning with continuous smooth acceleration and the appropriate initial conditions for position and velocity, a simpler sinusoidal trajectory will be used, i.e.,
Equation (14) can be rewritten as
Let equation (18) be equal to zero and get the result as follows:
If , then is the optimal stiffness that will make the output torque the smallest torsion spring.
For different trajectories, the specific positions of the main torsion springs for joint contact need different analysis, and it is necessary to ensure that the torsion spring contacts the deceleration when the joint driving torque is greatly reduced, and a symmetrical acceleration torque is required. When using different driving torque curves for acceleration and deceleration, the situation will become more complicated and changeable.
3. Energy Consumption Analysis
Without considering other factors such as friction loss, the energy consumed by a single joint model with a nonlinear elastic joint in one exercise cycle can be expressed as where is the joint drive power, is the joint drive torque, is the joint rotation angular velocity, and is the joint rotation angle. From equation (20), it can be seen that the energy consumption is related to the integral of the output torque and the angular velocity at this moment, and the output torque and the joint angular velocity are both functions of time. Equations (16) and (17) can be expressed as
If we want to obtain a stiffness that minimizes , then
Let to get . If , then is the demand.
Assume that the energy consumption ratio is an evaluation standard for energy consumption . This assessment of energy consumption is based on the metabolic effects of the organism [18, 19]. The energy consumption ratio means that the unit of weight requires a unit of energy to dissipate the distance. The smaller the value is, the higher the energy consumption can be obtained. where represents the total mass of the robot and is the standard of the robot’s moving distance.
The rotational movement of the joint is the main movement, so make some changes to equation (23) to conform to the simulation movement in this paper. Let the moving standard and the total weight be replaced with the moment of inertia about the rotation axis, equation (23) can be rewritten as
4. Simulation Experiments
4.1. Experiment Environment
Using Adams software for simulation experiments, the experimental model was modeled in Creo software and imported into Adams, as shown in Figure 4. After removing the components that are not related to the movement, add the parameters of friction and force of the restraint and the motion parameter. Table 2 is the setting of the frictional force parameter of the joint rotation; Table 3 lists the parameter setting of the contact force between the lever and the torsion spring; Table 4 is the setting of the secondary torsion spring physical parameters; Table 5 lists the parameters of the main torsion spring physical. The joint adopts a general point driving method. After inputting the endpoint trajectory, the required torque value of the joint is sampled. In order to keep the motor from no-load spinning, the end of the joint is loaded with an equivalent mass load. The trajectory function is shown in where is time.
The model with a nonlinear elastic joint was set as the experimental group, and the model without the elastic joint was the control group. The physical dimensions and simulation parameters of the two models were basically the same, as shown in Table 6.
4.2. Simulation Experiment Results
The simulation experiments found that under the same quality and external conditions, the output torque required by the experimental group model compared to the control group model was greatly reduced. The simulation results of the required output torque at the joint of the control group model are shown in Figure 5, and the simulation results of the required output torque at the joint of the experimental group are shown in Figure 6.
We changed the stiffness of the torsion spring and carried out the simulation experiment, as shown in Figure 7, which were the required output torque of the joint when the stiffness of the torsion spring was different.
It can be clearly seen from Figure 6 that due to the limitation of the function in equation (13), the joint torque is a normal output before contacting the main torsion spring. In Figures 5 and 6, there is a coincident torque output value under different stiffness. Figure 7 show the torsion spring has a stiffness ranging from to . The required output torque of the joint first slowly decreases to a certain value and then begins to rebound. This rebound phenomenon is most obvious when the joint gets close to the limit of rotation. The torsion spring in this case has an optimum stiffness that minimizes the required output torque. The trajectory parameters and the model physical parameters are brought into equation (19), where and , resulting in an optimum stiffness .
The above and nearby values were used as the stiffness of the simulation model. The results () are shown in Figure 8. The experimental results showed that the simulation experiment was not exactly the same as the theoretical value and there was a certain error.
After several simulations around , the simulation results showed that the required output torque was slightly compared with other stiffness at the same time around . The following numerical calculation will verify its correctness.
Matlab was used to process the discrete torque derived output torque from the Adams. After using the trapez function to integrate its data over a period, and then dividing it by the total time in one cycle, the average output torque under different stiffness per unit time was obtained. As shown in Figure 9, the simulation result around the theoretical derivation is shown in Figure 10.
Figure 10 shows that the average moment around is the lowest, and the maximum error between the simulated value and the theoretical value is about 3.7%.
The trajectory curve was brought into equation (21). So that it was integrated over the interval to , resulting in the torsional spring stiffness that optimized the energy. The simulation results of the required joint output torque under the stiffness around are shown in Figure 11.
Then, Matlab is used to multiply the derived angular velocity and the output torque discrete data at the sampling point, and the trapez function is used to integrate the data over a period of time. After bringing its value into equation (24), it got the energy ratio at different stiffness, as shown in Figure 12. The simulation result around the theoretical derivation value is shown in Figure 13.
Figure 13 shows that the average moment around is the lowest, and the maximum error between the simulated value and the theoretical value is about 1.95%.
According to the data of simulation experiments, the difference between the energy ratio in the case of nonlinear elastic joints and the energy ratio in the case of the optimal energy-saving torsion spring stiffness was first calculated. Then, calculate the ratio of the difference and the energy consumption when no elastic joint was added. It can be concluded that energy consumption of up to 52.81% can be reduced in the case of an optimal energy-saving stiffness.
4.3. Discussion of the Experimental Results
Chen et al. use a torsional spring to participate in the movement of the quadrupedal robot joint, assisting the hydraulic drive to significantly reduce the hydraulic output during walking, thereby reducing the energy consumption. Zhou and Fu propose a method based on biomechanics and bionic control strategy for the compliant quadruped robots. It mainly includes mechanical compliance element and control compliance element that can reduce the contact force both on hip and knee joints. Finally, the experiment results proved to effectively optimize joint torque and prevent damage to the robot. Ma et al. , inspired by small jumping animals, designed a leapfrog robot named “Grillo.” Its passive forelimbs can cushion the landing impact and can be converted into elastic potential energy and released during jumping to increase the peak power output of the motor. The above papers and the theoretical analysis of this paper and the simulation experiments indicate that adding elastic elements at certain positions of the robot joint can indeed reduce the energy consumption of the robot during walking, but may cause instability of the output torque. It may cause the decline of the stability and precision of endpoint of the robot, so it is necessary to weigh the advantages and disadvantages brought about before the introduction of the elastic joint.
(1)This paper designs a nonlinear elastic joint and deduces it theoretically. It is proved that this design can effectively reduce the energy consumption of the legged robot when walking. The simplification of the model may bring about errors between the theoretical analysis and the simulation experiment. Even if there is an error, it can reduce about 50% of the energy consumption compared with the full-rigid joint robot without considering external friction and gravity(2)This design will reduce the order of the robot’s control equation to a certain extent and increase the difficulty of control. However, for a more redundant robot, the reduction in energy consumption may increase the overall performance of the robot. This paper will be validated in later work. However, the position of the elastic joint in the overall mechanism is not flexible enough. If the joint rotation trajectory curve does not coincide with it, it may cause a reaction. That is to say, when the output torque of the joint motor accelerates the joint, the joint mechanism may come into contact with the torsion spring in advance, which causes the joint motor to drive to provide extra torque. And in the real world, there are constants of the acceleration of gravity. The upper and lower elastic joints cannot be assembled symmetrically. It is even more difficult to fit the trajectory to the elastic joints(3)The future work of this paper is to give the appropriate elastic joint assembly position and the optimal stiffness for the more general end trajectory and try to design a universal adjustable nonlinear elastic joint
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Conflicts of Interest
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article. This work was supported in part by the National Natural Science Foundation of China (Grant No. 51205075) and the Fundamental Research Funds for the Central Universities (Grant No. HEUCF180703).
A. Schneider, J. Schmitz, H. Cruse, and Ieee, “A bio-inspired joint controller for the decentral control of a closed kinematic chain consisting of elastic joints,” in Proceedings of the 44th IEEE Conference on Decision and Control, pp. 233–238, Seville, Spain, December 2005.View at: Publisher Site | Google Scholar
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G. R. Colborne, S. Naumann, P. E. Longmuir, and D. Berbrayer, “Analysis of mechanical and metabolic factors in the gait of congenital below knee amputees. A comparison of the SACH and Seattle feet,” American Journal of Physical Medicine & Rehabilitation, vol. 71, no. 5, pp. 272–278, 1992.View at: Publisher Site | Google Scholar
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The Design Of Fiscal Rules /1,2 Davide Lombardo International Monetary Fund Turkish Banks’ Association Seminar February 26, 2009 1/ The views expressed in this presentation are the author’s, and not necessarily the IMF’s. 2/ Based on the December 1-2 2008 workshop organized by the Treasury, SPO, and MOF together with the Fiscal Affairs Department of the IMF. In particular it borrows on the presentations by Debreu and Debreu and Kumar (available on the website of the Turkish Treasury—see press release dated 18 Dec 2008). Outline First principles: (Definition, Rationale, Function) Taxonomy Design trade-offs Basic properties of selected rules Monitoring and enforcement Fiscal rules around the world: EU sample Broader sample Lessons from stylized facts Conclusions Definition Mechanism placing durable constraints on fiscal discretion through numerical limits on budgetary aggregates (expenditure, revenue, budget balance and/or public debt). Any fiscal policy rule is made of 3 parts: Numerical target/ceiling on one or more specific fiscal indicators Explicit cost for non-complying policymakers A monitoring/enforcement procedure. What is not a fiscal policy rule? The annual budget is not a rule, as it only applies for one year. An IMF-supported program, as it is not expected to be a permanent feature of fiscal policy. The new UK ―rule‖: “To set policies to improve the cyclically-adjusted current budget each year, once the economy emerges from the downturn, so it reaches balance and debt is falling as a proportion of GDP once the global shocks have worked their way through the economy in full.” Rationale: why constraining discretion? Unconstrained fiscal policy may be perceived as systematically deviating from desirable policies. In practice: pro-cyclical and/or unsustainable policies. Reasons why unconstrained policies can be biased: Time-inconsistency Fiscal federalism/monetary union Political economy: Myopia, re-election concerns, fiscal illusion, distributive conflicts, and coordination failures. Other disciplining mechanisms? Markets: yes, but unreliable (comes too late and to strongly) Delegation to non-partisan technical agencies: poses considerable issues in terms of democratic accountability. Functions of fiscal rules Rules are commitment devices: they make deviations from socially desirable targets too costly for policymakers. Rules are signaling tools: they can help policymakers signal their genuine commitment to sustainable and stabilizing policies. Rules can also serve to anchor expectations, thus reducing uncertainties and risk premia. Principles of a taxonomy Many parameters enter the design of an actual fiscal policy rule: fiscal target(s) nature of costs in case of deviation monitoring/enforcement escape clauses, etc. A good rule must imply good policies in most (if not all) circumstances the policy response induced by the rule to a variety of shocks is therefore key. Key feature of the rules: response to shocks to discretion make sense, but they Limits should NOT: Prevent adequate responses (e.g., let automatic stabilizers play in bad times) Force inadequate responses (e.g., force a fiscal contraction in response to a temporary spike in interest rate, or depreciation of the exchange rate) Let the bias unchecked in specific circumstances (e.g., allow for procyclical expansions). The design of fiscal rules There is no one-size-fits-all fiscal policy rule. Much depends on: Constellation of shocks prevalent in the economy Nature and magnitude of policy bias under discretion. A good rule is (Kopits and Symansky, 1998): …simple …transparent …coherent with the final goal …but mindful of other goals of public policy: • Not discouraging structural reforms • Allowing for fiscal stabilization (time-frame, cyclical adjustment) • Avoiding low-quality adjustments (undue tax hikes, cuts in quality/priority spending). Two key trade-offs Credibility-flexibility: allowing for greater responsiveness to shocks could undermine credibility of attaining the final goal. Flexibility-simplicity: combinations of rules can relax somewhat the credibility-flexibility trade-off, at the cost of simplicity and transparency. The devil is in the details Other design issues Coverage of the government sector Policy coordination in federal/decentralized systems Legal foundations Need for adequate budgetary procedures (preparation, execution, and ex-post auditing of budgets) Need to limit scope for creative accounting Pros and cons of selected rules Simple rules: Debt rules Budget balance rules Expenditure rules Revenue rules Combined rules: e.g., debt objectives coupled with binding deficit ceilings, debt/budget balance objectives coupled with binding medium-term expenditure ceilings Deficit Rules Balanced budget and overall deficit limits Pros: pin down asymptotic properties of debt directly address the deficit bias can be simple and transparent (unless cyclical considerations, escape clauses, etc). Cons: procyclical (unless cyclically adjusted or ―over the cycle‖) could force cut in investment… • golden rule, but these open door to manipulations, and do not guarantee debt sustainability. Overall vs. primary balance? Primary is more robust to volatility in interest payments. Debt Rules Upper limit (or desirable time path) for gross or net public debt Pros: Directly tackle debt sustainability Can be transparent and simple Can accommodate large shocks if debt is well below the ceiling Cons: Lack controllability Can force procyclical and undesirable responses to interest rate and exchange rate shocks if debt is close to the limit Borrowing constraints generally applied at regional and local levels Expenditure Rules Set expenditure ceilings (nominal or in terms of GDP) or limits on growth (nominal or real). Pros: tackle one of main source of deficit bias translate directly into budget preparation minimizes pro-cyclicality risks fosters medium-term planning Cons: May leave room for discretionary tax cuts sustainability? complex to design (nominal vs. real, exclusions,…) Best used in combination with deficit or debt rules. [Revenue Rules] Rules imposing limits on revenues with a view to: Contain size of the public sector / tax burden Allocate ex-ante revenue windfalls (e.g., due to surprisingly high growth) Pros: Can reduce procyclicality in good times Cons: Limited impact on deficit bias if not coupled with other rules Can be procyclical in case the rule targets a given revenue-to- GDP ratio (due to the progresssivity of the tax system) Combination of rules Drawbacks with individual rules have led most countries to adopt combination of rules Most common combinations: Debt and overall deficit ceiling and overall balance target (e.g., Maastricht) Overall balance and expenditure ceilings (Sweden, Finland, Netherlands, etc). Monitoring Ex-ante auditing: Is the government proposed fiscal policy consistent with the rule and its objectives? Ex-post assessment: Was the rule implemented? • Analysis of underlying policies • Certification role: track attempts to resort to creative accounting • Assessment of ex-post adjustments to the target (CABs,…), activation of escape clauses, etc. A key pre-requisite Budgetary transparency: Fiscal data need to be accessible, timely, and reliable Comprehensive periodic reporting requirements Clarity about the budget preparation and execution procedures Clarity about roles and responsibilities of different levels of government Fiscal councils Independent institutions with country-specific mandates adequate and highly qualified staff guaranteed multi-year budget. Can contribute to greater transparency and accountability of fiscal policy. Can help monitoring and enforcing a rule. The case of Chile Fiscal institutional setup designed to buttress fiscal sustainability and dampen cyclical fluctuations. Rule: maintain a structural surplus of 1 percent of GDP for the central government. Fiscal expenditures follow the dynamics of structural revenue. Two independent expert panels provide key projections: 1. the inputs (growth of the labor force, real investment, and total labor productivity) for estimating trend GDP 2. ten-year forecasts of the price of copper. Independent panels enhance policy credibility, while allowing some policy flexibility. The case of the Netherlands The Central Planning Bureau plays a key role in the budgetary process: Provides projections and forecasts Estimates desired structural budget balance Vets the programs of all political parties (which are thus subject to reputational sanctions) Undertakes analysis of specific budgetary projects. High credibility borne out of tradition Fiscal Rules: International Experience Stylizedfacts for EU (EC database: annual data over 1990-2005), and the rest of the world (IMF-FAD database on the design and implementation of fiscal rules) Worldwide: 81 countries identified as having fiscal policy rules, with complete information for 77 of them Growing appetite for fiscal rules (EU) 0.90 Maastricht SGP EMU 0.80 0.70 0.60 Raw index 0.50 0.40 0.30 0.20 EU-15 Euro-11 0.10 NMS Big-4 0.00 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 Emerging Market and LICs are catching up with industrial countries Number of fiscal rules by category of countries: 1990-2008 80 Industrial EU-27 70 Emerging 60 LIC's 50 40 30 20 10 0 1990 1995 2000 2008 Budget-balance and debt rules dominate, but expenditure rules are increasingly popular Number of countries with at least one fiscal policy rule (by type of rule) 80 Budget balance 70 Debt Expenditure 60 Revenue 50 40 30 20 10 0 1990 1995 2000 2008 Expenditure and revenue rules are thus relative newcomers Median duration of existing fiscal rules (in years) 10 9 8 7 6 5 4 3 2 1 0 Budget balance Debt Expenditure Revenue Same trends in EU sample 60 Revenue Rules 50 Expenditure rules Debt Rules Budget balance rules 40 30 20 10 0 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 Budget balance rules appear stronger and have wider coverage Selected features of rules-based fiscal frameworks by type of rule (common features only) Budget balance 1.00 Debt 0.90 Revenue 0.80 Expenditure 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 Independent Independent Coverage (median Statutory basis (median enforcement (relative monitoring (relative relative to maximum relative to maximum frequency) frequency) possible score) possible score) This is reflected in synthetic measures of strength and coverage Indices of strength (max= 6) and of coverage (max=4) by type of rule 3 4.0 Strength Coverage 3.5 2.5 3.0 2 2.5 1.5 2.0 1.5 1 1.0 0.5 0.5 0 0.0 Budget balance Revenue Debt Expenditure Strength and coverage relatively similar across countries Indices of strength (max= 6) and of coverage (max=4) by country category 3.0 4.0 Strength Coverage 3.5 2.5 3.0 2.0 2.5 1.5 2.0 1.5 1.0 1.0 0.5 0.5 0.0 0.0 Industrial Resource rich LICs Emerging markets Features by country groups Selected features of rules-based fiscal frameworks (relative frequencies by country groups) 1.00 Industrial Emerging LIC's Resource-rich 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 Independent M TEF Independent FRL (all rules) Provision(s) Exclusion of Independent enforcement (relative to forecasts (all for fiscal high-quality monitoring (all procedure (all BBR and DR) rules) stabilization spending rules) rules) (relative to (relative to BBR) BBR and ER) Fiscal rules lower interest rates (EU-25) Table 2. Determinants of Long-Term Interest Rates in the EU-25 (1990–2005): Selected Sub-Samples Estimator: Arellano-Bond Fixed effects 1/ No Unstable EU-15 Non-EU15 Euro Non-euro High debt Low debt Delegation Stable gov. delegation gov. Long-term interest rate (lagged) 0.55 *** 0.35 *** -0.11 0.53 *** 0.47 *** 0.31 *** 0.39 *** 0.39 *** 0.56 *** 0.13 (8.77) (4.26) (-1.44) (5.67) (9.98) (6.35) (4.52) (4.58) (6.41) (1.53) Short-term interest rate 0.26 *** 0.58 *** 0.50 *** 0.23 *** 0.26 *** 0.50 *** 0.40 *** 0.23 *** 0.12 0.43 *** (4.23) (16.99) (10.52) (3.07) (11.32) (13.49) (4.29) (2.56) (0.92) (6.93) Cyclically-adjusted primary balance -0.08 ** -0.01 0.00 -0.03 -0.07 * -0.04 -0.07 -0.05 0.03 -0.04 (-2.44) (-0.03) (-0.26) (-0.39) (-1.77) (-0.82) (-1.34) (-0.88) (0.56) (-0.72) Public debt (lagged) 0.01 0.00 -0.01 0.00 0.01 0.00 0.02 ** 0.01 0.00 0.01 (0.73) (0.22) (-0.85) (0.27) (0.46) (0.08) (2.04) (0.61) (-0.29) (0.88) Real GDP growth 0.13 *** -0.01 0.04 ** 0.11 ** 0.26 *** 0.04 0.06 ** 0.09 0.07 0.11 * (3.04) (-0.06) (2.14) (2.38) (4.29) (0.91) (1.93) (1.39) (1.63) (1.80) Inflation 0.22 ** -0.07 * 0.11 ** 0.22 * 0.45 *** -0.04 0.01 0.33 *** 0.26 * 0.09 (2.10) (-1.77) (2.12) (1.92) (4.22) (-1.07) (0.17) (2.70) (1.88) (1.18) Enlargement (dummy) … … … 0.03 … … … -0.61 … … (0.07) (-1.40) Election year (dummy) … … 0.08 ** 0.38 … … 0.26 0.28 0.47 * 0.02 (2.17) (1.40) (1.48) (1.00) (1.76) (0.11) Stability and Growth Pact (dummy) … … … … … … … -0.29 0.13 -0.96 ** (-1.30) (0.56) (-2.13) Government stability … … -0.03 ** -0.17 * -0.21 *** 0.03 -0.18 *** -0.16 0.14 -0.08 (-1.89) (-1.85) (-3.75) (0.24) (-3.69) (-1.61) (0.59) (-1.19) Ideological range of governing coalition … … -0.02 0.13 * … … -0.14 0.11 *** 0.10 0.15 (-1.53) (1.90) (-1.42) (2.71) (1.52) (1.65) Fiscal Rule Index -0.16 0.10 -0.09 * -0.16 -0.17 -0.31 ** -0.10 -0.41 ** -0.06 -0.43 (-1.58) (0.10) (-1.63) (-0.93) (-1.37) (-2.48) (-1.06) (-2.03) (-0.50) (-1.59) Number of observations 213 47 72 155 107 153 72 148 148 104 Number of countries 15 10 12 25 15 22 9 19 24 21 Test for 2nd order autocorrelation (p-value) 2/ 0.93 0.16 0.10 0.38 0.25 0.21 0.55 0.45 … … Sargan test (p-value) 3/ 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 … … Note: The t-statistics are reported in parentheses (with superscripts *, **, and *** denoting statistical significance at the 10, 5 and 1 percent levels respectively). They are robust to cross-sectional heteroskedasticity. All models include a constant and country effects (not reported). The latter are jointly significant at conventional confidence levels in all equations. The enlargement dummy is equal to 1 for new member states, after they joined the EU. The SGP dummy is equal to 1 for euro area member states after 1998. 1/ Arellano-Bond estimation could not be obtained for one of the subsamples (lagged dependent variable highly insignificant and evidence of second-order autocorrelation). The comparison is thus based on fixed-effects estimates. 2/ Arellano-Bond test of the null hypothesis of no autocorrelation of residuals. 3/ Test of the null hypothesis that identifying restrictions are valid. Debrun and Joshi (2008) Fiscal rules lower interest rates (EU-25) Table 3. Impact of the Fiscal Rules on Long-Term Interest Rates: Specific Features of the Rule (EU-25, 1990–2005, Non-Excessive Public Debt) Specific dimensions of fiscal rules as captured by sub-indices: Statutory basis -0.47 ** (-2.07) All key dimensions Independent body monitoring the rule's implementation -0.48 ** (-2.07) matter Effective Independent body contributing to rule's enforcement -0.52 ** rules deliver lower (-2.01) interest rates. Strength of enforcement procedure -0.40 ** (-2.03) Media impact of the rule -0.47 ** (-2.23) Memorandum items: Overall fiscal rule index -0.45 ** (-2.09) Expenditure rule index 0.05 (0.29) Notes: The t-statistics are reported in parentheses (with superscripts *, **, and *** denoting statistical significance at the 10, 5 and 1 percent levels respectively). Given that subindices are only available for the overall fiscal rule index and the expenditure rule index, the results, obtained with model (6) in Table 1, use the overall index but control for the effect of expenditure rules. Debrun and Joshi (2008) Lessons Historical preference for deficit and debt rule But recently emergence of expenditure rules (in combination with BBRs and DRs) What explains the trends for more rules and more expenditure rules? Increasing realization that purely discretionary fiscal policies are inconsistent with sustainability (growing public debt) Mounting pressures from long-term issues: aging, global warming, etc. Regional monetary unions. Conclusions Well-designed fiscal rules can help improve fiscal performance on average… …thus lowering risk-premia and interest rates on long-term government bonds. But: political commitment is key As are adequate public financial management, monitoring and enforcing systems.
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CC-MAIN-2014-42
| 16,546 | 1 |
https://plainmath.net/82567/describe-the-gravitational-forces
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math
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A 730-N man stands in the middle of a frozen pond of radius 5.0 m. He is unable to get to the other side because of a lack of friction between his shoes and the ice. To overcome this difficulty, he throws his 1.2 kg physics textbook horizontally toward the north shore , at a speed of 5.0 m/s. How long does ittake him to reach the south shore?
A 15.0 kg block is dragged over a rough, horizontal surface by a70.0 N force acting at 20.0 degree angle above the horizontal. The block is displaced 5.0 m, and the coefficient of kinetic friction is 0.3. Find the work done on the block by ; a) the 70.0 N force,b) the normal force, and c) the gravitational force. d) what is the increase in the internal energy of the block-surface system due to friction? e) find the total change in the kinetic energy of the block.
An 80.0-kg skydiver jumps out of a balloon at an altitude of1000 m and opens the parachute at an altitude of 200 m. (a) Assuming that the total retarding force on the diver is constant at 50.0 N with the parachute closed and constant at 3600 N with the parachute open, what is the speed of the diver whenhe lands on the ground? (b) Do you think the skydiver will be injured? Explain. (c) At what height should the parachute be opened so that the final speed of the skydiver when he hits the ground is 5.00m/s? (d) How realistic is the assumption that the total retarding force is constant? Explain.
A model airplane of mass 0.750 kg flies in a horizontal circle at the end of a 60.0 m control wire, with a speed of 35.0 m/s. Compute the tension in the wire if it makes a constant angle of 20.0° with the horizontal. The forces exerted on the airplane are the pull of the control wire, the gravitational force, and aerodynamic lift, which acts at 20.0° inward from the vertical.
A student drops two metallic objects into a 120 g steel container holding 150 g of water at 25°C. One object is a240 g cube of copper that is initially at 85°C, and the other is a chunk of aluminum that is initially at 5.0°C. To the surprise of the student, the water reaches a final temperature of 25°C, exactly where it started. What is the mass of the aluminum chunk?
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CC-MAIN-2022-33
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http://eduqna.com/Homework-Help/2520-homework-help-11.html
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math
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1.) How do you factor this expression?
16x^2 - 36 + 9
2. A circular table is to be made that will own a top covered with fabric that costs $3.50 per square foot. The covering is to cost no more than $60. What is the maximum radius for the top of the table?
Answers: 2. area of circle = pi x r ^2 = 60/3.5
divide by pi to factor it out
later square root the result so as to achieve r
i infer this is right??
This article contents is post by this website user, EduQnA.com doesn't promise its accuracy.
More Questions & Answers...
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CC-MAIN-2017-30
| 527 | 9 |
https://www.jiskha.com/display.cgi?id=1346426735
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math
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posted by akshay .
if 2 white balls are drawn a box of 5,find the probability that all are white using bayes theorem
Okay - this is the first time I've ever tried to use Bayes Theorem...
P(A|B) = P(B|A).P(A) / Sum(P(B|Aj).P(Aj)
Let A be the event "All five balls are white".
Let Aj be the event "There are exactly j white balls". Let B be the event "I draw two white balls".
Then P(A|B) = P(All five balls are white, given that I draw two white balls).
P(B|A) = P(I draw two white balls, given that all five balls are white), and this obviously equals 1.
P(B|A0) = P(I draw two white balls, given that no balls are white) = 0.
P(B|A1) = P(I draw two white balls, given that exactly one is white) = 0.
P(B|A2) = P(I draw two white balls, given that exactly two are white) = (2/5)x(1/4) = (1/10).
P(B|A3) = P(I draw two white balls, given that exactly three are white) = (3/5)x(2/4) = (3/10).
P(B|A4) = P(I draw two white balls, given that exactly four are white) = (4/5)x(3/4) = (6/10).
P(B|A5) = P(I draw two white balls, given that all five are white) = 1.
Now, here's the subjective bit - because Bayes Theorem is all about subjectivity. I have no idea a priori how many white balls there are in the bag, so all possibilities are equally likely as far as I'm concerned. That means that P(A0) = P(A1) = P(A2) = P(A3) = P(A4) = P(A5) = (1/6). So in the original equation, all of the P(Aj)s in the numerator and the denominator are going to cancel out.
That leaves P(B|A) in the numerator, and (P(B|A0) + P(B|A1) + P(B|A2) + P(B|A3) + P(B|A4) + P(B|A5)) in the denominator. The numerator = P(B|A) = 1 (we worked it out above). The denominator is the sum of all those other fractions, which is 0 + 0 + (1/10) + (3/10) + (6/10) + 1 = (10/10) + 1 = 2. So the answer is (1/2), i.e. exactly one half.
Do you reckon I've got it right?
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| 1,827 | 17 |
https://e-booksdirectory.com/details.php?ebook=11303
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math
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A First Course in Linear Algebra
by Ken Kuttler
Publisher: Lyryx 2017
Number of pages: 608
The book presents an introduction to the fascinating subject of linear algebra. As the title suggests, this text is designed as a first course in linear algebra for students who have a reasonable understanding of basic algebra. Major topics of linear algebra are presented in detail, with proofs of important theorems provided.
Home page url
Download or read it online for free here:
by Edwin H. Connell
Covers abstract algebra in general, with the focus on linear algebra, intended for students in mathematics, physical sciences, and computer science. The presentation is compact, but still somewhat informal. The proofs of many theorems are omitted.
by W. Keith Nicholson - Lyryx
The aim of the text is to achieve a balance among computational skills, theory, and applications of linear algebra. It is a relatively advanced introduction to the ideas and techniques of linear algebra targeted for science and engineering students.
by Mohammed Kaabar - Arxiv.org
There are five chapters: Systems of Linear Equations, Vector Spaces, Homogeneous Systems, Characteristic Equation of Matrix, and Matrix Dot Product. It has also exercises at the end of each chapter above to let students practice additional problems.
by Sergei Treil
This book covers a first course of linear algebra, it introduces mathematically advanced students to rigorous proof and formal definitions. The author of the text tried to emphasize topics important for analysis, geometry and probability.
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s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571472.69/warc/CC-MAIN-20220811133823-20220811163823-00083.warc.gz
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CC-MAIN-2022-33
| 1,558 | 15 |
https://www.learntocalculate.com/factorize-a-number/
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math
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Factorization or factoring consists of writing a number or another mathematical object as a product of several factors.
We use factorization to perform arithmetic operations. Whereby we group the factors so that it is easier for us to multiply.
We also use factorization is to find the least common multiple and greatest common divisor.
Formula to factorize a number.
Factoring integers is pretty simple. You begin by ;
- Dividing the number starting with two and work your way up.
You should know that any quotient that does not have a remainder means that both the divisor and the quotient are factors of that number.
Suppose you are asked to factorize 20.
Your solution will be as follows, using the factoring tree.
Therefore, the factors of 20 are 2, 2, 5 and 1. This means that if you multiply these numbers you will get 20.
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CC-MAIN-2024-10
| 829 | 10 |
https://www.hackmath.net/en/example/1079
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math
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Calculate deceleration of car (as multiple of gravitational acceleration g = 9.81 m/s2) which occurs when a car in a frontal collision slows down uniformly from a speed 111 km/h to 0 km/h in 1.2 meters trajectory.
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At what horizontal distance reaches the car weight m = 1534 kg speed v = 85 km/h when the car engine develops a tensile force F = 1691 N. (Neglect resistance of the environment.)
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Stone was pushed into the abyss: 2 seconds after we heard hitting the bottom. How deep is the abyss (neglecting air resistance)? (gravitational acceleration g = 9.81 m/s2 and the speed of sound in air v = 343 m/s)
Flywheel turns 450 rev/min (RPM). Determine the magnitude of the normal acceleration of the flywheel point which are at a distance of 10 cm from the rotation axis.
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The train slowed down from 90 km/h to 72 km/h in 5 seconds. How long track travel?
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From Znojmo to Brno started truck with a trailer at an average speed 53 km per hour. Against him 14 minutes later from Brno started car with an average speed of 1.2-times greater than the truck. How long and how far from Znojmo, they meet if the distanc
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CC-MAIN-2018-43
| 3,229 | 30 |
https://www.csauthors.net/yongbin-ge/
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math
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According to our database1, Yongbin Ge authored at least 6 papers between 2010 and 2019.
Legend:Book In proceedings Article PhD thesis Other
A class of high-order compact difference schemes for solving the Burgers' equations.
Applied Mathematics and Computation, 2019
Partial semi-coarsening multigrid method based on the HOC scheme on nonuniform grids for the convection-diffusion problems.
Int. J. Comput. Math., 2017
A transformation-free HOC scheme and multigrid method for solving the 3D Poisson equation on nonuniform grids.
J. Comput. Physics, 2013
Multigrid method based on the transformation-free HOC scheme on nonuniform grids for 2D convection diffusion problems.
J. Comput. Physics, 2011
Multigrid method and fourth-order compact difference discretization scheme with unequal meshsizes for 3D poisson equation.
J. Comput. Physics, 2010
A high order finite difference method with Richardsonextrapolation for 3D convection diffusion equation.
Applied Mathematics and Computation, 2010
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CC-MAIN-2019-47
| 994 | 14 |
https://math.answers.com/Q/What_are_the_first_3_digits_of_pi
|
math
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There are 100230 '3's in the first million digits of pi.
The first 160 digits ( including the 3 ) are... 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128
Because March 14th are the first 3 digits of pi which are 3.14
The first eleven digits of pi are:3.141592653
3.14159265358979323846 are the first 20 digits of pi.
It is March 14th which are the first 3 digits of pi as 3.14
3.14159265358979323846264338327 are the first 30 digits of pi.
The first ten digits of pi are 3.141592653. If rounded to the nearest billionth, that last digit would be 4 instead of a 3.
Because March 14th or 3-14 are the first 3 digits of pi which are 3.14
3.141592653589793238462643383279502884197169399375105820974944 That's 60 digits not including the first 3.
Check out the Joy of Pi link, for the first 10000 digits.
The first quintillion digits of pi cannot, and will not, be posted here on WikiAnswers. That is too much to fit, and no one in the world has even counted to the quintillionth digit of pi yet, but the first 95 digits are 3. 14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211.
The USA because March 14 are the first 3 digits of pi which are 3.14
Because the first 3 digits of pi are 3.14 which is March 14th
To 30 digits, pi equals 3.141592653589793238462643383279.
there are 29 zeros in the first 200 digits of pi.
There are twelve twos in the first 100 digits of pi.
The first 250 digits of pi are: 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091
Not including the initial digit, the first 23 decimal digits of pi are 3.14159265358979323846264.
The first 55 digits of pi after the decimal point are: 3.1415926535897932384626433832795028841971693993751058209
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647 Those are the first 120 digits of Pi
The first digit of pi is 3, the first decimal is 1. Pi ≈ 3.14 The first fifty digits are 3.1415926535897932384626433832795028841971693993751
There are no last 14 digits in pi. The first 14 are 3.14159265358979.
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CC-MAIN-2021-43
| 2,384 | 23 |
https://www.hackmath.net/en/math-problem/301
|
math
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Calculate the edge of cube, if its surface and its volume is numerically equal number.
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CC-MAIN-2019-43
| 1,819 | 36 |
https://podcasts.bcast.fm/e/5nzzyz4n
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math
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#39 Scientific balloon missions
October 7, 2020
José V. Siles is a radio frequency engineer at NASA's Jet Propulsion Laboratory (JPL) in Pasadena, California. His research area is known as plantary science, i.e. the study of the celestial bodies that orbit stars, with a particular focus on our own solar system. José and his colleagues are trying to understand the life cycle of stars, how they form, how they die... because stars may hold the answer to the question: where do we come from?
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CC-MAIN-2024-10
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https://fixedpointtheoryandapplications.springeropen.com/articles/10.1186/1687-1812-2013-306
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math
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- Open Access
A Schauder fixed point theorem in semilinear spaces and applications
© Agarwal et al.; licensee Springer. 2013
- Received: 24 May 2013
- Accepted: 10 October 2013
- Published: 22 November 2013
In this paper we present existence and uniqueness results for a class of fuzzy fractional integral equations. To prove the existence result, we give a variant of the Schauder fixed point theorem in semilinear Banach spaces.
- fuzzy fractional differential equation
- Schauder fixed point theorem
- Ascoli-Arzelá-type theorem
- existence result
The topic of fuzzy differential equations has been extensively developed in recent years as a fundamental tool in the description of uncertain models that arise naturally in the real world. Fuzzy differential equations have become an important branch of differential equations with many applications in modeling real world phenomena in quantum optics, robotics, gravity, artificial intelligence, medicine, engineering and many other fields of science. The fundamental notions and results in the fuzzy differential equations can be found in the monographs and .
The concept of fuzzy fractional differential equations has been recently introduced in some papers [3–10]. In , the authors established the existence and uniqueness of the solution for a class of fuzzy fractional differential equations, where a fuzzy derivative is used in the sense of Seikkala. In , the authors proposed the concept of Riemann-Liouville H-differentiability which is a direct extension of strongly generalized H-differentiability (see Bede and Gal ) to the fractional literature. They derived explicit solutions to fuzzy fractional differential equations under Riemann-Liouville H-differentiability. In , the authors established an existence result for fuzzy fractional integral equations using a compactness-type condition. In this paper, we present an existence result for a class of fuzzy fractional integral equations without a Lipschitz condition. For this we use a variant of the Schauder fixed point theorem. Since the space of continuous fuzzy functions is a semilinear Banach space, we prove a variant of the Schauder fixed point theorem in semilinear Banach spaces.
The paper is organized as follows. Section 2 includes the properties and results which we will use in the rest of the paper. We present an example which shows that a fuzzy fractional differential equation is generally not equivalent to a fuzzy fractional integral equation. In Section 3, we establish the Schauder fixed point theorem in semilinear Banach spaces. In Section 4, we prove an existence result for a class of fuzzy fractional integral equations without a Lipschitz condition. Finally, using Weissinger’s fixed point theorem, we give an existence and uniqueness result.
for all , .
is a complete and separable metric space with respect to the Hausdorff-Pompeiu metric .
y is normal, i.e., there exists such that ;
- (iii)y is a convex fuzzy function, i.e., for all , and for all , we have
y is an upper semi-continuous function.
is called the α-level set of y. Then from (i)-(iv) it follows that the set for all .
for all , and .
where is the Hausdorff-Pompeiu metric. Then is a complete metric space (see ).
Proposition 2.1
for all ,
Define as the space of fuzzy sets with the property that the function is continuous with respect to the Hausdorff-Pompeiu metric on .
Let be an interval. We denote by the space of all continuous fuzzy functions on T.
A subset is said to be compact-supported if there exists a compact set such that for all .
A is level-equicontinuous on if A is level-equicontinuous at α for all .
Theorem 2.2
A is a relatively compact subset of ;
A is level-equicontinuous on .
Remark 2.3
Then is relatively compact in .
A continuous function is said to be compact if and is bounded imply that is relatively compact in .
provided the expression on the right-hand side is defined.
A function is called measurable (see ) if for all closed set , where ℬ denotes the Borel algebra of . A function is called integrably bounded if there exists a function such that for a.e. . If such F has measurable selectors, then they are also integrable and is nonempty.
A fuzzy function is integrably bounded if there exists a function such that for all . A measurable and integrably bounded fuzzy function is said to be integrable on if there exists such that for all .
Lemma 2.4
provided that the equation defines a fuzzy number . It is easy to see that , .
Lemma 2.5
define the α-level intervals of .
which is a fuzzy number for . However, it is not a fuzzy number for . Thus does not satisfy equation (2.1).
In this section, we prove the Schauder fixed point theorem for semilinear Banach spaces. First, we recall the Schauder fixed point theorem.
Theorem 3.1 (, Schauder fixed point theorem)
Let Y be a nonempty, closed, bounded and convex subset of a Banach space X, and suppose that is a compact operator. Then P has at least one fixed point in Y.
for all ,
for all and .
In this case, we can define a norm on S by , where is the zero element in S. If S is a semilinear metric space, then addition and scalar multiplication on S are continuous. If S is a complete metric space, then we say that S is a semilinear Banach space.
for all .
Theorem 3.2
for all and .
Theorem 3.3
Suppose that S is a semilinear metric space. Then the set all equivalence classes G, constructed above, is a metric vector space and j is an isometry.
Now, we are able to prove a variant of the Schauder fixed point theorem in semilinear Banach spaces.
Theorem 3.4 (Schauder fixed point theorem for semilinear spaces)
Let B be a nonempty, closed, bounded and convex subset of a semilinear Banach space S having the cancelation property, and suppose that is a compact operator. Then P has at least one fixed point in B.
Thus is a fixed point of P. □
Remark 3.5 The space of fuzzy sets is a semilinear Banach space S having the cancelation property. Therefore, the Schauder fixed point theorem holds true for fuzzy metric spaces.
where , and is continuous on .
holds for all .
then by Lemma 2.5 is a solution of (4.2), but the converse is not true.
In , the authors showed that the space can be embedded in , the Banach space of continuous real-valued functions defined on , where is the unit ball. In , an Ascoli-Arzelá-type theorem was proved. We use this theorem to establish an existence theorem for fuzzy fractional integral equations. Let be the zero function in .
is an equicontinuous subset of ;
is relatively compact in for each .
so when for all . This implies that is equicontinuous on . Now we show that is relatively compact in and by Theorem 2.2 this is equivalent to proving that is a level-equicontinuous and compact-supported subset of .
which proves that is compact-supported. Thus, T is a compact operator. Hence, by Theorem 3.4, it follows that T has a fixed point in Ω, which is a solution of integral equation (4.1). □
The following Weissinger fixed point theorem will be used to prove an existence and uniqueness result.
Theorem 4.3
for all and for all . Then the operator T has a unique fixed point . Furthermore, for any , the sequence converges to the above fixed point .
for all , where . Then there exists a unique solution to integral equation (4.1).
The series with is a convergent series (see Theorem 4.1 in ). Thus by Theorem 4.3 we deduce the uniqueness of the solution of our integral equation. □
The second author acknowledges the financial support of Higher Education Commission (HEC) of Pakistan.
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This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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| 11,704 | 94 |
http://www.docstoc.com/docs/115347648/The-Thickness-of-Aluminum-Foil-%EF%BF%BD-Density-Lab
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math
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Name: Date: Block:
Density Lab - Honors
The Thickness of Aluminum Foil
The volume of a regular object is found by using the formula V = L x W x H, where L = length, W =
width, and H = height. Imagine that the regular object is a rectangular-shaped piece of foil. Then the formula
might be revised to V = L x W x T, where T = thickness of the foil. Going one step further, the area of the foil
can be expressed as A = L x W, so the original formula for volume can be restated as V = A x T. Since this
experiment involves finding the thickness, it would be better to rearrange the formula again. Dividing both
sides of the equation by A, the new equation is: T = V / A.
The next problem will be to find the volume and area of a piece of aluminum foil. Remember that
density is a property that is expressed as D = M / V, where M = mass, and V = volume. The density of
aluminum is known, and the mass of a piece of aluminum foil can be measured with a balance. The volume of
the aluminum can then be calculated by using the rearranged equation V = M / D.
1) Measure three rectangular pieces of aluminum foil (you may wish to cut them or use the squares that have
been previously prepared).
2) Using a centimeter ruler, carefully measure the longest length and longest width of each piece of foil.
Record the measurements on the data table. How precise can the measurements be, keeping in mind the use of
significant figures? Think carefully before recording the results.
3) Using a balance find the mass of each piece of aluminum foil. Record the mass on the data table. Again, be
careful to be as precise as possible.
4) Clean up, add your data from both labs to the class spreadsheet, and begin calculations.
Length Density Volume Thickness
Foil Piece Width (cm) Area (cm2) Mass (g)
(cm) (g/cm3) (cm3) (cm)
Foil #1 2.6989
Foil #2 2.6989
Foil #3 2.6989
1) Calculate the area for each piece of foil. Show all calculations. Area = L x H.
2) Calculate the volume for each piece of foil. Show all calculations. Volume = Mass/Density.
3) Calculate the thickness of each piece of foil. Show all calculations. Thickness = Volume/Area.
Chemistry: Unit 1 – Matter Page 1 of 2
Name: Date: Block:
Conclusions: (To be completed on a separate piece of paper and attached)
Accuracy is the closeness of an experimental value to an accepted value. For example, the accepted
value for the length of a football field is 91.4 meters (100 yards). If an individual were to measure the field and
come close to 91.4 meters, then that measurement would be considered accurate. Precision is the closeness of
the repeated measurements to each other. Using the same example, if the individual measured the field three
times and each time found the length to be 89.0 meters, the measurements would be precise to the tenths, but
In this experiment, the accepted value for the thickness of aluminum foil is available from the instructor.
The closeness to these accepted values will determine the accuracy of the measurements. If more than one trial
is performed with the same piece of aluminum foil, then the precision of the measurements can be calculated.
1) Calculate your percent error (and your percent correctness) for this experiment and determine at least three
sources of error for each lab (this determines your accuracy).
2) How precise were the calculated values and your actual measurements?
3) A very thin layer of gold plating was placed on a metal tray that measured 25.22 cm by 13.22 cm. The gold
plating increased the mass of the plate by 0.0512 g. Calculate the thickness of the plating. The density of gold
is 19.32 g/cm3. Show all work.
4) By mistake, 1.00 quart of oil was dumped into a swimming pool that measures 25.0 m x 30.0 m. The density
of the oil was 0.750 g/cm3. How thick was the resulting oil slick? Some helpful conversions are that 1.06
quarts = 1.00 liter = 1000.00 cm3 and 100.0 cm = 1.00 m. Be careful with significant figures and exponential
notation. Show all work. Hint: Convert your volume of 1.00 quart into liters, then cm3, and convert your
meters into cm before you begin.
5) In the determination of the density of a rectangular metal bar, a student made the following measurements:
length, 8.53 cm; width, 2.4 cm; height, 1.0 cm; mass, 52.7064 g. Calculate the density of the metal to the
correct number of significant figures.
The following procedure was carried out to determine the volume of a flask. The flask was massed dry and
then filled with water. If the masses of the empty flask and filled flask were 56.12 g and 87.39 g, respectively,
and the density of water is 0.9976 g/cm3, calculate the volume of the flask in cubic centimeters.
Chemistry: Unit 1 – Matter Page 2 of 2
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CC-MAIN-2014-15
| 4,701 | 59 |
https://www.thestudentroom.co.uk/showthread.php?t=1850888
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math
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I have 2 questions that I'm stuck on.
The cumulative distribution function for X is
0, if x is less than or = 0
x, if x is between 0 and 1
1, if x is greater than or = 0
Then find the cumulative distribution function of the random variable Y = -2lnX
Find the probability density function of Y and identify the distribution.
Also determine variance of Y.
Suppose that X1, X2,... ,Xn is a random sample of size n, from a normal distribution N(0,theta).
State whether the following are biased or unbiased, justify this.
If they are biased, calculate the bias.
theta hat = sigma(from i=1 to n) (((Xi)^2)/n)
theta2 hat = n*[sigma(from i=1 to n) (Xi/n)^2 ]^2
Thanks for any help!
Turn on thread page Beta
Probability - cumulative distribution and bias watch
- Thread Starter
- 29-11-2011 19:40
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CC-MAIN-2018-43
| 787 | 18 |
https://lovebattery.net/sin-1-sqrt-3-2/
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math
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Just how do you do these inquiries without a calculator? You put on"t need to do them all-- I simply require to recognize exactly how to do a minimum of among them! I value any kind of recommendations and/or services.
There are a couple of worths of $\ transgression$ or $\ cos$ that you must recognize, e.g. $\ wrong(\ pi/3) = \ sqrt / 2$. Making use of the truth that $\ transgression$ is a weird feature, $\ wrong(-\ pi/3) = -\ sqrt 3/ 2$. So $\ transgression ^ -1 (-\ sqrt / 2) = -\ pi/3$.
An equilateral triangular has all angles equivalent to $60 ^ \ circ$. Allow this triangular have a side size of $2$. Separate the triangular right into 2 ideal triangulars. Among the legs of the best triangular is of size $1$, the hypotenuse is of size $2$, as well as from the Pythagorean thesis, the various other leg is of size $\ sqrt 3 $. From this we obtain $\ transgression(60 ^ \ circ)=\ frac 2 $. You can utilize this triangular to locate the worths of the trigonometric features of $30 ^ \ circ$ as well as $60 ^ \ circ$.
For trigonometric features of $45 ^ \ circ$, make use of an isosceles right triangular.
There are easy-to-remember vales of the tangent, cosine and also sine features for 0, 30, 45, 60 as well as 90 levels: Sine: 0, $\ frac12$, $\ frac \ sqrt2 2$, $\ frac \ sqrt3 2 $, 1 respectivelyCosine: turn around the above series of valuesTangent: 0, $\ frac1 \ sqrt3 $, 1, $\ sqrt3$, $\ infty$
Then you require to bear in mind just how each of their charts resembles, their linked balance connections and also their reciprocatory features. Then you can exercise the worths of these inverted features you have right here.
For instance, the initial inquiry requests for x where $\ wrong x=-\ frac \ sqrt3 2$. Given that the sine feature is weird, $\ wrong -LRB--x-RRB-=\ frac 2$ therefore x is $-60 °$ ($-\ frac \ pi3$ radians).
You are clear on exactly how to do these concerns. Consequently in this response, no description is essential. Only solutions will certainly be supplied. Right here there are:
a. $-\ frac $
b. $-\ frac \ pi $
c. $\ frac $
d. $\ frac 4 $
e. $\ frac $
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Inconsistency in separating: $y=\ tan ^ \ left(\ frac \ right)$
Variety of options to $\ cos \ left(A \ cos \ left(\ frac \ ideal)\ ideal) = \ wrong \ left(A \ cos \ left(\ frac \ ideal)\ best)$
Reviewing $\ wrong ^ -1 \ left <\ cot \ left(\ wrong ^ -1 \ sqrt \ right)+\ cos ^ -1 \ left(\ frac \ sqrt 12 4 \ right)+\ sec ^ \ sqrt \ ideal> $
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s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103626162.35/warc/CC-MAIN-20220629084939-20220629114939-00702.warc.gz
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| 3,893 | 29 |
https://bizfluent.com/how-4421050-calculate-operating-profit-margin.html
|
math
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Operating profit margin, also known as operating margin, is a comparison of a company's operating income to revenue in a given period. The formula for calculating it is simple.
Operating profit is what remains after all costs of goods sold and operating expenses are removed from revenue in a given period. Operating expenses encompass all of the costs incurred for selling, general and administrative functions of the company. If revenue equals $2 million and COGS plus SG&A expenses equal $1.25 million, for example, operating profit is $750,000. To calculate the operating profit margin, divide $750,000 by $2 million to get 0.375. To express the operating margin as a percentage, multiply the result by 100 to get a 37.5 percent margin.
Operating margin reveals the efficiency with which a business converts its revenue into operating profit. While non-cash expenses, currency fluctuations and other items are accounted for in net profit, the operating profit shows income earned from regular business activities. Therefore, higher operating margins are a good sign of financial health. Margins vary greatly by industry, so optimum margins are based on industry norms and company trends over time. Operating margins that remain stable or improve are favorable.
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CC-MAIN-2023-06
| 1,264 | 3 |
https://quickmathblog.blogspot.com/2016/10/fractions-are-no-problem-with-little-help.html
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math
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Math is the most dreaded subject in the world, and most people try to get away from it as much as possible. But the bad news is that you have to study math in your school because no academic curriculum is considered complete without math. In fact, there is no school in the world that would not teach math. Math is the art of every person’s everyday life. You have to deal with numbers no matter what. So, you must stop running away from it and start finding ways to deal with math.
When it comes to math, there are a few portions that are particularly tough to understand such as converting fraction to decimal.
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The main trick of solving such problems always remains basic. You first need to simplify the fraction as much as possible to convert it to decimal. If you have a fraction that is attached to a whole number, you will have to first make the whole thing into a simple fraction. This can be done by multiplying the whole number with the denominator and adding the numerator part of the fraction to it. Now the number thus obtained acts as a numerator, and the number that acted as a denominator in the original problem still serves as the denominator. Now the new numerator can be divided by the denominator to get the decimal, and that would be the answer.
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s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676590329.25/warc/CC-MAIN-20180718193656-20180718213656-00504.warc.gz
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| 2,702 | 8 |
http://www.ask.com/web?q=Mole+Conversions&o=2603&l=dir&qsrc=3139&gc=1
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math
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In a general chemistry class, you usually end up having to perform a lot of
conversions involving moles (mol). Whether you're converting from moles to
Nov 11, 2010 ... Mr. Andersen shows you how to convert moles to grams and moles to molecules.
Intro Music Atribution Title: I4dsong_loop_main.wav Artist: ...
By understanding the relationship between moles and Avogadro's number,
scientists can convert between number of moles and number of atoms.
How to use this mole bridge: 1. You are only allowed to go from left to right. 2.
The great divide side, you will divide. 3. On the Rabittville, side you will be ...
Gram/Mole/Volume Conversions. Multiple-choice exercise. Use a periodic table
to complete this review. DO NOT use a calculator, since you are not allowed one
Sep 28, 2015 ... So if I was to do a conversion where I had to find out how many moles are say 15
grams of lithium how would I find out? Would I just do 15 ...
Sep 5, 2004 ... Mole Conversions. Objectives: 1. Explain how to convert between the number. 2.
Define molar volume of a gas. 3. Determine how to determine ...
*Theme/Title: Mass-Mole Conversions. * Description/Instructions. Converting
from mass of a substance to moles of a substance requires simple division.
This worked example problem shows how to convert the number grams of a
molecule to the number of moles of the molecule.
There are at least three different mole-to-volume conversions you'll encounter in
general chemistry. Before you can make the conversion, ask: moles of what ?
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s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698544097.11/warc/CC-MAIN-20161202170904-00332-ip-10-31-129-80.ec2.internal.warc.gz
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| 1,518 | 20 |
https://www.hackmath.net/en/examples/9th-grade-(14y)?tag_id=148
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math
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9th grade (14y) + similarity of triangles - examples
Railways climb 7.4 ‰. Calculate the height difference between two points on the railway distant 3539 meters.
There is regular hexagon ABCDEF. If area of the triangle ABC is 22, what is area of the hexagon ABCDEF? I do not know how to solve it simply....
On the road sign, which informs the climb is 8.7%. Car goes 5 km along this road. What is the height difference that car went?
- Similarity coefficient
The ratio of similarity of two equilateral triangles is 3.5 (ie 7:2). The length of the side of smaller triangle is 2.4 cm. Calculate the perimeter and area of the larger triangle.
Road has climbing 1:27. How big is a angle corresponds to this climbing?
- MO - triangles
On the AB and AC sides of the triangle ABC lies successive points E and F, on segment EF lie point D. The EF and BC lines are parallel and is true this ratio FD:DE = AE:EB = 2:1. The area of ABC triangle is 27 hectares and line segments EF, AD, and DB se
- Triangle KLB
It is given equilateral triangle ABC. From point L which is the midpoint of the side BC of the triangle it is drwn perpendicular to the side AB. Intersection of perpendicular and the side AB is point K. How many % of the area of the triangle ABC is area o
A force of 300 kg (3000 N) is required to pull a boat up a ramp inclined at 14° with horizontal. How much does the boat weight?
- Sun rays
If the sun's rays are at an angle 60° then famous Great Pyramid of Egypt (which is now high 137.3 meters) has 79.3 m long shadow. Calculate current height of neighboring chefren pyramid whose shadow is measured at the same time 78.8 m and the current hei
- Euclid theorems
Calculate the sides of a right triangle if leg a = 6 cm and a section of the hypotenuse, which is located adjacent the second leg b is 5cm.
- Trapezoid IV
In a trapezoid ABCD (AB||CD) is |AB| = 15cm |CD| = 7 cm, |AC| = 12 cm, AC is perpendicular to BC. What area has a trapezoid ABCD?
- Shadow and light
Nine meters height poplar tree has a shadow 16.2 meters long. How long shadow have at the same time Joe if he is 1,4m tall?
- Rectangular triangles
The lengths of corresponding sides of two rectangular triangles are in the ratio 2:5. At what ratio are medians relevant to hypotenuse these right triangles? At what ratio are the contents of these triangles? Smaller rectangular triangle has legs 6 and 8 c
Calculate the cosine of the smallest internal angle in a right-angled triangle with cathetus 3 and 8 and with the hypotenuse 8.544.
- Diagonal in rectangle
In that rectangle ABCD is the center of BC point E and point F is center of CD. Prove that the lines AE and AF divide diagonal BD into three equal parts.
There are two laths in the garage opposite one another: one 2 meters long and the second 3 meters long. They fall against each other and stay against the opposite walls of the garage and both laths cross 70 cm above the garage floor. How wide is the garag
ABCD is a rhombus, ABD is an equilateral triangle and AC is equal to 4. Find the area of the rhombus.
- Traffic laws
Under traffic regulations, car lights can illuminate the road up to a maximum of 30 m. To check the reach of the dipped-beam lights of their car, Peter stopped car at 1.5 m from the wall. The dipped-beam headlights are 60 cm high. At what height on the wa
- Tree shadow
The shadow of the tree is 16 meters long. Shadow of two meters high tourist sign beside standing is 3.2 meters long. What height has tree (in meters)?
- MO Z9–I–2 - 2017
In the VODY trapezoid, VO is a longer base and the diagonal intersection K divides the VD line in a 3:2 ratio. The area of the KOV triangle is 13.5 cm2. Find the area of the entire trapezoid.
See also our trigonometric triangle calculator.
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s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560628000613.45/warc/CC-MAIN-20190627035307-20190627061307-00306.warc.gz
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CC-MAIN-2019-26
| 3,747 | 34 |
https://nrich.maths.org/public/leg.php?code=5039&cl=3&cldcmpid=575
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math
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The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.
Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?
Can you beat the computer in the challenging strategy game?
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates?
Use Excel to explore multiplication of fractions.
Help the bee to build a stack of blocks far enough to save his friend trapped in the tower.
A collection of resources to support work on Factors and Multiples at Secondary level.
Use Excel to investigate the effect of translations around a number grid.
A group of interactive resources to support work on percentages Key Stage 4.
Cellular is an animation that helps you make geometric sequences composed of square cells.
Practise your skills of proportional reasoning with this interactive haemocytometer.
Can you discover whether this is a fair game?
This set of resources for teachers offers interactive environments to support work on loci at Key Stage 4.
An environment that enables you to investigate tessellations of regular polygons
There are thirteen axes of rotational symmetry of a unit cube. Describe them all. What is the average length of the parts of the axes of symmetry which lie inside the cube?
Discover a handy way to describe reorderings and solve our anagram in the process.
Can you give the coordinates of the vertices of the fifth point in the patterm on this 3D grid?
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.
The interactive diagram has two labelled points, A and B. It is designed to be used with the problem "Cushion Ball"
Use an interactive Excel spreadsheet to explore number in this exciting game!
Match the cards of the same value.
This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.
Match pairs of cards so that they have equivalent ratios.
A metal puzzle which led to some mathematical questions.
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
This game challenges you to locate hidden triangles in The White Box by firing rays and observing where the rays exit the Box.
An environment that simulates a protractor carrying a right- angled triangle of unit hypotenuse.
A simple file for the Interactive whiteboard or PC screen, demonstrating equivalent fractions.
A tool for generating random integers.
A simple spinner that is equally likely to land on Red or Black. Useful if tossing a coin, dropping it, and rummaging about on the floor have lost their appeal. Needs a modern browser; if IE then at. . . .
A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
Use an Excel to investigate division. Explore the relationships between the process elements using an interactive spreadsheet.
To avoid losing think of another very well known game where the patterns of play are similar.
A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand. . . .
P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?
A collection of our favourite pictorial problems, one for each day of Advent.
Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.
An Excel spreadsheet with an investigation.
Use Excel to practise adding and subtracting fractions.
Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?
Use an interactive Excel spreadsheet to investigate factors and multiples.
Here is a chance to play a fractions version of the classic Countdown Game.
Use an Excel spreadsheet to explore long multiplication.
A java applet that takes you through the steps needed to solve a Diophantine equation of the form Px+Qy=1 using Euclid's algorithm.
A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .
A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.
Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.
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s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257648431.63/warc/CC-MAIN-20180323180932-20180323200932-00483.warc.gz
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CC-MAIN-2018-13
| 5,688 | 50 |
https://studysoup.com/note/2322307/asu-chm-113-week-5-fall-2016
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math
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Week 5 Notes
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This 4 page Class Notes was uploaded by Andrew Notetaker on Wednesday September 14, 2016. The Class Notes belongs to CHM 113 at Arizona State University taught by Cabirac in Fall 2016. Since its upload, it has received 8 views. For similar materials see General chemistry 1 in Science at Arizona State University.
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Date Created: 09/14/16
Chapter 3: 9-7, 9-12 & 9-14 Wednesday, September 7, 2016 3:10 PM Chemical Reactions and Reaction Stoichiometry 3.1 Chemical Equations Chemical reactions are represented by writing chemical equations Conservation of mass= reactant and product sides must have same number of atoms Subscripts tell the number of atoms in a molecule Coefficients tell the number of molecules General Guidelines for balancing equations 1. Write correct formulas for reactants and products 2. Can only adjust coefficients, not subscripts 3. Start with more complex molecules 4. Balance polyatomic ions as a single unit 5. Check to verify coefficients Notes from reading: Changing subscript changes identity and properties Changing coefficient changes amount CH 4 O -2> CO +2H O 2Unbalanced) CH 4 O -2> CO +22H O 2balanced) A chemical equation needs to be completely accurate, it must include the state of matter for all reactants and products. (s)=solid (g)=gas (l)=liquid (aq)= aqueous (dissolved in water) Simple Patterns of Chemical Reactivity 1. Combination reactions 2. Decomposition reactions 3. Combustion reactions Combination reactions CHM 113 Lecture Page 1 Combination reactions A+B=C 2Mg (s) + O 2g) -> 2MgO (s) Decomposition reaction one or more substance breaks down into two or more substances C -> A + B 2NaN (3) -> 2Na (s) + 3N 2g) Combustion reactions most often involve hydrocarbons or carbohydrates reacting with oxygen. Complete combustion leads to production of CO an2 H O. 2 Formula Weights The quantitative significance of chemical formulas A formula weight is the sum of atomic weights for all atoms in a chemical formula. H 2 fw = 18 amu or g/mol A molecular weight is the sum of atomic weights of the atoms in a molecule C2H 5H mw = 46.07 amu or g/mol Percent Composition One can find the mass percentage of each element in a compound by using the equation: Avogadro's number and the mole 1 gram = 6.02 x 10 amu The mass of one C-12 atom is exactly 12 amu 1 mole = 6.022 x 10 C-12 atoms 1 mole = 6.022 x 10 of anything CHM 113 Lecture Page 2 1 mole = 6.022 x 10 of anything Molar mass is the mass of 1 mol of a substance g/mol Molar mass can be used as a conversion factor Mass in grams of 0.2500 mole of O 2 Grams -> Molar mass <- Moles -> Avogadros # <- Molecules Empirical Formulas from Analysis Assume 100g of compound 30.45% Nitrogen and 69.56% O -30.45g N -60.56g O Convert to moles Divide each by the lowest number of mol (2.173 mol) 2.173 mol N/2.173 = 1 mol 4.348 mol O/ 2.173 = 2 mol Molecular formula can be determined only if molecular weight is known. If needed, multiple to get whole numbers if empirical formula gives decimal numbers How to interpret a balanced chemical equation N 2 3H (2) -> 3NH 3 1:3:2 1 molecule: 3 molecule: 2 molecules Mole ratio for moles of N 2hat will react with 2.4 mol of H 2 2.4 mol H 2 (1 mol N )2= 0.8 mol N 2 (2 mol H ) 2 General scheme for calculating grams of a product if grams of a substance is known Limiting Reactants 2H 2 O -2 2H O 2 CHM 113 Lecture Page 3 2H 2 O -2 2H O 2 What is the limiting reactant? Hydrogen gas. How many moles of H O wi2l form when the following are mixed? 4 mol H 2nd 2 mol O 4 m2l H O 2 2 mol H 2nd 2 mol O 2 m2l H O 2 10 mol H a2d 4 mol O 8 m2l H O 2 How to determine limiting reactants: Method 1: Compare moles of each reactant required to consume all of other reactant. Use limiting reactant to calculate the amount of the product Method 2: 1. Calculate moles or grams of desired product from both (or all) starting reactants. Whichever produces less product is the limiting reagent. Easily calculate limiting reactant by calculating moles of each reactant and divide by the balanced equation coefficient. The smaller number is the limiting reactant. Limiting reactant-> moles of limiting reactant-> moles of product using mole ratio -> theoretical yield of product Theoretical Yields Theoretical yield is what's calculated from the limiting reactant. Actual yield is what's produced when the reaction is run. Percent yield= Actual/Theoretical x 100% CHM 113 Lecture Page 4
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CC-MAIN-2016-44
| 4,944 | 15 |
http://e-booksdirectory.com/details.php?ebook=11653
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math
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Around Kolmogorov Complexity: Basic Notions and Results
by Alexander Shen
Publisher: arXiv.org 2015
Number of pages: 51
Algorithmic information theory studies description complexity and randomness and is now a well known field of theoretical computer science and mathematical logic. This report covers the basic notions of algorithmic information theory: Kolmogorov complexity (plain, conditional, prefix), Solomonoff universal a priori probability, notions of randomness, effective Hausdorff dimension.
Home page url
Download or read it online for free here:
by Venkatesan Guruswami, Atri Rudra, Madhu Sudan - University at Buffalo
Error-correcting codes are clever ways of representing data so that one can recover the original information even if parts of it are corrupted. The basic idea is to introduce redundancy so that the original information can be recovered ...
by Neri Merhav - arXiv
Lecture notes for a graduate course focusing on the relations between Information Theory and Statistical Physics. The course is aimed at EE graduate students in the area of Communications and Information Theory, or graduate students in Physics.
by Robert H. Schumann - arXiv
A short review of ideas in quantum information theory. Quantum mechanics is presented together with some useful tools for quantum mechanics of open systems. The treatment is pedagogical and suitable for beginning graduates in the field.
by Renato Renner - ETH Zurich
Processing of information is necessarily a physical process. It is not surprising that physics and the theory of information are inherently connected. Quantum information theory is a research area whose goal is to explore this connection.
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| 1,676 | 15 |
https://electronics.stackexchange.com/questions/502597/torque-production-in-pmsm
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math
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Regarding your first query you cannot just choose any curve you like to calculate mmf. The mmf here is due the conductor so you have to include it in the curve while applying ampere's circuital law to calculate mmf due to that conductor.
And the 0 is all about chosen reference. Here the maximum of fundamental mmf waveform is chosen as 0 arbitrarily.
If you don't get it like this see the diagram (attached) at chosen zero. Consider the mmf due to single conductors,you will see that both the flux add up as they are in same direction in the the region around zero.
And as for the sinusoidal peak (assumed that you know about fourier series). The given waveform as we see is a even periodic waveform so the components are likely to contain cosine fundamental and harmonics.
In case you assume a distributed conductor (attached) you can easily see why there is a peak at the position shown. Don't get confused by the details of waveform just see them and get a fair idea of what is happening here.
As you can see in these images the reference is chosen at 0 mmf and not at maximum as before. It is chosen arbitrarily. What really matters is the shape of waveform and its relative position w.r.t to rotor mmf waveform.
For maximum torque.... see it like this, the torque acts in order to align the two mmf either parallel or antiparallel and we see the maximum misalignment is when the two mmf are perpendicular to each other.
Mathematically this is represented by the equation
$$Te = -(π/8)P2ØFs\sinδs$$ Nm
P = Number of poles
Ø = Resultant air gap flux per pole
Fr = Rotor mmf
Fs = Stator mmf
The torque is proportional to product of magnitude of two mmfs and sine of angle between them. The part is maximum at angle = 90.
I have tried to keep the answer as simple and as informative as possible. The answer is open to advice and content addition.
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CC-MAIN-2021-17
| 1,852 | 15 |
https://papergurus.net/2018/05/06/1-using-cost-plus-pricing-what-is-the-price-if-atc-14-50-and-the-target-rate/
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math
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1. Using cost plus pricing what is the price if ATC = $14.50 and the target rate of return is 4 percent?$15.10$49.34$14.5$22.10 2. If Tiger Toys faces a demand curve of P = 85 .25Q and a MC = ATC = 20 then the economic profits would be:$130.00.$6825.00.$2600.00.$4225.00. 3. Using the linear approximation system to estimate the profit maximizing price requires that the managers know the costs of production and:the production function.one price and quantity of demand.two prices and quantities of demand.decision-making process of the marketplace 4. (TCO 2) The purpose of stock valuation is: (Points : 3) to set the fair market price for a given common stock. to determine whether the common stocks value is fairly represented by its market price. of limited value since the efficient market hypothesis proves that all common stock is always fairly priced. to find common stocks whose market price equals the intrinsic value.
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s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141176922.14/warc/CC-MAIN-20201124170142-20201124200142-00530.warc.gz
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CC-MAIN-2020-50
| 928 | 1 |
https://howmuchisdoxycycline.com/qa/what-are-the-6-types-of-graphs.html
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math
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- What are the different types of graphs and their uses?
- What are the 3 main types of graphs?
- How do you name a bar graph?
- How do I choose the right graph?
- What type of graph is used for statistics?
- What is the best type of graph?
- What are the 7 parts of a graph?
- What is the legend of a graph?
- Is a pie chart a graph?
- What is a circle graph used for?
- What are the types of graph?
- What are the different types of line graphs?
- What are parts of a bar graph?
- What is the most used graph?
- What graph category should you avoid?
What are the different types of graphs and their uses?
Types of Graphs and Charts And Their UsesLine Graphs.
A line chart graphically displays data that changes continuously over time.
Bar charts represent categorical data with rectangular bars (to understand what is categorical data see categorical data examples).
Spline Chart.More items….
What are the 3 main types of graphs?
Three types of graphs are used in this course: line graphs, pie graphs, and bar graphs. Each is discussed below.
How do you name a bar graph?
Required bar graph titles include the name of the graph, the title of the vertical axes, and the title of the horizontal axes. It is important to title bar graphs carefully so the information makes sense and the graph is easy to read and understand. Name the bar graph.
How do I choose the right graph?
Selecting the right chart type Ask yourself how many variables do you want to show, how many data points you want to display and how you want to scale your axis. Line, bar and column charts represent change over time. Pyramids and pie charts display parts-of-a-whole.
What type of graph is used for statistics?
Statistical Graphs (bar graph, pie graph, line graph, etc.) Exponential Graphs. Logarithmic Graphs. Trigonometric Graphs.
What is the best type of graph?
Line graphs are used to track changes over short and long periods of time. When smaller changes exist, line graphs are better to use than bar graphs. Line graphs can also be used to compare changes over the same period of time for more than one group.
What are the 7 parts of a graph?
The following pages describe the different parts of a bar graph.The Title. The title offers a short explanation of what is in your graph. … The Source. The source explains where you found the information that is in your graph. … X-Axis. Bar graphs have an x-axis and a y-axis. … Y-Axis. … The Data. … The Legend.
What is the legend of a graph?
The legend of a graph reflects the data displayed in the graph’s Y-axis, also called the graph series. This is the data that comes from the columns of the corresponding grid report, and usually represents metrics. A graph legend generally appears as a box to the right or left of your graph.
Is a pie chart a graph?
A pie chart is a circular graph that is broken down into segments (i.e slices of pie). These segments represent each category’s contribution to display parts of a whole.
What is a circle graph used for?
A circle graph, or a pie chart, is used to visualize information and data. A circle graph is usually used to easily show the results of an investigation in a proportional manner. The arcs of a circle graph are proportional to how many percent of population gave a certain answer.
What are the types of graph?
Types of Charts The four most common are probably line graphs, bar graphs and histograms, pie charts, and Cartesian graphs. They are generally used for, and are best for, quite different things. You would use: Bar graphs to show numbers that are independent of each other.
What are the different types of line graphs?
They are:Simple Line Graph: Only one line is plotted on the graph.Multiple Line Graph: More than one line is plotted on the same set of axes. A multiple line graph can effectively compare similar items over the same period of time.Compound Line Graph: If information can be subdivided into two or more types of data.
What are parts of a bar graph?
Parts of a Bar Graph Title: The title is explains what the graph is about. Scale: The scale is the numbers that show the units used on the bar graph. Labels: Both the side and the bottom of the bar graph have a label that tells what kind of data is shown. … Bars: The bar is measures the data number.
What is the most used graph?
Line Graphs#1 Line Graphs The most common, simplest, and classic type of chart graph is the line graph.
What graph category should you avoid?
There are some kinds of graphs which must be avoided. Those are – Pie charts, 3D and different tricks, radar graphs, donuts and surface graphs. The worthy data for the Pie charts is not easily available. The 3 D graphs are little bit confusing.
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s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323588216.48/warc/CC-MAIN-20211027150823-20211027180823-00374.warc.gz
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CC-MAIN-2021-43
| 4,701 | 48 |
https://scholars.duke.edu/display/presentation38414
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math
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Riemannian manifolds with special holonomy and special curvature. Simons Center program "G2 manifolds". Simons Center for Geometry and Physics . September 30, 2014
While the local generality of metrics with special holonomy is now well-understood and the existence of complete or even compact examples is well-established, the construction of explicit examples remains a challenge. An approach that has been followed with great success is to look for examples with large symmetry groups, for this often reduces the problem to studying systems of ordinary differential equations. A different approach is to look for examples whose curvature tensors satisfy some algebraically natural conditions, for this imposes higher order (partial) differential equations on the solutions, and so the methods of exterior differential systems can be brought to bear to analyze the resulting overdetermined systems of equations. This can sometimes yield solutions with relatively high cohomogeneity and may be expected to provide interesting characterizations of some of the known solutions as well. In this talk, I will report on work in progress along these lines, particularly in the low dimensions, in which the holonomy can be either $SU(2)$, $SU(3)$, or $G_2$. I will also show how these methods can be used to study similar problems in certain geometries with torsion.
Service Performed By
Service or Event Name
Simons Center program "G2 manifolds"
Simons Center for Geometry and Physics
Location or Venue
Simons Center for Geometry and Physics, Stony Brook, NY
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s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141715252.96/warc/CC-MAIN-20201202175113-20201202205113-00656.warc.gz
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CC-MAIN-2020-50
| 1,552 | 8 |
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