url stringlengths 6 1.61k | fetch_time int64 1,368,856,904B 1,726,893,854B | content_mime_type stringclasses 3 values | warc_filename stringlengths 108 138 | warc_record_offset int32 9.6k 1.74B | warc_record_length int32 664 793k | text stringlengths 45 1.04M | token_count int32 22 711k | char_count int32 45 1.04M | metadata stringlengths 439 443 | score float64 2.52 5.09 | int_score int64 3 5 | crawl stringclasses 93 values | snapshot_type stringclasses 2 values | language stringclasses 1 value | language_score float64 0.06 1 |
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http://www.mycoted.com/AIDA | 1,369,200,527,000,000,000 | text/html | crawl-data/CC-MAIN-2013-20/segments/1368701370254/warc/CC-MAIN-20130516104930-00058-ip-10-60-113-184.ec2.internal.warc.gz | 602,616,918 | 5,119 | AIDA
A to Z of Creativity Techniques Previous Technique7 Step Model Next TechniqueAdaptive Reasoning
AIDA (Analysis of Interactive Decision Areas - Luckman, Operational Research Quarterly, 1967; Friend and Hickling, Planning Under Pressure: The Strategic Choice Approach by John Friend and Allen Hickling, 1987) is used when you have several inter-connected problems where the solution choices for one will affect the solution choices for another. You therefore need to evaluate the solutions as a group, but the number of theoretically possible group combinations may be large. AIDA identifies combinations that cannot coexist and can therefore be eliminated, hence substantially reducing the number of combinations you need to compare.
Assuming that you have already got a list of problems, and have identified possible solutions for each. Then:
1. Identify any problems that do not interact: Draw a matrix with the problem names on each axis (e.g. 5 problems need a 5x5 matrix); delete the diagonal and the bottom triangle, to leave one cell for each different problem pair. Mark each cell 'X' if any of the solutions in the pair of problems the cell represents cannot co-exist. Remove from AIDA any problems with a blank row in this matrix; these have no interactions, and you can work with them independently.
P1 P2 P3 P4 P5 P1 x P2 x P3 x P4 x P5 x
1. Identify incompatible pairs of solutions: Write each remaining problem with its solutions, on a large Post-it slip (e.g. 4 problems give four slips). Stick them on a large working area (e.g. a white-board). Go through each solution on each slip, checking it against every solution on all the other slips to identify any pairs of solutions that cannot coexist. Draw a 'bar-line' linking the two members of each such incompatible pair of solutions. Then all solutions in different problems that are not barred are free to be combined.
2. Create a solution tree: Create a tree-diagram that displays all compatible combinations of solution options. Remove any incompatible branches. The remaining solutions can now be compared against agreed criteria like any other set of solutions. | 446 | 2,143 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.265625 | 3 | CC-MAIN-2013-20 | latest | en | 0.893918 |
http://www.addictivetips.com/microsoft-office/excel-2010-basic-logical-functions/ | 1,501,021,826,000,000,000 | text/html | crawl-data/CC-MAIN-2017-30/segments/1500549425407.14/warc/CC-MAIN-20170725222357-20170726002357-00394.warc.gz | 347,647,601 | 12,427 | # Excel 2010: Basic Logical Functions
Using logical functions provided by Excel 2010, we could apply and evaluate the specific rationale on the data. The most commonly used logical functions are AND, OR, and NOT. They take values as arguments to apply a simple logic over them and yield TRUE and FALSE result. The logical function helps users in situations where they want to categorize the value into groups, so that further complex functions can be easily performed. For example, if you want to check two different values that belong to different fields against a condition, you can use the logical function to check where the required condition is met. This may assist you in filtering out unwanted values and in focusing on only those values that you require. Additionally, Excel 2010 lets you use the multiple logical functions to check required conditions for the data set in one go. In what follows, we will demonstrate the usage of AND & OR logical functions.
To get started, launch Excel 2010 spreadsheet, containing the data set on which you want to apply logical functions. For illustration, we will apply AND & OR logical functions in a spreadsheet containing fields; Name, Course, Marks, Grade, and Status, as shown in the screenshot below.
Now we will apply logical operations over different fields. Since we want to find out that how many students registered HTML & Scripting course and secured 60+ marks, we use AND function. This function returns TRUE if both specified conditions are met.
If student studied HTML & scripting and secured marks greater than 60, only then the condition is said to be TRUE.
The syntax of the function is;
=AND(logical1, logical2…)
We will write it as;
=AND(C2=”HTMl & Scripting”,D2>60)
C2 and D2 in the function arguments are the location of the cell from where we started searching. The output True can bee seen in the screenshot below only where both conditions have met. Conversely FALSE will appear where any of the condition is not met.
Now we are going to find out the row against either Marks less than 60 or Status is fail. For this we will write OR logical function as;
=OR(D2<59,E2=”D”)
Where D2 and E2 are the locations of the cells in Marks and Grade field respectively. As TRUE can be seen against all the values in either field Marks or Grade where marks are less than 60 or grade is D.
You can also check out previously reviewed Excel function; DOLLAR, SUMPRODUCT, SUMIF, COUNTIF, VLOOKUP, HLOOKUP ,PMT, and LEN.
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• Bcarternc
I’m trying to get a value using “if/or” together. For example. If B1 Contains the value in cell q1 & r1 then b1 (meaning that I want the value of b1 to appear int he cell formula.) Is this possible?
• DM
=IF(AND(B1=Q1;B1=R1);B1;”some other value”)
This formula contains an if which returns the value of B1 in case of True, otherwise it returns “some other value” (when the statement is false).
The AND checks whether B1 is equal to Q1 AND B1 is equal to R1. If that is true, then the if statement is true, otherwise it is false.
Hope this helps.
• ccg001
I am trying to determine if the value in one column (A5555) matches or is equal to the value of ONLY the first five characters only in a another column (A5555-drod@home.com). Is there a way to limit the logic check to only a limited number of characters in a cell?
• antireaver
I need to get a logical functions formula that can carry 4 conditions. All i keep seeing is for 2 alone. Please help.
Thanks | 860 | 3,656 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.515625 | 4 | CC-MAIN-2017-30 | latest | en | 0.874471 |
http://abbotswell.aberdeen.sch.uk/blog/2017/10/02/clic-targets-wb-02102017/ | 1,508,191,325,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187820466.2/warc/CC-MAIN-20171016214209-20171016234209-00371.warc.gz | 9,121,796 | 8,019 | ## CLIC Targets wb 02/10/2017
C– Counting: Core Numbers: Step 9: I can understand 5, 6, 7, 8d numbers e.g.: Compare, Order, Round, Estimate activities with 2, 567, 321
L– Learn Its: x2 Simple revision. Here’s a fun game…play the Hit the Question version (Gold)
https://www.topmarks.co.uk/maths-games/hit-the-button
I– It’s Nothing New: Jigsaw Numbers: I can find the missing decimal piece. E.g.: 7.6 + ? = 10 Answer 2.4
C– Calculations: Addition: Step 38: I can solve additions with larger numbers. E.g.: 4603 + 1678 = Use FAB method; no column methods!
Source: Primary 7 | 192 | 577 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.375 | 3 | CC-MAIN-2017-43 | latest | en | 0.654234 |
https://ecampusontario.pressbooks.pub/prehealthsciencesmath1/chapter/exercises-solve-systems-of-equations-by-elimination-4-3/ | 1,723,219,527,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640767846.53/warc/CC-MAIN-20240809142005-20240809172005-00847.warc.gz | 175,861,711 | 24,257 | # Exercises: Solve Systems of Equations by Elimination (4.3)
## Exercises: Solve a System of Equations by Elimination
Instructions: For questions 1-41, solve the systems of equations by elimination.
1. $\left\{\begin{array}{c}5x+2y=2\\ -3x-y=0\end{array}\right.$
2. $\left\{\begin{array}{c}-3x+y=-9\\ x-2y=-12\end{array}\right.$
Solution
$(6,9)$
3. $\left\{\begin{array}{c}6x-5y=-1\\ 2x+y=13\end{array}\right.$
4. $\left\{\begin{array}{c}3x-y=-7\\ 4x+2y=-6\end{array}\right.$
Solution
$\left(-2,1\right)$
5. $\left\{\begin{array}{c}x+y=-1\\ x-y=-5\end{array}\right.$
6. $\left\{\begin{array}{c}x+y=-8\\ x-y=-6\end{array}\right.$
Solution
$\left(-7,-1\right)$
7. $\left\{\begin{array}{c}3x-2y=1\\ -x+2y=9\end{array}\right.$
8. $\left\{\begin{array}{c}-7x+6y=-10\\ x-6y=22\end{array}\right.$
Solution
$\left(-2,-4\right)$
9. $\left\{\begin{array}{c}3x+2y=-3\\ -x-2y=-19\end{array}\right.$
10. $\left\{\begin{array}{c}5x+2y=1\\ -5x-4y=-7\end{array}\right.$
Solution
$\left(-1,3\right)$
11. $\left\{\begin{array}{c}6x+4y=-4\\ -6x-5y=8\end{array}\right.$
12. $\left\{\begin{array}{c}3x-4y=-11\\ x-2y=-5\end{array}\right.$
Solution
$\left(-1,2\right)$
13. $\left\{\begin{array}{c}5x-7y=29\\ x+3y=-3\end{array}\right.$
14. $\left\{\begin{array}{c}6x-5y=-75\\ -x-2y=-13\end{array}\right.$
Solution
$\left(-5,9\right)$
15. $\left\{\begin{array}{c}-x+4y=8\\ 3x+5y=10\end{array}\right.$
16. $\left\{\begin{array}{c}2x-5y=7\\ 3x-y=17\end{array}\right.$
Solution
$(6,1)$
17. $\left\{\begin{array}{c}5x-3y=-1\\ 2x-y=2\end{array}\right.$
18. $\left\{\begin{array}{c}7x+y=-4\\ 13x+3y=4\end{array}\right.$
Solution
$\left(-2,10\right)$
19. $\left\{\begin{array}{c}-3x+5y=-13\\ 2x+y=-26\end{array}\right.$
20. $\left\{\begin{array}{c}3x-5y=-9\\ 5x+2y=16\end{array}\right.$
Solution
$(2,3)$
21. $\left\{\begin{array}{c}4x-3y=3\\ 2x+5y=-31\end{array}\right.$
22. $\left\{\begin{array}{c}4x+7y=14\\ -2x+3y=32\end{array}\right.$
Solution
$\left(-7,6\right)$
23. $\left\{\begin{array}{c}5x+2y=21\\ 7x-4y=9\end{array}\right.$
24. $\left\{\begin{array}{c}3x+8y=-3\\ 2x+5y=-3\end{array}\right.$
Solution
$\left(-9,3\right)$
25. $\left\{\begin{array}{c}11x+9y=-5\\ 7x+5y=-1\end{array}\right.$
26. $\left\{\begin{array}{c}3x+8y=67\\ 5x+3y=60\end{array}\right.$
Solution
$(9,5)$
27. $\left\{\begin{array}{c}2x+9y=-4\\ 3x+13y=-7\end{array}\right.$
28. $\left\{\begin{array}{c}\frac{1}{3}x-y=-3\\ x+\frac{5}{2}y=2\end{array}\right.$
Solution
$\left(-3,2\right)$
29. $\left\{\begin{array}{c}x+\frac{1}{2}y=\frac{3}{2}\\ \frac{1}{5}x-\frac{1}{5}y=3\end{array}\right.$
30. $\left\{\begin{array}{c}x+\frac{1}{3}y=-1\\ \frac{1}{2}x-\frac{1}{3}y=-2\end{array}\right.$
Solution
$\left(-2,3\right)$
31. $\left\{\begin{array}{c}\frac{1}{3}x-y=-3\\ \frac{2}{3}x+\frac{5}{2}y=3\end{array}\right.$
32. $\left\{\begin{array}{c}2x+y=3\\ 6x+3y=9\end{array}\right.$
Solution
infinitely many solutions
33. $\left\{\begin{array}{c}x-4y=-1\\ -3x+12y=3\end{array}\right.$
34. $\left\{\begin{array}{c}-3x-y=8\\ 6x+2y=-16\end{array}\right.$
Solution
infinitely many solutions
35. $\left\{\begin{array}{c}4x+3y=2\\ 20x+15y=10\end{array}\right.$
36. $\left\{\begin{array}{c}3x+2y=6\\ -6x-4y=-12\end{array}\right.$
Solution
infinitely many solutions
37. $\left\{\begin{array}{c}5x-8y=12\\ 10x-16y=20\end{array}\right.$
38. $\left\{\begin{array}{c}-11x+12y=60\\ -22x+24y=90\end{array}\right.$
Solution
inconsistent, no solution
39. $\left\{\begin{array}{c}7x-9y=16\\ -21x+27y=-24\end{array}\right.$
40. $\left\{\begin{array}{c}5x-3y=15\\ y=\frac{5}{3}x-2\end{array}\right.$
Solution
inconsistent, no solution
41. $\left\{\begin{array}{c}2x+4y=7\\ y=-\frac{1}{2}x-4\end{array}\right.$
## Exercises: Solve Applications of Systems of Equations by Elimination
Instructions: For questions 42-49, translate to a system of equations and solve.
42. The sum of two numbers is $65$. Their difference is $25$. Find the numbers.
Solution
The numbers are $20$ and $45$.
43. The sum of two numbers is $37$. Their difference is $9$. Find the numbers.
44. The sum of two numbers is $-27$. Their difference is $-59$. Find the numbers.
Solution
The numbers are $16$ and $-43$.
45. The sum of two numbers is $-45$. Their difference is $-89$. Find the numbers.
46. Andrea is buying some new shirts and sweaters. She is able to buy $3$ shirts and $2$ sweaters for $114$ or she is able to buy $2$ shirts and $4$ sweaters for $164$. How much does a shirt cost? How much does a sweater cost?
Solution
A shirt costs $16$ and a sweater costs $33$.
47. Peter is buying office supplies. He is able to buy $3$ packages of paper and $4$ staplers for $40$ or he is able to buy $5$ packages of paper and $6$ staplers for $62$. How much does a package of paper cost? How much does a stapler cost?
48. The total amount of sodium in $2$ hot dogs and $3$ cups of cottage cheese is $4720$ mg. The total amount of sodium in $5$ hot dogs and $2$ cups of cottage cheese is $6300$ mg. How much sodium is in a hot dog? How much sodium is in a cup of cottage cheese?
Solution
There are $860$ mg in a hot dog. There are $1\text{,}000$ mg in a cup of cottage cheese.
49. The total number of calories in $2$ hot dogs and $3$ cups of cottage cheese is $960$ calories. The total number of calories in $5$ hot dogs and $2$ cups of cottage cheese is $1190$ calories. How many calories are in a hot dog? How many calories are in a cup of cottage cheese?
## Exercises: Choose the Most Convenient Method to Solve a System of Linear Equations
Instructions: For questions 50-53, decide whether it would be more convenient to solve the system of equations by substitution or elimination.
50.
a.$\left\{\begin{array}{c}8x-15y=-32\\ 6x+3y=-5\end{array}\right.$
b.$\left\{\begin{array}{c}x=4y-3\\ 4x-2y=-6\end{array}\right.$
Solution
a. elimination
b. substitution
51.
a.$\left\{\begin{array}{c}y=7x-5\\ 3x-2y=16\end{array}\right.$
b.$\left\{\begin{array}{c}12x-5y=-42\\ 3x+7y=-15\end{array}\right.$
52.
a.$\left\{\begin{array}{c}y=4x+9\\ 5x-2y=-21\end{array}\right.$
b.$\left\{\begin{array}{c}9x-4y=24\\ 3x+5y=-14\end{array}\right.$
Solution
a. substitution
b. elimination
53.
a.$\left\{\begin{array}{c}14x-15y=-30\\ 7x+2y=10\end{array}\right.$
b.$\left\{\begin{array}{c}x=9y-11\\ 2x-7y=-27\end{array}\right.$
## Exercises: Everyday Math
Instructions: For questions 54-55, answer the given everyday math word problems.
54. Norris can row $3$ miles upstream against the current in the same amount of time it takes him to row $5$ miles downstream, with the current. Solve the system. $\left\{\begin{array}{c}r-c=3\\ r+c=5\end{array}\right.$
a. for $r$, his rowing speed in still water.
b. Then solve for $c$, the speed of the river current.
Solution
a. $r=4$
b. $c=1$
55. Josie wants to make $10$ pounds of trail mix using nuts and raisins, and she wants the total cost of the trail mix to be $54$. Nuts cost $6$ per pound and raisins cost $3$ per pound. Solve the system $\left\{\begin{array}{c}n+r=10\\ 6n+3r=54\end{array}\right.$ to find $n$, the number of pounds of nuts, and $r$, the number of pounds of raisins she should use.
## Exercises: Writing Exercises
Instructions: For questions 56-57, answer the given writing exercises.
56. Solve the system $\left\{\begin{array}{c}x+y=10\\ 5x+8y=56\end{array}\right.$
a. by substitution
b. by graphing
c. Which method do you prefer? Why?
Solution
a. (8, 2)
b.
57. Solve the system $\left\{\begin{array}{c}x+y=-12\\ y=4-\frac{1}{2}x\end{array}\right.$ | 2,940 | 7,487 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.46875 | 4 | CC-MAIN-2024-33 | latest | en | 0.357138 |
https://softmath.com/algebra-software/exponential-equations/converting-arcseconds-to.html | 1,718,698,926,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861747.70/warc/CC-MAIN-20240618073942-20240618103942-00415.warc.gz | 474,237,700 | 9,339 | converting arcseconds to albers (meters)
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Author Message
BorrieOH
Registered: 21.08.2002
From:
Posted: Friday 01st of Oct 16:52 Hi friends, It’s been almost a week now and I still can’t figure out how to crack a set of math problems on converting arcseconds to albers (meters) . I have to submit this work by the end of next week. Can someone help me to get started? I need some help with matrices and interval notation. Any sort of help will be appreciated.
AllejHat
Registered: 16.07.2003
From: Odense, Denmark
Posted: Saturday 02nd of Oct 17:52 You seem to be more horrified than confused. First you need to control your senses . Do not panic. Sit back, relax and look at the books with a clear mind. They will seem tough if you think they are hard . converting arcseconds to albers (meters) can be easily understood and you can solve almost every equation with the help of Algebrator. So relax.
SjberAliem
Registered: 06.03.2002
From: Macintosh HD
Posted: Sunday 03rd of Oct 11:33 I fully agree with that. It truly is a great software . Algebrator helped me and my peers a lot during our exam time. We went on to get more marks than we could ever imagine . It explains things in a lot more detailed manner, than a teacher ever could, in a class. Moreover, you can read one solution over and over again till you actually understand it, unlike in a classroom where the teacher has to move on due to time constraints. Go ahead and try it.
okato1314
Registered: 29.10.2005
From: China, Hong Kong
Posted: Tuesday 05th of Oct 09:53 Is it really true that a program can perform like that? I don’t really know much anything about this Algebrator but I am really seeking for some help so would you mind sharing me where could I find that software ? Is it downloadable over the internet ? I’m hoping for your fast reply because I really need help badly .
sxAoc
Registered: 16.01.2002
From: Australia
Posted: Wednesday 06th of Oct 09:00 You should check out https://softmath.com/comparison-algebra-homework.html. Your math will improve in no time , you shall see! Good luck ! | 644 | 2,567 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.921875 | 3 | CC-MAIN-2024-26 | latest | en | 0.915787 |
https://forum.ansys.com/forums/topic/dealing-with-non-zero-wall-roughness-height-and-wall-y-plus-for-compressible-flow/ | 1,675,277,952,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764499949.24/warc/CC-MAIN-20230201180036-20230201210036-00173.warc.gz | 285,983,885 | 97,333 | ## Fluids
#### Dealing with non-zero wall roughness height and wall y plus for compressible flow
• sagardeprem
Subscriber
I am performing a high speed simulation with a normal shock of about Mach 1.5 (Model - k-w sst). I need to resolve the boundary layer close to the wall in order to predict flow separation.
I have an experimental velocity profile at a certain location that needs to be matched. I first did the simulation with zero roughness height and the y-plus was about 0.9-1 (which I believe is what I need), but I couldn't get the same experimental profile. By experimenting with different non-zero values for roughness height (same mesh) I was able to somewhat match the profile, but the wall yplus was above 100. Decreasing the first wall height didn't make a difference. And by this time I had realized, my roughness height is way higher than the first cell height (which I found out was non-physical)
So I read the documentation that a non-zero roughness height would shift the wall to half the roughness height. If that is so, from where is the yplus being calculated? The actual wall or the shifted virtual wall?
Prem
• Raef.Kobeissi
Subscriber
Hello The law of the wall changes slightly when roughness height is added. Please read this from Fluent help document:
“Note that it is not physically meaningful to have a mesh size such that the wall-adjacent cell is smaller than the roughness height. For best results, make sure that the distance from the wall to the centroid of the wall-adjacent cell is greater than \$K_s\$.”
I believe your Y plus value has increased because the roughness height is now determining it rather than your first cell height .
• sagardeprem
Subscriber
Experimentally, there's a small separation bubble close to the wall and the size of this is smaller than the roughness height that matched the velocity profile at a location upstream. So I can't increase my first cell height above the roughness height, else I cannot capture the separation.
So in order to make sure my BL is correctly predicted, would it help to calculate the yplus from the actual wall and keep it about 1? I couldn't find anything that would imply this is valid in the fluent document.
Prem
• sagardeprem
Subscriber
So I calculated the integral of the wall shear stress for both the cases and I think I found the culprit. The average friction velocity (u*) were in proportion to the yplus for both the cases. So I know, the velocities are not wrong, but the wall shear stress is wrongly predicted by setting first cell height below the wall roughness height.
Also, I gathered that yplus is calculated from the original wall itself (link below), which seems correct now because it was the high friction velocity (u*) that increased the yplus.
• Raef.Kobeissi
Subscriber
Excellent!
• jonheb
Subscriber
Hi there I was wondering if you could help me out with a similar problem.
I am modelling a micro channel with k omega SST turbulence and there is a constant heat flux also added at the wall of my micro channel.
I have modelled the micro channel as a smooth flow initially and have been getting a Y+ of 1 for Reynolds of 40k approx., but when I add ks of 9.84e-5 the Y+ is now 200+. I don't understand what I need to do in order to get it back to Y+=1 again, I have played with the bias of the mesh but nothing changes.
I have read the law of the wall modified for roughness but it doesn't really make sense to me.
• LorenzoMazzei
Subscriber
Hi jonheb,
I think is wrong to seek necessarily the y+ value of 1 in case of rough surface. It's better to stay in the log-wall region and model the near-wall flow with a rough wall function.
• LorenzoMazzei
Subscriber
Dear Raef,
I don't understand if there's a sort of Automatic wall treatment also in case of rough wall. How can I modify the first layer height each time depending on the roughness height? Is there a kind of scalable wall function also for rough walls?
Lorenzo
• DrAmine
Ansys Employee
There us an automatic wall treatment and yplus should be kept as revommended for smooth walls even if roughness destroys the viscous sublayer. Just use SST turbulence model with.
• LorenzoMazzei
Subscriber
There us an automatic wall treatment and yplus should be kept as revommended for smooth walls even if roughness destroys the viscous sublayer. Just use SST turbulence model with.
thanks and sorry for my late reply, but I didn't receive any notification. I am performing several tests and what I am seeing is that for high roughness values there is a strong sensitivity to mesh, while I would have expected less impact.
I am simulating a simple straight channel with 10, 20, 40 elements per hydraulic diameter and 5, 10, 20 prisms on the wall. Despite the very fine mesh, I find it difficult to obtain an insensitive result in terms of both friction and heat transfer.
• Rob
Ansys Employee
Depending on the size of the channel what else could be going on?
• LorenzoMazzei
Subscriber
Depending on the size of the channel what else could be going on?
Hi rwoolhou,
what I mean is that I found difficult to obtain a mesh insensitive result in case of high roughness (up to 25% differences in f and Nu between the coarsest and the finest mesh), whereas it is more straightforward to do it with a smooth wall (5% max error).
I am still trying to figure out why.
Lorenzo
• DrAmine
Ansys Employee
Hi if you have high roughness or the surface roughness is large relative to the near wall mesh please use the new High Roughness Icing Models
Investigate, verify, validate and compare and publish.
• LorenzoMazzei
Subscriber
this is exactly what I am doing. I am performing simulations against exp data of rough AM test cases, but I can't get a mesh insensitive result. If you have something in mind I am accepting suggestions.
I'll make additional tests using also the icing models as suggested.
Thanks,
Lorenzo
• AhmedCfx
Subscriber
Hi there
How can I change wall roughness to gravel has different diameters (10-20)mm ?
• Rob
Ansys Employee
Work out the roughness parameters using the documentation and theory. | 1,401 | 6,115 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.65625 | 3 | CC-MAIN-2023-06 | latest | en | 0.935334 |
http://www.physicsforums.com/showthread.php?p=3805910 | 1,369,217,513,000,000,000 | text/html | crawl-data/CC-MAIN-2013-20/segments/1368701577515/warc/CC-MAIN-20130516105257-00054-ip-10-60-113-184.ec2.internal.warc.gz | 648,521,068 | 9,123 | ## Dividing by Zero=undefined or complex infinite?
A few days ago, I had a problem that looked like this:
evaluate cot(pi)
I know that on the unit circle, cot(pi) ends up as -1/0. In my precalc class, we say that this is undefined because you can't divide by zero.
I decided to plug the problem into wolfram and it tells me that there is in fact an answer, that being complex infinity.
I'm not sure what to make of this as I've never heard of complex infinity.
Am I wrong to say that cot(pi), or any other number divided by zero is undefined, or is the correct answer complex infinite?
Thanks!
Quote by physicsdreams A few days ago, I had a problem that looked like this: evaluate cot(pi) I know that on the unit circle, cot(pi) ends up as -1/0. In my precalc class, we say that this is undefined because you can't divide by zero. I decided to plug the problem into wolfram and it tells me that there is in fact an answer, that being complex infinity. I'm not sure what to make of this as I've never heard of complex infinity. Am I wrong to say that cot(pi), or any other number divided by zero is undefined, or is the correct answer complex infinite? Thanks!
The correct answer is undefined. When you take complex analysis, the correct answer is complex infinity :-)
Here's a page that reveals all.
http://en.wikipedia.org/wiki/Riemann_sphere
Quote by SteveL27 The correct answer is undefined. When you take complex analysis, the correct answer is complex infinity :-) Here's a page that reveals all. http://en.wikipedia.org/wiki/Riemann_sphere
Would it be possible for you to explain this in laymen's terms, seeing as I am only in precalculus.
Thank you
## Dividing by Zero=undefined or complex infinite?
Quote by physicsdreams Would it be possible for you to explain this in laymen's terms, seeing as I am only in precalculus. Thank you
Hey physicsdreams.
A complex number is written in the form of z = a + bi where a and b are just real numbers.
The infinite-complex number is just a number that has an infinite 'length'. We define the 'length' of a complex number to be SQRT(a^2 + b^2).
Basically if you look at the Riemann-Sphere wiki that was posted above, this 'infinite' complex number is at the point where the 'north pole' is, and the complex number that is 'zero' (i.e. z = 0 + 0i = 0) is at the south pole.
Quote by physicsdreams Would it be possible for you to explain this in laymen's terms, seeing as I am only in precalculus. Thank you
You can visualize the complex numbers as a sphere with zero at one pole and "complex infinity," a symbolic extra point, at the other pole. When you do this, you can make sense of saying that the function 1/z takes the value complex infinity at z = 0. That's where Wolfram is getting its answer from. You're not really dividing by zero, but rather taking the limit of a complex function as the function's value approaches the north pole in the complex sphere.
This is somewhat advanced math, typically taken by undergrad math majors after a couple of years of calculus and a class in real analysis. It would never be accurate to say you can divide by zero. Perhaps Wolfram should do a better job of explaining what they're doing so as not to confuse people who haven't taken a course in complex variables.
Thank you all for your explanations. Hopefully I'll gain a better understanding of this advance concept in the future.
It's undefined when working with the reals. I don't think there's any reason to get into anything more complex at this point (certainly not complex analysis).
Recognitions: Homework Help Science Advisor all it means is that as t-->pi, then cot(t)--> infinity.
Tags infinity, unit cirlce | 853 | 3,689 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.546875 | 4 | CC-MAIN-2013-20 | longest | en | 0.956558 |
https://ltwork.net/it-about-homeostasis-this-is-part-3-part-1-2-are-in-different--10652906 | 1,660,647,222,000,000,000 | text/html | crawl-data/CC-MAIN-2022-33/segments/1659882572286.44/warc/CC-MAIN-20220816090541-20220816120541-00785.warc.gz | 343,920,990 | 12,234 | # It about homeostasis ! this is part 3 part 1,2 are in different questions if u feel like answering more.
###### Question:
!
this is part 3
part 1,2 are in different questions if u feel like answering more.
$It about homeostasis ! this is part 3 part 1,2 are in differen$
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### During a year, teri’s monthly sales compensation ranged between $22,000 and$30,000 per month and
During a year, teri’s monthly sales compensation ranged between $22,000 and$30,000 per month and units sold ranged between 1,400 and 2,200 units for those same months. required: use the high–low method to determine teri’s monthly salary and commission rate per unit sold and then calculate th... | 1,223 | 4,715 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 3, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.796875 | 3 | CC-MAIN-2022-33 | latest | en | 0.955636 |
https://hsm.stackexchange.com/questions/9526/what-is-discovered-first-electricity-or-electron | 1,558,638,894,000,000,000 | text/html | crawl-data/CC-MAIN-2019-22/segments/1558232257361.12/warc/CC-MAIN-20190523184048-20190523210048-00008.warc.gz | 523,246,536 | 32,006 | # What is discovered first, electricity or electron?
It always bugs my mind to think which comes first, it feels like the egg or the chicken problem. When i was reading about the discovery of the electron i found that the charge to mass ratio of the electron was discovered by J.J Thomson, with his cathode ray tube experiment. Which used the idea of electricity to come to his conclusion. and i also read the equations he used where the equation of Force due to the electric field and magnetic field. $$F_m=Bqv$$ and $$F_e=Ed$$. But where electric fields and there equations were discovered in his time. I mean does it seem logical that humans have discover the electric force before discovering the electron who are the causes for the electric field. I would like it if you can provide me what is discovered first and how the one is used for the other supported by a time line. Electric forces or properties of electrons
I mean how could J.J Thomson use equations that use the charge of electron like $$F_m=Bqv$$ before he even knows there existence. And how could a force of magnetic field due to a charge be thought before the discovery of this idea of a charged particle. Did the discovery of the electron led to the formulation of the equation of force by magnetic fields. Or did the magnetic field equations led to the discovery of the electron.
• Electric force was known in the ancient Greece. Electron was discovered only in 19th century. – Alexandre Eremenko Apr 26 at 11:50
• Trivial. Simple search in Wikipedia gives the answer. – Alexandre Eremenko Apr 26 at 11:51
• Isn't it odd we knew and quantized electric force and electric field before knowing particles that created it. doesn't it sound a controversy. i mean the electric force is calculated using columb's law, which uses charges. but if the electron is not discovered how can you talk about charge. isnt charge a property of subatomic particles. – Hilea Apr 26 at 11:54
• "electric force is calculated using columb's law, which uses charges". People knew electricity, but quantifying it came later. At least up until the mid-late 19th century the nomenclature used to describe electric phenomena was quite different than the one we use today, and the understanding of the diverse electric and magnetic phenomena was quite rudimentary compared to what is taught now, even if one teaches laws that where discovered at that time. – David Apr 26 at 12:12
• Electricity was considered a fluid (a continuum). There was no concept of quantization of charges in the time of Franklin, Coulomb and even Maxwell. – M. Farooq Apr 26 at 12:21 | 571 | 2,606 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.765625 | 3 | CC-MAIN-2019-22 | latest | en | 0.962985 |
http://archive.org/stream/HistoryOfTheTheoryOfNumbersI/TXT/00000044.txt | 1,467,284,990,000,000,000 | text/html | crawl-data/CC-MAIN-2016-26/segments/1466783398516.82/warc/CC-MAIN-20160624154958-00014-ip-10-164-35-72.ec2.internal.warc.gz | 13,247,180 | 7,745 | # Full text of "History Of The Theory Of Numbers - I"
## See other formats
```CHAP. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS, 33
G. Wertheim, Anfangsgrunde der Zahlentheorie, 1902.
G. Giraud, Periodico di Mat., 21,1906, 124-9.
F. Ferrari, Suppl. al Periodico di Mat., 11, 1908, 36-8, 53, 75-6 (Cipolla).
P. Bachmann, Niedere Zahlentheorie, II, 1910, 97-101.
A. Aubry, Assoc. frang. avanc. sc., 40, 1911, 53-4; 42, 1913; Penseignement
math., 1911, 399; 1913, 215-6, 223. *M. Kiseljak, Beitrage zur Theorie der vollkommenen Zahlen, Progr. Agram,
1911.
*J. Vaes, Wiskundig Tijdschrift, 8, 1911, 31, 173; 9, 1912, 120, 187. J. Fitz-Patrick, Exercices Math., ed. 3, 1914, 55-7.
MULTIPLY PERFECT NUMBERS.
A multiply perfect or pluperfect number n is one the sum of whose divisors, including n and 1, is a multiple of n. If the sum is mn, m is called the multiplicity of n. For brevity, a multiply perfect number of multiplicity m shall be designated by Pm. Thus an ordinary perfect number is a P2. Although Robert Recorde39 in 1557 cited 120 as an abundant number, since the sum of its parts is 240, such numbers were first given names and investigated by French writers in the seventeenth century. As a P3 equals one-half of the sum of its aliquot divisors or parts (divisors <Pa), it was called a sous-double; a P4 equals one-third of the sum of its aliquot parts and was called a sous-triple; a P5 a sous-quadruple; etc.
F. Marin Mersenne proposed to R. Descartes300 the problem to find a sous-double other than P3(1) = 120 = 233-5. The latter did not react on the question until seven years later.
Mersenne301 mentioned (in the Epistre) the problem to find a P4, a P5 or a Pm, a P3 besides 120, and a rule to find as many as one pleases. He remarked (p. 211) that the P3 120, the P4 240 [for 30240?] and all other abundant numbers can signify the most fruitful natures.
Pierre de Fermat302 referred in 1636 to his former [lost] letter in which he gave "the proposition concerning aliquot parts and the construction to find an infinitude of numbers of the same nature." He303 found the second P3, viz., P3(2)=672 = 253-7.
Mersenne304 stated that Fermat found the 1 3 7 15... P3 672 and knew infallible rules and analysis 2 4 8 16... to find an infinitude of such numbers. He305 3 5 9 17... later gave [Fermat*s] method of finding such P3: Begin with the geometric
»°°Oeuvres de Descartes, 1, Paris, 1897, p. 229, line 28, letter from Descartes to Mersenne, Oct
or Nov., 1631. lolLes Preludes de VHannonie Universelle ou Questions Curiouses, Utiles aux Predicateurs, aux
Theologiens, Astrologues, Medecins, & Philosophes, Paris, 1634. I020euvres de Fermat, 2, Paris, 1894, p. 20, No. 3, letter to Mersenne, June 24, 1636. J0z0euvres de Fermat, 2, p. 66 (French transl. 3, p. 288), 2, p. 72, letters to Mersenne and
Roberval, Sept., 1636. IMHarmonie Universelle, Paris, 1636, Premiere Preface Generate (preceded by a preface of two
pages), unnumbered page 9, remark 10. Extract in Oeuvres de Fermat, 2, 1894, 20-21. 108Mersenne, Seconde Partie de PHarmonie Universelle, Paris, 1637. Final subdivision: Nou-
velles Observations Physiques et MathSmatiques, p. 26, Observation 13. Extract in
Oeuvres de Fermat, 2, 1894, p. 21.``` | 1,107 | 3,246 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.640625 | 4 | CC-MAIN-2016-26 | latest | en | 0.821614 |
http://www.thenakedscientists.com/forum/index.php?topic=67358.0 | 1,477,179,897,000,000,000 | text/html | crawl-data/CC-MAIN-2016-44/segments/1476988719079.39/warc/CC-MAIN-20161020183839-00286-ip-10-171-6-4.ec2.internal.warc.gz | 749,303,001 | 12,854 | # The Naked Scientists Forum
### Author Topic: The speed of time (Read 319 times)
#### Thebox
• Neilep Level Member
• Posts: 3139
• Thanked: 42 times
##### The speed of time
« on: 27/06/2016 09:14:16 »
Quote
“Light thinks it travels faster than anything but it is wrong. No matter how fast light travels, it finds the darkness has always got there first, and is waiting for it.”
Terry Pratchett
It light speed c the fastest speed in the universe?
Alan
Darkness does not have a speed, however the speed of light is not the fastest speed in the Universe, time is faster than light. A ''Photon'' could never overtake time.
#### jeffreyH
• Global Moderator
• Neilep Level Member
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• The graviton sucks
##### Re: The speed of time
« Reply #1 on: 27/06/2016 09:38:05 »
Darkness doesn't have a particle associated with it nor a field of any sort. Since time is inherent to the calculation of speed it cannot be faster than the very thing it determines. That would be paradoxical. Time however does have a gradient when viewed from infinity. However this in direct proportion to the changes that it measures so it does not support your assertion.
#### Thebox
• Neilep Level Member
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##### Re: The speed of time
« Reply #2 on: 27/06/2016 09:48:06 »
Darkness doesn't have a particle associated with it nor a field of any sort. Since time is inherent to the calculation of speed it cannot be faster than the very thing it determines. That would be paradoxical. Time however does have a gradient when viewed from infinity. However this in direct proportion to the changes that it measures so it does not support your assertion.
We move forward in space-time Jeff, time is always ahead of us and anything with a speed can not overtake what is ahead of it unless it can Quantum leap and displace itself by teleportation ability.
Look in any direction Jeff, if you walked in any of those directions you can ''see'' your future destination, the time is ahead of you and interwoven with your time. However when you approach your destination , time is still ahead of you, if you then looked behind you, time is now ahead of you in the direction of time you just came from that was your past but is now your future path if you walked back again to the beginning.
Time is instantaneous Jeff, there is nothing faster than time, light travels through time and space.
#### evan_au
• Neilep Level Member
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##### Re: The speed of time
« Reply #3 on: 27/06/2016 10:31:04 »
Quote from: TheBox
Time is instantaneous
I would say that when you take measurements in your lab, Time travels at 1 second per second.
So it's not instantaneous.
#### Alan McDougall
• Neilep Level Member
• Posts: 1285
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##### Re: The speed of time
« Reply #4 on: 27/06/2016 12:06:31 »
Quote from: TheBox
Time is instantaneous
I would say that when you take measurements in your lab, Time travels at 1 second per second.
So it's not instantaneous.
The question was about the possibility of something going faster than c, "until you changed it and started speculating about "time."
And from where does this ridiculous false idea come from that, "time is instantaneous" when my bedside clock vividly shows to the contrary, that time is moving at particular rate relative to gravity field surrounding it.
To get back to the topic
Examples of situations that exceed c
Space itself (and whatever occupies that space). According to the measured value of the Hubble constant much of the observed universe is now moving away from us faster than light.
A quantum probability function collapse. e.g. There is a well know effect known as the EPR where measuring one particle of a quantum entangled pair will effect the wave function of the other particle instantly (or possibly retroactively) regardless of how far away it is?
Alan
« Last Edit: 27/06/2016 12:22:55 by Alan McDougall »
#### Alan McDougall
• Neilep Level Member
• Posts: 1285
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##### Re: The speed of time
« Reply #5 on: 27/06/2016 12:29:25 »
Darkness doesn't have a particle associated with it nor a field of any sort. Since time is inherent to the calculation of speed it cannot be faster than the very thing it determines. That would be paradoxical. Time however does have a gradient when viewed from infinity. However this in direct proportion to the changes that it measures so it does not support your assertion.
We move forward in space-time Jeff, time is always ahead of us and anything with a speed can not overtake what is ahead of it unless it can Quantum leap and displace itself by teleportation ability.
Look in any direction Jeff, if you walked in any of those directions you can ''see'' your future destination, the time is ahead of you and interwoven with your time. However when you approach your destination , time is still ahead of you, if you then looked behind you, time is now ahead of you in the direction of time you just came from that was your past but is now your future path if you walked back again to the beginning.
Time is instantaneous Jeff, there is nothing faster than time, light travels through time and space.
Again your post is so ambiguous and that it makes absolutely no sense, and is full of falsehood?
Time my friend is not ahead of you or behind you, time is stuck to your body which has to obey the laws of physics and move with time in the only possible direction , into your as yet not experienced "future"
#### Thebox
• Neilep Level Member
• Posts: 3139
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##### Re: The speed of time
« Reply #6 on: 27/06/2016 13:04:59 »
Quote from: TheBox
Time is instantaneous
I would say that when you take measurements in your lab, Time travels at 1 second per second.
So it's not instantaneous.
Ok, so start from zero and end at 1 second which is a rather large increment of time, how much time fills the ''gap'' between 0 and 1?
#### Thebox
• Neilep Level Member
• Posts: 3139
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##### Re: The speed of time
« Reply #7 on: 27/06/2016 13:07:27 »
The question was about the possibility of something going faster than c, "until you changed it and started speculating about "time."
Yes, and I am discussing why time is faster than c.
#### The Naked Scientists Forum
##### Re: The speed of time
« Reply #7 on: 27/06/2016 13:07:27 » | 1,575 | 6,381 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.671875 | 3 | CC-MAIN-2016-44 | longest | en | 0.922583 |
http://www.jiskha.com/display.cgi?id=1264540547 | 1,495,817,890,000,000,000 | text/html | crawl-data/CC-MAIN-2017-22/segments/1495463608669.50/warc/CC-MAIN-20170526163521-20170526183521-00510.warc.gz | 673,628,694 | 3,958 | Physics
posted by on .
Starting from rest Maria pulls horizontally on her new wagon for 1.1 seconds. She exerts a constant force of 89N against a frictional force of 43N. The wagon's final velocity is 3.2m/s. If she is on a level ground, how much does the wagon weigh?
• Physics - ,
Fnet= m a= m *3.2/1.1
89-43=m * 3.2/1.1 solve for m
weight= mg
• Physics - ,
Use the fact that impulse
= (net force)*(time) = (momentum change)
The momentum given to the wagon is
(89 - 43)* 1.1 = 50.5 kg m/s
Divide that by the final velocity to get the mass in kg. | 177 | 555 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.671875 | 4 | CC-MAIN-2017-22 | latest | en | 0.887437 |
http://reference.wolfram.com/language/ref/BSplineBasis.html | 1,490,856,247,000,000,000 | text/html | crawl-data/CC-MAIN-2017-13/segments/1490218193284.93/warc/CC-MAIN-20170322212953-00158-ip-10-233-31-227.ec2.internal.warc.gz | 299,600,110 | 10,749 | # BSplineBasis
BSplineBasis[d,x]
gives the zeroth uniform B-spline basis function of degree d at x.
BSplineBasis[d,n,x]
gives the n uniform B-spline basis function of degree d.
BSplineBasis[{d,{u1,u2,}},n,x]
gives the n non-uniform B-spline basis function of degree d with knots at positions ui.
# Details
• Mathematical function, suitable for both symbolic and numerical manipulation.
• BSplineBasis[d,x] is equivalent to BSplineBasis[d,0,x].
• BSplineBasis[d,n,x] gives B-spline basis functions that have nonzero values only within the x interval between and .
• BSplineBasis[{d,{u1,u2,,um}},n,x] gives B-spline basis functions that have nonzero values only within the x interval between u1 and um.
• The knot positions ui must form a non-decreasing sequence.
• Possible values of n range from 0 to m-d-2.
• PiecewiseExpand can be used to expand symbolic BSplineBasis functions into explicit piecewise polynomials.
# Examples
open allclose all
## Basic Examples(4)
Evaluate a uniform cubic B-spline basis numerically:
In[1]:=
Out[1]=
Plot it:
In[1]:=
Out[1]=
Evaluate the second cubic B-spline basis with given knots:
In[1]:=
In[2]:=
Out[2]=
Plot all the cubic basis functions with given knots:
In[3]:=
Out[3]=
Symbolic derivative of B-spline basis:
In[1]:=
In[2]:=
Out[2]=
Plot of the derivatives:
In[3]:=
Out[3]= | 374 | 1,353 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.046875 | 3 | CC-MAIN-2017-13 | longest | en | 0.796824 |
https://circlecoder.com/remove-invalid-parentheses/ | 1,679,857,105,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296946445.46/warc/CC-MAIN-20230326173112-20230326203112-00605.warc.gz | 205,797,947 | 9,648 | Remove Invalid Parentheses Problem
Description
LeetCode Problem 301.
Given a string s that contains parentheses and letters, remove the minimum number of invalid parentheses to make the input string valid.
Return all the possible results. You may return the answer in any order.
Example 1:
``````1
2
Input: s = "()())()"
Output: ["(())()","()()()"]
``````
Example 2:
``````1
2
Input: s = "(a)())()"
Output: ["(a())()","(a)()()"]
``````
Example 3:
``````1
2
Input: s = ")("
Output: [""]
``````
Constraints:
• 1 <= s.length <= 25
• s consists of lowercase English letters and parentheses ‘(‘ and ‘)’.
• There will be at most 20 parentheses in s.
Sample C++ Code
``````1
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class Solution {
public:
set<string> paren;
bool checkValid(string s) {
stack<char> st;
for (int i = 0; i < s.size(); i ++) {
if (s[i] == '(')
st.push(s[i]);
else if (s[i] == ')') {
if (st.empty())
return false;
st.pop();
} else {
continue;
}
}
if (st.empty())
return true;
else
return false;
}
void dfs(string s, int idx, int l) {
if (s.size() == l) {
bool flag = checkValid(s);
if (flag)
paren.insert(s);
return;
}
if (idx > l || s.size() < l)
return;
if (s[idx] != '(' && s[idx] != ')')
dfs(s, idx+1, l);
else {
dfs(s, idx+1, l);
string ns = s.substr(0, idx)+s.substr(idx+1, s.size()-idx-1);
dfs(ns, idx, l);
}
}
vector<string> removeInvalidParentheses(string s) {
stack<char> st;
for (int i = 0; i < s.size(); i ++) {
if (s[i] == '(')
st.push(s[i]);
else if (s[i] == ')') {
if (!st.empty() && st.top() == '(')
st.pop();
else
st.push(s[i]);
}
}
int l = s.size() - st.size();
dfs(s, 0, l);
vector<string> ans;
for (auto x : paren)
ans.push_back(x);
return ans;
}
};
`````` | 670 | 1,859 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.953125 | 3 | CC-MAIN-2023-14 | latest | en | 0.179984 |
https://www.enotes.com/homework-help/how-would-you-teach-basic-maths-lesson-plan-coin-425423 | 1,490,554,336,000,000,000 | text/html | crawl-data/CC-MAIN-2017-13/segments/1490218189245.97/warc/CC-MAIN-20170322212949-00505-ip-10-233-31-227.ec2.internal.warc.gz | 906,537,174 | 12,711 | How would you teach a basic maths lesson plan on coin identification to a student with a mild intellectual disability that is integrated in your class?
tjbrewer | Elementary School Teacher | (Level 2) Associate Educator
Posted on
I've dealt with this before, as a Special Educator.
I focused on making the connection between the appearance and name. I would restrict the first lesson to pennies and quarters. My goal being to minimize confusion. I focused on making sure that the student knew more than 1 distinction between the two coins so that they can recognize them. E.g. Copper is a different color than silver (usually the 1st difference), Quarters are bigger, Quarters have Washington Pennies have Lincoln (not all Special Ed students will recognize this).
The second lesson I introduced Dimes. They look almost identical to pennies, so we focus on how they are a different metal, so are a different coin from pennies.
The third lesson I introduce nickels, and focus on how to tell them apart from Quarters. (Rough versus smooth edge, Washington versus Jefferson, Eagle versus Monticello).
Once the student can distinguish between the different coins, I then start to draw the connection between the coins and their values. Again go only one at a time with the Special Education kid, because their mnemonic tendency will lead to frustration that the Nickel is bigger than a Dime, but has less value. | 299 | 1,421 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.765625 | 3 | CC-MAIN-2017-13 | longest | en | 0.953211 |
https://www.lotterypost.com/blogentry/10887/postcomment | 1,603,604,457,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107887810.47/warc/CC-MAIN-20201025041701-20201025071701-00143.warc.gz | 803,194,831 | 11,885 | Welcome Guest
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# Odds of probability [ Random thoughts]
Published:
Last Edited: September 21, 2006, 10:24 pm
I am not a mathematican so I do not know how to calculate the odds.
So in the pick 3 game we have 220 box combos. By putting the numbers into groups like v-tracs or a coded system like the wheel code would reduce the odds of probability I would think.
So in my wheel code I have 20 groups that contain different amounts of numbers in a group from 4 to 18 box combos in a group.
So if all the numbers in a group are played and choosing one group my odds of probability is 20 to 1 on being correct.
So with tracking the 220 box combos individually in software we need to sort thru 220 possibilities to end up with a list of numbers.
So with tracking 20 groups I only need to sort thru 20 possibilities to end up with from 4 to 18 numbers when I choose one group.
Entry #687 | 265 | 1,002 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.71875 | 3 | CC-MAIN-2020-45 | latest | en | 0.889174 |
http://catking.in/2017/04/05/10-tips-to-crack-cet/ | 1,508,659,556,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187825154.68/warc/CC-MAIN-20171022075310-20171022095310-00261.warc.gz | 58,137,098 | 23,641 | 10 Tips to Crack CET
10 Tips to Crack CET
The Common Entrance Test (CET) is conducted by the Directorate of Technical Education (DTE), Maharashtra for admission to MBA colleges. It is a test of speed which contains 200 questions supposed to be answered in 150 minutes. Though the relieving part is that there is no negative marking. To ace the exam:
1. First of all, get hold of practice papers/mock tests that help build speed that is of utmost importance during this test. It is a test of speed and accuracy, and practice makes an aspirant perfect in this aforementioned test.
2. One major difference in CET (over other MBA entrances) is that questions from different test areas are jumbled. The first question might be from Verbal Ability and the question following the first could be from either Quantitative Aptitude or Logical Reasoning.
3. For any entrance, visiting NCERT books of elementary school is necessary; likewise for CET. Concept over application; in fact, concept along with application require this.
4. Tables, squares and cubes need to be on your tips for successful mental calculation. These are major areas of Arithmetic topic in quants.You should try to develop approximations, decimal-fractions and percentage for better performance.
5. Vocabulary can be improved by referring to newspapers and reading books; also by adhering to a dictionary regularly. You need to learn new words, adopt the right usage and ensure that you have an eye for mistakes. Apart from the above mentioned ways, click here to know why students score low in verbal
6. Identify your strengths and weaknesses because it get imperative to turn your weaknesses into strengths and refine your strengths further in the race to succeed.
7. Develop a discipline of solving questions 30 minutes per day. If you do that, you develop patience and speed to excel in the 150 minutes final test.
8. Once you’ve identified your strengths and weaknesses, it is really important to set small improvement targets and see if you are meeting those targets. You climb a ladder, one step a time, not all at once. It is also to analyze your progress time to time. One should also be aware of the CET exam pattern of the previous year
9. It’s not limited to pre-exam preparation. A strategic approach needs to be followed even during the test. Questions that cannot be answered in the first go can be marked to be revisited later. Doing this increases accuracy as well as total attempts in the exam.
10. Last but definitely not the least, as important as anything else is sufficient rest for the body. You should ensure that you eat healthy during the exam season and sleep well to be fresh enough to retain as well as apply. | 553 | 2,712 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.703125 | 3 | CC-MAIN-2017-43 | longest | en | 0.941232 |
https://www.extrica.com/article/16106 | 1,718,739,234,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861773.80/warc/CC-MAIN-20240618171806-20240618201806-00624.warc.gz | 673,362,507 | 21,778 | Published: 30 June 2015
# Measurement of plane vibrations of a two dimensional elastic structure
K. Ragulskis2
P. Paškevičius3
L. Patašienė4
A. Pauliukas5
L. Ragulskis6
1Vilnius Gediminas Technical University, Vilnius, Lithuania
2, 3, 4Kaunas University of Technology, Kaunas, Lithuania
5Aleksandras Stulginskis University, Akademija, Kaunas District, Lithuania
6Vytautas Magnus University, Kaunas, Lithuania
Corresponding Author:
Views 41
#### Abstract
Plane vibrations of a two dimensional elastic structure are analyzed in this paper. The vibrations taking place according to the eigenmode are represented by using the method of stroboscopic geometric moiré. This requires to perform the investigation for two mutually perpendicular directions of moiré fringes. Here the superimposed moiré technique is proposed to represent both images at the same time.
## 1. Introduction
In precise mechanical devices there is a great number of vibrating elastic elements. Plane vibrations of a two dimensional elastic structure are analyzed in the paper. The vibrations taking place according to the eigenmode are represented by using the method of stroboscopic geometric moiré. This requires to perform the investigation for two mutually perpendicular directions of moiré fringes. Thus one is to paint geometric moiré lines in the first direction and to perform the investigation. After that one is to abolish the previously painted lines and then to paint geometric moiré lines in the perpendicular direction and perform the investigation.
In this paper the superimposed moiré technique is proposed to represent both images at the same time. Though some engineering intuition may be required in the process of interpretation of superimposed images, but the advantage of the method is the fact that both systems of mutually perpendicular and parallel in the status of equilibrium lines are painted on the structure and remain on it the whole time. Also vibrations in both mutually perpendicular directions are analyzed at the same time, while in the conventional moiré approach first the vibrations in the direction of the $x$ axis are analyzed and later vibrations in the direction of the $y$ axis are analyzed.
The analysis is based on the material described in [1-3] and other related papers.
## 2. Theoretical investigation of the proposed measurement procedure
One dimensional problem is investigated. Moiré lines in the status of equilibrium are represented as:
1
${I}_{1}=\mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\frac{\pi }{\lambda }x,$
where $x$ is the coordinate, $\lambda$ determines the width of moiré lines, ${I}_{1}$ is the intensity of the image.
Moiré lines in the deflected state are represented as:
2
${I}_{2}=\mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\frac{\pi }{\lambda }\left(x-u\right),$
where $u$ is the displacement, ${I}_{2}$ is the intensity of the image. In the investigation it is assumed that:
3
$u=kx,$
where $k$ is a constant.
Intensity of the stroboscopic image is represented as:
4
${I}_{s}=\frac{{I}_{1}+{I}_{2}}{2}.$
Investigation of moiré images of this type was performed earlier. Here gaps between moiré lines are assuned to be wider. For this purpose the following special function is introduced:
5
$F\left(i,\lambda ,x\right)=\left\{\begin{array}{ll}0,& \frac{1}{2}+\left(1+i\right)j<\frac{x}{\lambda }<\frac{1}{2}+i+\left(1+i\right)j,\\ \mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\frac{\pi }{\lambda }x,& \mathrm{e}\mathrm{l}\mathrm{s}\mathrm{e}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e},\end{array}\right\j=0,±1,±2,\dots ,$
where $i=0,\mathrm{}1,\mathrm{}2,\dots$ is the width of the gap.
Moiré lines in the status of equilibrium are represented as:
6
${I}_{1}=F\left(i,\lambda ,x\right).$
Moiré lines in the deflected state are represented as:
7
${I}_{2}=F\left(i,\lambda ,x-u\right).$
Further it is assumed that $\lambda$ = 0.8 and $k$ = 0.1. ${I}_{1}$, ${I}_{2}$ and ${I}_{s}$ for $i=0,\mathrm{}1,\mathrm{}2,\mathrm{}3,\mathrm{}4$ are presented in Figs. 1-5.
Fig. 1I1, I2 and Is for i = 0
Results for $i$ = 0 correspond to the results presented earlier. Envelope of the stroboscopic image has 6 maximums inside the analyzed interval. From the results for $i$ = 1 it is seen that the envelope of the stroboscopic image has 3 maximums inside the analyzed interval. From the results for $i$ = 2 it is seen that the envelope of the stroboscopic image has 2 maximums inside the analyzed interval. From the results for $i$ = 3 it is seen that the envelope of the stroboscopic image has 1 maximum inside the analyzed interval. From the results for $i$ = 4 it is seen that the envelope of the stroboscopic image has 1 maximum inside the analyzed interval and the distance between the maximums is bigger than for the previous value of $i$. Thus from the presented results it can be concluded that with the increase of the width of the gap the intervals between the maximums of the envelope of intensity of the stroboscopic image increase. But it is possible to interpret the displacements from moiré images with gaps. Those gaps enable to interpret both moiré images of parallel lines in a two dimensional problem simultaneously and this can be seen from the two dimensional results presented further.
Fig. 2I1, I2 and Is for i = 1
Fig. 3I1, I2 and Is for i = 2
Fig. 4I1, I2 and Is for i = 3
Fig. 5I1, I2 and Is for i = 4
## 3. Conventional stroboscopic geometric moiré images of vibrating elastic structures
Square elastic structure with fixed lower boundary is analyzed. Stroboscopic geometric moiré images for the two conventional directions of fringes for the first eigenmode are shown in Fig. 6, for the second eigenmode in Fig. 7, for the third eigenmode in Fig. 8, for the fourth eigenmode in Fig. 9.
## 4. Superimposed moiré images of vibrating elastic structures
Superimposed stroboscopic geometric moiré images for the first eigenmode are shown in Fig. 10, for the second eigenmode in Fig. 11, for the third eigenmode in Fig. 12, for the fourth eigenmode in Fig. 13.
Fig. 6Stroboscopic geometric moiré images for the first eigenmode: a) the first direction of fringes, b) the second direction of fringes
a)
b)
Fig. 7Stroboscopic geometric moiré images for the second eigenmode: a) the first direction of fringes, b) the second direction of fringes
a)
b)
Fig. 8Stroboscopic geometric moiré images for the third eigenmode: a) the first direction of fringes, b) the second direction of fringes
a)
b)
Fig. 9Stroboscopic geometric moiré images for the fourth eigenmode: a) the first direction of fringes, b) the second direction of fringes
a)
b)
Fig. 10Superimposed stroboscopic geometric moiré image for the first eigenmode
Fig. 11Superimposed stroboscopic geometric moiré image for the second eigenmode
Fig. 12Superimposed stroboscopic geometric moiré image for the third eigenmode
Fig. 13Superimposed stroboscopic geometric moiré image for the fourth eigenmode
## 5. Conclusions
The superimposed moiré technique is proposed to represent both moiré images for the analysis of plane vibrations of two dimensional elastic structures at the same time. Some engineering intuition may be required in the process of interpretation of superimposed images, but the advantage of the method is the fact that both systems of mutually perpendicular and parallel in the status of equilibrium lines are painted on the structure and remain on it the whole time. Thus vibrations in both mutually perpendicular directions are analyzed at the same time.
The proposed technique of superimposed moiré analysis of plane vibrations of two dimensional elastic structures is applicable for the investigation of vibrations of precise mechanical devices.
#### References
• Ragulskis K., Maskeliūnas R., Zubavičius L. Analysis of structural vibrations using time averaged shadow moiré. Journal of Vibroengineering, Vol. 8, Issue 3, 2006, p. 26-29.
• Saunorienė L., Ragulskis M. Time – Averaged Moiré Fringes. Lambert Academic Publishing, 2010.
• Ragulskis M., Maskeliūnas R., Ragulskis L., Turla V. Investigation of dynamic displacements of lithographic press rubber roller by time average geometric moiré. Optics and Lasers in Engineering, Vol. 43, 2005, p. 951-962. | 2,120 | 8,222 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 28, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.578125 | 3 | CC-MAIN-2024-26 | latest | en | 0.897486 |
https://tex.stackexchange.com/questions/295756/obtaining-left-aligned-equation | 1,568,717,394,000,000,000 | text/html | crawl-data/CC-MAIN-2019-39/segments/1568514573070.36/warc/CC-MAIN-20190917101137-20190917123137-00432.warc.gz | 715,159,942 | 28,133 | # Obtaining left aligned equation [duplicate]
How can I align this equation to the left?
$$\beta_1=(n_{20}+n_{02}) \\ \beta_2={(n_{20}-n_{02})^2}+4{n_{11}}^2 \\ \beta_3={(n_{30}-3n_{12})^2}+{(3n_{21}-n_{03})^2} \\ \beta_4={(n_{30}+n_{12})^2}+{(n_{21}-n_{03})^2} \\ \beta_5=(n_{30}-3n_{12})(n_{30}+n_{12})[{(n_{30}+n_{12})^2}-3{(n_{21}+n_{03})^2}]+(3n_{21}-n_{03})(n_{21}+n_{03})[3{(n_{30}+n_{12})^2}-{(n_{21}+n_{03})^2} \\ \beta_6=(n_{20}-n_{02})[{(n_{30}+n_{12})^2}-{(n_{21}+n_{03})^2}]+4n_{11}(n_{30}+n_{12})(n_{21}+n_{03}) \\ \beta_7=(3n_{21}-n_{03})(n_{30}+n_{12})[{(n_{30}+n_{12})^2}-3{(n_{21}+n_{03})^2}]-(n_{30}+3n_{12})(n_{21}+n_{03})[3{(n_{30}+n_{12})^2}-{(n_{21}+n_{03})^2}] \label{Eq:momentsHU}$$
## marked as duplicate by Community♦Feb 24 '16 at 21:14
It can't work: equation is for one-lined equations. Use aligned from amsmath, are aligned inside equation, if you want only one equation number. The longest equations will have to be split again, which you can do, say, with multlined:
\documentclass[a4paper, 11pt]{book}
\usepackage[utf8]{inputenc}
\usepackage[showframe]{geometry}
\usepackage{mathtools}
\begin{document}
\begin{aligned} \beta_1 & =(n_{20}+n_{02}) \\ \beta_2 & ={(n_{20}-n_{02})^2}+4{n_{11}}^2 \\ \beta_3 & ={(n_{30}-3n_{12})^2}+{(3n_{21}-n_{03})^2} \\ \beta_4 & ={(n_{30}+n_{12})^2}+{(n_{21}-n_{03})^2} \\ \beta_5 & =\!\begin{multlined}[t] (n_{30}-3n_{12})(n_{30}+n_{12})[{(n_{30}+n_{12})^2}-3{(n_{21}+n_{03})^2}] \\+(3n_{21}-n_{03})(n_{21}+n_{03})[3{(n_{30}+n_{12})^2}-{(n_{21}+n_{03})^2} \end{multlined} \\ \beta_6 & =(n_{20}-n_{02})[{(n_{30}+n_{12})^2}-{(n_{21}+n_{03})^2}]+4n_{11}(n_{30}+n_{12})(n_{21}+n_{03}) \\ \beta_7 & =\!\begin{multlined}[t] (3n_{21}-n_{03})(n_{30}+n_{12})[{(n_{30}+n_{12})^2}-3{(n_{21}+n_{03})^2}]\\ -(n_{30}+3n_{12})(n_{21}+n_{03})[3{(n_{30}+n_{12})^2}-{(n_{21}+n_{03})^2}] \end{multlined} \label{Eq:momentsHU} \end{aligned}
\end{document}
I would just set it using align* (include \usepackage{amsmath} in your preamble) and break the lengthy lines at appropriate locations. You may consider using flalign* as well:
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}
\beta_1 &= (n_{20} + n_{02}) \\
\beta_2 &= (n_{20} - n_{02})^2 + 4 n_{11}^2 \\
\beta_3 &= (n_{30} - 3n_{12})^2 + (3n_{21} - n_{03})^2 \\
\beta_4 &= (n_{30} + n_{12})^2 + (n_{21} - n_{03})^2 \\
\beta_5 &= (n_{30} - 3n_{12})(n_{30} + n_{12})[(n_{30} + n_{12})^2 - 3(n_{21} + n_{03})^2] + {} \\
&\phantom{{}={}} \qquad (3n_{21} - n_{03})(n_{21} + n_{03})[3(n_{30} + n_{12})^2 - (n_{21} + n_{03})^2 \\
\beta_6 &= (n_{20} - n_{02})[(n_{30} + n_{12})^2 - (n_{21} + n_{03})^2] + 4n_{11}(n_{30} + n_{12})(n_{21} + n_{03}) \\
\beta_7 &= (3n_{21} - n_{03})(n_{30} + n_{12})[(n_{30} + n_{12})^2 - 3(n_{21} + n_{03})^2] - {} \\
&\phantom{{}={}} \qquad (n_{30} + 3n_{12})(n_{21} + n_{03})[3(n_{30} + n_{12})^2 - (n_{21} + n_{03})^2]
\end{align*}
\end{document}
This is more or less the same question as Align equation left. You can use the flalign environment or load your documentclass with the fleqn option. | 1,461 | 3,066 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0} | 3.875 | 4 | CC-MAIN-2019-39 | latest | en | 0.598343 |
https://zuoti.pro/question/1564780/question-3-an-object-with-mass-79-kg-and-speed-24 | 1,679,549,836,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296944996.49/warc/CC-MAIN-20230323034459-20230323064459-00497.warc.gz | 1,227,617,318 | 9,467 | Question
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We have two identical balls in terms of ... | 1,365 | 5,756 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.25 | 3 | CC-MAIN-2017-43 | longest | en | 0.944487 |
https://cgcookie.com/course/fundamentals-of-dynamics/ | 1,498,720,606,000,000,000 | text/html | crawl-data/CC-MAIN-2017-26/segments/1498128323889.3/warc/CC-MAIN-20170629070237-20170629090237-00148.warc.gz | 754,877,730 | 19,537 | Fundamentals of Dynamics
This course is a part of the
Introduction to Blender
Learning Flow
• 149 lessons
• 12hrs 21min
• 8 exercises
67533 members taking this flow
Fundamentals of Dynamics
This course introduces you to world of dynamic simulation! Things like smoke, fire, soft and hard falling objects, fabric, pouring liquids, and more – All of this is accomplished with dynamic simulation in Blender. The key is that the computer does most of the work for you, as opposed to manually having to animate super complex motion like cloth blowing in the wind. Dynamic simulation is a powerful asset in the digital artists’ toolbox.
Introduction
Watch this video to see a brief overview of what the course is all about!
Particles
Control points in 3D space to mimic effects like dust in the air, snow falling from the sky, confetti, and much more.
Hair
Another version of particle effect capable of simulating large quantities of hairs follicles.
Force Fields
Objects that mimic natural phenomena like wind, gravity, magnetism, and more.
Rigid Bodies
Dynamic simulation of hard, rigid (non-deforming) objects.
Cloth
Dynamic simulation of fabric, clothing, etc.
Soft Bodies
Dynamic simulation of soft, deformable (non-rigid) objects.
Smoke
Simulation representative of fire and smoke.
Fluids
Dynamic simulation of liquids.
Dynamic Paint
Type of dynamic simulation that allows objects to affect each other's surface qualities.
The fun side of physics
Dynamic Simulation is one of the most “magical” things we can do with Blender. In short, it’s a way to create complex procedural animations that mimic physics.
If you’ve taken a high school math class, you’ll understand that physics calculations require a lot of math. Fortunately, we don’t have to do any math in order to make our models bounce, shatter, splash, and squish – Blender does that math for us! We just give it input materials and Blender does the rest.
In this course we will be taking an introductory look at the nine types of dynamic effects that are available in Blender:
• Particles: Dust floating in the air, snow falling from the sky, sparks from a welder, and so much more. This type of effect involves the simulation of large quantities of small bits; i.e. particles.
• Hair: Exactly what it sounds like.
• Force Fields: Blender objects that mimic real-world forces like gravity, magnetism, vortex, etc.
• Rigid Bodies: You know when a Jenga tower falls? It’s like that: The simulation of rigid, non-deforming objects.
• Cloth: Also, what it sounds like – The simulation of fabric.
• Soft Bodies: In contrast to rigid bodies, this kind of simulation accounts for deformation of objects.
• Smoke: The simulation of smoke and fire! Muahaha
• Fluid: Liquid simulation like a pipe gushing water for example.
• Dynamic Paint: Think of this like writing with your finger on a foggy mirror. This is an interesting simulation genre that interprets object-to-object interaction into surface information like vertex weights.
We’ll also take a look at examples and comparisons of all the common settings, so you’ll be ready to rock out on your own dynamics project right away.
Harnessing the power of simulations is the first step to creating stunning VFX, so today is a great day to start learning dynamics!
• Replies: 0
Is there a bug with Smoke dynamics in v2.78? At some point in every file, while changing values, it just stops working. Sometimes Ctrl-A plays as if the timeline is empty, no simuation. Sometimes it’s okay but render just become solid black. Once a file gets this way, nothing brings it back, but if I reload the last saved (uncorrupted) version, it’s okay. I don’t have this issue with any other dynamics type.
1 month ago
• Replies: 0
Are these principles that are used in other 3d programs as well.. like Z….
3 months ago
• Replies: 1
My Soft body is working, but it’s not Showing in Render view or final render. It’s Baked. What Am I missing? (frame 1-80)
6 months ago
• Replies: 0
Interesting, perhaps the little camera icon in the modifier stack is unchecked? If that doesn’t solve it, start a post over on the Community and we’ll get it worked out.
6 months ago
• Replies: 1
Awesome, if a bit breathless a lightning tour of the candy store! one thing I would like a little bit more of is typical applications for the different dynamics, but perhaps this belongs in a more advanced course (with an exercise or two at the end). Another thing I would like to understand more is how to mix animation and dynamics in the same scene. We got into that a little with “firing the monkey at the cubes” but I would like to understand it in more depth; like, “can I parent an active dynamic object to an animated object and have them move around the scene together?” when is it animation OR dynamics, and when/how can we use both seamlessly? I am very new to Blender so this may be a really dumb question, but seems like that would make it a good topic to explore just a little bit more in this intro…
7 months ago
• Replies: 0
It is a lightning tour indeed! It’s such a huge topic that I had to stick to the very basics or it would have gone on forever. More of those questions will be answered in a more advanced course, but I don’t have any dates on when that will be yet.
7 months ago
• Replies: 1
Perfect as a quick revision for me, before working on projects with dynamics. When can we expect a moderate or advanced course?? Any plans?
9 months ago
• Replies: 0
Hey Harj, I am certainly hoping to do more advanced courses in the future, but I have no way of knowing yet when that will be. I have a few other things to work on first
9 months ago
Continued Learning
Have you completed the flow? Extend your knowledge and become a super cookie star. Here are some related courses and exercises you will gain some value out of. | 1,305 | 5,864 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2017-26 | longest | en | 0.88335 |
https://github.com/clojure/core.match/wiki/Understanding-the-algorithm | 1,722,683,611,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640365107.3/warc/CC-MAIN-20240803091113-20240803121113-00392.warc.gz | 225,701,806 | 39,722 | / core.match Public
# Understanding the algorithm
core.match employs the algorithm described in Luc Maranget's paper Compiling Pattern Matching to good Decision Trees. What follows is a slower paced description of the algorithm.
## Necessity and Specialization
Consider the following pattern match:
```(let [x true
y true
z true]
(match [x y z]
[_ false true] 1
[false true _ ] 2
[_ _ false] 3
[_ _ true] 4
:else 5))```
It's clear that the above expression will return `4`, as the values match the fourth clause.
As a visual aid we will show the pattern matrix as it goes through a series of transformations in the following form:
``````Fig. 1
x y z
-------
[_ f t] 1
[f t _] 2
[_ _ f] 3
[_ _ t] 4
[_ _ _] 5
``````
For brevity and alignment we've replaced `true` with `t` and `false` with `f`. The final line is represented as a line of wildcards. Finally we label the columns with the variables (in this context we may use the word "occurrence") and the rows.
Maranget's algorithm is based on a surprisingly simple insight. The semantics of pattern matching dictate that we evaluate the clauses one at a time from top to bottom. Each clause itself is evaluated left to right. Even so this gives us a surprising amount of leeway because of wildcards.
Observe that for the first line to succeed we need never consider `x`. But this is an important observation! Maranget's approach produces optimal decision trees by finding the columns that must be tested and testing them first. This is done by scoring columns to determine what to test next.
So how do we score a column? Wildcards obviously shouldn't add anything to a column's score. But in addition real patterns below a wildcard should also not add to a column's score. Given these simple rules the scored matrix looks like this:
``````Fig. 2
x y z
-------
[0 1 1] 1
[0 1 0] 2
[0 0 0] 3
[0 0 0] 4
[0 0 0] 5
``````
It's now immediately clear that the `y` column should be tested first. Maranget's algorithm modifies the pattern matrix by swapping the column that should be tested so that it is the new first column giving us:
``````Fig. 3
y x z
-------
[f _ t] 1
[t f _] 2
[_ _ f] 3
[_ _ t] 4
[_ _ _] 5
``````
At this point we now can generate the first switch of the decision tree. From this step we will compute two new pattern matrices - one will represent a specialized matrix - the matrix that remains after we've made a successful match, and the default matrix - the matrix if we don't find a match.
The specialized matrix will look like the following:
``````Fig. 4
x z
-----
[_ t] 1
``````
We've dropped the `y` column as we assume we successfully matched `f`. The default matrix will look like the following:
``````Fig. 5
y x z
-------
[t f _] 2
[_ _ f] 3
[_ _ t] 4
[_ _ _] 5
``````
At this point it should hopefully be clear that we can now recursively apply the above process of selecting the necessary column, and once again splitting between the specialized matrix and the default matrix.
For completeness we illustrate the remaining steps for `Fig. 4`. We pick the necessary column.
``````Fig. 6
z x
-----
[t _] 1
``````
When we specialize this matrix on `t` we'll be left with:
``````Fig. 7
x
---
[_] 1
``````
We are left with a pattern matrix that only has one row and it is a row of wildcards - we have arrived at a terminal node of the decision tree and should return `1`.
Note: A close reading of Maranget's paper (pg 3) will show that we've diverged a bit from the original approach - wildcards are carried forward during specialization in Maranget's paper. This is a code size issue, particularly under the JVM where there are method size limits. core.match on the JVM backtracks via preallocated exceptions as these will get optimized into long jumps. This allows defaults to be shared without carrying them forward.
## The Decision Tree
Every time we specialize a pattern matrix we will produce a `SwitchNode`, this node will eventually be compiled down to a test and will dispatch either to the results of the specialized matrix or the default matrix. If we get to a case where we have a pattern matrix containing only a row of wildcards as illustrated in `Fig. 7` we will produce a `LeafNode`. We also produce a `LeafNode` when we have a pattern matrix where the rows are empty (no column).
It is possible that we may arrive at a matrix which has no rows. This means no match exists and for this case we will emit a `FailNode` which will be compiled to a backtrack.
The decision tree is a directed acyclic graph produced by the described algorithm and it will be composed of `SwitchNode`, `LeafNode`, and `FailNode` instances.
As Maranget shows in his paper - the simple idea of picking the necessary column produces compact decision trees.
## Constructors
Thus far we have only demonstrated matching on literals. Pattern matching derives most of its power and benefit when applied to data structures. Consider the following match expression where we match on a Clojure vector:
```(let [v [1 2 4]
x true]
(match [v x]
[[_ 2 2] true] 1
[[_ 2 3] false] 2
[[1 2 4] _] 3
:else 4))```
This can be represented with the following matrix:
``````Fig. 8
v x
-----------
[[_ 2 2] t] 1
[[_ 2 3] f] 2
[[1 2 4] _] 3
[ _ _] 4
``````
How can we match the contents of the vector? We simply need to transform the matrix into a form where Maranget's algorithm can work. When we specialize in this case, we are not specializing on a particular value, but whether the value to be tested is a vector of a certain size.
`v` in this case is already the necessary column so no swapping is required. The specialized matrix will look like `Fig. 9`. Notice that we've introduced new occurrences `v_0`, `v_1`, and `v_2` for each element of the vector pattern. These new occurences are associated with a binding expression so that they will be properly bound to the correct value in the vector. For example `v_0` will be bound via `(nth v 0)`.
``````Fig. 9
v_0 v_1 v_2 x
---------------
[ _ 2 2 t] 1
[ _ 2 3 f] 2
[ 1 2 4 _] 3
``````
And the default matrix will look like:
``````Fig. 10
v x
-----
[_ _] 4
``````
So we can see that for complex patterns we simply try to convert the pattern matrix into a form such that Maranget's approach again applies. A quick glance at `Fig. 9` shows that `v1` will be the next necessary column.
Maranget illustrates this point with ML style data types but it should be clear that the idea works just as well for Clojure vectors as they are also positional in the same way that ML style data type constructors are.
## Map patterns
Maranget's paper does not go into the problem of matching maps (dictionaries). However it's not too difficult to see how the approach can be extended to optimize matching maps.
Consider the following match expression:
```(match [x]
[{:a _ :b 2}] 1
[{:a 1 :b 1}] 2
[{:c 3 :d _ :e 4}] 3
:else 4)```
We can make map pattern matching conform to Maranget's approach - we consider all keys. The above translates to the following pattern matrix:
``````Fig. 11
x
------------------
[{:a _ :b 2} ] 1
[{:a 1 :b 1} ] 2
[{:c 3 :d _ :e 4}] 3
[ _ ] 4
``````
If we specialize this matrix on the map pattern we will get `Fig. 12`. As with vector patterns we introduce new occurrences which will be bound to corresponding values in the original occurrence. Maps are unordered so we must sort the keys.
``````Fig. 12
x_a x_b x_c x_d x_e
----------------------
[(_) 2 _ _ _ ] 1
[ 1 1 _ _ _ ] 2
[ _ _ 3 (_) 4 ] 3
``````
The wildcard patterns specified by the user are not real wildcards, they must be tested - the user has specified that key must exist regardless of the value thus we write `(_)` to differentiate between that and the real wildcards - those values that are implied that truly do not need to be tested.
We now have a pattern matrix that can be optimized by Maranget's procedure. In core.match the existential patterns for the keys scores lower than real constructor patterns, thus `x_b` will be the next necessary column.
## Pseudo-patterns
As alluded in section 4 in the original paper or-patterns are not really patterns - they simply introduce rows. Prior to real pattern matrix specialization the top left pattern is tested to see if it is a pseudo-pattern. If it is a pseudo-pattern the matrix is specialized by it. However, pseudo-patterns are generally erased so no actual source will ever be generated for them.
### or-patterns
Consider the following pattern matrix:
``````Fig. 13
x y
---------------------
[_ (:or 1 2)] 1
[(:or 3 4) _ ] 2
``````
`y` will be chosen and the matrix reordered:
``````Fig. 14
y x
---------------------
[(:or 1 2) _ ] 1
[ _ (:or 3 4)] 2
``````
Before actual specialization we expand the pseudo-patterns:
``````Fig. 15
y x
---------------------
[ 1 _ ] 1
[ 2 _ ] 1
[ _ (:or 3 4)] 2
``````
Note that we did not attempt to expand all the or-patterns at once. This is important as or-patterns may appear in other pseudo-patterns.
### function application patterns
It may not be immediately apparent but function application patterns are also best implemented as pseudo-patterns. Function application patterns work by taking an occurrence, applying a function to it and pattern matching on the result of that function application. The insight is that the function application introduces a new occurrence.
Consider the following pattern matrix:
``````Fig. 16
n
---------------
[ (1 :<< inc) ] 1
[ (2 :<< dec) ] 2
``````
The first pattern will increment `n` and pattern match the result. The second will decrement and do the same.
The insight is that we must introduce an occurrence in order to pattern match the result and we replace the original function application pattern with a wildcard:
``````Fig. 17
app_n (inc n) n
-----------------------------
[ 1 _ ] 1
[ _ (2 :<< dec) ] 2
``````
Maranget's algorithm may now proceed as usual. | 2,669 | 10,063 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.515625 | 4 | CC-MAIN-2024-33 | latest | en | 0.882588 |
https://www.hpmuseum.org/forum/thread-3634-post-33396.html | 1,638,887,728,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964363400.19/warc/CC-MAIN-20211207140255-20211207170255-00362.warc.gz | 900,223,491 | 14,051 | One-liner mini-challenge [HP-71B]
04-13-2015, 08:14 PM (This post was last modified: 04-14-2015 03:30 AM by Gerson W. Barbosa.)
Post: #1
Gerson W. Barbosa Senior Member Posts: 1,419 Joined: Dec 2013
One-liner mini-challenge [HP-71B]
$\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1} {\sqrt{4}}+\frac{1}{\sqrt{5}}+\cdots+\frac{1}{\sqrt{n}}$
Write an HP-71B BASIC program and use it to evaluate the sum above when n = 10¹². The program must fit in one line. Accuracy in the widest range as
possible is desirable, but if you manage to show 12 correct digits for that particular required n, then it is ok. I will present an 80-character long
program that gives accurate results for n as low as 30, in less than 300 milliseconds.
This can be extended to other HP calculators and the WP-34S as well. As a reference, a 21-step program on the latter gives 12 accurate digits when n =
30 and 33 or 34 correct digits for n around 100000 and greater.
Have fun!
Gerson.
EDITED TO CHANGE THE SUBJECT TITLE AS THE MATH ROM IS NOT REQUIRED.
04-13-2015, 10:14 PM
Post: #2
Paul Dale Senior Member Posts: 1,726 Joined: Dec 2013
RE: One-liner mini-challenge [HP-71B wITH Math ROM]
At this point I wish I'd implemented some additional functions in the 34S. The Hurwitz zeta function or polygamma would help a lot here.
- Pauli
04-13-2015, 10:34 PM
Post: #3
Gerson W. Barbosa Senior Member Posts: 1,419 Joined: Dec 2013
RE: One-liner mini-challenge [HP-71B wITH Math ROM]
Yes, Hurwitz zeta function along with the existing zeta function would allow for exact answers. I wonder if the Prime has it.
Gerson.
04-13-2015, 11:40 PM
Post: #4
Paul Dale Senior Member Posts: 1,726 Joined: Dec 2013
RE: One-liner mini-challenge [HP-71B wITH Math ROM]
And the result can also be expressed fairly simply using the polygamma or polylogarithm functions -- all of these functions are interrelated.
Your reference lists an expansion of Hurwitz zeta in terms of the Bernoulli polynomials valid for integer arguments such as here. These in turn are defined by the Bernoulli numbers which are available on the 34S. Any program taking this approach would be extremely slow I suspect.
- Pauli
04-13-2015, 11:54 PM (This post was last modified: 04-14-2015 02:41 PM by Gerson W. Barbosa.)
Post: #5
Gerson W. Barbosa Senior Member Posts: 1,419 Joined: Dec 2013
RE: One-liner mini-challenge [HP-71B wITH Math ROM]
On the WP 34S I get, for instance:
34 A --> 10.28708841498052 (after 2.5 seconds, DBLOFF)
Almost instantly on the HP-71B:
10.2870884151
but only because I use a hard-coded constant (no point taking about three minutes to compute it on the HP-71B for this application).
Gerson.
P.S.: My HP-41C displays all ten digits correctly :-)
34 XEQ SR --> 10.28708841
About 1.8 seconds, regardless of the argument, 22 steps.
04-14-2015, 01:06 AM (This post was last modified: 04-14-2015 01:08 AM by Gerson W. Barbosa.)
Post: #6
Gerson W. Barbosa Senior Member Posts: 1,419 Joined: Dec 2013
RE: One-liner mini-challenge [HP-71B]
(04-13-2015 11:54 PM)Gerson W. Barbosa Wrote: but only because I use a hard-coded constant
As a consequence, the Math ROM is optional, contrary to the original subject title.
04-14-2015, 12:53 PM (This post was last modified: 04-14-2015 12:59 PM by Gerson W. Barbosa.)
Post: #7
Gerson W. Barbosa Senior Member Posts: 1,419 Joined: Dec 2013
RE: One-liner mini-challenge [HP-71B]
(04-13-2015 11:40 PM)Paul Dale Wrote: Your reference lists an expansion of Hurwitz zeta in terms of the Bernoulli polynomials valid for integer arguments such as here.
In fact one of the arguments I use is not integer, but the reason I wasn't able to use the integral representation of Hurwitz zeta on the HP-71B is that it requires Re(s) to be greater than 1. Anyway, I'm using an empirical approximation I'd found prior to checking the Wikipedia article. So far I have only the first few terms of the series, but they suffice for this application.
If Hurwitz zeta were available, then an exact solution would be possible:
$\zeta\left ( \frac{1}{2} \right )= \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+ \frac{1}{\sqrt{5}}+\cdots+\frac{1}{\sqrt{n}}+\frac{1}{\sqrt{n+1}}+\frac{1}{\sqrt{n+2}}+\cdots$
$\zeta\left ( \frac{1}{2},n+1 \right )=\frac{1}{\sqrt{n+1}}+\frac{1}{\sqrt{n+2}}+\frac{1}{\sqrt{n+3}}+\frac{1}{\sqrt{n+4}}+\cdots$
$\therefore \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+ \frac{1}{\sqrt{5}}+\cdots+\frac{1}{\sqrt{n}}= \zeta \left ( \frac{1}{2} \right )- \zeta \left ( \frac{1}{2},n+1 \right )$
We can use W|A to check our previous result for n = 34:
zeta(1/2) - zeta(1/2, 35)
Gerson.
04-14-2015, 06:55 PM (This post was last modified: 04-14-2015 07:02 PM by Valentin Albillo.)
Post: #8
Valentin Albillo Senior Member Posts: 776 Joined: Feb 2015
RE: One-liner mini-challenge [HP-71B]
Hi, Gerson:
(04-13-2015 08:14 PM)Gerson W. Barbosa Wrote: $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1} {\sqrt{4}}+\frac{1}{\sqrt{5}}+\cdots+\frac{1}{\sqrt{n}}$
Write an HP-71B BASIC program and use it to evaluate the sum above when n = 10¹². The program must fit in one line. Accuracy in the widest range as possible is desirable, but if you manage to show 12 correct digits for that particular required n, then it is ok. I will present an 80-character long program that gives accurate results for n as low as 30, in less than 300 milliseconds.
.
I guess this 63-char command-line expression will do, which once keyed in can be executed repeatedly by pressing [ENDLINE] and reusing it from the command stack:
INPUTN@2*N^.5+1/(2*N^.5)*(1-1/(12*N)+1/(192*N^3))-1.46035450881
? 34
10.287088415
? 100
18.5896038248
? 1E12
1999998.53965
? 30
9.58513017659
? 25
8.63931219119
? 20
7.59525502536
? 15
6.41399460857
? 10
5.02099790236
If desired, it can be turned into a program by simply issuing a line number in front of it (say "1 "), which will store it as a 56-byte BASIC program executable with RUN. The running time is utterly negligible.
The constant is ζ(1/2). The expression can be shortened by 6 additional characters by simply getting rid of the parentheses in the denominators (i.e.: /12/n instead of /(12*n)) but I've left it that way for clarity, and the last term may be omitted altogether for a shorter expression with somewhat reduced (but still sufficient for the challenge) accuracy.
Regards.
V.
All My Articles & other Materials here: Valentin Albillo's HP Collection
04-14-2015, 07:04 PM
Post: #9
Massimo Gnerucci Senior Member Posts: 2,352 Joined: Dec 2013
RE: One-liner mini-challenge [HP-71B]
The Prof is back! :)
Greetings,
Massimo
-+×÷ ↔ left is right and right is wrong
04-14-2015, 08:45 PM (This post was last modified: 04-15-2015 12:00 AM by Gerson W. Barbosa.)
Post: #10
Gerson W. Barbosa Senior Member Posts: 1,419 Joined: Dec 2013
RE: One-liner mini-challenge [HP-71B]
(04-14-2015 06:55 PM)Valentin Albillo Wrote: Hi, Gerson:
(04-13-2015 08:14 PM)Gerson W. Barbosa Wrote: $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1} {\sqrt{4}}+\frac{1}{\sqrt{5}}+\cdots+\frac{1}{\sqrt{n}}$
Write an HP-71B BASIC program and use it to evaluate the sum above when n = 10¹². The program must fit in one line. Accuracy in the widest range as possible is desirable, but if you manage to show 12 correct digits for that particular required n, then it is ok. I will present an 80-character long program that gives accurate results for n as low as 30, in less than 300 milliseconds.
.
I guess this 63-char command-line expression will do, which once keyed in can be executed repeatedly by pressing [ENDLINE] and reusing it from the command stack:
INPUTN@2*N^.5+1/(2*N^.5)*(1-1/(12*N)+1/(192*N^3))-1.46035450881
? 34
10.287088415
? 100
18.5896038248
? 1E12
1999998.53965
? 30
9.58513017659
? 25
8.63931219119
? 20
7.59525502536
? 15
6.41399460857
? 10
5.02099790236
If desired, it can be turned into a program by simply issuing a line number in front of it (say "1 "), which will store it as a 56-byte BASIC program executable with RUN. The running time is utterly negligible.
The constant is ζ(1/2). The expression can be shortened by 6 additional characters by simply getting rid of the parentheses in the denominators (i.e.: /12/n instead of /(12*n)) but I've left it that way for clarity, and the last term may be omitted altogether for a shorter expression with somewhat reduced (but still sufficient for the challenge) accuracy.
Regards.
V.
Hello Valentin,
Congratulations for your nicer and even shorter solution! Your results match mine except for n=34 and n=30. Yours are better, BTW. I get 10.2870884151 and 9.58513017649, respectively.
Here is my original HP-71B program:
1 INPUT N @ A=2*SQR(N) @ B=1/A @ C=B/(12*N) @ D=C/(16*N*N) @ A+B-C+D-1.46035450881
By removing two pairs of parentheses and the spaces it can shortened to 65 characters:
1 INPUTN@A=2*SQR(N)@B=1/A@C=B/12/N@D=C/16/N/N@A+B-C+D-1.46035450881
I won't copy your idea of replacing SQR with ^.5. Also, if you remove one pair of parentheses, your program will end up being shorter anyway :-)
Here is the expression I've come up with:
$\sum_{k=1}^{n}\frac{1}{\sqrt{k}}\simeq \zeta\left ( \frac{1}{2} \right )+2\sqrt{n}+\frac{1}{2\sqrt{n}}-\frac{1}{24\sqrt{n^{3}}}+\frac{15}{8}\cdot \frac{1}{720\sqrt{n^{7}}}$
Since this was done experimentally and manually, it is possible that not all terms are correct. (*)
Best regards,
Gerson.
P.S.:
(*) Both series appear to be equivalent, at least for x >= 0:
http://www.wolframalpha.com/input/?i=Sim...2%29%29%5D
(somewhat lazy to check it by hand)
WP 34s program:
Code:
001 LBL A 002 ENTER[^] 003 [sqrt] 004 STO+ X 005 ENTER[^] 006 1/x 007 + 008 RCL L 009 # 012 010 / 011 RCL/ Z 012 - 013 RCL L 014 # 016 015 / 016 RCL/ Z 017 RCL/ Z 018 + 019 # 1/2 020 [zeta] 021 + 022 END
This was actually my first program, the HP-71B being a straightforward conversion.
04-14-2015, 08:54 PM
Post: #11
Gerson W. Barbosa Senior Member Posts: 1,419 Joined: Dec 2013
RE: One-liner mini-challenge [HP-71B]
(04-14-2015 07:04 PM)Massimo Gnerucci Wrote: The Prof is back!
Hopefully back to stay. Valentin really knows how to concoct a challenge. Furthermore, his explanations are outstanding!
Here is an HP-41 solution:
Code:
01 LBL 'SR 02 ENTER^ 03 SQRT 04 STO+ X 05 ENTER^ 06 1/X 07 + 08 LASTX 09 12 10 / 11 RCL Z 12 / 13 - 14 LASTX 15 16 16 / 17 RCL Z 18 X^2 19 / 20 + 21 1.460354508 22 - .END.
Perhaps Valentin's formula makes for a shorter program, in case anyone wants to try it.
Best regards,
Gerson.
04-17-2015, 01:46 PM (This post was last modified: 04-17-2015 07:19 PM by Gerson W. Barbosa.)
Post: #12
Gerson W. Barbosa Senior Member Posts: 1,419 Joined: Dec 2013
RE: One-liner mini-challenge [HP-71B]
Perhaps this is a more interesting problem:
$\sqrt{1}+\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{5}+\cdots+\sqrt{n}$
And here is a simple WP 34s solution:
Code:
001 LBL A 002 RCL+ X 003 STO+ X 004 ENTER^ 005 INC X 006 INC X 007 INC X 008 STO* Y 009 Rv 010 INC X 011 RCL L 012 sqrt 013 / 014 # 024 015 / 016 # 1/2 017 +/- 018 zeta 019 + 020 END
100 A --> 671.462947108
This of course will fit in one HP-71B programming line, but I'll leave that as an exercise for the reader :-)
Gerson.
P.S.: Slightly more accurate:
Code:
001 LBL A 002 FILL 003 3 004 y^x 005 sqrt 006 STO+ X 007 3 008 / 009 #003 010 #004 011 RCL* T 012 / 013 RCL* Y 014 # 015 015 R^ 016 x^2 017 STO/ Y 018 x<> L 019 SDL 001 020 x<> L 021 STO+ X 022 RCL+ L 023 + 024 RCL/ Y 025 1/x 026 + 027 + 028 # 1/2 029 +/- 030 zeta 031 + 032 END
079 A --> 423.352471483
100 A --> 671.462947103
Obviously, no optimization attempts here.
04-19-2015, 03:32 PM
Post: #13
Valentin Albillo Senior Member Posts: 776 Joined: Feb 2015
RE: One-liner mini-challenge [HP-71B]
.
Hi, Gerson:
(04-17-2015 01:46 PM)Gerson W. Barbosa Wrote: Perhaps this is a more interesting problem:
$\sqrt{1}+\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{5}+\cdots+\sqrt{n}$
.
This 69-char HP-71B one-liner will probably do as well:
INPUTN@2/3*N^1.5+N^.5/2-0.207886224977+(1-1/80/N^2+1/384/N^4)/N^.5/24
which gives 12-digit precision for arguments as low as 7 or 8 and even 6-digit precision (save for a few ulps) for arguments as low as 1 ! (1,00003...).
The constant is Zeta(-1/2). Of course one or several final terms can be omitted if desired for a shorter expression with somewhat reduced accuracy.
Regards.
V.
.
All My Articles & other Materials here: Valentin Albillo's HP Collection
04-20-2015, 04:41 PM
Post: #14
Gerson W. Barbosa Senior Member Posts: 1,419 Joined: Dec 2013
RE: One-liner mini-challenge [HP-71B]
(04-19-2015 03:32 PM)Valentin Albillo Wrote: .
Hi, Gerson:
(04-17-2015 01:46 PM)Gerson W. Barbosa Wrote: Perhaps this is a more interesting problem:
$\sqrt{1}+\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{5}+\cdots+\sqrt{n}$
.
This 69-char HP-71B one-liner will probably do as well:
INPUTN@2/3*N^1.5+N^.5/2-0.207886224977+(1-1/80/N^2+1/384/N^4)/N^.5/24
which gives 12-digit precision for arguments as low as 7 or 8 and even 6-digit precision (save for a few ulps) for arguments as low as 1 ! (1,00003...).
The constant is Zeta(-1/2). Of course one or several final terms can be omitted if desired for a shorter expression with somewhat reduced accuracy.
Regards.
V.
.
Hello Valentin,
Thank you for your ongoing interest in this fascinating subject!
At page 81 of his book Prime Obsession, John Derbyshire says:
"...if you take the trouble to actually work out the sums and add them up, you see that the first ten terms add up to 5.020997899..., the first hundred add up to 18.589603824..., the first thousand add up to 61.801008765..., the first ten thousand to 198.544645449... and so on."
The first problem was an attempt to overcome the trouble of doing these tasks which would become impracticable for very large numbers, even with the help of calculators or computer. The reading of that book has also prompted me to write these programs to compute the Riemann's zeta funcion on various RPL calculators which work for complex arguments, albeit |Im(s)| being limited to values no greater than 10, for full accuracy. The odd-numbered chapters become become heavier and heavier (at least for me) past half of the book.
But I digress. Back to your very short solution, I would like to point out I get your exact result for n = 1, using this series, disregarding the last term:
$\zeta \left ( -\frac{1}{2} \right )+n\left ( \frac{2\sqrt{n}}{3}+\frac{1}{2\sqrt{n}}+\frac{1}{24\sqrt{n^{3}}}-\frac{3}{8}\cdot \frac{1}{720\sqrt{n^{7}}}+\frac{35}{8}\cdot \frac{1}{40320\sqrt{n^{11}}}-\frac{31185}{128}\cdot \frac{1}{3628800\sqrt{n^{15}}}+\cdots \right )$
By the time I posted my first WP 34s program, I had only the first three terms of the series. The sixth term doesn't help much for n = 1, but it of help as n gets larger:
---n----------------------Sum---------------------number of terms
0001-------1.000034781967089878427137719047395------------5
0001-------0.9999676432952148784271377190473951-----------6
0005-------8.382332349275680487306898205740514------------5
0005-------8.382332347354059569143016435388880------------6
actual:-----8.382332347441762038738308734446847
0700-------12359.86189681703238439605016939615------------5
0700-------12359.86189681703238439605014782688------------6
actual:-----12359.86189681703238439605014782693
My second program is flawed, that is, it's inaccurate for small arguments (I don't remember what I did).
Best regards,
Gerson.
04-20-2015, 09:23 PM
Post: #15
Valentin Albillo Senior Member Posts: 776 Joined: Feb 2015
RE: One-liner mini-challenge [HP-71B]
.
Hi again, Gerson:
(04-20-2015 04:41 PM)Gerson W. Barbosa Wrote: Thank you for your ongoing interest in this fascinating subject!
You're welcome. Thanks to you for issuing this particular one-liner mini-challenge, it's indeed a fascinating subject as you say.
Quote:At page 81 of his book Prime Obsession, John Derbyshire says:
"...if you take the trouble to actually work out the sums and add them up, you see that the first ten terms add up to 5.020997899..., the first hundred add up to 18.589603824..., the first thousand add up to 61.801008765..., the first ten thousand to 198.544645449... and so on."
I followed the link but regrettably page 81 is nowhere to be seen, it belongs in a page gap which isn't visualized so I couldn't check the reference.
Quote:But I digress. Back to your very short solution, I would like to point out I get your exact result for n = 1, using this series, disregarding the last term: [...] The sixth term doesn't help much for n = 1, [...]
It's an asymptotic series (and thus divergent) so using more terms improves accuracy only up to some point, adding more terms after that actually degrades the precision. This can be seen in your example for x=1, where the first five terms do help while adding yet another one (or more) does not.
Regards.
V.
.
All My Articles & other Materials here: Valentin Albillo's HP Collection
04-22-2015, 10:57 PM (This post was last modified: 04-24-2015 04:14 PM by Gerson W. Barbosa.)
Post: #16
Gerson W. Barbosa Senior Member Posts: 1,419 Joined: Dec 2013
RE: One-liner mini-challenge [HP-71B]
Hello again, Valentin,
(04-20-2015 09:23 PM)Valentin Albillo Wrote:
(04-20-2015 04:41 PM)Gerson W. Barbosa Wrote: At page 81 of his book Prime Obsession, John Derbyshire says:
"...if you take the trouble to actually work out the sums and add them up, you see that the first ten terms add up to 5.020997899..., the first hundred add up to 18.589603824..., the first thousand add up to 61.801008765..., the first ten thousand to 198.544645449... and so on."
I followed the link but regrettably page 81 is nowhere to be seen, it belongs in a page gap which isn't visualized so I couldn't check the reference.
It's still working here. There are links that lead to chapters 5 and 6 in page vii (Contents), but I cannot copy them:
5 Riemann’s Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6 The Great Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
The other day links to more chapters were available though.
(04-20-2015 09:23 PM)Valentin Albillo Wrote:
(04-20-2015 04:41 PM)Gerson W. Barbosa Wrote: But I digress. Back to your very short solution, I would like to point out I get your exact result for n = 1, using this series, disregarding the last term: [...] The sixth term doesn't help much for n = 1, [...]
It's an asymptotic series (and thus divergent) so using more terms improves accuracy only up to some point, adding more terms after that actually degrades the precision. This can be seen in your example for x=1, where the first five terms do help while adding yet another one (or more) does not.
You are right! Although the sixth term improves the result for n = 1, albeit only slightly, the seventh term makes it equal 1.00005029507, which is slightly worse than the two previous results. Curve fitting is an option if we want more accurate results for small numbers, as you can see in the (not so small) HP-42S program.
Best regards,
Gerson.
HP-42S program:
Code:
00 { 138-Byte Prgm } 01>LBL "S" 02 ENTER 03 RCL+ ST X 04 LASTX 05 SQRT 06 × 07 3 08 ÷ 09 3 10 RCL× ST Y 11 RCL÷ ST Z 12 4 13 ÷ 14 STO+ ST Y 15 12 16 ÷ 17 RCL÷ ST Z 18 STO+ ST Y 19 80 20 ÷ 21 RCL÷ ST Z 22 RCL÷ ST Z 23 STO- ST Y 24 24 25 ÷ 26 5 27 × 28 RCL÷ ST Z 29 RCL÷ ST Z 30 + 31 2.07886224977E-1 32 - 33 X<>Y 34 LN 35 -8.03012316627 36 × 37 1.19449787182 38 RCL÷ ST T 39 - 40 9.07189622971 41 - 42 E^X 43 LN1+X 44 1 45 + 46 ÷ 47 .END.
Examples:
1 .... 1.............(.....1....)
2 .... 2.41421356235 (.....6237 )
3 .... 4.14626436995 (.....6994 )
4 .... 6.14626436989 (.....6994 )
5 .... 8.38233234743 (.....4744 )
10.... 22.4682781862 (.....1862 )
100... 671.462947104 (.....7103 )
1.E12. 6.66666666668E17 (....67 )
1.E333 2.10818510678E499 (...78 )
a := 2*n*sqrt(n)/3
b := 3*a/(4*n)
c := b/(12*n)
d := c/(80*n^2)
e := 5*d/(24*n^2)
s := (a + b + c - d + e - 0.207886224977355)/(1 + ln(1 + exp(-8.03012316626854*ln(n) - 9.07189622971032 - 1.19449787181792/n)))
P.S.: On the HP-71B, this hardly fits in two lines:
Code:
1 INPUTN@A=2*N*SQR(N)/3@B=3*A/4/N@C=B/12/N@D=C/80/N/N@E=5*D/24/N/N@S=A+B+C-D+E 2 (S-.207886224977)/(1+LOGP1(EXP(-8.03012316627*LN(N)-9.07189622971-1.19449787182/N)))
? 1
1
? 2
2.41421356235
? 3
4.14626436995
? 4
6.14626436989
? 5
8.38233234743
? 10
22.4682781862
? 100
671.462947104
? 1E12
6.66666666668E17
? 1E333
WRN L1:Underflow
WRN L2:Underflow
2.10818510678E499
P.P.S.: The constant 9.07189622971 in step 40 of the the HP-42S program and in line 2 of the HP-71B program has how been corrected. Previously the second '9' was missing: 9.0718622971. As a result, both programs now return 1 exactly for n = 1. I ought to have noticed 0.99999999882 was way off!
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https://math.stackexchange.com/questions/3288612/find-entropy-given-that-conditional-proability | 1,580,229,870,000,000,000 | text/html | crawl-data/CC-MAIN-2020-05/segments/1579251779833.86/warc/CC-MAIN-20200128153713-20200128183713-00106.warc.gz | 549,637,451 | 31,138 | # find entropy given that conditional proability
city B
city A\begin{align} \ P (city B|city A)&& B=sunny && B=Rain \\A=sunny &&\frac{4}{5} && \frac{1}{5} \\ A=Rain &&\frac{1}{2}&&\frac{1}{2} \\ \end{align}
i need to calculate the entropy for $$H(City A)$$ , $$H(city B|City A=sunny)$$, $$H(city A|City b)$$ and $$H(city a,city b)$$
but im not sure i how can i find $$p(A,B)$$ from the table such as $$p(A=sunny,B=sunny)$$, $$p(A=sunny,B=rain)$$ ,$$p(A=rain,B=sunny)$$, $$p(A=rain,B=rain)$$
i know that $$P(city A=sunny)=p(B=rain).p(A=sunny|B=rain)+p(B=sunny).p(A=sunny|B=sunny)$$
is this related to bayes rule in joint conditional probability? im used to tabel that is given joint probability such as $$p(A,B)$$ but in this table it is given the conditional probability what rule should i use
and also $$P(B=sunny|A=sunny)=0.8=\frac{p(city A=sunny|city B=sunny).p(city B=sunny)}{P(city A=sunny)}$$
i try to draw the tree diagram but im not sure how can i relate two probability of two city?
EDIT: there is similar problem in book that said $$a=P(city A=sunny)=P(city B=sunny=b$$ but im not sure is this right or not
$$\color{brown}{\frac4{5}\cdot a+\frac1{2}\cdot (1-a)=b }$$
from a=b
$$\color{brown}{\frac4{5}\cdot a+\frac1{2}\cdot (1-a)=a}$$
solving $$a=\frac{5}{7}$$
$$P(a=S,b=S)=\frac{5}{7} * 0.8=\frac{4}{7}$$
$$P(a=S,b=R)=\frac{1}{7}$$
$$P(a=R,b=S)=\frac{1}{7}$$
$$P(a=R,b=R)=\frac{1}{7}$$
We know that $$P(X=x|Y=y)\cdot P(Y=y)=P(X=x\cap Y=y)$$. Then let me first define some denotations.
$$A$$=City a is sunny ($$a$$), $$\overline A$$=City a is sunny $$((1-a))$$, $$B$$=City a is sunny ($$b$$), $$\overline B$$=City a is sunny $$((1-b))$$.
$$\small{\textrm{The small letters in the brackets will replace the corresponding probabilities}}$$
1.1 The proability that City A and City B are sunny is
$$P(B|A)\cdot P(A)=P(A\cap B)\Rightarrow \frac4{5}\cdot a=P(A\cap B)$$
1.2 The proability that City A is rainy and City B is sunny is
$$P(B|\overline A)\cdot P(\overline A)=P(\overline A\cap B)\Rightarrow \frac1{2}\cdot (1-a)=P(\overline A\cap B)$$
We can sum up both equations and get $$\color{brown}{\frac4{5}\cdot a+\frac1{2}\cdot (1-a)=b \quad (1)}$$
2.1 The proability that City A and City B are sunny is
$$P(B|A)\cdot P(A)=P(A\cap B)\Rightarrow \frac4{5}\cdot a =P(A\cap B)$$
2.2 The proability that City A is sunny and City B is rainy is
$$P(\overline B|A)\cdot P( A)=P( A\cap \overline B)\Rightarrow \frac1{5}\cdot (1-b)=P( A\cap \overline B)$$
Summing up both equations again and get
$$\color{brown}{\frac4{5}\cdot a+\frac1{5}\cdot (1-b)=a \quad (2)}$$
You just have to solve this little equation system and then use the very first equation to calculate the joint distribution. I got
$$\large{\begin{array}{|c|c|c|} \hline P(B \cap A) & B & \overline B \\ \hline A &\frac{4}{13} & \frac{1}{13} \\ \hline \overline A &\frac{4}{13}&\frac{4}{13} \\ \hline \end{array}}$$
• thankyou so much, i edit my post, there is similar probelm in my book, that said $a=P(city A=sunny)=P(city B=sunny=b)$, but im not sure why is that.. i tried to solve the equation that lead to different result from your answer, i dont know which one is right(?) – fiksx Jul 11 '19 at 15:46
• Have you posted the similar problem? – callculus Jul 11 '19 at 15:49
• posted the problem here? – fiksx Jul 11 '19 at 15:51
• If you ask me, then it is good for me to know what the problem is. – callculus Jul 11 '19 at 15:53
• Is still not clear what your problem is. Please state clear what the (new) problem is. A new question would be the best place. – callculus Jul 11 '19 at 16:14 | 1,282 | 3,618 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 38, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.03125 | 4 | CC-MAIN-2020-05 | latest | en | 0.650608 |
https://dir.md/wiki/Trigonometry?host=en.wikipedia.org | 1,652,726,203,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662512229.26/warc/CC-MAIN-20220516172745-20220516202745-00037.warc.gz | 272,050,986 | 12,467 | # Trigonometry
Trigonometry (from Ancient Greek τρίγωνον (trígōnon) 'triangle', and μέτρον (métron) 'measure')[1] is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.[2] The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine.[3]
Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation.[4]
Trigonometry is known for its many identities. These trigonometric identities[5][6] are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation.[7]
Sumerian astronomers studied angle measure, using a division of circles into 360 degrees.[9] They, and later the Babylonians, studied the ratios of the sides of similar triangles and discovered some properties of these ratios but did not turn that into a systematic method for finding sides and angles of triangles. The ancient Nubians used a similar method.[10]
In the 3rd century BC, Hellenistic mathematicians such as Euclid and Archimedes studied the properties of chords and inscribed angles in circles, and they proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. In 140 BC, Hipparchus (from Nicaea, Asia Minor) gave the first tables of chords, analogous to modern tables of sine values, and used them to solve problems in trigonometry and spherical trigonometry.[11] In the 2nd century AD, the Greco-Egyptian astronomer Ptolemy (from Alexandria, Egypt) constructed detailed trigonometric tables (Ptolemy's table of chords) in Book 1, chapter 11 of his Almagest.[12] Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today.[13] (The value we call sin(θ) can be found by looking up the chord length for twice the angle of interest (2θ) in Ptolemy's table, and then dividing that value by two.) Centuries passed before more detailed tables were produced, and Ptolemy's treatise remained in use for performing trigonometric calculations in astronomy throughout the next 1200 years in the medieval Byzantine, Islamic, and, later, Western European worlds.
The modern sine convention is first attested in the Surya Siddhanta, and its properties were further documented by the 5th century (AD) Indian mathematician and astronomer Aryabhata.[14] These Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry.[15][16] The Persian polymath Nasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right.[17][18][19] Nasīr al-Dīn al-Tūsī was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form.[20] He listed the six distinct cases of a right-angled triangle in spherical trigonometry, and in his On the Sector Figure, he stated the law of sines for plane and spherical triangles, discovered the law of tangents for spherical triangles, and provided proofs for both these laws.[21] Knowledge of trigonometric functions and methods reached Western Europe via Latin translations of Ptolemy's Greek Almagest as well as the works of Persian and Arab astronomers such as Al Battani and Nasir al-Din al-Tusi.[22] One of the earliest works on trigonometry by a northern European mathematician is De Triangulis by the 15th century German mathematician Regiomontanus, who was encouraged to write, and provided with a copy of the Almagest, by the Byzantine Greek scholar cardinal Basilios Bessarion with whom he lived for several years.[23] At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond.[24] Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts.
Driven by the demands of navigation and the growing need for accurate maps of large geographic areas, trigonometry grew into a major branch of mathematics.[25] Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595.[26] Gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry. The works of the Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series.[27] Also in the 18th century, Brook Taylor defined the general Taylor series.[28]
Trigonometric ratios are the ratios between edges of a right triangle. These ratios are given by the following trigonometric functions of the known angle A, where a, b and h refer to the lengths of the sides in the accompanying figure:
The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. See below under Mnemonics.
Since any two right triangles with the same acute angle A are similar,[29] the value of a trigonometric ratio depends only on the angle A.
The reciprocals of these functions are named the cosecant (csc), secant (sec), and cotangent (cot), respectively:
The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-".[30]
With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines.[31] These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known.
A common use of mnemonics is to remember facts and relationships in trigonometry. For example, the sine, cosine, and tangent ratios in a right triangle can be remembered by representing them and their corresponding sides as strings of letters. For instance, a mnemonic is SOH-CAH-TOA:[32]
One way to remember the letters is to sound them out phonetically (i.e. SOH-kə-TOH, similar to Krakatoa).[33] Another method is to expand the letters into a sentence, such as "Some Old Hippie Caught Another Hippie Trippin' On Acid".[34]
Indication of clockwise and counterclockwise amounts of key rotations in degrees, in the unit circle.
Using the unit circle, one can extend the definitions of trigonometric ratios to all positive and negative arguments[37] (see trigonometric function).
The following table summarizes the properties of the graphs of the six main trigonometric functions:[38][39]
Because the six main trigonometric functions are periodic, they are not injective (or, 1 to 1), and thus are not invertible. By restricting the domain of a trigonometric function, however, they can be made invertible.[40]: 48ff
The names of the inverse trigonometric functions, together with their domains and range, can be found in the following table:[40]: 48ff [41]: 521ff
When considered as functions of a real variable, the trigonometric ratios can be represented by an infinite series. For instance, sine and cosine have the following representations:[42]
With these definitions the trigonometric functions can be defined for complex numbers.[43] When extended as functions of real or complex variables, the following formula holds for the complex exponential:
This complex exponential function, written in terms of trigonometric functions, is particularly useful.[44][45]
Trigonometric functions were among the earliest uses for mathematical tables.[46] Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy.[47] Slide rules had special scales for trigonometric functions.[48]
Scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes cis and their inverses).[49] Most allow a choice of angle measurement methods: degrees, radians, and sometimes gradians. Most computer programming languages provide function libraries that include the trigonometric functions.[50] The floating point unit hardware incorporated into the microprocessor chips used in most personal computers has built-in instructions for calculating trigonometric functions.[51]
In addition to the six ratios listed earlier, there are additional trigonometric functions that were historically important, though seldom used today. These include the chord (crd(θ) = 2 sin( θ/2)), the versine (versin(θ) = 1 − cos(θ) = 2 sin2( θ/2)) (which appeared in the earliest tables[52]), the coversine (coversin(θ) = 1 − sin(θ) = versin( π/2θ)), the haversine (haversin(θ) = 1/2versin(θ) = sin2( θ/2)),[53] the exsecant (exsec(θ) = sec(θ) − 1), and the excosecant (excsc(θ) = exsec( π/2θ) = csc(θ) − 1). See List of trigonometric identities for more relations between these functions.
For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions,[54] predicting eclipses, and describing the orbits of the planets.[55]
In modern times, the technique of triangulation is used in astronomy to measure the distance to nearby stars,[56] as well as in satellite navigation systems.[16]
Historically, trigonometry has been used for locating latitudes and longitudes of sailing vessels, plotting courses, and calculating distances during navigation.[57]
Trigonometry is still used in navigation through such means as the Global Positioning System and artificial intelligence for autonomous vehicles.[58]
In land surveying, trigonometry is used in the calculation of lengths, areas, and relative angles between objects.[59]
On a larger scale, trigonometry is used in geography to measure distances between landmarks.[60]
The sine and cosine functions are fundamental to the theory of periodic functions,[61] such as those that describe sound and light waves. Fourier discovered that every continuous, periodic function could be described as an infinite sum of trigonometric functions.
Even non-periodic functions can be represented as an integral of sines and cosines through the Fourier transform. This has applications to quantum mechanics[62] and communications,[63] among other fields.
Trigonometry is useful in many physical sciences,[64] including acoustics,[65] and optics.[65] In these areas, they are used to describe sound and light waves, and to solve boundary- and transmission-related problems.[66]
Other fields that use trigonometry or trigonometric functions include music theory,[67] geodesy, audio synthesis,[68] architecture,[69] electronics,[67] biology,[70] medical imaging (CT scans and ultrasound),[71] chemistry,[72] number theory (and hence cryptology),[73] seismology,[65] meteorology,[74] oceanography,[75] image compression,[76] phonetics,[77] economics,[78] electrical engineering, mechanical engineering, civil engineering,[67] computer graphics,[79] cartography,[67] crystallography[80] and game development.[79]
Trigonometry has been noted for its many identities, that is, equations that are true for all possible inputs.[81]
Identities involving only angles are known as trigonometric identities. Other equations, known as triangle identities,[82] relate both the sides and angles of a given triangle.
In the following identities, A, B and C are the angles of a triangle and a, b and c are the lengths of sides of the triangle opposite the respective angles (as shown in the diagram).[83]
The law of sines (also known as the "sine rule") for an arbitrary triangle states:[84]
The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles:[84]
The law of tangents, developed by François Viète, is an alternative to the Law of Cosines when solving for the unknown edges of a triangle, providing simpler computations when using trigonometric tables.[85] It is given by:
Given two sides a and b and the angle between the sides C, the area of the triangle is given by half the product of the lengths of two sides and the sine of the angle between the two sides:[84]
Heron's formula is another method that may be used to calculate the area of a triangle. This formula states that if a triangle has sides of lengths a, b, and c, and if the semiperimeter is
The following trigonometric identities are related to the Pythagorean theorem and hold for any value:[87]
Other commonly used trigonometric identities include the half-angle identities, the angle sum and difference identities, and the product-to-sum identities.[29] | 2,993 | 13,484 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.625 | 4 | CC-MAIN-2022-21 | latest | en | 0.931993 |
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## Which edge is which when tessellating?
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### #1cephalo Members
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Posted 22 April 2013 - 03:15 PM
Ok, so I have 10 control points for a 3 order bezier triangle. It is arbitrary which points are which, I know who they are and how to use them, but there is no obvious relation to these points from the edges to tessellate. In my hull shader, I decide how much to tessellate each of the three edges, but how do I know which edge is edge zero, one, two etc?
For example, if cp[0], cp[1] and cp[2] are the control points to my bezier triangle corners, how do I decide how much to tessellate each edge based on the screen size of the edge?
### #2Jason Z Members
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Posted 22 April 2013 - 07:53 PM
You know the orientation of the control points relative to one another, and you know the world transformation that they are being transformed with, so you should have the information that you need. If you are currently doing the transformation in the domain shader, then you may have to move that back to the hull shader. Then you can transform each control point, and then using whatever metric you want you can adjust the tessellation accordingly.
In general this is better than doing the transformation in the domain shader, since it is running after the amplification from tessellation. Always try to push the computation upstream if possible!
Jason Zink :: DirectX MVP
Direct3D 11 engine on CodePlex: Hieroglyph 3
Games: Lunar Rift
### #3cephalo Members
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Posted 23 April 2013 - 07:29 AM
Here is my current hull shader:
PatchConstants PatchHS(InputPatch<VertexOutput,10> cp, uint patchID : SV_PrimitiveID)
{
PatchConstants pt;
pt.EdgeTess[0] = 6;
pt.EdgeTess[1] = 6;
pt.EdgeTess[2] = 6;
pt.InsideTess = 6;
return pt;
}
How do I know which cp's are associated with pt.EdgeTess[0]? I have to make sure that shared edges are tessellated the same way. My shader pipeline currently supports two forms of triangle input sets, one pointing up and the other pointing down. After translating them with my matrices, they could be oriented any old way. If one of my edges on my triangle is short on the screen, how do I know which element in the EdgeTess array to decrease?
### #4cephalo Members
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Posted 23 April 2013 - 07:50 AM
In general this is better than doing the transformation in the domain shader, since it is running after the amplification from tessellation. Always try to push the computation upstream if possible!
In my case, the domain shader is calculating patch surface normals according to world space. If transform my control points to screen space in the hull shader, I wouldn't be able to calculate normals any more, because at that point all the geometry is co-planar correct?
Edited by cephalo, 23 April 2013 - 07:54 AM.
### #5cephalo Members
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Posted 23 April 2013 - 08:16 AM
Ok, I have discovered that by setting one of my tessellation factors to 3, I can find out by visual experimentation which edge is which. It appears to be a stable association that is NOT dependent on the view space. In my case, the edge from cp[0] to cp[1] is edge index 2, edge cp[1] to cp[2] is edge index 0 and edge cp[2] to cp[0] is edge index 1. In my case, they also seem to line up by accident so I don't have to worry about mismatched shared edges.
Strange. Is there any way to predict this without experimentation? There must be some rule it would seem.
### #6unbird Members
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Posted 23 April 2013 - 01:14 PM
It's fixed, and your result looks correct. edgen is opposite to vertexn. Haven't found that in the docs, only in this paper, page 4. Don't ask me about quad domains, haven't tried that.
### #7Jason Z Members
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Posted 24 April 2013 - 04:45 AM
Here is my current hull shader:
PatchConstants PatchHS(InputPatch<VertexOutput,10> cp, uint patchID : SV_PrimitiveID)
{
PatchConstants pt;
pt.EdgeTess[0] = 6;
pt.EdgeTess[1] = 6;
pt.EdgeTess[2] = 6;
pt.InsideTess = 6;
return pt;
}
How do I know which cp's are associated with pt.EdgeTess[0]? I have to make sure that shared edges are tessellated the same way. My shader pipeline currently supports two forms of triangle input sets, one pointing up and the other pointing down. After translating them with my matrices, they could be oriented any old way. If one of my edges on my triangle is short on the screen, how do I know which element in the EdgeTess array to decrease?
It actually doesn't matter (I know - that isn't intuitive...). As long as neighboring patches share the same control points, and the control point to tessellation factor algorithm will produce the same output for the given input control points, then the order doesn't make any difference.
But for the same reason you are having trouble to understand it, it would be impossible to have a standard notation to say which control point goes where since it depends on the primitive topology, the number of control points, and so on. I believe there is a way to identify where a control point is located within a control patch, but I don't think there is one for the oriented control patches.
Jason Zink :: DirectX MVP
Direct3D 11 engine on CodePlex: Hieroglyph 3
Games: Lunar Rift
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PDF is category theory to FM as calculus is to physics University of cs toronto edu ~zdiskin Pubs googleDiscussion pdf PDF THE 1r CALCULUS IN FM Springer Link link springer content pdf 10 1007 978
## Calculus Solutions Manual 2
testbanklive sample calculus 2nd edition Calculus 2nd Edition Briggs Solutions Manual Full download all chapters instantly please go to Solutions Manual, Test Bank site testbanklive nightwitchbodyart Student Solutions Manual Part 2 Download Free Student Solutions Manual Part 2 For University Calculus Pt
## Calculus, Student Solutions Manual - Anton, Bivens & Davis
### Howard Anton Calculus 10th Edition Solution Manual
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## Calculus Success in 20 Minutes a Day2ndEdition[1]
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PDF Read Book \\ Calculus Success in 20 Minutes a Day data accessplan mysydneycbd nsw gov au 9781576858899 calculus calculus success in 20 minutes a day pdf PDF AP Calculus at JF has always been a very challenging and bedfordjfhs sharpschool
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### The Calculus Tutoring Book By Carol Ash Robert B Ash - Ebooks
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## Calculus With Analytic Geometry SM Yusuf (Solution Manual)
### Course Number and Title: CALCULUS AND ANALYTICAL GEOMETRY-NS
eqexnaka files wordpress 2015 07 calculus Calculus And Analytic Geometry By Sm Yusuf Pdf Download EBook Calculus With Analytic Geometry SM Yusuf (Solution Manual) for freePublisher Imprint Ilmi Kitab Beautiful Code Download Free PDF eBook susotaran files wordpress 2015 07 solution Solution Manual Of Calculus With Analytic
## Calculus
### Time to conquer Calculus! - Photomath
PDF Calculus ocw mit edu ans7870 resources Strang Edited Calculus Calculus pdf PDF Calculus Volume 1 cloudfront d3bxy9euw4e147 cloudfront oscms CalculusVolume1 OP pdf PDF Calculus Volume 2 cloudfront d3bxy9euw4e147
1. calculus pdf | 1,318 | 5,131 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.59375 | 3 | CC-MAIN-2020-50 | latest | en | 0.42719 |
https://ask.truemaths.com/question/o-is-the-centre-of-a-circle-of-radius-8-cm-the-tangent-at-a-point-a-on-the-circle-cuts-a-line-through-o-at-b-such-that-ab-15-cm-find-ob/ | 1,670,663,293,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710421.14/warc/CC-MAIN-20221210074242-20221210104242-00406.warc.gz | 148,614,859 | 29,473 | • 1
O is the centre of a circle of radius 8 cm. The tangent at a point A on the circle cuts a line through O at B such that AB = 15 cm. Find OB.
• 1
we have to find that line from that tangent AB
Share | 61 | 205 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.75 | 3 | CC-MAIN-2022-49 | longest | en | 0.935698 |
http://www.yougonumbers.com/roman-numeral/xxxiv | 1,529,472,923,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267863463.3/warc/CC-MAIN-20180620050428-20180620070428-00179.warc.gz | 536,773,348 | 5,681 | ### Read and Write Roman Numeral XXXIV
Roman numeral symbols are written and read from left to right, from highest to lowest values. If XXXIV is text, it is the number 34. Find XXXIV in numbers and convert XXXIV to decimal and binary..
### Miscellaneous & Conversions For 34
Text Thirty-Four Roman Numeral: 34 XXXIV RGB Colour (# 022 or #000022)
Decimal / Number 34 Binary 100010 Ternary 1021 Hexadecimal (Hex) 0x22 Quaternary 202 Quinary 114 Senary (Heximal) 54 Octal 42 Duodecimal (Dozenal) 2a
### Mathematics
Prime Factors: 2 x 17 = 34 Factorial: 34! 2.9523279904E+38 Triangular Number Of 34 595 Square Root: √34 5.83095189485 Cube Root (Root 3): 3√34 3.23961180128 Tesseract Root (Root 4): 4√34 2.41473640277 Quintal / nth Root. (Root 5) 5√34 2.0243974585 ¾ Root 14.0802118033 Fibonnaci # 34 5702887 Recipricol 1/ 34 0.0294117647059 Quarter 1/4 8.50 or 8.50/34 Half 1/2 17 or 17/34 Third 1/3 11.33 or 11.33/34
### Roman Numeral Chart. Read & Write 34
Reading and writing roman numerals isn't common knowledge. However, this roman numeral chart should help you read and write numbers in roman numerals from 1 to 1,000,000. Using an overline symbol is times 1,000. ( Example: X means to add 3 zeros to your number. So X which is 5, but written as X with the overline is 5,000.
• When a symbol appears after a larger symbol it is added: Example: VI = V + I = 5 + 1 = 6
• When a symbol appears before a larger symbol it is subtracted: Example: IX = X - I = 10 - 1 = 9
• Do not use the same symbol more than 3 times. But, IIII is sometimes used for 4
1 I 10 x 500 D 1,001 MI 2 II 20 XX 501 DI 1,002 MII 3 III 30 XXX 530 DXXX 1,030 MXXX 4 IV 40 XL 550 DL 1,550 MDL 5 V 50 L 600 DC 1,555 MDLV 6 VI 60 LX 650 DCL 5,000 V 7 VII 70 LXX 700 DCC 10,000 X 8 VIII 80 LXXX 800 DCCC 50,000 L 9 IX 90 XC 890 DCCCXC 100,000 C 10 X 95 XCV 900 CM 500,000 D 11 XI 99 XCIX 950 CML 1,000,000 M 12 XII... 100 C 1000 M
31 xxxi 34 xxxiv 32 xxxii 1,031 mxxxi 131 cxxxi 38 xxxviii 3,037 mmmxxxvii 35 xxxv
2% Fibonacci
38% Numbers
20% Binary | 755 | 2,031 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.0625 | 4 | CC-MAIN-2018-26 | latest | en | 0.648637 |
https://www.matlabhelponline.com/how-to-plot-sin-x-in-matlab-28411 | 1,674,972,477,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764499700.67/warc/CC-MAIN-20230129044527-20230129074527-00273.warc.gz | 889,763,155 | 25,740 | # How To Plot Sin X In Matlab
How To Plot Sin X In Matlab/Unicode? It is my hope that this post I wrote will provide the same answers in Matlab as you, even if it won’t really give an answer it site link As I am aware you do not have to know all the various steps before you plot the coordinates of a dot. They may take any number of steps. That does need a bit of explanation or data, but it is very important for understanding where to start in the first place. If you want to cut into the time in such a way as easily one can just compute \$r_0\$ which can be done in C and you will be able to do it in python. If you want to follow one of the later steps I suggest you to use Plotly, open a text file and reference it whenever reading it. If you want to look at how to plot a dot or point you can usually do it with a function called dotplot and I’ve just found the Matlab/Unicode code. Plotly has many methods of plotting from very basic plotting files (click here for example for ImageMagick and Googlesia) and is one of the most developed and widely used tools in Matlab. You can learn more about Plotly as well as get a feel for how it works in more detail in Wikipedia. Can I see the steps my plot my answer? 1. Download the Matlab/Unicode spreadsheet and paste the code where I mention how to plot the coordinates of a dot (X,Y) Let it be that way and save the same code in one file (plot.matlabx). Give it a name and paste all of my steps as it is the way. b. If I’m not mistaken it should be like this 1/ 2/ 2/ 3/ 3/ 4/ 4/ 4/ Like I said above (basically what I am now more familiar with) but in general it isn’t that simple because you are having to have each of the two files in one place and this one you declare as a “image”. Basically in your code how to calculate the coordinates of a dot in image2.gosh and plot with a dotplot. To do this I think the first step of learning Matlab as in this get redirected here are bit-map and path functions. f. Now I need to generate my code as an example.
## Plot Options In Matlab
If I use a function called dotplot, it would be in the simplest form. I will use a variable called “dopies” to generate a dotplot, I just need only two lines with the r values of 1 and 2 Next I need to generate a little variable of function: g. Start with something useful, like a function that should be callable in some editor in this workspace \$ To generate a dotplot I have come to the following code, that you will be writing in the image command-line. The function itself: In this example the dotplot is provided by M. Mathieu, who is a very good lecturer for Matlab. I will give a couple examples to explain what’s going on that is happening, where it is helpful and when and how to use the function. image2How To Plot Sin X In Matlab Under YCAS For years it’s been an alien take over for Apple. What’s even more awesome is that I love the combination of Mac and Windows. Let me tell you a very simple one – Mac and Windows on the same thing. The main difference is that Mac vs Windows can’t compare much, so why not just convert yours/our files into mac and if we can’t we can put a Mac file onto one of the windows (and simultaneously change the command line option) and we’re good to go, even if not 100% sure it’ll match Mac on the Mac side, my whole experience with MySQL, in both the Mac and Windows versions of Mac seems pretty to dissapointive if I ever need to know. If you’re a huge Mac user, you’re going to probably download Mac’s Mac installer to install mac and start it up very easily. If you already have Mac on your computer, you’ll need to get it from a third party, like Adobe or Apple’s Mac AppStore. Once you have the Mac and working Mac on the Mac, you’ll actually need to purchase a Mac app/instacademy to get your project to work. Of course, Mac is much more basic than most projects on the web, or I might be mistaken. If you’re only using Windows I’d recommend installing Mac, and then using any Windows app if you don’t have a macOS install, if you have an iOS app or if you use an iCloud for iCloud. Anyway, if you’re not familiar with Linux, you may have a few years to learn how to use Mac. I’m sure you’ll wish you were, but a lot depends on your business needs. Make sure to read a few posts on Linuxicular about Mac and GUI programming. My Mac Screen First up is see this website Mac window. Over the years my MacBook Pro often has my screen in it, even though the screen is my MacBook, it seems to work just fine on all platforms except iOS.
## Matlab Stacked Line Plot
After booting up everything is installed before the screen. When this happens I get a little frustrated, and also if it doesn’t boot, it’s almost impossible to run the display. I found the file imglace on the Mac screen under the contents of the file, and when I downloaded the Xcode builder, it did this: The Xcode build went live very quick in the morning (around 2:30 or so), and I had a lot of time to edit my programs. This is Mac apps only. This is a Mac app only. The main part is to actually create a new Mac app for every useful content you use in your business, or just leave it for another day. Are you sure it is a Mac app only? No Windows app I have made was a Mac app only in the past ages. Here you will see a blank, white, shiny Mac screen. Perhaps just a smaller screen. And I am happy to describe the Mac screen with a few pictures in the comments. Lets know my screen and keyboard layout in Python: What is a Mac Screen? Mac screen for Mac is a small screen with a single line, made with the Mac screen editor, where the desktop is divided into mac, windows and home. It’s usually on the top of the Mac screen and is the width of the screen. The screen is a bit soft, so the smaller Mac window is able to move around and have varying effects– all around the desktop. I note that you can only have 15, 16 or 17 Mac screens on Mac or Windows or wherever else you might happen. If you have only one mac and another one installed, then you’ll have several Macs – I find that each Mac Mac mini does exactly the same thing every time I run it. So be sure to read lots of more web articles about Mac Screen and Viewing Screens. I apologize for the mess. Keep the screenshot so you know that what you get from a Mac app only in the Mac App Source is from the Mac App. What is a Screen Window?#Mac Screen? When you upgrade from OS X, you usuallyHow website link Plot Sin X In Matlab This is a post for matlab 2019 2, where I’d love to help you improvematlab! Follow me for this post with multiple solutions and links to go find and write your own solution and explain why it works without breaking our code. I know there’s page lot to gain by messing with this part but that’s been done to some extent already on my site.
## Use Plot Command To Plot Sin Over A Range
Starting with the examples was looking at the syntax for several of my macros, but didn’t really find next workable way to get this piece going anymore. We decided to create our own syntax in the Matlab code by cleaning the file and see this page that as often as we could. We weren’t sure what we were doing before we started writting this code – changing the functions to make it look like our code would match so well or so clean-ups would be more beneficial. We didn’t find much of a way to get this code doing basic arithmetic, so the magic came from seeing the results in one file. Still wanted to find out how we were more information the resulting check out here to stand out from the rest of the code. Here are the key words, the short and the long: int main( pop over here n, char *argc, const char **argv ){ Some simple mathematical functions : a *b, ¬b, sqrt(b) *a; b *a, sqrt(a); (That’s how we’ve designed the Matlab code) a *b, ¬c, bz(a) *a; (That’s how we’ve designed the Matlab code) b *b, sqrt(a * a) *b, ½a *b; (That’s how we’ve designed our code) a *c, ¬c, bz(a) *c; (That’s how we’ve designed our code) c *d, ´z, ´n, ´m, a *c, ¬d, z(c); sqrt(d); (That’s how we’ve designed our code) c *z, ´z { ½a } (That’s how we’ve designed our code) d *a, ´z { ½a } c *c, ´z { ¬d } (That’s how we’ve designed our code) (That’s how we’ve designed our code) (That’s how we’ve designed our code)) (That’s how we’ve designed our code) Here are all the variables under the *s where you click the button on the matlab buttons: [1-3] +a -b +c -d +w 2} y = ¬x *x**3; (If an x* has a y*, so should also ¬x**3, as there’s a distance of x-y in it) Z = µ / (¬x *x**3). Next: adding the square to each set of seven variables being used for the last one: [3-8] = µ / (¬x * (¬x *(¬x *ªx + 1))); Then adding in the whole image to the input file, we only include a single space inside each individual variable: [1-3] +a -b +c -d -W /2; (One) we have to consider the part of our code that was trying to do triangle-like operation to work appropriately today. First; take 3 y as example. Now turn on all three variables in the previous equation. bz(x * z) = µ/ (¬x * (¬x * (¬x *(¬x *(¬x *(¬x + 1))));); And then | 2,258 | 9,033 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.375 | 3 | CC-MAIN-2023-06 | latest | en | 0.946874 |
https://www.huffingtonpost.co.uk/2015/12/11/-gchq-cryptic-christmas-card-puzzle-_n_8781508.html | 1,642,440,171,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320300574.19/warc/CC-MAIN-20220117151834-20220117181834-00617.warc.gz | 880,522,220 | 58,512 | Tech
# The Internet Attempts GCHQ's Cryptic Christmas Card Puzzle
The concept of sending Christmas cards to your loved (and not-so-loved) ones during the festive season has taken an interesting twist, after the GCHQ sent a series of mysterious messages to the British public.
The spy agency's director Robert Hannigan posted a puzzle in his annual message, giving recipients till 31 January 2016 to send their answers in.
However, Hanningan and team may have underestimated human intelligence as code-breakers have posted their versions of the answers on Reddit.
The first puzzle posted by GCHQ
According to instructions posted on the agency's site, "each row or column is labelled with a string of numbers. The numbers indicate the length of all consecutive runs of black squares, and are displayed in the order that the runs appear in that line. For example, a label "2 1 6" indicates sets of two, one and six black squares, each of which will have at least one white square separating them."
Responding to the challenge, Reddit user oneweirdglobe, posted:
What you are essentially seeing is a QR code, which can be scanned by a mobile phone, moving the code-cracker to the next stage of the puzzle.
Other questions in this category included:
"I was looking at a man on top of a hill using flag semaphore to send a message, but to me it looked like a very odd message. It began "ZGJJQ EZRXM" before seemingly ending with a hashtag. Which hashtag?" The options? #SGM #SEM #SEN #SGN #TEN
#TGN.
Getting the right answers then unlocks part 3:
If you're still keen to power through the rest of the brain teasers, parts 4 and 5 look like this:
When we asked GCHQ about the number of correct submissions they've had so far, they were equally cryptic and told us results would only be announced in January.
The winner will be drawn from all the successful entries and players are invited to make a donation to the 'National Society for the Prevention of Cruelty to Children.'
Well, at least we all have options if the conversation around the dinner table gets too boring on Christmas Day. | 464 | 2,098 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.671875 | 3 | CC-MAIN-2022-05 | latest | en | 0.958589 |
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One more problem A fluid is just like a gas creating outward pressure in all directions.
Pressure can be represented as Pressure= Force/Area Fluids in its simplest form Fluids exerts a net upward force on any object. Archimedes Principle and Buoyancy Fluid Flow and Continuity Food for thought... When you turn on a hose the water flowing through it is slow... What do you do at the end of the hose to make it flow faster. IMPORTANT! Fluids are now applied to the work energy theorem. Relationship between pressure, speed and height of the fluid. The Adventures of Pressure Fluids Important as well is the density of a fluid. known as the letter p or Rho. Density=Mass/Volume Hydrostatic pressure or still fluid is determined by Depth and Density. key idea: The deeper in a fluid the more pressure is felt.
Equation P=Po+pgh This is called absolute pressure. The pgh is gauge pressure. Or P-Po Example: A water bed is 2.0 m on a side an 30.0 cm deep.
(a) Find its weight if the density of water is 1000 kg/m3.
(b) Find the pressure the that the water bed exerts on the floor. Assume that the
entire lower surface of the bed makes contact with the floor. Example of pressure varying at depths is a barometer.
Key Concept: Water likes to seek its own level. U shaped tube. Pascals Principle: An external pressure is applied to an inclosed incompressible static fluid. The energy is transmitted undiminished through the fluid.
Example: Hydraulic lift. Use P=F/A to find the pressure in the first cylinder then insert that into the second P=F/A for the opposing cylinder. Advantage to this is a small area piston can move a large mass with little pressure. If the area is 100 times greater on one side through the equations we can see that it would mean our force was magnified 100 times. EXAMPLE This is buoyancy The force can be found by the equation F=pgV Archimedes' principle An object fully submerged in a fluid will experience an upward force equal to the weight of the fluid displaced. Example: A cube of steel that measures 5.0 cm on each side is immersed in water. The density of steel is 9.0 x 103 kg/m3. The density of water is 1.0 x 103 kg/m3. What is the (a) buoyant force acting on the cube and what is (b) its apparent weight? Ideas:
The buoyancy formula can be used to find the density of a body and then used to find the percentage of body fat.
Also through the equation Vsub=Vs(ps/pf) the amount of an object floating below the fluid level can be found.
Floatation: The block example.
The metal example.
Change buoyancy the density needs to change.
Apparent weight is the difference between the actual weight and the buoyant force. This is know as fluid flow. In a straight system the fluid will not be gained or lost.
The equation AV=A2V2 can be used to find different portions of incompressible fluid passing through certain parts of the system. The concept of continuity can only be used in a system that has
no bends in the system. Example: You turn on the hose to water your sassafras garden. When the hose is turned on the water starts to flow at a rate of 5 m/s. The hose goes from .005 m to .0025m at the end of the hose. What is the new speed of the water. Bernoulli Key idea: The faster a fluid flows the less pressure there is. Reasoning the fire hose. Key Idea: As height increases in a pipe the pressure decreases. Combine the two ideas and you get the complete Bernoulli equation: Applications of Bernoulli:
Paper example.
Air Planes force faster air on top of the wing and slower on the bottom of the wing to create lift. There are basically two ways to make fluid flow through a pipe. One way is to tilt the pipe so the flow is downhill, this means gravitational kinetic energy is transformed to kinetic energy. The second way is to make the pressure at one end of the pipe larger than the pressure at the other end. A pressure difference is like a net force, producing acceleration of the fluid.
As long as the fluid flow is steady, and the fluid is non-viscous and incompressible, the flow can be considered on an energy perspective. Example: Consider a geyser that shoots water 25 m into the air. How fast is the water traveling when it emerges from the ground? If the water originates in a chamber 35 m below the ground, what is the pressure there? Key Formulas Pressure= p=f/a
hydrostatic pressure= Po+pgh
Pascals= F'=FA'/A
Archimedes= Fb=W displaced
Continuity= Av=Av
1/2pv^2+pgh+p
Full transcript | 1,160 | 4,942 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.53125 | 4 | CC-MAIN-2019-09 | latest | en | 0.923825 |
https://study.com/academy/lesson/x-coordinates-definition-examples-quiz.html | 1,569,162,601,000,000,000 | text/html | crawl-data/CC-MAIN-2019-39/segments/1568514575515.93/warc/CC-MAIN-20190922135356-20190922161356-00290.warc.gz | 672,318,493 | 42,501 | # X-Coordinates: Definition & Examples
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Instructor: Kimberlee Davison
Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.
X-coordinates are an essential part of plotting points on graphs. Learn how an x-coordinate relates to direction on 2-dimensional mathematical grids. After, test yourself with the quiz.
## What Is an X-Coordinate?
An x-coordinate is the x value in an ordered pair, which is just two mathematical objects, such as numbers, paired together. Usually, the x-coordinate is the first number in that pair. For instance, in the ordered pair (2,5), the x-coordinate would be '2.' This value tells you how far a point is from the origin, or starting point, in the x direction on a 2-dimensional graph.
## Example
Let's look at an example. On this graph, the x-coordinates are shown in red. They're nearly always the first value in an ordered pair and the horizontal direction on a Cartesian plane, or a 2-dimensional mathematical graph.
The x-coordinate of the heart is '4', the x-coordinate of the star is '3', and the x-coordinate of the smiley face is '-4'.
## Directions and Coordinates
The x-direction, however, needs to be labeled clearly, so everyone who looks at a graph knows exactly what you mean.
Let's suppose you are giving your second cousin directions to your house for a birthday bash you are holding for your great-grandmother. You might say, 'Drive 3 miles north on Main Street, turn left, and go three more blocks.' Then, your great-grandmother's sister calls for directions and you say, 'Drive 10 miles south on Main Street, turn right, and go three more blocks.' Why did you give different directions? Well, because these two relatives were coming from different starting points. My point is this: directions are relative.
In mathematics, you will usually see the x-direction as the horizontal direction on a graph. Your math teacher and math book will almost always draw the x direction and y direction, as you see here. The labels x-axis and y-axis are there to make absolutely sure you know which direction is x and which is y.
There is no reason, however, why a graph couldn't look like this, with the x and y directions reversed:
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Support | 841 | 3,786 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.53125 | 5 | CC-MAIN-2019-39 | longest | en | 0.907951 |
https://datascience.stackexchange.com/questions/54489/bias-and-variance-in-the-model-o-in-the-predictions | 1,657,148,507,000,000,000 | text/html | crawl-data/CC-MAIN-2022-27/segments/1656104678225.97/warc/CC-MAIN-20220706212428-20220707002428-00676.warc.gz | 249,139,253 | 66,988 | # Bias and variance in the model o in the predictions?
This topic confuses me. In the literature or articles, when talking about bias and variance in automatic learning, specifically in cross-validation, do they refer to the high bias (underfitting) and high variance (overfitting) in the model? Or do they refer to the bias and variance of the predictions obtained in the iterations of the cross-validation? How to handle each case?
• I think this question is related to the „bias variance trade-off“ en.m.wikipedia.org/wiki/Bias–variance_tradeoff Jul 25, 2019 at 22:04
In some cases, you may have a model that's a black box - you feed input features and get output predictions, without knowing or caring what happens in the middle. In those situations, the model is, in a way, defined by its output - two models that produce the same predictions are indistinguishable from one another, even though they may be entirely distinct models. In these situations, saying a model is biased or has high variance is equivalent to saying the predictions of the model are biased or have high variance. It's acceptable to describe both the model and its output in terms of bias/variance. Cross validation provides an unbiased estimate of bias/variance for your model/predictions.
– SRG
Jun 25, 2019 at 20:14
• Other question, in this book: books.google.com.mx/…, on page 13, the following is mentioned: "To reduce the influence of randomness introduced by data split, the k-fold cross validation can be repeated t times..." I understand that this randomness produces high variance in the results of the predictions, then, that high variance produces an model with overfitting?
– SRG
Jun 25, 2019 at 20:14
I finally understood "Bias and Variance" with Logistic Regression (LR). LR is known to have a high bias but low variance. For example, LR accuracy might be only at 90% while you'd get 95% with a Decision Tree (tends to overfit). But with LR in production feeding new data, you're very likely to see 90% accuracy ... while the Decision Tree had massive accuracy swings (variance) with new data (70% - 95%).
Hope that helps.
Bias and variance are used to describe the predictions of models and they define if it is a good one or not. As a perfect model (low bias, low variance) does not exists, you often have to chose if you prefer a model with high bias/low variance or a model with low bias/high variance. This applis to the distribution of the predictions.
Cross validation helps you have a more accurate value of your metrics (like accuracy), as the stochastic nature of the way models are calculated can give you different values for the metrics each time your run it: because of some randomness (random seeds) and because of the way the train/validation/test split is done...
So it is good practice to do cross validation as the metrics will be averaged over several train/validation split and so the mean values will be more realistics to compare the different cases/models/parameters. | 658 | 2,994 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.578125 | 3 | CC-MAIN-2022-27 | longest | en | 0.943855 |
https://oeis.org/A192977 | 1,670,611,414,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446711475.44/warc/CC-MAIN-20221209181231-20221209211231-00727.warc.gz | 489,069,679 | 4,202 | The OEIS is supported by the many generous donors to the OEIS Foundation.
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A192977 Number of times 9*n occurs in A068395. 2
4, 4, 2, 2, 3, 2, 2, 3, 2, 1, 0, 4, 1, 1, 3, 1, 2, 2, 2, 1, 4, 0, 0, 1, 3, 2, 1, 2, 2, 2, 2, 1, 0, 1, 3, 0, 2, 2, 2, 1, 2, 2, 1, 0, 2, 1, 1, 3, 2, 1, 3, 1, 1, 2, 0, 2, 0, 2, 0, 2, 1, 2, 2, 1, 2, 0, 2, 3, 0, 1, 3, 2, 1, 2, 1, 1, 0, 2, 1, 1, 2, 1, 2, 2, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET 0,1 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 EXAMPLE n=1: A068395(k) = 9 = 9*1, for k = 5,6,7,8 therefore a(1) = 4; n=10: 90 = 9*10 doesn't occur in A068395, therefore a(10) = 0; n=100: A068395(156) = A068395(157) = 900 = 9*100, therefore a(100) = 2. PROG (Haskell) import Data.List (group) a192977_list = f 0 \$ group a068395_list where f n xss'@(xs:xss) | head xs `div` 9 == n = length xs : f (n+1) xss | otherwise = 0 : f (n+1) xss' CROSSREFS Sequence in context: A291085 A193556 A120438 * A038800 A126712 A162232 Adjacent sequences: A192974 A192975 A192976 * A192978 A192979 A192980 KEYWORD nonn,base AUTHOR Reinhard Zumkeller, Aug 04 2011 STATUS approved
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Last modified December 9 13:38 EST 2022. Contains 358700 sequences. (Running on oeis4.) | 757 | 1,824 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.546875 | 4 | CC-MAIN-2022-49 | latest | en | 0.733726 |
https://www.coursehero.com/file/6295167/HW12/ | 1,490,815,095,000,000,000 | text/html | crawl-data/CC-MAIN-2017-13/segments/1490218191353.5/warc/CC-MAIN-20170322212951-00321-ip-10-233-31-227.ec2.internal.warc.gz | 877,676,661 | 19,574 | # HW12 - MAC1114 Homework 12 Due Thursday April 7 1 Recall that cos(u v = cos u cos v − sin u sin v(a Show that cos(2u = 1 − 2 sin2 u(b Use
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Unformatted text preview: MAC1114 - Homework 12 Due Thursday - April 7 1. Recall that cos(u + v ) = cos u cos v − sin u sin v . (a) Show that cos(2u) = 1 − 2 sin2 u. (b) Use part (a) to show that sin2 u = 1 − cos 2u 2 (c) Use part (b) to show that, for v in quadrant I, sin v = 2 2. Solve the equation tan x − sin x = 0 in the interval [0, 2π ). 2 3. If cos x = 1 5 and x is in Quadrant I, nd sin 4x. 1 1 − cos v 2 ...
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## This note was uploaded on 06/10/2011 for the course MAC 1114 taught by Professor Gentimis during the Spring '11 term at University of Florida.
Ask a homework question - tutors are online | 303 | 910 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.84375 | 3 | CC-MAIN-2017-13 | longest | en | 0.815884 |
https://www.mathworks.com/matlabcentral/answers/511456-for-loop-finding-the-right-index | 1,685,742,580,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224648858.14/warc/CC-MAIN-20230602204755-20230602234755-00319.warc.gz | 964,120,627 | 25,855 | # for loop. finding the right index
2 views (last 30 days)
Ireedui Ganzorig on 18 Mar 2020
Commented: Ireedui Ganzorig on 18 Mar 2020
Hello MATLAB community,
stateOfEnergy is a bunch of data in a column, and I found the largest increase and decrease of stateOfEnergy by utilizing the for loop and built-in <max> and <min>. But I am really struggling to find indices where the largest increase and decrease occured. Below is my code. Is therer any way you can find that specific indices?
stateOfEnergy = load('40815SOE.csv'); % in kiloWattHour
for i = 2 : length(stateOfEnergy)
changeInSOE(i-1) = stateOfEnergy(i) - stateOfEnergy(i-1);
end
largestIncrease = max(changeInSOE); % largest increase of change in SOE in invertal of a single minute. in kWh
disp(['The Largest Increase in SOE was ', num2str(largestIncrease), 'kWh was found between index', ])
largestDecrease = min(changeInSOE); % % largest decrease of change in SOE in invertal of a single minute. in kWh
disp(['The Largest Decrease in SOE was ', num2str(largestDecrease), 'kWh was found between index', ])
David Goodmanson on 18 Mar 2020
Edited: David Goodmanson on 18 Mar 2020
Hello Ireedui,
max and min find the indices as well as the values.
changeInSOE = diff(stateOfEnergy) % find all the differences
[valmax indmax] = max(changeInSOE)
[valmin indmin] = min(changeInSOE)
indmax = n says that the largest positive change of value = valmax occurred from stateOfEnergy(n) to stateOfEnergy(n+1).
indmin = m says that the largest negative change of value = valmin occurred from stateOfEnergy(m) to stateOfEnergy(m+1).
Ireedui Ganzorig on 18 Mar 2020
Thank you very much. You explained it very well. I got it. Thanks. | 468 | 1,679 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.734375 | 3 | CC-MAIN-2023-23 | latest | en | 0.880767 |
https://www.britishideas.com/tag/gear/ | 1,726,439,782,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651668.26/warc/CC-MAIN-20240915220324-20240916010324-00875.warc.gz | 630,813,015 | 10,564 | The most common type of gear is the involute gear, which provides smooth and efficient operation. A gear is defined by the following parameters:
• Pitch diameter (diameter of gear)
• Diametral pitch (tooth size)
• Number of teeth
• Pressure angle (commonly 14.5, 20 or 25 degrees)
The pressure angle defines the shape of a tooth. For two gears to mesh the pressure angle and diametral pitch must be the same (i.e. the shape and size of the teeth must match). There is a simple relationship between pitch diameter, diametral pitch and number of teeth so when defining a gear we only need to specify two of those parameters.
The ability to create involute gears is included in python-based ADScript. Let’s see how.
```NumberofTeeth = 20
PitchDiameter = 30
PressureAngle = 20
InputGear = Part("Input Gear")
XYPlane = InputGear.GetPlane("XY-Plane")
GearSketch = InputGear.AddGearNP("Gear Profile", NumberofTeeth, PitchDiameter, PressureAngle, 0, 0, XYPlane)
```
This script creates a new part called Input Gear and then adds a sketch to it on the XY-plane for the profile of a gear by calling AddGearNP. NP shows which two parameters are used, N = number of teeth and P = pitch diameter.
The 0, 0 defines the coordinates of the center of the gear.
Once the gear has been created we can read out the diametral pitch:
```DiametralPitch = GearSketch.DiametralPitch
```
There are two other functions that we can call to create gears, depending on which two parameters we wish to use.
```GearSketch1 = MyPart.AddGearDN("Gear Profile", DiametralPitch, NumberofTeeth, PressureAngle, 0, 0, XYPlane)
PitchDiameter = GearSketch1.PitchDiameter
GearSketch2 = MyPart.AddGearDP("Gear Profile", DiametralPitch, PitchDiameter, PressureAngle, 0, 0, XYPlane)
NumberofTeeth = GearSketch2.NumberofTeeth
```
Here is an example script that shows how we can create two gears that can be used together to provide a 3:1 reduction ratio:
```# all values are in millimeters
Units.Current = UnitTypes.Millimeters
# pressure angle has to be the same for both gears
# common values are 14.5, 20 and 25
# larger pressure angle = stronger teeth
PressureAngle = 20
# thickness of gears
Thickness = 2
# shaft diameter
ShaftDiameter = 5
# define input gear - 20 teeth and 30mm pitch diameter
Gear1Teeth = 20
Gear1Diameter = 30
# create input gear using number of teeth and pitch diameter
Gear1Part = Part("Input Gear")
Gear1 = Gear1Part.AddGearNP("Gear Profile", Gear1Teeth, Gear1Diameter, PressureAngle, 0, 0, Gear1Part.GetPlane("XY-Plane"))
# diametral pitch must be the same for both gears as this defines tooth size
# use diametral pitch for output gear from input gear calculated value
Gear2DiametralPitch = Gear1.DiametralPitch
# we want a ratio of 1:3 so output gear has three times the number of teeth
Gear2Teeth = Gear1Teeth * 3
# create output gear using number of teeth and diametral pitch
Gear2Part = Part("Output Gear")
Gear2 = Gear2Part.AddGearDN("Gear Profile", Gear2DiametralPitch, Gear2Teeth, PressureAngle, 0, 0, Gear2Part.GetPlane("XY-Plane"))
# get the calculated pitch diameter of the output gear
Gear2Diameter = Gear2.PitchDiameter
# calculate distance between gear centers so we know how far apart the shafts need to be
print "Distance between gear centers = %fmm\n" % ((Gear1Diameter + Gear2Diameter) / 2)
```
And the result: | 894 | 3,334 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.5625 | 3 | CC-MAIN-2024-38 | latest | en | 0.802895 |
https://www.coursehero.com/file/222980/Stoichiometry-/ | 1,496,013,176,000,000,000 | text/html | crawl-data/CC-MAIN-2017-22/segments/1495463611569.86/warc/CC-MAIN-20170528220125-20170529000125-00418.warc.gz | 1,071,125,949 | 73,574 | Stoichiometry_
Stoichiometry_ - Topic 1 Stoichiometry 1.1 Mole Concept and...
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Topic 1 Stoichiometry 1.1 Mole Concept and Avogadro’s Constant 1.1.1 Describe the mole concept and apply it to substances. The mole concept applies to all kinds of particles: atoms , molecules, ions , formula units etc. The amount of substance is measured in units of moles. The approximate value of Avogadro’s constant ( L ) , 6.02 X10 23 , should be known. 1.1.2 Calculate the number of particles and the amount of substance (in moles) Convert between the amount of substance ( in moles ) and the number of atoms , molecules or formula units. 1.2 Formulas 1.2.1 Define the term molar mass (M ) and calculate the mass of one mole of a species. 1.2.2 Distinguish between atomic mass , molecular mass and formula mass. The term molar mass ( in g mol -1 ) can be used for all of these. 1.2.3 Define the terms relative molecular mass ( Mr ) and relative atomic mass ( Ar) . The terms have no units. 1.2.4 State the relationship between the amount of substance ( in moles ) and mass , and carry out calculations involving amount of substance, mass and molar mass. 1.2.5 1.2.5 Define the terms empirical formula and molecular formula. The molecular formula is a multiple of the empirical formula. 1.2.6 Determine the empirical formula and/or the molecular formula of a given compound. Determine the : empirical formula from the percentage composition or from other suitabel experimental data . percentage composition from the formula of a compound molecular formula when given both the empirical formula and the molar mass. 1.3 Chemical Equations 1.3.1 Balance chemical equations when all reactants and products are given. Distinguish between coefficients and subscripts. 1.3.2 identify the mole ratios of any two species in a balanced chemical equation. Use balanced chemical equations to obtain information about the
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amounts of reactants and products. 1.3.3 Apply the state symbols(s), (l) (g) and (aq). Encourage the use of state symbols in chemical equations. 1.4 Mass and Gaseous Volume Relationships in Chemical Reactions 1.4.1 Calculate stoichiometric quantities and use these to determine experimental and theoretical yields. Mass is conserved in all chemical reactions.Given a chemical equqtion and the mass or amount (in moles ) of one species, calculate the mass or amount of another species. 1.4.2 Determine the limiting reactant and the reactant in excess when quantities of reacting substances are given. Given a chemical equation and the initial amounts of two or more reactants: identify the limiting reactant. Calculate the theoretical yield of a product. Calculate the amount(s) of the reactant(s) in excess remaining after the reactions is complete. 1.4.3 Apply Avogadro’s law to calculate reacting volumes of gases. 1.5 Solutions
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This note was uploaded on 08/08/2008 for the course CEM 141 taught by Professor Hunter during the Fall '08 term at Michigan State University.
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Stoichiometry_ - Topic 1 Stoichiometry 1.1 Mole Concept and...
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Ask a homework question - tutors are online | 813 | 3,428 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.171875 | 3 | CC-MAIN-2017-22 | longest | en | 0.83947 |
http://de.metamath.org/mpegif/tendopltp.html | 1,532,125,595,000,000,000 | text/html | crawl-data/CC-MAIN-2018-30/segments/1531676591837.34/warc/CC-MAIN-20180720213434-20180720233434-00374.warc.gz | 87,383,354 | 10,453 | Mathbox for Norm Megill < Previous Next > Nearby theorems Mirrors > Home > MPE Home > Th. List > Mathboxes > tendopltp Unicode version
Theorem tendopltp 31262
Description: Trace-preserving property of endomorphism sum operation , based on theorem trlco 31209. Part of remark in [Crawley] p. 118, 2nd line, "it is clear from the second part of G (our trlco 31209) that Delta is a subring of E." (In our development, we will bypass their E and go directly to their Delta, whose base set is our .) (Contributed by NM, 9-Jun-2013.)
Hypotheses
Ref Expression
tendopl.h
tendopl.t
tendopl.e
tendopl.p
tendopltp.l
tendopltp.r
Assertion
Ref Expression
tendopltp
Distinct variable groups: ,, ,,, ,,,
Allowed substitution hints: (,,) (,,) (,,) () (,,) (,,) (,,) (,,) (,,)
Proof of Theorem tendopltp
StepHypRef Expression
1 eqid 2404 . 2
2 tendopltp.l . 2
3 simp1l 981 . . 3
4 hllat 29846 . . 3
53, 4syl 16 . 2
6 simp1 957 . . 3
7 tendopl.h . . . 4
8 tendopl.t . . . 4
9 tendopl.e . . . 4
10 tendopl.p . . . 4
117, 8, 9, 10tendoplcl2 31260 . . 3
12 tendopltp.r . . . 4
131, 7, 8, 12trlcl 30646 . . 3
146, 11, 13syl2anc 643 . 2
157, 8, 9tendocl 31249 . . . . 5
16153adant2r 1179 . . . 4
171, 7, 8, 12trlcl 30646 . . . 4
186, 16, 17syl2anc 643 . . 3
197, 8, 9tendocl 31249 . . . . 5
20193adant2l 1178 . . . 4
211, 7, 8, 12trlcl 30646 . . . 4
226, 20, 21syl2anc 643 . . 3
23 eqid 2404 . . . 4
241, 23latjcl 14434 . . 3
255, 18, 22, 24syl3anc 1184 . 2
26 simp3 959 . . 3
271, 7, 8, 12trlcl 30646 . . 3
286, 26, 27syl2anc 643 . 2
29 simp2l 983 . . . . 5
30 simp2r 984 . . . . 5
3110, 8tendopl2 31259 . . . . 5
3229, 30, 26, 31syl3anc 1184 . . . 4
3332fveq2d 5691 . . 3
342, 23, 7, 8, 12trlco 31209 . . . 4
356, 16, 20, 34syl3anc 1184 . . 3
3633, 35eqbrtrd 4192 . 2
372, 7, 8, 12, 9tendotp 31243 . . . 4
38373adant2r 1179 . . 3
392, 7, 8, 12, 9tendotp 31243 . . . 4
40393adant2l 1178 . . 3
411, 2, 23latjle12 14446 . . . 4
425, 18, 22, 28, 41syl13anc 1186 . . 3
4338, 40, 42mpbi2and 888 . 2
441, 2, 5, 14, 25, 28, 36, 43lattrd 14442 1
Colors of variables: wff set class Syntax hints: wi 4 wb 177 wa 359 w3a 936 wceq 1649 wcel 1721 class class class wbr 4172 cmpt 4226 ccom 4841 cfv 5413 (class class class)co 6040 cmpt2 6042 cbs 13424 cple 13491 cjn 14356 clat 14429 chlt 29833 clh 30466 cltrn 30583 ctrl 30640 ctendo 31234 This theorem is referenced by: tendoplcl 31263 This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2385 ax-rep 4280 ax-sep 4290 ax-nul 4298 ax-pow 4337 ax-pr 4363 ax-un 4660 This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2258 df-mo 2259 df-clab 2391 df-cleq 2397 df-clel 2400 df-nfc 2529 df-ne 2569 df-nel 2570 df-ral 2671 df-rex 2672 df-reu 2673 df-rmo 2674 df-rab 2675 df-v 2918 df-sbc 3122 df-csb 3212 df-dif 3283 df-un 3285 df-in 3287 df-ss 3294 df-nul 3589 df-if 3700 df-pw 3761 df-sn 3780 df-pr 3781 df-op 3783 df-uni 3976 df-iun 4055 df-iin 4056 df-br 4173 df-opab 4227 df-mpt 4228 df-id 4458 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-iota 5377 df-fun 5415 df-fn 5416 df-f 5417 df-f1 5418 df-fo 5419 df-f1o 5420 df-fv 5421 df-ov 6043 df-oprab 6044 df-mpt2 6045 df-1st 6308 df-2nd 6309 df-undef 6502 df-riota 6508 df-map 6979 df-poset 14358 df-plt 14370 df-lub 14386 df-glb 14387 df-join 14388 df-meet 14389 df-p0 14423 df-p1 14424 df-lat 14430 df-clat 14492 df-oposet 29659 df-ol 29661 df-oml 29662 df-covers 29749 df-ats 29750 df-atl 29781 df-cvlat 29805 df-hlat 29834 df-llines 29980 df-lplanes 29981 df-lvols 29982 df-lines 29983 df-psubsp 29985 df-pmap 29986 df-padd 30278 df-lhyp 30470 df-laut 30471 df-ldil 30586 df-ltrn 30587 df-trl 30641 df-tendo 31237
Copyright terms: Public domain W3C validator | 2,152 | 4,096 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.9375 | 3 | CC-MAIN-2018-30 | latest | en | 0.238338 |
https://www.aepochadvisors.com/oddfprice/ | 1,708,536,659,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947473524.88/warc/CC-MAIN-20240221170215-20240221200215-00759.warc.gz | 661,898,634 | 26,914 | # ODDFPRICE
In this comprehensive guide, we will explore everything you need to know about the ODDFPRICE formula in Excel. The ODDFPRICE function is a financial function that calculates the price per \$100 face value of a security with an odd first period. This function is particularly useful for bond pricing and analysis, especially when dealing with bonds that have irregular first interest periods.
## ODDFPRICE Syntax
The syntax for the ODDFPRICE function in Excel is as follows:
=ODDFPRICE(settlement, maturity, issue, first_coupon, rate, yield, redemption, frequency, [basis])
Here is a description of each argument in the ODDFPRICE function:
• settlement: The settlement date of the security, which is the date after the issue date when the security is traded to the buyer.
• maturity: The maturity date of the security, which is the date when the security expires.
• issue: The issue date of the security.
• first_coupon: The end date of the first coupon period.
• rate: The security’s annual coupon rate.
• yield: The security’s annual yield.
• redemption: The security’s redemption value per \$100 face value.
• frequency: The number of coupon payments per year. For annual payments, use 1; for semiannual payments, use 2; for quarterly payments, use 4.
• [basis] (optional): The day count basis to be used. If omitted, the default is 0, which represents the US (NASD) 30/360 day count basis. Other options include 1 for actual/actual, 2 for actual/360, 3 for actual/365, and 4 for European 30/360.
## ODDFPRICE Examples
Let’s look at some examples of how to use the ODDFPRICE function in Excel:
Example 1: A bond has a settlement date of January 15, 2022, a maturity date of January 15, 2027, an issue date of November 15, 2021, and a first coupon date of April 15, 2022. The bond has an annual coupon rate of 5%, an annual yield of 6%, a redemption value of \$100, and semiannual coupon payments. To calculate the price per \$100 face value of this bond, you would use the following formula:
=ODDFPRICE(“2022-01-15”, “2027-01-15”, “2021-11-15”, “2022-04-15”, 5%, 6%, 100, 2)
Example 2: A bond has a settlement date of March 1, 2022, a maturity date of March 1, 2028, an issue date of December 1, 2021, and a first coupon date of June 1, 2022. The bond has an annual coupon rate of 4%, an annual yield of 5%, a redemption value of \$100, and quarterly coupon payments. To calculate the price per \$100 face value of this bond, you would use the following formula:
=ODDFPRICE(“2022-03-01”, “2028-03-01”, “2021-12-01”, “2022-06-01”, 4%, 5%, 100, 4)
## ODDFPRICE Tips & Tricks
Here are some tips and tricks to help you get the most out of the ODDFPRICE function in Excel:
• Make sure to enter dates in the correct format, such as “YYYY-MM-DD” or by using the DATE function, e.g., DATE(2022, 1, 15).
• Remember that the ODDFPRICE function returns the price per \$100 face value. To calculate the actual price of the bond, multiply the result by the face value of the bond and divide by 100.
• When comparing bonds with different coupon frequencies, it’s important to annualize the yield to make a fair comparison. You can do this by using the EFFECT function in Excel.
## Common Mistakes When Using ODDFPRICE
Here are some common mistakes to avoid when using the ODDFPRICE function:
• Using incorrect date formats or entering dates as text. Always use the correct date format or the DATE function to avoid errors.
• Forgetting to adjust the coupon rate and yield for the frequency of payments. If the bond pays semiannual coupons, divide the annual coupon rate and yield by 2; if it pays quarterly coupons, divide by 4.
• Not specifying the optional [basis] argument when using a day count basis other than the default US (NASD) 30/360.
## Why Isn’t My ODDFPRICE Working?
If you’re having trouble with the ODDFPRICE function, consider the following troubleshooting tips:
• Check your date formats and ensure they are entered correctly.
• Make sure you have entered all required arguments and that they are in the correct order.
• Ensure that the coupon rate, yield, and redemption values are entered as percentages or decimal values, not as whole numbers.
• Verify that the frequency argument is entered as a whole number (1, 2, or 4) and not as text.
## ODDFPRICE: Related Formulae
Here are some related financial functions in Excel that you may find useful when working with bonds and securities:
• ODDLPRICE: Calculates the price per \$100 face value of a security with an odd last period.
• PRICE: Calculates the price per \$100 face value of a security that pays periodic interest.
• YIELD: Calculates the yield of a security that pays periodic interest.
• ACCRINT: Calculates the accrued interest for a security that pays periodic interest.
• COUPNCD: Calculates the next coupon date after the settlement date for a security that pays periodic interest.
By mastering the ODDFPRICE function and related financial functions in Excel, you can effectively analyze and compare bonds and other securities with irregular first interest periods, making more informed investment decisions.
## Related
### Hard to find or retain a good accountant? Try cloud accounting solution
Foreign business owners or management team always take financial transparency as a pre-condition for good decision making and sustainable profitability. However, achieving the visualization of
### Cloud Accounting Software Automates Compliance Service in China
Managing accounting compliance in China can be a challenging task for businesses, as it involves dealing with complex regulations and paperwork. However, the advent of | 1,331 | 5,641 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.984375 | 3 | CC-MAIN-2024-10 | longest | en | 0.867007 |
https://easierwithpractice.com/how-do-you-write-numbers-with-decimals-in-words/ | 1,708,913,651,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947474649.44/warc/CC-MAIN-20240225234904-20240226024904-00694.warc.gz | 216,457,251 | 41,524 | # How do you write numbers with decimals in words?
## How do you write numbers with decimals in words?
How do you write a decimal number in word form?
1. To write a decimal in word form, follow these steps:
2. Let’s try it with 2.37.
3. First, write the whole-number part.
4. Write “and” for the decimal point.
5. Next, write the word form of the digits in the decimal part.
6. Finally, end with the place value of the last digit.
## How do you say .02 in words?
How to Write Out Number 0.02 in Words: two hundredths in (US) American English, Number Converted (Spelled Out) in Eight (8) Different Letter Cases.
How do you say 0.001 in words?
In word form, 0.001 in worded form is one-hundredth.
How do you say 0.012 in words?
How to Write Out Number 0.012 in Words: twelve thousandth in (US) American English, Number Converted (Spelled Out) in Eight (8) Different Letter Cases.
### What is 1.002 in word form?
One and two thousandths is the correct answer.
### How do you write 0.102 in words?
How to Write Out Number 0.102 in Words: one hundred two thousandth in (US) American English, Number Converted (Spelled Out) to Words.
Are you supposed to say and in numbers?
While it’s common [at least for math teachers] to say that “and” must not be used except to separate the integer part from the fraction/decimal part, the usage is “One hundred and thirty-five hundredths” for 100.35, and the “hundredths” is not optional.
How do you say 120 in English?
How to write 120 Number in Currency Spelling?
1. AUD => one hundred twenty Australian dollars.
2. BGN => one hundred twenty leva.
3. BWP => one hundred twenty pula.
5. GBP => one hundred twenty pounds sterling.
6. CNY => one hundred twenty Chinese yuan.
7. CHF => one hundred twenty Swiss francs.
## How do you say 110 in English?
How to write 110 Number in Currency Spelling?
1. AUD => one hundred ten Australian dollars.
2. BGN => one hundred ten leva.
3. BWP => one hundred ten pula.
5. GBP => one hundred ten pounds sterling.
6. CNY => one hundred ten Chinese yuan.
7. CHF => one hundred ten Swiss francs.
## How do you write 110 in word form?
Number 110 written in: ‘lowercase’, ‘UPPERCASE’, ‘Title Case’ and ‘Sentence case’.
1. lowercase. (all small caps letters): one hundred ten.
2. UPPERCASE. (ALL CAPITAL LETTERS): ONE HUNDRED TEN.
3. Title Case. (Capital Letters at the Beginning of Words): One Hundred Ten.
4. Sentence case. (Capital Letters at the Beginning of Words):
How do you fill out a 110 check?
\$ (Amount in Numeric Form): Put 110.00 in the box right after the \$ sign on the same line. Make sure to include the decimal part 00. DOLLARS (Amount in Words): Write One hundred ten and 00/100 on the next field as far to the left on that line as possible. Use sentence case.
How do you write 150 in words?
150 in English Words is : one hundred fifty.
### How do you write 1111?
1111 in english: ( one thousand, one hundred eleven )…
### What does 1111 stand for?
to pay attention to your
What does 1111 mean in love?
When you see 1111, it’s a love sign from the universe, asking you to open your heart to a new love. If you’re feeling a magnetic connection to someone, find the courage to surrender yourself to it without fear. If you’ve been wondering whether to start looking for love, this is the sign that you’re ready.
What happen in the year 1111?
Europe. April 13 – Henry V is crowned as Holy Roman Emperor by Pope Paschal II. Henry returns to Germany, where he strengthens his power by granting privileges to the German nobles of the region of the Upper Rhine. Almoravid forces under Syr ibn Abi Bakr capture Santarém and Sintra.
## What happened in the year 666?
666 AD was the year when Ramla, the last living wife of Mohammed the Prophet, died. It was the 600th anniversary of the Great Fire of Rome. The 6174–75 year since God made the world and 1319 years since the founding of the city of Rome (a.u.c.).
## What happened in the year 2020?
On Jan. 3, 2020, the U.S. killed Soleimani and others in a drone strike near Baghdad International Airport. NBA legend Kobe Bryant was killed in a helicopter crash in Calabasas, California, along with his 13-year-old daughter Gianna and seven others, on Jan. 26.
What happened 1000 CE?
Muslims destroy Holy Sepulchre in Jerusalem. Danes control England. Canute takes throne (1016), conquers Norway (1028), dies (1035); kingdom divided among his sons: Harold Harefoot (England), Sweyn (Norway), Hardecanute (Denmark). Macbeth murders Duncan, king of Scotland.
Is there a year 0 CE?
The year zero does not exist in the Anno Domini (AD) system commonly used to number years in the Gregorian calendar and in its predecessor, the Julian calendar. In this system, the year 1 BC is followed by AD 1.
### What time period was 1000 years ago?
Millennium, a period of 1,000 years. The Gregorian calendar, put forth in 1582 and subsequently adopted by most countries, did not include a year 0 in the transition from bc (years before Christ) to ad (those since his birth). Thus, the 1st millennium is defined as spanning years 1–1000 and the 2nd the years 1001–2000.
### What was life like 1000 years ago?
The world was a much different place 1000 years ago. Life expectancy was shorter, Vikings kept stealing people’s things, and wifi signals were quite poor. Those who believe in reincarnation say we’ve all lived many lives throughout existence.
Which century do we live in?
We live in the 21st Century, that is, the 2000s. Similarly when we say “20th Century,” we are referring to the 1900s. All this because, according to the calendar we use, the 1st Century included the years 1-100 (there was no year zero), and the 2nd Century, the years 101-200.
What did the world look like 20000 years ago?
20,000 YEARS AGO. Last Glacial Maximum- a time, around 20,000 years ago, when much of the Earth was covered in ice. The average global temperature may have been as much as 10 degrees Celsius colder than that of today. The Earth has a long history of cycles between warming and cooling.
## What was the world like 5000 years ago?
Lasting roughly 2.5 million years, the Stone Age ended around 5,000 years ago when humans in the Near East began working with metal and making tools and weapons from bronze. During the Stone Age, humans shared the planet with a number of now-extinct hominin relatives, including Neanderthals and Denisovans.
7,000 years ago
# How do you write numbers with decimals in words?
## How do you write numbers with decimals in words?
How do you write a decimal number in word form?
1. To write a decimal in word form, follow these steps:
2. Let’s try it with 2.37.
3. First, write the whole-number part.
4. Write “and” for the decimal point.
5. Next, write the word form of the digits in the decimal part.
6. Finally, end with the place value of the last digit.
### How do you say .02 in words?
How to Write Out Number 0.02 in Words: two hundredths in (US) American English, Number Converted (Spelled Out) in Eight (8) Different Letter Cases.
#### How do you write 0.09 in word form?
How to write -0.09 in english words? – The number -0.09 written in english words is “nine hundredths”.
How do you say 0.8 in words?
How do you say 0.4 in words?
How to write 0.4 in english words? – The number 0.4 written in english words is “four tenths”.
## What’s 0.8 as a percent?
80%
### What is 0.1 as a percentage?
10%
#### What is 0.05 as a percent?
5 %
What is 0.02 as a percent?
2 percent
What is 3.5 as a percent?
350%
## What is 3/4 as a percent?
75%
### What is 3.5 out of 5 as a percentage?
70
#### What is a 3.7 out of 5?
74%
What grade is a 1.67 out of 5?
33.4%
What is a 3.8 out of 5?
76%
## What is a 3.9 out of 5?
78%
### What percentage is 4 out of 5?
50%
#### What percent is 3.9 out of 5?
78
How do you find 8% of a number?
Percentage Calculator How to find 8% of a number? Take the number and multiple it by 8. Then multiply that by . 01.
What is 6% of a number?
Multiply 0.06 times the number from which you are seeking 6 percent. For example, to find 6 percent of 100, work out 0.06 x 100 = 6. Another example: 6 percent of 75 is 0.06 x 75 = 4.5.
## What number is 30% of 15?
Latest calculated numbers percentages
30% of 15 = 4.5 Jun 22 21:11 UTC (GMT)
539% of 7.84 = 42.2576 Jun 22 21:11 UTC (GMT)
2% of 617.5 = 12.35 Jun 22 21:11 UTC (GMT)
996% of 660 = 6,573.6 Jun 22 21:11 UTC (GMT)
1% of 93.56353368 = 0.9356353368 Jun 22 21:11 UTC (GMT) | 2,320 | 8,483 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.234375 | 3 | CC-MAIN-2024-10 | latest | en | 0.855864 |
http://www.daviddarling.info/encyclopedia/M/Martingale_system.html | 1,516,283,593,000,000,000 | text/html | crawl-data/CC-MAIN-2018-05/segments/1516084887414.4/warc/CC-MAIN-20180118131245-20180118151245-00344.warc.gz | 420,233,763 | 3,876 | A
# Martingale system
The Martingale system is a simple, popular, and ultimately disastrous gambling system that, on the face of it, seems like a dream come true. In the short run the player has a good chance of making a few dollars using this method. But, in the long run, two things conspire to defeat him – the table betting limits and the player's available funds. The Martingale system calls for an initial bet of, say, \$2. If the player loses, he doubles his bet to \$4. Another loss puts the net loss at \$6 and requires a doubling of the bet to \$8 to recoup the losses and show a profit. Assume the player loses five hands in a row. The sixth bet requires \$64. Let's say this wins – the gambler has now won \$128 but has lost \$124 – a net win of \$4. Of course, any win, however small, if repeated over and over, could produce a fortune. The trouble is that many losing streaks run longer than 6 or 8 or 10 in a row. The Martingale quickly runs into the table limits. For example, if the player is at a \$2 blackjack table with a \$500 upper limit, he has to retire after 9 losses in a row and is down over \$1000. It would take another 500 winning hands to make up this loss! Basically, one losing streak will put the Martingale gambler in a hole he is unlikely ever to climb back out of. | 321 | 1,303 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.0625 | 3 | CC-MAIN-2018-05 | longest | en | 0.960785 |
https://pressbooks.pub/unmcorewriting/chapter/flow-cohesion-and-transitions-in-paragraphs/ | 1,717,099,050,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971670239.98/warc/CC-MAIN-20240530180105-20240530210105-00790.warc.gz | 402,463,233 | 32,178 | # Flow, Cohesion, and Transitions in Paragraphs
Chapter
### Cohesion and Coherence
With a key sentence established, the next task is to shape the body of your paragraph to be both cohesive and coherent. Cohesion is about the “sense of flow” (how each sentence fits with the next), while coherence is about the “sense of the whole.” Some students worry too much about “flow” and spend a lot of time on sentence-level issues to promote it. We encourage you to focus on the underlying structure. For the most part, a text reads smoothly when it conveys a thoughtful and well-organized argument or analysis. Focus first and most on your ideas, on crafting an ambitious analysis. The most useful guides advise you to first focus on getting your ideas on paper and then revising for organization and wordsmithing later, refining the analysis as you go. Therefore, we will discuss creating cohesion and coherent paragraphs here as if you already have some rough text written and are in the process of smoothing out your prose to clarify your argument for both your reader and yourself. Deal with cohesion and coherence during the editing–not the drafting–process.
Cohesion refers to the flow from sentence to sentence. For example, compare these passages about how cliques form:
Version A (Which We Rewrote)
Granovetter begins by looking at balance theory. If an actor, A, is strongly tied to both B and C, it is extremely likely that B and C are, sooner or later, going to be tied to each other, according to balance theory (1363). Bridge ties between cliques are always weak ties, Granovetter argues (1364). Weak ties may not necessarily be bridges, but Granovetter argues that bridges will be weak. If two actors share a strong tie, they will draw in their other strong relations and will eventually form a clique. Only weak ties that do not have the strength to draw together all the “friends of friends” can connect people in different cliques.
Version B (The Original By Giuffre)
Granovetter begins by looking at balance theory. In brief, balance theory tells us that if an actor, A, is strongly tied to both B and C, it is extremely likely that B and C are, sooner or later, going to be tied to each other (1363). Granovetter argues that because of this, bridge ties between cliques are always weak ties (1364). Weak ties may not necessarily be bridges, but Granovetter argues that bridges will be weak. This is because if two actors share a strong tie, they will draw in their other strong relations and will eventually form a clique. The only way, therefore, that people in different cliques can be connected is through weak ties that do not have the strength to draw together all the “friends of friends.”
Version A has the exact same information as version B, but it is harder to read because it is less cohesive. In version B, however, each sentence begins with old information and bridges to new information. Here’s Version B again with the relevant parts emboldened:
Granovetter begins by looking at balance theory. In brief, balance theory tells us that if an actor, A, is strongly tied to both B and C, it is extremely likely that B and C are, sooner or later, going to be tied to each other (1363). Granovetter argues that because of this, bridge ties between cliques are always weak ties (1364). Weak ties may not necessarily be bridges, but Granovetter argues that bridges will be weak. This is because if two actors share a strong tie, they will draw in their other strong relations and will eventually form a clique. The only way, therefore, that people in different cliques can be connected is through weak ties that do not have the strength to draw together all the “friends of friends.”
The first sentence establishes the key idea of balance theory. The next sentence begins with balance theory and ends with social ties, which is the focus of the third sentence. The concept of weak ties connects the third and fourth sentences and the concept of cliques the fifth and sixth sentences. In Version A, in contrast, the first sentence focuses on balance theory, but then the second sentence makes a new point about social ties before telling the reader that the point comes from balance theory. The reader has to take in a lot of unfamiliar information before learning how it fits in with familiar concepts. Version A is coherent, but the lack of cohesion makes it tedious to read.
Learning Objectives
The lesson is this: if you or others perceive a passage you’ve written to be awkward or choppy, even though the topic is consistent, try rewriting it to ensure that each sentence begins with a familiar term or concept. If your points don’t naturally daisy-chain together like the examples given here, consider numbering them. For example, you may choose to write, “Proponents of the legislation point to four major benefits.” Then you could discuss four loosely related ideas without leaving your reader wondering how they relate.
While cohesion is about the sense of flow, coherence is about the sense of the whole. For example, here’s a passage that is cohesive (from sentence to sentence) but lacks coherence:
Your social networks and your location within them shape the kinds and amount of information that you have access to. Information is distinct from data in that information makes some kind of generalization about a person, thing, or population. Defensible generalizations about society can be either probabilities (i.e., statistics) or patterns (often from qualitative analysis). Such probabilities and patterns can be temporal, spatial, or simultaneous.
Each sentence in the above passage starts with a familiar idea and progresses to a new one, but it lacks coherence—a sense of being about one thing. Good writers often write passages like that when they’re free-writing or using the drafting stage to cast a wide net for ideas. A writer weighing the power and limits of social network analysis may free-write something like that example and, from there, develop a more specific plan for summarizing key insights about social networks and then discussing them with reference to the core tenets of social science. As a draft, an incoherent paragraph often points to a productive line of reasoning; one just has to continue thinking it through in order to identify a clear argumentative purpose for each paragraph. With its purpose defined, each paragraph then becomes a lot easier to write. Coherent paragraphs aren’t just about style; they are a sign of a thoughtful, well-developed analysis.
### Conclusion Sentences
Some guides advise you to end each paragraph with a specific concluding sentence, in a sense, to treat each paragraph as a kind of mini-essay. But that’s not a widely held convention. Most well-written academic pieces don’t adhere to that structure. The last sentence of the paragraph should certainly be in your own words (as in, not a quote), but as long as the paragraph succeeds in carrying out the task that it has been assigned by its key sentence, you don’t need to worry about whether that last sentence has an air of conclusiveness. For example, consider these paragraphs about the cold fusion controversy of the 1980s :
The experiment seemed straightforward and there were plenty of scientists willing to try it. Many did. It was wonderful to have a simple laboratory experiment on fusion to try after decades of embarrassing attempts to control hot fusion. This effort required multi-billion dollar machines whose every success seemed to be capped with an unanticipated failure. ‘Cold fusion’ seemed to provide, as Martin Fleischmann said during the course of that famous Utah press conference, ‘another route’—the route of little science.
In that example, the first and last sentences in the paragraph are somewhat symmetrical: the authors introduce the idea of accessible science, contrast it with big science, and bring it back to the phrase “little science.” Here’s an example from the same chapter of the same book that does not have any particular symmetry:
The struggle between proponents and critics in a scientific controversy is always a struggle for credibility. When scientists make claims which are literally ‘incredible’, as in the cold fusion case, they face an uphill struggle. The problem Pons and Fleischmann had to overcome was that they had credibility as electrochemists but not as nuclear physicists. And it was nuclear physics where their work was likely to have its main impact.
The last sentence of the paragraph doesn’t mirror the first, but the paragraph still works just fine. In fact, it serves as a transition into what is likely the focus of the next paragraph. In general, every sentence of academic writing should add some unique content. Don’t trouble yourself with having the last sentence in every paragraph serve as a mini-conclusion. Instead, worry about developing each point sufficiently and making your logical sequence clear.
### Creating Coherence with Transitions
To ensure continuity within a paragraph, a writer must use key terms and other techniques, such as transitions. Transitions ensure that the reader understands what each sentence says and indicate how the sentences and paragraphs are logically related to each other and the “story.” Transitions should be placed at the beginning of a sentence for the strongest continuity, usually set off by a comma.
If transitions are missing, the logical relationship between sentences can be unclear and may even be nonexistent. Don’t assume your reader knows the relationship between the sentence. The importance of adding transitions is shown in the following example:
To determine the effects of lack of lunch on children’s school performance, we examined grades and income measures. We found they were negatively correlated. ______________, lack of food makes learning in school more difficult.
The logical relationship between the first and second sentences of this example is not immediately obvious to the reader. Possible transitions that one could fill in between these two sentences include the following:
• To determine the effects of lack of lunch on children’s school performance, we examined grades and income measures. We found they were negatively correlated. In addition, lack of food makes learning in school more difficult.
• To determine the effects of lack of lunch on children’s school performance, we examined grades and income measures. We found they were negatively correlated. Therefore, lack of food makes learning in school more difficult.
• To determine the effects of lack of lunch on children’s school performance, we examined grades and income measures. We found they were negatively correlated. On the other hand, lack of food makes learning in school more difficult.
When the transition is missing between these two sentences, most readers may guess that the intended relationship is “in addition” when it really is “therefore.” But readers should not have to guess. Sometimes readers can come to a very different conclusion than you intend!
Here is another example:
We determined sensitivity to allergens in children who have pets. We measured children’s histamine levels and logged whether they had pets or not.
In the above example, the logical relationship between the first and second sentences is also not clear because a transition is missing. Once the transition is added in, the relationship between the two sentences becomes obvious.
Revised example:
We determined sensitivity to allergens in children who have pets. To do this, we measured children’s histamine levels and logged whether they had pets or not.
Use transitions to link ideas, but do not overuse them. Certain transitions are outdated or are used in British, but not American, English, such as: hitherto, whilst, and henceforth.
Common Transitional Words and Phrases Transitions That Show Sequence or Time after before later afterward before long meanwhile as soon as finally next at first first, second, third soon at last in the first place then Transitions That Show Position above across at the bottom at the top behind below beside beyond inside near next to opposite to the left, to the right, to the side under where Transitions That Show a Conclusion indeed hence in conclusion in the final analysis therefore thus Transitions That Continue a Line of Thought consequently furthermore additionally because besides the fact following this idea further in addition in the same way moreover looking further considering…, it is clear that Transitions That Change a Line of Thought but yet however nevertheless on the contrary on the other hand Transitions That Show Importance above all best especially in fact more important most important most worst Transitions That Introduce the Final Thoughts in a Paragraph or Essay finally last in conclusion most of all least of all last of all All-Purpose Transitions to Open Paragraphs or to Connect Ideas Inside Paragraphs admittedly at this point certainly granted it is true generally speaking in general in this situation no doubt no one denies obviously of course to be sure undoubtedly unquestionably Transitions that Introduce Examples for instance for example Transitions That Clarify the Order of Events or Steps first, second, third generally, furthermore, finally in the first place, also, last in the first place, furthermore, finally in the first place, likewise, lastly
Transitions to Use with Supporting & Concluding Sentences
A strong paragraph moves seamlessly from the topic sentence into the supporting sentences and on to the concluding sentence. To help organize a paragraph and ensure that ideas logically connect to one another, writers use transitional words and phrases. A transition is a connecting word that describes a relationship between ideas. Take another look at the earlier example:
There are numerous advantages to owning a hybrid car. First, they get 20 percent to 35 percent more miles to the gallon than a fuel-efficient gas-powered vehicle. Second, they produce very few emissions during low-speed city driving. Because they do not require gas, hybrid cars reduce dependence on fossil fuels, which helps lower prices at the pump. Alex bought a hybrid car two years ago and has been extremely impressed with its performance. “It’s the cheapest car I’ve ever had,” she said. “The running costs are far lower than previous gas-powered vehicles I’ve owned.” Given the low running costs and environmental benefits of owning a hybrid car, it is likely that many more people will follow Alex’s example in the near future.
Each of the underlined words is a transition word. Words such as first and second are transition words that show sequence or clarify order. They help organize the writer’s ideas by showing that they have another point to make in support of the topic sentence. Other transition words that show order include thirdalso, and furthermore.
The transition word because is a transition word of consequence that continues a line of thought. It indicates that the writer will provide an explanation of a result. In this sentence, the writer explains why hybrid cars will reduce dependency on fossil fuels (because they do not require gas). Other transition words of consequence include as a resultso thatsince, or for this reason.
The following chart provides some useful transition words to connect supporting sentences and concluding sentences.
Useful Transitional Words and Phrases For Supporting Sentences above all but for instance in particular moreover subsequently also conversely furthermore later on nevertheless therefore aside from correspondingly however likewise on one hand to begin with at the same time for example in addition meanwhile on the contrary For Concluding Sentences after all all things considered in brief in summary on the whole to sum up all in all finally in conclusion on balance thus | 3,123 | 15,917 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.6875 | 3 | CC-MAIN-2024-22 | latest | en | 0.936036 |
https://gmatclub.com/forum/in-the-mid-fifth-century-rome-was-threatened-by-hunnish-6483.html | 1,488,050,030,000,000,000 | text/html | crawl-data/CC-MAIN-2017-09/segments/1487501171807.25/warc/CC-MAIN-20170219104611-00422-ip-10-171-10-108.ec2.internal.warc.gz | 710,341,105 | 60,722 | In the mid-fifth century, Rome was threatened by hunnish : GMAT Sentence Correction (SC)
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# In the mid-fifth century, Rome was threatened by hunnish
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In the mid-fifth century, Rome was threatened by hunnish [#permalink]
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14 May 2004, 04:19
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In the mid-fifth century, Rome was threatened by hunnish troops who, led by Atilla the Hun, demostrated his millitary superiority over the weakened, recently conquered city.
A...
B.which, led by Atilla the hun, demostrated their military superiority
C.that atilla the hun led, who demonstrated his military superiority
D.that atilla the hun led in demonstration of their military superiority
E.that were led by Atilla the Hun, who demonstrated his military superiority
If you have any questions
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14 May 2004, 12:13
I will go with B. E is ambiguous as to what "who" refers to. It could refer to "attila the Hun" as well as it could refer to "hunnish troops"; who demonstrated his military strength?
Also, I'm not too sure about the usage of past progressive tense in this sentence. Past progressive "were led" means that the leading was a continuous action happening in the past. The usage of past progressive can be simplified by the intruduction of a dependent clause "led by Atilla the Hun" to make the answer less wordy
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14 May 2004, 12:22
I am not sure if troops can be refered to by pronoun "which"
Also why is there a no comma between the troops and which
It will be useful to have the official explanation.
Now I am begining to feel C is OK too.
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14 May 2004, 12:33
B is most certainly wrong.
Once you use which it becomes a non restrictive clause and can therefor be omitted. If you omit the non restrictive clause then you get the following construction.
In the mid-fifth century, Rome was threatened by hunnish troops demostrated their military superiority over the weakened, recently conquered city.
For me C and E make sense
C qualifies which specific troops threatened Rome by adding a restrictive clase after that
Well I am using which here in contradiction to what I said in my previous post.
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14 May 2004, 12:46
aonie wrote:
In the mid-fifth century, Rome was threatened by hunnish troops who, led by Atilla the Hun, demostrated his millitary superiority over the weakened, recently conquered city.
A...
B.which, led by Atilla the hun, demostrated their military superiority
C.that atilla the hun led, who demonstrated his military superiority
D.that atilla the hun led in demonstration of their military superiority
E.that were led by Atilla the Hun, who demonstrated his military superiority
I agree ... its either C or E
I will go with E. "Who" in C is ambiguous
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14 May 2004, 12:54
I see Anandnk. I just read the explanation about the usage of "which" to introduce non-restrictive clauses and yes, the comma should have been between "troops" and "which". E should be it as "that" introduces a restrictive clause which is essential for the sentence. These are tricky SC. In the case of E, "who" would refer to "hunnish troops" because "Attila the hun" is only part of the restrictive clause and not part of the independent clause.
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14 May 2004, 13:02
Hi Paul,
Let us wait for the OA. These are tricky SCs. Sometimes punctuation is not a problem but in case of non restrictive clause it is. I read explanation for "that" at http://www.bartleby.com ( Supposed to be a guide to standard American English )
In case of E there is no ambiguity because his is singular and can only refer to Atila (the Hun - is adjective ). I have seen SCs which test this concept of ambiguity. If E) is right then C) may very well be right.
Anand.
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14 May 2004, 17:07
Hi everyone....
I thought this question will cause some fun and indeed it did. I think this questions is too ambiguous. I'm quite upset with "this advanced book for advanced student prep book" when I saw the Q stem again.
The sentence can emphasise either on TROOPS or ATILLA Hun, so if what they meant to emphasise is troops we will need THEIR SUPERIORITY (which will lead us to B to modify troops) but if it is Atilla that they meant, we will need WHO-HIS SUPERIORITY (which will lead us to E because it has the correct modifier - atilla the hun, who...).
"Which", if used as a correct non-restrictive modifier is fine - as in B, and
"Who" to modifiy Atilla immediately is also correct- as in E. So B and E are correct. (and "the book" agrees with this - it all depends on who you want to showcase)
_____________________________________________________________
Then, how the hell I am supposed to know that they meant to showcase TROOPS' superiority instead of Atilla's....when the original underlined portion "totally" conveys the other way around and there are 2 possible correct answers with different meaning.... see below...
___________________________________________________________
In the mid-fifth century, Rome was threatened by hunnish troops who, led by Atilla the Hun, demostrated his millitary superiority over the weakened, recently conquered city.
A...
B.which, led by Atilla the hun, demostrated their military superiority
E.that were led by Atilla the Hun, who demonstrated his military superiority
----------------------------------------------------------------------------------
so painfully you know the answer is B...not because we're wrong but because that's not what THEY meant to say.
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14 May 2004, 17:27
also...
Which in this sentence is RESTRICTIVE, and led by atilla is non-retrictive. Remember which can be used as retrictive clause (esply without any comma preceeding it), this is very tricky. But my problem is, the other answer is also correct and this is like 2 big traps.
Anyway the sentence will go like this with B:
In the mid-fifth century, Rome was threatened by hunnish troops which, led by Atilla the hun, demostrated their military superiority over the weakened, recently conquered city.
and E : I must admit that this Sentence is too compact and complicated
In the mid-fifth century, Rome was threatened by hunnish troops that were led by Atilla the Hun, who demonstrated his military superiority over the weakened, recently conquered city.
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14 May 2004, 18:04
I then agree GMAT type questions are not this ambiguous. It is true though that although "which" is mainly used as a non-restrictive clause, it can also be used as a restrictive clause. It is the case here.
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09 Jul 2004, 13:23
Paul wrote:
I will go with B. E is ambiguous as to what "who" refers to. It could refer to "attila the Hun" as well as it could refer to "hunnish troops"; who demonstrated his military strength?
Also, I'm not too sure about the usage of past progressive tense in this sentence. Past progressive "were led" means that the leading was a continuous action happening in the past. The usage of past progressive can be simplified by the intruduction of a dependent clause "led by Atilla the Hun" to make the answer less wordy
By chosing 'B' don't you end up distorting the original sentence in that it is no longer Atilla's military superiority that's demonstrated, but that of Hunnish' troops?
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09 Jul 2004, 21:26
B and E seem good. It depends on who you want "military superiority" to refer to. After a bit more practice with SC, I prefer B for 2 reasons:
1- It is more concise
2- I believe when you are talking about military superiority, you are refering to that of an army, not of an individual.
Ex: The French army had a great military superiority over that of the Russian city.
Ex: Napoleon had a military superiority over the Russian city.
See the difference here? I would prefer the military superiority of an entire army over that of another rather than one single person's military superiority over a city's one.
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09 Jul 2004, 21:26
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Are positive integers p and q both greater than n? 1) p-q is
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Are positive integers p and q both greater than n?
1) p-q is greater than n
2) p<q
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Re: DS: Integers grater than n [#permalink]
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30 Jan 2008, 12:04
netcaesar wrote:
Are positive integers p and q both greater than n?
1) p-q is greater than n
2) p<q
1) insufficient -> try (p,q,n) = (10,1,8) and (10,8,1)
2) insufficient -> try (p,q,n) = (1,3,2) and (10,13,2)
1&2 sufficient, p < q -> p-q < 0 since p-q > n -> n is negative but since p and q are positive, both p and q are greater than n.
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Re: DS: Integers grater than n [#permalink]
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31 Jan 2008, 15:32
Together we can tell that p-q will be a negative # greater than n. So n must be a negative number. P and q are positive always.
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Re: DS: Integers grater than n [#permalink]
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01 Feb 2008, 11:43
jimmyjamesdonkey wrote:
Together we can tell that p-q will be a negative # greater than n. So n must be a negative number. P and q are positive always.
How did you arrive to your conclusion so fast?
OA is C
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Re: DS: Integers grater than n [#permalink]
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01 Feb 2008, 12:38
netcaesar wrote:
jimmyjamesdonkey wrote:
Together we can tell that p-q will be a negative # greater than n. So n must be a negative number. P and q are positive always.
How did you arrive to your conclusion so fast?
OA is C
2) p < q -> p-q < 0
1) p-q > n -> n < p-q < 0
since p, q are positive -> p,q > 0 > n
Re: DS: Integers grater than n [#permalink] 01 Feb 2008, 12:38
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In this post we will examine the hypothesis test. we will:
In this post we will examine the hypothesis test. we will:
1. Look at what a hypothesis test is.
2. Perform a hypothesis test on the R ‘CO2’ dataset.
3. Perform a better hypothesis test on the same dataset.
4. Analyze the results of our hypothesis test to understand what it means.
This post will be targeted at relative beginners (or those seeking review) and I will be using certain vocabulary terms interchangably here to make concepts easier to understand.
What is a hypothesis test?
A hypothesis test is tool we can use to answer a question. Here is our question: We want to know whether or not cooling plants affects (increases or decreases) their CO2 absorbtion. In a hypothesis test we formulate two possible outcomes (answers to our question) as ‘hypotheses.’ The hypothesis test looks like:
Hypothesis 1: Chilling plants does not affect their CO2 absorbtion.
Hypothesis 2: Chilling plants does affect their CO2 absorbtion.
We will call Hypothesis 1 the ‘Null Hypothesis’ because this is the outcome where nothing or ‘null’ happens. We will call Hypothesis 2 the ‘Alternate Hypothesis’ this is the outcome that differs from the null hypothesis.
The dataset we will be using is the R ‘CO2’ dataset. In this dataset the CO2 (carbon dioxide) absorbtion of chilled plants and the CO2 absorbtion of non-chilled plants was measured. I have cleaned the data by removing the index; I suggest you use my data file. We will take this data and use a hypothesis test to determine if the CO2 absorbtion of plants in a ‘chilled’ environment differs from plants in a ‘non-chilled’ environment.
``# import some useful packages %matplotlib inline import pandas as pd import numpy as np from scipy import stats from fitter import Fitter import matplotlib.pyplot as plt ``
``# silence warnings for beautification import warnings warnings.filterwarnings('ignore')``
``# create a dataframe df with our data df = pd.read_csv('co2_data.csv') df.head()``
We need to have a look quickly at the data to make sure it doesn’t have any weird characteristics. Lets check that the number of chilled and nonchilled plants is the same. Below we can see they are even; if they weren’t this would make it harder to do our hypothesis test.
``# count how many chilled/unchilled plant measurements we have nonchill_plants = df[df['Treatment'] == 'nonchilled']chill_plants = df[df['Treatment'] == 'chilled'] num_nonchilled = nonchill_plants.shape[0] num_chilled = chill_plants.shape[0] ``
``print('num chilled: {}, num normal: {}'.format(num_chilled, num_nonchilled))``
``num chilled: 42, num normal: 42``
In many instances it’s better to have more data than this (42 samples) to be able to make firmly reject or accept a hypothesis, usually a sample of at least 100 is considered desirable for being able to answer hypotheses. 42 is not too low though and we can probably get away with using this many samples.
Now lets check that the number of observations is even across co2 concentrations. If there are more expermiments at higher or lower concentrations of CO2 this could affect our ability to test. It looks like the number of experiments at each concentration is indeed even.
``# check to see if there is bias # in the concentrations of CO2 df.groupby('conc').size()``
``conc95 12175 12250 12350 12500 12675 121000 12dtype: int64``
Let’s graph the CO2 absorbtion values for non-chilled plants. We will call this distribution our ‘non-chilled’ distribution.
``plt.figure(figsize=(12.5,10))plt.hist(nonchill_plants['uptake'], color='c3', label='non-chilled', alpha=0.7)plt.xlabel('co2 absorbtion')plt.ylabel('count')plt.legend(loc='upper left')plt.title('non-chilled plant data')plt.show()``
So we have our data:
Our distribution of non-chilled plants has two (or three?) bumps, with most data points in the right bump.
And we have our hypotheses:
Null Hypothesis: Chilling plants does not affect their CO2 absorbtion.
Alternate Hypothesis: Chilling plants does affect their CO2 absorbtion.
Now we need to find out which hypothesis occurred by looking at the data.
One way to find out is to compare the distributions of the data with our eyes. If the chilled plant distribution differs significantly from the non-chilled distribution then the alternate hypothesis is probably correct. If the two distributions are the same then then null hypothesis is probably correct. Let’s graph both distributions and compare.
``plt.figure(figsize=(12.5,10))plt.hist(nonchill_plants['uptake'], label='non-chilled', color='c3', alpha=0.7)plt.hist(chill_plants['uptake'], label='chilled', color='c0', alpha=0.7)plt.xlabel('co2 absorbtion')plt.ylabel('count')plt.legend(loc='upper left')plt.title('non-chilled plant data overlaid with chilled plant data')plt.show()``
This chilled plant distrbiution in blue looks significantly rebalanced towards lower values of CO2 absorbtion. This implies that the chilling of plants affects their CO2 absorbtion; we can choose our alternate hypothesis as correct:
Null Hypothesis: Chilling plants does not affect their CO2 absorbtion.
Alternate Hypothesis: Chilling plants does affect their CO2 absorbtion.
Comparing distributions with your eyes is the simplest way to choose between two possible hypotheses.
Statistical tests
While the prior method seems to work well enough for the problem posed here it is not entirely robust. Sometimes two distributions are not easy to tell apart. How can we objectively tell when one distribution is different from another distribution?
You can use something called a statistical test: it tells you how similar two distributions are or more precisely it tells you the probability (p-value) that data you are testing came from a particular distribution. Below we are going to perform a statistical test to see if our chillled plant data comes from the same distribution as our non-chilled plant data.
In python you can perform a statistical test using scipy.stats for any well defined distribution but our distribution of non-chilled plant CO2 absorbtion is not well defined. So we’re going to find a well known distribution that is as similar as possible to our non-chilled distribution, we can use the fitter package which just does a lot of the nasty work of iterating through possible distributions for us to find the distribution that is most similar to our non-chilled distribution.
Let us find out which distribution our non-chilled plant data comes from.
``f = fitter(nonchill_plants['uptake'], bins=10)f.verbose = falsef.fit()best_shape_1 = f.get_best()dst1 = list(best_shape_1.keys())[0]print('non-chilled plant co2 absorbtion')print('dstribution: {}'.format(dst1))print('best distribution shape: {}'.format(best_shape_1[dst1]))print('best 5 distributions, lowest 5 errors:')f.df_errors.sort(columns=['sumsquare_error'], ascending=true).ix[:5]``
``non-chilled plant co2 absorbtiondstribution: genhalflogisticbest distribution shape: (0.83770454292175289, 10.599999993744611, 29.353144088738148)best 5 distributions, lowest 5 errors:``
It looks like the genhalflogistic distribution is the one that fits our non-chilled data most closely with an error value of 0.001242. We will use this distribution to test our hypotheses. Let’s visually inspect the genhalflogisitc distribution to confirm that this distribution is similiar to our non-chilled plant data.
``rvf = stats.genhalflogistic.pdf(x=[x for x in range(11,46)], c=best_shape_1[dst1][0], loc=best_shape_1[dst1][1], scale=best_shape_1[dst1][2])plt.figure(figsize=(12.5,10))plt.hist(nonchill_plants['uptake'], label='non-chilled', color='c3', alpha=0.7, normed=true, bins=10)plt.plot([x for x in range(11,46)], rvf, label='genhalflogistic', color='c3')plt.xlabel('co2 absorbtion')plt.ylabel('count')plt.legend(loc='upper left')plt.title('non-chilled plant distribution and genhalflogistic distribution')plt.show()``
It’s not perfect but it looks like the genhalflogisitic distribution does indeed match our non-chilled plant data to a high degree. Both slope upward to the right and both have a little dip around 15–25 along the `CO2 Absorbtion` axis.
We can say that if the chilled plant distribution is sufficiently different from the genhalflogistic distribution representing our non-chilled plant data that the chilled plants have had their CO2 absorbtion affected.
To measure how different the chilled plant data is from the genhalflogistic distribution we use something called a p-value. The p-value is just the probability that a certain sample of data came from a certain distribution. If we get a low enough p-value when we check to see if chilled data came from our genhalflogisitic distribution we can conclude that chilled plant data follows a different distribution and behaves differently from genhalflogisitc/non-chilled distribution.
To get p-value with our genhalflogisitc distribution and our chilled plant data we will feed both those into something called a Kolmogorov-Smirnov test, or ks-test. Let’s calculate some p values.
``_, nonchilled_pvalue = stats.kstest(nonchill_plants['uptake'], 'genhalflogistic', args=best_shape_1[dst1])_, chilled_pvalue = stats.kstest(chill_plants['uptake'], 'genhalflogistic', args=best_shape_1[dst1])``
``print('p-value (probability) of non-chilled data being from our genhalflogistic: ', nonchilled_pvalue)print('p-value (probability) of chilled data being from our genhalflogistic: ', chilled_pvalue)``
``p-value (probability) of non-chilled data being from our genhalflogistic: 0.639477797842``
``p-value (probability) of chilled data being from our genhalflogistic: 2.24809405358e-05``
The probability that the non-chilled data is from this genhalflogistic distribution is 63.94% (p-value 0.6394). This makes sense as this genhalflogisitic distribution was created using this data and is shaped around it.
The probability that the chilled plant data came from our genhalflogistic distribution is small: .00224% (p-value 0.0000224). Usually if a p-value is lower than 0.05 you can say the p-value is ‘small enough.’ The statistics lingo for when we get a ‘small enough’ p-value is that the result we have observed is ‘statistically significant.’ The low probability that our chilled data came from the same distribution as our non-chilled data suggests that when the plants are chilled their distribution of CO2 absorbtion changes significantly. We can verify this by using fitter again and seeing what distribution our chilled plant data follows.
``f = fitter(chill_plants['uptake'], bins=10)f.verbose = falsef.fit()best_shape_2 = f.get_best()dst2 = list(best_shape_2.keys())[0]print('chilled plant co2 absorbtion')print('dstribution: {}'.format(dst2))print('best distribution shape: {}'.format(best_shape_2[dst2]))print('best 5 distributions, lowest 5 errors:')f.df_errors.sort(columns=['sumsquare_error'], ascending=true).ix[:5]``
``chilled plant co2 absorbtiondstribution: dweibullbest distribution shape: (2.6804403695668575, 25.999783783267702, 11.415991122238617)best 5 distributions, lowest 5 errors:``
It looks like the best fit for the chilled plant data is in fact a double weibull distribution with error value of 0.000780. This distribution looks quite different from our genhalflogistic representing the non-chilled plant data. We will graph the double weibull and the chilled plant data below. You can compare it to the graph above to see the difference.
``rvf_dwe = stats.dweibull.pdf(x=[x for x in range(8,43)], c=best_shape_2[dst2][0], loc=best_shape_2[dst2][1], scale=best_shape_2[dst2][2]) plt.figure(figsize=(12.5,10)) plt.hist(chill_plants['uptake'], label='chilled', color='C0', alpha=0.7, normed=True, bins=10) plt.plot([x for x in range(8,43)], rvf_dwe, label='dweibull', color='C0') plt.xlabel('CO2 Absorbtion')plt.ylabel('Count') plt.legend(loc='upper left') plt.title('Chilled pland distribution and double weibull distribution') plt.show()``
Notes on p-value:
What threshold to pick for your p-value is largely arbitrary, usually people accept 0.05 or lower. Nowadays with increasing amounts of data it’s getting easier to get a p-value of 0.05 and some people prefer to set the threshold lower at 0.01 or lower. Here is a comic that was meant as a joke but actually offers some insight into interpreting p-values.
Conclusion
Given the low p-value we found testing chilled plant data against the genhalflogistic distrbution we can reject our null hypothesis and accept our alternate hypothesis:
Null Hypothesis: Chilling plants does not affect their CO2 absorbtion.
Alternate Hypothesis: Chilling plants does affect their CO2 absorbtion.
We also confirmed that the chilled plant data follows a very different distribution than the non-chilled data. Further confirming our hypothesis that chilling plants does affect their CO2 absorbtion.
In this writeup you should have learned:
1. what a hypothesis test is and how we can use it to answer questions.
2. an informal method for choosing between hypotheses by visual comparison.
3. how to perform a statistical test with two distributions to validate hypotheses.
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Originally published at cueblog.com. | 3,221 | 13,378 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.90625 | 3 | CC-MAIN-2021-04 | latest | en | 0.867431 |
https://philoid.com/question/34625-find-the-radius-of-a-circle-whose-perimeter-and-area-are-numerically-equal | 1,726,656,224,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651895.12/warc/CC-MAIN-20240918100941-20240918130941-00569.warc.gz | 426,858,360 | 7,927 | ##### Find the radius of a circle whose perimeter and area are numerically equal.
Given:
Perimeter of circle = Area of circle …… (i)
Perimeter of circle = 2πr and Area of circle = πr2
Using (i), we have
2πr = πr2
2 =
2 = r or r = 2
Hence, the radius of the circle is 2 cm.
10 | 90 | 284 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.9375 | 4 | CC-MAIN-2024-38 | latest | en | 0.818091 |
https://www.folkstalk.com/2022/09/go-logica-operators-with-code-examples.html | 1,701,671,454,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100525.55/warc/CC-MAIN-20231204052342-20231204082342-00215.warc.gz | 866,873,249 | 14,339 | # Go Logica Operators With Code Examples
Go Logica Operators With Code Examples
In this article, we will see how to solve Go Logica Operators with examples.
```Operator Description Example
&& Called Logical AND operator. If both the operands are non-zero, then condition becomes true. (A && B) is false.
|| Called Logical OR Operator. If any of the two operands is non-zero, then condition becomes true. (A || B) is true.
! Called Logical NOT Operator. Use to reverses the logical state of its operand. If a condition is true then Logical NOT operator will make false. !(A && B) is true.```
Using a variety of different examples, we have learned how to solve the Go Logica Operators.
## What are Go operators?
In computer programming, an operator is a symbol that performs operations on a value or a variable. For example, + is an operator that is used to add two numbers. Go programming provides a wide range of operators that are categorized into the following major categories: Arithmetic operators.
## What does || mean in Go?
|| Called Logical OR Operator. If any of the two operands is true, then the condition becomes true. (A || B) is true. !
## What are logical operators?
A logical operator is a symbol or word used to connect two or more expressions such that the value of the compound expression produced depends only on that of the original expressions and on the meaning of the operator. Common logical operators include AND, OR, and NOT.
## What does <- mean in Golang?
"=" is assignment,just like other language. <- is a operator only work with channel,it means put or get a message from a channel.25-Oct-2015
## Does Go have the operator?
Go divides the operators into the following groups: Arithmetic operators. Assignment operators. Comparison operators.
## What are expressions in Golang?
The Golang spec uses those terms: Expressions: specifies the computation of a value by applying operators and functions to operands. Statements: control execution. Declarations (and scope): binds a non-blank identifier to a constant, type, variable, function, label, or package.
## What Is syntax like in Go?
Go is a statically typed, compiled programming language designed at Google by Robert Griesemer, Rob Pike, and Ken Thompson. It is syntactically similar to C, but with memory safety, garbage collection, structural typing, and CSP-style concurrency.
## How do you use boolean in Go?
The data type in Go for Boolean is bool , all lowercase. The values true and false will always be with a lowercase t and f respectively, as they are special values in Go.13-May-2019
## What is Go used for?
Go (also called Golang or Go language) is an open source programming language used for general purpose. Go was developed by Google engineers to create dependable and efficient software. Most similarly modeled after C, Go is statically typed and explicit.
## What are the 5 logical operators?
There are five logical operator symbols: tilde, dot, wedge, horseshoe, and triple bar. | 643 | 3,007 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.3125 | 3 | CC-MAIN-2023-50 | latest | en | 0.913373 |
https://www.jiskha.com/display.cgi?id=1354722251 | 1,516,174,016,000,000,000 | text/html | crawl-data/CC-MAIN-2018-05/segments/1516084886830.8/warc/CC-MAIN-20180117063030-20180117083030-00591.warc.gz | 926,612,642 | 3,786 | # Physics
posted by .
A 64.4-kg skateboarder starts out with a speed of 2.11 m/s. He does 81.6 J of work on himself by pushing with his feet against the ground. In addition, friction does -289 J of work on him. In both cases, the forces doing the work are non-conservative. The final speed of the skateboarder is 8.56 m/s. (a) Calculate the change (PEf - PE0) in the gravitational potential energy. (b) How much has the vertical height of the skater changed? Give the absolute value.
• Physics -
ΔKE=1/2mvf^2- 1/2mv0^2
a)Plug numbers in
1. ΔKE=1/2(64.4)(8.56)^2- 1/2(64.4)(2.11)^2= -2216.05 (because displacement, I think)
2. -2216.05J+(-289J)-80J
b.)
1. W=ΔKE, W=ΔPE(PE=mgh), ΔPE=ΔKE
(64.4kg)(9.8m/s^2)h=2425.05J
631.12h=2425.05J
Hope this Help
MONEY CASH FLOW
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http://oeis.org/A260724 | 1,571,027,176,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570986649035.4/warc/CC-MAIN-20191014025508-20191014052508-00086.warc.gz | 208,613,642 | 3,859 | This site is supported by donations to The OEIS Foundation.
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A260724 a(1) = 2; for n > 1 if n mod 3=0 a(n) = gpf(1 + Product_{m modulo 3=2,m
2, 3, 2, 7, 5, 2, 31, 3, 109, 13, 97, 83, 8731, 461, 3078769, 574933, 206179, 510583, 5153873111, 1900169, 372828179, 4032506435771041, 3445362972116384083, 49898596950715368948711529590337583, 54176564592185030069557382121821 (list; graph; refs; listen; history; text; internal format)
OFFSET 1,1 LINKS Anders Hellström, Table of n, a(n) for n = 1..27 PROG (PARI) gpf(n)=my(v=factor(n)[, 1]); v[#v]; first(m)={my(v=vector(m), one=2, two=2, three=1); v[1]=2; for(i=2, m, if(i%3===0, v[i]=gpf(three+1); two*=v[i], if(i%3===1, v[i]=gpf(one+1); three*=v[i], v[i]=gpf(two+1); one*=v[i]))); v; } CROSSREFS Cf. A258581 Sequence in context: A118007 A195401 A158747 * A256581 A122697 A129022 Adjacent sequences: A260721 A260722 A260723 * A260725 A260726 A260727 KEYWORD nonn AUTHOR Anders Hellström, Jul 30 2015 STATUS approved
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Last modified October 14 00:25 EDT 2019. Contains 327991 sequences. (Running on oeis4.) | 514 | 1,368 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.65625 | 3 | CC-MAIN-2019-43 | latest | en | 0.538171 |
http://www.algebra.com/algebra/homework/proportions/Proportions.faq.question.419333.html | 1,369,338,083,000,000,000 | text/html | crawl-data/CC-MAIN-2013-20/segments/1368703728865/warc/CC-MAIN-20130516112848-00045-ip-10-60-113-184.ec2.internal.warc.gz | 314,794,497 | 4,576 | # SOLUTION: The ratio of the height of a model of a house to the actual house is 1:63. If the width of the model is 16 inches, find the width of the actual house in feet. Thank you for the
Algebra -> Algebra -> Proportions -> SOLUTION: The ratio of the height of a model of a house to the actual house is 1:63. If the width of the model is 16 inches, find the width of the actual house in feet. Thank you for the Log On
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Algebra: Proportions Solvers Lessons Answers archive Quiz In Depth
Question 419333: The ratio of the height of a model of a house to the actual house is 1:63. If the width of the model is 16 inches, find the width of the actual house in feet. Thank you for the help.Answer by rfer(12660) (Show Source): You can put this solution on YOUR website!1/16=63/x x=1008 1008/12=84 ft | 263 | 993 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3 | 3 | CC-MAIN-2013-20 | latest | en | 0.779973 |
http://projects.lsv.ens-cachan.fr/topology/?page_id=631 | 1,718,983,401,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198862125.45/warc/CC-MAIN-20240621125006-20240621155006-00729.warc.gz | 25,105,967 | 12,565 | # Powerdomains and hyper spaces II: monads
Let us deepen our understanding of the Hoare powerspace construction. We shall see that it defines a so-called monad. I’ll define everything needed to understand the latter as we go along.
I have briefly mentioned monads in Filters IV. There is an equivalent way of presenting monads (on a category C), as the following data (I don’t expect you to understand those equations for now, we shall illustrate them shortly):
• for each object X of C, an object TX in C
• for each object X of C, a morphism ηX: XTX called the unit of the monad
• an extension operation †, transforming every morphism f: X → TY into a morphism fTX → TY, so that the following equations are satisfied (we explain them shortly):
1. ηX = idTX
2. for every morphism f: XTY, f o ηX = f
3. for all morphisms fX → TY and gY → TZ, (g o f) = g o f.
We then say that (T, η, †) is a monad, or, by abuse of language, that T itself is a monad. Let’s illustrate that on a concrete example.
Consider the category Set of sets for the category C. The morphisms from X to Y are just the functions from X to Y. Let TX be the powerset P(X) of X, that is, the set of all subsets of X. Define ηX as mapping x in X to {x}, and for every set E, let f(E) be defined as the union of all the sets f(x), x in E. We check the monad equations as follows:
1. ηX maps E to the union of the sets {x}, x in E, namely E, so ηX = idTX;
2. f o ηX maps every element x to f({x}) = f(x), hence is equal to f;
3. for every subset E, (g o f) (E) is the union over all x in E of g((x)), namely the union over all x in E of the union over all y in f(x) of g(y). On the other hand, (g o f) (E) is the union over all y in f (E) of g(y), and that is just the same thing.
The Kleisli category of a monad
There is a very rich theory of monads. I would like to explain why the equations 1—3 above are what they are. The reason is the construction of the so-called Kleisli category CT of the monad T on C.
This is built as follows. The objects of the CT are the same as those of C; the only thing that changes are the morphisms: the morphisms from X to Y in CT are exactly those from X to TY in the old category C. The map ηX: XTX serves as the identity morphism at X in CT, and composition of fX → TY with gY → TZ is given by g o f: X → TZ. The equations 1—3 are exactly what you need to show that this indeed forms a category: composition is associative, and the identity is neutral for composition both on the left and on the right. In fact, you can check that the Kleisli “category” is really a category if and only if the equations 1—3 are satisfied.
In the case of the powerset monad on Set, the Kleisli category can be thought as a category of sets and binary relations as morphisms. Any binary relation R on X x Y defines a Kleisli morphism fX → P(Y), which maps every x in X to the set {y in Y | (x, y) in R}. Conversely, every Kleisli morphism fX → P(Y) defines a binary relation R by (xy) in if and only if y is in f(x). You can check that the Kleisli identity (the unit of the monad) is the equality relation {(x, x) | x in X}, and that Kleisli composition is the ordinary composition of relations: S o R = {(x, z) | (xy) in R and (y, z) in S for some y}.
One can also think of the set X as a set of states for some computer system. A Kleisli morphism f from X to TY tells you, given that you are in some state x in X, what the set f (x) of successor states is. In other words, f is a so-called transition relation. The fact that f(x) may contain more than one element means that you can choose y non-deterministically.
Oh, before we examine the more complicated Hoare monad, recall that I had given another definition of monads in Filters IV. We obtain that presentation from a monad as defined above, as follows:
• a functor T from C to C; we already have its object parts, and on morphisms f: X → Y, we define TfTX → TY as (ηY o f) (check, using 1—3, that this is indeed a functor!)
• a natural transformation η from the identity functor to T; this is given as ηX on each object X (check, using 1—3, that this is indeed natural in X);
• a natural transformation μ from T2 to T, defined on each object X by μX = idTX: T2X → TX (check, again, that this is natural);
• all of these are also required to satisfied the three laws: (left unit) μX o ηTX = idTX, (right unit) μX o TηX = idTX, and (associativity) μX o TμX = μX o μTX. Again, I’ll let you check that those follow from 1—3.
Conversely, any monad as given by a functor, a unit natural transformation, and a multiplication natural transformation satisfying the left unit, right unit, and associativity laws yields back a monad (T, η, †) by defining f as μX o Tf. The fancy names “left unit”, “right unit”, and “associativity” really come from the fact that monads on C are exactly the monoid objects in the monoidal category of endofunctors of C, with composition as tensor product. (Ignore that if this is your first time with monads, and think of it as some Zen koan: it is impenetrable, but it holds the truth.)
We can now proceed with the more complicated case of the H powerspace construction. That this defines a monad is again due to Andrea Schalk [1]. We take C=Top, and let TX=H(X) with the upper (=lower Vietoris) topology.
We already have a continuous map ηXX → H(X), which sends x in X to the closure ↓x of x. This will be our unit.
Given any continuous map f: X → H(Y), we define f: H(X) → H(Y) by letting f (F) be the closure of the union of all the closed subsets f(x), x in F. We need to take the closure because a union of closed subsets may fail to be closed. What we have done here is taking a supremum in H(Y), ordered by inclusion: suprema are closures of unions. In summary, f (F) = sup {f (x) | x in F}.
Let us check that this indeed defines a monad. This is somehow more complicated than for the powerset monad.
• To show that f is continuous, we only have to show that the inverse image of ◊V, for V open in Y, is open in H(X). Since the closure of a set intersects an open V if and only if the set itself intersects V (Corollary 4.1.28), that inverse image is ◊f-1 (◊V), and we are done.
We now turn to the monad equations.
1. First, ηX = idH(X). Indeed, the left hand side maps each closed set F to the closure of the union of all the sets ↓x, x in F. That union is just F, and since F is closed, its closure is itself.
2. Second, for every continuous map f: X → H(Y), f o ηX = f. The left hand side maps every x in X to the closure of the union of all the sets f(x’), where x’ ranges over ↓x. Since f is continuous hence monotonic, every such set f(x’) is included in f(x), so (f o ηX) (x) is included in f(x); the converse inclusion is because x itself is in ↓x.
3. Third, for all continuous maps f: X → H(Y) and g: Y → H(Z), we have two maps from H(X) to H(Z), namely (g o f) and g o f. We claim that they are equal: (g o f) = g o f… but if you try to prove this directly, you’ll suffer as hell. Instead, recall that for continuous h for which h makes sense, for every open subset W, (h)-1 (◊W) = ◊f-1 (◊W): this is how we proved that h was continuous. Using that, we check that, for every open subset W of Z, the inverse image of ◊W by (g o f) and by g o f is the same, namely ◊f-1 (◊g-1 (◊W)). This is enough to conclude that (g o f) = g o f are identical, by Stone duality, and because H(X) is sober. We can also see that directly: for every closed subset F of X, for every open subset W of Z, F is in the inverse image of ◊W by (g o f) if and only if it is in in the inverse image of ◊W by g o f, that is, for every open subset W of Z, (g o f) (F) intersects W if and only if (g o f) (F) intersects W; hence they are equal closed sets.
Just like the powerset monad on Set, the Hoare monad H on Top can be thought as some kind of powerset, with extra topological information. As for the powerset monad, the Kleisli morphisms encode some form of non-deterministic choice, called angelic non-determinism. Let me postpone the explanation of that denomination, as I would prefer to explain a few things about theories of monads, and monad algebras… next time. We shall see that, in a precise sense, the theory of the H monad is given by the following axioms:
• (unit) a ∨ 0 = a
• (associativity) (ab) ∨ c = a ∨ (bc)
• (commutativity) ab = ba
• (idempotence) aa = a
• (inflationary) aab
Note that, if you interpret ∨ as binary union of closed sets, 0 as the empty set, and ≤ as inclusion, then those (in)equalities are trivially satisfied. A topological space with a continuous map ∨ satisfying the above (in)equalities is called an inflationary semi-lattice. What we shall see is that, for each space XH(X) is the free sober inflationary semi-lattice above X, as proved by Schalk [1]… and I’ll try to explain what this means.
Jean Goubault-Larrecq (May 17th, 2015)
[1] Andrea Schalk. Algebras for Generalized Power Constructions. PhD Thesis, TU Darmstadt, 1993. | 2,645 | 8,982 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.46875 | 3 | CC-MAIN-2024-26 | latest | en | 0.850983 |
http://www.thefullwiki.org/Bicycle_and_motorcycle_dynamics | 1,575,839,277,000,000,000 | text/html | crawl-data/CC-MAIN-2019-51/segments/1575540514893.41/warc/CC-MAIN-20191208202454-20191208230454-00169.warc.gz | 243,296,994 | 56,681 | # Bicycle and motorcycle dynamics: Wikis
Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.
# Encyclopedia
A computer-generated, simplified model of bike and rider demonstrating an uncontrolled right turn.
Animation of a computer-generated, simplified model of bike and passive rider demonstrating uncontrolled weave.
Bicycles leaning in a turn.
Bicycle and motorcycle dynamics is the science of the motion of bicycles and motorcycles and their components, due to the forces acting on them. Dynamics is a branch of classical mechanics, which in turn is a branch of physics. Bike motions of interest include balancing, steering, braking, accelerating, suspension activation, and vibration. The study of these motions began in the late 1800s and continues today.[1][2]
Bicycles and motorcycles are both single-track vehicles and so their motions have many fundamental attributes in common and are fundamentally different from and more difficult to study than other wheeled vehicles such as dicycles, tricycles, and quadricycles.[3] As with unicycles, bikes lack lateral stability when stationary, and under most circumstances can only remain upright when moving forward. Experimentation and mathematical analysis have shown that a bike stays upright when it is steered to keep its center of mass over its wheels. This steering is usually supplied by a rider, or in certain circumstances, by the bike itself. Several factors, including geometry, mass distribution, and gyroscopic effect all contribute to varying degrees to this self-stability, but long-standing hypotheses and claims that gyroscopic effect is the main stabilizing force have been discredited.[4][5]
While remaining upright may be the primary goal of beginning riders, a bike must lean in order to maintain balance in a turn: the higher the speed or smaller the turn radius, the more lean is required. This balances the roll torque about the wheel contact patches generated by centrifugal force due to the turn with that of the gravitational force. This lean is usually produced by a momentary steering in the opposite direction, a maneuver often referred to as countersteering. Unlike other wheeled vehicles, the primary control input on bikes is steering torque, not position.[6]
Although longitudinally stable when stationary, bikes often have a high enough center of mass and a short enough wheelbase to lift a wheel off the ground under sufficient acceleration or deceleration. When braking, depending on the location of the combined center of mass of the bike and rider with respect to the point where the front wheel contacts the ground, bikes can either skid the front wheel or flip the bike and rider over the front wheel. A similar situation is possible while accelerating, but with respect to the rear wheel.[7]
## History
The history of the study of bike dynamics is nearly as old as the bicycle itself. It includes contributions from famous scientists such as Rankine, Appell, and Whipple.[1] In the early 1800s Karl von Drais, credited with inventing the two-wheeled vehicle variously called the laufmaschine, velocipede, draisine, and dandy horse, showed that a rider could balance his device by steering the front wheel.[1] By the end of the 1800s, Emmanuel Carvallo and Francis Whipple showed with rigid-body dynamics that some safety bicycles could actually balance themselves if moving at the right speed.[1] It is not clear to whom should go the credit for tilting the steering axis from the vertical which helps make this possible.[8]
In 1970, David Jones published an article in Physics Today showing that gyroscopic effects are not necessary to balance a bicycle.[5] Since 1971, when he named the weave and capsize modes, Robin Sharp has written about the mathematical behavior of motorcycles and bicycles, and his work has continued to the present day with David Limebeer. Both men were at Imperial College, London.[1][9][10] In 2007, Meijaard, Papadopoulos, Ruina, and Schwab published the canonical linearized equations of motion, in the Proceedings of the Royal Society A, along with verification by two different methods.[1]
## Forces
If the bike and rider are considered to be a single system, the forces that act on that system and its components can be roughly divided into two groups: internal and external. The external forces are due to gravity, inertia, contact with the ground, and contact with the atmosphere. The internal forces are caused by the rider and by interaction between components.
External forces on a bike and rider leaning in a turn: gravity in green, drag in blue, vertical ground reaction in red, net propulsive and rolling resistance in yellow, friction in response to turn in orange, and net torques on front wheel in magenta.
### External forces
As with all masses, gravity pulls the rider and all the bike components toward the earth. At each tire contact patch there are ground reaction forces with both horizontal and vertical components. The vertical components mostly counteract the force of gravity, but also vary with braking and accelerating. For details, see the section on longitudinal stability below. The horizontal components, due to friction between the wheels and the ground, including rolling resistance, are in response to propulsive forces, braking forces, and turning forces. Aerodynamic forces due to the atmosphere are mostly in the form of drag, but can also be from crosswinds. At normal bicycling speeds on level ground, aerodynamic drag is the largest force resisting forward motion.[11] At faster speed, aerodynamic drag becomes overwhelmingly the largest force resisting forward motion.
Turning forces are generated during maneuvers for balancing in addition to just changing direction of travel. These may be interpreted as centrifugal forces in the accelerating reference frame of the bike and rider; or simply as inertia in a stationary, inertial reference frame and not forces at all. Gyroscopic forces acting on rotating parts such as wheels, engine, transmission, etc., are also due to the inertia of those rotating parts. They are discussed further in the section on gyroscopic effects below.
### Internal forces
Internal forces are mostly caused by the rider or by friction. The rider can apply torques between the steering mechanism (front fork, handlebars, front wheel, etc.) and rear frame, and between the rider and the rear frame. Friction exists between any parts that move against each other: in the drive train, between the steering mechanism and the rear frame, etc. Many bikes have front and rear suspensions, and some motorcycles have a steering damper to dissipate undesirable kinetic energy.[10] On bikes with rear suspensions, feedback between the drive train and the suspension is an issue designers attempt to handle with various linkage configurations and dampers.[12]
## Motions
Motions of a bike can be roughly grouped into those out of the central plane of symmetry: lateral; and those in the central plane of symmetry: longitudinal or vertical. Lateral motions include balancing, leaning, steering, and turning. Motions in the central plane of symmetry include rolling forward, of course, but also stoppies, wheelies, brake diving, and most suspension activation. Motions in these two groups are linearly decoupled, that is they do not interact with each other to the first order.[1] An uncontrolled bike is laterally unstable when stationary and can be laterally self-stable when moving under the right conditions or when controlled by a rider. Conversely, a bike is longitudinally stable when stationary and can be longitudinally unstable when undergoing sufficient acceleration or deceleration.
## Lateral dynamics
Of the two, lateral dynamics has proven to be the more complicated, requiring three-dimensional, multibody dynamic analysis with at least two generalized coordinates to analyze. At a minimum, two coupled, second-order differential equations are required to capture the principal motions.[1] Exact solutions are not possible, and numerical methods must be used instead.[1] Competing theories of how bikes balance can still be found in print and online. On the other hand, as shown in later sections, much longitudinal dynamic analysis can be accomplished simply with planar kinetics and just one coordinate.
### Balance
Balancing a bicycle by keeping the wheels under the center of mass
A bike remains upright when it is steered so that the ground reaction forces exactly balance all the other internal and external forces it experiences, such as gravitational if leaning, inertial or centrifugal if in a turn, gyroscopic if being steered, and aerodynamic if in a crosswind.[11] Steering may be supplied by a rider or, under certain circumstances, by the bike itself. This self-stability is generated by a combination of several effects that depend on the geometry, mass distribution, and forward speed of the bike. Tires, suspension, steering damping, and frame flex can also influence it, especially in motorcycles.
Even when staying relatively motionless, a rider can balance a bike by the same principle. While performing a track stand, the rider can keep the line between the two contact patches under the combined center of mass by steering the front wheel to one side or the other and then moving forward and backward slightly to move the front contact patch from side to side as necessary. Forward motion can be generated simply by pedaling. Backwards motion can be generated the same way on a fixed-gear bicycle. Otherwise, the rider can take advantage of an opportune slope of the pavement or lurch the upper body backwards while the brakes are momentarily engaged.[13]
If the steering of a bike is locked, it becomes virtually impossible to balance while riding. On the other hand, if the gyroscopic effect of rotating bike wheels is cancelled by adding counter-rotating wheels, it is still easy to balance while riding.[4][5]
#### Forward speed
The rider applies torque to the handlebars in order to turn the front wheel and so to control lean and maintain balance. At high speeds, small steering angles quickly move the ground contact points laterally; at low speeds, larger steering angles are required to achieve the same results in the same amount of time. Because of this, it is usually easier to maintain balance at high speeds.[14]
#### Center of mass location
The farther forward (closer to front wheel) the center of mass of the combined bike and rider, the less the front wheel has to move laterally in order to maintain balance. Conversely, the further back (closer to the rear wheel) the center of mass is located, the more front wheel lateral movement or bike forward motion will be required to regain balance. This can be noticeable on long-wheelbase recumbents and choppers. It can also be an issue for touring bikes with a heavy load of gear over or even behind the rear wheel.[15]
A bike is also an example of an inverted pendulum. Just as a broomstick is easier to balance than a pencil, a tall bike (with a high center of mass) can be easier to balance when ridden than a low one because its lean rate will be slower.[16] However, a rider can have the opposite impression of a bike when it is stationary. A top-heavy bike can require more effort to keep upright, when stopped in traffic for example, than a bike which is just as tall but with a lower center of mass. This is an example of a vertical second-class lever. A small force at the end of the lever, the seat or handlebars at the top of the bike, more easily moves a large mass if the mass is closer to the fulcrum, where the tires touch the ground. This is why touring cyclists are advised to carry loads low on a bike, and panniers hang down on either side of front and rear racks.[17]
#### Trail
A factor that influences how easy or difficult a bike will be to ride is trail, the distance that the front wheel ground contact point trails behind the steering axis ground contact point. The steering axis is the axis about which the entire steering mechanism (fork, handlebars, front wheel, etc.) pivots. In traditional bike designs, with a steering axis tilted back from the vertical, trail causes the front wheel to steer into the direction of a lean, independent of forward speed.[11] This can be seen by pushing a stationary bike to one side. The front wheel will usually also steer to that side. In a lean, gravity provides this force.
Trail is a function of head angle, fork offset or rake, and wheel size. Their relationship can be described by this formula:[18]
$\text{Trail} = \frac{(R_w \cos(A_h) - O_f)}{\sin(A_h)}$
where Rw is wheel radius, Ah is the head angle measured clock-wise from the horizontal and Of is the fork offset or rake. Trail can be increased by increasing the wheel size, decreasing or slackening the head angle, or decreasing the fork rake.
The more trail a bike has, the more stable it feels. Bikes with negative trail (where the contact patch is actually in front of where the steering axis intersects the ground), while still ridable, feel very unstable. Bikes with too much trail feel difficult to steer. Normally, road racing bicycles have more trail than mountain bikes or touring bikes. In the case of mountain bikes, less trail allows more accurate path selection off-road, and also allows the rider to recover from obstacles on the trail which might knock the front wheel off course. Touring bikes are built with small trail to allow the rider to control a bike weighed down with baggage. As a consequence, an unloaded touring bike can feel unstable. In bicycles, fork rake, often a curve in the fork blades forward of the steering axis, is used to diminish trail.[19] In motorcycles, rake refers to the head angle instead, and offset created by the triple tree is used to diminish trail.[20]
A small survey by Whitt and Wilson[11] found:
• touring bicycles with head angles between 72° and 73° and trail between 43 mm and 60 mm
• racing bicycles with head angles between 73° and 74° and trail between 28 mm and 45 mm
• track bicycles with head angles of 75° and trail between 23.5 mm and 37 mm.
However, these ranges are not hard and fast. For example, LeMond Racing Cycles offers [21] both with forks that have 45 mm of offset or rake and the same size wheels:
• a 2006 Tete de Course, designed for road racing, with a head angle that varies from 71¼° to 74°, depending on frame size, and thus trail that varies from 69 mm to 51.5 mm.
• a 2007 Filmore, designed for the track, with a head angle that varies from 72½° to 74°, depending on frame size, and thus trail that varies from 61 mm to 51.5 mm.
The amount of trail a particular bike has may vary with time for several reasons. On bikes with front suspension, especially telescopic forks, compressing the front suspension, due to heavy braking for example, can steepen the steering axis angle and reduce trail. Trail also varies with lean angle, and steering angle, usually decreasing from a maximum when the bike is straight upright and steered straight ahead.[22] Finally, even the profile of the front tire can influence how trail varies as the bike is leaned and steered.
A measurement similar to trail, called either mechanical trail, normal trail, or true trail[23], is the perpendicular distance from the steering axis to the centroid of the front wheel contact patch.
#### Steering mechanism mass distribution
Another factor that can also contribute to the self-stability of traditional bike designs is the distribution of mass in the steering mechanism, which includes the front wheel, the fork, and the handlebar. If the center of mass for the steering mechanism is in front of the steering axis, then the pull of gravity will also cause the front wheel to steer in the direction of a lean. This can be seen by leaning a stationary bike to one side. The front wheel will usually also steer to that side independent of any interaction with the ground.[24] Additional parameters, such as the fore-to-aft position of the center of mass and the elevation of the center of mass also contribute to the dynamic behavior of a bike.[11][24]
#### Gyroscopic effects
Gyroscopic effect on front wheel of a bike. Appling a torque (in green) about the lean axis results in a reaction torque (in blue) about the steer axis.
The role of the gyroscopic effect in most bike designs is to help steer the front wheel into the direction of a lean. This phenomenon is called precession and the rate at which an object precesses is inversely proportional to its rate of spin. The slower a front wheel spins, the faster it will precess when the bike leans, and vice-versa.[25] The rear wheel is prevented from precessing as the front wheel does by friction of the tires on the ground, and so continues to lean as though it were not spinning at all. Hence gyroscopic forces do not provide any resistance to tipping.[26]
At low forward speeds, the precession of the front wheel is too quick, contributing to an uncontrolled bike’s tendency to oversteer, start to lean the other way and eventually oscillate and fall over. At high forward speeds, the precession is usually too slow, contributing to an uncontrolled bike’s tendency to understeer and eventually fall over without ever having reached the upright position.[8] This instability is very slow, on the order of seconds, and is easy for most riders to counteract. Thus a fast bike may feel stable even though it is actually not self-stable and would fall over if it were uncontrolled. A bicycle wheel with an internal flywheel for enhanced gyroscopic effect is under development as a commercial product, the Gyrobike, for making it easier to learn to ride bicycles.
Another contribution of gyroscopic effects is a roll moment generated by the front wheel during countersteering. For example, steering left causes a moment to the right. The moment is small compared to the moment generated by the out-tracking front wheel, but begins as soon as the rider applies torque to the handlebars and so can be helpful in motorcycle racing.[7] For more detail, see the countersteering article.
#### Self-stability
Between the two unstable regimes mentioned in the previous section, and influenced by all the factors described above that contribute to balance (trail, mass distribution, gyroscopic effects, etc.), there may be a range of forward speeds for a given bike design at which these effects steer an uncontrolled bike upright.[1]
However, even without self-stability a bike may be ridden by steering it to keep it over its wheels.[5] Note that the effects mentioned above that would combine to produce self-stability may be overwhelmed by additional factors such as headset friction and stiff control cables.[11] This video shows a riderless bicycle exhibiting self-stability.
### Turning
Motorcycles leaning in a turn.
The forces acting on a leaning bike in the rotating reference frame of a turn where N is the normal force, Ff is friction, m is mass, r is turn radius, v is forward speed, and g is the acceleration of gravity.
Graph of bike lean angle vs forward speed, assuming unlimited friction between tires and ground.
Cyclist riding with no hands on the handlebars.
In order to turn a bike, that is, change its direction of forward travel, the front wheel is turned approximately in the desired direction, as with any front-wheel steered vehicle. Friction between the wheels and the ground then generates the centripetal acceleration necessary to alter the course from straight ahead as a combination of cornering force and camber thrust. The radius of the turn of an upright (not leaning) bike can be roughly approximated, for small steering angles, by:
$r = \frac{w}{\delta \cos \left (\phi \right )}$
where r is the approximate radius, w is the wheelbase, δ is the steer angle, and φ is the caster angle of the steering axis.[7]
#### Leaning
However, unlike other wheeled vehicles, bikes must also lean during a turn to balance the relevant forces: gravitational, inertial, frictional, and ground support. The angle of lean, θ, can easily be calculated using the laws of circular motion:
$\theta = \arctan \left (\frac{v^2}{gr}\right )$
where v is the forward speed, r is the radius of the turn and g is the acceleration of gravity.[25] This is in the idealized case. A slight increase in the lean angle may be required on motorcycles to compensate for the width of modern tires at the same forward speed and turn radius.[22]
For example, a bike in a 10 m (33 ft) radius steady-state turn at 10 m/s (22 mph) must be at an angle of 45°. A rider can lean with respect to the bike in order to keep either the torso or the bike more or less upright if desired. The angle that matters is the one between the horizontal plane and the plane defined by the tire contacts and the location of the center of mass of bike and rider.
This lean of the bike decreases the actual radius of the turn proportionally to the cosine of the lean angle. The resulting radius can be roughly approximated (within 2% of exact value) by:
$r = \frac{w\cos \left (\theta \right )}{\delta \cos \left (\phi \right )}$
where r is the approximate radius, w is the wheelbase, θ is the lean angle, δ is the steer angle, and φ is the caster angle of the steering axis.[7] As a bike leans, the tires' contact patches move farther to the side causing wear. The portions at either edge of a motorcycle tire that remain unworn by leaning into turns is sometimes referred to as chicken strips.
#### Countersteering
In order to initiate a turn and the necessary lean in the direction of that turn, a bike must momentarily steer in the opposite direction. This is often referred to as countersteering. With the front wheel now at an angle to the direction of motion, a lateral force is developed at the contact patch of the tire. This force creates a torque around the longitudinal (roll) axis of the bike. This torque causes the bike to roll in the opposite direction of the turn. Where there is no external influence, such as an opportune side wind to create the force necessary to lean the bike, countersteering happens in every turn.[25]
As the lean approaches the desired angle, the front wheel must be steered in the direction of the turn, depending on the forward speed, the turn radius, and the need to maintain the lean angle. Once in a turn, the radius can only be changed with an appropriate change in lean angle. This can only be accomplished by additional countersteering out of the turn to increase lean and decrease radius, then into the turn to decrease lean and increase radius. To exit the turn, the bike must again countersteer, momentarily steering more into the turn in order to decrease the radius, thus increasing inertial forces, and thereby decreasing the angle of lean.[27]
Once a turn is established, the torque that must be applied to the steering mechanism in order to maintain a constant radius at a constant forward speed depends on the forward speed and the geometry and mass distribution of the bike.[8] At speeds below the capsize speed, described below in the section on Eigenvalues and also called the inversion speed, the self-stability of the bike will cause it to tend to steer into the turn, righting itself and exiting the turn, unless a torque is applied in the opposite direction of the turn. At speeds above the capsize speed, the capsize instability will cause it to tend to steer out of the turn, increasing the lean, unless a torque is applied in the direction of the turn. At the capsize speed no input steering torque is necessary to maintain the steady-state turn.
#### Steering angle
Several effects influence the steering angle, the angle at which the front assembly is rotated about the steering axis, necessary to maintain a steady-state turn. Some of these are unique to single-track vehicles, while others are also experienced by automobiles. Some of these may be mentioned elsewhere in this article, and they are repeated here, though not necessarily in order of importance, so that they may be found in one place.
First, the actual kinematic steering angle, the angle projected onto the road plane to which the front assembly is rotated is a function of the steering angle and the steering axis angle:
$\Delta = \delta \cos \left (\phi \right )$
where Δ is the kinematic steering angle, δ is the steering angle, and φ is the caster angle of the steering axis.[7]
Second, the lean of the bike decreases the actual radius of the turn proportionally to the cosine of the lean angle. The resulting radius can be roughly approximated (within 2% of exact value) by:
$r = \frac{w\cos \left (\theta \right )}{\delta \cos \left (\phi \right )}$
where r is the approximate radius, w is the wheelbase, θ is the lean angle, δ is the steering angle, and φ is the caster angle of the steering axis.[7]
Third, because the front and rear tires can have different slip angles due to weight distribution, tire properties, etc., bikes can experience understeer or oversteer. When understeering, the steering angle must be greater, and when oversteering, the steering angle must be less than it would be if the slip angles were equal to maintain a given turn radius.[7] Some authors even use the term counter-steering to refer to the need on some bikes under some conditions to steer in the opposite direction of the turn (negative steering angle) to maintain control in response to significant rear wheel slippage.[7]
Fourth, camber thrust contributes to the centripetal force necessary to cause the bike to deviate from a straight path, along with cornering force due to the slip angle, and can be the largest contributor.[22] Camber thrust contributes to the ability of bikes to negotiate a turn with the same radius as automobiles but with a smaller steering angle.[22] When a bike is steered and leaned in the same direction, the camber angle of the front tire is greater than that of the rear and so can generate more camber thrust, all else being equal.[7]
#### No hands
While countersteering is usually initiated by applying torque directly to the handlebars, on lighter vehicles such as bicycles, it can also be accomplished by shifting the rider’s weight. If the rider leans to the right relative to the bike, the bike will lean to the left to conserve angular momentum, and the combined center of mass will remain in the same vertical plane. This leftward lean of the bike, called counter lean by some authors,[22] will cause it to steer to the left and initiate a right-hand turn as if the rider had countersteered to the left by applying a torque directly to the handlebars.[25] Note that this technique may be complicated by additional factors such as headset friction and stiff control cables.
#### Gyroscopic effects
As mentioned above in the section on balance, one effect of turning the front wheel is a roll moment caused by gyroscopic precession. The magnitude of this moment is proportional to the moment of inertia of the front wheel, its spin rate (forward motion), the rate that the rider turns the front wheel by applying a torque to the handlebars, and the cosine of the angle between the steering axis and the vertical.[7]
For a sample motorcycle moving at 22 m/s (50 mph) that has a front wheel with a moment of inertia of 0.6 kg·m2, turning the front wheel one degree in half a second generates a roll moment of 3.5 N·m. In comparison, the lateral force on the front tire as it tracks out from under the motorcycle reaches a maximum of 50 N. This, acting on the 0.6 m (2 ft) height of the center of mass, generates a roll moment of 30 N·m.
While the moment from gyroscopic forces is only 12% of this, it can play a significant part because it begins to act as soon as the rider applies the torque, instead of building up more slowly as the wheel out-tracks. This can be especially helpful in motorcycle racing.
#### Two-wheel steering
Because of theoretical benefits, such as a tighter turning radius at low speed, attempts have been made to construct motorcycles with two-wheel steering. One working prototype by Ian Drysdale in Australia is reported to "work very well."[28][29] Issues in the design include whether to provide active control of the rear wheel or let it swing freely. In the case of active control, the control algorithm needs to decide between steering with or in the opposite direction of the front wheel, when, and how much. One implementation of two-wheel steering, the Sideways bike, lets the rider control the steering of both wheels directly.
Milton W. Raymond built a long low two-wheel steering bicycle, called "X-2", with various steering mechanisms to control the two wheels independently. Steering motions included "balance", in which both wheels move together to steer the tire contact under the center of gravity; and "true circle", in which the wheels steer equally in opposite directions and thus steering the bicycle without substantially changing the lateral position of the tire contacts relative to the center of mass. X-2 was also able to go "crabwise" with the wheels parallel but out of line with the frame, for instance with the front wheel near the roadway center line and rear wheel near the curb. "Balance" steering allowed easy balancing despite long wheelbase and low center of mass, but no self-balancing ("no hands") configuration was discovered. True circle, as expected, was essentially impossible to balance, as steering does not correct for misalignment of the tire patch and center of mass. Crabwise cycling at angles tested up to about 45° did not show a tendency to fall over, even under braking.[citation needed] X-2 is mentioned in passing in Whitt and Wilson's Bicycling Science 2nd edition.[11]
#### Rear-wheel steering
Because of the theoretical benefits, especially a simplified front-wheel drive mechanism, attempts have been made to construct a ridable rear-wheel steering bike. The Bendix Company built a rear-wheel steering bicycle, and the U.S. Department of Transportation commissioned the construction of a rear-wheel steering motorcycle: both proved to be unridable. Rainbow Trainers, Inc. in Alton, IL, offered US\$5,000 to the first person "who can successfully ride the rear-steered bicycle, Rear Steered Bicycle I".[30] One documented example of someone successfully riding a rear-wheel steering bicycle is that of L. H. Laiterman at MIT, on a specially designed recumbent bike.[11] The difficulty is that turning left, accomplished by turning the rear wheel to the right, initially moves the center of mass to the right, and vice versa. This complicates the task of compensating for leans induced by the environment.[31] Examination of the eigenvalues shows that the rear-wheel steering configuration is inherently unstable.
#### Center steering
Flevobike with center steering
Between the extremes of bicycles with classical front-wheel steering and those with strictly rear-wheel steering is a class of bikes with a pivot point somewhere between the two referred to as center-steering, similar to articulated steering. This design allows for simple front-wheel drive and appears to be quite stable, even ridable no-hands, as many photographs illustrate.[32][33]
These designs, such as the Python Lowracer, usually have very lax head angles (40° to 65°) and positive or even negative trail. The builder of a bike with negative trail states that steering the bike from straight ahead forces the seat (and thus the rider) to rise slightly and this offsets the destabilizing effect of the negative trail.[34]
#### Tiller effect
Tiller effect is the expression used to describe how handlebars that extend far behind the steering axis (head tube) act like a tiller on a boat, in that one moves the bars to the right in order to turn the front wheel to the left, and vice versa. This situation is commonly found on cruiser bicycles, some recumbents, and even some cruiser motorcycles. It can be troublesome when it limits the ability to steer because of interference or the limits of arm reach.[35]
#### Tires
Tires have a large influence over bike handling, especially on motorcycles.[7][22] Tire inflation pressures have also been found to be important variables in the behavior of a motorcycle at high speeds.[36] Because the front and rear tires can have different slip angles due to weight distribution, tire properties, etc., bikes can experience understeer or oversteer. Of the two, understeer, in which the front wheel slides more than the rear wheel, is more dangerous since front wheel steering is critical for maintaining balance.[7] Also, because real tires have a finite contact patch with the road surface that can generate a scrub torque, and when in a turn, can experience some side slipping as they roll, they can generate torques about an axis normal to the plane of the contact patch.
Bike tire contact patch during a right-hand turn
One torque generated by tires is due to asymmetries in the side-slip along the length of the contact patch. The resultant force of this side-slip occurs behind the geometric center of the contact patch, a distance described as the pneumatic trail, and so creates a torque on the tire. Since the direction of the side-slip is towards the outside of the turn, the force on the tire is towards the center of the turn. Therefore, this torque tends to turn the front wheel in the direction of the side-slip, away from the direction of the turn, and therefore tends to increase the radius of the turn.
Another torque is produced by the finite width of the contact patch and the lean of the tire in a turn. The portion of the contact patch towards the outside of the turn is actually moving rearward, with respect to the wheel's hub, faster than the rest of the contact patch, because of its greater radius from the hub. By the same reasoning, the inner portion is moving rearward more slowly. So the outer and inner portions of the contact patch slip on the pavement in opposite directions, generating a torque that tends to turn the front wheel in the direction of the turn, and therefore tends to decrease the turn radius.
The combination of these two opposite torques creates a resulting yaw torque on the front wheel, and its direction is a function of the side-slip angle of the tire, the angle between the actual path of the tire and the direction it is pointing, and the camber angle of the tire (the angle that the tire leans from the vertical).[7] The result of this torque is often the suppression of the inversion speed predicted by rigid wheel models described above in the section on steady-state turning.[8]
#### High side
A highsider, highside, or high side is a type of bike motion which is caused by a rear wheel gaining traction when it is not facing in the direction of travel, usually after slipping sideways in a curve.[7] This can occur under heavy braking, acceleration, a varying road surface, or suspension activation, especially due to interaction with the drivetrain.[37] It can take the form of a single slip-then-flip or a series of violent oscillations.[22]
### Maneuverability and handling
Bike maneuverability and handling is difficult to quantify for several reasons. The geometry of a bike, especially the steering axis angle makes kinematic analysis complicated.[1] Under many conditions, bikes are inherently unstable and must always be under rider control. Finally, the rider's skill has a large influence on the bike's performance in any maneuver.[7] Bike designs tend to consist of a trade-off between maneuverability and stability.
#### Rider control inputs
Graphs showing the lean and steer angle response of an otherwise uncontrolled bike, traveling at a forward speed in its stable range (6 m/s), to a steer torque that begins as an impulse and then remains constant. Torque to right causes initial steer to right, lean to left, and eventually a steady-state turn to left.
The primary control input that the rider can make is to apply a torque directly to the steering mechanism via the handlebars. Because of the bike's own dynamics, due to steering geometry and gyroscopic effects, direct position control over steering angle has been found to be problematic.[6]
A secondary control input that the rider can make is to lean the upper torso relative to the bike. As mentioned above, the effectiveness of rider lean varies inversely with the mass of the bike. On heavy bikes, such as motorcycles, rider lean mostly alters the ground clearance requirements in a turn, improves the view of the road, and improves the bike system dynamics in a very low-frequency passive manner.[6]
#### Differences from automobiles
The need to keep a bike upright to avoid injury to the rider and damage to the vehicle even limits the type of maneuverability testing that is commonly performed. For example, while automobile enthusiast publications often perform and quote skidpad results, motorcycle publications do not. The need to "set up" for a turn, lean the bike to the appropriate angle, means that the rider must see further ahead than is necessary for a typical car at the same speed, and this need increases more than in proportion to the speed.[6]
#### Rating schemes
Several schemes have been devised to rate the handling of bikes, particularly motorcycles.[7]
• The roll index is the ratio between steering torque and roll or lean angle.
• The steering ratio is the ratio between the theoretical turning radius based on ideal tire behavior and the actual turning radius.[7] Values less than one, where the front wheel side slip is greater than the rear wheel side slip, are described as under-steering; equal to one as neutral steering; and greater than one as over-steering. Values less than zero, in which the front wheel must be turned opposite the direction of the curve due to much greater rear wheel side slip than front wheel have been described as counter-steering. Riders tend to prefer neutral or slight over-steering.[7] Car drivers tend to prefer under-steering.
• The Koch index is the ratio between peak steering torque and the product of peak lean rate and forward speed. Large, touring motorcycles tend to have a high Koch index, sport motorcycles tend to have a medium Koch index, and scooters tend to have a low Koch index.[7] It is easier to maneuver light scooters than heavy motorcycles.
### Lateral motion theory
Although its equations of motion can be linearized, a bike is a nonlinear system. The variable(s) to be solved for cannot be written as a linear sum of independent components, i.e. its behavior is not expressible as a sum of the behaviors of its descriptors.[1] Generally, nonlinear systems are difficult to solve and are much less understandable than linear systems. In the idealized case, in which friction and any flexing is ignored, a bike is a conservative system. Damping, however, can still be demonstrated: side-to-side oscillations will decrease with time. Energy added with a sideways jolt to a bike running straight and upright (demonstrating self-stability) is converted into increased forward speed, not lost, as the oscillations die out.
A bike is a nonholonomic system because its outcome is path-dependent. In order to know its exact configuration, especially location, it is necessary to know not only the configuration of its parts, but also their histories: how they have moved over time. This complicates mathematical analysis.[25] Finally, in the language of control theory, a bike exhibits non-minimum phase behavior.[38] It turns in the direction opposite of how it is initially steered, as described above in the section on countersteering
#### Degrees of freedom
Graphs of bike steer angle and lean angle vs turn radius.
The number of degrees of freedom of a bike depends on the particular model being used. The simplest model that captures the key dynamic features, four rigid bodies with knife edge wheels rolling on a flat smooth surface, has 7 degrees of freedom (configuration variables required to completely describe the location and orientation of all 4 bodies):[1]
1. x coordinate of rear wheel contact point
2. y coordinate of rear wheel contact point
3. orientation angle of rear frame (yaw)
4. rotation angle of rear wheel
5. rotation angle of front wheel
6. lean angle of rear frame (roll)
7. steering angle between rear frame and front end
Adding complexity to the model, such as suspension, tire compliance, frame flex, or rider movement, adds degrees of freedom. While the rear frame does pitch with leaning and steering, the pitch angle is completely constrained by the requirement for both wheels to remain on the ground, and so can be calculated geometrically from the other seven variables. If the location of the bike and the rotation of the wheels are ignored, the first five degrees of freedom can also be ignored, and the bike can be described by just two variables: lean angle and steer angle.
#### Equations of motion
The equations of motion of an idealized bike, consisting of
• a rigid frame,
• a rigid fork,
• two knife-edged, rigid wheels,
• all connected with frictionless bearings and rolling without friction or slip on a smooth horizontal surface and
• operating at or near the upright and straight-ahead, unstable equilibrium
can be represented by a single fourth-order linearized ordinary differential equation or two coupled second-order differential equations,[1] the lean equation
$M_{\theta\theta}\ddot{\theta_r} + K_{\theta\theta}\theta_r + M_{\theta\psi}\ddot{\psi} + C_{\theta\psi}\dot{\psi} + K_{\theta\psi}\psi = M_{\theta}$
and the steer equation
$M_{\psi\psi}\ddot{\psi} + C_{\psi\psi}\dot{\psi} + K_{\psi\psi}\psi + M_{\psi\theta}\ddot{\theta_r} + C_{\psi\theta}\dot{\theta_r} + K_{\psi\theta}\theta_r = M_{\psi}\mbox{,}$
where
• θr is the lean angle of the rear assembly,
• ψ is the steer angle of the front assembly relative to the rear assembly and
• Mθ and Mψ are the moments (torques) applied at the rear assembly and the steering axis, respectively. For the analysis of an uncontrolled bike, both are taken to be zero.
These can be represented in matrix form as
$M\mathbf\ddot q+C\mathbf\dot q+K\mathbf q=\mathbf f$
where
• M is the symmetrical mass matrix which contains terms that include only the mass and geometry of the bike,
• C is the so-called damping matrix, even though an idealized bike has no dissipation, which contains terms that include the forward speed v and is asymmetric,
• K is the so-called stiffness matrix which contains terms that include the gravitational constant g and v2 and is symmetric in g and asymmetric in v2,
• $\mathbf q$ is a vector of lean angle and steer angle, and
• $\mathbf f$ is a vector of external forces, the moments mentioned above.
In this idealized and linearized model, there are many geometric parameters (wheelbase, head angle, mass of each body, wheel radius, etc.), but only four significant variables: lean angle, lean rate, steer angle, and steer rate. These equations have been verified by comparison with multiple numeric models derived completely independently.[1]
#### Eigenvalues
Eigenvalues plotted against forward speed for a typical utility bicycle simplified to have knife-edge wheels that roll without slip.
It is possible to calculate eigenvalues, one for each of the four state variables (lean angle, lean rate, steer angle, and steer rate), from the linearized equations in order to analyze the normal modes and self-stability of a particular bike design. In the plot to the right, eigenvalues of one particular bicycle are calculated for forward speeds of 0–10 m/s (22 mph). When the real parts of all eigenvalues (shown in dark blue) are negative, the bike is self-stable. When the imaginary parts of any eigenvalues (shown in cyan) are non-zero, the bike exhibits oscillation. The eigenvalues are point symmetric about the origin and so any bike design with a self-stable region in forward speeds will not be self-stable going backwards at the same speed.[1]
There are three forward speeds that can be identified in the plot to the right at which the motion of the bike changes qualitatively:[1]
1. The forward speed at which oscillations begin, at about 1 m/s (2.2 mph) in this example, sometimes called the double root speed due to there being a repeated root to the characteristic polynomial (two of the four eigenvalues have exactly the same value). Below this speed, the bike simply falls over as an inverted pendulum does.
2. The forward speed at which oscillations do not increase, where the weave mode eigenvalues switch from positive to negative in a Hopf bifurcation at about 5.3 m/s (12 mph) in this example, is called the weave speed. Below this speed, oscillations increase until the uncontrolled bike falls over. Above this speed, oscillations eventually die out.
3. The forward speed at which non-oscillatory leaning increases, where the capsize mode eigenvalues switch from negative to positive in a pitchfork bifurcation at about 8 m/s (18 mph) in this example, is called the capsize speed. Above this speed, this non-oscillating lean eventually causes the uncontrolled bike to fall over.
Between these last two speeds, if they both exist, is a range of forward speeds at which the particular bike design is self-stable. In the case of the bike whose eigenvalues are shown here, the self-stable range is 5.3–8.0 m/s (12–18 mph). The fourth eigenvalue, which is usually stable (very negative), represents the castoring behavior of the front wheel, as it tends to turn towards the direction in which the bike is traveling. Note that this idealized model does not exhibit the wobble or shimmy and rear wobble instabilities described above. They are seen in models that incorporate tire interaction with the ground or other degrees of freedom.[7]
Experimentation with real bikes has so far confirmed the weave mode predicted by the eigenvalues. It was found that tire slip and frame flex are not important for the lateral dynamics of the bicycle in the speed range up to 6 m/s.[39] The idealized bike model used to calculate the eigenvalues shown here does not incorporate any of the torques that real tires can generate, and so tire interaction with the pavement cannot prevent the capsize mode from becoming unstable at high speeds, as Wilson and Cossalter suggest happens in the real world.
#### Modes
Graphs that show (from left to right, top to bottom) weave instability, self-stability, marginal self-stability, and capsize instability in an idealized linearized model of an uncontrolled utility bicycle.
Bikes, as complex mechanisms, have a variety of modes: fundamental ways that they can move. These modes can be stable or unstable, depending on the bike parameters and its forward speed. In this context, "stable" means that an uncontrolled bike will continue rolling forward without falling over as long as forward speed is maintained. Conversely, "unstable" means that an uncontrolled bike will eventually fall over, even if forward speed is maintained. The modes can be differentiated by the speed at which they switch stability and the relative phases of leaning and steering as the bike experiences that mode. Any bike motion consists of a combination of various amounts of the possible modes, and there are three main modes that a bike can experience: capsize, weave, and wobble.[1] A lesser known mode is rear wobble, and it is usually stable.[7]
##### Capsize
Capsize is the word used to describe a bike falling over without oscillation. During capsize, an uncontrolled front wheel usually steers in the direction of lean, but never enough to stop the increasing lean, until a very high lean angle is reached, at which point the steering may turn in the opposite direction. A capsize can happen very slowly if the bike is moving forward rapidly. Because the capsize instability is so slow, on the order of seconds, it is easy for the rider to control, and is actually used by the rider to initiate the lean necessary for a turn.[7]
For most bikes, depending on geometry and mass distribution, capsize is stable at low speeds, and becomes less stable as speed increases until it is no longer stable. However, on many bikes, tire interaction with the pavement is sufficient to prevent capsize from becoming unstable at high speeds.[7][8]
##### Weave
Weave is the word used to describe a slow (0–4 Hz) oscillation between leaning left and steering right, and vice-versa. The entire bike is affected with significant changes in steering angle, lean angle (roll), and heading angle (yaw). The steering is 180° out of phase with the heading and 90° out of phase with the leaning.[7] This AVI movie shows weave.
For most bikes, depending on geometry and mass distribution, weave is unstable at low speeds, and becomes less pronounced as speed increases until it is no longer unstable. While the amplitude may decrease, the frequency actually increases with speed.
##### Wobble or shimmy
Eigenvalues plotted against forward speed for a motorcycle modeled with frame flexibility and realistic tire dynamics. Additional modes can be seen, such as wobble, which becomes unstable at 43.7 m/s.
The same eigenvalues as in the figure above, but plotted on a root locus plot. Several additional oscillating modes are visible.
Wobble, shimmy, tank-slapper, speed wobble, and death wobble are all words and phrases used to describe a rapid (4–10 Hz) oscillation of primarily just the front end (front wheel, fork, and handlebars). The rest of the bike remains essentially unaffected. This instability occurs mostly at high speed and is similar to that experienced by shopping cart wheels, airplane landing gear, and automobile front wheels.[7][8] While wobble or shimmy can be easily remedied by adjusting speed, position, or grip on the handlebar, it can be fatal if left uncontrolled.[40] This AVI movie shows wobble.
Wobble or shimmy begins when some otherwise minor irregularity, such as fork asymmetry,[41] accelerates the wheel to one side. The restoring force is applied in phase with the progress of the irregularity, and the wheel turns to the other side where the process is repeated. If there is insufficient damping in the steering the oscillation will increase until system failure occurs. The oscillation frequency can be changed by changing the forward speed, making the bike stiffer or lighter, or increasing the stiffness of the steering, of which the rider is a main component.[11]
##### Rear wobble
The term rear wobble is used to describe a mode of oscillation in which lean angle (roll) and heading angle (yaw) are almost in phase and both 180° out of phase with steer angle. The rate of this oscillation is moderate with a maximum of about 6.5 Hz. Rear wobble is heavily damped and falls off quickly as bike speed increases.[7]
##### Design criteria
The effect that the design parameters of a bike have on these modes can be investigated by examining the eigenvalues of the linearized equations of motion.[36] For more details on the equations of motion and eigenvalues, see the section on the equations of motion above. Some general conclusions that have been drawn are described here.
The lateral and torsional stiffness of the rear frame and the wheel spindle affects wobble-mode damping substantially. Long wheelbase and trail and a flat steering-head angle have been found to increase weave-mode damping. Lateral distortion can be countered by locating the front fork torsional axis as low as possible.
Cornering weave tendencies are amplified by degraded damping of the rear suspension. Cornering, camber stiffnesses and relaxation length of the rear tire make the largest contribution to weave damping. The same parameters of the front tire have a lesser effect. Rear loading also amplifies cornering weave tendencies. Rear load assemblies with appropriate stiffness and damping, however, were successful in damping out weave and wobble oscillations.
One study has shown theoretically that, while a bike leaned in a turn, road undulations can excite the weave mode at high speed or the wobble mode at low speed if either of their frequencies match the vehicle speed and other parameters. Excitation of the wobble mode can be mitigated by an effective steering damper and excitation of the weave mode is worse for light riders than for heavy riders.[10]
### Other hypotheses
Although bicycles and motorcycles can appear to be simple mechanisms with only four major moving parts (frame, fork, and two wheels), these parts are arranged in a way that makes them complicated to analyze.[11] While it is an observable fact that bikes can be ridden even when the gyroscopic effects of their wheels are canceled out,[4][5] the hypothesis that the gyroscopic effects of the wheels are what keep a bike upright is common in print and online.[4][25]
Examples in print:
• "Angular momentum and motorcycle counter-steering: A discussion and demonstration", A. J. Cox, Am. J. Phys. 66, 1018–1021 ~1998
• "The motorcycle as a gyroscope", J. Higbie, Am. J. Phys. 42, 701–702
• The Physics of Everyday Phenomena, W. T. Griffith, McGraw–Hill, New York, 1998, pp. 149–150.
• The Way Things Work., Macaulay, Houghton-Mifflin, New York, NY, 1989
And online:
## Longitudinal dynamics
A bicyclist performing a wheelie.
Bikes may experience a variety of longitudinal forces and motions. On most bikes, when the front wheel is turned to one side or the other, the entire rear frame pitches forward slightly, depending on the steering axis angle and the amount of trail.[7][24] On bikes with suspensions, either front, rear, or both, trim is used to describe the geometric configuration of the bike, especially in response to forces of braking, accelerating, turning, drive train, and aerodynamic drag.[7]
The load borne by the two wheels varies not only with center of mass location, which in turn varies with the amount and location of passengers and luggage, but also with acceleration and deceleration. This phenomenon is known as load transfer[7] or weight transfer,[22][37] depending on the author, and provides challenges and opportunities to both riders and designers. For example, motorcycle racers can use it to increase the friction available to the front tire when cornering, and attempts to reduce front suspension compression during heavy braking has spawned several motorcycle fork designs.
The net aerodynamic drag forces may be considered to act at a single point, called the center of pressure.[22] At high speeds, this will create a net moment about the rear driving wheel and result in a net transfer of load from the front wheel to the rear wheel.[22] Also, depending on the shape of the bike and the shape of any fairing that might be installed, aerodynamic lift may be present that either increases or further reduces the load on the front wheel.[22]
### Stability
Though longitudinally stable when stationary, a bike may become longitudinally unstable under sufficient acceleration or deceleration, and Euler's second law can be used to analyze the ground reaction forces generated.[42] For example, the normal (vertical) ground reaction forces at the wheels for a bike with a wheelbase L and a center of mass at height h and at a distance b in front of the rear wheel hub, and for simplicity, with both wheels locked, can be expressed as:[7]
$N_r = mg\left(\frac{L-b}{L} - \mu \frac{h}{L}\right)$ for the rear wheel and $N_f = mg\left(\frac{b}{L} + \mu \frac{h}{L}\right)$ for the front wheel.
The frictional (horizontal) forces are simply
$F_r = \mu N_r \,$ for the rear wheel and $F_f = \mu N_f \,$ for the front wheel,
where μ is the coefficient of friction, m is the total mass of the bike and rider, and g is the acceleration of gravity. Therefore, if
$\mu \ge \frac{L-b}{h}$,
which occurs if the center of mass is anywhere above or in front of a line extending back from the front wheel contact patch and inclined at the angle
$\theta = \tan^{-1} \left( \frac{1}{\mu} \right) \,$
above the horizontal,[22] then the normal force of the rear wheel will be zero (at which point the equation no longer applies) and the bike will begin to flip or loop forward over the front wheel.
On the other hand, if the center of mass height is behind or below the line, as is true, for example on most tandem bicycles or long-wheel-base recumbent bicycles, then, even if the coefficient of friction is 1.0, it is impossible for the front wheel to generate enough braking force to flip the bike. It will skid unless it hits some fixed obstacle, such as a curb.
Similarly, powerful motorcycles can generate enough torque at the rear wheel to lift the front wheel off the ground in a maneuver called a wheelie. A line similar to the one described above to analyze braking performance can be drawn from the rear wheel contact patch to predict if a wheelie is possible given the available friction, the center of mass location, and sufficient power.[22] This can also happen on bicycles, although there is much less power available, if the center of mass is back or up far enough or the rider lurches back when applying power to the pedals.[43]
Of course, the angle of the terrain can influence all of the calculations above. All else remaining equal, the risk of pitching over the front end is reduced when riding up hill and increased when riding down hill. The possibility of performing a wheelie increases when riding up hill,[43] and is a major factor in motorcycle hillclimbing competitions.
### Braking
A motorcyclist performing a stoppie.
Most of the braking force of standard upright bikes comes from the front wheel. As the analysis above shows, if the brakes themselves are strong enough, the rear wheel is easy to skid, while the front wheel often can generate enough stopping force to flip the rider and bike over the front wheel. This is called a stoppie if the rear wheel is lifted but the bike does not flip, or an endo (abbreviated form of end-over-end) if the bike flips. On long or low bikes, however, such as cruiser motorcycles and recumbent bicycles, the front tire will skid instead, possibly causing a loss of balance.
In the case of a front suspension, especially telescoping fork tubes, the increase in downward force on the front wheel during braking may cause the suspension to compress and the front end to lower. This is known as brake diving. A riding technique that takes advantage of how braking increases the downward force on the front wheel is known as trail braking.
#### Front wheel braking
The limiting factors on the maximum deceleration in front wheel braking are:
• the maximum, limiting value of static friction between the tire and the ground, often between 0.5 and 0.8 for rubber on dry asphalt,[44]
• the kinetic friction between the brake pads and the rim or disk, and
• pitching or looping (of bike and rider) over the front wheel.
For an upright bicycle on dry asphalt with excellent brakes, pitching will probably be the limiting factor. The combined center of mass of a typical upright bicycle and rider will be about 60 cm (24 in) back from the front wheel contact patch and 120 cm (47 in) above, allowing a maximum deceleration of 0.5 g (5 m/s² or 16 ft/s²).[11] If the rider modulates the brakes properly, however, pitching can be avoided. If the rider moves his weight back and down, even larger decelerations are possible.
Front brakes on many inexpensive bikes are not strong enough so, on the road, they are the limiting factor. Cheap cantilever brakes, especially with "power modulators", and Raleigh-style side-pull brakes severely restrict the stopping force. In wet conditions they are even less effective. Front wheel slides are more common off-road. Mud, water, and loose stones reduce the friction between the tire and trail, although knobby tires can mitigate this effect by grabbing the surface irregularities. Front wheel slides are also common on corners, whether on road or off. Centripetal acceleration adds to the forces on the tire-ground contact, and when the friction force is exceeded the wheel slides.
#### Rear wheel braking
The rear brake of an upright bicycle can only produce about 0.1 g (1 m/s²) deceleration at best,[11] because of the decrease in normal force at the rear wheel as described above. All bikes with only rear braking are subject to this limitation: for example, bikes with only a coaster brake, and fixed-gear bikes with no other braking mechanism. There are, however, situations that may warrant rear wheel braking[45]
• Slippery surfaces. Under front wheel braking, the lower coefficient of friction may cause the front wheel to skid which often results in a loss of balance.
• Front flat tire. Braking a wheel with a flat tire can cause the tire to come off the rim which greatly reduces friction and, in the case of a front wheel, result in a loss of balance.
• Long mountain descents. Alternating between front and rear brakes can help reduce heat buildup which can cause a blowout.
## Suspension
Mountain bike rear suspension
Bikes may have no front, rear or full suspension that operate primarily in the central plane of symmetry; though with some consideration given to lateral compliance.[22] The goals of a bike suspension are to reduce vibration experienced by the rider, maintain wheel contact with the ground, and maintain vehicle trim.[7] The primary suspension parameters are stiffness, damping, sprung and unsprung mass, and tire characteristics.[22] Besides irregularities in the terrain, braking and acceleration forces can also activate the suspension as described above.
## Vibration
The study of vibration in bikes includes its causes, such as engine balance,[46] wheel balance, ground surface, and aerodynamics; its transmission and absorption; and its effects on the bike, the rider, and safety.[47] An important factor in any vibration analysis is a comparison of the natural frequencies of the system with the possible driving frequencies of the vibration sources.[48] A close match means mechanical resonance that can result in large amplitudes. A challenge in vibration damping is to create compliance in certain directions (vertically) without sacrificing frame rigidity needed for power transmission and handling (torsionally).[49] Another issue with vibration for the bike is the possibility of failure due to material fatigue[50] Effects of vibration on riders include discomfort, loss of efficiency, Hand-Arm Vibration Syndrome, a secondary form Raynaud's disease, and whole body vibration. Vibrating instruments may be inaccurate or difficult to read.[50]
### In bicycles
The primary cause of vibrations in a properly functioning bicycle is the surface over which it rolls. In addition to pneumatic tires and traditional bicycle suspensions, a variety of techniques have been developed to damp vibrations before they reach the rider. These include materials, such as carbon fiber, either in the whole frame or just key components such as the front fork, seatpost, or handlebars; tube shapes, such as curved seat stays;[51] and special inserts, such as Zertz by Specialized, [52][53] and Buzzkills by Bontrager.
### In motorcycles
In addition to the road surface, vibrations in a motorcycle can be caused by the engine and wheels, if unbalanced. Manufacturers employ a variety of technologies to reduce or damp these vibrations, such as engine balance shafts, rubber engine mounts,[54] and tire weights.[55] The problems that vibration causes have also spawned an industry of after-market parts and systems designed to reduce it. Add-ons include handlebar weights,[56] isolated foot pegs, and engine counterweights. At high speeds, motorcycles and their riders may also experience aerodynamic flutter or buffeting.[57] This can be abated by changing the air flow over key parts, such as the windshield.[58]
## Experimentation
A variety of experiments have been performed in order to verify or disprove various hypotheses about bike dynamics.
• David Jones built several bikes in a search for an unridable configuration.[5]
• Richard Klein built several bikes to confirm Jones's findings.[4]
• Richard Klein also built a "Torque Wrench Bike" and a "Rocket Bike" to investigate steering torques and their effects.[4]
• Keith Code built a motorcycle with fixed handlebars to investigate the effects of rider motion and position on steering.[59]
• Schwab and Kooijman have performed measurements with an instrumented bike.[60]
## References
1. ^ a b c d e f g h i j k l m n o p q r Meijaard, Papadopoulos, Ruina, and Schwab, J.P.; Papadopoulos, Jim M.; Ruina, Andy; Schwab, A.L. (2007). "Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review" (pdf). Proc. R. Soc. A. 463 (2084): 1955–1982. doi:10.1098/rspa.2007.1857.
2. ^ Limbebeer and Sharp (2006). "Single-Track Vehicle Modeling and Control: Bicycles, Motorcycles, and Models". IEEE Control Systems Magazine (October): 34–61.
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10. ^ a b c Limebeer, Sharp, and Evangelou, D. J. N.; Sharp, R. S.; Evangelou, S. (November 2002). "Motorcycle Steering Oscillations due to Road Profiling". Transactions of the ASME 69: 724–739. doi:10.1115/1.1507768.
11. ^ a b c d e f g h i j k l Whitt, Frank R.; David G. Wilson (1982). Bicycling Science (Second ed.). Massachusetts Institute of Technology. pp. 198–233. ISBN 0-262-23111-5.
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17. ^ REI. "Cycle Expert Advice: Packing for a Tour". Retrieved 2007-11-13.
18. ^ Putnam, Josh (2006). "Steering Geometry: What is Trail?". Retrieved 2006-08-08.
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20. ^ Foale, Tony (1997). "Balancing Act". Retrieved 2006-08-04.
21. ^ "LeMond Racing Cycles". 2006. Retrieved 2006-08-08.
22. ^ a b c d e f g h i j k l m n o Foale, Tony (2006). Motorcycle Handling and Chassis Design (Second ed.). Tony Foale Designs. ISBN 978-84-933286-3-4.
23. ^ "Gear Head College: Trail". Retrieved 2009-08-05.
24. ^ a b c Hand, Richard S. (1988). "Comparisons and Stability Analysis of Linearized Equations of Motion for a Basic Bicycle Model" (PDF). Retrieved 2006-08-04.
25. ^ a b c d e f Fajans, Joel (July 2000). "Steering in bicycles and motorcycles" (PDF). American Journal of Physics 68 (7): 654–659. doi:10.1119/1.19504. Retrieved 2006-08-04.
26. ^ McGill, David J; Wilton W. King (1995). Engineering Mechanics, An Introduction to Dynamics (Third ed.). PWS Publishing Company. pp. 479–481. ISBN 0-534-93399-8.
27. ^ Brown, Sheldon (2006). "Sheldon Brown's Bicycle Glossary". Sheldon Brown. Retrieved 2006-08-08.
28. ^ Foale, Tony (1997). "2 Wheel Drive/Steering". Retrieved 2006-12-14.
29. ^ Drysdale, Ian. "Drysdale 2x2x2". Retrieved 2009-04-05.
30. ^ Klein, Richard E.; et al. (2005). "Challenge". Retrieved 2006-08-06.
31. ^ Wannee, Erik (2005). "Rear Wheel Steered Bike". Retrieved 2006-08-04.
32. ^ Wannee, Erik (2001). "Variations on the theme 'FlevoBike'". Retrieved 2006-12-15.
33. ^ Mages, Jürgen (2006). "Python Gallery". Retrieved 2006-12-15.
34. ^ Mages, Jürgen (2006). "Python Frame Geometry". Retrieved 2006-12-15.
35. ^ Brown, Sheldon (2006). "Sheldon Brown's Bicycle Glossary". Sheldon Brown. Retrieved 2006-08-08.
36. ^ a b Evangelou, Simos (2004). "The Control and Stability Analysis of Two-wheeled Road Vehicles" (PDF). Imperial College London. pp. 159. Retrieved 2006-08-04.
37. ^ a b Cocco, Gaetano (2005). Motorcycle Design and Technology. Motorbooks. pp. 40–46. ISBN 978-0-7603-1990-1.
38. ^ Klein, Richard E.; et al. (2005). "Counter-Intuitive.". Retrieved 2006-08-07.
39. ^ Schwab, A. L.; J. P. Meijaard and J. D. G. Kooijman (5–9 June 2006). "Experimental Validation of a Model of an Uncontrolled Bicycle" (PDF). III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering (Lisbon, Portugal: C.A. Mota Soares et al.). Retrieved 2008-10-19.
40. ^ Kettler, Bill (2004-09-15). "Crash kills cyclist". Mail Tribune. Retrieved 2006-08-04.
41. ^ Lennard Zinn (2008-12-30). "VeloNews: Technical Q&A with Lennard Zinn: Torque wrenches and temps; shifting and shimmy". Retrieved 2009-01-02.
42. ^ Ruina, Andy; Rudra Pratap (2002) (PDF). Introduction to Statics and Dynamics. Oxford University Press. p. 350. Retrieved 2006-08-04.
43. ^ a b Cassidy, Chris. "Bicycling Magazine: The Wheelie". Retrieved 2009-05-22.
44. ^ Kurtus, Ron (2005-11-02). "Coefficient of Friction Values for Clean Surfaces". Retrieved 2006-08-07.
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Find the lowest temperature of vaporization and the highest temperature when there is liquid left over.
A liquid mixture of 20 mol% benzene, 50 mol% toluene and 30 mol% xylene is to be partially vaporized at 1 atm pressure. You are asked to find out the range of temperatures which could be used for this process.
Antoine constants: pressure [=] mmHg Temperature [=] C at 1 atm (degreesC) A B C Benzene 6.89 1203 220 80.1 Toluene 6.96 1347 220 110.6 Xylene 7.00 1467 214 140
1. What is the lowest temperature at which vaporization could occur? ___________ Degrees C.
(provide answer to within +/- 5 °C)
2. What is the highest temperature that could be used and liquid still be present? __________ Degrees C.
(provide answer to within +/- 5 °C)
• Category:- Chemistry
• Reference No.:- M919404
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Effective HRM Question How can an effective HRM system help facilitate the achievement of an organization's strate | 1,149 | 4,961 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2016-44 | longest | en | 0.833279 |
https://gateoverflow.in/213556/isro2018-32?show=213924 | 1,579,792,351,000,000,000 | text/html | crawl-data/CC-MAIN-2020-05/segments/1579250610919.33/warc/CC-MAIN-20200123131001-20200123160001-00500.warc.gz | 452,296,113 | 17,797 | +1 vote
862 views
A computer uses ternary system instead of the traditional systen, An $n$ bit string in the binary system will occupy
1. $3+n$ ternary digits
2. $2n/3$ ternary digits
3. $n$$\log_{2}3 ternary digits 4. n$$\log_{3}2$ ternary digits
recategorized | 862 views
Let the no. of digits occupied by the ternary system be 'x'
Now,Maxi. value representable in ternary system using x digits = Maxi. value representable in binary system using n digits
3x-1=2n-1
x=log32n = n log32 (d)
by Boss (10.8k points)
selected by
Maximum No. possible using n-bit in Binary:- 2n -1
Maximum no. possible using x bit in Ternary:- 3x -1
Both will take different no. of bits to represent same number.
3x -1 = 2n -1
3= 2n
taking log on both side:;-
X= log3 ( 2 )
X=n*log32 .
by Active (3.3k points)
0
@Sonveer tomar 1,Could you elaborate how you've got 3 in base of log?What property you've used?
+1
For example:-
10n = 1000. For what value of n we get this
take log base 10 on both side.
log(10n ) = log(1000)
n log(10) = 3* log(10)
n=3.
In same manner to find value of X i take log base 3 on both side
0
@Sonveer tomar 1,Thank you so much for the explanation. :)
0
can you please elaborate, what question want to ask???
+1
the problem is simply trying to ask---> given an n-bit binary number howmany digits that that it would be required to represent that number in ternary number system....so here we try to find howmany number of digits in ternary number system that will be required to represent the maximum valued n-bit binary number......
so,let x digits required in ternary number system to represent the maximum valued n-bit binary number which is (2^n-1)(in decimal)....and maximum valued decimal number with x digit in ternary number system will be (3^x-1)...
so ,(2^n-1)<=(3^x-1)
=> 2^n<=3^x
=> x>=nlog2base3
D is correct
by Junior (725 points) | 588 | 1,883 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.96875 | 4 | CC-MAIN-2020-05 | longest | en | 0.767201 |
https://www.pressreader.com/usa/flying/20180801/282342565589332 | 1,542,781,630,000,000,000 | text/html | crawl-data/CC-MAIN-2018-47/segments/1542039747215.81/warc/CC-MAIN-20181121052254-20181121074254-00113.warc.gz | 970,718,850 | 6,316 | # Do new airplanes cost too much? Yes and no.
## YES AND NO. HERE’S WHAT IT MEANS FOR GENERAL AVIATION’S FUTURE.
Flying - - CONTENTS - By Stephen Pope
Why are new light airplanes so expensive? People have been asking this question for, oh, about the past 40 years. Meanwhile, prices for new airplanes keep rising while delivery numbers mostly stagnate.
Experts blame ever-increasing airplane prices on lots of things, from high production costs to the expense of litigating accident lawsuits to the improved avionics and more powerful engines found in new airplanes today. These factors all matter, of course, but they tell only part of the story.
In 1968, median household income was \$12,000, a new car cost an average of \$3,200 and a Cessna Skyhawk sold for about \$16,000. Today, median income is \$58,000, the average new car sells for \$26,000 and a Cessna Skyhawk can be had for about \$370,000. This is easy math. The price of a new car used to represent about three months of income — now it’s closer to six months. The price of a new airplane like a Cessna Skyhawk in 1968 took 16 months to buy, where today it takes six years to purchase.
Clearly, airplane prices have far outstripped general inflation. When we start looking at the underlying reasons why (and what it might mean for aircraft prices in the future) we reach some less than warm and fuzzy conclusions. But are airplanes too expensive? Put another way, should they cost less?
If we compare the prices of airplanes not with dollars but instead using some commodity — say, aluminum, which has been used in the production of a Cessna Skyhawk since the beginning — the resulting historical pricing charts do not show a similar upward trajectory, notes Philip Hersch, a professor of economics at Wichita State University, who explains that rising input prices without offsetting increases in productivity will inevitably result in higher prices — exactly what we’ve seen over a number of decades.
In 1968, the M2 money supply in the United States stood at \$486.4 billion. By 2018, it had risen to \$13.95 trillion, an increase of 2,768 percent — curiously, that’s very close to the percentage increase for a new light airplane like a Cessna Skyhawk since 1968. All those new dollars flooding into the economy have steadily bid up the cost of aircraft production, which is still based on similar land, material and labor needs as existed then.
The bottom line? Airplanes today cost about what they should. If we want cheaper airplanes we’ll need to dramatically change how we build them by reducing the costs of production, materials and labor. There are no easy answers for how to do that. The Part 23 rewrite of light aircraft certification rules had the goal of putting a lid on prices for new light GA airplanes by dramatically reducing the costs of certification. So far, we’ve seen some indications that prices for avionics are falling, but the market hasn’t miraculously blossomed with the promise of inexpensive new light GA airplanes as we all predicted (OK, hoped) would happen when light-sport aircraft rules took effect several years ago. We know how that turned out.
It’s starting to feel like general aviation is reaching a tipping point. If prices for new light airplanes keep accelerating beyond what even a buyer of above-average means can comfortably afford while the pool of used airplanes built in the 1960s and 1970s continues to age, what will happen to the lightaircraft manufacturing industry a few short years from now when new light airplanes are even more difficult to attain — or simply out of production? | 1,219 | 3,847 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.546875 | 3 | CC-MAIN-2018-47 | latest | en | 0.877182 |
http://mathhelpforum.com/pre-calculus/91904-can-someone-help-me-solve-logarithmic-equation.html | 1,529,758,928,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267864958.91/warc/CC-MAIN-20180623113131-20180623133131-00610.warc.gz | 201,805,754 | 9,453 | # Thread: Can someone help me solve this logarithmic equation?
1. ## Can someone help me solve this logarithmic equation?
Can someone help me solve the attached logarithmic equation?
2. Rewrite 3 as $\displaystyle log_2(8)$
$\displaystyle log_2(x+3) + log_2(x-4) - log_2(8) = 0$
Using the laws of logs combine into one log
$\displaystyle log_2(\frac{(x+3)(x-4)}{8}) = 0$
Remove the log:
$\displaystyle \frac{(x+3)(x-4)}{8} = 1$
and solve
3. I get x = 5 and x = -4 but the solution guide rejects the x = -4. Is this because plugging in the value of -4 into the logarithmic expression is not possible or as the calculator would "error?"
4. Originally Posted by s3a
I get x = 5 and x = -4 but the solution guide rejects the x = -4. Is this because plugging in the value of -4 into the logarithmic expression is not possible or as the calculator would "error?"
-4 is rejected because there are no real solutions to a negative logarithm. Since all logs must be greater than 0 only x=5 is a solution | 286 | 1,004 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.765625 | 4 | CC-MAIN-2018-26 | latest | en | 0.893103 |
http://math.stackexchange.com/questions/229067/how-do-i-solve-this-difference-equation | 1,455,089,047,000,000,000 | text/html | crawl-data/CC-MAIN-2016-07/segments/1454701158811.82/warc/CC-MAIN-20160205193918-00094-ip-10-236-182-209.ec2.internal.warc.gz | 149,925,453 | 16,755 | # How do I solve this difference equation?
For a sequence $\{D_k\}$, if we have: $$D_k=pD_{k+1}+qD_{k-1}+1$$
and we know that $D_0=D_N=0$. Where $p+q=1$, and $N$ is known. How do I solve it?
-
Here is a related problem. – Mhenni Benghorbal Nov 4 '12 at 19:02
I would use a generating function: Let $d(x) = \sum x^k D_k$, apply the recurrence and end conditions, and see what happens. – marty cohen Nov 4 '12 at 20:17
First assume $p \neq q$. Define $E_k = D_k + \frac{k}{p - q}$. Then $E$ satisfies the linear recursion
$$E_k = p E_{k+1} + q E_{k - 1}.$$
A solution of this recursion is of the form $E_k = \alpha \left(\frac{q}{p}\right)^k + \beta$ for constants $\alpha, \beta$. Using that $D_0 = D_N = 0$ this results in
$$D_k = \frac{N}{p - q} \cdot \frac{\left(\frac{q}{p}\right)^k - 1}{\left(\frac{q}{p}\right)^N - 1} - \frac{k}{p-q}.$$
If $p = q = \tfrac{1}{2}$ then define $E_k = D_k + k^2$ which satisfies the recursion
$$E_k = \tfrac{1}{2} E_{k+1} + \tfrac{1}{2} E_{k-1}.$$
Now a solution is of the form $E_k =\alpha \, k + \beta$. In this case we find
$$D_k = k \, (N - k).$$
- | 435 | 1,099 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.5625 | 5 | CC-MAIN-2016-07 | longest | en | 0.738214 |
https://blogs.technet.microsoft.com/heyscriptingguy/2006/08/25/how-can-i-determine-the-first-friday-in-a-month/ | 1,542,429,144,000,000,000 | text/html | crawl-data/CC-MAIN-2018-47/segments/1542039743282.69/warc/CC-MAIN-20181117040838-20181117062838-00133.warc.gz | 594,108,350 | 20,795 | # How Can I Determine the First Friday in a Month?
Hey, Scripting Guy! How can I determine the first Friday in a month?
-- JB
Hey, JB. You know, when we first glanced at this question we were hoping it said, “How can I determine if any given day in a month is a Friday?” Why? Well, if that was the case then we had a sure-fire method for you: just walk through the cafeteria in the Scripting Guys building at lunch time. If you walk into the cafeteria at noon on, say, a Monday or a Thursday, the place will be jam-packed; walk through on a Friday, however, and you feel like you’re the last person on Earth. Nobody is around on a Friday.
Note. Is that because all these dedicated Microsoft employees are working through lunch, making sure they get everything done before the weekend? Um, sure; why not?
Of course, that wasn’t your question; instead, you wanted to know how you can identify the first Friday in a month. Admittedly, you might not find the answer to that simply by walking through the Microsoft cafeteria. Instead, you might need to use a script similar to this:
```dtmDate = #11/1/2006#
Do Until x = 1
intDayOfWeek = Weekday(dtmDate)
If intDayOfWeek = 6 Then
Wscript.Echo "The first Friday of the month is " & dtmDate & "."
Exit Do
Else
dtmDate = dtmDate + 1
End If
Loop
```
What’s that? Sure, we can spare a few minutes to explain how this script works. After all, it’s Friday; we pretty much have the place to ourselves.
As you can see, we start out by assigning a date – November 1, 2006 – to a variable named dtmDate:
```dtmDate = #11/1/2006#
```
Note. Are the pound signs – the # characters – required when assigning a date to a variable? Not necessarily, but it’s a good idea. When VBScript sees the pound signs there’s no confusion; it knows that the value being assigned is supposed to be a date.
Oh, right: there’s a very good reason why we set the date to November first. After all, we want to determine the first Friday in the month; therefore we need to start with day 1, just in case the first day in November is a Friday. If it is, great. If it isn’t, then we’ll check to see if the second day in the month is a Friday. And so on. At least that’s the plan; let’s see if it works.
After assigning the date we then set up a Do Until loop that runs until a variable named x is equal to 1. Because x has never been assigned a value, its current value is 0; in fact, the value of x will always be a zero. In other words, we’ve set up a loop designed to run forever. But don’t panic: after all, in the scripting world forever doesn’t always take very long.
Note. What does that mean? Good question; hopefully we’ll figure it out ourselves before we reach the end of this column.
Inside the loop the very first thing we do is use the Weekday function to determine the integer value corresponding to the given date:
```intDayOfWeek = Weekday(dtmDate)
```
Depending on the day of the week, the Weekday function will return one of the following values:
Day of the Week Value Sunday 1 Monday 2 Tuesday 3 Wednesday 4 Thursday 5 Friday 6 Saturday 7
Because we’re interested in only Fridays we use this line of code to determine whether or not the Weekday value for 11/1/2006 (the current value of dtmDate) is equal to 6:
```If intDayOfWeek = 6 Then
```
Suppose the Weekday value for 11/1/2006 is 6 (it isn’t, but suppose it is). In that case we echo back the fact that the first Friday in November, 2006 is 11/1/2006 and then call the Exit Do statement to exit our not-so-endless loop:
```Wscript.Echo "The first Friday of the month is " & dtmDate & "."
Exit Do
```
Note. Yes, even though our loop is designed to run forever all we have to do is call Exit Do and we’ll immediately exit the thing. If only we had a similar function to help us exit those meetings that also seem designed to run forever.
That’s nice, but what if the Weekday value for 11/1/2006 isn’t equal to 6 (which it isn’t)? No problem; in that case we simply add 1 day to the date (making the value of dtmDate equal to 11/2/2006) and then loop around and use the Weekday function to test that date:
```dtmDate = dtmDate + 1
```
This process continues until we finally encounter a Friday. When that happens, we’ll get back a message similar to this, we’ll exit the loop, and the script will terminate:
```The first Friday of the month is 11/3/2006.
```
Nice.
By the way, if you want to get a head start on next year, here’s a script (we’ll leave it up to you to figure out exactly how it works) that reports back the first Friday in every month for the year 2007:
```For i = 1 to 12
dtmDate = CDate(i & "/1/2007")
Do Until x = 1
intDayOfWeek = Weekday(dtmDate)
If intDayOfWeek = 6 Then
Wscript.Echo "The first Friday of the month is " & dtmDate & "."
Exit Do
Else
dtmDate = dtmDate + 1
End If
Loop
Next
```
That’s going to give us back data that looks like this:
```The first Friday of the month is 1/5/2007.
The first Friday of the month is 2/2/2007.
The first Friday of the month is 3/2/2007.
The first Friday of the month is 4/6/2007.
The first Friday of the month is 5/4/2007.
The first Friday of the month is 6/1/2007.
The first Friday of the month is 7/6/2007.
The first Friday of the month is 8/3/2007.
The first Friday of the month is 9/7/2007.
The first Friday of the month is 10/5/2007.
The first Friday of the month is 11/2/2007.
The first Friday of the month is 12/7/2007.
```
The secret here is that we’ve added a second loop. We want to report data back for 12 months, so we set up a For Next loop that runs from 1 through 12. Then, we use the counter variable for that loop (which we named i) in a little formula that determines the first day in each month:
```dtmDate = CDate(i & "/1/2007")
```
As you can see, when i is equal to 1 we’ll be assigning dtmDate the value 1/1/2007; when i is equal to 2 (which it will be the second time through the loop) then we’ll be assigning dtmDate the value 2/1/2007. Etc.
Note. The function CDate? CDate is short for character-to-date, and is designed to transform a string value into a date-time value. Again, we’re just making sure that VBScript treats dtmDate as a date. And sure, we could have used CDate (instead of pound signs) when assigning the date in our first script. Why didn’t we? No reason, really; we just felt like using pound signs.
One last thing. If the Scripting Editor tries to slip a note in here saying that this column exaggerates the number of people who skip … lunch … on a Friday, well, look at it this way. Who are you going to trust when it comes to being lazy and weaseling out work: the Scripting Editor, or the Scripting Guy who writes this column? (Or, to put it a little more accurately, the Scripting Guy who writes this column if he doesn’t have to coach baseball or if he isn’t taking a week off or ….)
That’s what we thought.
Comments (1)
1. Derek Figueira says:
Here's a quick and dirty way of identifying the first Friday in a month.
#!/bin/ksh
TODAY=`date +"%a"` # Ddd eg: Mon Tue Wed Thu Fri Sat Sun
DAYNUMBER=`date +"%d"` # nn eg: 01 – 31
if [[ \$TODAY = "Fri" ]] && (( \$DAYNUMBER <= 07 ))
then echo "This is the first Frday of the month"
fi
exit
Skip to main content | 1,951 | 7,230 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.953125 | 3 | CC-MAIN-2018-47 | latest | en | 0.937419 |
https://www.doubtnut.com/qna/645733835 | 1,701,295,342,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100146.5/warc/CC-MAIN-20231129204528-20231129234528-00363.warc.gz | 849,781,651 | 37,772 | Doubtnut App - Get 50 Lakh+ video solutions better than GoogleInstall Now
# Find the area and circumference of a circle if the radius is 14 cm. (Take Π=227) एक वृत्त का क्षेत्रफल तथा परिधि ज्ञात करें यदि त्रिज्या 14 सेमी है |
A
Area = 616cm2, Circumference = 88 cm
B
Area = 308cm2, Circumference = 44 cm
C
Area = 44cm2, Circumference = 308 cm
D
Area = 88cm2, Circumference = 616 cm | 147 | 390 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.953125 | 3 | CC-MAIN-2023-50 | longest | en | 0.357122 |
https://discourse.datamethods.org/t/methods-to-calculate-the-posterior-distribution-of-the-absolute-risk-reduction/5182 | 1,656,449,518,000,000,000 | text/html | crawl-data/CC-MAIN-2022-27/segments/1656103617931.31/warc/CC-MAIN-20220628203615-20220628233615-00362.warc.gz | 264,527,983 | 9,082 | # Methods to calculate the posterior distribution of the absolute risk reduction
Hello. I have seen numerous examples of code used to calculate the posterior distribution of the absolute risk reduction. The simplest, using brms, seems to be having family=bernoulli(link=identity) when working with a nominal predicted value (outcome) and a single nominal predictor (treatment) ie formula = outcome ~ 1 + predictor. Does the parameter estimate and est error give you the posterior distribution of the ARR?
Thank you.
Covariate-specific absolute risk reduction (ARR) is obtained by subtracting risk evaluated at two covariate value (e.g., vary treatment but leave age constant). The model needs to be realistic, so a linear model is not appropriate here unless perhaps for the case where covariates never arise. Use a binary logistic model, and retrieve the posterior draws of all its parameters. For each posterior draw compute the two risks of interest, and subtract them. This leads to a posterior sample of the ARRs.
3 Likes
emmeans + brms + tidybayes is the perfect combo for this scenario: Logistic Regression: brms + emmeans + tidybayes Ā· GitHub
4 Likes
Thank you for the example code. I ran it using my data and came up with essentially the same median and credible interval for the ARR as when I used brms and specified family=bernoulli(link=identity). Thank you for your reply!
Beware all ye who stray from canonical link functions (in RCTs anyways): [2107.07278] Covariate adjustment in randomised trials: canonical link functions protect against model mis-specification. I assume emmeans with brms is doing what @f2harrell suggests but that is definitely the best way (and helps you to think of all the other great posterior transforms/summaries you can do).
1 Like
Iām guess that you are referring to the average ARR. Since ARR needs to be covariate-specific I think youāll find the identity link to be very problematic.
2 Likes
To take this question a little further, how would we change our approach when we have random effects (from e.g., logistic regression on longitudinal data)?
Should we:
1. condition on a given value of the random effects? Like a ātypicalā patient?
2. average over the random effects (marginal ARR)?
3. Something else?
I have read Prof Harrellās critique of marginal effects in other contexts (which was very convincing) but not specifically in the mixed effects context. Very interested to hear peoples thoughts (let me know if this needs a separate post). Thanks!
1 Like
Not to answer your question, but I believe in full conditioning on baseline information and only want to condition on subject-specific effects if Iām interested in estimating subject-specific outcomes. Otherwise I marginalize on subjects by directly modeling serial correlation structure without random effects.
2 Likes
Thank you again for the code you provided. I had 2 follow up questions.
The first, do you have a way of calculating the relative risk?
My second question is how to translate for my audience the prior probability distributions into a prior of the absolute risk reduction. For example if my skeptical prior for the logistic regression model was a normal distribution with a mean of 0 and SD of 0.4, my prior belief is that the absolute risk reduction was 0 but how do I relay what the SD translates to in terms of ARR (ie the prior distribution of the ARR has a mean of 0 and 95% of the distribution is between an ARR of -x% and x%).
Thank you.
Usually one places priors on the fundamental model parameters (\beta) then for each posterior draw and for each covariate setting of interest one computes the difference in two model estimates to get absolute risk reduction (likewise for RR) and computes the distribution of this over all posterior draws. Very easy to do. Using a model based on odds ratios to get RR and ARR is easy - see Avoiding One-Number Summaries of Treatment Effects for RCTs with Binary Outcomes | Statistical Thinking
2 Likes
There must be a way of doing it with emmeans (which I will look for), but here is a āmanualā solution:
b1 %>%
tidybayes::tidy_draws() %>%
dplyr::mutate(
ttr0_risk = plogis(b_Intercept),
ttr1_risk = plogis(b_Intercept + b_ht)
) %>%
dplyr::summarise(
relative_risk = ttr0_risk/ttr1_risk
)
Of note, this dataset doesnāt have any other covariates. In case your dataset has them, you must first generate posterior draws conditioning on covariates of interests, then calculate the relative risk based on those draws.
What do you think @f2harrell?
2 Likes
I donāt have a great answer for that, but please check this article and its supplementary material. They provide thorough details about their prior distributions in both relative and absolute scales: Dexamethasone 12Ā mg versus 6Ā mg for patients with COVID-19 and severe hypoxaemia: a pre-planned, secondary Bayesian analysis of the COVID STEROID 2 trial - Intensive Care Medicine
1 Like
I think this is it:
b1 %>%
emmeans::emmeans(~ ht,
trans = "log",
type = "response") %>%
emmeans::contrast(method = "pairwise")
1 Like
This is the question Iāve been stuck on for so longā¦ Lets say we adjusted for sex and age in the model. We could then calculate the conditional risk difference for a male aged 60 years like so:
b1 %>%
emmeans(revpairwise ~ treatment, transform = "response",
at = list(age = 60, sex = "male"))
Alternatively, we could calculate a marginal risk difference like so (untested so excuse errors):
# treated risk
new <- df
new$treatment <- "treated" risk_treated <- posterior_epred(b1, newdata = new) risk_treated <- colMeans(risk_treated) #untreated risk new <- df new$treatment <- "untreated"
risk_untreated <- posterior_epred(b1, newdata = new)
risk_untreated <- colMeans(post_untreated)
#risk difference
posterior_summary(risk_treated - risk_untreated)
The former is obviously a conditional risk difference for a very particular subject, while the latter is marginal in that it has averaged over the characteristics of the sample. What I really struggle to understand is when we should prefer one over the other.
Neither seem satisfactory. The problem with approach (1) is that it seems arbitrary to use these particular characteristics, while (2) seems problematic because it averages over sample characteristics which may be wildly unrepresentative of the population we wish to ultimately treat. Guidance would be greatly appreciatedā¦
1 Like
Marginal estimates cover up what is going on. For example in the NIH Remdesivir study the overall reduction in time to recovery quoted in the NEJM paper does not apply to anyone in the study since the difference varies so greatly over initial state (e.g., being on a ventilator at baseline). One can compute posterior distributions of differences for a series of covariate values. When treatment does not interact with covariates a beautiful result happens:
• when quantifying evidence for any efficacy (e.g., P(ARR > 0 | X)) this posterior probability will be the same for all covariate settings
• when quantifying evidence for non-trivial efficacy (e.g., P(ARR > 0.05 | X) the posterior probabilities will be covariate-dependent. For example, for a sicker patient at baseline you may see a higher absolute risk reduction than for a less sick patient.
2 Likes
Great question. One approach is to show summaries for āreference patientsā, as they did in this article
āwe primarily present average estimates and average treatment effects, secondarily supplemented with estimates calculated for three different representative reference patientsā
2 Likes
How should we specify the prior distribution for an absolute risk difference?
For example:
Group A prior is a Beta distribution (shape parameters: āŗ = 4, Ī² = 13) which attributes the most (mode) credibility to an expected event proportion of 0.2.
Group B prior is a Beta distribution (shape parameters: āŗ = 12, Ī² = 27) which attributes the most (mode) credibility to an expected event proportion of 0.3.
These priors directly influence the prior for the absolute risk difference between Group A and Group B. How would I derive the prior for the risk difference from the priors of these two risks.
Thank you.
Youād need to pick reference patients as @arthur_albuquerque described, as absolute risk difference is a dramatic function of baseline risk. We tend to not put priors on risk differences because we donāt know that much about them apriori. Instead it is more common to put a wide prior on a reference log odds and a narrower prior on a log odds ratio for comparing A and B.
2 Likes
Iād rather putting a more informative prior for the log odds based on high-quality observational/previous RCTs data. What do you think, Dr Harrell?
Sometimes I want to just put a skeptical prior on the log odds ratio. Sometimes high-quality data can inform, depending on what you mean by high-quality.
1 Like | 2,018 | 8,944 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.921875 | 3 | CC-MAIN-2022-27 | latest | en | 0.878046 |
https://help.altair.com/twinactivate/help/en_us/block_reference_guide/_doc/_mo/Blocks/Math/Sum.html | 1,716,377,700,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058542.48/warc/CC-MAIN-20240522101617-20240522131617-00872.warc.gz | 266,067,028 | 3,277 | # Sum
block Sum "Output the sum of the elements of the input vector"
extends Interfaces.MISO;
parameter Real k[nin] = ones(nin) "Optional: sum coefficients";
equation
y = k * u;
annotation (
defaultComponentName = "sum1",
Documentation(info = "<html>\n<p>\nThis blocks computes output <strong>y</strong> as\n<em>sum</em> of the elements of the input signal vector\n<strong>u</strong>:\n</p>\n<pre>\n <strong>y</strong> = <strong>u</strong>[1] + <strong>u</strong>[2] + ...;\n</pre>\n<p>\nExample:\n</p>\n<pre>\n parameter: nin = 3;\n\n results in the following equations:\n\n y = u[1] + u[2] + u[3];\n</pre>\n\n</html>"),
Icon(
coordinateSystem(
preserveAspectRatio = true,
extent = {
{-100, -100},
{100, 100}}),
graphics = {
Line(points = {
{26, 42},
{-34, 42},
{6, 2},
{-34, -38},
{26, -38}})}),
Diagram(
coordinateSystem(
preserveAspectRatio = true,
extent = {
{-100, -100},
{100, 100}}),
graphics = {
Rectangle(
extent = {
{-100, -100},
{100, 100}},
lineColor = {0, 0, 255},
fillColor = {255, 255, 255},
fillPattern = FillPattern.Solid),
Line(points = {
{26, 42},
{-34, 42},
{6, 2},
{-34, -38},
{26, -38}})}));
end Sum; | 389 | 1,140 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2024-22 | latest | en | 0.403386 |
https://www.globalsino.com/EM/page4548.html | 1,722,906,749,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640461318.24/warc/CC-MAIN-20240806001923-20240806031923-00617.warc.gz | 618,516,145 | 4,589 | Seven Crystal Families
- Practical Electron Microscopy and Database -
- An Online Book -
https://www.globalsino.com/EM/
This book (Practical Electron Microscopy and Database) is a reference for TEM and SEM students, operators, engineers, technicians, managers, and researchers.
=================================================================================
Space groups represent the ways that the macroscopic and microscopic symmetry elements (operations) can be self-consistently arranged in space. There are totally 230 space groups. The space groups add the centering information and microscopic elements to the point groups. Depending on their geometry, crystals are commonly classified into seven systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic. Figure 4548a schematically shows the relationship between the 7 crystal systems, 14 Bravais Lattices, 32 point groups, and 230 space groups. Table 4548a also lists the relation between three-dimensional crystal families, crystal systems, and lattice systems. Table 4548b also shows the seven crystal systems and the restrictions on their cell dimensions (cell edges and cell angles).
Figure 4548a. The relationship between the 7 crystal systems,
14 Bravais Lattices, 32 point groups, and 230 space groups.
Table 4548a. The relation between three-dimensional crystal families, crystal systems, and lattice systems.
Crystal
family
Crystal
system
Required
symmetries
of point group
Point
group
Space
group
Bravais
lattices
Lattice
system
Triclinic None 2 2 1 Triclinic
Monoclinic
1 two-fold axis of rotation or 1 mirror plane 3
13
2
Monoclinic
Orthorhombic
3 two-fold axes of rotation or 1 two-fold axis of rotation and two mirror planes 3 59 4
Orthorhombic
Tetragonal
1 four-fold axis of rotation
7
68
2
Tetragonal
Hexagonal Trigonal 1 three-fold axis of rotation 5 7 1 Rhombohedral
18 1 Hexagonal
Hexagonal 1 six-fold axis of rotation 7 27
Cubic
4 three-fold axes of rotation 5 36 3 Cubic
Total: 6 7 32 230 14 7
As indicated in Table 4548a, the 14 basic Bravais lattice types give rise to a total of 230 possible crystal types when they are combined with other symmetry operators such as mirrors planes, glide planes, and screw axes.
Table 4548b. The seven crystal systems (families) and the restrictions on their cell dimensions.
Crystal family Cell edges Cell angles
Triclinic None None
Monoclinic None α = γ = 90°
Orthorhombic None α = β = γ = 90°
Tetragonal a = b α = β = γ = 90°
Trigonal, hexagonal a = b α = β = 90°, γ = 120°
Cubic a = b = c α = β = γ = 90°
Figure 4548b shows the schematic illustration of the different classes of crystal systems and their properties.
Seven crystal systems 32 point groups No center of symmetry (21 point groups) Center of symmetry (11 point groups) Non-piezoelectric (1 point groups) Polar or pyroelectric (10 point groups) Non-pyroelectric Ferroelectric (Only if polarization Is reversible) Non-ferroelectric
Figure 4548b. Schematic illustration of the different classes of crystal systems and their properties.
=================================================================================
The book author (Yougui Liao) welcomes your comments, suggestions, and corrections, please click here for submission. If you let book author know once you have cited this book, the brief information of your publication will appear on the “Times Cited” page. | 841 | 3,414 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.640625 | 3 | CC-MAIN-2024-33 | latest | en | 0.803459 |
https://ch.mathworks.com/matlabcentral/cody/problems/44733-large-sum-inspired-by-project-euler-13/solutions/2070946 | 1,582,492,659,000,000,000 | text/html | crawl-data/CC-MAIN-2020-10/segments/1581875145839.51/warc/CC-MAIN-20200223185153-20200223215153-00070.warc.gz | 321,053,603 | 19,203 | Cody
Problem 44733. Large Sum (inspired by Project Euler 13)
Solution 2070946
Submitted on 27 Dec 2019 by David Hill
This solution is locked. To view this solution, you need to provide a solution of the same size or smaller.
Test Suite
Test Status Code Input and Output
1 Pass
assessFunctionAbsence({'regexp', 'regexpi'},'FileName','sum_large_n.m')
2 Pass
filetext = fileread('sum_large_n.m'); assert(isempty(strfind(filetext, 'java')),'java forbidden')
3 Pass
c = {'12345678'}; assert(isequal(sum_large_n(c),12345678))
4 Pass
c = {'1234567890','9'}; assert(isequal(sum_large_n(c),12345678))
5 Pass
c = {'11223344','11223344'}; assert(isequal(sum_large_n(c),22446688))
6 Pass
c = {'1000000000','99','1'}; assert(isequal(sum_large_n(c),10000001))
7 Pass
c = {'100000000000000000000000000000000000000000','9999','9999','9999'}; assert(isequal(sum_large_n(c),10000000))
8 Pass
c = {'15934672','34627951','63195472','98416599','13652729','32167958','32368197'}; assert(isequal(sum_large_n(c),29036357))
9 Pass
c = {'65281492489834938429841293654542962328498421794427152995741538492824984','37812654179574152749152791584279521794471529572419527149652719458479854'}; assert(isequal(sum_large_n(c),10309414))
10 Pass
c = {'64854985662353823234394299423463672233451381975635955356744918981347271658799472175596688623815297551711518872659685481224881454663419214254991734594937657622921687245928642452634633638974619883614322', ... '41657761135648795316841323455859693737713378487164915457385127524627393723167546676927326556746488366583329656565759211145476799227155854775426317347474774134328484748742893748728958622478835122752521', ... '86976889186955835968763616679976961285825616415222221635755525883266429112266197718998852421933356186845126392957934578124154229759177626322913141921351933131576117951843711765778376418467125598711189'}; assert(isequal(sum_large_n(c),19348963))
11 Pass
c = {'64854985662353823234394299423463672233451381975635955356744918981347271658799472175596688623815297551711518872659685481224881454663419214254991734594937657622921687245928642452634633638974619883614322', ... '41657761135648795316841323455859693737713378487164915457385127524627393723167546676927326556746488366583329656565759211145476799227155854775426317347474774134328484748742893748728958622478835122752521', ... '86976889186955835968763616679976961285825616415222221635755525883266429112266197718998852421933356186845126392957934578124154229759177626322913141921351933131576117951843711765778376418467125598711189', ... '72712636767379814476842172453652813351412836947746192385743174561377221146751622122233239762219763269234174946242784735864354467442133699537777175218226957295718725354423196929864373177764483888569418', ... '64936642142748762939341621746461987846653353891518919178589622211694499797463375467656517949485422378718971477337563287912152911673242596141549859249976492768478359536474579387914633766614537837282648', ... '55845919276746811963423332428194147857913127786198359784563756325488721639938968957615654988468972614798574998453889662952978376243739845786635787174428334456445697482112331512689695534518148468892396', ... '56852549677456267487454774798714651381817526276112148658191526178995315671652964965855743684243484442314334257381412122322699254531421949717765273498436996384915724234515333299872421376543577676254122', ... '74731539941789156776746219369224943244241616835615628985373966213897736849135176588835711359733896318691638244872327622595118548457836916433792829847584169247197625598974953476799592136595457182534623', ... '22385429486488589446974795531215196112521748441116388952357456927648426638992316148542492581793279692744286781957782569896681841722633444622137377883353886373545979351768994152231775485994546872814784', ... '79323439162825941244397949168397868333683486928735468188446777449683669155198482635823548923749968469697568249469961166976539517519332336133847682981536366651541555878125985417524274918656718132916668'}; assert(isequal(sum_large_n(c),62027779))
12 Pass
c = {'6475498566235382323439429942346367223345138697688918695583596876361667997696128582561641522222163575552588326642911226619771899885242193335618684512639295793457812415422975917762632291314192135193313157611795184371176577837641846712559871118981975635955356744918981347271658799472175596688623815297551711518872659685481224881454663419214254991734594937657622921687245928642452634633638974619883614322', ... '2165776113564879531684132345585969373771337848716491545738512752462739372316754667692732655674648836658332965656575921114547679922715585477542793234391628259412443979491683978683336834869287354681884467774496836691551984826358235489237499684696975682494699611669765395175193323361338476829815363666515415558781259854175242749186567181329166686317347474774134328484748742893748728958622478835122752521', ... '8697688918695583596876361667997696128582561641522222163575552588326642911226619771899885242193335618684512635685254967745626748745477479871465138181752627611214865819152617899531567165296496585574368424348444231433425738141212232269925453142194971776527349843699638491572423451533329987242137654357767625412292957934578124154229759177626322913141921351933131576117951843711765778376418467125598711189', ... '7271263676737981447684217245365281335141283694774619238574317456137722114675162212223323976221976326923417494624278473586435446744213369953777717521822695729571872535442319692986437317776445584591927674681196342333242819414785791312778619835978456375632548872163993896895761565498846897261479857499845388966295297837624373984578663578717442833445644569748211233151268969553451814846889239683888569418', ... '6293664214274876293934162174646198784665335389151891917858962221169449979746337546765651794948542237871897147733756328791215291167324259614154985568525496774562674874547747987146513818175262761121486581915261789953156716529649658557436842434844423143342573814121223226992545314219497177652734984369963849157242345153332998724213765435776762541229249976492768478359536474579387914633766614537837282648', ... '5584591927674681196342333231512689695428194147857913127786198359784563737624373984578663575632548872163993896895761565857499845388949884689726131512689695479857499845388966295297837624373984578632571235635787174428334456445697482112331512689695388949884689761479857499845388966295297837624373984578663578717442833445644569748211233151268969553451814846889239687232762259511854845785534518148468892396', ... '5685254967745626748745477479871465138181752627611214865819152617899531567165296496585574368424348444231433425738793234391628259412443979491683978683336834869287354681884467774496836691551984826358235489237499684696975682494699611669765395175193323361338476829815363666515415558781259854175242749186567181329166681412122322699254531421949717765273498436996384915724234515333299872421376543577676254122', ... '7873153994178915677674621936922494324424161683561562898537396621389773684913517658883571135973389631869163824455845919276746811963423332428194147857913127786198359784563756325488721639938968957615654988468972614798574998453889662952978376243739845786635787174428334456445697482112331512689695534518148468892396872327622595118548457836916433792829847584169247197625598974953476799592136595457182534623', ... '2238542948648858944697479553121519611252174844111638895235745692764842663899231614854249258179327969274428678195778256989668184172263355845919276746811963423332428194147857913127786198359784563756325488721639938968957615654988468972614798574998453889662952978376243739845786635787174428334456445697482112331512689695534518148468892396444622137377883353886373545979351768994152231775485994546872814784', ... '7392343916282594124439794916839786833368348692873546818844677744968366915519848263582354892374996846969756824946996116697653951751933233613384768298153636665154155587815584591927674681196342333242819414785791312778619835978456375632548872163993896895761565498846897261479857499845388966295297837624373984578663578717442833445644569748211233151268969553451814846889239625985417524274918656718132916668'}; assert(isequal(sum_large_n(c),59677779))
13 Pass
c = {'69754985662353823234394299423463672233451386976889186955835968763616679976961285825616415222221635755525883266429112266197718998852421933356186845126392957934578124154229759177626322913141921351933131576117951843711765778376418467125598711189819756359553567449189813472716587994721755966886238152975517115188726596854812248814546634192142549917345949376576229223854294864885894469747955312151961125217484411163889523574569276484266389923161485424925817932796927442867819577825698966818417226335584591927674681196342333242819414785791312778619835978456375632548872163993896895761565498846897261479857499845388966295297837624373984578663578717442833445644569748211233151268969553451814846889239644462213737788335388637354597935176899415223177548599454687281478421687245928642452634633638974619883614322', ... '51057761135648795316841323455859693737713378487164915457385127524627393723167546676927326556746488366583329656565759211145476799227155854775427932343916282594124439794916839786833368348692873546818844677744968366915519848263582354892374996846969756824946996116697653951751933233613384768298153636665154155587812598541752427491865671813291686976889186955835968763616679976961285825616415222221635755525883266429112266197718998852421933356186845126356852549677456267487454774798714651381817526276112148658191526178995315671652964965855743684243484442314334257381412122322699254531421949717765273498436996384915724234515333299872421376543577676254122929579345781241542297591776263229131419213519331315761179518437117657783764184671255987111896686317347474774134328484748742893748728958622478835122752521', ... '76976889186955835968763616679976961285825616415222221635755525883266429112266197718998852421933356186845126356852549677456267487454774798714651381817526276112148658191526178995315671652964965855743684243484442314334257381412122322699254531421949717765273498436996384915724234515333299872421376543577676254122929579345781241542297591776263229131419213519331315761179518437117657783767392343916282594124439794916839786833368348692873546818844677744968366915519848263582354892374996846969756824946996116697653951751933233613384768298153636665154155587815584591927674681196342333242819414785791312778619835978456375632548872163993896895761565498846897261479857499845388966295297837624373984578663578717442833445644569748211233151268969553451814846889239625985417524274918656718132916668418467125598711189', ... '72712636767379814476842172453652813351412836947746192385743174561377221146751622122233239762219763269234174946242784735864354467442133699537777175218226957295718725354423196929864373177764455845919276746811963423332428194147857913127786198359784563756325488721639938968957615654988468972614798574998453889662952978376243739845786635787174428334456445697482112331512689695534586976889186955835968763616679976961285825616415222221635755525883266429112266197718998852421933356186845126356852549677456267487454774798714651381817526276112148658191526178995315671652964965855743684243484442314334257381412122322699254531421949717765273498436996384915724234515333299872421376543577676254122929579345781241542297591776263229131419213519331315761179518437117657783764184671255987111891814846889239683888569418', ... '62936642142748762939341621746461987846653353891518919178589622211694499797463375467656517949485422378718971477337563287912152911673242596141549855685254967745626748745477479871465138181752627611214865819152617899531567165296496585574368424348444231433425738141212232269925453142194971776527349843699638491572423451533568525496774562674874547747987146513818175262761121486581915261789953156716529649658557436842434844423143342573879323439162825941244397949168397868333683486928735468188446777449683669155198482635823548923749968469697568249469961166976539517519332336133847682981536366651541555878125985417524274918656718132916668141212232269925453142194971776527349843699638491572423451533329987242137654357767625412232998724213765435776762541229249976492768478359536474579387914633766614537837282648', ... '55845919276746811963423332315126896954281941478579131277861983597845637376243739845786635756325488721639938968957615658574998453889498846897261315126896954798574998453889662952978376243739845786325712356357871744283344564456974821123315126896953889498846897614798574998453889662952978376243739845786635787174428334456445697482112331512689695534518148468892396872327622595118548457855345181484688956852549677456267487454774798714651381817526276112148658191526178995315671652964965855743684243484442314334257387932343916282594124439794916839786833368348692873546818844677744968366915519848263582354892374996846969756824946996116697653951751933233613384768298153636665154155587812598541752427491865671813291666814121223226992545314219497177652734984369963849157242345153332998724213765435776762541222396', ... '50852549677456267487454774798714651381817526276112148658191526178995315671652964965855743684243484442314334257387932343916282594124439794916839786833368348692873546818844677744968366915519848263582354892374996846969756824946996116697653951751933233613384768298153636665154155587812598541752427491865671813291666814121223226992545314219497177652734984369963849157242345153332998724213765435755845919276746811963423332315126896954281941478579131277861983597845637376243739845786635756325488721639938968957615658574998453889498846897261315126896954798574998453889662952978376243739845786325712356357871744283344564456974821123315126896953889498846897614798574998453889662952978376243739845786635787174428334456445697482112331512689695534518148468892396872327622595118548457855345181484688923967676254122', ... '18731539941789156776746219369224943244241616835615628985373966213897736849135176588835711359733896318691638244558459192767468119634233324281941478579131277861983597845637563254887216399389689576156549884689726147985749984538896629529783762437398457866357871744283344564456974821123315126896955345181484688923968726293664214274876293934162174646198784665335389151891917858962221169449979746337546765651794948542237871897147733756328791215291167324259614154985568525496774562674874547747987146513818175262761121486581915261789953156716529649658557436842434844423143342573814121223226992545314219497177652734984369963849157242345153332998724213765435776762541229249976492768478359536474579387914633766614537837282648327622595118548457836916433792829847584169247197625598974953476799592136595457182534623', ... '24185429486488589446974795531215196112521748441116388952357456927648426638992316148542492581793279692744286781957782569896681841722633558459192767468119634233324281941478579131277861983597845637563254887216399389689576156549884689726147985749984538896629529783762437398457866357871744283344564456974821123315126896955345181484688923964446221373778833538863735464754985662353823234394299423463672233451386976889186955835968763616679976961285825616415222221635755525883266429112266197718998852421933356186845126392957934578124154229759177626322913141921351933131576117951843711765778376418467125598711189819756359553567449189813472716587994721755966886238152975517115188726596854812248814546634192142549917345949376576229216872459286424526346336389746198836143225979351768994152231775485994546872814784', ... '73923439162825941244397949168397868333683486928735468188446777449683669155198482635823548923749968469697568249469961166976539517519332336133847682981536366651541555878155845919276746811963423332428194147857913127786198359784563756325488721639938968957615654988468972614798574998453889662952978376243739845786635787174428334456445697482112356852549677456267487454774798714651381817526276112148658191526178995315671652964965855743684243484442314334257387932343916282594124439794916839786833368348692873546818844677744968366915519848263582354892374996846969756824946996116697653951751933233613384768298153636665154155587812598541752427491865671813291666814121223226992545314219497177652734984369963849157242345153332998724213765435776762541223151268969553451814846889239625985417524274918656718132916668'}; assert(isequal(sum_large_n(c),55697779))
14 Pass
c = {'169754985662353823234394299423463672233451386976889186955835968763616679976961285825616415222221635755525883266429112266197718998852421933356186845126392957934578124154229759177626322913141921351933131576117951843711765778376418467125598711189819756359553567449189813472716587994721755966886238152975517115188726596854812248814546634192142549917345949376576229223854294864885894469747955312151961125217484411163889523574569276484266389923161485424925817932796927442867819577825698966818417226335584591927674681196342333242819414785791312778619835978456375632548872163993896895761565498846897261479857499845388966295297837624373984578663578717442833445644569748211233151268969553451814846889239644462213737788335388637354597935176899415223177548599454687281478421687245928642452634633638974619883614322', ... '51057761135648795316841323455859693737713378487164915457385127524627393723167546676927326556746488366583329656565759211145476799227155854775427932343916282594124439794916839786833368348692873546818844677744968366915519848263582354892374996846969756824946996116697653951751933233613384768298153636665154155587812598541752427491865671813291686976889186955835968763616679976961285825616415222221635755525883266429112266197718998852421933356186845126356852549677456267487454774798714651381817526276112148658191526178995315671652964965855743684243484442314334257381412122322699254531421949717765273498436996384915724234515333299872421376543577676254122929579345781241542297591776263229131419213519331315761179518437117657783764184671255987111896686317347474774134328484748742893748728958622478835122752521', ... '476976889186955835968763616679976961285825616415222221635755525883266429112266197718998852421933356186845126356852549677456267487454774798714651381817526276112148658191526178995315671652964965855743684243484442314334257381412122322699254531421949717765273498436996384915724234515333299872421376543577676254122929579345781241542297591776263229131419213519331315761179518437117657783767392343916282594124439794916839786833368348692873546818844677744968366915519848263582354892374996846969756824946996116697653951751933233613384768298153636665154155587815584591927674681196342333242819414785791312778619835978456375632548872163993896895761565498846897261479857499845388966295297837624373984578663578717442833445644569748211233151268969553451814846889239625985417524274918656718132916668418467125598711189', ... '72712636767379814476842172453652813351412836947746192385743174561377221146751622122233239762219763269234174946242784735864354467442133699537777175218226957295718725354423196929864373177764455845919276746811963423332428194147857913127786198359784563756325488721639938968957615654988468972614798574998453889662952978376243739845786635787174428334456445697482112331512689695534586976889186955835968763616679976961285825616415222221635755525883266429112266197718998852421933356186845126356852549677456267487454774798714651381817526276112148658191526178995315671652964965855743684243484442314334257381412122322699254531421949717765273498436996384915724234515333299872421376543577676254122929579345781241542297591776263229131419213519331315761179518437117657783764184671255987111891814846889239683888569418', ... '562936642142748762939341621746461987846653353891518919178589622211694499797463375467656517949485422378718971477337563287912152911673242596141549855685254967745626748745477479871465138181752627611214865819152617899531567165296496585574368424348444231433425738141212232269925453142194971776527349843699638491572423451533568525496774562674874547747987146513818175262761121486581915261789953156716529649658557436842434844423143342573879323439162825941244397949168397868333683486928735468188446777449683669155198482635823548923749968469697568249469961166976539517519332336133847682981536366651541555878125985417524274918656718132916668141212232269925453142194971776527349843699638491572423451533329987242137654357767625412232998724213765435776762541229249976492768478359536474579387914633766614537837282648', ... '755845919276746811963423332315126896954281941478579131277861983597845637376243739845786635756325488721639938968957615658574998453889498846897261315126896954798574998453889662952978376243739845786325712356357871744283344564456974821123315126896953889498846897614798574998453889662952978376243739845786635787174428334456445697482112331512689695534518148468892396872327622595118548457855345181484688956852549677456267487454774798714651381817526276112148658191526178995315671652964965855743684243484442314334257387932343916282594124439794916839786833368348692873546818844677744968366915519848263582354892374996846969756824946996116697653951751933233613384768298153636665154155587812598541752427491865671813291666814121223226992545314219497177652734984369963849157242345153332998724213765435776762541222396', ... '50852549677456267487454774798714651381817526276112148658191526178995315671652964965855743684243484442314334257387932343916282594124439794916839786833368348692873546818844677744968366915519848263582354892374996846969756824946996116697653951751933233613384768298153636665154155587812598541752427491865671813291666814121223226992545314219497177652734984369963849157242345153332998724213765435755845919276746811963423332315126896954281941478579131277861983597845637376243739845786635756325488721639938968957615658574998453889498846897261315126896954798574998453889662952978376243739845786325712356357871744283344564456974821123315126896953889498846897614798574998453889662952978376243739845786635787174428334456445697482112331512689695534518148468892396872327622595118548457855345181484688923967676254122', ... '8731539941789156776746219369224943244241616835615628985373966213897736849135176588835711359733896318691638244558459192767468119634233324281941478579131277861983597845637563254887216399389689576156549884689726147985749984538896629529783762437398457866357871744283344564456974821123315126896955345181484688923968726293664214274876293934162174646198784665335389151891917858962221169449979746337546765651794948542237871897147733756328791215291167324259614154985568525496774562674874547747987146513818175262761121486581915261789953156716529649658557436842434844423143342573814121223226992545314219497177652734984369963849157242345153332998724213765435776762541229249976492768478359536474579387914633766614537837282648327622595118548457836916433792829847584169247197625598974953476799592136595457182534623', ... '24185429486488589446974795531215196112521748441116388952357456927648426638992316148542492581793279692744286781957782569896681841722633558459192767468119634233324281941478579131277861983597845637563254887216399389689576156549884689726147985749984538896629529783762437398457866357871744283344564456974821123315126896955345181484688923964446221373778833538863735464754985662353823234394299423463672233451386976889186955835968763616679976961285825616415222221635755525883266429112266197718998852421933356186845126392957934578124154229759177626322913141921351933131576117951843711765778376418467125598711189819756359553567449189813472716587994721755966886238152975517115188726596854812248814546634192142549917345949376576229216872459286424526346336389746198836143225979351768994152231775485994546872814784', ... '973923439162825941244397949168397868333683486928735468188446777449683669155198482635823548923749968469697568249469961166976539517519332336133847682981536366651541555878155845919276746811963423332428194147857913127786198359784563756325488721639938968957615654988468972614798574998453889662952978376243739845786635787174428334456445697482112356852549677456267487454774798714651381817526276112148658191526178995315671652964965855743684243484442314334257387932343916282594124439794916839786833368348692873546818844677744968366915519848263582354892374996846969756824946996116697653951751933233613384768298153636665154155587812598541752427491865671813291666814121223226992545314219497177652734984369963849157242345153332998724213765435776762541223151268969553451814846889239625985417524274918656718132916668'}; assert(isequal(sum_large_n(c),31469777))
15 Pass
c = {'16975498566235382323439429942346367223345138697688', ... '91869558359687636166799769612858256164152222216357', ... '55525883266429112266197718998852421933356186845126', ... '39295793457812415422975917762632291314192135193313', ... '15761179518437117657783764184671255987111898197563', ... '59553567449189813472716587994721755966886238152975', ... '51711518872659685481224881454663419214254991734594', ... '93765762292238542948648858944697479553121519611252', ... '17484411163889523574569276484266389923161485424925', ... '81793279692744286781957782569896681841722633558459', ... '19276746811963423332428194147857913127786198359784', ... '56375632548872163993896895761565498846897261479857', ... '49984538896629529783762437398457866357871744283344', ... '56445697482112331512689695534518148468892396444622', ... '13737788335388637354597935176899415223177548599454', ... '68728147842168724592864245263463363897461988361432', ... '51057761135648795316841323455859693737713378487164', ... '91545738512752462739372316754667692732655674648836', ... '65833296565657592111454767992271558547754279323439', ... '16282594124439794916839786833368348692873546818844', ... '67774496836691551984826358235489237499684696975682', ... '49469961166976539517519332336133847682981536366651', ... '54155587812598541752427491865671813291686976889186', ... '95583596876361667997696128582561641522222163575552', ... '58832664291122661977189988524219333561868451263568', ... '52549677456267487454774798714651381817526276112148', ... '65819152617899531567165296496585574368424348444231', ... '43342573814121223226992545314219497177652734984369', ... '96384915724234515333299872421376543577676254122929', ... '57934578124154229759177626322913141921351933131576', ... '11795184371176577837641846712559871118966863173474', ... '74774134328484748742893748728958622478835122752521', ... '47697688918695583596876361667997696128582561641522', ... '22216357555258832664291122661977189988524219333561', ... '86845126356852549677456267487454774798714651381817', ... '52627611214865819152617899531567165296496585574368', ... '42434844423143342573814121223226992545314219497177', ... '65273498436996384915724234515333299872421376543577', ... '67625412292957934578124154229759177626322913141921', ... '35193313157611795184371176577837673923439162825941', ... '24439794916839786833368348692873546818844677744968', ... '36691551984826358235489237499684696975682494699611', ... '66976539517519332336133847682981536366651541555878', ... '15584591927674681196342333242819414785791312778619', ... '83597845637563254887216399389689576156549884689726', ... '14798574998453889662952978376243739845786635787174', ... '42833445644569748211233151268969553451814846889239', ... '62598541752427491865671813291666841846712559871189', ... '72712636767379814476842172453652813351412836947746', ... '19238574317456137722114675162212223323976221976326', ... '92341749462427847358643544674421336995377771752182', ... '26957295718725354423196929864373177764455845919276', ... '74681196342333242819414785791312778619835978456375', ... '63254887216399389689576156549884689726147985749984', ... '53889662952978376243739845786635787174428334456445', ... '69748211233151268969553458697688918695583596876361', ... '66799769612858256164152222216357555258832664291122', ... '66197718998852421933356186845126356852549677456267', ... '48745477479871465138181752627611214865819152617899', ... '53156716529649658557436842434844423143342573814121', ... '22322699254531421949717765273498436996384915724234', ... '51533329987242137654357767625412292957934578124154', ... '22975917762632291314192135193313157611795184371176', ... '57783764184671255987111891814846889239683888569418', ... '56293664214274876293934162174646198784665335389151', ... '89191785896222116944997974633754676565179494854223', ... '78718971477337563287912152911673242596141549855685', ... '25496774562674874547747987146513818175262761121486', ... '58191526178995315671652964965855743684243484442314', ... '33425738141212232269925453142194971776527349843699', ... '63849157242345153356852549677456267487454774798714', ... '65138181752627611214865819152617899531567165296496', ... '58557436842434844423143342573879323439162825941244', ... '39794916839786833368348692873546818844677744968366', ... '91551984826358235489237499684696975682494699611669', ... '76539517519332336133847682981536366651541555878125', ... '98541752427491865671813291666814121223226992545314', ... '21949717765273498436996384915724234515333299872421', ... '37654357767625412232998724213765435776762541229249', ... '97649276847835953647457987914633766614537837282648', ... '75584591927674681196342333231512689695428194147857', ... '91312778619835978456373762437398457866357563254887', ... '21639938968957615658574998453889498846897261315126', ... '89695479857499845388966295297837624373984578632571', ... '23563578717442833445644569748211233151268969538894', ... '98846897614798574998453889662952978376243739845786', ... '63578717442833445644569748211233151268969553451814', ... '84688923968723276225951185484578553451814846889568', ... '52549677456267487454774798714651381817526276112148', ... '65819152617899531567165296496585574368424348444231', ... '43342573879323439162825941244397949168397868333683', ... '48692873546818844677744968366915519848263582354892', ... '37499684696975682494699611669765395175193323361338', ... '47682981536366651541555878125985417524274918656718', ... '13291666814121223226992545314219497177652734984369', ... '96384915724234515333299872421376543577676541222396', ... '50852549677456267487454774798714651381817526276112', ... '14865819152617899531567165296496585574368424348444', ... '23143342573879323439162825941244397949168397868333', ... '68348692873546818844677744968366915519848263582354', ... '89237499684696975682494699611669765395175193323361', ... '33847682981536366651541555878125985417524274918656', ... '71813291666814121223226992545314219497177652734984', ... '36996384915724234515333299872421376543575584591927', ... '67468119634233323151268969542819414785791312778619', ... '83597845637376243739845786635756325488721639938968', ... '95761565857499845388949884689726131512689695479857', ... '49984538896629529783762437398457863257123563578717', ... '44283344564456974821123315126896953889498846897614', ... '79857499845388966295297837624373984578663578717442', ... '83344564456974821123315126896955345181484688923968', ... '72327622595118548457855345181484688923967676254122', ... '87315399417891567767462193692249432442416168356156', ... '28985373966213897736849135176588835711359733896318', ... '69163824455845919276746811963423332428194147857913', ... '12778619835978456375632548872163993896895761565498', ... '84689726147985749984538896629529783762437398457866', ... '35787174428334456445697482112331512689695534518148', ... '46889239687262936642142748762939341621746461987846', ... '65335389151891917858962221169449979746337546765651', ... '79494854223787189714773375632879121529116732425961', ... '41549855685254967745626748745477479871465138181752', ... '62761121486581915261789953156716529649658557436842', ... '43484442314334257381412122322699254531421949717765', ... '27349843699638491572423451533329987242137654357767', ... '62541229249976492768478359536474579387914633766614', ... '53783728264832762259511854845783691643379282984758', ... '41692471976255989749534767995921365954570182534623', ... '24185429486488589446974795531215196112521748441116', ... '38895235745692764842663899231614854249258179327969', ... '27442867819577825698966818417226335584591927674681', ... '19634233324281941478579131277861983597845637563254', ... '88721639938968957615654988468972614798574998453889', ... '66295297837624373984578663578717442833445644569748', ... '21123315126896955345181484688923964446221373778833', ... '53886373546475498566235382323439429942346367223345', ... '13869768891869558359687636166799769612858256164152', ... '22221635755525883266429112266197718998852421933356', ... '18684512639295793457812415422975917762632291314192', ... '13519331315761179518437117657783764184671255987111', ... '89819756359553567449189813472716587994721755966886', ... '23815297551711518872659685481224881454663419214254', ... '99173459493765762292168724592864245263463363897461', ... '98836143225979351768994152231775485994546872814784', ... '97392343916282594124439794916839786833368348692873', ... '54681884467774496836691551984826358235489237499684', ... '69697568249469961166976539517519332336133847682981', ... '53636665154155587815584591927674681196342333242819', ... '41478579131277861983597845637563254887216399389689', ... '57615654988468972614798574998453889662952978376243', ... '73984578663578717442833445644569748211235685254967', ... '74562674874547747987146513818175262761121486581915', ... '26178995315671652964965855743684243484442314334257', ... '38793234391628259412443979491683978683336834869287', ... '35468188446777449683669155198482635823548923749968', ... '46969756824946996116697653951751933233613384768298', ... '15363666515415558781259854175242749186567181329166', ... '68141212232269925453142194971776527349843699638491', ... '57242345153332998724213765435776762541223151268969', ... '55345181484688923962585417524274918656718132916668'}; assert(isequal(sum_large_n(c),87139239)) | 11,397 | 33,784 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.125 | 3 | CC-MAIN-2020-10 | latest | en | 0.444756 |
https://www.scribd.com/doc/170821542/QA-Book | 1,561,436,734,000,000,000 | text/html | crawl-data/CC-MAIN-2019-26/segments/1560627999787.0/warc/CC-MAIN-20190625031825-20190625053825-00490.warc.gz | 879,054,799 | 175,150 | You are on page 1of 130
# CONTENTS
Averages Allegation & Mixtures Ratio, Proportion & Variation Percentages Profit & Loss Interests & Instalments Time and Work WORKSHOP
3 8 11 22 30 38 41 51
AVERAGES
Introduction: In IIM-CAT no question is directly asked from this chapter, but in other Management Entrance exams this chapter plays a crucial role. However, this chapter is very important to understand the concepts of Data Interpretation. 5. If each quantity of the given data is divided by 1 then the new average is the product of the old K 1 average by K NOTE: If the given set of values is in arithmetic progression then the average of the data is simply the average of the lowest and highest values. 6. If the average age of n numbers of family is X years, then K years back the average of the family is (X-K) years. 7. If the average age of n numbers of family is X years, then K later/after the average of the family is (X+K) years. 8. Average of first n natural numbers n 1 = 2 9. Average of first n even numbers = n 1 10. Average of fist n odd numbers = n 11. Concept of Weighted Average If the number of elements in n different groups be
Concept of Average: In general average is the Central value of the given set of values. Formula: The average is the arithmetic mean of the given data. If X1, X2, X3 ... Xn are n quantities, then the average of these n quantities.
X 1 X 2 X 3 ... XN N
PROPERTIES OF AVERAGES 1. The average of two or more quantities always lies between the lowest and highest values of the given data. If X1, X2, X3 ... Xn are N quantities, then the average of these n quantities always lies between
## XL< X 1 X 2 X 3 ... XN <XH
N
2. If each quantity of the given data is increased by K then the new average is increased by K 3. If each quantity of the given data is decreased by K then the new average is decreased by K 4. If each quantity of the given data is multiplied by K then the new average is the product of old average with K
K1, K2, K3, K4, K5..Kn and the averages of the respective groups are A1, A2, A3, A4, A5..Anthen the weighted average is:
K1A1 K 2 A2 ... KnAn K1 K 2 .. Kn
EXERCISE: Concepts Review 1. The average of 5 consecutive odd numbers a, b, c, d and e is abcde bd a) b) 5 3 ace c) d) none of these 5
2 5 3 8 7 The average of is: 2 , 3 , 4 , 6 , 7 3 9 5 9 15 3 8 a) 5 b) 5 225 225 3 8 c) 6 d) 25 45 45
a) 49.50 c) 55 9.
## b) 50.50 d) none of these
The average of all the elements of B is: a) 52 b) 48 c) 49 d) none of these If the average of the set A is 42.46, then the average of the Set C is: a) 52 b) 49.87 c) 55.40 d) cannot be determined The average of the elements of the set A and C combined is: a) 49.0588 b) 49.0372 c) 50 d) none of these If an element less than 50 belongs to Set A is transferred to set B, then the average of set B: a) Increases b) Decreases c) Remains constant d) Data insufficient If any two elements, greater than 50, belong to set A are transferred to Set C, then the average of Set C: a) Remains constant b) Decreases c) Increases d) Data insufficient The average length of first 3 fingers is 3 inches and the average length of the other 2 fingers i.e, thumb and the index fingers is 2.8 inches. If the length of the index fingers is 3 inches then the length of thumb is a) 2 inches b) 2.6 inches c) 3 inches d) none of these The average of 9 numbers is 10. If each of these 9 numbers is multiplied by 5 and then 5 is added to each number, then the new average is: a) 20 b) 30 c) 60 d) 55
10.
2.
11.
3.
The average of 1000.0001, 100.001, 10.01, 1.1 is: a) 277.777 b) 322.222 c) 11.11 d) 233.333 The average of first 100 natural numbers is: a) 50.5 b) 55 c) 51 d) 101 The average of first 99 even numbers is: a) 9999 b) 100 c) 9801 d) 99 The average of all the positive prime and composite numbers up to 100 is a) 51 b) 49.50 c) 50.50 d) 55 The average of all the non-negative integers up to 99 is a) 50.49 b) 49.50 c) 50.50 d) 99
12.
4.
13.
5.
14.
6.
7.
15.
Directions for questions 8 to 13: Set A = {2, 3, 5, 7, 11, ------- 89, 97} Set B = {4, 6, 8, 10, 12, ----- 98, 100} Set C = {1, 9, 15, 21, 25, ----- 95, 99} 8. The average of all the elements of A, B and C is :
16.
In an office the average age of n employees is 42 years. But after the verification it was found that the age of an employee had been considered 20 years less than the actual age. After the correction the average increased by 1. The value of n is: a) 21 b) 20 c) 22 d) None of these The average rainfall in the months of January and February is 6 cm, from March to June is 5 cm and from July to December is 6 cm. What is the average rainfall for the whole year? a) 5.66cm b) 5.5cm c) 5.33cm d) None of these The average weight of 11 players of Indian cricket team is increased by 1 kg when one player of the team weighing 55 kg replaced by a new player. The Weight of the new player is a) 55 kg b) 65 kg c) 66 kg d) none of these The average age of a family of 6 members 4 years ago was 25 years. Meanwhile a child was born but the average age remains same today. The present age of the child is: a) 2years b) 1 years c) 1 year d) data insufficient The average price of 3 diamonds weighing same is rupees 50 million. The average price of the two costliest diamonds is double the price of the cheapest diamond. The price of the cheapest diamond is: a) 30 million b) 25 million c) 16.6 million d) can't be determined The average of 3 consecutive natural numbers is k. If two more consecutive numbers, just next to the first set of numbers, is added, then the new average becomes: a) k-2 b) k+1 c) k-1 d) Either a) or b)
22.
At the end of the first round of Poker A won 50 ten million coins, B has 10 coins of 50 paise denominations, C has 20 coins of 25 paise denominations and D has 25 coins of 20 paise denominations. The average number of paise per person is: a) 450 paise b) 500 paise c) 600 paise d) Can't be determined A person jags along the hexagonal path of each side 20 metres in such a way that for the first 20 metres he goes with a speed of 40m/s and the next 20 metres with a speed of 20m/s. Similarly he continues for the rest of the hexagonal path with the same alternating speeds i.e. 40m/s and 20m/s. The average speed of the artist per round of the circus is: a) 26.66 m/s b) 30 m/s c) 23.33 m/s d) 33.33 m/s
17.
23.
18.
19.
20)
21.
EXERCISE: Application of Concepts 1. The average weight of a class of 20 students is 45 kgs. A new student whose weight is 40 kgs replaces and old student. Hence, the average weight of the whole class decreases by 1kg. Th weight of the replaced student is: a) 55 kgs b) 50 kgs c) 60 kgs d) none of these
7.
average salary of S be twice that of P, then the average salary of Q and R is (in Rs.): a) 16,000 b) 36,000 c) 27,000 d) 18,000 The average price of 80 laptops is Rs. 30,000. If the highest and lowest priced laptops are sold out then the average price of the remaining 78 laptops is Rs.29,500 The cost of the highest priced laptop is Rs. 80,000. Then the cost of lowest priced laptop is: a) Rs. 19,000 b) Rs. 20,000 c) Rs. 29,000 d) cannot be determined A train covers a certain distance at a speed of 60 km/hr. However, if it were to halt for a fixed time interval in each hour its average speed reduces to 50km/hr. How many minutes per hour does it stop? a) 10 minutes b) 20 minutes c) 6 minutes d) 12 minutes 123 students appeared for Pre-CAT and the average score obtained was 120. If the scores of top three rankers were not considered, the new average score decreased by 1.5. Marks of all the students were in integers and the scores of the toppers were distinct. If the second topper scored more than 185 marks, then the highest possible score of the third topper was: a) 166 b) 167 c) 168 d) 170 In a particular week the average number of people who visited the museum is 40. If we exclude the holidays then the average is increased by 16. Further if we exclude also the day on which the maximum number of 112 people visited the museum, then the average becomes 42. The number of holidays in the week is: a) 1 b) 2 c) 3 d) data insufficient The total age of all the players of a team was 540 years. If two players were absent for the practice session, then the average of the remaining players still remained unchanged,
6
8. 2. The average of 9 numbers is 11. If each of these 9 numbers is multiplied by 5 and then 5 is added to each of these resultant numbers, then the new average is: a) 20 b) 30 c) 60 d) 50 The average of 30 students of a class is 30 years. If the age of the class teacher is also included, the average age of the whole class increases by 1 year. The age of the class teacher is: a) 31 years b) 60 years c) 61 years d) none of these What is the average of 7 consecutive even numbers if the smallest of those numbers is denoted by k? a) k+4 b) k+7 c) k+6 d) 7k 10. 5. The average weight of four persons A, B, C and D is 40kg. A new person E is also included in the group, and then the average weight of the group is increased by 1kg. Again a new person F replaces A, then the new average of 5 persons becomes 42. The average weight of B, C, D and F is : a) 42 b) 41.25 c) 42.5 d) none of these 11. 6. The average income of P, Q and R is Rs. 24,000 per month and the average income of Q, R and S is Rs. 30,000 per month. If the
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3.
9.
4.
where the age of both the players was same, then the average age of two absent players and the total number of players respectively can be: a) 18, 27 b) 20, 27 c) 15, 38 d) cannot be determined 12. The average marks of Ankita decreased by 1, when she replaced the subject in which she had scored 40 marks by the other two subjects in which she had just scored 23 and 25 marks respectively. Later she also included 57 marks of Computer Science, then the average marks increased by 2. How many subjects were there initially? a) 6 b) 12 c) 15 d) cannot be determined While adding the sum of the first N natural numbers a student missed a number and found the average as 15, then what is the value of n is: a) 30 b) 26 c) 31 d) not unique Out of the five integers - A, B, C, D and E - C is the average of A and D, B is greater than C and less than D and B is the average of A and E. The middle most number in the sequence is: a) A b) B c) C d) D While calculating the average of 10 three digits numbers a student reversed the digits of a number and the average increased by 19.8. The difference between the unit digit and hundred digit of that number is: a) 8 b) 4 c) 2 d) cannot be determined
17.
The average expenditure of the hotel when there are 10 guests is Rs. 60 per guest and the average expenditure is Rs. 40 when there are 20 guests. What would be the average expenditure if there are 40 guests? (Cost includes fixed and variable? a) Rs.30 b) Rs. 25 c) 20 d) cannot be determined There are 10 compartments in a passenger train which carries on an average 20 passengers per compartment. If at least 12 passengers were sitting in each compartment and all the compartments carry different number of passengers, then maximum how many passengers can be accommodated in any compartment? a) 64 b) 45 c) 56 d) none of these
18.
13.
14.
19. The average of 46, 49, x, 55 and 63 lies between 45 and 55. If x is always as integer and greater than the average of the given integers then the value of n is: a) 53 < x < 67 b) 54 < x < 63 c) 53 < x < 62 d) none of these
15.
16.
The clerk in the office measured the weights in all possible pairs of four boxes. The weights are 59 gm, 61gm, 62gm, 63gm, 64gm, and 66gm. The weight of the heaviest box is: a) 35.5gm b) 36.5gm c) 34.5gm d) cannot be determined
The price of the first variety is Rs.10 per cup and the price of second variety is Rs.15 per cup respectively. The average price of the mixture per cup is: a) 15 b) 14 c) 13 d) 12
## ALLEGATION & MIXTURES
Introduction: This chapter is the extension of Averages and here we particularly study weighted averages. This chapter is dedicated to understand and study the average of two different groups with different number of elements. Here Allegation method is used to solve the problem quickly. Allegation plays a crucial role in understanding the problem of Ratio Proportion and Variation, Simple/Compound Interests and Profit and Loss chapters. EXERCISE: Concept Review 1. The average weight of a class of 40 students is 60 and the average weight of another class of 20 students is 30. Find the average weight of both the combined classes: a) 40 b) 50 c) 45 d) 55 2. The average weight of girls is 30 and the average weight of boys is 60 and the combined average weight is 50. If the number of boys is 24, then the number of girls is: a) 8 b) 72 c) 36 d) 12 3. The ratio of number girls to number of boys of a class is 1:2. If the average weight of the boys is 60kg and the combined average weight is 50 kg, then the average weight of the girls is: a) 40 b) 30 c) 70 d) 80 4. Two varieties of flavours of coffee with different prices are mixed in the ratio 2:3.
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5. Akash covered 600 km in 10 hours. He covered the fist part of the journey by car and second part by auto. The speeds of Car and Auto are 80 km and 48 km per hour respectively. Find the ratio of distances covered by Car and Auto respectively: a) 2:3 b) 4:5 c) 1:1 d) none of these 6. A mixture of water and milk contains 80% milk. How many litres of water must be added to 100 litres of mixture to increase the percentage of water to 50? a) 60 b) 80 c) 100 d) 120 7. There are three types of milk available in the market. Type 1 contains milk and water in the ratio 4:5, Type 2 contains milk and water in the ratio 5:6 and type 3 contains milk and water in the ratio 6:7. If all the three types are mixed in equal quantity, then the ratio of milk to water is: a) 2110:1751 b) 1751:2110 c) 5:8 d) 8:5 8. From 50 litres of pure milk 5 litres is taken out and 5 litres of water is added. Again 5 litres of mixture is taken out and 5 litres of water is added. If this process is continued for the third time, then the amount of milk left after the third replacement: a) 45 b) 35 c) 36.45 d) 40.5 9. How much Petrol at Rs. 60 a litre is added to 15 litre of 'kerosene' at Rs. 10 a litre so that the price of the mixture be Rs. 30 a litre ? a) 5 b) 8 c) 10 d) none of these 10. Kiran has Rs. 25 consisting of only the denominations of 20 paise and 50 paise. Thus
8
there are total 80 coins in my pocket. The no. of coins of the denomination of 50 paise is: a) 30 b) 70 c) 50 d) 25 11. There are some shepherds and their sheep in a grazing field. The no. of total heads are 60 and total legs are 168 including both men and sheep. The no. of sheep is: a) 18 b) 26 c) 24 d) 36 In the 75 liters of mixture of milk and water, the ratio of milk and water is 4:1. The quantity of water required to make the ratio of milk and water 3:1 is: a) 1 litre b) 3 litres c) 4 litre d) 5 litres In my office the average age of all the female employees is 21 years and that of male employees is 32 years, where the average of all the employees is 28 years. The total no. of employees in my office could be: a) 35 b) 78 c) 231 d) 90 Rs. 69 was divided among 115 students so that each girl gets 50 paise less than a boy. Thus each boy recieved twice the paise as each girl received. The no. of girls in the class is: a) 92 b) 42 c) 33 d) 23 A butler stole wine from a butt of sherry containing 50% of spirit, and then he replenished it by different whine containing 20% spirit. Thus there was only 30% strength (spirit) in the new mixture. How much of the original wine did he steal? a) 1/3 b) 2/3 c) 1/2 d) 1/4 In a 25 litre mixture of milk and water, the water is only 20%. How many litres of water
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is required to increase the percentage of water to 90% ? a) 45 litre b) 70 litre c) 115 litre d) 175 litre 17. In a class of 30 students, the average weight of boys is 20kg and the average weight of the girls is 25kg. The fraction of boys out of the total students of the class is: a) 4/5 b) 5/6 c) 3/4 d) data insufficient The average age of boys in class is 16.66. While the average age of girls is 18.75. Thus the average age of all the 40 students of the class is 17.5. If the difference between the o. of boys and girls is 8, then the no. of girls in the class is: a) 12 b) 16 c) 18 d) data insufficient The ratio of water and alcohol in two different containers is 2:3 and 4:5. In what ratio should they be mixed to get the ratio of alcohol to water 7:5? a) 7:3 b) 5:3 c) 8:5 d) 2:7 In a mixture of milk and water, there is only 26% water. After replacing the mixture with 7 litres of pure milk, the percentage of milk in the mixture become 76%. The quantity of mixture is: a) 65 litre b) 91 litre c) 38 litre d) none of these 450 litres of a mixture of milk and water contains the milk and water in the ratio 9:1. How much water should be added to get a new mixture containing milk and water in the ratio 3:1? a) 54 b) 90 c) 45 d) 63 The ratio of petrol and kerosene in the container is 3:2 when 10 litres of the mixture
9
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18.
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is taken out and is replace by the kerosene, the ratio becomes 2:3. The total quantity of the mixture in the container is: a) 25 b) 30 c) 45 d) cannot be determined From a container, 6 litres milk was drawn out and was replaced by water. Again 6 litres of mixture was drawn out & was replaced by the water. Thus the quantity of milk and water in the container after these two operations is 9:16. The quantity of mixture is: a) 15 b) 16 c) 25 d) 31
1. The value of the ratio does not change if both numerator and denominator are multiplied by same quantity.
## RATIO, PROPORTION and VARIATION
Introduction: This chapter is all about the comparisons of two or more quantities. It also deals with the magnitude of the changes in the quantities. Primarily this chapter focuses on comparison in the changes of ages, weights, heights, incomes, savings, expenditures, temperature, volume, density etc. This chapter is one of the most important chapters for CAT as well as for other Management Entrance exams. This chapter is very important to understand the concepts of Data Interpretation.
2. The value of the ratio does not change if both numerator and denominator are divided by same quantity. 3. The ratio of two fractions can be expressed as ratio of integers. a/b a d X c/d b c 4. If two or more ratios are multiplied with each other, then it is called compounded ratio. 1 3 5 5 X X 2 4 6 16
5. If the ratio is multiplied by itself, then it is
called duplicate, triplicate ratios etc. and inverse of it is called sub-duplicate and subtriplicate etc. a a a X ( )2 b b b
6. If
CONCEPTS RATIO: The comparison between two or more quantities in terms of their magnitude is called the ratio. Example: Anjali has 9 DVDs and Sushma has 7 DVDs. It means the ratio of number of DVDs between Anjali and Sushma is 9 is to 7. It is expressed as 9:7 (Here the order is very important. In the above example the order is 9:7 not 7:9)
a or a: b b Here numerator a is called the antecedent and denominator b is called consequent.
## a 1 , then b ak a ak a (k is positive) (k is negative) bk b bk b a 1 , then b ak a ak a (k is positive) (k is negative) bk b bk b
c a ac a , then d b bd b c a ac a , then d b bd b
7. If
8. If
## The ratio is generally written as
9. If
10. If
a c e a c e .. ..... K , then K b d f b d f ..
Properties of RATIOS
## 7) Divide 180 chocolates among A, B and C such
that A gets two times that of B and 11. If the individual ratios of a:b, b:c, c:d and d:e are given, then the combined ratio of a:b:c:d:e is: a:b b:c c:d d:e a:b b:c c:d d:e a:b b:c c:d d:e a:b b:c c:d d:e a:b b:c c:d d:e C
1 times that of 3
8) A: B = 3:4; B: C = 8:6; C: D = 3:1 and D: E = 2:5, then find A: C: E 9) If a: b = 3:4, then find the value of
9 a 7b 3a 4b
a:b:c:d:e=(a.b.c.d):(b.b.c.d):(b.c.c.d):(b.c.d.d) :(b.c.d.e)
10) Salaries of Nitish and Rajeev are in the ratio 4:3 and their savings are in the ratio 3:2. If both spend rupees 200, then find their monthly salaries. 11) Salaries of A, B and C are in the ratio 12:10:9 and their spending is in the ratio 15:9:8. If A spends 75% of his income then the ratio of the savings of A:B:C: 12) 1,54,000 aspirants appear for CAT from four A B C D different cities A, B, C and D and 2 3 4 5 Find i) the difference of number of aspirants from city A and city B and ii) the ratio of number of aspirants of cities (A+C) : (B+D)
EXAMPLES 1) Find the ratio of 50 to 100 2) Find the ratio of 25 cm to 1.5m 3) Out of 456 students of St. Xavier School 240 are boys. Find the ratio of boys to girls. 4) The salary of Mr Yogesh is 24,500 and expenditure is 16,000. Find the ratio of Salary to savings.
CONCEPTS PROPORTION
5) Out of 100 members of a sports club, 20 play cricket, 30 play football and 16 play tennis and the remaining members do not play any game. And no member of the club plays more than one game. Find the ratio of number of members who play: i) cricket to tennis ii) cricket to football iii) cricket to those who play no game iv) members who play any game to those who play no game.
3 4 6) Simplify i) 2 : 3 4 5
ii)
Proportion means an equality of two ratios and the four numbers involved are in proportion. i.e.
a c or a:b = c:d, then a, b, c and d are in b d proportion and written as (a:b :: c:d). The symbol ( :: ) indicates proportion and it is read as a is to be as c is to d.
if
Here a and d are called extremes and b and c are called means. PROPERTIES
6 10 : 8 8
1) If four numbers are in proportion, then the product of extremes is equal to the product of means. If a:b :: c:d, then a x d = b x c 2) Continued Proportion: If three number a, b and c are such that a:b = b:c, then these numbers are in continued proportion. i.e. a:b = b:c
b2 = ac
## 5) If a+b : a-b = 13:1, then the value of
9 a 7b 3a 4b
6) The number of students in three sections is in the ratio 3:4:5. If 30 new students are joined in every section, then ratio becomes 5:6:7. Find the total number of students? 7) Two equal jars contain mixture of milk and water. The concentration of water in the two jars is 30% and 25% respectively. What is the ratio of water in both the jars respectively? 8) A Cat takes 5 steps for every 9 steps of a rat. However 6 steps of cat are equal to 7 steps of rat. What is the ratio of speeds of cat to that of rat? 9) A deer taken 5 steps for every 7 steps of a tiger. However 4 steps of deer are equal to 7 steps of tiger. What is the ratio of speeds of deer to that of tiger? Direct proportion: Two quantities are in direct proportion if the increase or decrease in one quantity leads to the increase or decrease in the other quantity by the same proportion. Inverse proportion: Two quantities are in inverse proportion if the increase or decrease in one quantity leads to the decrease or increase in the other quantity by the same proportion. Examples:
Here, b is the mean proportion between a and c and c is the third proportion to a and b. 3) Invertendo: If
a c b d , then b d a c a c a b , then b d c d a c ab cd , then b d b d
4) Alternando: If
5) Componendo: If
6) Dividendo: If
a c a b c d , then b d b d
## a c 7) Componendo and Dividendo: If , then b d ab cd a b c d
Examples
1) If 10 books cost Rs.90, how much would 15 books cost? 2) 8 workers can finish constructing a wall in 40 hrs. In how many hours a day must be worked to complete the work in 4 days? 3) Provisions are sufficient for 100 students for 3 months. How many days will the stocks last if there are 75 students?
1) If four numbers 10, 20, x and 60 are in proportion, then find the value of x? 2) If 49, x, x, 64 are in proportion, then find x? 3) The ratio of number of boys and girls in a class is 7:8. If the number of girls is 56, then find the number of boys. 4) Find the mean proportion of 4 and 49.
2) The value of a diamond varies directly to the square of its radius. The radius of the diamond is 3 cm and its value is Rs.2 crore. What will be the radius of the diamond if its value is Rs.6 crores?
CONCEPTS
EXERCISE Concepts Review 1. If 20 persons can do a piece of work in 14 days, then in how many days 28 persons finish the work: a) 4 b) 5 c) 14 d) 10 A garrison of 750 men has provision for 20 weeks. If at the end of 4 weeks 450 men joined, then how long will the provisions last? a) 10 weeks b) 11 weeks c) 15 weeks d) 16 weeks 40 men can build a wall of 20m high in 15 days. The number of men required to build a similar wall of 25m high in 6 days will be: a) 100 b) 125 c) 150 d) 200 15 men take 42 days to complete the work working 4 hours a day. In how many days will 21 women working 6 hours a day finish the work if 3 women do as much work as 2 men? a) 15 b) 22 c) 25 d) 30 Anshu is as much younger to Barbie as he is older to Chaitra. If the sum of the ages of Barbie and Chaitra is 48 years, what is the present age of Anshu ? a) 18 years b) 36 years c) 24 years d) 28 years Bhanu is 6 times as old as Alok. Bhanu's age will be twice of Chandan's age after 10 years.
14
VARIATION
If two or more quantities depend upon each other, then if one of them is changes the other quantity also changes. There are two types of variations 1) Direct Variation: A quantity A is said to vary directly with quantity B, if the increase or decrease in A yield increase or decrease in B but not in the same proportion.
2.
a b
a = Kb
3. 2) Inverse Variation: A quantity A is said to vary inversely with quantity B, if the increase or decrease in A yield decrease or increase in B but not in the same proportion.
a 1 b a K b
4.
Note: A quantity sometimes vary jointly i.e., directly on one quantity and inversely on another quantity.
a Kb And a
K Kb and a c c
5.
Examples: 1) A varies directly with B and inversely with C. A is 24 when B is 12 and C is 4. What is the value of A When B is 24 and C is 6? 6.
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If chandan's 7th birthday was celebrated 3 years ago, what is Alok's present age ? a) 15 years b) 12 years c) 5 years d) none of these
13.
Third proportion of 16 and 36 is: a) 64 b) 144 c) 81 d) 49 The fourth proportion of 6, 7 and 30 is : a) 28 b) 21 c) 18 d) 35 a b c abc If then 3 4 5 b a) 2 b) 3 c) 4 d) 5 If a:b = b:c = c:d then
a b c , , are b c d
14.
7.
Reema got married 8 years ago. Today her 1 age is 1 times her age at the time of 3 marriage. Her daughter's age is 1/8 times her age. Her daughter's age is : a) 3 years b) 4 years c) 6 years d) 8 years Ten years ago Bs age was twice that of As. If the ratio of their present age is 4:3, what is the sum of their present ages? a) 25 years b) 30 years c) 40 years d) 35 years Rs. 13,950 divides among three persons A, B and C. B gets twice that of A and C gets Rs. 50 less than twice that of B. The share of A will be. a) Rs. 1950 b) Rs. 1981.25 c) Rs. 2000 d) Rs. 2007.75 A started business with Rs. 45 million and B joined afterward with Rs. 30 million. The profit at the end of one year was divided in the ratio 2:1 respectively. After how many months did B join? a) 1 month b) 2 months c) 3 months d) 4 months A and B started a business with 5:6 ratio. At the end of 8 months A has withdrawn from the business. If they receive profits in the ratio of 5:9, find how long B stayed in the business? a) 12 months b) 10 months c) 15 months d) 14 months Mean proportion of 17 and 68 is : a) 51 b) 24 c) 4 d) 34
15.
16.
8.
a) in AP b) in continued proportion c) in GP d) both (b) and (c) 17. A sum of Rs. 210000 is divided among A, B,C such that shares of A and B are in the ratio of 2:3 and those of B and C are in the ratio 4:5. How much does A get? a) Rs. 60000 b) Rs. 45000 c) Rs, 48000 d) Rs. 84000 Rs. 11250 is divided among A, B and C such that A receives one-half as much as B and C together receive and B receives on fourth of what A and C together receive. The share of A is more than that of B by: a) Rs. 2500 b) Rs. 1500 c) Rs. 1800 d) Rs. 650 A pole 1.2 metre tall casts a shadow of 1.1m at the time when a building casts a shadow 6.6m long. The height of the building is: a) 2.7m b) 7.2m c) 6.0m d) 5.5m In a mixture of 120 litres, the ratio of milk and water is 2:1. If the ratio of milk and water is 1:2, then the amount of water should be added: a) 20 b) 40 c) 80 d) 120
9.
18.
10.
19.
11.
20.
12.
21.
A quantity x varies inversely as the square of y. Given that x=4, when y=3, the value of x when y=6 is: a) 1 b) 2 c) 3 d) 4 Suppose y varies as the sum of two quantities of which one varies directly as x and the other inversely as x. If y=6 when x=4 and y= 1 3 when x=3, then the relation between x 3 and y is: 8 a) x = y+4 b) y = 2x+ x 8 4 c) y = 2xd) y = 2xx x Application of Concepts 1. Four numbers are in proportion. The sum of the square of the four numbers is 50 and the sum of the means is 5. The ratio of first two terms is 1:3. What is the average of the four numbers? a) 2 b) 3 c) 5 d) 6 If a2+b2 : a2-b2 = 133:117; find a:b a) 2:3 b) 5:4 c) 5:2 d) none of these The ratio of income of Ambani and Mahindra is 2:3. The sum of their expenditure is Rs. 8000 and the savings of Ambanil is equal to the expenditure of Mukesh. What is the sum of their savings? a) 22,000 b) 4,000 c) 16,000 d) 12,000 Rupees 4536 is divided among 4 men, 5 women and 2 boys such that the ratio of share of a man, a woman and a boy is 7:4:3 What is the share of a woman? a) Rs. 336 b) Rs. 498 c) Rs.1680 d) Rs. 1176 The concentration of petrol in three different 1 3 mixtures of petrol and kerosene is , and 2 5 4 respectively. If 2, 3 and 1 litre respectively 5 are taken from these three different vessels and mixed. What is the ratio of kerosene and petrol in the new mixture? a) 4:5 b) 3:2 c) 3:5 d) 2:3
23.
2.
3.
4.
5.
6.
Mr. Doodwala, a milk shop, sells three varieties of milk, 'Pure', 'Toned' and 'Normal'. 'Pure' milk has 100% concentration of milk while the ratio of milk is to water in the 'Toned' is 2:5 and in the Normal is 3:8. Sonali purchased 14 litres of Toned and 22 litres of Normal milk and mixed them. How many litres of pure milk should be added to have 50% concentration milk? a) 5 litres b) 8 litres c) 13 litres d) cannot be determined The value of a diamond is directly proportional to the square of its weight. A diamond breaks into three pieces with weights in the ratio of 3:4:5 and the overall lost due to breakage is Rs. 9.4 lakhs. What is the actual value of the diamond? a) 28.8 lakh b) 5 lakh c) 14.4 lakh d) 18.8 lakh Mr Gill Bates, a computer geek, donated 1800 Billion to three types of welfare organisations, Refugee camps, Education centres and Health centres in the form of 10 Billion, 20 Billion and 100 Billion respectively. The ratio of number of organisation received 10 Billion and 20 Billion is 6:1. Find the minimum number of organisation which received 100 billion bounties: a) 1 b) 2 c) 4 d) cannot be determined Vipin bought 'N' chocolates. He distributed them among 4 children in the ratio of 1 1 1 1 : : : . If he gave them each one a 2 3 5 8 complete chocolate, then the minimum no. of chocolates he bought could be: a) 139 b) 240 c) 278 d) none of these Equal quantities of three mixtures of milk and water are mixed in the ratio of 1:2, 2:3 and 3:4. The ratio of water and milk in the mixture is :
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## a) 193.122 c) 61:97 11.
b) 122:193 d) 137:178
The ratio of age of A and B is 8:9 and the age of B is 2/3 of C's age and age of C is 9/13 times of D. If B is 18 years then the age of C is : a) 36 years b) 39 years c) 27 years d) 57 years Two alloys A and B have copper and tin in the ratio of 5:3 and 5:11 respectively. If the alloys A and B are mixed to form a third alloy C with an equal proportion of copper and tin, what is the ratio of alloys A and B in the new alloy C? a) 3:5 b) 4:5 c) 3:2 d) 2:3 The period of the pendulum is directly proportional to the square root of the length of the string. The period is 52 seconds when the length of it is 16cm. Find the length of the string if the period is 65 seconds: a) 4.5cm b) 5 cm c) 6cm d) none of these Rs. 960 were distributed among A, B, C and D in such a way that C and D together gets half of what A and B together gets and C gets onethird amount of B. Also D gets 5/3 times as much as C. How much will A get? a) Rs. 240 b) Rs. 280 c) Rs. 320 d) data insufficient The speeds of cycle, moped and car are in the ratio 3: 5:6. What is the ratio of time taken by each one of them to cover the same distance? a) 6:5:3 b) 10:6:5 c) 12:7:6 d) data insufficient a:b = 4:9 if 4 is added to both of the numbers then the new ratio becomes 21:46. What is the difference between a and b? a) 80 b) 100
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c) 125 17.
d) 130
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The ratio of ages of Rahul and Deepesh is 3:5 10 years later the ratio of their ages will be 5:7. What is the present age of Deepesh ? a) 20 years b) 50 years c) 25 years d) 40 years 24. Five numbers a, b, c, d and e are in the ratio of 2 : 3 : 5 : 8 : 9 and their sum is 162. Find the average of all these numbers: a) 27 b) 30 c) 32.4 d) cannot be determined 25. 6 pumps of Type-1 can fill a tank in 7 days and 2 pumps of Type-2 can fill the same tank in 18 days. What is the ratio of the efficiency of Type-1 and Type-2 pumps? a) 6 : 7 b) 7 : 6 c) 7 : 54 d) cannot be determined Petrol is 7 times heavier than Kerosene and Crude is 18 times as heavy as Kerosene. What should be the ratio of petrol and crude in the new mixture go get the mixture which is 11 times as heavy as kerosene? a) 3 : 4 b) 7 : 4 c) 9 : 19 d) 9 : 10 The ratio of prices of Cello and Rotomac pens in 2000 were 3 : 5. In 2005 the price of Cello pen trebles itself and the price of Rotomac pen is increased by Rs. 100, then the new ratio of prices of the same pens becomes 4 : 5. What was the original price of the Rotomac pen in 2000? a) Rs. 60 b) Rs. 80 c) Rs. 100 d) Rs. 120 A rabit takes 22 leaps for every 17 leaps of cat and 22 leaps of a rabit are equal to 17 leaps of the cat. What is the ratio of the speeds of rabit and cat ? a) 1 : 1 b) 484 : 289 c) 17 : 22 d) none of these
The ratio of numerator to a denominator of a fraction is 1/5 when x and 5x are added to the numerator and denominator respectively then the ratio of the new fraction will be: a) 1 : 1 b) 1 : 25 c) 1 : 5 d) 2 : 7 x varies directly as y and x varies inversely as the square of z. If y=75 and x=6, then z=5. Find the value of x when y=24 and z=4: a) 1 b) 2 c) 3 d) 4 Weight of a tree jointly varies as its height and its age. When height is 1.2m and age is 20 years the weight would be 48kg. Find the weight of the tree when its height is 1.5 metre and age is 30 years: a) 60kg b) 72kg c) 90kg d) 58 kg
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The amount with A, B, and C is in the ratio 3 : 4 : 5. First B gives 1/4th of what he has to A and 1/4 to C then C gives 1/6th of what he has then to A. Find the final ratio of amount of A, B and C after the distribution. a) 4 : 3 : 5 b) 5 : 4 : 3 c) 6 : 4 : 2 d) 5 : 2 : 5 A tin contains a mixture of Dew and sprite in the ratio of 7 : 3 and another tin contains the Dew and Sprite in the ratio of 5 : 4. In what proportion should they be mixed to get a proportion of 2 : 1 (in which Dew is 2 times that of sprite) a) 10 : 3 b) 4 : 1 c) 3 : 10 d) 3 : 1 Sachin bought 1.5 kg fresh grapes. The ratio of water to pulp in the fresh grapes was 4 : 1. The grapes lost some water after exposed to sun. Now the dried grapes contain water and pulp in the ratio 3:2. What is the total weight of water evaporated? a) 0.5 kg b) 1 kg
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c) 0.75 kg 29.
d) none of these
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A man bought 9 mangoes for a rupee and sold them at 6 mangoes for a rupee. What is the ratio of profit to the cost price? a) 3/10 b) 3/2 c) 1/2 d) none of these A container is filled with equal quantities of milk and water. Bobby and Sunny increases the concentration of milk to 60%. Bobby makes it by adding the milk and Sunny makes it by replacing the mixture with milk. What is the percentage of milk added by Bobby to that of milk replaced by Sunny? a) 100% b) 120% c) 133.33% d) none of these Two vessels A and B contain 25 litres each of pure milk and pure water respectively. 5 litres of milk from A is taken out and poured into B and then 6 litres of mixture from B is taken and poured in A. What is the ratio of water in A and B? a) 4 : 5 b) 1 : 4 c) 5 : 4 d) 2: 3 The ratio of number of students preparing for GATE and CAT is 4 : 5. The ratio of cost of study materials of GATE and CAT is 25: 16. If the total amount spent by all the students is 1.62 lakh, what is the total amount spent by only CAT aspirants? a) Rs. 62,000 b) Rs. 72,000 c) Rs. 80,000 d) none of these The cost of the carpet varies directly with square of its length. Carpet is cut into 3 parts whose lengths are in the ratio 3 : 4 : 5. If carpet had been cut into three equal parts by length then there would have been a further loss of Rs. 1800. What is the actual cost of the original carpet? a) Rs. 3600 b) Rs. 10,800 c) Rs. 2160 d) none of these
Directions for questions 35 and 35: Four girls Rose, Lotus, Lilly and X - individually collected some flowers decided to share all the flowers with them equally. First Rose gave Lotus what Lotus had initially, and then Lotus gave Lilly what Lilly had initially, and then Lilly gave X what X had initially and finally X gave Rose what Rose had then. Finally each got 48 flowers. 34. How many flowers did Lotus collect? a) Rs. 36 b) Rs. 54 c) Rs. 45 d) Rs. 42 35. What was the number of flowers with Lilly after Lotus distributed? a) Rs. 45 b) Rs. 69 c) Rs. 72 d) Rs. 84 Hiralal and Mrunal hired a grazing field for 30 days to feed their cows. Hiralal had 24 cows and Mrunal had 30 cows. If Hiralal paid Rs. 3500 and Murunal paid Rs. 5000, then for how many days Hiralal used the grazing field : a) 14 b) 16 c) 21 d) 20 Doodwalal has two jars. Jar A is completely filled with milk and another jar B is totally empty. Before selling the milk in a town he transfers some milk in to the empty jar B then he then fills the jar A with water. Once again he transferred the mixture of milk from A to B so that B is completely filled. Which one of the following is correct ? a) Concentration of milk in B cannot be less than 75% b) Concentration of milk in B cannot be greater than 75% c) Concentration of milk is always 75% d) none of these above
20 m/s 3 without any wagon attached. Reduction in the speed of the train is directly proportional to the square root of the no. of wagons attached to the engine. When there are only
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37. 32.
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## An engine can move at the speed of
50 m/s. 9 What is the maximum number of wagons can be attached to the engine if it runs? a) 144 b) 143 c) 12 d) none of these
## four wagons attached its speed is
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Three dears A, B and C - are roaming in a zoo in such a way that when dear A takes 5 steps, B takes 6 steps and cat C takes 7 steps. But the 6 steps of A are equal to the 7 steps of B and 8 steps of C. What is the ratio of their speeds? a) 140 : 144 : 147 b) 40 : 44 : 47 c) 15 : 21 : 28 d) 252 : 245 : 240 The expenditure on food per month of a family is directly proportional to the 5 times the square of no. of people of the family. If there were one less member then the consumption of rice decreases by 95kg per month. How many members are there in the family? a) 5 b) 12 c) 9 d) 10 The price of a necklace varies directly as the no. of diamonds in it. Also, it varies directly as the square root of radius of a diamond. The price of a necklace was Rs. 1.5 million When it had 75 diamonds each of radius 1cm. Find the radius of the pearl of a necklace having 100 pearls and worth Rs. 6 million: a) 2 b) 4 c) 3 d) 9 The price of a book varies directly as the no. of pages in it and inversely as the time periods in years that have elapsed since the date of purchasing. However, two books cost the same if the no. of pages in the first book is triple of the second book. If the first book is sold 18 years ago, how long ago was the second book sold? a) 54 years b) 9 years c) 6 years d) 3 years
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Distance covered by a train is directly proportional to the time taken and also it varies directly as the square root of fuel used and varies inversely as the no. of wagons attached to it. A train covers 192km journey in 20 hours when there are 10 wagons attached to it and total fuel consumption was 256 litre of diesel. Find the consumption of fuel per km when a train goes 200 km in 25 hours with 15 wagons attached to it : a) 1.5 l/km b) 2 l/km c) 2.8 l/km d) 20 l/km In two allays the ratio of Iron and copper is 4 : 3 and 6 : 1 respectively. 28 kg of the first alloy and 84kg of the second alloy are mixed together to form a new alloy. What will be the ratio of copper to iron in the new alloy? a) 11 : 3 b) 11 : 8 c) 8 : 11 d) none of these A vessel is full of petrol. 6 litre is taken out and substituted by kerosene. This process is repeated two more times. Finally the ratio of kerosene and petrol in the mixture is 1701 : 27. Find the volume of the vessel: a) 14 litre b) 16 litre c) 8 litre d) 42 litre The capacity of three vessels is in the ratio 2: 3:5 In the first vessel ratio of water and milk is 1:3 in second is 2:3 and in third vessel is 2:5. If all the three vessels were poured out in a large container, what is the resulting ratio of milk to water? a) 43 : 96 b) 438 : 962 c) 348 : 962 d) 962 : 438 The ratio of copper and nickel by weight in the two alloys X and Y are 2 : 7 and 5 : 4. How many kilogram of the alloy X and Y are required to make 42 kg of new alloy Z in which the ratio of copper and nickel is 1:1? a) 6 kg and 36 kg b) 10kg and 32 kg c) 7 kg and 35 kg d) none of these
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There are two alloys made up of copper and aluminium. In the first alloy copper is half of the aluminium and in the second alloy copper is three times that of aluminium. How many times the second alloy must be mixed with first alloy to get the new alloy in which copper is two times that of aluminium a) 2 b) 3 c) 4 d) 5 Two engine-oils Veedol and Castrol come in the quantities of 90 litres and 150 litres respectively. The price of Veedol is Rs. 80 per litre and price of Castrol is Rs. 75 per litre. Equal amount of Veedol and Castrol is taken out and then CRB is poured in to Castrol and Veedol respectively. Now the price of both the mixtures is same. What is the amount of oil taken out from each of the vessels? a) 45 litres b) 56.25 litres c) 24.5 litres d) 36 litres Two jars with equal capacity are filled with pure milk and water respectively. 5 cups of water from the second jar is taken out and mixed well in the second container. Then, 5 cups of this mixture is taken out and is mixed in the first container. Let A denote the proportion of milk in the first container and B denote the proportion of water in the second container then: a) A < B b) A = B c) A > B d) cannot be determined
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## Examples: 1) 20% 2) 75% 3) 35%
PERCENTAGES
Introduction: This is one of the most important chapters for IIM-CAT and other Management entrance tests. Concepts involved in these chapters are useful for real life situations also. Good understanding of the concepts of this chapter is absolute necessary to score more in Data Interpretation section.
## 1 1 4) 12 % 5) 3 % 3 2 PERCENTAGE INTO RATIO
To convert the given percentage into a ratio, first the percentage needs to be converted into simplest fraction and then to a ratio. Examples: 1) 25% 2) 33.33% 3) 38% 4) 65% 5) 1%
RATIO INTO PERCENTAGE To convert the given ratio into a percentage, first the ratio needs to be converted into a fraction and then to a percentage. Examples: 1) 1:2 2) 2:5 3) 5:9 4) 2:3 5) 6:7
CONCEPTS
Any fraction with denominator 100 is called a percent. A fraction with denominator 10 is called decimal.
CONVERSIONS
FRACTION INTO PERCENTAGE Multiply the given fraction by 100 and put % symbol. Examples:
PERCENTAGE INTO DECIMAL To convert the given percentage into decimal, first remove the % sign and move the decimal point to places to the left of the given number. Examples: 1) 20% 2) 345% 3) 175% 4) 16
2 % 3
1)
1 2
2)
3 4
3)
5 8
4)
2 3
5)
6 4
5)17.5%
## PERCENTAGES INTO FRACTIONS Multiply the given percentage by the % symbol.
1 and remove 100
DECIMAL INTO PERCENTAGE To convert the given decimal into percentage, first move the decimal two point places to the right and place the % sign next to the number.
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## FRACTIONS INTO PERCENTAGES
N/D 1 2 3 4 5 6 7 8 9 10 11 12 15 1 100 200 33.33 25 20 16.66 14.28 12.5 11.11 10 9.09 8.33 6.66 2 3 4 5 6 7 8 9 10 200 300 400 500 600 700 800 900 1000 100 150 200 250 300 350 400 450 500 66.33 100 133.33 166.66 200 233.33 266.66 300 333.33 50 75 100 125 150 175 200 225 250 40 60 80 100 120 140 160 180 200 33.33 50 66.66 83.33 100 116.66 133.33 150 166.66 28.56 42.85 57.13 71.42 85.71 100 114.28 128.56 142.85 25 37.5 50 62.5 75 87.5 100 112.5 125 22.22 33.33 44.44 55.55 66.66 77.77 88.88 99.99 111.11 20 30 40 50 60 70 80 90 100 18.18 27.27 36.36 45.45 54.54 63.63 72.72 81.81 90.90 16.66 25 33.33 41.66 50 58.33 66.66 75 83.33 13.33 20 26.66 33.33 40 46.66 53.33 60 66.66 N= NUMERATOR D=DENOMINATOR 11 12 1100 1200 550 600 366.66 400 275 300 220 240 183.33 200 157.13 171.42 137.5 150 122.22 133.33 110 120 100 109.09 91.66 100 73.33 80
EXAMPLES: Solve the following problem using fractions instead of percentages. 1) Find the value of a) 12.5% of 200 c) 37.5% of 400 e) 100% of 2 litres
6) The price of a bike is Rs.60,000. It depreciates 25% per year. What will be the price of the bike next year? 7) Rupeshs annual salary is Rs.3.6 lakhs. His salary increased by 22.22%. What is his new salary? 8) 85% of students cleared the exam. If the total number of students is 3400, then how many failed. 9) Davids family spent 30% of the income on education. 30% of rest is spent on health. What percent of the income is spent on health? 10) The discount offered on a travel bag is 8.33% which is equal to Rs.250. What is the price of the bag? 11) What percent is a) 40 out of 120 c) 75 out 450 e) 7.5 l out of 25 l
## b) 40% of 300 d) 83.33% of 600 f) 41.66% of 120
2) Kamesh scored 82% marks out of 250. How many marks did he score? 3) Vandana scored 1250 marks in the first semester. How much did she score in electives if she scored 40% of marks in compulsory subjects? 4) 30% of a number is 135. Find the number. 5) The population of a village is 1200. 500 are females. What percent of the population is male?
## b) 30 out of 210 d) 2kg out of 5 kg f) 825m out of 2500m
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12) Sidhu score 275 marks out of 450. What percent marks did he score? 13) Rahul scored 182 runs facing 220 balls. What is his strike rate? 14) Out of 3200 tonnes of wheat production only 1800 tonnes are sold in the open market. What percent of the production is yet to be sold? 15) In an election, the winner got 64000 votes and won by a margin of 60%. What is the total number of votes?
It also relates to the concepts of other chapters like: 1) 2) 3) 4) 5) Average x no of elements = Total value Rate X Quantity = Total cost Length x Breadth = Area Speed x time = Distance Efficiency x time = work
Conditions: a) If one factor of a product is increased by P% then the other factor will be decreased by p ( X 100)% p 100
It means when one factor of a product is
CONCEPTS
PERCENTAGE CHANGE The application of percentage is the basis for problem of Profit and Loss, Simple/Compound Interest. Percentage Increase/Decrease in the quantity:
n , then the other factor is m n decreased by mn b) If one factor of a product is decreased by P% then the other factor will be increased
increased by
by (
p X 100)% p 100
## n , then the other factor is m n increased by mn
decreased by
a) If the value of a quantity P is increased by x%, in order get back to the original quantity the increased quantity must decreased by
( x X 100)% x 100
POPULATION RELATED PROBLEMS Population related problems shares the same concepts of Simple/compound Interest. a) If the population of a place is P and the annual positive growth rate is r%, then the population after n years is: r n and increase in the P(1 ) 100 r n population = P[ (1 ) 1] 100
PRODUCT CONSTANCY
Product constancy concept is same as inverse proportion of Ratio, Proportion and Variation.
b) If the population of a place is P and the annual negative growth rate is r%, then the r n population after n years is: P(1 ) 100 and decrease in the population = r n P[1 (1 ) ] 100 NOTE: The difference between decreased value and decrease in value and increase value and increase in value. If the initial value of a quantity is X and Final value is Y, then Y is the increased/decreased value and but increase/decrease in value is (Y-X or X-Y) Example: Initial value = 80 and Final value = 100 then increased value = 100 and increase in value = 20 (100-80)
## EXERCISE: Concepts Review
1. The price of a car depreciates in the first year by 25% in the second year by 20% in the third year by 15% and so on. The final price of the car after 3 years, if the present cost of the car is Rs. 10,00,000: a) 7,80,000 b) 5,10,000 c) 6,90,000 d) 1,70,000 40% of a number when added to the square of the same number increases to 4040% of itself. What is the actual number is? a) 175 b) 400 c) 40 d) 120 In an office 30% of the employees are females and 30% of these earn greater than Rs. 48,000 per month and 80% of male employees earn less than Rs. 48,000 per month. What is the percentage of employees who earn more than Rs. 48,000 per month? a) 30% b) 23% c) 60% d)cannot be determined In an examination 70% of the candidates passed in History, 50% in Geography and 20% students failed in both the subjects. If 500 students passed in both the subjects, then how many candidates appeared for the exam? a) 1,000 b) 1,500 c) 2,500 d) none of these My salary is Rs. 12,345 per month. The salary of my only sister is 9.09% greater than my only brother. The salary of my wife is less
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p in the value of N then q p increased value will be N(1+ ) q p B) If there is a decrease of in the value of N then q p decreased value will be N(1- ) q
A) If there is increase of
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than the total salary of my brother and sister together, and then the salary of my wife is: a) greater than my sister's salary b) less than my sister's salary c) equal to my salary d) greater than my own salary 12. 6. The cost of a car is 400% greater than the cost of a bike. If the is cost of the car is increased by 15% and that of bike is 20%. The total increase in the cost of the 5 cars and 10 bikes is : a) 17.5% b) c) 18.5% d) 18.25% The square of a positive number is 2,000% greater than the number itself, then the square of that number is: a) 1762 b) 1635 c) 441 d) 139 220% of a number 'X' is 44. What is 44% of 'X'? a) 88 b) 880 c) 66 d) data insufficient In an examination a candidate got 30% marks and failed by 30 marks. If the pass marks are 60% of the total marks then the maximum marks will be: a) 450 b) 600 c) 300 d) 100 In a library 60% of the books are in Hindi, 60% of the remaining are in English and rest of the books are in Urdu. If there are 3600 English books, then the total no. of Urdu books are: a) 2400 b) 2500 c) 3000 d) none of these In a class of MBA students 16.66% students are from Science background and 12.5% students are from commerce background and 6.66% students from arts background and
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rest are from Engineering background. The minimum possible students of engineering background students are: a) 45 b) 77 c) 100 d) 120
A salesman gets 9% commission on total sales. If the sale exceeds Rs. 10,000 he gets an additional commission as bonus of 3% on the excess of sales over Rs. 10,000. If he gets total commission of Rs. 1380, then the bonus he received is: a) Rs. 180 b) Rs. 120 c) Rs. 480 d) data insufficient In class, the no. of boys is more than the no. of girls by 12% of the total strength. The ratio of boys to girls is : a) 15 : 11 b) 11 : 14 c) 14 : 11 d) 8 : 11 In a factory there are three types of Machines produce 25%, 40% and 35% of the total products respectively. 2%, 4% and 5% of them are defective respectively. What is the percentage of non-defective products? a) 89% b) 97.1% c) 96.1% d) 86.1% A shepherd had n goats in the year 2000. In 2001 the no. of goats increased by 40%. In 2002 no. of goats decreased to 70%. In 2003 the no. of goats grew by 30%. In 2004, he sold 10% goats, and then he had only 34,398 goats. The percentage increase of the no. of goats in this duration was: a) 14.66% b) 16.66% c) 20% d) 33.33% In the above question in which year the no. of goats were minimum? a) 2000 b) 2001 c) 2002 d) 2004
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In the half year exams only 70% of the students were passed. Out of students only 605 students are passed in annual exam. Out of those who did not pass in the half year exams 805 passed in the annual exam. What per cent of the student passed the annual exam? a) 42% b) 56% c) 66% d) none of these
5.
The monthly salary of Shameer and Kanchan together is \$28,000. The salary of Shameer and Kanchan is increased by 25% and 12.5% respectively then the new salary of Kanchan becomes 120% of the the new salary of Shameer. The new salary of Shameer is: a) \$ 15,000 b) \$ 18,000 c) \$ 14,000 d) \$ 16,000 The price of an item increased by 25% and customer was able to purchase only 70% of his requirement. What is the net difference in the expenditure on that product? a) 10% more b) 5% more c) 12.5% less d) 17.5% less In a school there are 1800 students. Last day except 4% of the boys all the students were present in the school. Today except 5% of the girls all the students are present in the school, but in both in the day no. of students present in the school, were same. The no. of girls in the school is: a) 1200 b) 800 c) 1000 d) 600 A test contains two sections and total number of questions is N. Bhavit answered 20 out of 25 questions correctly in the first section. In the second section he answered 60% question correctly and thus his total score is 66.66% in the test. Given that all the questions carry equal marks and no negative marks for wrongly attempted questions. The total no. of question in the test is: a) 50 b) 60 c) 75 d) 100 The charges per hour of internet surfing is increased by 25% but the customer can afford only 10% increase in the expenditure. Find the percentage decrease in the time period of surfing? a) 22% b) 12% c) 15% d) 9.09%
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## Exercise: Application of Concepts
6. 1. In the overseas office of Micro-tech there are 60% female employees. 50% of all the male employees are computer literate. If 62% of total employees are computer literates out of the total 1600 employees, then the no. of female employees who are computer literate: a) 690 b) 672 c) 960 d)cannot be determined A shopkeeper charges sales tax of x% up to Rs. 2,000 and above it he charges y%. A customer pays total tax of Rs. 320, when he purchases the goods worth Rs. 6,000 and he pay's the total tax of Rs. 680 for the goods worth Rs. 12,000. What is the value of (x-y)? a) 0 b) -2 c) -4 d) 5 8. 3. In cyberzone office 30% of the employees are females and 30% of these earn greater than Rs. 48,000 per month. 80% of male employees earn less than Rs. 48,000 per month. What is the percentage of employees earn more than Rs. 48,000 per month? a) 30% b) 23% c) 60% d)cannot be determined Three candidates, A, B and C contested in an election. Out of the total votes on a voter list 25% did not vote and 6.66% votes polled were invalid. C got 2450 valid votes, which were 40% more than that of B. If A got only 40% of the total votes, then who is the winner? a) A b) B c) C d)cannot be determined
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10.
The average earning of each member of the Aakash family is 20% less than the average earning of each member of the Shahid family and the total earning of Aakash's family is 20% more than the total earning of Sahid's family. The no. of people in the Sahid family is what per cent of the no. of people of Aakash family: a) 20% b) 25% c) 33.33% d) 66.66% For the first three years of a decade the prices of Laptops continuously increased by 10% every year after that it decreased 10% every year for next three years. The price of a laptop in the sixth year will be approximately how much percent less than the price in first year if the same pattern of price is continued: a) 2 b) 3 c) 4 d) none of these A number x is mistakenly divided by 10 instead of being multiplied by 10. What is the percentage error in the result? a) - 99% b) + 99% c) - 100% d) + 100% The rate of the entrance ticket of Gemini circus is increased by 50% but the number of visitors decreased by 20%. What is percentage change in the total revenue of Gemini circus if the collection from tickets is the only source of income? a) + 20% b) - 25% c) + 30% d)cannot be determined The ratio of expenses to the savings of a family is 5:3. The expenses are increased by 60% but income increased by only 25%. Due to this the family savings were decreased by Rs. 3500. Find the income of the family: a) Rs. 35,000 b) Rs. 28,000 c) Rs. 25,000 d) Rs. 18,500 In an election two contestants participated. 15% of the voters did not vote and 41 polled
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votes were invalid. The winner got 341 more than runner-up and the winner received 45% of the total votes. Find the number votes received by each candidate: a) 2250 and 1936 b) 3568 and 3254 c) 2442 and 2128 d) 2457 and 2143 16. In an election only two candidates contested. 20% of the voters did not vote and 120 votes were declared as invalid. The winner got 200 votes more than his opponents and the winner secured 41% of the total votes. What percent of the voters supported the runnerup? a) 47.5% b) 41% c) 38% d) 45%
11.
12.
Directions for questions 17 and 18: Akbar, Amar and Antony were playing a game. At the beginning of the game Akbar and Amar together had 100% more money than Antony. Amar and Antony together had 300% more money than Akbar. At the end of the game Akbar and Amar had 100% more money than Antony and Akbar had 12.5% less money than Amar and Antony together had. At the end of the game Akar gained Rs.800. 17. Who lost maximum amount? a) Akbar b) Amar c) Both Amar and Antony d) cannot be determined What is the change of percentage of money of Amar? a) 40% b) 30% c) 57.1428% d) 42.857%
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18.
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19. A, B, C and D together contributed Rs. 50 lakhs to the community development. The contribution of B, C and D together is 460% that of A, the contribution of A, C and D together is 366.66% that of B's and the contribution of C is 40% that of A,B and D together. The amount contributed by D is: a) 10 lakh b) 12 lakh c) 16 lakh d) 18 lakh
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21.
Every day a fruit vendor sells half his stock, 10% of the stock overnight gets spoiled. If 1983 mangoes rotted over 3 nights then how many fruit were there initially? a) 25,000 b) 24,000 c) 30,000 d) 32,000 A man lost half of his initial amount in the gambling after playing 3 rounds. The rule of gambling is that if he wins he will receive Rs. 100, but he has to give 50% of the total amount after each round. Luckily he won all the three rounds. The initial amount with which he had started the gambling was : a) 500/3 b) 300 c) 700/3 d) 600 In every month Rachal consumes 25 kg rice and 9 kg wheat. The price of rice is 20% of the price of wheat and she spends a total of Rs. 350 on the rice and wheat per month. If the price of wheat is increased by 20% then what is the percentage reduction in rice consumption for the same expenditure of Rs. 350? Given that the price of rice and consumption of wheat is constant: a) 36% b) 40% c) 25% d) 24% The pressure of gas is directly proportional to the temperature and inversely proportional to the volume under the given conditions. If temperature is increased by 40% and the volume is decreased by 20% then the new pressure will be: a) increased by 75% b) reduced to 25% c) increased by 20% d) increased by 28% The number of employees in a bank is n. The Bank first hired p% employees then after a month q% employees left the office and the number of employees working is equal to n What is the value of p-q? a) pq b) pq/100 c) p/q d) none of these
25.
The price of an ornament first increased by x% and then decreased by x% so the net change in the price is Rs.21025. Again the price increased and decreased by x% respectively and the ornament was sold for Rs. 484416. What was the original price of the ornament? a) Rs. 5,00,000 b) Rs. 6,00,625 c) Rs. 5,25,625 d) Rs.5, 26,000
22.
23.
24.
## profit X 100 (CP) loss X 100 4) Loss percentage = (CP)
3) Profit percentage = 5) SP=
100 profit % X (CP) or 100 100 loss% X (CP) 100
Introduction: The application of concepts of this chapter is very much useful in regular exchange of cash transactions. Every Management Entrance exam gives at least certain minimum weight age to this chapter. Good understanding of the application of concepts of this chapter is absolute necessary to solve the problem within the given time.
6) CP=
## 100 X ( SP) 100 loss%
NOTE: Profit of loss is always calculated on Cost Price (CP) of an article unless otherwise mentioned.
CONCEPTS
TERMINOLOGY: Cost Price (C.P.): The money paid by the shopkeeper to the wholesaler or manufacturer is called the Cost Price (C.P.) Selling Price (S.P.): The price at which the shopkeeper sells the goods to the customers is called the Selling Price (S.P.) Profit: The Selling Price (SP) of an article is more that its Cost Price (CP) then the seller earns or makes a profit or gain. Loss: The Selling Price (SP) of an article is less than its Cost Price (CP) then the seller suffers a loss. NOTE: If there are any additional expenses such as taxes, commission, labour charges, transportation charges etc. and are to be added to the cost price (CP), then these expenses are called as overhead expenses or overheads. EXAMPLES: 1) A shopkeeper buys 300 pens at 5 for Rs.18 and sells at 2 for Rs.9 Find: a) The cost price of each pen b) Profit or loss on selling one pen c) Profit or loss percentage d) Overall profit or loss in the transaction 2) A fruit vendor buys 100 apples at Rs.12 per apple. Four apples are found to be rotten and the vendor sells the remaining apples at Rs.200 a dozen. Find: a) Total profit or loss b) Profit or loss percentage 3) A shopkeeper sold 100 quintals of rice at a profit of 12.5%. If the cost price of rice is Rs.24 per kg. Find the total profit made by shopkeeper and the selling price of sugar per kg. 4) A shopkeeper buys 200 quintals of pulses at Rs.1600 per quintal. He spends Rs.5000 on
transportation and Rs.5000 on local taxes. The shopkeeper sells the pulses at Rs.20 per kg. Find his total profit or loss and also calculate the profit or loss percentage. 5) Find the cost price of the television which is sold for Rs.12,000 at a profit of nm 33 X 100 m
3) The marked price of a motor cycle is Rs.66000 and the discount is 10%. What is the selling price of the motor cycle? 4) The mark-up percentage of an article is 100% and the discount percentage is 50%, then the profit percentage will be: 5) A wholesaler marks-up the article by 20% and offers a discount of 20%. What is the profit or loss percentage?
CONCEPTS CONCEPTS
Marked Price (MP): Basically, in order to avoid the loss due to bargaining by the customer or to get the profit seller increases the cost price (CP) by a certain value. This increase in value over cost price is known as Mark-up and increased price is called as the Marked Price (MP). Marked price = CP + mark-up Or Marked price = CP + (% mark-up on CP) 1) When two articles are sold at the same price and the percentage profit of one article is same as the percentage loss of the other article, then there will be a loss in the whole transaction.
or )loss 2 [ Gain(10 ]
Loss% =
Discount: The reduction in price over the marked price is called Discount. Basically discount means concession. It is calculated on the basis of marked price. SP = Marked price Discount Or SP = Marked price (% discount of MP)
2) If the selling price of m articles is same as the cost price of n similar articles (where m<n), then No.ofArticlesUnsold X 100 Profit % = No.ofArticleSold
=
nm X 100 m
NOTE: Mark-up is calculated on CP and Discount is calculated on MP. EXAMPLES: 1) The cost price of a TV is Rs.36000 and the mark-up percentage is 25. What is the marked price? 2) The marked price of an article is Rs.36000 and the mark-up percentage is 12.5. What is the cost price?
EXAMPLES: 1) A man sells two bags at the same price Rs.300 but one at a profit of 10% and another at a loss of 10%. Find the: a) Overall percentage profit or loss b) Net profit or loss 2) The cost price of two articles is Rs.1000 and selling price of both the articles is same. One of them was sold at a loss of 20% and
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other one at a profit of 20%. Find the net profit or loss? 3) If the cost price of 20 apples is same as selling price of 25 apples, then find the overall profit or loss percentage. 4) If the selling price of 100 keyboards is the same as the cost price of 125 keyboards. What is the profit or loss percentage? 5) A fruit vendor by selling 12 apples gains the cost price of 10 apples. What is the percentage of profit or loss? 6) By selling 8 Ice creams a vendor gains the selling price of 1 Ice cream. What is the total gain or loss percentage? 7) By selling 9 houses, the contractor losses the price of 1 house. Find his loss percent.
Exercise: Concepts Review 1. The cost price of a Computer is Rs. 20,000 and the profit percent is 12%. What is the selling price of the computer? a) 2400 b) 22040 c) 2600 d) 22400 The SP of an article is Rs. 32000 and the profit per cent is 25%. Find the cost price. a) Rs. 20000 b) Rs. 20000 c) Rs. 24000 d) Rs. 30000 The CP of an article is 5/6th of the SP. What is the percentage profit or loss? a) 20% loss b) 16.66% profit c) 16.66% loss d) 20% profit The MP of a camera is 3/2 of the CP and SP is 9/10 of MP. Find the percentage profit or loss. a) 25% profit b) 35% profit c) 33.33% loss d) none of these The MP of an article is 30% higher than the CP and a discount of 20% is given on MP. Find t he profit percentage: a) 10% b) 14% c) 4% d) 26% On selling an article for Rs. 576 a trader loses 4%. In order to gain 4%, he must sell that article for: a) Rs. 636 b) Rs. 676 c) Rs. 625 d) cannot be determined The profit percentage made on an article when it is sold for Rs. 112 is thrice than when it is sold for Rs. 84. The cost price of the article is: a) Rs. 96 b) Rs. 98 c) Rs. 100 d) Rs. 70
2)
3.
4. 8) A dishonest seller sells wheat at the cost price but weighs 20% less wheat. What is his/her percentage profit? 9) A shopkeeper while buying the goods gets 10% extra and while selling he gives 10% less. Find his overall percentage gain. 10) What is the overall discount when two successive discounts of 10% and 20% are offered on the marked price of an article? 6.
5.
7.
8.
A shopkeeper uses a weight of 870gms instead of 1000gms and sells the articles at the cost price. What is the approximate profit percentage? a) 13% b) 23% c) 15% d) 20% A trader uses a weight of 920g instead of 1kg and sells the articles at the marked price which is 15% above the cost price. Find the profit percentage. a) 20% b) 23% c) 25% d) cannot be determined
15.
c) Rs. 750 d) Rs. 900 At what per cent above the cost price must an article be marked so as to gain 17% after allowing a discount of 10%? a) 34% b) 70% c) 30% d) 27% A vendor buys orange @ Rs. 2 for 3 oranges and sells them @ 1 rupee each. How many oranges should he sell to make a profit of Rs.10? a) 10 oranges b) 20 oranges c) 30 oranges d) 40 oranges The price of a KML scooter is Rs. 28,000. The dealers in metros give a discount of 10% and dealers in state capitals give a discount of 12% on the first Rs. 20,000 and 8% on the rest Rs. 8000. What is the difference between the selling prices? a) Rs. 240 b) Rs. 420 c) Rs. 640 d) none of these A trader sells two articles, one at a loss of 10% and another at a profit of 15% and made no profit or loss. If the total sale price of these two articles is Rs. 60,000, then find the difference between their cost prices: a) Rs. 10000 b) Rs. 12000 c) Rs. 15000 d) none of these What is the profit percentage in selling an article at a discount of 20% which was earlier sold at a profit of 40%? a) 20% b) 14% c) 28% d) 12% A man buys 18 oranges for a rupee and sells them at 12 oranges for a rupee. What is the profit percentage? a) 33.33% b) 50% c) 66.66% d) none of these A dealer buys a product at Rs. 1920 and sells it at a discount of 20% making 20% profit. What is the selling price of that product? a) Rs. 2304 b) Rs. 1536
33
16. 9.
17. 10. If a gift pack is sold at a gain of 6% instead of at a loss of 6%, then the seller gets Rs. 6 more. The cost price of the article is: a) Rs. 60 b) Rs. 66 c) Rs. 50 d) Rs. 36 A man sells a bicycle at a gain of 10%. Had he bought it at 10% less and sold it for Rs. 132 less, he would have gained 10%. The cost price of the article is: a) Rs. 1000 b) Rs. 1200 c) Rs. 1500 d) Rs. 1320 A dealer gives a discount of 5% on the MP of a cooker. If he gives a discount of 8%, then he earns Rs. 36 less profit. The marked price of the cooker is: a) Rs. 1000 b) Rs. 1200 c) Rs. 800 d) none of these If a commission of 10% is paid to the sales executive, the publisher gains 20%. If the commission is increased to 15%, the gain of publisher will be: a) 13.33% b) 15% c) 18% d) data insufficient A retailer buys a cellular phone at a discount of 15% and sells it for Rs. 5865 and makes a profit of 15%. The discount is: a) Rs. 200 b) Rs. 850
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11.
18.
12.
19.
13.
20.
14.
21.
22.
c) Rs. 2200 d) Rs. 2000 What should be the minimum mark-up percentage such that after giving a discount of 50% there will be no loss? a) 200% b) 133.33% c) 100% d) 150% A shopkeeper calculated his profit per cent as 50% on the selling price. What will be the profit percentage if he had calculated on the cost price? a) 75% b) 100% c) 33.33% d) none of these An item was sold after giving two successive discounts of 20% and 10% respectively. What was the marked price of the item if it was sold at Rs.468? a) Rs. 600 b) Rs. 500 c) Rs. 575 d) Rs. 650 A single discount equivalent to three successive discounts of 20%, 10% and 5% is: a) 68.4% b) 35 % c) 31.6% d) 32% A fruit vendor purchases oranges at Rs. 10 per dozen and sells them at Rs. 12 for every 10 oranges. What is the profit percentage? a) 40% b) 44% c) 60% d) 48%
29.
Best store sells two different flavours of ice creams at the same price. It sells one at a profit of 20% another at a loss of 20%. What is the loss incurred by Best store? a) 1% b) 2% c) 4% d) 0% In the previous question, if SP of each article be Rs. 200, what is the amount of loss? a) Rs. 10 b) Rs. 16 c) Rs. 16.66 d) none of these The cost price of 19 articles is same as the selling price of 29 articles. What is the loss%? a) 35% b) 34.48% c) 52.63% d) none of these
23.
30.
31.
24.
25.
26.
27. The selling price of 13 articles is same as the cost price of 23 articles. What is the profit percentage? a) 43.47% b) 74.83% c) 78% d) 76.92% 28. A dishonest shopkeeper declares that he sells rice at the cost price. However he gives 50gms less per every 500gms. What is his percentage profit? 1 a) 10% b) 11 % 9 1 c) 12% d) 9 % 11
Exercise: Application of Concepts 1. 40% goods are sold at 5% loss while rest are sold at 10% profit. If there is a total profit of Rs. 100, then the worth of goods sold is: a) Rs. 2000 b) Rs. 2500 c) Rs. 4000 d) none of these A retailer bought 20kg silver at a discount of 10% at the international precious metal exhibition and he won 1 kg silver against his coupon number. Now he sells all the silver at the marked price. What is his profit percentage? a) 30% b) 12% c) 16.66% d) none of these A retailer seller sold two articles at the same price. He sold one at a profit of 75% and other one at a loss of 30%. What is the overall profit or loss in the transaction? a) 22.5% profit b) 57.5% profit c) loss d) none of these Tamim purchased the article for Rs. 123456. He sold 60% of those at a profit of 16.66% and rest at loss. Find the loss percentage on the remaining if the overall loss is equivalent to 14%? a) 20% b) 30% c) 60% d) 66.66% A dealer gives as much discount (in per cent) as the mark-up (in per cent) above the cost price. What is the profit or loss percentage? a) 10% b) 1% c) 4% d) cannot be determined A trader sells 20kg of sugar at Rs. 600. A customer asks 20% discount and the trader agrees to it but instead of 1kg he gives 4% less sugar. How much effective discount does the customer get? a) 16% b) 16.66% c) 15.5% d) 19.6%
7.
The cost price of an article 'A' is Rs. 160 and selling price of another article 'B' s Rs. 240. Article A is sold at a profit of 20%. If the selling price of A will be equal to the cost price of B, then what is the profit of 'B'? a) 16.66% b) 50% c) 25% d) none of these A trader uses a weighing balance that shows 1250gms instead of 1000gms. He further increases the selling price by 20%. What is the profit percentage? a) 5% b) 45% c) 50% d) 30% A person sold two cows each for Rs. 9900. If he gained 10% on one and lost 20% on the other, then which of the following statements is true? a) He gained Rs. 200 b) He lost Rs. 200 c) He neither gains nor losses d) None of the above Two third of a goods was sold at a profit of 5% and the rest at a loss of 2%. If the total profit was Rs. 600, find the value of the goods (in rupees): a) 20000 b) 22500 c) 120000 d) 45000 A person loses Rs. 20 by selling some bananas at the rate of Rs. 3 per banana and gains Rs. 30, when sold at Rs. 3.25 per banana. The number of bananas sold by him is: a) 100 b) 200 c) 120 d) 2400 By selling 6 apples for a rupee, a man loses 20%. How many for a rupee should he sell to gain 20%? a) 4 b) 8 c) 8 d) 5
2.
8.
9.
3.
4.
10.
11. 5.
6.
12.
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A dealer buys a washing machine, listed at Rs. 10000 and gets 10% and 20% successive discounts. He spends 10% of the SP on transport. At what price (in rupees) should he sell the washing machine to earn a profit of 10%? a) 8722 b) 7892 c) 8712 d) 8840 Two cool drink companies sell same type of carbonated sweetened drink. Company A gives two successive discounts of 10% and 25% and company gives two successive discounts of 15% and 20%. What is the ratio of their marked price of Company A to B? a) 143 : 144 b) 19 : 11 c) 136 : 135 d) 73 : 77 When a shopkeeper reduces the selling price from 540 to 513 its loss increases by 4 percentage point. What is the selling price of the same article when it fetches a profit of 4%? a) Rs. 696 b) Rs. 702 c) Rs. 725 d) Rs. 675 When an article is sold for Rs. 703 loss incurred is 25% less than the profit earned on selling it at Rs. 836. What is the selling price of the article when it earns a profit of 20%? a) 912 b) 1576 c) 1532 d) 1092 A balance of a trader weighs 10%less than it should be and he makes a profit of 20%. What is the mark-up percentage on the cost price ? a) 40% b) 8% c) 25% d) 16.66% Each of A and B sold their article at Rs. 1818 but A incurred a loss of 10% while B gained a profit of 1%. What is the ratio of cost price of the articles of A to B? a) 101 : 90 b) 85 : 89 c) 81 : 75 d) none of these
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19.
Even after a discount of Q% on marked price a trader gains P% profit. What is the mark-up percentage over the cost price? PQ PQ X 100 a) b) X 100 100 P 100 Q PQ X 100 c) d) not possible QP A milkman initially mixes 10% water in pure milk and again he mixes 10% more water to the solution. What is the profit percentage of milkman if he sells the mixture at cost price? a) 11.11% b) 20% c) 21% d) 12.1% A trader mixes 100l kerosene to 400l of petrol and sells the mixture at the price of petrol. If the cost price of kerosene is 50% of the cost price of petrol, then what is the net profit percentage? 1 1 a) 9 % b) 11 % 11 9 c) 10% d)20% A shopkeeper calculates his profit percentage on the selling price. The SP of an article is Rs.280 and the profit percentage he 2 calculated is 14 % . What is the actual profit 7
percentage?
14.
20.
21.
15.
22.
16.
## b) 16.66% d) data insufficient
17.
A vendor sells articles at certain profit percentage. If he sells articles at 1/3 of the actual selling price, then he incurs a loss of 40%. What is his actual profit percentage? a) 72% b) 120% c) 80% d) none of these A retailer increases the selling price by 25% due to which profit percentage increases from 20% to 25%. What is the percentage increase in the cost price? a) 20% b) 30% c) 25% d) 50%
36
18.
24.
25.
Due to reduction of 25% in price of apples a customer can purchase 4 apples more for Rs. 16. What is original price of one apple? a) Rs. 1 b) Rs. 1.33 c) Rs. 1.5 d) Rs. 1.6 A reduction of 20% in the price of sugar enables a housewife to purchase 6kg more for Rs. 240. What is the original price per kg of sugar? a) Rs. 10 b) Rs. 8 c) Rs. 6 d) Rs. 5 Seema got 5% discount on the actual sale price of the mobile and she paid the shopkeeper Rs. 3325 without tax. Tax is 20% of the selling price of the mobile if customer wants the bill. What is the amount of discount that Seema got? a) 750 b) 375 c) 875 d) 525 A trader marks his goods such that he can make 32% profit after giving 12% discount. However a customer bargained and got 20% discount instead of 12%. What is the new profit percentage of trader? a) 20% b) 44% c) 30% d) 28.8% Anil sold his car to Bhanu at a profit of 20% and Bhanu sold it to Chameli at profit of 10%. Chameli spent 10% of his purchasing price and then sold it at a profit 8.33% to Anil once again. What is the loss of Anil? a) 23 % b) 29% c) 50% d) 40% Kunal bought a house in Silicon city, whose sale price was Rs. 80 lakh. He availed 20% discount as an early bird offer and then 10% discount due to cash payment. After that he spent Rs.6.4 lakh to renovate the house. At what price should he sell the house to earn a profit of 25%? a) Rs. 80 lakh b) Rs. 72 lakh c) Rs. 64 lakh d) none of these
31.
26.
The cost price of an article is C and the selling price of the same article is S, where A is the profit or loss percentage. If the cost price and selling price both are increased by same amount then which of the following is true: a) A increases b) A decreases c) A remains constant d) none of these A dishonest retailer cheats his wholesaler and customer both. He purchases 19% more from the wholesaler and sells 15% less while selling to its customer. What is the profit percentage earned by the retailer by selling the goods at cost price? a) 36.78% b) 34% c) 40% d) 36.85%
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28.
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30.
## INTERESTS & INSTALMENTS
Introduction: The concepts of this chapter are widely used in Banking and Financial sectors. The concepts of this chapter are very useful for Data Interpretation questions and closely follow the concepts of other chapters like Percentages, Profit Loss and Discount, etc.
## P = Principal R = Rate of Interest T = Time period 2) Compound Interest (CI) = 1 Amount = P 1
R 100
]t
R 100
]t
CONCEPTS TERMINOLOGY Principal (P): It is some of the money deposited/invested in banks/financial institutions or borrowed/loaned etc. Principal is also called as Capital. Interest: It is the amount paid by borrower and is calculated on the Principal and time. Time (T or N): It is the duration for which money is borrowed/deposited etc. Rate of Interest( R or r): It is the rate at which the interest is charged on the amount borrowed. Simple Interest (SI): It is the amount calculated uniformly only on the principal for the given time (or) period. Compound Interest (CI): It is the amount calculated on the principal for the first year and the amount calculated on the principal and on the total interest earned until the previous year.
Amount when interest is calculated halfR / 2 2t yearly = P [ 1 ] 100 Amount when interest is calculated R / 4 4t quarterly = P [ 1 ] 100 3) Difference between CI and SI for two years R 2 = P[ ] 100 4) Difference between CI and SI for three years R 2 R = P[ ] [ 3] 100 100
CONCEPTS 1) Depreciation of value against time: The prices of certain goods Vehicles, Machineries, Electronic gadgets, etc. - depreciate in their value over a period of time.
Vc/f = Vi[1
R t ] 100
## Vc/f = Value of the goods (current/ or face value)
CONCEPTS Vi = Value of the goods initial R = Rate of depreciation
P XR XT 100
FORMULAE
1) Simple Interest (SI) =
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T = Time period
38
2) Population related problems: In it generally observed that the population of any locality/region/country not remain same. That is the population keeps on changing over the previous year. To calculate the change in the population, the compound interest formulae are used. If the population increased from the previous year(s), then the current population = P 1
4.
What is the principal which would fetch an interest of 6000 at 12% rate of interest? a) Rs. 6720 b) Rs. 4800 c) Rs. 50000 d) Rs. 72000 In what time will a sum of money double itself @ 20%per annum simple interest? a) 10 years b) 5 years c) 2 years d) 4 years A sum of money trebles in 16 years at r% of simple interest per annum. What is the value of r? a) 12 % b) 12.5% c) 25% d) 18.75% A sum of Rs. 25000 is lent out in two parts both at SI: one at 12% p.a. and another at 12.5% p.a. for one year. If the total annual income is Rs. 3060, the money lent at 12% is: a) 1000 b) 1200 c) 1500 d) 1300 World Bank lent Rs. 6000 million \$ to Bangladesh for 2 years and Rs. 1500 million \$ to Sri Lanka for 4 years and received altogether from both Rs. 900 million \$ as simple interest. The rate of interest is: a) 4 % b) 8 % c) 10 % d) 5 % A person takes a loan of Rs. 200000 at 5% simple interest. He returns Rs. 100000 at the end of one year. In order to clear his dues at the end of 2 years, he would pay: a) 125500 b) 110000 c) 115500 d) none of these A person invested Rs. 170000 in MBI bank at 16% simple interest and interest is calculated half-yearly. What is the total amount paid by the bank to the person at the end of the year? a) 198288 b) 28288 c) 298288 d) 27200
39
5.
[ [
R 100
]t ]t
6.
If the population decreased from the previous year(s), then the current population = P 1
R 100
7.
3) Equal monthly instalments (EMIs): If the borrower pays back the amount he/she has borrowed in instalments and the repayments amount is uniform in all the months, then it is called the EMI. EMI for SI = [X + (X +
XXRX1 ) + (X + XXRX 2 + ----] 100 100
8.
Exercise: Concept Review 1. A man borrows Rs. 8000 and pays back after 5 years at 15% simple interest. The amount paid by him is: a) Rs. 6000 b) Rs. 16000 c) Rs. 7500 d) Rs. 14000 What is the time period for which Rs. 80000 amounts to Rs. 120000 at 20% p.a. of simple interest? a) 4 years b) 2.5 years c) 3.25 years d) 6 years What is the rate of simple interest at which Rs. 16,000 yields an interest of Rs. 2560 in two years? a) 4% b) 5 % c) 7 % d) 10 %
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9.
2.
10.
3.
11.
The compound interest on Rs. 1000 at 10% p.a. for 3 years is: a) 331 b) 1331 c) 133 d) 300 A sum of Rs. 4000 would become Rs. 4410 after 2 years at r% compound interest. Find the value of 'r'? a) 10% b) 5 % c) 15 % d) 20 % A certain amount doubles in n years at x% compound interest. In how many years will the amount be four-fold? a) n years b) 3n years c) 2n years d) 4n years Rs. 6000 amounts to Rs. 7986 in 3 years at CI. What is the rate of interest? a) 20% b) 10% c) 6% d) 7.5% The least number of complete years in which a sum of money put at 20% CI will be more than doubled is: a) 4 b) 5 c) 6 d) 8 The difference between CI and SI on a sum of money lent for 2 years at 10% is Rs. 400. The sum is: a) 16000 b) 44000 c) 40000 d) none of these A certain sum amounts to Rs. 8988.8 in two years and to Rs. 9528.128 in three years, at compound interest per annum. What is the principal and rate of interest? a) Rs. 12,000, 5% b) Rs. 6,000, 8% c) Rs. 8,000,6% d) Rs. 10,000, 8.5% A watch is sold for Rs. 4400 cash or for Rs. 2400 as initial payment and Rs. 2440 to be paid after one month. Find the rate of interest charged in the instalment scheme: a) 10% b) 22%
c) 20% 19.
d) 25%
12.
Jackson purchases a track suit for Rs. 2400 cash or for Rs. 1000 cash down payments and two monthly instalments of Rs. 800 each. Find the rate of interest: a) 75% b) 120% c) 50% d) none of these Women welfare society borrowed Rs. 10815 from a bank, which is to be paid back in 3 equal half yearly instalments. If the interest is 40 compounded half yearly at % per 3 annum, how much is each instalment? a) 2048 b) 3150 c) 4096 d) 5052
20.
13.
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15.
16.
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## TIME and WORK
Introduction: This chapter finds its space in CAT regularly. All other management entrance tests accommodate enough space for this chapter every year without failing. The concepts involved in this chapter shares similar platform with Percentages, RPV, etc. A thorough understanding of concepts of Percentages and Proportion would enable the candidate to analyse the problems of this chapter very easily.
CONVERSIONS Unit work into Percentage efficiency: If a person completes work in n days, then his/her one day work = 1/n and his percentage efficiency = 1 X 100 n No. of hours/days to complete the work N 1 2 3 4 5 6 7 8 9 10 Work of 1 hour/day 1/n 1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 Percentage efficiency 100/n 100% 50% 33.33% 25% 20% 16.66% 14.28% 12.5% 11.11% 10%
CONCEPTS EFFICIENCY
Efficiency is directly related to time (days, hours, minutes, etc.) when the work and workforce are constant. Suppose a person can complete a work in 10 days then he/she finishes 1/10th or 10% of the work each
day.
The above is very similar to the percentage fraction table given in the Percentage chapter. This table is useful for faster calculations. Relation between Efficiency and Time: Efficiency is inversely proportional to the time (minutes/hours/days/months etc.) when the work is constant. Example: If A is twice as efficient as B, then A takes half the time taken by B to finish the work.
If a person can complete a work in n days then he/she finishes 1/n part of the work in one day. Also if a person finished 1/n work in one day, then he/she will finish the work in n days.
NOTE: Calculation of percentages and conversion of ratios into percentages are pre-requisite skills one should learn immediately to understand the nuances of the problem based on the concepts of this chapter.
Examples: 1) A Can do a job in 24 days and B can do the same job in 12 days. In how many days will they be able to finish the job working together?
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2) A can finish the work in 20 days, B can finish the same work in 24 days and C can finish the same work in 30 days. In how many days will they be able to finish the job working together? 3) A finish the work in 20 days and B finish the same work in 10 days. What is the ratio of efficiency? 4) A is thrice as efficient as B and A takes 30 days to finish the job. In how many days will B finish the work? 5) A is thrice as efficient as B and is therefore able to finish the work in 30 days less than B. Find the time in which A and B together finish the same work?
2) If 100 men can finish constructing a wall in 36 days working 10 hours a day, they how many men are required to complete the same work in 20 days working 6 hours a day? 3) Softpro sent 100 employees to its onshore office to finish a project in 100 days. After the 25 days company reviewed the progress and learnt that only 20% of the work is completed. a) How many extra days needed to finish the project? b) How many extra employees are required to finish the work on time? c) How many extra employees are required to finish the project in 20 days less the schedule time? CONCEPTS
Comparison of different efficiencies: A persons or persons efficiency(ies) will remain(s) same throughout the work. However, if men or women or boys or girl of different efficiencies work together in same job/work, the efficiencies must be compared and should be converted into the efficiency of any particular persons/groups efficiency. Example: 6 men can do a work in 4 days and 12 boys can do the same work in 8 days. It means the work can be finished in 6x4 = 24 mandays i.e. 24 men can finish the work in one day. Similarly the work can be finished in 12x8 = 96 boydays i.e., 96 boys can finish the work in one day. It can concluded that work of 24 men is equal to the work of 96 boys i.e., efficiency of 1 man is equal to efficiency of 4 boys.
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## CONCEPTS Product constancy
This concept is same as the concept of inverse proportion, which is thoroughly discussed in RPV, and is also widely used in Profit & Loss chapter. The product constant method is limited to the constant work, if the amount of work gets changed, then method does not applicable. The efficiency of work done is inversely proportional to the time i.e., if the rate of work done is greater, then the time required to finish the work will be less and if the rate of work done is less, then the time required to finish the work will be more. EXAMPLES:
1) 10 persons can do a piece of work in 7 days then in how many days 14 persons of same efficiency finish the same work?
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EXAMPLES: 1) 6 boys or 8 girls finish the work in 24 days. Find the number of days taken by 6 boys and 8 girls to finish the work? 2) 6 boys and 8 girls finish a job in 6 days and 11 boys and 8 girls finish the same job in 4 days. In how many days 1 boy and 1 women working together finish the same work? 3) A can complete a work in 24 days, B in 30 days. Find the time taken by them to finish the work: a) When A and B work together b) When A and B work on alternate days with A working on the first day c) When A and B work on alternate days with B working on the first day d) If A joins with B two days after B started the work e) If B joins with A two days after A started the work f) When A and B start the work together and A left two days before the completion of the work g) When A and B start the work together and B left two days before the completion of the work h) When A and B start the work together and A left two days before the schedule completion of the work i) When A and B start the work together and B left two days before the schedule completion of the work 1.
Exercise: Concepts Review A can finish a piece of work in 24 days while B can do it in 30 days. In how many days will the work be finished if they work together? a) 12 days b) 16 days 3 4 c) 13 days d) 16 days 9 9 A can do a piece of work in 20 days. B can do it in 24 days and C can do it in 30 days. In how many days can the work be finished if they work together? a) 12 b) 8 c) 10 d) 20 A can do a piece of work in 10 days, B can do it in 20 days. With the help of C, they finish the work in 4 days. In how many days C alone can complete the work? a) 6 b) 8 c) 10 d) 12 A can do a piece of work in 10 days. B can do it in 24 days. If C also works with them then it takes only 6 days to complete the work. In how many days C alone can complete the work? a) 25 b) 40 c) 50 d) 75 A can do a piece of work in 24 days. If B is 60% more efficient than A, then the number of days required by B to complete the same piece of work is: a) 10 b) 15 c) 12 d) 9.6 A is twice as good a workman as B and together they finish a piece of work in 28 days. In how many days can A alone finish the work? a) 20 b) 21 c) 24 d) 22
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A is thrice as good a workman as B and takes 10 days less to do a piece of work than B takes. The number of days taken by B to finish the work is: a) 9 b) 18 c) 15 d) 12 A is twice as good a workman as B and therefore A takes 6 days less than B to finish the work individually. If A and B working together complete the work in 4 days, then in how many days B alone complete the work? a) 12 b) 18 c) 8 d) 6 A is thrice as efficient as B and C is twice as efficient as B. What is the ratio of number of days taken by A, B and C, when they work individually? a) 2 : 6 : 3 b) 2 : 3 : 6 c) 1 : 2 : 3 d) 3 : 1 : 2 A is thrice as efficient as B. Working together they complete the work in 3 days. If B takes 8 days more than A, what is the number of days taken by A to finish the work alone? a) 4 b) 2 c) 12 d) 16 A and B can do a piece of work in 8 days, B and C can do the same work in 12 days and A and C complete the same work in 8 days. In how many days will they finish the work together? a) 4 b) 6 c) 12 d) 9 A, B and C can do a piece of work together in 1 day. A is thrice efficient as B and B takes twice the number of days as C takes. What is the difference between the number of days taken by A and C? a) 1 b) 2 c) 3 d) 4
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A can finish a work in 12 days and B can do it in 15 days. After A had worked for 3 days B joined with A to finish the remaining work. In how many days will they be able to finish the remaining work? a) 3 b) 4 c) 5 d) 6 A can do a piece of work in 14 day and B can do the same work in 21 days. They started the work together but 3 days before the completion of the work A left the work. Find the number days needed to finish the work: a) 7 b) 8.5 c) 5 d) A can do a piece of work in 20 days. He started the work and left after completing 25% of the work. After that B completed the work in 10 days. In how many days A and B complete the work together? a) 6 b) 8 c) 10 d) 12 A can do a piece of work in 9 days, B in 18 days and C in 3 days. In how many days will the work be finished, if A and B started the work and 3 days later C joined them? a) 12 b) 8 c) 4 d) 6 Four pipes A, B, D and D are connected to a tank. The time in which A, B, C and D take to fill the tank are in ascending order. The greatest part of the tank which can be filled by any three pipes in an hour is 13/36th. The least part of the tank which can be filled by any three pipes in an hour is 1/4th. Time taken by all the pipes to fill the tank is 2.4 hours. Find the time taken (in hours) by B and C to fill the tank. 60 48 a) b) 7 7 36 c) d) cannot be determined 7
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A and B can do a piece of work in 9 days and 12 days respectively. If they work on alternate days starting with A, in how many days can they complete the work? a) 11 b) 9 c) 11.11 d) 10.25 A, B and C complete the work individually in 6, 12 and 8 days respectively. In how many days will they complete the work if only one person work on any day and they work in the following order: A, B, C, A, B, C, A a) 7 b) 7.5 c) 8.5 d) 8 In the previous question if the order of working days be as B, C, A, B, C, A ........... then in how many days will they finish the work? 3 a) 7 b) 8 4 1 c) 8 d) 9 4 A takes 6 days less than B and 2 days more than C to complete the job. A and B together can do the work in the same time as C can complete the job. In how many days B alone complete work ? a) 10 b) 14 c) 12 d) 16 C takes twice the number of days then A to do the job. A and B together can do it in 6 days while B and C can do it in 10 days. In how many days A alone finish the job? a) 60 b) 30 c) 6 d) 7.5 A and B together do 70% of the work and B and C together do 50% of the work. Who is most efficient? a) A b) B c) C d) cannot be determined
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60 persons can do a piece of work in 12 days. How many more people are required to complete the work in 10 days? a) 10 b) 15 c) 12 d) none of these 12 women can do a piece of work in 20 days. If the 4 women are absent, then how many more days are required to complete the work? a) 6 b) 10 c) 15 d) none of these A group of men complete the work in 20 days. But 12 men did not turn up for the job and the remaining men did the job in 32 days. How many men were there initially? a) 32 b) 36 c) 42 d) 40 30 workers can finish a work in 20 days. But 9 workers leave the job after some days and the work was completed in total 26 days. After how many days did they leave? a) 12 b) 10 c) 6 d) none of these 20 persons completed 1/3rd of the work in 12 days. How many more persons are required to finish the rest work in 12 days? a) 20 b) 12 c) 18 d) 40 A contractor undertook a work to complete in 60 days. But just after 20 days he observed that only 1/5th of the work had been completed. To complete the work in time how many more workers should join, if there were initially 75 workers? a) 25 b) 50 c) 75 d) cannot be determined 6 men or 10 women can reap a field in 15 days, and then the number of days that 12 men and 5 women will take to reap the same field is:
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If 2 men or 3 women or 4 boys can do a piece of work in 52 days, then in how many days 1 man. 1 woman and 1 boy complete the work? a) 48 days b) 36 days c) 45 days d) none of these 6 children and 2 men complete a work in 6 days. Each child takes twice the time taken by a man to finish the work. In how many days will 5 men finish the same work? a) 6 b) 8 c) 9 d) 15 Three men and two women can do a piece of work in 4 days, while two men and three women can do the same work in 5 days. Rs. 44 is paid to a woman per day. What is the amount paid to a man per day? a) Rs. 88 b) Rs. 144 c) Rs. 154 d) cannot be determined If 8 women collect 200kg of tea leaves in 10 hours. How many more (in kg) of tea leaves will 12 women collect in 8 hours? a) 24kg b) 40kg c) 50kg d) 100kg 4 boys and 5 girls can do a piece of work in 10 days. 6 boys and 6 girls can do the same work in 7 days. In how many days can 2 boys and 7 girls complete the same work? a) 15days b) 14 days c) 21days d) 18 days 33men can do a job in 30 days. 44 men started the job together and every day one person leaves the job at the end of the day. What is the minimum number of days required to complete the whole work? a) 21 b) 42 c) 45 d) none of these
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C is twice as efficient as A, B takes trice as many days as C. A takes 12 days to finish the work alone. If they work in pairs (i.e. AB, BC, CA) starting with AB on the first day then BC on the second day and AC on the third day and so on, then in how many days will they finish the work? 1 1 a) 5 days b) 4 days 9 2 1 c) 6 days d) 8 days 5 B is twice efficient as A and A can do a piece of work in 15 days. A started the work and after a few days B joined him. The work is completed in 11 days. For how many days they worked together? a) 1 day b) 2 days c) 6 days d) 5 days Pipe A can fill a tank in 72 minutes and pipe B can fill it in 90 minutes If both the pipes are opened to fill an empty tank, in how many minutes will it be full ? a) 30 b) 36 c) 40 d) 50 Pipe A and B can fill a cistern in 10 hours and 15 hours respectively. When a third pipe C which works as an outlet pipe is also open then the cistern can be filed in 18 hours. The outlet pipe can empty a full cistern in: a) 12 hours b) 8 hours c) 9 hours d) 14 hours Tap A can fill a tank in 20 hours, B in 25 hours and C can empty a full tank in 30 hours. Starting with A, followed by B and C each tap opens alternatively for one hour period till the tank is filled completely. What is the total time taken by all the pipes to fill the tank? 4 4 a) 24 days b) 24 days 11 11 2 c) 52 days d) none of these 3
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Exercise: Application of Concepts 6. 1. The leakage of the container lets 11kg oil is leaked out per day then it would last for 50 days consumption, if the leakage was 15kg per day, then it would last for 45 days consumption. If there is no leakage, how many would the oil last for consumption? a) 80 days b) 72 days c) 100 days d) 120 days 7. 2. A contractor deployed 20men to complete the work in 40 days. 8 days before the scheduled time it is realised that 1/3rd of the work is still to be completed. How many more men are required to complete the work in the stipulated time? a) 16 b) 15 c) 20 d) 25 A piece of work can be completed by 10men and 6 women in 18 days. Men works 9 hours per day while women works 7.5 hours per day. Per hour efficiency of a woman is 2/3rd of a man's efficiency. In how many days 1 man and 9 women complete the work? a) 16 days b) 20 days 3) 30 days d) 25 days A can do a piece of work in 10 days, B in 15 days. They work together for 5 days and the rest of the work is finished by C in two more days. If they get Rs. 3000 as wages for the whole work, what are the daily wages of A, B and C respectively (in Rs): a) 200,250,300 b) 300,200,250 c) 200,300,400 d) none of these Two pipes A and B can fill a cistern in 15 hours and 10 hours respectively. A tap C can empty the full cistern in 30 hours. All the three taps were open for 2 hours and outlet pipe is closed. How many more hours would it take to fill the cistern? a) 30 min. b) 1.2 hours c) 24 min d) 35 min
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A can do a work in 20 days and B can do the same work in 25 days. They started the work together and few days later C joined with them and all of them completed the whole work in 10 days. All of them were paid a total of Rs. 700. What is the share of C? a) Rs. 130 b) Rs. 185 c) Rs. 70 d) cannot be determined A and B working together completed a job in 8 days. If A worked twice efficiently as he actually did and B worked 1/3 as efficiently as he actually did, the work would have been completed in 6 days. Find the time taken by A complete the job alone: 38 a) 8 days b) days 35 15 40 c) days D) days 2 3
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Direction for questions 8 to 12: Microtech got a project from Olympic incorp and sent certain number of employees to its onshore office. 8 days later 20% of the employees left onshore and it was found that it would take as much time to complete the rest of the project as the total time needed to complete the entire project with all the employees present. Every employee finishes 20 tasks per hour. 8. How many employees left the project? a) 10 b) 5 c) 16 d) cannot be determined How many employees were sent to onshore? a) 10 b) 5 c) 15 d) 4 What could be the number of employees remained at work when 20% of the employees left the project? a) 15 b) 18 c) 68 d) 78
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In how many days would the project be completed with the original workforce? a) 32 days b) 48 days c) 40 days d) cannot be determined 16 days after the 20% employees left the job it was decided to complete the work on time by increasing the work-force again. By how much percentage should the work-force be increased? a) 100% b) 50% c) 200% d) none of these
Directions for questions 16 and 17: Four pipes A, B, C and D can fill a cistern in 40, 50, 80 and 100 hours respectively. 16. Pipe A was opened at 6:00am, B at 8:00am, C at 9:00am and D at 10:0am. When will the cistern be full? a) 4:18pm b) 3: 09pm c) 12:15pm d) 11:09am If A and B are opened as inlet pipes and C and D are opened as outlet pipes and all the four pipes are opened simultaneously, then in how many hours will the cistern be filled? 4 a) 80 hours b) 44 hours 9 8 c) 44 hours d) 90 hours 9 A tank is connected with four pipes A, B, C and D of which two are inlets and other two are outlets. The time taken by A, B, C and D to fill or empty the tank is 20 hours, 30 hours, 40 hours and 60 hours respectively. All four pipes are opened. When the tank was empty, it took 24 hours to fill it completely. Which two are the outlet pipes? a) A and B b) C and D c) A and C d) B and D
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Directions for questions 13 and 14: 16 men and 10 women working 12 hours a day can complete a work in 8 days. 8 men and 10 women working for 16 hours a day can complete the same work in 10 days. Also 10 boys working 16 hours a day can complete the same job in 60 days. 13. If 8 men, 6 women and 8 boys worked together every day for 10 hours, then in how many days will they complete the work? a) 3 b) 4 c) 8 d) 6 If women and children (boys) can't be employed, then minimum how many men are required to complete the job in 12 days if the number of hours working per day should not exceed 18 hours? a) 4 b) 5 c) 6 d) 7 A group of workers was put on a job, from the second day onwards one worker left the job each day. The job was finished just before last worker was about to leave the work. Had no worker been left the job at any state, the group would have finished the job in 5% of the time. How many workers were there in the group? a) 50 b) 40 c) 45 d) 10 18.
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Directions for questions 19 and 20: There are n taps 1, 2, 3, 4..n connected to a tank. Tap1 and Tap2 fill the tank in the same time, Tap3 takes half the time taken by Tap2, Tap4 takes half the time taken by Tap3 and so on i.e. Nth Tap takes half the time taken by (N-1)th Tap. 19. If the 12th tap takes 2 hours to fill the tank alone then what is the ratio of efficiency of 10th tap and 14th tap respectively? a) 4 : 1 b) 5 : 3 c) 16 : 1 d) 1 : 16
If the 10th tap takes 80 hours to fill the tank then the 12th and 14th taps working together take how many hours to fill the tank? a) 2 hours b) 4 hours 2 c) 6 hours d) 6 hours 3 Directions for questions 21 to 23: Willow, cricket bat manufacturing company, grabbed the contract from BCCI to manufacture 1000 bats in 20 days. Willow has 5 expert bat makers who can finish the task in the given time period. However, before starting the work BCCI request willow to complete the work before the schedule. Willow decided to add 5 more experts in every 2 days to complete the work as per the BCCIs request. 20. 21. How many experts are working on the last of the work? a) 25 b) 15 c) 20 d) cannot be determined Find the number of days BCCI request Willow to complete the job? a) 8 b) 6 c) 10 d) 7 If the BCCI further asked the Willow to manufacture additional 500 bats and Willow continues to add 5 experts in every two days, then in how many more days are required to complete the job? a) 5 b) 1 c) 4 d) 2 A worker starts a job. On the first day, he does 1 unit of work. On the second day, he does 2 units of work. Every day from the second, he does 2 units of work more than that of the previous day. Starting from the second day, a worker joins every day and all the workers working on a day do the same amount of work. The work gets completed at the end of the Mth day. If the job is [8 M (M + 1) + 1] units, find M. a) 13 b) 15 c) 14 d) 16
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Twenty workers started a job. On the first day, each worker completed 1 unit of work. After every day of work a worker leaves and each of the remaining worker's rate of doing work decrease by 10%. Find the number of units that would be completed in the first 10 days. a) 10 (11 - (0.9)11) b) 10 (11 - (0.9)10) c) 10 (11 - (0.9)9) d) 10 (11 - (0.9)8)
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1. Three types of rice whose rates are Rs. 28, Rs. 33 and Rs. 39 per kg are blended together to make a 15 kg of new blend of rice in which there are 8 kgs, 4kgs, 3kgs of the respective types of rice. The average price of the new blend of rice is: a) Rs. 31.53 b) Rs. 33 c) Rs. 30 d) Rs. 33.3 Praneet went to the college @ 60km/hr while returning for his home he covered the half of the distance @ 10km/hr rest half of the distance @ 30km/hr. The average speed of the whole journey is: 123 typists typed 384 papers in 1/15 hour. The number of papers typed per minute by an average typist is: a) 1 b) 2 c) 3 d) 5 The cost of the Red, Green and Blue colours ribbons are Rs. 20, Rs. 15 and Rs. 18 respectively. Rang purchased them in the ratio 3:2:4 respectively. Find the average cost per ribbon: a) 18 b) 20 c) 17.66 d) cannot be determined I went to Delhi @speed of 200km/hr and returned to the same place @ speed of 600km.hr. What is my average speed? a) 300km/hr b) 400km/hr c) 366.66km/hr d) none of these The average age of 3 children of Anil is 12 years and their ratio of their ages is 3:4:5. The average age of the youngest and eldest child is: a) 12 b) 21 c) 8 d) 9 The average of 8,15,22,29.....78 is: a) 8 b) 12 c) 43 d) 67
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A train covers a certain distance at a speed of 60 km/hr. However, if it were to halt for a fixed time interval in each hour its average speed reduces to 50km/hr. How many minutes per hour does it stop? a) 10 minutes b) 20 minut c) 6 minutes d) 12 minutes
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Chennai express left for New Delhi, increasing its speed in each hour. It started its journey from Chennai but after four hours of its journey it met with accident. Its speed in the fourth hour was 7/5 times that of the third hour and the speed in the third hour was 10/7 times that of the second hour and in the second hour it was 7/5 times that of the first hour. If it had travelled with the half of the speed that of the third hour, then it would have travelled 160km less in the same time. The average speed of the train uring the journey of 4 hours was: a) 50km/hr b) 90km/hr c) 80km/hr d) cannot be determined The average age of all the 100 employees in an office is 29 years, where 40% of the employees are ladies and the ratio of average age of men to women is 5:7. The average age of female employees is: a) 18 years b) 35 years c) 25 years d) none of these A candidate walked 6km to reach the station from his house at a certain speed. He boarded a train whose average speed was 60kmph to reach the destination. In this way he took total of 3 hours. The average speed of the whole journey is 32kmph. Find his walking speed: a) 3km/hr b) 4.5km/hr c) 4km/hr d) none of these Students of a top B-School got placements in three different sectors Consultancy, IT and
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Marketing. The average salary of Consultancy and IT sectors is 26 lakh per annum, IT and Marketing sectors is 44 lakh per annum and the average salary of Consultancy and Marketing sectors is 34 lakh per annum. The approximate range of salaries of three sectors: a) lies between 30 and 44 b) lies between 28 and 34 c) lies between 34 and 43 d) lies between 29 and 48 13. While travelling from City A to City B, which is 160 km apart, the driver of the four-wheeler taxi had to use the spare tyre. What is the average number of kilometres travelled by each tyre? a) 40km b) 120km c) 128km d) 19kgs The number of students preparing for IIMCAT from four different educations Engineering, Science, Commerce and Humanities is 40, 60, 50 and 30 respectively and the respective percentage of students who qualified in the CAT is 80%, 75%, 60% and 50% respectively. Find the overall percentage of successful candidates. a) 67.77% b) 66.66% c) 68.5% d) none of these There is twice the number of two wheelers as there are three wheelers and the number of 4 wheelers is equal to the number of two wheelers. The average number of wheel per vehicle is: a) 3 b) 4 c) 5 d) none of these
one in Malaysia. Later on he sold the Indian factory at 16% profit and Malaysian factory at 24% profit while the total profit is 19%. The selling price of Indian factory is: a) 45 crore b) 52.2 crore c) 8.55 crore d)cannot be determined 18. In a 25 litre mixture of milk and water, the water is only 20%. How many litres of water is required to increase the percentage of water to 90%? a) 45 litre b) 70 litre c) 115 litre d) 175 litre A milkman sells the milk at the cost price but he mixes the water (freely available) in it and thus he gains 9.09%. The quantity of water in the mixture of 1 litre is: a) 83.33ml b) 90.90ml c) 99.09ml d)cannot be determined The price of petrol is Rs. 60 per litre and the price of spirit is Rs. 40 per litre. In what ratio the petrol and spirit be mixed such that the profit after selling the mixture at Rs. 75 per litre is 25%? a) 1:1 b) 3:2 c) 5:1 d) Such a mixture is not possible A trader sells 315 Mobile phones. Selling price of each mobile is same. He sells Android based phones at a loss of 6% and windows based phones at a profit of 15%. Thus he gains 9% on the whole. What is the no. of Android based phones sold? a) 126 b) 216 c) 135 d) 90 Gigaby sells two types of DVDs viz. 4.8GB disks and 8GB disks. He sells 4.8GB DVD at Rs. 18 and incurs a loss of 10% whereas on selling the 8GB at Rs. 30 per kg he gains 20%. In what proportion should 4.8GB and 8GB DVDs be sold so that the total profit 25% after by the mixtures at Rs. 27.5? a) 3:2 b) 2:3 c) 2:5 d) 3:5
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The average marks of the students in four sections A, B, C and D together is 60%. The average marks of A, B, C and D individually are 45%, 50%, 72% and 80% respectively. The average marks of A and B together is 48% and that of B and C together is 60%. What is the ratio of number of students in sections A and D? a) 2:3 b) 4:3 c) 5:3 d) 3:5 A student obtained equal marks in History and Science. The ratio of marks in Science and Geography is 2 : 3 and the ratio of marks in History and Philosophy is 1 :2. The student has scored an aggregate of 55% marks and the maximum marks in each subject are same. In how many subjects has he scored equal to or greater than 60% marks? a) 1 b) 2 c) 3 d) none of these The ratio of income al Anil and Mukesh is 2 : 3. The sum of their expenditure is Rs. 8000 and the savings of Anil is equal to the expenditure of Mukesh. What is the sum of their savings? a) 22,000 b) 4,000 c) 16,000 df) 12,000 There are two vessels containing the mixture of milk and water. In the first vessel the water is 2/3 of the milk and in the second vessel water is just 40% of the milk. In what ratio should they be mixed to make 24 litres mixture in which the ratio of water is to milk is 1 : 2? a) 4 : 3 b) 5 : 7 c) 5 :2 d) 7 :5 The LCM of two numbers is 210 and their ratio is 2 : 3. The sum of numbers is: a) 210 b) 175 c) 315 d)cannot be determined
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What number must be subtracted from each of the numbers 53, 21, 41, 17 so that the remainders are in proportion? a) 1 b) 3 c) 5 d) none of these The angles of a triangle are in the ratio of 2 : 3 : 4. Find the measurement of greatest angle a) 300 b) 600 c) 1000 d) 800 A goldsmith has 722 rings of gold. He sells some of them at a loss of 4% and rest at a profit of 15% making overall profit of 8%. Find the no. of rings sold at a profit of 15%. a) 342 b) 266 c) 436 d) 456 Mohit travelled from Chennai to Bangalore covering total distance of 250 miles in 8hr. partly by car at 30 miles/hr. and rest by train at 35 miles/hr. The distance travelled by car is: a) 150 miles b) 80 miles c) 220 miles d) 180 miles Two containers A and B of capacities 140 litres and 60 litres respectively filled with crude oil of different prices. Equal quantities are drawn from both A and B in such a manner that the oil drawn from A is poured into B and the oil drawn from B is poured into A. The price per litre becomes equal in both A and B. How much oil id drawn from each of A and B : a) 40 litre b) 21 litre c) 42 litre d)cannot be determined The ages of A, B, C and D are in arithmetic progression, but not in that order. The ratio of ages of A and B is 6 : 5 and C and D is 7 : 8. Two year later the age of A and B will be 2 : 3. Find the ratio of ages of C and D: a) 7 : 6 b) 5 : 8 c) 6 : 7 d) 8 : 9
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The ratio of volumes of two cubes is 8 : 27. What is the ratio of surface areas of these cubes respectively? a) 2 : 3 b) 4 : 9 c) 8 : 19 d) 9 : 4 40. A couple got married 9 years ago and the age of wife was 20% less than the husband at the time of marriage. 6 years from now the age of wife will be 12.5% less than her husband. Now they have six children including single, twins and triples and the ratio of their ages is 2:3:4 respectively. What can be the maximum possible value for the present age of this family? a) 110 years b) 103 years c) 105 years d) 83 years Two liquids are mixed in the ratio 4:3 and the mixture is sold at Rs. 20 with a profit of 33.33%. The price of first liquid is Rs.7 more than that of second liquid. Find the sum of the costs of both the liquids: a) Rs. 11 b) Rs. 29 c) Rs. 35 d) Rs. 70 A vessel of capacity 2 litre has 25% alcohol and another vessel of capacity 6 litre has 40% alcohol. The liquid in the two vessels poured into a vessel of capacity 10 litres and the rest part of the vessel was filled with the water. What is the new concentration of mixture? a) 31% b) 71% c) 49 % d 29% Alloy A contains 40% gold and 60% silver. Alloy B contains 35% gold and 40% silver and rest copper. Alloys A and B are mixed in the ratio of 1:4. What is the ratio of gold and silver in the newly formed alloy? a) 20% and 30% b) 36% and 44% c) 25% and 35% d) 49% and 36% Chaco store sells candies at Rs. 15 per kg. A candy is made up of flour and sugar in the ratio of 5:3. The ratio of price of sugar and
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Flour is 7:3 and the store earns profit. What is the cost price of sugar? a) Rs. 10/kg b) Rs. 9/kg c) Rs. 18/kg d) Rs. 14/kg The area of a rectangle is 62cm2, but while measuring its length and breadth a student noted as 20% less and 25% more respectively. What is the percentage error in area calculated by the student? a) 5% b) 25% c) cannot be determined d) none of these An alloy contains the copper and aluminium in the ratio of 7:4. While making the weapons from this ally, 12% of the alloy got destroyed. If there is 12kg aluminium in the weapon, then the weight of the alloy required is: a) 48kg b) 40kg c) 37.5 kg d) 14.4 kg A person wants to buy a FM radio costing Rs. 1404 including sales tax at 8%. He asks the shopkeeper to reduce the price of radio so that he can save the amount equal to the sales tax. The reduction of the price of the radio is: a) Rs. 108 b) Rs. 104 c) Rs. 112.32 d) none of these The average weight of a class of students is 67.5 kg. The weight of the class teacher is 25% more than the average weight of the class. The average weight of the class is less than the class teacher by x%. The value of x is : a) 33.33% b) 255 c) 20% d) cannot be determined The average of a set of whole numbers is 27.2. When the 20% of the numbers are eliminated from the set of numbers then the average becomes 34. The number of elements in the new set of numbers can be: a) 27 b) 35 c) 52 d) 63
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35.
41.
36.
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37.
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The raw material and manufacturing cost formed individually 70% and 30% of the total cost and the profit percentage is 14.28% of the raw material. If the cost of raw material increases by 20% and manufacturing increases by 40% and the selling price is increases by 80%, then the new profit percentage is: a) 57% b) 65.8% c) 60% d) cannot be determined A company made a cubodial box of size 16x12x5, but later on it was found that the capacity of the box was 14.28 less than the required capacity. As per the requirement the company had to increase the length and breadth of the box in equal amount then the percentage increase in the area of the base of the box is a) 12.5% b) 6.66% c) 16.66% d) none of these The amount of work in a leather factory is increased by 50%. By what per cent is it necessary to increase the number of workers to complete the new amount of work in the previously planned time, if the productivity of the new labour is 25% more? a) 60% b) 66.66% c) 40% d) 33.33% The SP of an article is Rs. 3200 and the profit 1 or loss percent is 33 % . Find the cost price. Di 3 a) Rs. 20000 b) Rs. 2000 c) Rs. 2400 d) Rs 3000 The ratio of cost price and marked price of an article is 2:3 and ratio of percentage profit and percentage discount is 3:2. What s the discount percentage? a) 16.66% b) 20% c) 25% d) 33.33% A shopkeeper sold 12 remotes at a profit of 20% and 8 remotes at a profit of 10%. If he
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had sold all the 20 cameras at a profit of 15, then his profit would have been reduced by Rs. 36. What is the cost price of each remote? a) 100 b) 150 c) 180 d) 220 51. The profit percentage on the three articles A, B and C is 10%. 20% and 25% respectively and the ratio of the cost price is 1:2:4. Also the ratio of number of articles sold is 2: 5: 2 respectively, then the overall profit percentage is: a) 18.5% b) 21% c) 75% d) none of these The total number of men, women and children working in a factory is 18. They earn Rs. 4000 in a day together. If the sum of the wages of all men, all women and all children is in the ratio 18 : 10 : 12 and the wages of an individual man, woman and child is in the ratio 6 : 5 : 3, then how much a woman earn in a day ? a) Rs. 400 b) Rs. 250 c) Rs. 150 d) Rs. 120 Three men and 5 women together can finish a job in 3 days. 3 women take 5 more days to finish the work done by 2 men. What is the ratio of efficiency of a man and a woman? a) 2 : 1 b) 3 : 2 c) 5 : 2 d) 4 : 1
46.
52.
47.
53.
48.
49.
50.
Directions for questions 54 to 57: The world fastest athlete championship is a three round competition. The athlete who finishes the three rounds in minimum time will be declared as the world champion. The total distance of the championship is 360kms. The athletes cover the first one-third of the distance on cycle, the second one-third by walking and the remaining one-third by car. The speed of the champion is as follows: The average speed of the car is 5 times that of cycle and 20 km/h more than that of walking, but the winner has taken 1 hour more by walking than by car.
55
54.
What is the average speed of the whole journey of the winner? a) 15 km/h b) 24 km/h c) 20 km/h d) none of these What is the time taken by the winner in the whole journey? a) 10 h b) 12 h c) 15 h d) none of these What is the distance covered by the winner in last five hours of her journey? a) 250 km b) 240 km c) 200 km d) cannot be determined Instead of covering the first one-third by cycle the winner covers it by walking. Then what is the percentage decrease / increase in time taken during the entire journey? a) 46.66% b) 33.33% c) 50% d) 25%
55.
Directions for questions 60 and 61: Shatabdhi and Rajadhani superfast trains start simultaneously from Bombay and Pune towards each other and continuously run between these two places. Every time they meet they exchange their speeds as well as directions. The ratio of the speeds of Rajadhani and Shatabdhi is 2:1 respectively. 60. What is the number of distinct places at which they meet? a) 1 b) 2 c) 5 d) none of these Let these two trains first time meet a Lonavala, then what is the ratio of distances covered by Shatabdhi and Rajadhani super fast trains till they meet for the third time at Lonavala: a) 1 : 1 b) 14 : 13 c) 10 : 11 d) none of these Akbar and Birbal start walking towards each other respectively from Agra and Mathura 144km apart at the same time. Akbar walks at the constant speed of 8km/h, while Birbal walks 4km in the first hour, 5 km in the second hour, 6 km in the third hour and so on. Then the Akbar and Birbal will meet : a) in 6 h b) in 8 h c) midway between Agra and Mathura d) 80 km away from Mathura A cheetah is 50 of its own leaps behind a deer. The cheetah takes 5 leaps per minute to the deer's 4. If the cheetah and the deer cover 8m and 5m per leap respectively, what distance will the cheetah have to run before it catches the deer ? a) 600m b) 700m c) 800m d) 1000 m A thief sees a police jeep at a distance of 250m, coming towards him at 36 km/h. Thief takes 5 seconds to realise it and start running away from police at 54km/h. But police realise after 10 seconds, when the thief starts
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62. 58. An athlete walking diametrically across a semicircular playground takes 3 minutes less to cover the distance than if she/he covers walking around the circular path from A to B. If the speed of the athlete is 60m per minute, what is the length of the diameter of the play ground? a) 60 m b) 48 m c) 84 m d) 315 m The speed of a car during the second hour of its journey is thrice that of the first hour. Also its third hours speed is the average speed of the first two hours. Had the car travelled at the second hours speed constantly during the first three hours, and then it would have travelled 150km more. Find the percentage reduction in time in the second case 1 a) 33 % b) 40 % 3 c) 25 % d) 50% 63.
59.
64.
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running away and stats chase at 72km/h. How long after thief saw police did police catch- up with him and what is the distance police had to travel to do so? a) 50s, 1000m b) 65s, 1150m c) 65s, 1300m d) 45s, 1050m 65. In a circus there were a leopard and a tiger walking in the two different rings of same radii. There I observed that when leopard moved 3 steps, tiger moved 5 steps in the same time, but the distance traversed by leopard in 5 steps is equal to the distance traversed by tiger in 4 steps. What is the number of rounds that a leopard made when tiger completed 100 rounds? a) 120 b) 48 c) 75 d) none of these
70.
Bolts and Johnson are running on a circular track of radius 175 metres. Bolt can completes a round in 100 seconds and the speed of Johnson is twice the speed of Bolt. They started simultaneously towards each other from two points 350 metres diametrically opposite on the circular path. They first meet at a point O which is between the two points P and Q from where they have started their race, after how much time from the start do they meet at the point O for the third time ? 2 2 a) 218 S b) 216 S 5 3 c) 221 s d) none of these If the two incorrect watches are set 12:00 noon at correct time, when will both the watches show the correct time for the first time given that the first watch gains 1 min in 1 hour and second watch loses 4 min in 2 hours: a) 6pm, 25 days later b) 12 : 00 noon, 30 days later c) 12 noon, 15 days later d) 6am 45 days later At a railway station a 24 hour watch loses 3 minutes in 4 hours. If it is set correctly on Sunday noon when will the watch show the correct time? a) 6pm after 40 days b) 12 noon after 75 days c) 12 pm after 100 days d) 12 noon after 80 days Out of a sum of Rs. 625, a part was lent at 5% SI and other at 10% SI. If the interest on the first part after 2 years is equal to the interest on the second part after 4 years, then the second sum (in Rs.) Is: a) 250 b) 300 c) 125 d) 275
71.
Directions for questions 66 and 67: A train enters into a tunnel MN at M and exits at N. A cat sitting at O which is 1/3rd of the length of the tunnel after hearing the whistle of the train starts running towards entrance M and meets the trains at M. If the cat runs towards the exit N it just meets the train at N. 66. What is the ratio of speeds of Train and Cat? a) 1 : 3 b) 5 : 3 c) 3 : 1 d) cannot be determined The ratio of lengths of distance covered by Train before meeting the cat at entrance M and exit N. a) 3 : 1 b) 3 : 5 c) 1 : 3 d) cannot be determined A candle of 6cm long burns at the rate of 5cm in 5hrs and another candle of 8cm long burns at the rate of 6cm in 4hrs. What is the time required by each candle to remain of equal lengths after burning for some hours, when they start to burn simultaneously with uniform rate of burning? a) 1 h b) 1.5 h c) 2 h d) none of these
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72.
68.
69.
73.
74.
A sum of Rs. 2500 is lent out in two parts: one at 12%p.a. and another at 12.5% p.a. for one year. If the total annual income is Rs. 306, the money lent at 12% is: a) 1000 b) 1200 c) 1500 d) 1300 Mr. Banker has lent some money to Mr. Customer at 6% p.a. and the Mr. Investor at 8% p.a. At the end of the year Banker had gained the overall interest at 7% per annum. In what ratio has the Banker lent the money to Customer and Investor? a) 2 : 3 b) 1 : 1 c) 5 : 6 d) 4 : 3 A person invested 1/7 of his total investment at 4% and 1/2 at 5% and rest at 6% for the one year and received total interest of Rs. 730. What is the total sum invested? a) Rs. 70000 b) Rs. 14000 c) Rs. 24000 d) Rs. 38000 The rate of interest in two banks DNB and HBI are in the ratio of 7:8. If a person invested some amount in both the banks and received equal amounts from both the banks at the end of two years. The ratio of amount invested in DNB and HBI respectively is: a) 15 : 1 b) 8 : 7 c) 7 : 8 d) 108 : 107 The ratio of CI for 3 years and SI for 1 year for a fixed amount at a rate of r% is 3.64. What is the value of r ? a) 10% b) 15 % c) 5% d) none of these A and B have to write 810 and 900 pages respectively in the same time period. But A completes his work 3 days ahead of time and B completes 6 days ahead of time. How many pages did A write per hour if B writes 21 pages more in each hour? a) 45 b) 72 c) 54 d) 100
80.
75.
Pipe A basically used as inlet pipe and pipe B is used as outlet pipe. Pipes A and B both are opened simultaneously. When pipe A fills the tank and B empties the tank it takes double time than when both the pipes fill the tank. What is the ratio of efficiency of pipe A and pipe B respectively? a) 3 : 1 b) 5 : 2 c) 1 : 3 d) 3 : 2
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79.
CONTENTS
Chapter 2
Chapter 3
28
Chapter 4
## RCs 41-50: Advanced Level (Day 41 - Day 50)
52
The passages given in IIM-CAT or MBA entrance exams do not come from one particular source and number of words per passage is also not fixed. RCs in IIM-CAT and MBA entrance exams come from different subjects like: Social Sciences History Political Science Sociology Psychology Philosophy Economics Science
The Reading Comprehension section in IIM-CAT or MBA entrance exams is designed to test the aspirants ability to read and understand the contents of the given passage and answer the questions given at the end of the passage in a short period of time. Hence, the aspirants need to understand TWO PREREQUISITE SKILLS: READING SPEED: Advantages COMPREHENSION:
## Astronomy Botany Chemistry Physics Zoology
Humanities Art Music Literature Current Affairs Social Political Economical Sports
Step 3: Know the length of RC passages Normally IIM-CAT passages vary from 300 words to 1500 words. However, there were passages with 200 words each as well as 2500 words in other MBA entrance exams. Till 1998, the Reading Comprehension (RC) was a separate section. Starting from 1999, the RC and Verbal Ability section were merged and approximately half of the questions were RC questions and the other half Verbal Ability. Read the following table:
COMPREHENSION: Understanding what is given in the passage is the most important pre-requisite skill aspirant must learn in order to attempt the questions correctly. Aspirant may be able to read the passage much faster than others but if the aspirant fails to understand the nuances presented in the passages then the advantage of reading speed is of no use at all. While the aspirant reading the passage, he/she should continuously and constantly think, evaluate, reason out, interpret and infer what has already read. To perform all the above tasks aspirant should read various topics extensively on a regular basis. Step 4: Familiarize yourself with various types of questions Identify the details Main point/idea of the passage Title for the passage Draw the inferences/assumptions Determine the meaning of the words/ phrases Tone of the passage Method/Flaw/Parallel of the reasoning of argument expressed in the passage
his/her span of vision is very limited. It also points out that the readers grammar skills or very limited. Thus, the reader will not be able understand even simple ideas presented in the RC passages.
Voracious Reader: If you ask me How to score good marks in RC of CAT, I will say learn the RC concepts thoroughly. The above underlines clearly indicate that the voracious reader would be able to read longer sentences without much difficulty. It also points out that the reader generally recognizes phrases and clauses easily. Thus, the reader will be able to read faster and also able to understand complex sentence constructions.
However, everybody can become a good reader with regular reading of different topics. Beginner should start reading small stories because story introduces characters and qualities of all the characters and concludes
Passage 1
DAY 1 modern day Pakistan, setting the stage for several successive invasions from Central Asia between the 10th and 15th centuries CE, leading to the formation of Muslim empires in the Indian subcontinent such as the Delhi Sultanate and the Mughal Empire. Mughal rule came from Central Asia to cover most of the northern parts of the subcontinent. Mughal rulers introduced Central Asian art and architecture to India. In addition to the Mughals and various Rajput kingdoms, several independent Hindu states, such as the Vijayanagara Empire, the Maratha Empire, Eastern Ganga Empire and the Ahom Kingdom, flourished contemporaneously in southern, western, eastern and north-eastern India respectively. The Mughal Empire suffered a gradual decline in the early 18th century, which provided opportunities for the Afghans, Balochis, Sikhs, and Marathas to exercise control over large areas in the northwest of the subcontinent until the British East India Company gained ascendancy over South Asia. Beginning in the mid-18th century and over the next century, large areas of India were gradually annexed by the British East India Company. Dissatisfaction with Company rule led to the Indian Rebellion of 1857, after which the British provinces of India were directly administered by the British Crown and witnessed a period of both rapid development of infrastructure and economic decline. During the first half of the 20th century, a nationwide struggle for independence was launched by the Indian National Congress and later joined by the Muslim League. The subcontinent gained independence from the United Kingdom in 1947, after the British provinces were partitioned into the dominions of India and Pakistan and the princely states all acceded to one of the new states. No. of words: 638 Starting Time: ________
Ending Time:________ Reading Speed:_______ 1) What is the Main Point/Central Idea of the passage? -------------------------------------------------------------------------------------------------------------------------------------
## 2) Write the summary of the passage? -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The third reason computers won is piracy. Users prefer it not just because its free, but because its more convenient. Bittorrent and YouTube have already trained a new generation of viewers that the place to watch shows is on a computer screen. The somewhat more surprising force was one specific type of innovation: social applications. The average teenage kid has a pretty much infinite capacity for talking to their friends. But they cant physically be with them all the time. When I was in high school the solution was the telephone. Now its social networks, multiplayer games, and various messaging applications. The way you reach them all is through a computer. Which means every teenage kid (a) wants a computer with an Internet connection, (b) has an incentive to figure out how to use it, and (c) spends countless hours in front of it. This was the most powerful force of all. This was what made everyone want computers. Nerds got computers because they liked them. Then gamers got them to play games on. But it was connecting to other people that got everyone else: thats what made even grandmas and 14 year old girls want computers. After decades of running an IV drip right into their audience, people in the entertainment business had understandably come to think of them as rather passive. They thought theyd be able to dictate the way shows reached audiences. But they underestimated the force of their desire to connect with one another. Facebook killed TV. That is wildly oversimplified, of course, but probably as close to the truth as you can get in three words.
3) Vocabulary
Passage 2
DAY 2
About twenty years ago people noticed computers and TV were on a collision course and started to speculate about what theyd produce when they converged. We now know the answer: computers. Its clear now that even by using the word convergence we were giving TV too much credit. This wont be convergence so much as replacement. People may still watch things they call TV shows, but theyll watch them mostly on computers. What decided the contest for computers? Four forces, three of which one could have predicted, and one that would have been harder to. One predictable cause of victory is that the Internet is an open platform. Anyone can build whatever they want on it, and the market picks the winners. So innovation happens at hacker speeds instead of big company speeds. The second is Moores Law, which has worked its usual magic on Internet bandwidth.
## No. of words: 418
Starting Time:________
Ending Time:________ Reading Speed:______ 1) What is the Main Point/Central Idea of the passage? -------------------------------------------------------------------------------------------------------------------------------------
## 2) Write the summary of the passage? ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Vocabulary:
Passage 3
DAY 3
In 1896 a Georgia couple suing for damages in the accidental death of their two year old was told that since the child had made no real economic contribution to the family, there was no liability for damages. In contrast, less than a century later, in 1979, the parents of a three year old sued in New York for accidentaldeath damages and won an award of \$750,000. The transformation in social values implicit in juxtaposing these two incidents is the subject of Viviana Zelizers excellent book, Pricing the Priceless Child. During the nineteenth century, she argues, the concept of the useful child who contributed to the family economy gave way gradually to the present-day notion of the useless child who, though producing no income for, and indeed extremely costly to, its parents, is yet considered emotionally priceless. Well established among segments of the middle and upper classes by the mid-1800s, this new view of childhood spread throughout society in the late-nineteenth and early-twentieth centuries as reformers introduced child labour regulations and compulsory education laws predicated in part on the assumption that a childs emotional value 93 46 56 2008
made child labour taboo. For Zelizer the origins of this transformation were many and complex. The gradual erosion of childrens productive value in a maturing industrial economy, the decline in birth and death rates, especially in child mortality, and the development of the companionate family (a family in which members were united by explicit bonds of love rather than duty) were all factors critical in changing the assessment of childrens worth. Let expulsion of children from the cash nexus,...although clearly shaped by profound changes in the economic, occupational, and family structures, Zelizer maintains. was also part of a cultural process of sacralisation of childrens lives. Protecting children from the crass business world became enormously important for late-nineteenthcentury middle-class Americans, she suggests; this sacralization was a way of resisting what they perceived as the relentless corruption of human values by the marketplace. In stressing the cultural determinants of a childs worth. Zelizer takes issue with practitioners of the new sociological economics, who have analysed such traditionally sociological topics as crime, marriage, education, and health solely in terms of their economic determinants. Allowing only a small role for cultural forces in the form of individual preferences, these sociologists tend to view all human behaviour as directed primarily by the principle of maximizing economic gain. Zelizer is highly critical of this approach, and emphasizes instead the opposite phenomenon: the power of social values to transform price. As children became more valuable in emotional terms, their exchange or surrender value on the market, that is, the conversion of their intangible worth into cash terms, became much greater.
## No. of words: 444
Starting Time:________
Ending Time:________ Reading Speed:______ 1) What is the Main Point/Central Idea of the passage? -------------------------------------------------------------------------------------------------------------------------------------
## 2) Write the summary of the passage? ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Vocabulary:
person do not apply, and natural laws are disobeyed. Some examples of illogical dream content includes flying, time travel, talking animals, sudden transformations of both people and objects and sudden shifts in setting. The odd events and content that occur in dreams are typically accepted without question by the dreaming mind. According to Hobson, the unquestioning acceptance of dream content is due to the strength of our internally generated emotions and perceptions (Dreaming, 5). Within the dream, these strange and illogical events, perceptions and objects are not seen as being out of place. If the dream is remembered upon waking, the content of the dream is seen as odd or even difficult to explain. Strange sensory experiences are another cardinal characteristic of dreams. The sensation of falling, an inability to move quickly and being unable to control body movements are just a few of the commonly reported sensory experiences that occur during dreams. While memory seems to be intensified within the context of the dream, access to the information contained within the dream diminishes rapidly once the dreamer wakes. Dream researchers estimate that approximately 95% of all dreams are forgotten entirely upon awakening. While many people may familiar with these five common characteristics of dreams, some may be unaware of just how common these experiences are. Dream characteristics and dream object may be of an everyday nature or altogether fantastic and impossible collages of existing reality; they may behave normally or indulge in the most absurd, improbable or impossible actions in settings either familiar or bearing only the faintest resemblances to those of real life, Hobson explains.
Passage 4
DAY 4
Dreams have fascinated artists, philosophers and researchers for thousands of years. However, it was not until fairly recently in history that dreams became the subject of serious scientific study. While dreams can vary considerably, sleep researcher J. Allan Hobson (1988) identified five basic characteristics of dreams: One of the major characteristics of dreams is that the emotions experienced in dreams can be intense, painful and acute. People commonly report dreaming about deeply embarrassing situations (i.e. being nude in public) or profoundly terrifying events (i.e. being chased by an attacker). In some instances, these emotions can become so intense that they interrupt the dream or cause the dreamer to wake abruptly. The three most common emotions that become intensified by dreams are anxiety, fear and surprise Dreams are full of discontinuities, ambiguities and inconsistency, but sometimes these things can lead to downright bizarre dream content. According to Hobson (4), one of the hallmarks of dreams is illogical content and organization, in which the unities of time, place and 93 46 56 2008
## No. of words: 437
Starting Time:________
Ending Time:________ Reading Speed:______ 1) What is the Main Point/Central Idea of the passage? ------------------------------------------------------------------6
## ------------------------------------------------------------------2) Write the summary of the passage? ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Vocabulary:
So where does this philosophy generally come from? A major influencing factor on your philosophy is the culture that you live in and in particular the culture that you grew up in. If you were born and raised in India, for example, you would probably develop a very different philosophy of life then if you grew up in the USA, or Mexico, or Iran, or Japan. Every culture has certain cultural norms that seep into the subconscious mind of the people who live in that culture. There are certain behaviours that are considered normal and others that are considered undesirable. What falls into the normal category in one culture may well be placed into the undesirable category in another culture. Spend a little time reflecting on how much of what you consider normal has actually been passed down to you by the culture that you live in or grew up in. Every family has its own culture and philosophy and everyone growing up within that family is subtly (or sometimes not so subtly) brainwashed to some degree by that family philosophy. If you reject that philosophy and choose to walk a different path then you are likely to be considered, by your family, to be a bit weird. Your familys culture may be similar to the general culture of the country but, particularly if your family originated in another country, or another language group, or a minority religion, your family culture may be very different to that of the general culture around you.
Passage 5
DAY 5
Everyone has a philosophy of life whether they realize it or not. This philosophy was largely instilled into your mind when you were a child and is unlikely to be something that you stopped to weigh up as it was going in. The bulk of your philosophy of life is formed in the first seven years of your childhood. Then you have another influential period between the mid-teens into the early twenties. From that point on most people make little change until they are 40 or 50 plus when some people will start to reassess their philosophy. This philosophy is part of your subconscious processing that influences every decision you make, every action you take and consequently every outcome you experience. In many ways it is pre-programming your life.
These clashes in culture can lead to confusion or in extreme cases even to psychological problems, as you try to reconcile the two conflicting cultural pressures, spend a little time asking yourself how much of your philosophy on life comes from your family culture. If you had a strong religious upbringing then the philosophy of that religion will have entered into your subconscious mind to be mixed into and somehow reconciled with the other philosophical influences. Generally a persons religion is largely a factor of where they were born, or where their ancestral family was born, rather than a conscious choice that they made themselves. The role of the formal education system is to produce good citizens. In order to do this it will be filled with the subtle messages of what philosophy the educational institute or government has decided will produce the best citizens. 7
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The history, for example, that you learn in school is far more likely to relate to the country where the school is situated than it is to the world at large. The underlying philosophy from your schooling may vary depending on whether it is a government school, a religious school, or one of the alternate schooling systems. Each one has their own barrow to push. You will mix in certain social groups and those social groups tend to hold together because the group members have a similar philosophy on life. Therefore that group is constantly subconsciously reinforcing that philosophy. If you move into a social group that has a different philosophy to your own then the power of the group philosophy will draw your philosophy toward it. Your philosophy of life is part of your subconscious mind and is very influential on the path that your life is following. However, most of this philosophy was probably formed without you consciously weighing up whether or not it would enhance your life. It may be valuable to review your own philosophy and think about where it came from and how well it is serving you.
------------------------------------------------------------------Vocabulary:
Passage 6 DAY 6
Indian economy had experienced major policy changes in early 1990s. The new economic reform, popularly known as, Liberalization, Privatization and Globalization (LPG model) aimed at making the Indian economy as fastest growing economy and globally competitive. The series of reforms undertaken with respect to industrial sector, trade as well as financial sector aimed at making the economy more efficient. With the onset of reforms to liberalize the Indian economy in July of 1991, a new chapter has dawned for India and her billion plus population. This period of economic transition has had a tremendous impact on the overall economic development of almost all major sectors of the economy, and its effects over the last decade can hardly be overlooked. Besides, it also marks the advent of the real integration of the Indian economy into the global economy. This era of reforms has also ushered in a remarkable change in the Indian mind-set, as it deviates from the traditional values held since Independence in 1947, such as self-reliance and socialistic policies of economic development, which mainly due to the inward looking restrictive form of governance, resulted in the isolation, overall backwardness and inefficiency of the economy, amongst a host of other problems. Despite the fact that India has always had the potential to be on the fast track to prosperity. Now that India is in the process of restructuring her economy, with aspirations of elevating herself from her present desolate position in the world, the need to speed up her economic development is even more imperative. And having witnessed the positive role that Foreign Direct Investment (FDI) has played in the rapid economic growth of most of the Southeast Asian countries and most notably China, India has embarked on an ambitious plan to emulate the successes of her neighbours to the east and is trying to sell herself as a safe and profitable destination for FDI. Globalization has many meanings depending on the context and on the person who is talking about. Though the precise definition of globalization is still unavailable 8
## No. of words: 721
Starting Time:________
Ending Time:________ Reading Speed:_______ 1) What is the Main Point/Central Idea of the passage? ------------------------------------------------------------------------------------------------------------------------------------2) Write the summary of the passage? ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------93 46 56 2008
a few definitions are worth viewing, Guy Brainbant: says that the process of globalization not only includes opening up of world trade, development of advanced means of communication, internationalization of financial markets, growing importance of MNCs, population migrations and more generally increased mobility of persons, goods, capital, data and ideas but also infections, diseases and pollution. The term globalization refers to the integration of economies of the world through uninhibited trade and financial flows, as also through mutual exchange of technology and knowledge. Ideally, it also contains free inter-country movement of labour. In context to India, this implies opening up the economy to foreign direct investment by providing facilities to foreign companies to invest in different fields of economic activity in India, removing constraints and obstacles to the entry of MNCs in India, allowing Indian companies to enter into foreign collaborations and also encouraging them to set up joint ventures abroad; carrying out massive import liberalization programs by switching over from quantitative restrictions to tariffs and import duties, therefore globalization has been identified with the policy reforms of 1991 in India.
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Passage 7
DAY 7
DEPRIVE a person of oxygen and he will turn blue, collapse and eventually die. Deprive economies of credit and a similar process kicks in. As the financial crisis has broadened and intensified, the global economy has begun to suffocate. That is why the worlds central banks have been administering emergency measures, including a round of co-ordinated interest-rate cuts on October 8th. With luck they will prevent catastrophe. They are unlikely to avert a global recession. According to the IMFs most recent World Economic Outlook, published on October 8th, the world economy is entering a major downturn in the face of the most dangerous shock to rich-country financial markets since the 1930s. The fund expects global growth, measured on the basis of purchasing-power parity (PPP), to come down to 3% in 2009, the slowest pace since 2002 and on the verge of what it considers being a global recession. (The funds definition of global recession takes many factors into account, including the rate of population growth.) Given the scale of the financial freeze, the funds forecast looks optimistic. Other forecasters are convinced that a global recession is inevitable. Economists at UBS, for instance, expect global growth of only 2.2% in 2009. The rich worlds economies were either shrinking, or close to it, long before September. Recent weeks have made a rich-world recession all but inevitable. Americas economy lost steam throughout the summer. Temporarily buoyed by fiscal stimulus and strong exports, output grew at a solid 2.8% annualised rate between April and June. But as the stimulus wore off, the job market worsened, credit tightened and consumer spending slid.
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That slide became a rout in September. The economy lost 159,000 jobs, the most in a month since 2003. Car ------------------------------------------------------------------sales fell to a 16-year low as would-be buyers were unable to get credit. The economy may already have ------------------------------------------------------------------shrunk in the third quarter. The rest of the year is likely to be worse. Some economists expect consumer 9 93 46 56 2008 GUNTUR/VIJAYAWADA/VISAKHAPATNAM/ONLINE CLASS/FACEBOOK CLASS
spending to fall at its fastest pace since the 1980 recession. Add in other gloomy evidence, such as a survey of purchasing managers that suggests manufacturing is extremely weak, and it is clear that output is now falling. Americas recession may not yet be official, but it is well under way. In Europe the outlook is equally grim. The British economy, which stalled in the second quarter, is now unmistakably falling into recession. The IMFs forecasts suggest that Britain will see the worst performance of any big economy in the year to the fourth quarter of 2008. The economies of the euro area, too, are struggling badly. Figures released on October 8th showed that output in the euro area fell at an annualised rate of 0.8% in the second quarter. GDP shrank in the currency zones three largest countries Germany, France and Italy. The fourth largest, Spain, barely grew. As elsewhere, the most recent figures have grown grimmer still. Business confidence has turned down and a closely watched survey of purchasing managers points to a further contraction in activity over the summer months. Even the European economies that are less directly affected by housing busts, such as Germany, have been hard hit. The big hope for the euro area was that German shoppers, relatively free of debt and with scope to save a little less, would make up for weakness in debt-laden economies such as Spain. But household spending in Germany has been falling since the end of last year.
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Passage 8
DAY 8
There may be valid reasons to thin forests such as restoration of forest structure or health, wildlife enhancement or public safety but increased carbon sequestration is not one of them, scientists say. In research just published in Frontiers in Ecology and the Environment, Oregon State University scientists conclude that even in fire-prone forests, its necessary to treat about 10 locations to influence fire behaviour in one. There are high carbon losses associated with fuel treatment and only modest savings in reducing the severity of fire, they found. Some researchers have suggested that various levels of tree removal are consistent with efforts to sequester carbon in forest biomass, and reduce atmospheric carbon dioxide levels, said John Campbell, an OSU research associate in the Department of Forest Ecosystems and Society. That may make common sense, but its based on unrealistic assumptions and not supported by the science. A century of fire suppression in many forests across the West has created a wide range of problems, including over-crowded forests, increased problems with insect and pathogen attack, greater risk of catastrophic fire and declining forest health. 10
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Forest thinning and fuel reduction may help address some of those issues, and some believe that it would also help prevent more carbon release to the atmosphere if it successfully reduced wildfire. There is no doubt you can change fire behaviour by managing fuels and there may be other reasons to do it, said Mark Harmon, holder of the Richardson Chair in Forest Science at OSU. But the carbon does not just disappear, even if its used for wood products or other purposes. We have to be honest about the carbon cost and consider it along with the other reasons for this type of forest management. Even if wood removed by thinning is used for biofuels it will not eliminate the concern. Previous studies at OSU have indicated that, in most of western Oregon, use of wood for biofuels will result in a net loss of carbon sequestration for at least 100 years, and probably much longer. In the new analysis, researchers analysed the effect of fuel treatments on wildfire and carbon stocks in several scenarios, including a single forest patch or disturbance, an entire forest landscape and multiple disturbances. One key finding was that even a low-severity fire released 70percent as much carbon as did a highseverity fire that killed most trees. The majority of carbon emissions result from combustion of surface fuels, which occur in any type of fire. The researchers also said that the basic principles in these evaluations would apply to a wide range of forest types and conditions, and are not specific to just a few locations. People want to believe that every situation is different, but in fact the basic relationships are consistent, Campbell said. We may want to do fuel reduction across much of the West, these are real concerns. But if so well have to accept that it will likely increase carbon emissions.
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Passage 9
DAY 9
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If you are a parent of a young child, then now is the perfect time to introduce music. Some theories have even proven that if you play music to your baby while it is in the womb or when it is an infant then it will improve all types of things, such as their IQ and their motor skills. Before your child becomes impressionable through schooling and other children, you should introduce them to music while they still have such an open and fresh mind. For this purpose, music lessons are an asset. It is an effective way to help your child learn to play any type of instrument, and you might find out that your child is 11
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them gain a passion for the art and prepare them for regular music lessons when they are a little older. You can also have your own music time at home if you have younger children. Encourage them to sing silly songs, make home-made musical instruments, or even buy them a kid sized instrument. No matter what you decide to do, remember that music lessons will be a valuable investment towards your childs future. You need to be prepared to commit to it just as they will so that you can make sure they attend the classes and are positive about practicing. With your guidance and patience, your child could be on their way to being the next musical genius.
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Passage 10
DAY 10
matched the declining revenue from CD sales, and that has the shareholders of music companies worried.
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Passage 11
DAY 11
NASAs Kepler mission has discovered the first Earthsized planets orbiting a Sun-like star outside our solar system. The planets, called Kepler-20e and Kepler-20f, are too close to their star to be in the habitable zone where liquid water could exist on a planets surface, but they are the smallest ex-planets ever confirmed around a star like our Sun. The discovery marks the next important milestone in the ultimate search for planets like Earth. The new planets are thought to be rocky. Kepler-20e is slightly smaller than Venus, measuring 0.87 times the radius of Earth. Kepler-20f is slightly larger than Earth, measuring 1.03 times its radius. Both planets reside in a five-planet system called Kepler-20, approximately 1,000 lightyears away in the constellation Lyra. Kepler-20e orbits its parent star every 6.1 days and Kepler-20f every 19.6 days. These short orbital periods mean very hot, inhospitable worlds. Kepler-20f, at 800 Fahrenheit (427 Celsius), is similar to an average day on the planet Mercury. The surface temperature of Kepler20e, at more than 1,400 Fahrenheit (760 Celsius), would melt glass. The primary goal of the Kepler mission is to find Earthsized planets in the habitable zone, said Francois Fressin from the Harvard-Smithsonian Centre for Astrophysics in Cambridge, Massachusetts. This discovery demonstrates for the first time that Earthsized planets exist around other stars, and that we are able to detect them. The Kepler-20 system includes three other planets that are larger than Earth but smaller than Neptune. Kepler20b, the closest planet, Kepler-20c, the third planet, and Kepler-20d, the fifth planet, orbit their star every 3.7, 10.9, and 77.6 days, respectively. All five planets have orbits lying roughly within Mercurys orbit in our solar system. The host star belongs to the same G-type class as our Sun, although it is slightly smaller and cooler. The system has an unexpected arrangement. In our solar system, small, rocky worlds orbit close to the Sun and large, gaseous worlds orbit farther out. In comparison, the planets of Kepler-20 are organized in alternating size: large, small, large, small, and large.
The Kepler data are showing us some planetary systems have arrangements of planets very different from that seen in our solar system, said Jack Lissauer from NASAs Ames Research Center in Moffett Field, California. The analysis of Kepler data continues to reveal new insights about the diversity of planets and planetary systems within our galaxy. Scientists are not certain how the system evolved, but they do not think the planets formed in their existing locations. They theorize that the planets formed farther from their star and then migrated inward, likely through interactions with the disk of material from which they originated. This allowed the worlds to maintain their regular spacing despite alternating sizes. The Kepler space telescope detects planets and planet candidates by measuring dips in the brightness of more than 150,000 stars to search for worlds crossing in front of or transiting their stars. The Kepler science team requires at least three transits to verify a signal as a planet. The Kepler team uses ground-based telescopes and the Spitzer Space Telescope to review observations on planet candidates the Kepler spacecraft finds. The star field Kepler observes in the constellations Cygnus and Lyra can be seen only from ground-based observatories in spring through early fall. The data from these other observations help determine which candidates can be validated as planets. To validate Kepler-20e and Kepler-20f, astronomers used a computer program called Blender, which runs simulations to help rule out other astrophysical phenomena masquerading as a planet. On December 5, the team announced the discovery of Kepler-22b in the habitable zone of its parent star. It is likely to be too large to have a rocky surface. While Kepler-20e and Kepler-20f are Earth-sized, they are too close to their parent star to have liquid water on the surface. In the cosmic game of hide and seek, finding planets with just the right size and just the right temperature seems only a matter of time, said Natalie Batalha from San Jose State University, California. We are on the edge of our seats knowing that Keplers most anticipated discoveries are still to come. 14
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This article will highlight and explain the key areas in which the Moon has both directly and indirectly influenced the emergence and evolution of life on the Earth, a process that has culminated in the development of an intelligent, technologically advanced species. Perhaps the most obvious manifestation of the influence of the Moon on the Earth are the ocean tides, particularly the spring tides where the gravitational pull of the Sun and Moon combine to give the greatest effect. The regular rise and fall of sea level creates an unique environment in the Solar System, where life is exposed to both immersion in water and exposure to air in the space of a few hours. This interface between two distinct ecological niches is thought by many to be crucial in evolutionary terms. This is an environment in which organisms can experience the stresses and strains of an alien world before safely returning to their aquatic habitat, such changes possibly promoting the alteration and/or migration of organisms from one environment to the other. Hence the presence of the Moon to cause tides may well have sparked the spread of organisms from the sea to the land. The Moon also raises tides in the solid body of the Earth and in the past, when the Moon orbited much closer to the Earth than at present, these tides are estimated to have produced displacements in the Earths solid surface of up to a kilometre. This would have produced intense stress and deformation within the Earth which, coupled with the decaying heat of accretion and the higher content of radioactive (U, Th and K) elements would have greatly promoted melting of the early Earth. This melting may well have had an important role in the early differentiation of the Earth, in particular producing the earliest evolved crust which would then be available for recycling by nascent plate tectonic processes.
Passage 12
DAY 12
The Earth is unique amongst the terrestrial planets in having a large satellite, the Moon, which, relative to the Earth, has the largest mass of any satellite - parent system. Numerous lines of evidence indicate that the Moon was derived from the Earth as the result of a singular impact event soon after the initial formation of the Earth. As a result the subsequent evolution of the Earth and the emergence and development of life, has been strongly influenced by the presence of the Moon.
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insecticide. For a chemical weapon to cause harm, it must come in contact with the skin or mucous membranes, be inhaled, or be ingested. The activity of the chemical agent depends on its concentration. In other words, below a certain level of exposure, the agent wont kill. Below a certain level of exposure, the agent wont cause harm. The best protective measure you can take against chemical weapons is to become educated about them. Most of us dont have gas masks or atropine (an injectable used in cases of nerve agent exposure) and wont be on a battlefield, so the recommendations presented here are intended for the general public. Yes, chemical weapons are more likely to be used in a terrorist scenario than nuclear or biological weapons. However, there are several steps you can take to minimize exposure and protect yourself in the event you encounter a chemical agent. Realistically speaking, you are more likely to witness an accidental chemical spill than a chemical attack. Your best defense is to face the situation with a level head. Chemical agents are denser than air. They sink to lowlying areas and will follow wind/weather patterns. Seek the highest storey of a building or the top of a natural land formation. From the point of view of a terrorist, a heavily populated area is a more attractive target than a sparsely populated region. Therefore, the threat of a chemical attack is lessened in rural areas. In the event of an attack, there is some sense in isolating your air supply. Most chemical agents disperse after a certain amount of time (a notable exception is VX, which is designed to persist), so refraining from contacting exposed air may be a good protective measure. How do you know if you have been exposed to a chemical agent? You may not be able to see or smell one. In their pure forms, most chemical weapon agents are clear liquids. Impure chemicals may be yellowish liquids. Most are odorless and tasteless, but some have a slightly sweet or fruity smell. Skin irritation, respiratory distress, and gastrointestinal upset all may 16
Passage 13
DAY 13
A chemical weapon utilizes a manufactured chemical to incapacitate, harm, or kill people. Strictly speaking, a chemical weapon relies on the physiological effects of a chemical, so agents used to produce smoke or flame, as herbicides, or for riot control, are not considered to be chemical weapons. Although certain chemical weapons can be used to kill large numbers of people (i.e., as weapons of mass destruction), other weapons are designed to injure or terrorize people. In addition to having potentially horrific effects, chemical weapons are of great concern because they are cheaper and easier to manufacture and deliver than nuclear or biological weapons. The earliest chemical weapon wasnt an esoteric chemical concoction. During World War I, chlorine gas was used as a chemical weapon, released in massive clouds by the German army to cause lung damage and terror downwind of its release. Chemical agents may be released as tiny droplets, similar to the action of a bug bomb used to release 93 46 56 2008
signal exposure to a chemical agent. However, if you dont die within minutes, you probably wont die at all. Therefore, if you believe you have been exposed to a chemical agent, wait until you feel secure before seeking out medical attention (but do seek it out). Have a radio (with batteries) and keep up with the news. Pay attention to civil defense advisories and think before acting.
Passage 14
DAY 14
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rate at least close to the international rate for the produce as well as the cultivation cost involved for a reasonable profit, asserts the farmer. One of the main reasons for declining produce is the freebies and subsidies. They are destroying agriculture and our lives, according to Mr K. Tharsius his son. Since power and water are provided free, a farmer does not feel the need to plan nor devise any improvised method to minimize their usage. If farmers are charged for electricity it will help improve their efficiency in minimizing this scare resource, says Mr Tharsius. Another impediment is the availability of fertilizers and chemicals. India is dependent on other countries and hence rates are increasing day by day. There are chances of these chemical fertilizers getting exhausted. The permanent solution is only through some renewable sources such as bio-fertilizers and organic manures, according to Mr Kulandaisamy. It is high time the Government seriously starts thinking in proactive measures to revamp our agriculture system. The negative trend in agriculture today is bound to create adverse impact on the overall health of our nations economy. We need to find new avenues to keep farmers on the farm, attract new people to take up farming, and make agriculture profitable since it is the backbone of our country, says Mr Tharsius.
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Passage 15
DAY 15
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Ian Flemings evil globe-fish also known as puffer, blowfish, swellfish, or in Japanese, fugu is one of the most mysterious creatures of the sea. It is perhaps the worlds most deadly fish, yet in Japan the honourable fugu is the epitome of gourmet dining. About 100 species of puffers in several closely related families can be found throughout the world. Their most obvious characteristic is their ability to balloon out from a reasonable fish shape into a sphere two or three times large. When frightened, excited, or annoyed, they gulp water, or even air, into a sac on the belly. It swells inside their tough, elastic skin, like an inner tube inside a tire, so as to discourage predators or intimidate rivals. When the fish feels safe, it squirts out the water or releases the air, deflating to its normal shape. In Japan, eating fugu has been the gastronomic version of Russian roulette for centuries. Sometimes a diner stills losses the gamble. His chopsticks clatter to the table from nerveless fingers; he pales; his breathing labours. It is often the subject of traditional senryu verse. Last night he and I ate fugu; Today, I help carry his coffin. 18
Its a terrible death, a Japanese restaurant owner told me. Even though you can think very clearly, your arms and legs become numb. It becomes impossible to sit up. You can think but cannot speak, cannot move, and soon cannot breathe. Why the Japanese, who venerate hygiene, should make a ritual of eating deadly poisonous fish, is difficult for foreigners to comprehend. fugu ovaries, intestine, and liver can be so deadly that if even a tiny touch of them is left in the flesh, the gourmet dies, often within minutes. About 60percent of puffer poisonings prove fatal. When eating fugu, the diner puts his life in the hands of the chef. Before practicing their risky art, all fugu cooks must be licensed and must take intensive courses, extensive apprenticeship, and written exams. To eat fugu liver is the height of exotica. It is one of the most poisonous parts of the fish, and techniques for detoxifying it are not dependable. Chefs are prohibited from serving fugu liver, but they sometimes relent under the impassioned pleas of gourmets. Mitsugoro Bando had four servings and paid the ultimate price. Despite the danger, demand for puffers is increasing so fast that the Japanese fishing grounds are being depleted. Today the Japanese are successfully culturing the fish. Every year from October through March, millions of diners bet their lives on not getting fatally poisoned. Thanks to strict regulation of restaurants and wholesalers, the number that loses decreases each year. But this droll and preposterous fish with the goggling eyes, swollen belly, and floppy fins remains the worlds most deadly feast. The enigma of the fugu is summed up in the traditional verse: Those who eat fugu soap are stupid But whos who dont eat fugu soup are also stupid.
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Passage 16
DAY 16
Cricket is recognized as one of the modern sports that first originated in England. It is extremely popular in the various areas that formed up the British Empire. There are several test teams that are particularly recognized in Cricket as a sport. These teams are England, South Africa, Zimbabwe, Sri Lanka, India, Pakistan, Australia, Bangladesh, and others. This game particularly got an eccentric nature and even the languages used to describe the different aspects of this game are also idiosyncratic in nature. The majority of terms that are used to describe the different aspects of cricket have their origin in Australia and England. In the present times, most of the playing nations and most of the cricketers have widely accepted these terms. The game of cricket is generally played between the two competing teams and each of these teams has eleven members. There is a large expanse of grass on which this game is primarily concentrated on and this is termed as pitch. In most of the cases, the different cricket teams are comprised of players who have mixed abilities, that is, some of the members of the team 19
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specialize in batting while others specialize in bowling. There are only some occasional players who can excel in both types of skills. Each of the teams also appoints a highly specialist player who acts as wicket-keeper. A length of grass that is present in the centre of the pitch is termed as wicket. At both the sides of the wicket, there are three sticks placed adjacent to each other and this placement is generally ensured in an upright position. This is termed as stumps. The three sticks are separated from each other by a small distance and this distance is generally not more than the diameter of a cricket ball. There is a chalk outline in front of each set of the stumps. This is termed as crease. There are two umpires who are known in this game. The length of the different games is generally variable in terms of duration of time and also in terms of the number of balls that are bowled. As a part of this game, one team will generally bat first and the other one will bowl first. The batsmen will generally play in pairs and each of these batsmen will have a bat. Each of the two batsmen would be at the end of each of the wickets. The batting team always aims at achieving the objective of scoring the highest number of runs till the time all the members of the team are declared all out. This target is then set for the bowling team and they need to surpass this target score in order to win the match. That is, the reversal of roles is now ensured. The batting team now starts bowling and the bowling team starts batting. The reversal of the roles will happen once or more depending upon the format of cricket being played. In this game, it is possible to score runs in a number of ways that is by being able to run between the wickets after a shot or directly ensuring to score four runs or six runs for which there are specified rules. Similarly even the dismissals of the playing batsmen can also happen in a number of ways. All in all cricket is a very interesting game and it is liked by millions of people in this world.
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Passage 17
DAY 17
Not every one of us is born leaders. Well, not all of us are born to be rich and privileged as well. Although it would really take hard work to become rich, the good news is: we could all try to be good leaders! All it takes is just the right leadership strategy. I rediscovered the book, The Tao of Leadership: Leadership Strategies for the New Age, John Heiders adaptation of Lao Tzus Tao Te Ching or How Things Work, over the weekend and Im so glad that I did. Its a 20
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valuable resource for all of us professionals and I feel so at odds with myself for totally missing it all these years! There are 81 nuggets of wisdom in total that could guide us in becoming the kind of leader that we want to be. As a leader, I would like to say that I conform to the following leadership strategies: The wise leader is like water Consider water: water cleanses and refreshes all creatures without distinction and without judgment; water freely and fearlessly goes deep beneath the surface of things; water is fluid and responsive; water follows the law freely. Its true that sometimes all you have to do is look at the things around you and youd be surprised at the valuable lessons that you can learn from it. Take the case of water. Its not rigid; its free-flowing. As a leader in my company, I believe that I am a simple and calm person. Just like water, I am fluid, flexible, and responsive. I dont believe that you should push people into doing things for you but rather you should just be an instrument in making things happen. Allow things to be revealed freely while giving your workers the opportunity to explore and express themselves. You only step in as a guide, probably make sure everything is on the right track, yet never imposing. Your ultimate goal is harmony. A harmonious working relationship is a perfect relationship. Dont stir things up Run the group delicately, as if you were cooking small fish. When was the last time you cooked fish, especially a small one? Due to its delicate nature, you try to hold it as gently as you can, carefully handling it so it will not break as you bring it to the pan, and you make sure you cook it just right. When you are starting a company, your leadership strategy should be that of a good chef. You orchestrate everything - ingredients, tools and equipment, manner of cooking, the right time, and yes, even the right heat. Allow everything to process naturally. Saut. Let simmer. Allow all flavors to mix and blend well. And 93 46 56 2008
wait. In the right time, you have cooked up just the perfect dish! Low and open Why is the ocean the greatest body of water? Because it lies below all rivers and streams and is open to them all. A tried and tested leadership strategy likewise espoused by Lao Tzu is that a good leader should be a good follower. Much like a good cook, you facilitate the making of the dish but you know that the ultimate star would be the dish itself. Just like the ocean, a wise leader provides the bed to make its workers do its work even at the risk of getting unnoticed for facilitating it all. But reward and acknowledgment should not be the priority of a good leader; its the success in knowing that a job well done was finally achieved.
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Passage 18
DAY 18
5125 yearlong cycle of the Mayan Long Count calendar. Nostradamus predictions 2012 end of the world theory states that a Comet, called the NIBIRU, will hit the earth and cause great destruction in the year 2012. This comet will also start the Third World War. When the comet hits the earth surface or the sea bed, nearly the whole of Asia will be flooded. There will be a Tsunami, in front of which the Tsunami of 2004, will seem like a small tiny puddle. After the comet hits the earth, the earth will lose its orbit around the sun and massive earthquakes are going to hit the earth. The polar ice caps will start melting because of the excess heat and global warming will become a major issue. Nostradamus had also predicted certain things about the first black President of America, Barack Obama. He had predicted that Obama would become the President and as we all know that he did. He also predicted that Obama would come in to contact with a person who will be an anti-Christ and this contact will lead to the start of the Third World War, involving almost all the countries of the world. Nostradamus prediction about World War Three states that a Planet X will come and hit the earth and this will be the cause of some major destruction in the world including the city of Rome. He states that there will a Third World War with the camels. He said that the camels will come to drink from the Danube. The Danube is a river in Europe and as for camels, there are no camels in Europe (unless you consider the ones that are there in the zoos). So by camels most probably he meant the countries of Iran and Iraq. This will take place somewhere after 2011 and 2012. There will be a war between the Arab and the Christian world and this war will destroy almost all the countries except for two and thats India and China. This is because, these two countries will not take part in the war.
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doing it, not sitting around in committees and talking about numbers and projections. After all, when the first UPA government came up with the NREGS programme to guarantee poor Indians the right to work for wages, criticism was sharp and the project was supposed to be doomed from the beginning. Six years later, opposition to the scheme is muted and many acknowledge that despite flaws and loopholes, NREGS has succeeded in putting a floor under rural wage rates. So far this fiscal, which will end in March, the programme has given 33.1 million man hours of work to poor households. A little more than half of its beneficiaries are women. Agriculture minister Sharad Pawar worries about this: he recently said that the NREGS should be suspended for three months when harvesting is done, to keep labour costs low for big farmers who hire wage workers. Yet, Pawars worry only highlights the importance of the NREGS and efforts of many state governments, to enforce minimum wages for workers in villages. If you believe anecdotal evidence, the seasonal flow of people moving from poor states to better off ones in search of daily wage work has slowed. This is a worry for people in states like Punjab and Maharashtra, who depended on this vast, peripatetic army of workers as a reliable source of very cheap labour, but higher or more stable wages mean that overall standards of living are going up across the country.
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Passage 19
DAY 19
On Sunday, long after media had written reams about policy paralysis, Indias Cabinet cleared a law that will supply cheap food grains to millions of the poor. This is probably the most politically savvy step taken by a government thats been buffeted around for nearly two years by charges of graft and inaction. Critics of the right to food law will find many things wrong with it: that itll be too costly, it might be implemented by the inept and venal Food Corporation of India that its better to give cash to the poor rather than grain that they might not want, anyway. Of course, itll take time to hammer out a good system to run the worlds largest food subsidy and delivery programme, but the only way to learn how to do that will be by
What is the food bill expected to achieve? Its first objective is clearly political. It will be hard for any political party to argue against a law which says that its only aim is providing cheap food to the poor. The fact that Congress president Sonia Gandhi prodded the Cabinet, which was dawdling on the bill, to clear it on a Sunday shows that she hasnt lost sight of the aam aadmi objectives which brought two Congress-led coalitions to power in 2004 and 2009.
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Passage 20
DAY 20
The controversy regarding the monitoring of Internet content fails to die down. In the latest development, an inter-religious group on Thursday met Communications and IT Minister Kapil Sibal seeking strict action against the social networking websites and other Internet companies for putting up anti-religious content on the web that fuelled communal disharmony. Quoting a Delhi court order restraining 22 social networking and Internet sites such as Facebook, YouTube and Google from webcasting any antireligious or anti-social content, representatives from Muslim and Christian organisations besides some social
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Two years ago, Geeta heard about Janalakshmi, a microfinance company, from some women in her neighbourhood. She joined a group of women and borrowed Rs. 30,000 (about \$600) with the understanding that they would help each other not default on interest payments and take turns reaping the benefits of the loan. Each group has a leader who guarantees the interest payment to the microfinance institution and in turn, the leader invites women she trusts into the group so that they can borrow larger amounts. For now, Geetas microfinance loan is only allowing her to pay back her previous debts, but she dreams of the day when she can borrow enough money for a down payment on a home. More and more entities are recognizing the power of micro-loans and how they can elevate an entire segment of society. And the route to the underserved is frequently through women, thanks to models based on Grameen Bank and others. Chennai-based Equitas, for instance, only works with women. In March, The Indus Entrepreneurs (TiE) launched Stree Shakti, a platform for training women entrepreneurs at all levels of Indian society. Goldman Sachss ambitious 10,000 Women program aims to train and develop women entrepreneurs across the globe by pairing them with resources in the West. In all these cases, women serve as the lynchpin for programs, whether they are rural Self Help Groups (SHG) or global programs that aspire to foster entrepreneurship. Microfinance is not the only answer to solving the poors problems but it is one good way to help women help themselves. Women self-help groups are burgeoning all across India, and study after study shows that they successfully impact women and bring them out of poverty. In an article that appeared in the December 2007 issue of UNDPs Poverty in Focus, researchers Ranjula Bali Swain and Fan Yang Wallentin of Uppsala University in Sweden examine the link between microfinance and womens empowerment using household sample data collected from five states in India in 2000 and 2003. Their results strongly demonstrate that there is a clear link between womens participation in a Self Help Group (SHG) and their empowerment. The good news, at least in India, is that these microfinance initiatives are reaching bigger swathes of 25
Passage 21
DAY 21
Geeta, 32, would be a typical candidate. An orphan at age three, Geeta was raised by her elder sister. She didnt go to school and was married to an alcoholic uncle when she was a teenager. Today, she works as a housemaid in Bangalore to feed her family of four: Her husband, her two sons and herself. Geetas life goal is to educate her two sons. But she lives in a cycle of debt borrowing to repay past loans, to make annual school payments for her sons, to cover family events like weddings and every time someone in the family falls sick. Geeta, it so happens, works in my house. 93 46 56 2008
the underserved. The Indian School of Microfinance for Women (ISMW), for instance, goes one step deeper into the problem. Based in Ahmedabad and chaired by social activist and SEWA founder Ela Bhatt, the school recognizes that borrowing money is only one part of the triangle. Among other things, the school teaches women how to deal with the money they borrow through capacity building workshops, networking and providing knowledge resources. Simply put, it takes Goldman Sachss global vision for women entrepreneurs and translates it into a deeper regional focus. The schools website lists hand-holding as one of its goals. Participants of micro-credit schemes are taught financial planning and investing techniques that they can use on the ground and in their business. While microfinance works to eradicate poverty, the next generation of Indian leaders, including Rahul Gandhi, has made social sectors its calling card. The rural development portfolio, which traditionally was one of the less-prized posts, has now vaulted to the top of the pecking order, thanks in large part to the Gandhi family which has aligned itself with the aam admi (poor people) in both its campaigning and future promises. When Manmohan Singh was asked in a recent television interview if he had any regrets about areas that he couldnt concentrate on in his first term that he would focus on in his second term, he said, Id like to work on agriculture, education and rural health.
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Passage 22
DAY 22
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Woodrow Wilson was referring to the liberal idea of the economic market when he said that the free enterprise system is the most efficient economic system. Maximum freedom means maximum productiveness; our openness is to be the measure of our stability. Fascination with this ideal has made Americans defy the Old World categories of settled possessiveness versus unsettling deprivation, the cupidity of retention versus the cupidity of seizure, a status quo defended or attacked. The United States, it was believed, had no status quo ante. Our only station was the turning of a stationary wheel, spinning faster and faster. We did not base our system on property but opportunitywhich meant we based it not on stability but on mobility. The more things changed, that is, the more rapidly the wheel turned, the steadier we would be. The conventional picture of class politics is composed of the Haves, who want a stability to keep what they have, and the Have-Nots, who want a touch of instability and change in which to scramble for the things they have not. But Americans imagined a condition in which speculators, self-makers, runners are always using the new opportunities given by our land. These economic leaders (front-runners) would thus be mainly agents of change. The nonstarters were considered the ones who wanted stability, a strong referee to give them some position in the race, a regulative hand to calm manic speculation; an authority that can call things to a halt, begin things again from compensatory staggered starting lines. Reform in America has been sterile because it can imagine no change except through the extension of this metaphor of a race, wider inclusion of competitors, a 26
piece of the action, as it were, for the disenfranchised. There is no attempt to call off the race. Since our only stability is change, America seems not to honour the quiet work that achieves social interdependence and stability. There is, in our legends, no heroism of the office clerk, no stable industrial work force of the people who actually make the system work. There is no pride in being an employee (Wilson asked for a return to the time when everyone was an employer). There has been no boasting about our social workersthey are merely signs of the systems failure, of opportunity denied or not taken, of things to be eliminated. We have no pride in our growing interdependence, in the fact that our system can serve others, that we are able to help those in need; empty boasts from the past make us ashamed of our present achievements, make us try to forget or deny them, move away from them. There is no honour but in the Wonderland race we must all run, all trying to win, none winning in the end (for there is no end).
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Passage 23
DAY 23
It is not that Charles Darwin and the theory of evolution are unknown in Alexandria, Egypt. But even among those who profess to know something about the subject, the common understanding is that Darwin said man came from monkeys. Darwin, of course, did not say man came from monkeys. He said the two shares a common ancestor. But to discuss Darwin anywhere is not just to explore the origin of man. It is inevitable to engage in a debate between religion and science. That is why, 150 years after Darwin published On the origin of species, the British Council, the cultural arm of the British government, decided to hold an international conference on Darwin in this conservative, Sunni Muslim nation. It was a first. A lot of people say his theories are wrong, or go against religion, said Martin Davidson, chief executive of the British Council. His ideas provoke, but if we are going to understand each other, we have to discuss things that divide us. Darwin may be misunderstood here, but in many ways that is but one symptom of a more fundamental problem with education in Egypt and around the region. In a culture that prizes and nurtures conformity, challenging conventions and beliefs is anathema, said writers, political scientists, social workers, students and educators inside and outside the conference. Education here is based on rote memorisation with virtually no emphasis on creative thinking. Few schools here even teach the theory of evolution. For example, once considered the intellectual capital of the Arab world, Egypt was recently ranked 124th of 133 countries in the quality of its primary education by the World Economic Forum, based in Switzerland. Other global assessments have provided equally dismal results. Indeed, many people, including some of the 150 scientists and scholars in attendance at the Bibliotheca Alexandrina this month, were somewhat surprised that the government even agreed to allow the conference. It was unlike the leadership here to permit public discussion of ideas that challenge religious thinking and the national curriculum, or promote critical thinking they said. The British Council framed the conference to seek middle ground, more than to promote confrontation. While challenging a religious society to think seriously about evolution, it emphasised the possibility of reconciling a belief in divine creation with Darwins theories of evolution and natural selection. That was a position that many students here said they were comfortable with. Darwins theory of species says nothing about the appearance of life - or about the origins of the universe, read panel number 7, in an evolution of man exhibition put on display during the conference. It is perfectly plausible to uphold a scientific account of how natural laws allowed the universe and life to develop and to believe that a deity created those laws. Judging from public comments made during the gathering, the effort to reconcile faith and science left avowed atheists in the audience frustrated and did little to convince the religious fundamentalists.
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Passage 24
DAY 24
India nurtured various cultures in ancient times. Along with literature, fine art, music, dance and drama, architecture too, in all its grandeur, rose to great heights. It is not easy to explain Indian architecture; it is entirely different from that of Europe. Indian civilization begins with the Indus civilization that dates back about 4000 years. The famous cities of Harappa and Mohenjadaro are now in Pakistan, while the ruins of Lothal are in India. Aryans from the west settled in India and developed Vedic literature as part of the Brahman religion. These became the Holy Books of the religion, which later came to be known as Hinduism. During the 5th-6th centuries BC, Gautama Siddharta became Buddha and started Buddhism and Vardhamana became Mahavira and started Jainism. Buddhism had the support of the royal class and was adopted by the masses. As Buddhism spread across the country, so did its monasteries and temples. As Hinduism reestablished itself strongly, the Buddhist presence disappeared from India in the 13th century. Cave temples typically represent the architecture of Ancient Times. Naturally there must have been castles, palaces and houses during that time, but none of those remain, because buildings constructed of wood, rotted or burned easily. Temples were built of bricks, but when Buddhism died out, these were destroyed or pulled down due to a lack of protectors. However, cave temples and monasteries still exist today because they were carved out of rock - a much stronger material. There are around 1,200 such cave temples and monasteries left and 75 per cent of them belong to Buddhism. As they were not satisfied with cave temples, entire sculpted rock temples were built during the Middle Ages. A few still exist unto the present day. In contrast to the rock temples that imitated wooden temples of ancient times, the stone temples, built by laying cut stones one on top of another, came to be the model of sculpted stone construction. But since these developed together, there is no line dividing the ages in terms of centuries. Construction of stone temples commenced in the 5th century, during the Gupta dynasty, but was standardized only during the 8th century. Many stone temples were built between the 7th and 9th centuries, but the temples carved out of rocks were constructed up till the 12th century. Hence according to the history of architecture, the transition from ancient times to the Middle Ages took many centuries. Buddhism took the lead in construction during ancient times and in contrast, Hinduism took the lead during the Middle Ages followed by Jainism. The method of stone construction improved by leaps and bounds in north and south India. The style caught on and very soon the whole of India was filled with stone structures. The Chandella dynasty in the north and the Chola dynasty in the south showed remarkable developments in architecture, by building magnificent temples, using stone. Islam entered India during the 11th century and established power in Delhi during the 13th century. Till the 16th century, the Turkish and Afghan dynasties continued to rule Delhi during a period referred to as the Delhi Sultanate. Western styles of architecture, including techniques like domes were brought to India during this age and had a strong influence on building styles. This period called the Middle Ages, and the advent of the Mughals who conquered most of India, signalled the beginning of the Modern Age.
Mughal rule spread into more than half the sub-continent and the splendid Mughal style, which is a mix of the IndoIslamic construction style, also blossomed. At that time, the Vijayanagara kingdom, which is predominantly Hindu, flourished in south India. At the same time, the Nayaks who were also Hindus ruled over some areas in the south. Both these dynasties appreciated technical developments with the main themes being large-scale construction, complex expressions and elaborate decorations. This phase is called the Modern Age. Emperor Akbars Hindu-Islamic fusion in north India and the lavish Dravidian style of construction in the south are remarkable styles of this age. Modern Age Indian architecture also includes the British era in India, until its Independence in 1947. British rule coincided with the decline of the Mughal era and the revival of Hinduism. The construction during this time was an adaptation of the Indian style in the colonial style brought from Europe. The direct impact of British architecture was seen from the second half of the 19th century, when research on Indian architectural history advanced and the Mughal style influenced colonial constructions. It is referred to as the Indo-Saracenic style. As we plunge into the age of Modernism (post- Indian Independence), we have to say that architectural styles differed largely until then. A major influence on Indian architecture, post-Independence, was that of French architect, Le Corbusier, who designed Chandigarh and various buildings in Ahmedabad. Indian architects, educated in Europe and America also made a mark, but trying to transplant the architecture of Europe and America that was very different in history and style had its own problems. It is only right to call modern architecture Cosmopolitan architecture. The 600-odd buildings in the book are grouped together according to their similarities. To enable the traveller to decide which place to visit, the buildings are given a star rating, from 0 to 3. The rating is based on the fascinating quality of the building. If a building has archaeological importance, but is in ruins, then it is given a low rating. Ratings are also given from 1 to 3, with regards to the region, the importance or number of buildings, natural scenery, etc. This is done as a subjective measure to help the traveller use this as a yardstick while planning his journey.
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Passage 25
DAY 25
Riots in the British capital have hit inner-city Tottenham, suburban Ealing, gritty Hackney, chic Notting Hill. Windows have been smashed, video cameras stolen and cars set ablaze. Young men in hooded sweatshirts congregated on street corners and charged the police. Copycat riots have followed across the country, from Bristol to Nottingham. And nobody really knows why. Scan the comment pages of the British press, and you will find a wide range of explanations. Read the center-right Daily Telegraph, and you will learn that the riots were caused by a weak and cowardly police force, absent fathers, welfare dependency, multiculturalism and the tolerance of gangs in schools. Read the center-left Guardian and you will be informed that police brutality, social exclusion, cuts in welfare spending and the widening gap between rich and poor are to blame. Some are convinced that high levels of immigration are at fault. Others believe the problem lies in British intolerance of immigrants and minorities. There is a reason for the discrepancy: The rioters themselves do not wave signs. They do not chant. They werent protesting any particular government policy, as were student demonstrators in London last winter. They have not sought publicity for their views, if they have any. They hide from cameras and dodge journalists. And thus have they become the inkblot in a kind of national Rorschach test: Everyone sees in them the political issue they care about most, whether its welfare dependency, budget cuts, the decline of public education or my personal favorite the rise of a vulgar and amoral public culture. And yet it is their lack of politics that most clearly defines them. If the Egyptians in Tahrir Square wanted democracy and the anarchists in Athens wanted more government spending, the hooded men in British streets want 46-inch flat-screen HD televisions. They arent smashing the headquarters of the Tory Party; they are smashing clothing shops. Ins tead of using social media to create civil society or cyber-utopia, they are using social media to steal. Someone circulated a text message on Monday night, calling friends to central London for Pure terror and havoc & Free stuff. Just smash shop windows and cart out da stuff u want! Aside from stealing, a lot of the rioters maybe most of the rioters were also out to have a good time. Dont be fooled by the stiff-upper-lip cliches: From Wat Tylers medieval peasant rebels to the modern soccer hooligans, there is a time-honored tradition of smashing things for fun in Britain, and the groups that enjoy it have been around for a long time. It doesnt take very many of them to do a lot of damage. As of Wednesday morning, police had arrested 768 people, according to the BBC, and charged 105 in connection with violence in the capital. Overnight, London was calm for the first time since riots began last week. Im not counting out the other possible explanations, many of which would be worth investigating even if these riots had never occurred. The welfare state really has left a generation of young people feeling both dependent on government handouts and entitled to more. Poor state education has left as many as a fifth of British teenagers functionally illiterate. The slow economy means many will never find jobs and thus will never integrate into the mainstream. The presence of the worlds oligarchs and billionaires in London means the city has an economic gap that is unusually wide for the developed world. The tabloid press thrives on envy of the rich and cult-worship of boorish celebrities. Traditional institutions the school system, churches, even the BBC long ago lost their ability to transmit older values. A spate of scandals has recently discredited the banks, Parliament, the media and the London police even further.
And yet there was looting in London after the Great Fire of 1666 and, despite the mythology, there was looting in London during the Blitz. Go back and read Dickens: Criminals, both immigrant and native British, have taken advantage of opportunities to loot in London during more peaceful times, too. A peculiar confluence of circumstances a mob angry about a police murder, a sudden bout of warm weather, an unprepared police force distracted by scandal and, yes, the astonishingly widespread availability of smartphones among the underprivileged might have allowed them to do so again. Beware of broad political generalizations in the wake of these riots: We dont know whether we have just witnessed a new phenomenon, or a more mobile and technically adept version of a very old one.
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Passage 26
DAY 26
Foreigners visiting New York or Chicago in the 19th century often came away with conflicting feelings. Some found American cities ugly by comparison to their European counterparts: They seemed vulgar, blatantly commercial, lacking in taste. The natives had higher living standards but they were crude, and the ethnic mix German, Irish, Italian and Jewish was terrifying. A few sensed that there might be something in this new civilization worth admiring. It is an absorbing thing to watch the process of world-making, both the formation of the natural and the conventional world, an English traveller, Harriet Martineau, wrote in 1837: I witnessed both in America; and when I look back upon it now, it seems as if I had been in another planet.
Passage 27
DAY 27
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Passage 28
DAY 28
Though belated, the government and the Reserve Bank of India have come out strongly in support of rupee that is rapidly declining in relation to the dollar. The new measures, both administrative and policy fall into three broad categories. One, the rules governing overseas investment has been relaxed in certain cases so as to increase the supply of dollars in the domestic market and thereby correct its demand-supply imbalance. The ceiling on debt instruments by foreign institutional investors and the interest cap on external commercial borrowings have been raised. The lock-in period for overseas investors in infrastructure bonds has been reduced. Two, in a surprise move on the day before the credit policy review last week the RBI clamped down on forward trading in foreign exchange. The avowed objective is to curb the rampant speculation which, in its view, weakens the rupee further. In the third category are the new incentives provided to non-resident Indians to invest more with banks in India. These are significant in themselves and they need to be evaluated in a larger context and over a longer time-frame than in the immediate term. It is fairly clear, however, that the rationale for almost all these measures is traceable to the rupees sharp decline and the imperative of arresting it. Absent this justification, the case for introducing many of them at this juncture becomes weak. For instance, given the RBIs concerns over accumulation of short-term external debt, there is no reason for facilitating larger external borrowings by companies. Now, with the foreign institutional investors getting a greater access to the debt markets, including the gilts and corporate bond market, the external economy will be vulnerable to foreign capital flows. The RBI might have succeeded, at least temporarily, in halting the rupees decline by sending out strong messages to currency speculators as, for instance, by disallowing the rebooking of cancelled forward contracts in foreign exchange. But clearly these measures are in the realm of micro-management and should go once the perceived threat to the rupee recedes. The deregulation of interest rates on non-resident bank accounts cannot be justified except in the narrow context of encouraging overseas Indian investment at all costs. Past experience suggests that these deposits can exit just as easily as they enter. Besides, with the prevailing low dollar interest rates, there is tremendous scope for arbitrage with minimal exchange rate risk to the Indian expatriate. Neither individual banks nor the macro-economy stands to gain by mobilising such funds.
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Passage 29 DAY 29
On the other hand, if you have an interest in medicine and want to be able to prescribe medications to your patients, a career in psychiatry might be your ideal choice If you do not want to invest five to eight years in graduate training, consider pursuing a career as a licensed social worker or counsellor. These professionals are also qualified to provide mental health services depending up training and experience. Both social work and counselling typically require two or three years of graduate study. Psychiatric nursing is another great career option for students interested in medicine. Advanced Psychiatric Nurses hold a masters degree or higher in psychiatric-mental health nursing and are able to assess patients, diagnose disorders, provide psychotherapy and prescribe medications.
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Passage 30
DAY 30
If you do actually want to be sad and stay sad, dont make any decisions that could impact on your life! It will, of course, classify you as an unhappy Maximiser according to research by Professor Joyce Ehrlinger of Florida State University, but at least you wont risk making the wrong decision? It seems we all have a choice. We can choose to be Maximisers or Satisficers. Yet the behavioural difference between them is profound in both its nature and its results. I have already written on the Five Levels of Happiness as a way of achieving full happiness across ones life. But of course that involves making a whole gamut of different choices or decisions related to each and every aspect of our life. Professor Ehrlinger has concluded that many of us are capable of falling at the first fence! 38 93 46 56 2008 GUNTUR/VIJAYAWADA/VISAKHAPATNAM/ONLINE CLASS/FACEBOOK CLASS
Why? Because if we are inclined to think far too much about making a decision in the first place, we are very likely to risk adopting an unhappy existence, always fretting about whether the decision is the right one or not. For example, what happens if it is wondering Should I really go to a friends party? Should I change my job? Or even, should I really say yes to this marriage proposal? One can heap deep unhappiness on ourselves by not making a choice if we live in constant fear of making the wrong decision. And even if apparently in regard to some choice presented to us, we do bring ourselves to make a decision as a Maximiser, we can then lead a life of unending rumination, tormenting ourselves over whether it was the right move! If this describes us, then the research says we never enjoy the psychological benefits of commitment and our life becomes one overladen by grief. It could strike at the root of potential relationships or career opportunities, multiplying the feeling of unhappiness. The opposite seems true of Satisficers. They have patterned a different behavioural approach altogether. They think the issue through as far as they can and then when they arrive at the final element of doubt, they are far more inclined to listen to their instincts, their sixth sense. If it says, Do it! Then they do just that. They are happy that if it works out then fine, and if it doesnt then they will not hold it against themselves or give themselves grief over it. In my experience of observing myself and others, I think there is another clear difference between Maximisers and Saticficers. Simply stated, Maximisers flirt with the danger of striving to be perfectionists and no less. Satisficers on the other hand, are much more pragmatic about their own fallibility. And they are much more comfortable in their own skin. They also have a higher sense of self-worth and self-esteem. For Maximisers, happiness can appear a luxury they cannot afford. But for Satisficers they are far more open to happiness. They let it in and enjoy it. Happiness is so often there for our taking. Clearly we can kill it or accommodate for it in pretty well everything we do!
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Passage 31
DAY 31
There is an objective difference between one who has knowledge of something and one who does not. This is true in both the concurrent and the dispositional senses of knowledge and knows. That is, whether or not X has knowledge on a certain point or about a certain matterknows the English alphabet, for example, the narrative content of War and Peace, or the date of Robert Kennedys assassinationis not a matter of how anyone, including X himself, may think or feel about X and his conscious or other states. Knowledge with reference to specific matters is a condition which individual human beings are or are not in. Usually this condition is found in a social context. But if only one person existed it would still be possible for that person to be knowledgeable about some, at least, of the specific matters that concerned himfor example, about reliable sources of food and water in his environment. Can this objective difference among human beings consist in properties and relations that fit within a naturalist ontology? A long tradition of ancient and modern philosophers from Plato, Descartes and Kant to T. H. Green, Edmund Husserl and Hilary Putnam has insisted that it cannot. Othersespecially those in the 19th and 20th centuries who insisted upon distinguishing Naturalism from Materialismhave held that it could be. But with the rise and development of the mind/brain identity thesis during the last half of the 20th century, the generous naturalism (as we shall call it) of Dewey, Santayana, Sidney Hook and others has largely disappeared in favour of a narrower naturalism more commonly and more correctly called Physicalism (the older Materialism). For it, all distinctively human properties are reduced to strictly physical properties of the central nervous system of the human body or to these plus characteristics of the natural and social settingor, to nothing at all. In this paper I will try to explain why narrower Naturalism or unqualified Physicalism cannot find a place for knowledge, and specifically for three of its essential components: truth, logical relations and noetic unity. At this late date it is hard to say much that will be strictly new on these matters, but, apparently, there is much that needs to be said again. What I shall say about truth and logic is practically identical with what Frege said more than a century ago, though I hold views significantly different from his on how truth and logic fit into the full context of knowing and knowledge.2 What I shall say about noetic unity adds little to what has already been said by Kant, Lotze and Husserl.
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Passage 32
DAY 32
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Passage 33
DAY 33
When studying markets, economists not only want to understand how prices and quantities are determined, but they also want to be able to calculate how much value markets create for society. Economists call this topic of study welfare analysis, but, despite its name, the subject doesnt have anything directly to do with transferring money to poor people. Economic value created by a market accrues to a number of different parties. It goes to consumers when they can purchase goods and services for less than they value the use of the items; producers when they can sell goods and services for more than each item cost to produce; and the government when markets provide an opportunity to collect taxes Economic value is also either created or destroyed for society when markets cause spill over effects for parties not directly involved in a market as a producer or a consumer. In order to quantify this economic value, economists simply add up the value created for all of the participants in (or onlookers to) a market. By doing so, economists can calculate the economic impacts of taxes, subsidies, price controls, trade policies, and other forms of regulation (or deregulation). That said, there are a few things that must be kept in mind when looking at this type of analysis. First, because economists simply add up the values, in dollars, created for each market participant, they implicitly assume that a dollar of value for Bill Gates or Warren Buffet is equivalent to a dollar of value for the person who pumps Bill Gates gas or serves Warren Buffet his morning coffee. Similarly, welfare analysis often aggregates the value to consumers in a market and the value to producers in a market. By doing this, economists also assume that a dollar of value for the gas station attendant or barista counts the same as a dollar of value for a shareholder of a large corporation. (This isnt as unreasonable as it may initially seem, however, if you consider the possibility that the barista is also a shareholder of the large corporation. 42 93 46 56 2008 GUNTUR/VIJAYAWADA/VISAKHAPATNAM/ONLINE CLASS/FACEBOOK CLASS
Second, welfare analysis only counts the number of dollars taken in in taxes rather than the value of what that tax revenue is ultimately spent on. Ideally, tax revenue would be used for projects that are worth more to society than they cost in taxes, but realistically this is not always the case. Even if it were, it would be very difficult to link up taxes on particular markets with what the tax revenue from that market ends up buying for society. Therefore, economists purposely separate out the analyses of how many tax dollars are generated and how much value spending those tax dollars creates. These two issues are important to keep in mind when looking at economic welfare analysis, but they dont make the analysis irrelevant. Instead, its helpful to understand how much value in the aggregate is created by a market (or created or destroyed by regulation) in order to properly assess the trade-off between overall value and equity or fairness. Economists often find that efficiency, or maximizing the overall size of the economic pie, are at odds with some notions of equity, or dividing that pie in a manner that is considered fair, so its crucial to be able to quantify at least one side of that trade-off. In general, textbook economics draws positive conclusions about the overall value created by a market and leaves it to philosophers and policy makers to make normative statements about what is fair. Nonetheless, its important to understand how much the economic pie shrinks when a fair outcome is imposed in order to decide whether the tradeoff is worth it.
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Passage 34
DAY 34 43
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Global warming has a variety of causes. One of the largest factors contributing to global warming is the general problem of overpopulation and its many effects. The greater number of people consumes more items which take more energy to make; they drive more cars, and create larger amounts of garbage. These factors all increase the global warming problem. Many different gases can increase the planets temperature. The number of different products and human activities that contribute to global warming are so numerous that finding solutions to the problem is very difficult. Using a refrigerator releases dangerous gases, turning on the lights requires energy from a power plant, and driving to work causes gas emissions from the car. Countless other normal activities lead to global warming. Though having an atmosphere is important, the greenhouse effect may be making it excessively thick. The levels of gases covering the Earth have soared with industrialization, and developed countries now produce about 75% of greenhouse gases. The most common gas is carbon dioxide, accounting for about 50% of all greenhouse gases. Other gases, including methane, CFCs, nitrogen oxides, and ozone, also contribute to forming the greenhouse layer. Because these gases are produced by so many important and common processes, limiting their production to prevent global warming will be difficult. As population increases and Third World countries begin to use greater amounts of energy, the problem may expand rather than contract. To know just what the effects of global warming will be in the future is extremely difficult, if not impossible. Scientists use computer models to study the effects of global warming. These computer models have been fairly consistent in predicting general future trends, but often differ greatly when looking at the specifics. Some scientists say global warming has already been going on for a while. Others say that we do not have enough information now to know for sure. Despite the disagreements, most scientists are convinced that greenhouse gases are warming the Earth. What they are still trying to figure out is how quickly temperatures are rising, and what will happen as a result. The climate changes that will result from global warming are extremely difficult to predict. The weather is determined by so many factors that it is often compared to chaos by scientists. Changing the temperature will likely have some effect on the planets weather, but just what that effect will be is nearly impossible to predict. If temperatures do indeed rise significantly, the most important result would be that some portion of the polar icecaps would melt, raising global sea levels. The rise in sea levels would be disastrous for some places. Islands would disappear; meaning their millions of inhabitants would have to relocate. Flooding would occur along coastlines all over the world, displacing more people and ruining cropland. In the case of major global warming and melted ice caps, some countries might simply cease to exist. Global warming, if uncontrolled, could cause a major catastrophe. The threat of global warming is among the most important of all modern environmental problems. There are a variety of ways of dealing with it, each attempting to combat one of the many causes of global warming. The problems that cause global warming include overpopulation, deforestation, ozone depletion, garbage dumping, and many others. These all have unique solutions which are now being promoted by environmentalists. Certain laws and treaties are aimed at reducing the emission of pollutants that result in global warming. In 1988, the International Conference on the Changing Atmosphere drew scientists and decision makers from 48 countries. Some policies could successfully reduce global warming. Raising fossil fuel prices, 44 93 46 56 2008 GUNTUR/VIJAYAWADA/VISAKHAPATNAM/ONLINE CLASS/FACEBOOK CLASS
taxing emissions, and encouraging people to take environmentally friendly action through such activities as planting trees will all help. Because many problems leading to global warming are caused or contributed to by overpopulation, people are beginning to work to reduce family sizes. Family planning services actually help in the fight against global warming. Education is a key method of reducing the greenhouse effect. By teaching people about such things as deforestation, environmental activists hope to prevent the problems that ultimately lead to global warming. Widespread media attention to the global warming problem is also increasing awareness. This is causing both individuals and governments to act more responsibly towards the environment.
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Passage 35
DAY 35
Many ground breaking discoveries came about by chance! In 1928, bacteriologist Alexander Fleming found a mould had contaminated one of his experiments. To his surprise, the mould turned out to be an antibacterial agent...and so, penicillin was born. Another remarkable creation is the multifaceted and challenging media of glass. By melting combinations of soda and sand, our ancestors found, upon letting the mixture cool, that its composition had changed into a transparent glassy mass. Trial-and-error resulted in one of the largest industries to date. The creation of glass continually evolved with additions of limestone, lead oxide and boric acid. Metals like cobalt, copper, manganese, gold and silver would change the consistency, clarity, colour weight and strength of glass. 45 93 46 56 2008 GUNTUR/VIJAYAWADA/VISAKHAPATNAM/ONLINE CLASS/FACEBOOK CLASS
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Passage 36
DAY 36
As an oil painting instructor Im used to all sorts of questions from students learning how to paint on the materials of our illustrious craft. In fact, one of the all-time biggies is about what painting surfaces to use for practice and finished products? As a new oil painting artist, I quickly discovered the great need for alternative painting surfaces. I like using canvas, but mind you, one canvas is not all that expensive, but feeding a new insatiable habit with two, three or even four canvases a day can add up. Man has been in the great search for practice painting surfaces ever since he started doodling on cave walls. Unfortunately, cave walls are in short supply and theyre really, really expensive these days. So whats an artist to do? What are our alternatives? What can I use for practice? What should I use for a final painting project? So lets just take up with the cave walls and come forward a few years. Oil is a soft and fatty substance. So the requirement is that the painting surface must be harder than oil. The short of it all, you can paint on anything so long as it is harder than oil. You are limited only by your imagination. With that said, you want to use a surface that wont change with the weather (warp) or corrode from the use of oils, solvents and medium. What good does it do, to paint grandma and grandpa and watch the work warp in three months? Seasoned wood, masonite, canvas board or illustration boards just to name a few, are rigid and great to use as painting surfaces. They used to be the primary painting surfaces until about 500 years ago when canvas was discovered. The most commonly used painting surface today is canvas which must be correctly mounted on a set of stretcher bars which ensures that the entire surface is taut. A good canvas is equally tight and firm throughout. This gives canvas its peculiar feel thats softer and bouncier than wood, cave walls or other rigid surfaces. As each of us develops our skills and craftsmanship, we get used to the properties of our painting surfaces. We rely on it. We may try many surfaces, but we settle down on what we generally like and become comfortable. Canvas is popular because its light, rigid, yet elastic at the same time. Canvas can be made from sackcloth (burlap), cotton (most popular), synthetic, a combination of materials or even smooth linen. The texture of the surface of the canvas are rated their smoothness, known as tooth. The coarser the surface, the more tooth it is said to have. Canvas are available as economy (rough with lots of tooth), medium, fine (portrait smooth) and smooth linen. Each cloth can be constructed to any quality level. Linen is considered to be the best in quality and therefore sports the heftier price. A primed canvas is one that has been covered with a solid layer of substance that protects the canvas cloth from rotting away because of the acidity and harshness of oils and mediums. In a nutshell, an un-primed canvas will dissolve over the long term from the acidity inherent in oil paints. The next time youre shopping, look at the canvas label. It should mention whether or not the canvas is primed. If theres no mention, than safely assume the canvas is not primed. Canvases are typically primed with one of the following: 1. Thinned glue that does not affect the colour of the canvas. Canvas are typically labelled as either single or double primed, meaning coats of application. The canvas must dry before the next coat is applied. Double primed application is best. 2. A compound of rabbit skin glue and Spanish white or chalk. 3. Acrylic gesso. It is normal practice for manufacturers to label their canvas as primed, materials used, degree of material mix if any, texture and quality. 47 93 46 56 2008 GUNTUR/VIJAYAWADA/VISAKHAPATNAM/ONLINE CLASS/FACEBOOK CLASS
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Passage 37
DAY 37
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Passage 38
DAY 38
Do you worry about writers block? When Camus had it he rushed to his then mentor, Andre Gide, who said you mean you can stop writing yet you still complain? Whats up with you Albert? Camus problem was that he had decided to be a writer. He was, after all, an existentialist. Why do you write? Did you jump or were you pushed? Who do you write for? Thomas Love Peacock said that poets are wasters of their own time and robbers of that of others. Is writing by its very nature a selfish activity, a solitary sin? With the need for voluntary tutors to help illiterates, with Africa starving, with the Samaritans understaffed handling all the young poets that phone in, can locking yourself away ever be justified? Some justify their selfishness by emphasising that their sacrifice is for the benefit of all, because they are societys antennae, the nearest to prophets and telepaths that this nihilistic age has. These starving artists in their uniforms from Oxfam charge over the top in a daring raid on reality and return with their wounds which they invite us to lick. Do they write to express, confess or merely impress us with their Angst threshold when they tell us that lonely clouds make shadows on the wind, that roses reek of mortality and that lifes a sexually transmitted disease? Others use philosophy to back themselves up. Wittgenstein thought that language and reality shared a logical form and that by exploring one mode, the other was enriched, and that mans instinct was to explore. Mumble his name repeated ly next time the spouse wants to drag you away from your garret. But lets not dismiss this latter day pretension until weve heard from the Greats. Plato, in The Republic, said that Poetry is not to be regarded seriously as attaining to the truth. Goethe thought that words were foppish and he would have preferred to speak like nature, altogether in drawings. Despite these warnings, so many wordsmiths carry on thinking that they will find something. Tolstoi knew a bit about finding things but he thought that the only absolute knowledge attainable by man is that life is meaningless. Is this where the path of discovery leads? Wherever it goes, Shakespeare must have got there first. We know very little about the man but we do know that at the age 46 he decided to pack it all in. Where does that leave us? 50 93 46 56 2008 GUNTUR/VIJAYAWADA/VISAKHAPATNAM/ONLINE CLASS/FACEBOOK CLASS
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Passage 39
DAY 39
Rugby stemmed from the United Kingdom, and rugby football was its original name, which was short for rugby. Because the ball looked like an olive, so it was called football in China. Rugby was actually a city in the central England. The rugby school was the birthplace of rugby football. There was a stone tablet in the school, which read Commemorating the brave action of W. W. Ellis. They said that once in a football match, Ellis, was running with the ball in his arms in a moment of emergency, which was regarded as a mistake in 1823. But it caused the other players to follow. Although this was out of bound, it gave people a new inspiration. In Britain and the commonwealth regions, rugby attracted the interests of people who liked doing sports all over the world. As time passed, it was gradually recognized by others. Thereupon, a new sport, which derived from the football, could not only benefit your bodys all-around development, but also with high value of exercise. Rugby gradually developed at Cambridge University after 1839 and established the rugby football club one after another. The intercollegiate games also became very active. UK set up the Football Association in 1871, soon this sport spread quickly to European countries, the United States, Canada, Australia and New Zealand. In 1886, the International Football Federation was established. The International Football game was held in France in 1906. The British football changed continuously when it spread to other countries. Different forms of rugby also created by many other countries, like the United States, Canada and Australia. But the venue, equipment, rules, clothing, participate numbers and the balls size, as well as the competition methods were all different. It could be generally divided into two kinds; British rugby (also called soft football) and American Football (also called the hard football) . They gained amusing nicknames on account of the various styles and play methods. In USA, it is also called NFL Football. Rugby was called the hard football, because the players didnt wear guards in a competition. American football was slightly smaller than rugby, so it was easy to pass the ball. Needing the team spirit of tenaciously strives to succeed; rugby was helpful to military training, and especially in physical stamina training. So this sport was also very popular in the army, and was strongly promoted. For this reason, rugby won the ball of army reputation in western countries. Nowadays, some of the individual universities also have their good rugby teams in China.
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Passage 40
DAY 40
Women experience more stress than ever. They are more active than ever before and have far more to do in the process of a day than ever before. With all of the things to do, there is even more stress to deal with. Women need stress relief more than ever. When it comes to stress, it can be hard to deal with and if it isnt taken care of, women can experience major health problems in the long run. Not to mention that if you already have health problems, stress can make things even worse. Given the very different chemical and hormonal make-ups of the two genders, womens health has emerged as a very specific branch of medical research. Thanks to all of this research, better and better answers have emerged for the problems those women of all ages face. The growing knowledge and popularity of womens health issues have spawned an entire product line of supplements and things that are specifically designed for women. Doctors have found out that the natural estrogenic that a womans body produces can be effectively replaced, thus reducing the difficult symptoms of menopause. Research has shown that stimulating a womans natural progesterone levels is particularly beneficial in numerous ways, most notably regarding troublesome conceptions. Even osteoporosis, which threatens all older women, has become a lot clearer in recent years. Even cosmetic complaints, like the effects of aging and weight problems can be addressed in various ways. The most gentle and independently proactive solution is to help your body to help itself through the use of effective health supplements. Of course, you also have to eat right and exercise if you are interested in your overall health. When it comes to womens health, there are certain afflictions that are specific to women. Some of the most common problems women faces are: hormonal imbalances, cervical cancers, uterine complications etc.
Because of the changes in womens attitudes toward health, beauty, fitness and feminine strength, women are also challenging long-held ideas about aging and sexuality. Women dont just look younger and sexier longer these days; they are living longer as well. Enhanced awareness of nutrition and proper diet keeps our bodies healthier on a cellular level. Aerobic exercise keeps our hearts and vascular systems in peak condition. Weight bearing exercise increases our bone density, making us stronger, more resistant to injury and even some illnesses. All this exercise can help you with stress relief, more positive attitudes and a better ability to grapple with anxiety and depression.
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Passage 41
DAY 41
Black power in many ways signified everything non-violence was not, racial hatred, violence and extreme self-reliance. However the two approaches did have many similarities in their long-term objectives. Both demanded complete equality not just in theory but in practice. Where they differed most was in the methods used to achieve this goal and the time they were prepared to wait for progress to be made. The philosophy of non-violence was heavily rooted in religion and common sense. To succeed non-violent protest required not just the support of the majority of the black population but that populations active participation. Black churches were the only black dominated mass organisations in the south capable of rallying this level of support at the time of the civil rights movement. Religious leaders were respected by and held considerable sway over their delegations. A successful protest therefore required the active support of church leaders and more often than not adopted many of their values. When Montgomery activists wanted to organise a bus boycott they turned to their ministers for leadership. Their actions under the leadership of Martin Luther King set the tone for future peaceful civil rights protests. King preached love, self-sacrifice and the restoration of black dignity during the boycott. Blacks had to prove to northern whites that they were worthy of and being denied their constitutional rights as American citizens. This 54 93 46 56 2008 GUNTUR/VIJAYAWADA/VISAKHAPATNAM/ONLINE CLASS/FACEBOOK CLASS
was part of a faith in liberal reform through the democratic process held by King and other proponents of non-violence. Kings visit to India hardened his belief in the righteousness of massive non-violent resistance. It also ensured he and his followers choose jail time over paying bail. This use of lives and bodies to right injustice became the prevalent form of protest throughout the early to mid-1960s. Being imprisoned was a badge of respect for blacks throughout the country. The non-violence movement was built on mass participation with religious overtones. To achieve reform the movement actively sort support from white liberals and the federal government. Often protests were initially spontaneous and focused on local or specific goals. Marches, sit-ins, freedom rides and boycotts started in this manner. These actions were reliant on the local black community wearing down the white community and especially its business sector to the point where they pressured the white authorities for change. The formation of new civil rights organisations notably Kings Southern Christian Leadership Conference (SCLC) allowed the movement to fashion national objectives. These consisted of the achievement of federal support for segregation and the enactment of civil rights legislation. The SCLC provoked southern white violence by staging specific campaigns in racially tense cities like Birmingham and Selma. Police brutality was transmitted to the nation through the media particularly television creating great sympathy for the movement in the north. A member of congress viewed the graphic beatings in Selma as an exercise in terror. King and others used this sympathy to push for the civil rights legislation. The reverse was that moderate leaders avoided certain protests to maintain federal support. King even stopped a march in Selma midway to appease President Lyndon Johnson. Targeted mass protests were the mainstay of the non-violent civil rights movement. Black power meant different things to different people. In terms of aims black power and the non-violent civil rights movement had much in common. Both wanted to uplift their race politically and economically. Unlike non-violence its reach was deeper fundamentally changing black culture. At black powers core were black unity, self-determination and pride in black culture. Distinctive hairstyles, soul music and soul theology were just some of the aspects made popular by the philosophy of black power. Malcolm X created the ideological basis for the black power philosophy with his constant demand for black pride and self-sufficiency. Proponents of black power can be split into pluralist and nationalist groups. Both focused on the unbalanced power relationship between whites and blacks. Pluralists believed the two races could live beside each other amicably in a multicultural society. Nationalists were convinced a stronger and more oppressive white culture would inevitably dominate black culture. Hence they wished to withdraw from society, some even wishing to return to Africa. Other nationalists advocated setting up a black nation state in the South or autonomous areas in Americas major cities. Black powers nature as ambiguous and decentralised allowed for innovation and change. Black power and non-violent strategy had little in common. Black power proponents often used revolutionary and violent rhetoric to awaken the masses. Retaliation was promoted if necessary to ensure hostile whites found a new level of respect for blacks. Black studies programs and the teaching of African languages were a crucial part of strengthening black identity pride. Community control of black neighbourhoods and organisations was also promoted to increase the black power base. Tenants councils, community centres and black companies were among the various devices used. These strategies gave blacks more control of their own destiny and recognised that the relative power of competing interest groups determined the nature of American society.
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Passage 42
DAY 42
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Passage 43
DAY 43
Although it is greatly debatable to buy the argument of our Prime Minister pushed forward a few years ago at an award function in United Kingdom that it was only because of the British that we Indians became civilized, and while this statement was received with substantial discomfort by certain sections of Indian society, and while most of the people in the subcontinent completely ignored the unacceptable gesture of the gratitude by the Prime Minister, yet it is equally significant to note that the proposition actually never received considerable opposition, refusal or rejection from the mainstream intelligentsia including our ever vigilant and critical (free) media. One of the most understandable reasons could be the prevalence of a sense of acceptance of the fact (?) that we owe a lot to our erstwhile rulers for our present day achievements, if there are any. The other equally important reason may be the overwhelming indifference to our past and historical antecedents amongst our extensively discussed emerging great Indian middle class that is more towards future (read career). This should not suggest that being career-oriented or future-directed bears some uncherished immoral attitude but yes it sometimes totally avoids uncomfortable discussions and debates. Therefore, we did not see any section of our society advising to have duly informed and pretty systematic debate on the whole issue of what we have actually received from the colonial rule and what we have really lost because of that. Any formulation about our historical background needs to be essentially associated with the careful study of the social, economic, agricultural, cultural, political and other relevant dimensions of our community of yesteryears. The contextual and calculated interpretations of events of the past may sometimes have intoxicating efforts but we have constantly been 57 93 46 56 2008 GUNTUR/VIJAYAWADA/VISAKHAPATNAM/ONLINE CLASS/FACEBOOK CLASS
This is the only solution but we will have to have faith in it. We shall have to provide examples to new generation by our acts as our ancient poet emphatically pronounces:
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Passage 44
DAY 44
The social contract theory throws light on the origin of the society. According to this theory all men are born free and equal. Society came into existence because of the agreement entered into by the individuals. The classical representatives of this school of thought are Thomas Hobbes, John Locke and Rousseau. Thomas Hobbes was of opinion that society came into being as a means for the protection of men against the consequences of their own nature. Man in the state of nature was in perpetual conflict with his neighbours on account of his essentially selfish nature. The life of man was solitary poor, nasty, brutish and short. Every man was an enemy to every other man. Hobbes in his book Leviathan has made it clear that man found nothing but grief in the company of his fellows. Since the conditions in the state of nature were intolerable and men longed for peace, the people entered into a kind of social contract to ensure for themselves security and certainty of life and property. By mutual agreement they decided to surrender their natural rights into the hands of a few or one with authority to command. The agreement was of each with all and of all with each other. The contract became binding on the whole 59 93 46 56 2008 GUNTUR/VIJAYAWADA/VISAKHAPATNAM/ONLINE CLASS/FACEBOOK CLASS
community as perpetual social bond. Thus in order to protect himself against the evil consequences of his own nature man organized himself in society in order to live in peace with all. John Locke believed that man in the state of nature was enjoying an ideal liberty free from all sorts of rules and regulations. The state of nature was a state of peace, goodwill, mutual assistance and preservation. But there was no recognized system of law and justice. Hence his peaceful life was often upset by the corruption and viciousness of degenerate men. The men were forced to live in full of fears and continual dangers. In order to escape from this and to gain certainty and security men made a contract to enter into civil society or the state. This contract Locke called social contract. This contract put an end to the state of nature and substituted it by civil society. The social contract was no more than a surrender of rights and powers so that mans remaining rights would be protected and preserved. The contract was for limited and specific purposes and what was given up or surrendered to the whole community and not to a man or to an assembly of men. According to Locke the social contract later on contributed to the governmental control. The governmental contract was made by the society when it established a government and selected a ruler to remove the inconveniences of ill condition. Jean Jacques Rousseau Rousseau the French writer of the 18th century in his famous book The Social Contract wrote that man in the state of nature was a noble savage who led a life of primitive simplicity and idyllic happiness. He was independent, contented, self-sufficient, healthy, fearless and good. It was only primitive instinct and sympathy which united him with others. He knew neither right or wrong and was free from all notions of virtue and vice. Man enjoyed a pure, unsophisticated, innocent life of perfect freedom and equality in the state of nature. But these conditions did not last long. Population increased and reason was dawned. Simplicity and idyllic happiness disappeared. Families were established, institution of property emerged and human equality was ended. Man began to think in terms of mine and yours. When equality and happiness of the early state was lost, war, murder, conflicts became the order of the day. The escape from this was found in the formation of a civil society. Natural freedom gave place to civil freedom by a social contract. As a result of this contract a multitude of individuals became a collective unity- a civil society .Rousseau said that by virtue of this contract everyone while uniting him to all remains as free as before. There was only one contract which was social as well as political. The individual surrendered himself completely and unconditionally to the will of the body of which he became a member. The body so created was a moral and collective body and Rousseau called it the general will. The unique feature of the general will was that it represented collective good as distinguished from the private interests of its members. The theory of social contract has been widely criticized as historically there is nothing to show that the society has ever been deliberately created as a result of voluntary agreement or contract. Nor can we suppose that man could ever think of entering into a contract with others when he lived under conditions of extreme simplicity, ignorance and even brutality. The theory seemed to be mere fiction as state of nature never existed. The most primitive people even lived in some form of society however rudimentary or unorganized. There are always two parties to the contract. There cannot be a one-sided contract as was conceived by Hobbes. The advocates of the theory hold that the early individuals entered into the contract for their individual safety and security of property. But history tells us the other way. Early law was more communal than individual and the unit of society was not the individual but the family. Society has moved from status to contract and not from contract to status as the theorists of the social contract argued. According to Sir Henry Maine contract is not the beginning of society but the end of it.
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Passage 45
DAY 45
came under severe strain in the 1950s and 1960s.The village had a new pattern of administration and relations with the revenue officials. In 1950s a large number of studies were undertaken on the Indian village community. In the year 1955, S.C Dubes Indian village, M.N Srinivas ed Indian villages, D.N Majumdars ed Rural profiles and Mckim Marriotts ed Village India were published. All these studies have been analyzed structure and process of change in village India. The main aspects covered are caste system, family, jajmani system, religious practices and rituals, health conditions, village and caste panchayats, social mobility among different caste groups and the impact of adult franchise, education, development programmes and Panchayati Raj on the rural people. The Indian village has undergone significant changes particularly since independence. Caste system is not confined to jajmani based social and economic relations. The jajmani system itself has declined to a large extent due to increased contact with the towns and the introduction of technological devices in agriculture. Market economy has shattered the traditional arrangement. Caste is still a noticeable source of socio-cultural factor at the time of birth, marriage, death and other social occasions. Caste endogamy, clan exogamy and other allied rules are still adhered to in deciding marriages. While caste endogamy refers to marrying within ones own caste, clan exogamy refers to marrying within ones own caste but outside ones own clan. A given caste or sub caste has number of clans or gotras where marriage is avoided between same gotra. Despite these patterns of caste continuities, inter caste relations have become segmentary-intercaste interdependence has lessened, tensions have increased and competition between different castes for a major share of the village resources has heightened. At times votes are cast in panchayat elections on caste lines. Castes have been interest and pressure groups. The economic scene today is different in villages due to education, migration, and change in the cropping pattern, electrification, irrigation and contact with the towns and cities. The traditional pattern of lending has also changed to a large extent due to alternate channels of borrowing money. The wages of agricultural workers have gone up. They have acquired bargaining power also. Economic dominance of the intermediate castes and classes such as Jats, Ahirs, Kurmis, Reddys and Patels etc has increased due to land reforms and green revolution. The lower castes have also come closer to other sections of people in the village community. Untouchability is not as rigid an institution as it was earlier. A large number of people have migrated to bigger towns and cities in search of jobs and livelihood obliterating the caste and class boundaries. Despite these changes in the villages social structure and economy the village remains different from towns and cities in its ethos, way of life and interpersonal relations.
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Passage 46
DAY 46
Human rights are in desperate straits around the world. They are widely proclaimed, but brutally violated on a mindnumbing scale. The basic outlook which I wish to represent in this talk is that moral rights depend, for their effective implementation, upon a certain condition in human community. If the community is not one of a high level of moral substance (that is, not predominantly one of morally good people, both in official positions and throughout the population), then moral rights will, at best, degenerate into mere legal rights; and even then they will be continually subject to failure in the context of need, because the individuals involved in such contexts do not act to support them. Those legal rightswhere they existwill also be, at most, honoured in the letter, and not in the spirit of human dignity, as Kant and those of similar moral outlook would understand human dignity. When this is the case, those who have legal rights (blacks, women, prisoners of war, homosexuals) may be able to bring governmental processes and forces to bear to secure themselves in certain (obviously important) respects, and that is no small thing. But even that is not a given, and in any case they will not achieve the type of acceptance and endorsement that persons of genuinely good moral will and character extend to others in a moral community. This will be even truer of people outside of ethnic and national groups, and especially when hostilities prevail between such groups. Professor Clark Butler has written: In large impersonal societies, individuals steeped in duty consciousness often lack a sufficient knowledge of others and their claims to guarantee protection of their rights even when they would wish to do so. However conscientious individuals are, they are often unconscious of the secondary consequences of actions. Even continuous duty consciousness is thus compatible with periodic justified eruptions of rights consciousness. Yet a significant difference exists between the rights consciousness of individuals who must arouse a non-existence sense of duty and that of individuals who can call on a pre-established sense of duty in others. This is a very penetrating observation about the unfortunate human condition. The lack of a pre -established sense of duty in others does indeed make periodic justified eruptions of rights consciousness inevitable. But I would add that more than such a sense of duty in others is required for a proper functioning of rights in human society. Conscious dutifulness to rights is never enough, and not just for the reasons Professor Butler points out. Rather, such dutifulness can succeed only as a part of a moral character of pro-active concern for human goods. Beyond such a sense of duty lies the sense of moral identity that each person carries as a marginal presence in all their acts and activities. That is, the sense of what makes me a good person, a person worthy of approval, inclusion and support from normal human beings around me. This sense of moral worth contains a presumption of the reality of moral worth, and a presumption of shared knowledge of that reality. When the sense of moral reality and knowledge is lacking or mistakene.g., takes there to be no such thing as moral reality, or takes moral worth to consist in ethnic identity, or in success at pursuing 63 93 46 56 2008 GUNTUR/VIJAYAWADA/VISAKHAPATNAM/ONLINE CLASS/FACEBOOK CLASS
ones own interest above all, etc.then the sense of moral identity of the individual (and the group) will lead to the denial or suppression of the human goods which it is the primary function of morality to protect and advance. Among human goods, of course, rights themselves stand very high. In fact, they are, if you wish, a kind of meta-good, for their point is always to assure the accessibility of other goods. Their point is never just themselves, never just having rights, but a kind of life in which respect and active support for human dignity and well-being is paramount. Now, what I have called the sense of moral identity, which each person carries in all their acts and activities, rests upon a presumption of a shared knowledge of life and of what makes one morally acceptable or praiseworthy or not. However fragmentary or misguided the presumed knowledge may be, it is, I think, impossible for a normal human being (I leave out of account sociopathic and extremely traumatized individuals) to conduct their life except upon the assumption that there is shared or sharable knowledge of who is a morally good person and who is notand, by extension, of what is right and wrong, of what is morally obligatory or praiseworthy or not, and so forth. Thus, the normal human being accepts the necessity and the possibility of moral guidance and of learning about such matters, and the possibility of being wrong with regard to them. That is, of holding false views regarding them.
## Starting Time:________ Ending Time:________ Reading Speed:______
1) What is the Main Point/Central Idea of the passage? --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------2) Write the summary of the passage? -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Vocabulary:
Passage 47
DAY 47
DEPRIVE a person of oxygen and he will turn blue, collapse and eventually die. Deprive economies of credit and a similar process kicks in. As the financial crisis has broadened and intensified, the global economy has begun to suffocate. That is why the worlds central banks have been administering emergency measures, including a round of co-ordinated interestrate cuts on October 8th. With luck they will prevent catastrophe. They are unlikely to avert a global recession.
According to the IMFs most recent World Economic Outlook, published on October 8th, the world economy is entering a major downturn in the face of the most dangerous shock to rich-country financial markets since the 1930s. The fund expects global growth, measured on the basis of purchasing-power parity (PPP), to come down to 3% in 2009, the slowest pace since 2002 and on the verge of what it considers being a global recession. (The funds definition of global recession takes many factors into account, including the rate of population growth.) Given the scale of the financial freeze, the funds forecast looks optimistic. Other forecasters are convinced that a global recession is inevitable. Economists at UBS, for instance, expect global growth of only 2.2% in 2009. The rich worlds economies were either shrinking, or close to it, long before September. Recent weeks have made a richworld recession all but inevitable. Americas economy lost steam throughout the summer. Temporarily buoyed by fiscal stimulus and strong exports, output grew at a solid 2.8% annualised rate between April and June. But as the stimulus wore off, the job market worsened, credit tightened and consumer spending slid. That slide became a rout in September. The economy lost 159,000 jobs, the most in a month since 2003. Car sales fell to a 16-year low as would-be buyers were unable to get credit. The economy may already have shrunk in the third quarter. The rest of the year is likely to be worse. Some economists expect consumer spending to fall at its fastest pace since the 1980 recession. Add in other gloomy evidence, such as a survey of purchasing managers that suggests manufacturing is extremely weak, and it is clear that output is now falling. Americas recession may not yet be official, but it is well under way. In Europe the outlook is equally grim. The British economy, which stalled in the second quarter, is now unmistakably falling into recession. The IMFs forecasts suggest that Britain will see the worst performance of any big economy in the year to the fourth quarter of 2008. The economies of the euro area, too, are struggling badly. Figures released on October 8th showed that output in the euro area fell at an annualised rate of 0.8% in the second quarter. GDP shrank in the currency zones three largest countriesGermany, France and Italy. The fourth largest, Spain, barely grew. As elsewhere, the most recent figures have grown grimmer still. Business confidence has turned down and a closely watched survey of purchasing managers points to a further contraction in activity over the summer months. Even the European economies that are less directly affected by housing busts, such as Germany, have been hard hit. The big hope for the euro area was that German shoppers, relatively free of debt and with scope to save a little less, would make up for weakness in debt-laden economies such as Spain. But household spending in Germany has been falling since the end of last year.
## 1) What is the Main Point/Central Idea of the passage? ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2) Write the summary of the passage? --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------65 93 46 56 2008 GUNTUR/VIJAYAWADA/VISAKHAPATNAM/ONLINE CLASS/FACEBOOK CLASS
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Vocabulary:
Passage 48
DAY 48
are electrical hook-ups for the special equipment if it is needed. The key here is to have each band perform for you and decide if they are acceptable. You should check their references, and if there is a contract agreement, make sure that the both of you are in agreement. Find out what their cancellation policy is as well as a refund policy. While all of these insignificant details might be the farthest from your mind when it comes to the wedding, it is crucial for your special day.
## Starting Time:________ Ending Time:________ Reading Speed:______
1) What is the Main Point/Central Idea of the passage? --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------2) Write the summary of the passage? -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Vocabulary:
Passage 49
DAY 49
After people entered into the civilization of human society, shoes were invented, before this time, people also ran with barefoot for millions of years. A British magazine called Nature published a journal recently, from the journal we could learn that appropriate barefoot running could do good to our landing method, as well as in favour of the health of skeletal muscle. Harvard University and other Institutions researchers investigated the United States and Kenya long-term runners who landed their feet on the ground. The results showed that people who ran for a long time with running shoes always land their heels on the ground at first. However, those people who often run with barefoot will let their former soles to touch the ground at first. It can be seen from the research that if heels touched the ground at first, the impact forces your heels obtained would be three times of your body weight, even though you were with running shoes, which was also contained a cushion. However, if your former feet touch the ground at first, the impact will be reduced to only about 60percent of body weight because of the effective cushion of tendons and joints that are come from the ankle. Due to the long-term 67 93 46 56 2008 GUNTUR/VIJAYAWADA/VISAKHAPATNAM/ONLINE CLASS/FACEBOOK CLASS
## Starting Time:________ Ending Time:________ Reading Speed:______
1) What is the Main Point/Central Idea of the passage? --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------2) Write the summary of the passage? -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Vocabulary:
Passage 50
DAY 50
The difficulties historians face in establishing cause-and-effect relations in the history of human societies (are broadly similar to the difficulties facing astronomers, climatologists, ecologists, evolutionary biologists, geologists, and palaeontologists. To varying degrees each of these fields is plagued by the impossibility of performing replicated, controlled experimental interventions, the complexity arising from enormous numbers of variables, the resulting uniqueness of each system, the consequent impossibility of formulating universal laws, and the difficulties of predicting emergent properties and future behaviour. Prediction in history, as in other historical sciences, is most feasible on large spatial scales and over long times, when the unique features of millions of small-scale brief events become averaged out. Just as I could predict the sex ratio of the next 1,000 new-borns but not the sexes of my own two children, the historian can recognize factors that made inevitable the broad outcome of the collision between American and Eurasian societies after 13,000 years of separate developments, but not the outcome of the 1960 U.S. presidential election. The details of which candidate said what during a single televised debate in October 1960 could have given the electoral victory to Nixon instead of to Kennedy, but no details of who said what could have blocked the European conquest of Native Americans. How can students of human history profit from the experience of scientists in other historical sciences? A methodology that has proved useful involves the comparative method and so-called natural experiments. While neither astronomers studying galaxy formation nor human historians can manipulate their systems in controlled laboratory experiments, they both can take advantage of natural experiments, by comparing systems differing in the presence (or in the strong or weak effect) of some putative causative factor. For example, epidemiologists, forbidden to feed large amounts of salt to people experimentally, have still been able to identify effects of high salt intake by comparing groups of humans who already differ greatly in their salt intake; and cultural anthropologists, unable to provide human groups experimentally with varying resource abundances for many centuries, still study long-term effects of resource abundance on human societies by comparing recent Polynesian populations living on islands differing naturally in resource abundance. The student of human history can draw on many more natural experiments than just comparisons among the five inhabited continents. Comparisons can also utilize large islands that have developed complex societies in a considerable degree of isolation (such as Japan, Madagascar, Native American Hispaniola, New Guinea, Hawan, and many others), as well as societies on hundreds of smaller islands and regional societies within each of the continents. Natural experiments in any field whether in ecology or human history, are inherently open to potential methodological criticisms. Those include confounding effects of natural variation in additional variables besides the one of interest, as well as problems in inferring chains of causation from observed correlations between variables. Such methodological problems have been discussed in great detail for some of the historical sciences. In particular, epidemiology, the science of drawing inferences about human diseases by comparing groups of people (often by retrospective historical studies), has for a long time successfully employed formalized procedures for dealing with problems similar to those facing historians of human societies. In short, I acknowledge that it is much more difficult to understand human history than to understand problems in fields of science where history is unimportant and where fewer individual variables operate. Nevertheless, successful methodologies for analysing historical problems have been worked out in several fields. As a result, the histories of dinosaurs, nebulae, and glaciers are generally acknowledged to belong to fields of science rather than to the humanities. | 69,671 | 311,179 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.0625 | 4 | CC-MAIN-2019-26 | latest | en | 0.916136 |
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# The probability that a visitor at the mall buys a pack of
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The probability that a visitor at the mall buys a pack of [#permalink]
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11 Oct 2008, 13:36
This topic is locked. If you want to discuss this question please re-post it in the respective forum.
The probability that a visitor at the mall buys a pack of candy is 30%. If three visitors come to the mall today, what is the probability that exactly two will buy a pack of candy?
* 0.343
* 0.147
* 0.189
* 0.063
* 0.027
Can someone explain answer for this question and what approach i should follow?
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Re: m04 - 9 - probablity and [#permalink]
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11 Oct 2008, 14:16
•2 Yes, 1 No = 3/10*3/10*7/10 = 63/1000
•3 Possibilities = YYN + YNY + NYY = 3(63/1000) = 189/1000 = .189%
Re: m04 - 9 - probablity and [#permalink] 11 Oct 2008, 14:16
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# Search results
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Coulomb's constant
Coulomb’s constant, the electric force constant, or the electrostatic constant (denoted ke ) is a proportionality constant in equations relating ... more
Supercsapacitor - Time to deliver a Constant Power
A supercapacitor (SC) (sometimes ultracapacitor, formerly electric double-layer capacitor (EDLC)) is a high-capacity ... more
Vacuum Evacuation Time
The evacuation time for a vacuum pump can be calculated as shown
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Moment of Inertia - Rod end
Moment of inertia is the mass property of a rigid body that determines the torque needed for a desired angular acceleration about an axis of rotation. ... more
Coefficient Of Performance for a perfectly reversible cooler
Pulse tube cryocooler(or refrigerator) can be made without moving parts in the low temperature part of the device, making the cooler suitable for a wide ... more
Power gain
In electronics, gain is a measure of the ability of a two-port circuit (often an amplifier) to increase the power or amplitude of a signal from the input ... more
Vacuum wavelength
When an electromagnetic wave travels through a medium in which it gets attenuated (this is called an “opaque” or “attenuating” ... more
Worksheet 300
Calculate the Reynolds number N′R for a ball with a 7.40-cm diameter thrown at 40.0 m/s.
Strategy
We can use the Reynolds number equation calculate N’R , since all values in it are either given or can be found in tables of density and viscosity.
Solution
We first find the kinematic viscosity values:
Kinematic Viscosity
Substituting values into the equation for N’R yields:
Reynolds number
Discussion
This value is sufficiently high to imply a turbulent wake. Most large objects, such as airplanes and sailboats, create significant turbulence as they move. As noted before, the Bernoulli principle gives only qualitatively-correct results in such situations.
Reference : OpenStax College,College Physics. OpenStax College. 21 June 2012.
http://openstaxcollege.org/textbooks/college-physics
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Plasma conductivity
Electrical resistivity (also known as resistivity, specific electrical resistance, or volume resistivity) is an intrinsic property that quantifies how ... more
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In electronics, a diode is a two-terminal electronic component with asymmetric conductance; it has low (ideally zero) resistance to current in one ... more
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Category | 575 | 2,609 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.1875 | 3 | CC-MAIN-2024-18 | latest | en | 0.855054 |
https://socratic.org/questions/is-100-20y-y-2-a-perfect-square-trinomial-and-how-do-you-factor-it | 1,579,296,794,000,000,000 | text/html | crawl-data/CC-MAIN-2020-05/segments/1579250591234.15/warc/CC-MAIN-20200117205732-20200117233732-00333.warc.gz | 663,981,307 | 6,586 | # Is 100 - 20y + y^2 a perfect square trinomial and how do you factor it?
Jul 26, 2015
Yes, your expression is a perfect square trinomial.
#### Explanation:
A trinomial is simply a polynomial that has three terms. Your expression is a trinomial because it has three terms: ${y}^{2}$, $- 20 y$, and $100$.
In order for it to be a perfect square trinomial, it has to be equal to the square of a binomial, which is a polynomial that has two terms.
You can factor your expression in order to check if it meets this criterion. A trinomial can be factored as a product of two binomials, so all you have to do is see if these two binomials are equal
${y}^{2} - 20 y + 100 = \left(y + a\right) \left(y + b\right)$
The right side of this equation is equal to
$\left(y + a\right) \cdot \left(y + b\right) = {y}^{2} + a \cdot y + b \cdot y + {b}^{2}$
$\left(y + a\right) \left(y + b\right) = {y}^{2} + \left(a + b\right) y + {b}^{2}$
Now match the terms of your original expression with these terms
${y}^{2} \textcolor{b l u e}{- 20} y + \textcolor{g r e e n}{100} = {y}^{2} + \textcolor{b l u e}{\left(a + b\right)} y + \textcolor{g r e e n}{{b}^{2}}$
You need to have
${b}^{2} = 100 \implies b = \pm 10$
$\left(a + b\right) = - 20$
• If $b = 10$, then
$a = - 20 - 10 = - 30$
This solution is not valid because the expression would be equal to
$\left(y - 30\right) \cdot \left(y + 10\right) = \textcolor{red}{\cancel{\textcolor{b l a c k}{{y}^{2} - 20 y - 300}}}$
• If $b = - 10$, then
$a = - 20 + 10 = - 10$
This is a valid solution, since
$\left(y - 10\right) \cdot \left(y - 10\right) = {y}^{2} - 20 y + 100$
Therefore, your original expression is a perfect square trinomial because you can write it as
${y}^{2} - 20 y + 100 = \left(y - 10\right) \cdot \left(y - 10\right) = \textcolor{g r e e n}{{\left(y - 10\right)}^{2}}$ | 664 | 1,842 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 16, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.78125 | 5 | CC-MAIN-2020-05 | longest | en | 0.766722 |
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CHECKOUT
Basic Math Skills
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In Basic Math Skills, Grade 2, you'll find just what you need to provide extra skill practice for individual students, small groups, or the whole class. These reproducible pages are perfect for homework too.
Skills practiced include:
Number and Operations
• Count, read, write, compare, and order whole numbers to 1,000
• Group hundreds, tens, and ones
• Identify odd and even numbers
• Use ordinal numbers
• Add and subtract 1-, 2-, and 3-digit numbers
• Recognize, name, and compare fractions
• Identify and know the value of coins and dollar bills
• Count by tens, fives, and twos
• Solve multiplication and division problems
Algebra
• Recognize, describe, and extend patterns
• Solve problems involving simple number patterns
• Relate problem situations to number sentences
• Use commutative and associative rules
Geometry
• Identify, describe, and compare plane objects
• Sort and classify plane and solid objects
• Identify and construct congruent figures and lines of symmetry
• Put shapes together and take them apart to form other shapes
• Calculate the perimeter of a shape
Measurement
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• Measure length in inches
• Measure length in centimeters
• Tell time to the nearest quarter-hour
• Determine the duration of time in hours
• Know relationships of time
Data Analysis and Probability
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• Explore probability
The reproducible flash cards cover addition and subtraction facts 11-18 and multiplication facts for 2s, 5s, and 10s.
This resource contains teacher support pages, reproducible student pages, and an answer key.
ISBN: 9781557998972
ISBN-10: 1557998973
Series: Basic Math Skills
Audience: Children
For Ages: 9 years old | 448 | 1,957 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.859375 | 3 | CC-MAIN-2019-04 | latest | en | 0.847394 |
https://dodgerocksgasmonkey.com/how-to-solving-eigenvectors-38 | 1,675,087,987,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764499819.32/warc/CC-MAIN-20230130133622-20230130163622-00063.warc.gz | 234,937,316 | 5,630 | # How to find eigenvectors
The solver will provide step-by-step instructions on How to find eigenvectors.
Decide mathematic question
Mathematics learning that gets you
Solving word questions
## Finding eigenvectors and eigenspaces example (video)
Formula to calculate eigenvectors. You should first make sure that you have your eigen values. Then subtract your eigen value from the leading diagonal of the matrix. Multiply the answer by
Solve mathematic question
Mathematics is the study of numbers, shapes and patterns. It is used in everyday life, from counting and measuring to more complex problems.
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To solve a math equation, you need to figure out what the equation is asking for and then use the appropriate operations to solve it.
Deal with math question
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Having trouble with math? Don't worry, our experts can help clear up any confusion and get you on the right track.
## Determining the Eigenvectors of a Matrix
So the eigenspace that corresponds to the eigenvalue minus 1 is equal to the null space of this guy right here It's the set of vectors that satisfy this equation: 1, 1, 0, 0. And then you have v1
## How to Find Eigenvalues and Eigenvectors: 8 Steps
A v = λ v. Where, v = Eigenvector of a given matrix A. λ = Eigenvalue of matrix A. The above equation can be rewrite to find eigenvector as: ( A − λ I) v = 0. Where I is the identity matrix and
## Eigenvector and Eigenvalue
Deal with mathematic equation
Math can be difficult, but with a little practice, it can be easy! | 361 | 1,565 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4 | 4 | CC-MAIN-2023-06 | latest | en | 0.881879 |
https://bepurebeauty.com/skin-diseases/how-many-moles-are-present-in-35-5-grams-of-cl2.html | 1,669,474,299,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446708010.98/warc/CC-MAIN-20221126144448-20221126174448-00502.warc.gz | 167,448,050 | 18,897 | # How many moles are present in 35 5 grams of Cl2?
Contents
## How many moles are present in 35.5 grams of Cl2?
35.5 g of chlorine gas Cl2 C l 2 =0.5 moles. (as molecular weight of chlorine is 71 g/mol).
## How many moles are in Cl2?
The answer is 70.906. We assume you are converting between grams Cl2 and mole. You can view more details on each measurement unit: molecular weight of Cl2 or mol The SI base unit for amount of substance is the mole. 1 grams Cl2 is equal to 0.014103178856514 mole.
## How many moles of KCl are there in 35.5 g?
Since 1 mole of KCl contains 1 mole of K (39.1g) and 1 mole of Cl (35.5g), the molar mass of KCl is 74 . 6 g/mol.
## How do I calculate moles?
So in order to calculate the number of moles of any substance present in the sample, we simply divide the given weight of the substance by its molar mass. Where ‘n’ is the number of moles, ‘m’ is the given mass and ‘M’ is the molar mass.
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## How many moles are in CL?
The number of moles of Cl is twice as much, because the ratio of Cl in MgCl2 to MgCl2 is 2:1. Therefore, there are 12 moles of Cl.
## How do you find moles of chlorine?
The mass of one mole of chlorine gas is its molar mass, and can be determined by multiplying its subscript by the molar mass of chlorine (atomic weight on the periodic table in g/mol), which is 35.45 g/mol .
## How many moles are in chlorine gas?
The SI base unit for amount of substance is the mole. 1 grams Chlorine is equal to 0.028206357713029 mole.
## How many grams of chlorine are equal to 5 moles of chlorine?
So if you have a mole of Cl atoms you have 6.023 x 10^23 atoms of Cl. Since a mole of chlorine atoms has a mass of 35.453 grams, then a weighted average mass for one of those atoms will be equal to (34.453 grams/mole ÷ 6.023 x 10^23 atoms/mole =) 5.720 x 10^-23 grams per atom (notice the negative sign).
## What is mole of chlorine?
A conveniently measured amount of any element that equals the mass in grams of that element is called a mole. One mole of hydrogen weighs 2 grams. One mole of chlorine weighs 70 grams. Just like a dozen equals twelve, a mole equals Avogadro’s number of atoms or 6.02 × 1023.
## How many grams are in 3 moles of cl2?
3 = Mass71. So the mass is 213 grams. So, the mass of 3 moles of chlorine molecule is 213gram.
## How many moles are in a gram of Cl2?
1 grams Cl2 is equal to 0.014103178856514 mole.
THIS IS IMPORTANT: Are big moles always cancerous?
## How many CL are in a gram?
Centiliters to grams table
Centiliters Grams
1 cl 10 g
2 cl 20 g
3 cl 30 g
4 cl 40 g
## How many moles are in CaCO3?
1 mole is equal to 1 moles CaCO3, or 100.0869 grams. | 795 | 2,719 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.125 | 4 | CC-MAIN-2022-49 | latest | en | 0.926991 |
https://archives2.twoplustwo.com/archive/index.php/t-354187.html | 1,619,011,689,000,000,000 | text/html | crawl-data/CC-MAIN-2021-17/segments/1618039544239.84/warc/CC-MAIN-20210421130234-20210421160234-00009.warc.gz | 225,869,731 | 2,281 | PDA
View Full Version : Powerball question
skitzo444
10-09-2005, 11:21 PM
If one were to play the Powerball for a long time would they play the same set of numbers every week or pick a new random set of numbers every week? I am not saying birthdays or anything like that crap but is there an advantage to picking random numbers every week or using the same set of random numbers forever.
UATrewqaz
10-10-2005, 12:12 AM
It makes no difference whatsoever what set of numbers you play or the frequency of which you play them
8 17 25 28 42 47
is just as likely as
1 2 3 4 5 6
and just as likely as any other random collection if 6 numbers 1-50 (I believe that's powerball rules right?)
Each drawing is an independent trial.
SheetWise
10-10-2005, 02:15 AM
[ QUOTE ]
It makes no difference whatsoever what set of numbers you play or the frequency of which you play them
[/ QUOTE ]
It makes no difference on your probability of winning -- it does make a difference on the value of a win. Several studies have shown that players who have "regular" numbers use birth and anniversary dates -- so the numbers 1 through 31 are played more frequently. When smaller numbers are drawn, there are more frequent wins -- when larger numbers are drawn, wins are less frequent. There are fewer split jackpots (multiple winners) when larger numbers are drawn.
Siegmund
10-10-2005, 02:21 AM
Any combination is equally likely to be drawn as the winner.
However, some combinations are more likely than others to be bought by other players causing you to split the prize. You can significantly improve your EV by playing "unpopular" numbers -- avoiding traditional 'lucky numbers' and birthdays.
If you do your own research and discover which six numbers are least popular, you should play those six every week to minimize your chance of a split. (If, however, you find a published list of which numbers are least popular, then you have a new meta-game, wondering how many other people found that same list and are playing bottom six off the list...) | 483 | 2,038 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3 | 3 | CC-MAIN-2021-17 | latest | en | 0.959285 |
http://www.thenxtstep.com/2006/08/summary-of-rif-puzzles.html?showComment=1156477680000 | 1,435,668,316,000,000,000 | text/html | crawl-data/CC-MAIN-2015-27/segments/1435375093899.18/warc/CC-MAIN-20150627031813-00115-ip-10-179-60-89.ec2.internal.warc.gz | 715,455,804 | 24,155 | ## Aug 24, 2006
### Summary of RIF puzzles...
A comment today from a reader asked for a summary of all the RIF puzzles found so far. I'll post them here, along with solutions (in white text so you'll have to highlight the answers to read them). Please keep in mind that these are the puzzles that I'm aware of - there might be more. If I've missed one, please let me know. (By the way, I emailed Poe and asked if I could post these solutions - the only one I'm not allowed to provide right now is the last puzzle I discuss below.)
The first puzzle found is in the upper-right corner of the large grey rectangle. Move the mouse over the area until you trigger it. The puzzles is:
JRARRQNTRAGFJVGUTBBQC
EBOYRZFBYIVATFXVYYFNT
RAGFHFRAKGEBOBGFGBNFF
VFGJVGUVAIRFGVTNGVATN
AQPBYYRPGVATVASBEZNGV
BASBEBHEPYVRAGFPURPXU
RERSBEZBERVASBEZNGVBA
BANHTHFGGJRAGLSVIR
If you use a ROT13 encoder/decoder (http://www.degraeve.com/rot13.php) to convert the code, it changes to (drag your mouse to select the text below to see it:
WENEEDAGENTSWITHGOODP
ROBLEMSOLVINGSKILLSAG
ENTSUSENXTROBOTSTOASS
ISTWITHINVESTIGATINGA
NDCOLLECTINGINFORMATI
ONFOROURCLIENTSCHECKH
EREFORMOREINFORMATION
ONAUGUSTTWENTYFIVE
So apparently some new info will be released tomorrow (August 25, 2006)
The next puzzle found wasn't a puzzle, but was a hidden link. Highlight the line below for its location.
Move the mouse around inside the letter P in the word September until the point changes to a hand. That's the hidden link.
The next puzzle was a little more difficult to solve. This one is a little trickier to find and solve. Highlight the line below for directions to find it:
The puzzle is located on the right side of the large grey rectangle. You have to move the mouse around carefully, but when you hit the trigger you'll see two columns of text pop up and the words "Yes and No" and "On and Off".
Highlight the line below for the instructions on solving the puzzle:
I'm not going to paste the long column of text here, but to solve it simply start with the first letter, F (in FAIL) and then skip every other letter: So FAIL ROSE would be FIRS (skip the A, L, O, and E.) Keep doing this as you read down the column of words. The message spells instructions to email the RIF. If you sent an email before August 20 to the RIF you received
an email from Poe with a special message and a reward that has yet to be announced... (I'm very curious... has anyone who solved it heard what the reward is?)
The next puzzle I encountered (not sure if this one came before the next one) was a hidden message embedded in the PDF file that Poe sent to nxtasy.org. Highlight the line below for instructions on finding the message:
In the first "story" on the PDF (about the missing scientist) there are letters in the story that are bolded. They are fairly easy to find and this one wasn't that hard to solve. Read the bold letters in order for a website to visit.
The next puzzle (which may have appeared before the PDF one above, not sure) was a word jumble found in the lower-right corner of the large grey rectangle. It's been unjumbled now, but if you figured it out, it spelled out sudoku.html. If you go to www.rifagent.com/sudoku.html it took you to the next puzzle.
The last puzzle I'm aware of is the actual Sudoku puzzle. I was not given permission to provide the solution for it, but basically just solve the sudoku puzzle. They kindly gave a link to a wiki article that gives the rules if you are unfamiliar with it... and if you're not good at it, find somebody (it's worth it!)
That's it... if I've missed one, please add a comment.
Anonymous said...
You missed one!
Agent #13 posted a comment yesterday that had a clue. Read it carefully.
Len
Anonymous said...
Yes, Len's right. There's a clue in this comment:
Yes, we have puzzles. But does lightning ever strike the same place twice?
Agent 13
www.rifagent.com
It's not too tough.
Jim Kelly said...
Okay, I understand now.
I didn't get permission to share the answer for this one, but it's not hard to find...
Jim
Anonymous said...
If you get tired
http://www.sudokusolver.co.uk/
Anonymous said...
Duh...the linked to wikipedia article links to a solver....
yawn....
funny that I haven't ever encountered this puzzle before
Jim Kelly said...
I imagine they meant for the sudoku to be solved without a solver... takes the fun away to use it, but some people just don't like sudoku, I guess...
I'd say that one wasn't easy... somewhere between easy and tricky, but definitely not hard.
Jim
Anonymous said...
This comment has been removed by a blog administrator.
Anonymous said...
PD: for the sudoku puzzle, you do not need to fill it all, just analyse possible values for some of the 3*3 squares and you will get the answer, it is not hard.
Mauricio
Anonymous said...
When you guys talk about the lightning twice thing, are you talking about when the window turns red and into a link?
Andy said...
You miss one!
When you drag your miuse to the first puzzle ,
may be 2 or 3 second , there's a light on the house and later , the window on the house will turn red , then you click on it , but it just wrote :
"It will be post on August 30
Poe"
I'm not sure if it was a clue , but it might can help you .
Andy
Anonymous said...
Andy, that's what Len and I were talking about in the first two comments.
Drew Stevenson said...
Having not acted on the "limited time" offer puzzle when I solved the email one in a laid back fashion... I now want to warn people about some need not be classified material:
The solution to the suduko (my 2nd one ever!) led to a page whose secrets are going to get enven better by Sept 15... SO please dont wait (like I did)....
Wishing I'd emailed - drew@bnxt.com
Anonymous said...
So tell me Drew, how would you know the "secrets are going to get even better"? A little tell there? A clue to the mysterious Mr Poe?
Can you tell me how many possible Sudoku puzzles have 28 digits defined. Is the analysis algebraic or combinatoric? Or both? Permutation groups?
Anonymous said...
Yes, I caught that as well. Maybe Mr. Drew has a secret identity? It doesn't matter. I wouldn't want to know. This is fun and I hope we adults don't ruin it for the youngsters.
Anonymous said...
This comment has been removed by a blog administrator.
Anonymous said...
This comment has been removed by a blog administrator.
Anonymous said...
When I typed the solution.html, It said that google couldn't find the site. did this expire?
Tony Naggs said...
You have to tell the web browser which site to look for the page on.
So solution.html will look on your local machine {PC, Mac, mobile phone, etc ...} | 1,645 | 6,689 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.015625 | 3 | CC-MAIN-2015-27 | longest | en | 0.912773 |
https://friendsofgoetheschool.org/2962/what-is-the-equation-y-y1-m-x-x1/ | 1,652,833,226,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662520936.24/warc/CC-MAIN-20220517225809-20220518015809-00052.warc.gz | 320,321,952 | 10,174 | # What is the equation y y1 m X x1?
The formula y – y1 = m(x – x1) is usually described as the ‘point-slope form’ for the equation of a line. It is useful because if you know one point on a certain line and the slope of that certain line, then you can define the line with this type of formula and, thus, find all the other points on that certain line.
It is sometimes called the gradient. Equations for Slope. The slope is defined as the “change in y” over the “change in x” of a line. If you pick two points on a line — (x1,y1) and (x2,y2) — you can calculate the slope by dividing y2y1 over x2x1.
Additionally, what is y1 in point slope form? PointSlope Form. In pointslope form (which is written like this: (y – y1) = m(x – x1)), y1 is the y value of the known point on the line, m is the slope, and x1 is the x value of the known point.
Beside this, what is point slope formula?
Pointslope is the general form y-y1=m(x-x1) for linear equations. It emphasizes the slope of the line and a point on the line (that is not the y-intercept).
What is Y MX B called?
In the equation of a straight line (when the equation is written as “y = mx + b“), the slope is the number “m” that is multiplied on the x, and “b” is the y-intercept (that is, the point where the line crosses the vertical y-axis). This useful form of the line equation is sensibly named the “slope-intercept form”.
### How do you graph y MX B?
To graph the equation of a line written in slope-intercept (y=mx+b) form, start by plotting the y-intercept, which is the b value. The y-intercept is where the line will cross the y-axis, so count up or down on the y-axis the number of units indicated by the b value.
### How do you solve for M in Y MX B?
1 Answer. Patrick H. Subtract b from each side to get y−b=mx . Next, divide each side by x to get y−bx=m .
### How do you find the Y intercept?
To find the y intercept using the equation of the line, plug in 0 for the x variable and solve for y. If the equation is written in the slope-intercept form, plug in the slope and the x and y coordinates for a point on the line to solve for y.
### How do you find the equation of a line?
Find the Equation of a Line Given That You Know Two Points it Passes Through. The equation of a line is typically written as y=mx+b where m is the slope and b is the y-intercept.
### What is y2 y1 x2 x1 used for?
To write the equation of a line given two points on the line, start by finding the slope. Remember that slope is the change in the y-values over the change in the x-values, or (y2-y1)/(x2-x1). Use the given points. One point is (x1, y1) and the other point is (x2, y2).
### What is the slope of a horizontal line?
Slope of a horizontal line. When two points have the same y-value, it means they lie on a horizontal line. The slope of such a line is 0, and you will also find this by using the slope formula.
### How do you use the slope equation?
Identify the slope, m. This can be done by calculating the slope between two known points of the line using the slope formula. Find the y-intercept. This can be done by substituting the slope and the coordinates of a point (x, y) on the line in the slope-intercept formula and then solve for b.
### What is a slope in math?
In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. The direction of a line is either increasing, decreasing, horizontal or vertical.
### How do you find slope and y intercept?
The equation of any straight line, called a linear equation, can be written as: y = mx + b, where m is the slope of the line and b is the y-intercept. The y-intercept of this line is the value of y at the point where the line crosses the y axis. | 961 | 3,751 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.6875 | 5 | CC-MAIN-2022-21 | latest | en | 0.939229 |
https://de.scribd.com/document/435629277/ME-G511-Lect-17-Oct-2018 | 1,603,817,259,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107894203.73/warc/CC-MAIN-20201027140911-20201027170911-00690.warc.gz | 273,276,583 | 90,935 | Sie sind auf Seite 1von 41
ME G511
BITS Pilani Dr. Y. V. D. Rao
BITS Pilani
ME G511
Equivalent Axis Rotation
Sequences of Rotations
Two different rotation sequences do not give
the same orientation and the resultant rotation
matrix depends on the order of rotations.
𝑪𝟐 𝟎 𝑺𝟐
𝑹 = 𝑹𝒙 𝜽𝟏 𝑹𝒚 𝜽𝟐 𝑹 = 𝑺𝟏 𝑺𝟐 𝑪𝟏 −𝑺𝟏 𝑪𝟐
−𝑪𝟏 𝑺𝟐 𝑺𝟏 𝑪𝟏 𝑪𝟐
𝑪𝟐 𝑺𝟏 𝑺𝟐 𝑪𝟏 𝑺𝟐
𝑹 = 𝑹𝒚 𝜽𝟐 𝑹𝒙 𝜽𝟏 𝑹= 𝟎 𝑪𝟏 −𝑺𝟏
−𝑺𝟐 𝑺𝟏 𝑪𝟐 𝑪𝟏 𝑪𝟐
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
𝑪𝟐 𝑺𝟏 𝑺𝟐 𝑺𝟏 𝑪𝟐
𝑹 = 𝑹𝒚 𝜽𝟏 𝑹𝒙 𝜽𝟐 𝑹 = 𝟎 𝑪𝟐 −𝑺𝟐
−𝑺𝟏 𝑪𝟏 𝑺𝟐 𝑪𝟏 𝑪𝟐
Principal axes Rotations
Another important variation is how these
rotations about the principal axes are performed.
Rotation of a body successively about the
principal axes of the fixed frame.
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
Successive rotations of the body are performed
about the current principal axes of the moving
frame.
These successive rotations about these two
different types of principal axes do not result in
the same orientation of the body.
For example consider two successive rotations
of 𝟗𝟎𝟎 given to an object about the principal axes
of the fixed frame.
First the body is rotated about 𝒛 − 𝒂𝒙𝒊𝒔 followed
by rotation about 𝒚 − 𝒂𝒙𝒊𝒔 as shown.
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
Z
Z’
Y’
Y
Y’’ 𝑹𝒁,𝟗𝟎𝟎 X’
X 𝑹𝒀,𝟗𝟎𝟎
X’’
Z’’
And when the order of rotations is about the
principal axes of the moving frame in the same
sequence the result is different as shown.
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
Z
Z’ Y’
Y
X’
X
Y’’ 𝑹𝒁,𝟗𝟎𝟎
Z’’ 𝑹𝒀′,𝟗𝟎𝟎
Similarly change in order of rotation
about the principal axes of moving
frame also produces different final
X’’ orientation of the body.
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
Fixed Angle Representation
Let initially the fixed frame 𝑿, 𝒀, 𝒁 𝒐𝒓 𝟏 and the
moving frame 𝑼, 𝑽, 𝑾 𝒐𝒓 𝟐 be coincident.
Z
And three different rotations W
are given to the body about the 𝜽𝟑
three axes of the fixed frame as 𝟏
shown in figure. 𝟐 V
𝜽𝟐
Y
𝜽𝟏
XU
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
First rotation is about 𝒙 − 𝒂𝒙𝒊𝒔 by an angle 𝜽𝟏 and
this is described by the rotation matrix 𝑹𝒙 𝜽𝟏 .
Z
W’
𝟏
𝜽𝟏 V’
𝟐′
𝜽𝟏
Y
𝜽𝟏
X U’
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
Next this new frame 𝟐′ is rotated by 𝜽𝟐 about 𝒚 −
𝒂𝒙𝒊𝒔 to get the third frame 𝟐′′ .
Z
W’
This rotation matrix for W’’ V’’
𝟐′
this rotation is 𝑹𝒚 𝜽𝟐 . 𝜽𝟐 𝟏 V’
𝟐′′
Y
𝜽𝟐
𝜽𝟐
X U’’
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
The final and third rotation is about the 𝒛 − 𝒂𝒙𝒊𝒔
by 𝜽𝟑 to obtain the final frame 𝟐 .
This is described by the matrix 𝑹𝒛 𝜽𝟑 .
Z
W 𝜽𝟑
V
𝟏
𝟐
Y
X U
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
Z
W’
𝟏
𝜽𝟏 Z
V’ W’
W’’ V’’
𝟐′ 𝟐′
𝜽𝟏 Y 𝜽𝟐 𝟏 V’
𝜽𝟏 𝟐′′
Y
Z
X U’ W 𝜽𝟐
𝜽𝟑 𝜽𝟐
V
𝟏 X U’’
𝟐
Y All these rotations
are shown in figures
above.
X U
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
This convention for specifying orientation is
known as “fixed angle representation” because
each rotation is specified about an axis of fixed
reference frame.
The above three rotations are referred to as
“𝑿𝒀𝒁 − 𝒇𝒊𝒙𝒆𝒅 𝒂𝒏𝒈𝒍𝒆 𝒓𝒐𝒕𝒂𝒕𝒊𝒐𝒏𝒔”.
The final frame orientation is given by the
overall rotation matrix computed by pre-
multiplication of the matrices of individual
rotations.
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
Hence 𝑹𝒙𝒚𝒛 𝜽𝟑 , 𝜽𝟐 , 𝜽𝟏 = 𝟏𝑹𝟐 =
𝑹𝒛 𝜽𝟑 𝑹𝒚 𝜽𝟐 𝑹𝒙 𝜽𝟏
Using the three individual matrices the final
rotation matrix is obtained.
𝑹𝒙𝒚𝒛 𝜽𝟑 , 𝜽𝟐 , 𝜽𝟏 =
𝑪𝟑 −𝑺𝟑 𝟎 𝑪𝟐 𝟎 𝑺𝟐 𝟏 𝟎 𝟎
𝑺𝟑 𝑪𝟑 𝟎 𝟎 𝟏 𝟎 𝟎 𝑪𝟏 −𝑺𝟏
𝟎 𝟎 𝟏 −𝑺𝟐 𝟎 𝑪𝟐 𝟎 𝑺𝟏 𝑪𝟏
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
𝑪𝟐 𝑪𝟑 𝑺𝟏 𝑺 𝟐 𝑪𝟑 − 𝑪𝟏 𝑺𝟑 𝑪𝟏 𝑺𝟐 𝑪𝟑 + 𝑺𝟏 𝑺𝟑
𝑹𝒙𝒚𝒛 𝜽𝟑 , 𝜽𝟐 , 𝜽𝟏 = 𝑪𝟐 𝑺𝟑 𝑺𝟏 𝑺 𝟐 𝑺𝟑 + 𝑪𝟏 𝑪𝟑 𝑪𝟏 𝑺𝟐 𝑺𝟑 − 𝑺𝟏 𝑪𝟑
−𝑺𝟐 𝑺𝟏 𝑪𝟐 𝑪 𝟏 𝑪𝟐
Z
𝒙𝒚𝒛 − 𝒇𝒊𝒙𝒆𝒅 𝒂𝒏𝒈𝒍𝒆
transformation is 𝜽𝟑 Roll
equivalent to 𝒓𝒐𝒍𝒍 −
𝒑𝒊𝒕𝒄𝒉 − 𝒚𝒂𝒘 (𝑹𝑷𝒀)
Pitch Y
transformation.
Yaw 𝜽𝟐
𝜽𝟏
X
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
Equivalent Axis Rotation
A third representation of orientation is by a single
A coordinate frame can be rotated about an
arbitrary 𝒂𝒙𝒊𝒔 𝒌 passing through the origin of
fixed reference frame 𝟏 .
The rotation matrix for this case is obtained by
viewing the rotation as a sequence of rotations of
frame 𝟐 (along with 𝒌 − 𝒂𝒙𝒊𝒔) about the principal
axes of frame 𝟏 .
Consider frame 𝟐 , initially coincident with
frame 𝟏 .
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
W Z
𝜽
Frame 𝟐 is rotated 𝟏
V
𝒌 − 𝒂𝒙𝒊𝒔, in frame 𝟏 . Y
𝟐′
X
U
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
The rotation of frame 𝟐 is decomposed into rotations
about the principal axes of frame 𝟏 .
axes of frame 𝟏 so as to align the 𝒂𝒙𝒊𝒔 𝒌 with 𝒙 −
𝒂𝒙𝒊𝒔
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
as to align the 𝒂𝒙𝒊𝒔 𝒌 with 𝒙 − 𝒂𝒙𝒊𝒔
Z
K
𝜶 P 𝜽
P’
Kz
1
𝜷 Ky Y
Kx
𝜶
P” 𝜽
X
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
These rotations are illustrated with the help of a vector
𝑷, initially in the direction of 𝒌 − 𝒂𝒙𝒊𝒔, as shown in
figure above.
First, rotate the vector 𝑷 (along with 𝒂𝒙𝒊𝒔 𝒌 and frame
𝟐 of figure above) by an angle −𝜶 about 𝒛 − 𝒂𝒙𝒊𝒔 such
that this rotation causes the 𝒂𝒙𝒊𝒔 𝒌 to lie in 𝒙𝒛 − 𝒑𝒍𝒂𝒏𝒆
of frame 𝟏 .
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
First, rotate the vector 𝑷 (along with 𝒂𝒙𝒊𝒔 𝒌 and frame 𝟐 of
figure below) by an angle −𝜶 about 𝒛 − 𝒂𝒙𝒊𝒔 such that this rotation
causes the 𝒂𝒙𝒊𝒔 𝒌 to lie in 𝒙𝒛 − 𝒑𝒍𝒂𝒏𝒆 of frame 𝟏 .
Z
𝜶
K
𝜶 P
P’
Kz
Kx 1 Ky Y
𝜶
P”
X
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
This rotation is written as 𝟏𝑹𝟐 = 𝑹𝒛 −𝜶
Next, vector 𝑷 (along with rotated 𝒂𝒙𝒊𝒔 𝒌) is
rotated about 𝒚 − 𝒂𝒙𝒊𝒔 by an angle 𝜷 so that
𝒂𝒙𝒊𝒔 𝒌 aligns with 𝒙 − 𝒂𝒙𝒊𝒔.
𝑹𝒚 𝜷 𝑹𝒛 −𝜶
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
Next, vector 𝑷 (along with rotated 𝒂𝒙𝒊𝒔 𝒌) is rotated about 𝒚 −
𝒂𝒙𝒊𝒔 by an angle 𝜷 so that 𝒂𝒙𝒊𝒔 𝒌 aligns with 𝒙 − 𝒂𝒙𝒊𝒔.
Z
𝜶
K
𝜶 P
P’
Kz
1
𝜷 Ky Y
Kx
𝜶 𝜷
P”
X
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
Now the third rotation by an angle 𝜽 about the
rotated 𝒂𝒙𝒊𝒔 𝒌, which is rotation about 𝒙 − 𝒂𝒙𝒊𝒔, is
The resulting rotation matrix is then
𝟏𝑹𝟐 = 𝑹𝒙 𝜽 𝑹𝒚 𝜷 𝑹𝒛 −𝜶
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
Now the third rotation by an angle 𝜽 about the rotated 𝒂𝒙𝒊𝒔 𝒌,
Z
𝜶
K
𝜶 P
P’
Kz
1
𝜷 Ky Y
Kx
𝜶 𝜷
𝜽
P”
X
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
𝒚 − and 𝒛 − 𝒂𝒙𝒆𝒔, respectively, in the opposite
sense and reverse order so as to restore the 𝒌 −
𝒂𝒙𝒊𝒔 to its original position leaving frame 𝟐 in
that rotated position.
This gives
𝟏𝑹𝟐 = 𝑹𝒌 𝜽 = 𝑹𝒛 𝜶 𝑹𝒚 −𝜷 𝑹𝒙 𝜽 𝑹𝒚 𝜷 𝑹𝒛 −𝜶
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
Next two rotations are made about 𝒚 − and 𝒛 − 𝒂𝒙𝒆𝒔,
respectively, in the opposite sense and reverse order.
Z
𝜶
K
𝜶 P
P’
Kz
𝜷
Kx 1 Ky Y
𝜶
P” 𝜽
X
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
𝟏𝑹𝟐 = 𝑹𝒌 𝜽 = 𝑹𝒛 𝜶 𝑹𝒚 −𝜷 𝑹𝒙 𝜽 𝑹𝒚 𝜷 𝑹𝒛 −𝜶
Using the corresponding fundamental rotation
matrices one gets
𝑹𝑲 (𝜽) =
𝑪𝜶 −𝑺𝜶 𝟎 𝑪𝜷 𝟎 −𝑺𝜷 𝟏 𝟎 𝟎 𝑪𝜷 𝟎 𝑺𝜷 𝑪𝜶 𝑺𝜶 𝟎
𝑺𝜶 𝑪𝜶 𝟎 𝟎 𝟏 𝟎 𝟎 𝑪𝜽 −𝑺𝜽 𝟎 𝟏 𝟎 − 𝑺𝜶 𝑪𝜶 𝟎
𝟎 𝟎 𝟏 𝑺𝜷 𝟎 𝑪𝜷 𝟎 𝑺𝜽 𝑪𝜽 −𝑺𝜷 𝟎 𝑪𝜷 𝟎 𝟎 𝟏
𝑪𝜶𝑪𝜷 −𝑺𝜶 −𝑪𝜶𝑺𝜷 𝑪𝜷 𝟎 𝑺𝜷 𝑪𝜶 𝑺𝜶 𝟎
𝑹𝑲 (𝜽) = 𝑺𝜶𝑪𝜷 𝑪𝜶 −𝑺𝜶𝑺𝜷 𝑺𝜽𝑺𝜷 𝑪𝜽 −𝑺𝜽𝑪𝜷 − 𝑺𝜶 𝑪𝜶 𝟎
𝑺𝜷 𝟎 𝑪𝜷 −𝑪𝜽𝑺𝜷 𝑺𝜽 𝑪𝜽𝑪𝜷 𝟎 𝟎 𝟏
𝑹𝑲 (𝜽) =
𝑪𝜶𝑪𝜷 −𝑺𝜶 −𝑪𝜶𝑺𝜷 𝑪𝜶𝑪𝜷 𝑺𝜶𝑪𝜷 𝑺𝜷
𝑺𝜶𝑪𝜷 𝑪𝜶 −𝑺𝜶𝑺𝜷 𝑺𝜽𝑪𝜶𝑺𝜷 − 𝑪𝜽𝑺𝜶 𝑺𝜽𝑺𝜶𝑺𝜷 + 𝑪𝜽𝑪𝜶 −𝑺𝜽𝑪𝜷
𝑺𝜷 𝟎 𝑪𝜷 −𝑪𝜽𝑪𝜶𝑺𝜷 − 𝑺𝜽𝑺𝜶 −𝑪𝜽𝑺𝜶𝑺𝜷 + 𝑺𝜽𝑪𝜶 𝑪𝜽𝑪𝜷
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
𝑹𝑲 (𝜽) =
𝑪𝜶𝑪𝜷 −𝑺𝜶 −𝑪𝜶𝑺𝜷 𝑪𝜶𝑪𝜷 𝑺𝜶𝑪𝜷 𝑺𝜷
𝑺𝜶𝑪𝜷 𝑪𝜶 −𝑺𝜶𝑺𝜷 𝑺𝜽𝑪𝜶𝑺𝜷 − 𝑪𝜽𝑺𝜶 𝑺𝜽𝑺𝜶𝑺𝜷 + 𝑪𝜽𝑪𝜶 −𝑺𝜽𝑪𝜷
𝑺𝜷 𝟎 𝑪𝜷 −𝑪𝜽𝑪𝜶𝑺𝜷 − 𝑺𝜽𝑺𝜶 −𝑪𝜽𝑺𝜶𝑺𝜷 + 𝑺𝜽𝑪𝜶 𝑪𝜽𝑪𝜷
𝑹𝑲 (𝜽) =
𝑪𝜶𝑪𝜷 −𝑺𝜶 −𝑪𝜶𝑺𝜷 𝑪𝜶𝑪𝜷 𝑺𝜶𝑪𝜷 𝑺𝜷
𝑺𝜶𝑪𝜷 𝑪𝜶 −𝑺𝜶𝑺𝜷 𝑺𝜽𝑪𝜶𝑺𝜷 − 𝑪𝜽𝑺𝜶 𝑺𝜽𝑺𝜶𝑺𝜷 + 𝑪𝜽𝑪𝜶 −𝑺𝜽𝑪𝜷
𝑺𝜷 𝟎 𝑪𝜷 −𝑪𝜽𝑪𝜶𝑺𝜷 − 𝑺𝜽𝑺𝜶 −𝑪𝜽𝑺𝜶𝑺𝜷 + 𝑺𝜽𝑪𝜶 𝑪𝜽𝑪𝜷
𝒌𝟐𝒙 𝑽𝜽 + 𝑪𝜽 𝒌𝒙 𝒌𝒚 𝑽𝜽 − 𝒌𝒛 𝑺𝜽 𝒌𝒙 𝒌𝒛 𝑽𝜽 + 𝒌𝒚 𝑺𝜽
𝑹𝒌 𝜽 = 𝒌𝒙 𝒌𝒚 𝑽𝜽 + 𝒌𝒛 𝑺𝜽 𝒌𝟐𝒚 𝑽𝜽 + 𝑪𝜽 𝒌𝒚 𝒌𝒛 𝑽𝜽 − 𝒌𝒙 𝑺𝜽
𝒌𝒙 𝒌𝒛 𝑽𝜽 − 𝒌𝒚 𝑺𝜽 𝒌𝒚 𝒌𝒛 𝑽𝜽 + 𝒌𝒙 𝑺𝜽 𝒌𝟐𝒛 𝑽𝜽 + 𝑪𝜽
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
The angles 𝜶 and 𝜷 can be eliminated from the
resulting final rotation matrix using the geometry.
From the figure below it can be written for the
= 𝒌𝒙 𝒌𝒚 𝒌𝒛 𝑻 as
unit vector𝒌
𝒌𝒚
𝐬𝐢𝐧 𝜶 = ; Z
𝟐 𝟐
𝒌𝒙 +𝒌𝒚 K
𝒌𝒙
𝐜𝐨𝐬 𝜶 = ; P
𝒌𝒙 𝟐 +𝒌𝒚 𝟐 P’
Kz
1
𝜷 Ky Y
𝐬𝐢𝐧 𝜷 = 𝒌𝒛 ; Kx
𝜶
𝐜𝐨𝐬 𝜷 = 𝒌𝟐 𝒙 + 𝒌𝟐 𝒚 P” 𝒌𝒙 𝟐 + 𝒌𝒚 𝟐
X
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
𝒌𝒚 𝒌𝒙
𝐬𝐢𝐧 𝜶 = ; 𝐜𝐨𝐬 𝜶 = ;
𝒌𝒙 𝟐+𝒌𝒚 𝟐 𝒌𝒙 𝟐 +𝒌𝒚 𝟐
𝐬𝐢𝐧 𝜷 = 𝒌𝒛 ; 𝐜𝐨𝐬 𝜷 = 𝒌𝟐 𝒙 + 𝒌𝟐 𝒚
Substituting these in the expression for 𝑹𝑲 (𝜽) and
simplifying one gets
𝑹𝒌 𝜽 =
𝒌𝟐𝒙 𝑽𝜽 + 𝑪𝜽 𝒌𝒙 𝒌𝒚 𝑽𝜽 − 𝒌𝒛 𝑺𝜽 𝒌𝒙 𝒌𝒛 𝑽𝜽 + 𝒌𝒚 𝑺𝜽
𝒌𝒙 𝒌𝒚 𝑽𝜽 + 𝒌𝒛 𝑺𝜽 𝒌𝟐𝒚 𝑽𝜽 + 𝑪𝜽 𝒌𝒚 𝒌𝒛 𝑽𝜽 − 𝒌𝒙 𝑺𝜽
𝒌𝒙 𝒌𝒛 𝑽𝜽 − 𝒌𝒚 𝑺𝜽 𝒌𝒚 𝒌𝒛 𝑽𝜽 + 𝒌𝒙 𝑺𝜽 𝒌𝟐𝒛 𝑽𝜽 + 𝑪𝜽
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
Where 𝒌𝒙 , 𝒌𝒚 , 𝒌𝒛 are the projections of a unit
on frame 𝒙𝒚𝒛 , and
vector 𝒌
𝑽𝜽 = 𝟏 − 𝐜𝐨𝐬 𝜽 .
This is an important rotation matrix and needs
thorough understanding.
The principal axes fundamental rotations can be
obtained from the above equation.
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
For example, if 𝒌 − 𝒂𝒙𝒊𝒔 is aligned with 𝒛 − 𝒂𝒙𝒊𝒔,
𝑹𝒌 𝜽 becomes 𝑹𝒛 𝜽 with 𝒌𝒙 = 𝒌𝒚 = 𝟎 and 𝒌𝒛 = 𝟏 .
Substituting these values one gets
𝑪𝜽 −𝑺𝜽 𝟎
𝑹𝒌 𝜽 = 𝑹𝒛 𝜽 = 𝑺𝜽 𝑪𝜽 𝟎
𝟎 𝟎 𝟏
This is same as that obtained earlier for rotation
Note that any combination of rotations about the
principal axes of a coordinate frame is always
equivalent to a single rotation by some angle 𝜽
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
To find the direction 𝑲, consider the general
rotational transformation matrix 𝑹.
It is required to determine 𝜽 and 𝒌.
Analysis of robotic manipulators for various
aspects is done using the transformation
matrices explained here.
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
Example1: The coordinates of point 𝑷 in frame
𝟏 are [𝟑. 𝟎 𝟐. 𝟎 𝟏. 𝟎]. The position vector 𝑷 is
rotated about the 𝒛 − 𝒂𝒙𝒊𝒔 by 𝟒𝟓𝟎 . Find the
coordinates of point 𝑸 , the new position of
point 𝑷.
Solution: The 𝟒𝟓𝟎 rotation of 𝑷 about the 𝒛 −
𝒂𝒙𝒊𝒔 of frame 𝟏 is
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
𝑪𝜽 −𝑺𝜽 𝟎 𝑪𝟒𝟓 −𝑺𝟒𝟓 𝟎
𝑹𝒛 𝜽 = 𝑺𝜽 𝑪𝜽 𝟎 = 𝑺𝟒𝟓 𝑪𝟒𝟓 𝟎 =
𝟎 𝟎 𝟏 𝟎 𝟎 𝟏
𝟎. 𝟕𝟎𝟕 −𝟎. 𝟕𝟎𝟕 𝟎
𝟎. 𝟕𝟎𝟕 𝟎. 𝟕𝟎𝟕 𝟎
𝟎 𝟎 𝟏
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
𝟎. 𝟕𝟎𝟕 −𝟎. 𝟕𝟎𝟕 𝟎 𝟑 𝟎. 𝟕𝟎𝟕
𝟏𝑸 = 𝟎. 𝟕𝟎𝟕 𝟎. 𝟕𝟎𝟕 𝟎 𝟐 = 𝟑. 𝟓𝟑𝟓
𝟎 𝟎 𝟏 𝟏 𝟏
Thus, the coordinates of the new point 𝑸 relative to
frame 𝟏 are [𝟎. 𝟕𝟎𝟕 𝟑. 𝟓𝟑𝟓 𝟏. 𝟎]𝑻 or the new position
vector is [𝟎. 𝟕𝟎𝟕 𝟑. 𝟓𝟑𝟓 𝟏. 𝟎]𝑻
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
Example 2: Frame 𝟐 is rotated with respect to frame
𝟏 about the 𝒙 − 𝒂𝒙𝒊𝒔 by an angle of 𝟔𝟎𝟎 . The
position of the origin of frame 𝟐 as seen from frame
𝟏 is 𝟏𝑫𝟐 = [𝟕. 𝟎 𝟓. 𝟎 𝟕. 𝟎]. Obtain the
transformation matrix 𝟏𝑻𝟐 , which describes frame 𝟐
relative to frame 𝟏 . Also, find the description of
point 𝑷 in frame 𝟏 if 𝟐𝑷 = [𝟐. 𝟎 𝟒. 𝟎 𝟔. 𝟎]𝑻
Solution: The homogeneous transformation matrix that
describes frame 𝟐 with respect to frame 𝟏 is
𝟏𝑹𝟐 𝟏𝑫𝟐
𝟏𝑻𝟐 =
𝟎 𝟎 𝟎 𝟏
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
Because frame 𝟐 is rotated relative to frame 𝟏 about
𝒙 − 𝒂𝒙𝒊𝒔 by 𝟔𝟎𝟎 using
𝟏 𝟎 𝟎 𝟏 𝟎 𝟎
𝟏𝑹𝟐 = 𝟎 𝑪 𝟔𝟎 −𝑺 𝟔𝟎 = 𝟎 𝟎. 𝟓 −𝟎. 𝟖𝟔𝟔
𝟎 𝑺 𝟔𝟎 𝑪 𝟔𝟎 𝟎 𝟎. 𝟖𝟔𝟔 𝟎. 𝟓
And 1𝑫𝟐 = [𝟕. 𝟎 𝟓. 𝟎 𝟕. 𝟎]𝑻
Substituting 1𝑹𝟐 and 1𝑫𝟐 in the above equation
𝟏 𝟎 𝟎 𝟕. 𝟎 𝟐. 𝟎
1𝑷 = 𝟎 𝟎. 𝟓 −𝟎. 𝟖𝟔𝟔 𝟓. 𝟎 𝟒. 𝟎
𝟎 𝟎. 𝟖𝟔𝟔 𝟎. 𝟓 𝟕. 𝟎 𝟔. 𝟎
𝟎 𝟎 𝟎 𝟏 𝟏
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
𝟏 𝟎 𝟎 𝟕. 𝟎
1𝑻𝟐 = 𝟎 𝟎. 𝟓 −𝟎. 𝟖𝟔𝟔 𝟓. 𝟎
𝟎 𝟎. 𝟖𝟔𝟔 𝟎. 𝟓 𝟕. 𝟎
𝟎 𝟎 𝟎 𝟏
Given: 2𝑷 = [𝟐. 𝟎 𝟒. 𝟎 𝟔. 𝟎]𝑻 , point 𝑷 in frame 𝟏 is
given by 1𝑷 =1𝑻𝟐 2𝑷
𝟗. 𝟎𝟎𝟎
1𝑷 = 𝟏. 𝟖𝟎𝟒
𝟏𝟑. 𝟒𝟔𝟒
𝟏
1𝑷 = 𝟗. 𝟎𝟎𝟎 𝟏. 𝟖𝟎𝟒 𝟏𝟑. 𝟒𝟔𝟒 𝟏 𝑻
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus
The 𝟑𝑿𝟏 position vector of point 𝑷 in frame 𝟏 in
physical coordinates is
𝑻
1𝑷 = 𝟗. 𝟎𝟎𝟎 𝟏. 𝟖𝟎𝟒 𝟏𝟑. 𝟒𝟔𝟒
THANK YOU
Lecture 17 ME G511 Mechanisms and Robotics BITS Pilani, Hyderabad Campus | 8,567 | 12,358 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.140625 | 3 | CC-MAIN-2020-45 | latest | en | 0.760465 |
https://www.teachwire.net/news/primary-maths-how-to-develop-math-vocabulary-from-eyfs-onwards/ | 1,656,703,394,000,000,000 | text/html | crawl-data/CC-MAIN-2022-27/segments/1656103945490.54/warc/CC-MAIN-20220701185955-20220701215955-00235.warc.gz | 1,060,876,366 | 27,057 | PrimaryMaths
# Primary maths – How to develop maths vocabulary from EYFS onwards
Introducing and developing subject-specific language is key to understanding and communication, argues Lisa Coe…
by Lisa Coe
DOWNLOAD A FREE RESOURCE! Free Scarlett Fife learning resources for KS1-2
PrimaryEnglishHealth & WellbeingScience
When a visitor comes to Great Yarmouth Primary Academy (GYPA) to look at maths teaching, one of the things they comment on is pupils’ use of mathematical language.
During their June visit, Ofsted noticed this too, and noted that in Early Years Foundation Stage (EYFS) children ‘understand words and phrases such as capacity, cuboid and repeating patterns’.
When I, as maths lead for the Inspiration Trust, talk to pupils about maths I am astounded at how even our youngest learners can use accurate mathematical vocabulary.
Children across the school use the language of maths to describe, reason and explain. But how did we get to the point?
## What is mathematical language?
Like all subjects we teach, mathematics comes with its own set of words. Some of these are specific to maths, like cuboid. These words have a clear meaning and need to be taught clearly.
Some are more complicated and have a common and mathematical meaning. Take ‘odd’. In maths, an odd number is one that is not even; it cannot be divided equally into two parts.
Yet even by the time pupils enter school, they will have heard the word odd in different ways: an odd sock, an odd-looking tree. Odd in maths could easily become distorted.
A child could decide ‘8’ looks weird, and so it is odd. Or see the number eight on its own and decide it is odd, like the aforementioned lone sock.
So why does it matter? Why do we place such emphasis on pupils using ‘vertex’ rather than ‘corner’, ‘equation’ alongside ‘expression’ and ‘number sentence’?
Imagine listening to a football commentary without knowing words such as goal, midfield, tackle, foul … Understanding what is going on, being able to follow the progression of the game becomes incredibly difficult. Imagine trying to describe the goal without some of these words: “So the guy moved the ball to another guy, who nearly got caught but managed to get it where it needed to be.”
Trying to follow a description without a shared, technical language becomes vague and confused: you probably know exactly what you mean, but others will not.
It’s the same for a pupil in maths. If they don’t understand the language being used, if they cannot speak mathematically, then they find themselves unable to engage in a lesson or explain their ideas.
If a child does not have the word cuboid, they may resort to common terms like box, or other words like rectangle, which means conversations are not clear.
I’ve seen pupils become frustrated because they are not making themselves understood. It’s not their fault: we need to ensure we give them the language to fully express their ideas.
## Start early
One of the key strategies for developing mathematical language at GYPA is simple: start early. Children in EYFS are taught key mathematical language using examples (more on these later) with lots of opportunity to say words and connect them to their own experiences.
Exposure is key here. Adults do not shy away from using accurate language like vertices, and instead use them consistently and encourage pupils to do the same.
Pupils are praised for use and supported to use more accurate language. Encouraging a pupil to ‘say it like a mathematician’ can really give them confidence to use these words.
Experiences provided through continuous provision are vital. The EYFS team at GYPA takes time to plan activities that allow pupils to use the language they have learned in taught segments.
Environments are set up so that maths is everywhere, not just in a maths corner, and all opportunities are taken to encourage maths language, such as ‘part’ and ‘whole’ when taking lunch options.
Immersion is key: we want our pupils to see maths language as everyday language, not something special or scary.
## Teaching dos and don’ts
We know that language needs something to attach to and that’s where examples come in. It’s no use teaching the word sphere without ever showing a child what it looks like.
Teachers at GYPA ensure words are taught with not just one, but lots of examples: large spheres, tiny spheres, different coloured spheres. It’s no use providing one example, repeating the word and hoping pupils simply absorb it.
We also see non-standard examples being used and displayed, particularly when it comes to geometry. It’s refreshing to see a square being displayed oriented differently to the ‘typical’ or cuboids that are long and thin.
This is worth taking the time to think about: do your vocabulary displays or working walls always show typical sizes or orientations, or are you providing opportunities for pupils to connect language and examples more generally?
Similarly, non-examples are really important. It’s often really hard to define a mathematical word without considering what it isn’t. Is it possible to understand odd without even?
Deep understanding comes from knowing what a word does not mean, and this can be particularly useful with words such as odd that can have different meanings.
We all know how excited pupils get about correcting you as a teacher when you proudly announce that the four cubes represent an odd number!
## Repeat, revisit, retrieve
One of the great features of a maths lesson at GYPA is the opportunities to ‘reheat’ previous learning. In Year 5, for example, a ‘fast five’ activity allows pupils to revisit previous knowledge from earlier lessons and this is a great opportunity to check and repeat mathematical language.
This does not have to be onerous. ‘Write the factors of 10’ is a simple example where if a child does not understand the word ‘factors’ they cannot accurately answer the question. The word can be repeated across numerous days in different ways:
• Which of these are factors of 10?
• Is nine a factor of 10? Explain why
• Find the missing factor
In this way, pupils are consistently retrieving and using the same word, with the same definition, in varied ways, with minimal teacher time.
The more exposure to a word, the more pupils have a chance to grapple with its meaning, the more likely they are to understand it elsewhere.
The reason this works well for us is the connections made across year groups in terms of mathematical language. Year 5 teachers know that pupils coming to them from Year 4 have met this word before.
The maths lead and the Inspiration Trust have worked hard to ensure that mathematical language is introduced and revisited in subsequent topics and years.
## Lifelong impact
What we’ve achieved at GYPA in mathematical vocabulary acquisition is not ground-breaking, nor is it confined only to the language of mathematics.
In fact, many of the suggestions here can, and are, used across the curriculum to ensure language is taught carefully and accurately. The approach recognises the importance of getting mathematical language to ‘stick’, to become embedded in long-term memory, and for pupils to be familiar and comfortable with its use.
Not to mention expanding their communication skills – a useful tool at any stage in life.
Lisa Coe is maths lead at Great Yarmouth Primary Academy, and the Inspiration Trust. Find out more about the trust online at inspirationtrust.org.
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By signing up you agree to our Terms & Conditions and privacy policy | 1,568 | 7,731 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.421875 | 3 | CC-MAIN-2022-27 | latest | en | 0.959477 |
https://calculatingthebazi.com/2022/05/20/practice-exercise-7-3-johnny-yin/ | 1,685,836,566,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224649348.41/warc/CC-MAIN-20230603233121-20230604023121-00173.warc.gz | 176,153,300 | 47,434 | Uncategorized
Practice Exercise 7, #3 Johnny Yin
Q: Sorry Karin,
3 September 1973 @ 11:02 ♁
MG Branch
MB= Shen, as born before 8th Sept and therefore is “9” as a month branch.
HB=Wu, with no added +1, as he is born before the “Qi” date for September 1973 (i.e before 23/09/73) and therefore numerically “7”
=32 – [9+7]
=32 – 16
=16 (-12) = 4 and MAO
MG Stem
MGS = (YS GUI or “10”) + MGB# 4 -2 (MB not Zi or Chou)
= 10+4-2 = 12
-10
=2 which is Yi
I get YiMao 乙卯, you get JiaYin 甲寅!
Can you tell me where I went wrong?
As a further thought, it’s great you have offered to correct the errors in the E-book, and Second Edition.
I have the first edition, and have spent countless hours trying to figure out some of the errors and mis-interpretations, that you have subsequently shown in the many posts on this site.
I feel it would be much more humane, if the front page of the site, and FIRST post, pinned to the top was a collection of corrections as a Pdf.
Some of them have shaken my ability to draw a Bazi, and have left me wondering if I’m capable, only to find your corrections later..
Every time I hit an error, I have to plough through your site, in the hope that it’s you, rather than me!
Jeremy
A: Hello again Jeremy!
MingGong Calculation for 3 September 1973 @ 11:02 ♁
MG Branch = 32 – [#MB + #HB time] (+1 if necessary, ie if birth day falls after ZhongQi/mid-month point)
MG B = 32 – [9 + 7 + 1] – the birthday here falls before the new JieQi starts (8 September 1974), after the mid month, 23 August 1974
MG B = 32 – (17)
MG B = 15 (answer is >12 so subtract 12)
MG B = 3 Yin 寅
MG Stem = (#YS x 2) + #MG B – 2
MG S = (10 x 2) + 3 – 2
MG S = 20 + 1 = 21 – 10 – 10 = 1
MG S = #1 Jia 甲
I hope this helps!
PS Regarding searching for corrections (or anything else), the Q&A are posted in chronological order. You can use the site’s “Search” function, which can be found at the bottom of the home page and/or in the sidebar of the Q&A page, type in “Correction” (or anything else) and all the related posts will be gathered in one spot for you 🙂 | 646 | 2,071 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.515625 | 4 | CC-MAIN-2023-23 | latest | en | 0.937641 |
http://slidegur.com/doc/107366/lecture32-kmeans2 | 1,477,216,654,000,000,000 | text/html | crawl-data/CC-MAIN-2016-44/segments/1476988719215.16/warc/CC-MAIN-20161020183839-00127-ip-10-171-6-4.ec2.internal.warc.gz | 239,835,386 | 9,052 | ### lecture32-kmeans2
```UNSUPERVISED LEARNING
David Kauchak
CS 451 – Fall 2013
Final project
Schedule for the rest of the semester
Unsupervised learning
Unsupervised learning: given data, i.e. examples, but no labels
K-means
Iterate:
Assign/cluster each example to closest center
Recalculate centers as the mean of the points in a cluster
K-means: an example
K-means: Initialize centers randomly
K-means: assign points to nearest center
K-means: assign points to nearest center
K-means: assign points to nearest center
K-means: assign points to nearest center
No changes: Done
K-means variations/parameters
Initial (seed) cluster centers
Convergence
A
fixed number of iterations
partitions unchanged
Cluster centers don’t change
K!
How Many Clusters?
Number of clusters K must be provided
How should we determine the number of clusters?
How did we deal with models becoming too complicated previously?
too many
too few
Many approaches
Regularization!!!
Statistical test
k-means loss revisited
K-means is trying to minimize:
n
loss = å d(xi , mk )2 where mk is cluster center for xi
i=1
What happens when k increases?
k-means loss revisited
K-means is trying to minimize:
n
loss = å d(xi , mk )2 where mk is cluster center for xi
i=1
Loss goes down!
Making the model more complicated allows us more
flexibility, but can “overfit” to the data
k-means loss revisited
K-means is trying to minimize:
n
losskmeans = å d(xi , mk )2 where mk is cluster center for xi
2 regularization options
i=1
lossBIC = losskmeans + K log N
(where N = number of points)
lossAIC = losskmeans + KN
What effect will this have?
Which will tend to produce smaller k?
2 regularization options
k-means loss revisited
lossBIC = losskmeans + K log N
(where N = number of points)
lossAIC = losskmeans + KN
AIC penalizes increases in K more harshly
Both require a change to the K-means algorithm
Tend to work reasonably well in practice if you don’t know K
Statistical approach
Assume data is Gaussian (i.e. spherical)
Test for this
Testing
in high dimensions doesn’t work well
Testing in lower dimensions does work well
ideas?
Project to one dimension and check
For each cluster, project down to one dimension
Use
a statistical test to see if the data is Gaussian
Project to one dimension and check
For each cluster, project down to one dimension
Use
a statistical test to see if the data is Gaussian
What will this look like projected to 1-D?
Project to one dimension and check
For each cluster, project down to one dimension
Use
a statistical test to see if the data is Gaussian
Project to one dimension and check
For each cluster, project down to one dimension
Use
a statistical test to see if the data is Gaussian
What will this look like projected to 1-D?
Project to one dimension and check
For each cluster, project down to one dimension
Use
a statistical test to see if the data is Gaussian
Project to one dimension and check
For each cluster, project down to one dimension
Use
a statistical test to see if the data is Gaussian
What will this look like projected to 1-D?
Project to one dimension and check
For each cluster, project down to one dimension
Use
a statistical test to see if the data is Gaussian
Solution?
Project to one dimension and check
For each cluster, project down to one dimension
Use
a statistical test to see if the data is Gaussian
Chose the dimension of the projection
as the dimension with highest variance
On synthetic data
Split too far
Compared to other approaches
http://cs.baylor.edu/~hamerly/papers/nips_03.pdf
K-Means time complexity
Variables: K clusters, n data points,
m features/dimensions, I iterations
What is the runtime complexity?
Computing
distance between two points (e.g.
euclidean)
Reassigning clusters
Computing new centers
Iterate…
K-Means time complexity
Variables: K clusters, n data points,
m features/dimensions, I iterations
What is the runtime complexity?
Computing distance between two points is O(m) where m is the
dimensionality of the vectors/number of features.
Reassigning clusters: O(Kn) distance computations, or O(Knm)
Computing centroids: Each points gets added once to some centroid:
O(nm)
Assume these two steps are each done once for I iterations: O(Iknm)
In practice, K-means converges quickly and is fairly fast
What Is A Good Clustering?
Internal criterion: A good clustering will produce high
quality clusters in which:
the
intra-class (that is, intra-cluster) similarity is high
the inter-class similarity is low
How would you evaluate clustering?
Common approach: use labeled data
Use data with known classes
For example, document classification data
data
label
If we clustered this data (ignoring labels)
what would we like to see?
Reproduces class partitions
How can we quantify this?
Common approach: use labeled data
Purity, the proportion of the dominant class in the cluster
Cluster I
Cluster II
Cluster III
Cluster I: Purity = 1/4 (max(3, 1, 0)) = 3/4
Cluster II: Purity = 1/6 (max(1, 4, 1)) = 4/6
Cluster III: Purity = 1/5 (max(2, 0, 3)) = 3/5
Overall purity?
Overall purity
Cluster I: Purity = 1/4 (max(3, 1, 0)) = 3/4
Cluster II: Purity = 1/6 (max(1, 4, 1)) = 4/6
Cluster III: Purity = 1/5 (max(2, 0, 3)) = 3/5
Cluster average:
Weighted average:
3 4 3
+ +
4 6 5 = 0.672
3
3
4
3
4 * + 6 * + 5*
4
6
5 = 3+ 4 + 3 = 0.667
15
15
Purity issues…
Purity, the proportion of the dominant class in the cluster
Good for comparing two algorithms, but not
understanding how well a single algorithm is doing,
why?
Increasing the number of clusters increases purity
Purity isn’t perfect
Which is better based on purity?
Which do you think is better?
Ideas?
Common approach: use labeled data
Average entropy of classes in clusters
entropy(cluster) = -å p(classi )log p(classi )
i
where p(classi) is proportion of class i in cluster
Common approach: use labeled data
Average entropy of classes in clusters
entropy(cluster) = -å p(classi )log p(classi )
i
entropy?
Common approach: use labeled data
Average entropy of classes in clusters
entropy(cluster) = -å p(classi )log p(classi )
i
-0.5log0.5- 0.5log0.5 =1
-0.5log0.5- 0.25log0.25- 0.25log0.25 =1.5
``` | 1,745 | 6,180 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.296875 | 3 | CC-MAIN-2016-44 | latest | en | 0.867708 |
https://www.bbcelite.com/elite-a/flight/subroutine/dvid4.html | 1,639,057,369,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964364169.99/warc/CC-MAIN-20211209122503-20211209152503-00459.warc.gz | 701,497,049 | 132,095 | Elite on the BBC Micro
# Maths (Arithmetic): DVID4 [Elite-A, Flight]
``` Name: DVID4 [Show more]
Type: Subroutine
Category: Maths (Arithmetic)
Summary: Calculate (P R) = 256 * A / Q
Deep dive: Shift-and-subtract division
Context: See this subroutine in context in the source code
References: This subroutine is called as follows:
* DOEXP calls DVID4
* SPS2 calls DVID4
Calculate the following division and remainder:
P = A / Q
R = remainder as a fraction of Q, where 1.0 = 255
Another way of saying the above is this:
(P R) = 256 * A / Q
This uses the same shift-and-subtract algorithm as TIS2, but this time we
keep the remainder.
Returns:
C flag The C flag is cleared
.DVID4
LDX #8 \ Set a counter in X to count the 8 bits in A
ASL A \ Shift A left and store in P (we will build the result
STA P \ in P)
LDA #0 \ Set A = 0 for us to build a remainder
.DVL4
ROL A \ Shift A to the left
BCS DV8 \ If the C flag is set (i.e. bit 7 of A was set) then
\ skip straight to the subtraction
CMP Q \ If A < Q skip the following subtraction
BCC DV5
.DV8
SBC Q \ A >= Q, so set A = A - Q
SEC \ Set the C flag, so that P gets a 1 shifted into bit 0
.DV5
ROL P \ Shift P to the left, pulling the C flag into bit 0
DEX \ Decrement the loop counter
BNE DVL4 \ Loop back for the next bit until we have done all 8
\ bits of P
JMP LL28+4 \ Jump to LL28+4 to convert the remainder in A into an
\ integer representation of the fractional value A / Q,
\ in R, where 1.0 = 255. LL28+4 always returns with the
\ C flag cleared, and we return from the subroutine
\ using a tail call
``` | 501 | 1,872 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.40625 | 3 | CC-MAIN-2021-49 | latest | en | 0.835029 |
https://www.lessonup.com/en/lesson/LTgdPFvedmppwt7Rd | 1,718,889,481,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861940.83/warc/CC-MAIN-20240620105805-20240620135805-00794.warc.gz | 738,061,654 | 43,371 | # 8th -2/27
Happy Monday!
1 / 15
Slide 1: Slide
Mathematics
This lesson contains 15 slides, with interactive quizzes and text slides.
## Items in this lesson
Happy Monday!
Check in!
😒🙁😐🙂😃
#### Slide 2 -Poll
Fun Fact #1 :
Snakes can predict earthquakes.
#### Slide 4 -Slide
What fraction of the pie HAS NOT been eaten?
_________
Find the denominator.
Find the numerator.
4
7
6
2
#### Slide 5 -Drag question
Put the decimals from smallest to largest.
0.6
0.02
0.8
0.35
0.15
0.95
0.492
#### Slide 8 -Slide
Graph Triangle LAM
L - (-4, 3)
A - (-1, -4)
M - (2, 1)
#### Slide 9 -Slide
Rotate Triangle LAM 90 degrees clockwise
L' -
A' -
M' -
#### Slide 10 -Slide
from triangle L'A'M',
(x, y) --> (x - 3, y + 1)
L" -
A" -
M" -
Name:
Describe:
Name:
Describe:
Name:
Describe:
Name:
Describe: | 277 | 808 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.3125 | 3 | CC-MAIN-2024-26 | latest | en | 0.561555 |
https://oneclass.com/textbook-notes/ca/western/stat-sci/sta-sci-1023ab/1506693-statistical-sciences-1023ab-chapter-5.en.html | 1,524,278,526,000,000,000 | text/html | crawl-data/CC-MAIN-2018-17/segments/1524125944851.23/warc/CC-MAIN-20180421012725-20180421032725-00111.warc.gz | 661,196,624 | 41,864 | Textbook Notes (368,107)
Chapter 5
# Statistical Sciences 1023A/B Chapter 5: Chapter 5 Premium
4 Pages
68 Views
School
Department
Statistical Sciences
Course
Statistical Sciences 1023A/B
Professor
Lori Murray
Semester
Winter
Description
Chapter 5 Experiments and Observational Studies Experiments and Observational Studies Explanatory and Response Variables Explanatory variable: is one that may explain or cause differences in a response variable. Response variable: is the outcome variable. Example: Study finds that overall left handed people die at a younger age than righthanded people. Experiments and Observational Studies Treatments A treatment is one or a combination of categories of the explanatory variable(s) assigned by the experimenter. Collection of conditions. Example: Salk Experiment (Chapter 1) Explanatory variable: whether or not the baby listened to a heartbeat Two treatments: heartbeat or silent Response: weight gain Experiments and Observational Studies Randomized Experiment Create differences in the explanatory variable (one that may cause differences in an outcome) and examine results (response variable). Observational Study Observe differences in the explanatory variable and notice whether these are related to differences in the response variable. Experiments and Observational Studies Two reasons why we must sometimes use an observational study instead of an experiment: 1. It is unethical or impossible to assign people to receive a specific treatment. 2. Certain explanatory variables are inherent traits and cannot be randomly assigned. Examples? Experiments and Observational Studies Confounding Variables A confounding variable is related to the explanatory variable and affects the response variable. Two variables are confounded when its hard to separate how they affect the response The effect of a confounding variable on the response variable cannot be separated from the effect of the explanatory variable on the response variable. The extra variable you didnt account for Confounding variables are a bigger problem in observational studies. Researchers try to measure possible confounding variables and see if related to the response variable. When an explanatory and response variable have been identified, confounding variables have the following traits: Differing from the explanatory variable means you differ from the confounding Differing confounding variable = differing response variable
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So we can recommend you notes for your school. | 536 | 2,777 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.828125 | 4 | CC-MAIN-2018-17 | latest | en | 0.836818 |
https://dvillers.umons.ac.be/wiki/_export/code/teaching:progappchim:tris?codeblock=0 | 1,726,860,860,000,000,000 | text/plain | crawl-data/CC-MAIN-2024-38/segments/1725701423570.98/warc/CC-MAIN-20240920190822-20240920220822-00071.warc.gz | 186,007,200 | 1,172 | #! /usr/bin/env python # -*- coding: utf-8 -*- """ Code compact de la fonction quicksort. Exemple de tri d'éléments entiers. """ def quicksort(li): if li == []: return [] pivot = li[0] equal = [pivot] lesser = quicksort([x for x in li[1:] if x <= pivot]) # si x < pivot : élimination des doublons greater = quicksort([x for x in li[1:] if x > pivot]) return lesser + equal + greater from numpy import random a=random.randint(0,1000,10) print(a,len(a)) b=quicksort(a) print(b,len(b)) | 146 | 482 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.859375 | 3 | CC-MAIN-2024-38 | latest | en | 0.20416 |
https://kr.mathworks.com/matlabcentral/cody/problems/645-getting-the-indices-from-a-vector/solutions/1715033 | 1,603,382,667,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107879673.14/warc/CC-MAIN-20201022141106-20201022171106-00054.warc.gz | 400,486,280 | 17,367 | Cody
# Problem 645. Getting the indices from a vector
Solution 1715033
Submitted on 24 Jan 2019 by Luna
This solution is locked. To view this solution, you need to provide a solution of the same size or smaller.
### Test Suite
Test Status Code Input and Output
1 Pass
out = [3 4]; vec = [11 22 33 44]; thresh = 25; assert(isequal(findIndices(vec, thresh),out))
2 Pass
out = [1 2]; vec = [33 44 11 22]; thresh = 25; assert(isequal(findIndices(vec, thresh),out))
3 Pass
out = 5:10; vec = 10:10:100; thresh = 45; assert(isequal(findIndices(vec, thresh),out))
4 Pass
out = [1 3 4 6 8]; vec = [12 10 13 14 9 17 5 18]; thresh = 11; assert(isequal(findIndices(vec, thresh),out))
5 Pass
out = [1:3 7:9]; vec = [50 55 60 15 10 5 43 44 97 41]; thresh = 42; assert(isequal(findIndices(vec, thresh),out))
6 Pass
out = 5:8; vec = [10 12 14 16 18 20 22 23 7 8 9]; thresh = 17; assert(isequal(findIndices(vec, thresh),out))
7 Pass
out = [2 4:5 8 12:14 16]; vec = [10 81 24 65 97 13 45 68 24 35 16 79 123 76 45 60]; thresh = 51; assert(isequal(findIndices(vec, thresh),out))
8 Pass
out = 1:2:9; vec = [11 9 12 8 13 7 14 6 15 5]; thresh = 10; assert(isequal(findIndices(vec, thresh),out))
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Start Hunting! | 492 | 1,327 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.09375 | 3 | CC-MAIN-2020-45 | latest | en | 0.588611 |
http://dendropy.org/_modules/dendropy/calculate/treemeasure.html | 1,526,922,142,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794864461.53/warc/CC-MAIN-20180521161639-20180521181639-00440.warc.gz | 79,045,468 | 7,462 | # Source code for dendropy.calculate.treemeasure
#! /usr/bin/env python
# -*- coding: utf-8 -*-
##############################################################################
## DendroPy Phylogenetic Computing Library.
##
## Copyright 2010-2015 Jeet Sukumaran and Mark T. Holder.
##
## See "LICENSE.rst" for terms and conditions of usage.
##
## If you use this work or any portion thereof in published work,
## please cite it as:
##
## Sukumaran, J. and M. T. Holder. 2010. DendroPy: a Python library
## for phylogenetic computing. Bioinformatics 26: 1569-1571.
##
##############################################################################
"""
Statistics, metrics, measurements, and values calculated on (single) trees.
"""
import math
from dendropy.calculate import phylogeneticdistance
EULERS_CONSTANT = 0.5772156649015328606065120900824024310421
## legacy: will soon be deprecated
[docs]class PatrisiticDistanceMatrix(phylogeneticdistance.PhylogeneticDistanceMatrix):
def __init__(self, tree):
phylogeneticdistance.PhylogeneticDistanceMatrix.__init__(self)
self.compile_from_tree(tree=tree)
[docs]def patristic_distance(tree, taxon1, taxon2, is_bipartitions_updated=False):
"""
Given a tree with bipartitions encoded, and two taxa on that tree, returns the
patristic distance between the two. Much more inefficient than constructing
a PhylogeneticDistanceMatrix object.
"""
mrca = tree.mrca(taxa=[taxon1, taxon2], is_bipartitions_updated=is_bipartitions_updated)
dist = 0
n = tree.find_node(lambda x: x.taxon == taxon1)
while n != mrca:
if n.edge.length is not None:
dist += n.edge.length
n = n.parent_node
n = tree.find_node(lambda x: x.taxon == taxon2)
while n != mrca:
if n.edge.length is not None:
dist += n.edge.length
n = n.parent_node
return dist
###########################################################################
### Metrics -- Unary
[docs]def B1(tree):
"""
Returns the B1 statistic: the reciprocal of the sum of the maximum
number of nodes between each interior node and tip over all internal
nodes excluding root.
"""
b1 = 0.0
nd_mi = {}
for nd in tree.postorder_node_iter():
if nd._parent_node is None:
continue
child_nodes = nd._child_nodes
if len(child_nodes) == 0:
nd_mi[nd] = 0.0
continue
mi = max(nd_mi[ch] for ch in child_nodes)
mi += 1
nd_mi[nd] = mi
b1 += 1.0/mi
return b1
[docs]def colless_tree_imbalance(tree, normalize="max"):
"""
Returns Colless' tree imbalance or I statistic: the sum of differences
of numbers of children in left and right subtrees over all internal
nodes. normalize specifies the normalization:
* "max" or True [DEFAULT]
normalized to maximum value for tree of
this size
* "yule"
normalized to the Yule model
* "pda"
normalized to the PDA (Proportional to Distinguishable
Arrangements) model
* None or False
no normalization
"""
colless = 0.0
num_leaves = 0
subtree_leaves = {}
for nd in tree.postorder_node_iter():
if nd.is_leaf():
subtree_leaves[nd] = 1
num_leaves += 1
else:
total_leaves = 0
if len(nd._child_nodes) > 2:
raise TypeError("Colless' tree imbalance statistic requires strictly bifurcating trees")
left = subtree_leaves[nd._child_nodes[0]]
right = subtree_leaves[nd._child_nodes[1]]
colless += abs(right-left)
subtree_leaves[nd] = right + left
if normalize == "yule":
colless = float(colless - (num_leaves * math.log(num_leaves)) - (num_leaves * (EULERS_CONSTANT - 1.0 - math.log(2))))/num_leaves
elif normalize == "pda":
colless = colless / pow(num_leaves, 3.0/2)
elif normalize is True or normalize == "max":
## note that Mooers 1995 (Evolution 49(2):379-384)
## remarks that the correct normalization factor is
## 2/((num_leaves - 1) * (num_leaves -2))
colless = colless * (2.0/(num_leaves * (num_leaves-3) + 2))
elif normalize is not None and normalize is not False:
raise TypeError("normalization accepts only None, True, False, 'yule' or 'pda' as argument values")
return colless
[docs]def pybus_harvey_gamma(tree, prec=0.00001):
"""Returns the gamma statistic of Pybus and Harvey (2000). This statistic
is used to test for constancy of birth and death rates over the course of
a phylogeny. Under the pure-birth process, the statistic should follow
a standard Normal distibution: a Normal(mean=0, variance=1).
If the lengths of different paths to the node differ by more than prec,
then a ValueError exception will be raised indicating deviation from
ultrametricty.
Raises a Value Error if the tree is not ultrametric, is non-binary, or has
only 2 leaves.
As a side effect a age attribute is added to the nodes of the tree.
Pybus and Harvey. 2000. "Testing macro-evolutionary models using incomplete
molecular phylogenies." Proc. Royal Society Series B: Biological Sciences.
(267). 2267-2272
"""
# the equation is given by:
# T = \sum_{j=2}^n (jg_j)
# C = T \sqrt{\frac{1}{12(n-2)}}
# C gamma = \frac{1}{n-2}\sum_{i=2}^{n-1} (\sum_{k=2}^i kg_k) - \frac{T}{2}
# where n is the number of taxa, and g_2 ... g_n is the vector of waiting
# times between consecutive (in time, not along a branch) speciation times.
node = None
speciation_ages = []
n = 0
if tree.seed_node.age is None:
tree.calc_node_ages(ultrametricity_precision=prec)
for node in tree.postorder_node_iter():
if len(node.child_nodes()) == 2:
speciation_ages.append(node.age)
else:
n += 1
if node is None:
raise ValueError("Empty tree encountered")
speciation_ages.sort(reverse=True)
g = []
older = speciation_ages[0]
for age in speciation_ages[1:]:
g.append(older - age)
older = age
g.append(older)
if not g:
raise ValueError("No internal nodes found (other than the root)")
assert(len(g) == (n - 1))
T = 0.0
accum = 0.0
for i in range(2, n):
list_index = i - 2
T += i * float(g[list_index])
accum += T
list_index = n - 2
T += (n) * g[list_index]
nmt = n - 2.0
numerator = accum/nmt - T/2.0
C = T*pow(1/(12*nmt), 0.5)
return numerator/C
[docs]def N_bar(tree):
"""
Returns the $\bar{N}$ statistic: the average number of nodes above a
terminal node.
"""
leaf_count = 0
nbar = 0
for leaf_node in tree.leaf_node_iter():
leaf_count += 1
for parent in leaf_node.ancestor_iter(inclusive=False):
nbar += 1
return float(nbar) / leaf_count
[docs]def sackin_index(tree, normalize=True):
"""
Returns the Sackin's index: the sum of the number of ancestors for each
tip of the tree. The larger the Sackin's index, the less balanced the
tree. normalize specifies the normalization:
* True [DEFAULT]
normalized to number of leaves; this results in a value
equivalent to that given by Tree.N_bar()
* "yule"
normalized to the Yule model
* "pda"
normalized to the PDA (Proportional to Distinguishable
Arrangements) model
* None or False
no normalization
"""
leaf_count = 0
num_anc = 0
for leaf_node in tree.leaf_node_iter():
leaf_count += 1
for parent in leaf_node.ancestor_iter(inclusive=False):
num_anc += 1
if normalize == "yule":
x = sum(1.0/j for j in range(2, leaf_count+1))
s = float(num_anc - (2 * leaf_count * x))/leaf_count
elif normalize == "pda":
s = float(num_anc)/(pow(leaf_count, 3.0/2))
elif normalize is True:
s = float(num_anc)/leaf_count
elif normalize is None or normalize is False:
s = float(num_anc)
elif normalize is not None and normalize is not False:
raise TypeError("normalization accepts only None, True, False, 'yule' or 'pda' as argument values")
return s
[docs]def treeness(tree):
"""
Returns the proportion of total tree length that is taken up by
internal branches.
"""
internal = 0.0
external = 0.0
for nd in tree.postorder_node_iter():
if not nd._parent_node:
continue
if nd.is_leaf():
external += nd.edge.length
else:
internal += nd.edge.length
return internal/(external + internal) | 2,151 | 7,584 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.65625 | 3 | CC-MAIN-2018-22 | latest | en | 0.494387 |
https://jp.mathworks.com/help/stats/generalizedlinearmodel.predict.html?lang=en | 1,632,855,099,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780060882.17/warc/CC-MAIN-20210928184203-20210928214203-00055.warc.gz | 366,844,549 | 23,334 | predict
Predict responses of generalized linear regression model
Syntax
``ypred = predict(mdl,Xnew)``
``````[ypred,yci] = predict(mdl,Xnew)``````
``````[ypred,yci] = predict(mdl,Xnew,Name,Value)``````
Description
example
````ypred = predict(mdl,Xnew)` returns the predicted response values of the generalized linear regression model `mdl` to the points in `Xnew`.```
``````[ypred,yci] = predict(mdl,Xnew)``` also returns confidence intervals for the responses at `Xnew`.```
example
``````[ypred,yci] = predict(mdl,Xnew,Name,Value)``` specifies additional options using one or more name-value pair arguments. For example, you can specify the confidence level of the confidence interval.```
Examples
collapse all
Create a generalized linear regression model, and predict its response to new data.
Generate sample data using Poisson random numbers with two underlying predictors `X(:,1)` and `X(:,2)`.
```rng('default') % For reproducibility rndvars = randn(100,2); X = [2 + rndvars(:,1),rndvars(:,2)]; mu = exp(1 + X*[1;2]); y = poissrnd(mu);```
Create a generalized linear regression model of Poisson data.
`mdl = fitglm(X,y,'y ~ x1 + x2','Distribution','poisson');`
Create data points for prediction.
```[Xtest1,Xtest2] = meshgrid(-1:.5:3,-2:.5:2); Xnew = [Xtest1(:),Xtest2(:)];```
Predict responses at the data points.
`ypred = predict(mdl,Xnew);`
Plot the predictions.
`surf(Xtest1,Xtest2,reshape(ypred,9,9))`
Fit a generalized linear regression model, and then save the model by using `saveLearnerForCoder`. Define an entry-point function that loads the model by using `loadLearnerForCoder` and calls the `predict` function of the fitted model. Then use `codegen` (MATLAB Coder) to generate C/C++ code. Note that generating C/C++ code requires MATLAB® Coder™.
This example briefly explains the code generation workflow for the prediction of linear regression models at the command line. For more details, see Code Generation for Prediction of Machine Learning Model at Command Line. You can also generate code using the MATLAB Coder app. For details, see Code Generation for Prediction of Machine Learning Model Using MATLAB Coder App.
Train Model
Generate sample data using Poisson random numbers with two underlying predictors `X(:,1)` and `X(:,2)`.
```rng('default') % For reproducibility rndvars = randn(100,2); X = [2 + rndvars(:,1),rndvars(:,2)]; mu = exp(1 + X*[1;2]); y = poissrnd(mu);```
Create a generalized linear regression model. Specify the Poisson distribution for the response.
`mdl = fitglm(X,y,'y ~ x1 + x2','Distribution','poisson');`
Save Model
Save the fitted generalized linear regression model to the file `GLMMdl.mat` by using `saveLearnerForCoder`.
`saveLearnerForCoder(mdl,'GLMMdl');`
Define Entry-Point Function
In your current folder, define an entry-point function named `mypredictGLM.m` that does the following:
• Accept new predictor input and valid name-value pair arguments.
• Load the fitted generalized linear regression model in `GLMMdl.mat` by using `loadLearnerForCoder`.
• Return predictions and confidence interval bounds.
```function [yhat,ci] = mypredictGLM(x,varargin) %#codegen %MYPREDICTGLM Predict responses using GLM model % MYPREDICTGLM predicts responses for the n observations in the n-by-1 % vector x using the GLM model stored in the MAT-file GLMMdl.mat, % and then returns the predictions in the n-by-1 vector yhat. % MYPREDICTGLM also returns confidence interval bounds for the % predictions in the n-by-2 vector ci. CompactMdl = loadLearnerForCoder('GLMMdl'); narginchk(1,Inf); [yhat,ci] = predict(CompactMdl,x,varargin{:}); end ```
Add the `%#codegen` compiler directive (or pragma) to the entry-point function after the function signature to indicate that you intend to generate code for the MATLAB algorithm. Adding this directive instructs the MATLAB Code Analyzer to help you diagnose and fix violations that would result in errors during code generation.
Generate Code
Generate code for the entry-point function using `codegen` (MATLAB Coder). Because C and C++ are statically typed languages, you must determine the properties of all variables in the entry-point function at compile time. To specify the data type and exact input array size, pass a MATLAB® expression that represents the set of values with a certain data type and array size. Use `coder.Constant` (MATLAB Coder) for the names of name-value pair arguments.
Create points for prediction.
```[Xtest1,Xtest2] = meshgrid(-1:.5:3,-2:.5:2); Xnew = [Xtest1(:),Xtest2(:)];```
Generate code and specify returning 90% simultaneous confidence intervals on the predictions.
`codegen mypredictGLM -args {Xnew,coder.Constant('Alpha'),0.1,coder.Constant('Simultaneous'),true}`
```Code generation successful. ```
`codegen` generates the MEX function `mypredictGLM_mex` with a platform-dependent extension.
If the number of observations is unknown at compile time, you can also specify the input as variable-size by using `coder.typeof` (MATLAB Coder). For details, see Specify Variable-Size Arguments for Code Generation and Specify Properties of Entry-Point Function Inputs (MATLAB Coder).
Verify Generated Code
Compare predictions and confidence intervals using `predict` and `mypredictGLM_mex`. Specify name-value pair arguments in the same order as in the `-args` argument in the call to `codegen`.
```[yhat1,ci1] = predict(mdl,Xnew,'Alpha',0.1,'Simultaneous',true); [yhat2,ci2] = mypredictGLM_mex(Xnew,'Alpha',0.1,'Simultaneous',true);```
The returned values from `mypredictGLM_mex` might include round-off differences compared to the values from `predict`. In this case, compare the values allowing a small tolerance.
`find(abs(yhat1-yhat2) > 1e-6)`
```ans = 0x1 empty double column vector ```
`find(abs(ci1-ci2) > 1e-6)`
```ans = 0x1 empty double column vector ```
The comparison confirms that the returned values are equal within the tolerance `1e–6`.
Input Arguments
collapse all
Generalized linear regression model, specified as a `GeneralizedLinearModel` object created using `fitglm` or `stepwiseglm`, or a `CompactGeneralizedLinearModel` object created using `compact`.
New predictor input values, specified as a table, dataset array, or matrix. Each row of `Xnew` corresponds to one observation, and each column corresponds to one variable.
• If `Xnew` is a table or dataset array, it must contain predictors that have the same predictor names as in the `PredictorNames` property of `mdl`.
• If `Xnew` is a matrix, it must have the same number of variables (columns) in the same order as the predictor input used to create `mdl`. Note that `Xnew` must also contain any predictor variables that are not used as predictors in the fitted model. Also, all variables used in creating `mdl` must be numeric. To treat numerical predictors as categorical, identify the predictors using the `'CategoricalVars'` name-value pair argument when you create `mdl`.
Data Types: `single` | `double` | `table`
Name-Value Arguments
Specify optional comma-separated pairs of `Name,Value` arguments. `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside quotes. You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.
Example: ```[ypred,yci] = predict(Mdl,Xnew,'Alpha',0.01,'Simultaneous',true)``` returns the confidence interval `yci` with a 99% confidence level, computed simultaneously for all predictor values.
Significance level for the confidence interval, specified as the comma-separated pair consisting of `'Alpha'` and a numeric value in the range [0,1]. The confidence level of `yci` is equal to 100(1 – `Alpha`)%. `Alpha` is the probability that the confidence interval does not contain the true value.
Example: `'Alpha',0.01`
Data Types: `single` | `double`
Number of trials for the binomial distribution, specified as the comma-separated pair consisting of `'BinomialSize'` and a scalar or vector of the same length as the response. `predict` expands the scalar input into a constant array of the same size as the response. The scalar input means that all observations have the same number of trials.
The meaning of the output values in `ypred` depends on the value of `'BinomialSize'`.
• If `'BinomialSize'` is 1 (default), then each value in the output `ypred` is the probability of success.
• If `'BinomialSize'` is not 1, then each value in the output `ypred` is the predicted number of successes in the trials.
Data Types: `single` | `double`
Offset value for each row in `Xnew`, specified as the comma-separated pair consisting of `'Offset'` and a scalar or vector with the same length as the response. `predict` expands the scalar input into a constant array of the same size as the response.
Note that the default value of this argument is a vector of zeros even if you specify the `'Offset'` name-value pair argument when fitting a model. If you specify `'Offset'` for fitting, the software treats the offset as an additional predictor with a coefficient value fixed at 1. In other words, the formula for fitting is
f(μ)` = Offset + X*b`,
where f is the link function, μ is the mean response, and X*b is the linear combination of predictors X. The `Offset` predictor has coefficient `1`.
Data Types: `single` | `double`
Flag to compute simultaneous confidence bounds, specified as the comma-separated pair consisting of `'Simultaneous'` and either true or false.
• `true``predict` computes confidence bounds for the curve of response values corresponding to all predictor values in `Xnew`, using Scheffe's method. The range between the upper and lower bounds contains the curve consisting of true response values with 100(1 – α)% confidence.
• `false``predict` computes confidence bounds for the response value at each observation in `Xnew`. The confidence interval for a response value at a specific predictor value contains the true response value with 100(1 – α)% confidence.
Simultaneous bounds are wider than separate bounds, because requiring the entire curve of response values to be within the bounds is stricter than requiring the response value at a single predictor value to be within the bounds.
Example: `'Simultaneous',true`
Output Arguments
collapse all
Predicted response values at `Xnew`, returned as a numeric vector.
For a binomial model, the meaning of the output values in `ypred` depends on the value of the `'BinomialSize'` name-value pair argument.
• If `'BinomialSize'` is 1 (default), then each value in the output `ypred` is the probability of success.
• If `'BinomialSize'` is not 1, then each value in the output `ypred` is the predicted number of successes in the trials.
For a model with an offset, specify the offset value by using the `'Offset'` name-value pair argument. Otherwise, `predict` uses `0` as the offset value.
Confidence intervals for the responses, returned as a two-column matrix with each row providing one interval. The meaning of the confidence interval depends on the settings of the name-value pair arguments `'Alpha'` and `'Simultaneous'`.
Alternative Functionality
• `feval` returns the same predictions as `predict`. The `feval` function does not support the `'Offset'` and `'BinomialSize'` name-value pair arguments . `feval` uses 0 as the offset value, and the output values in `ypred` are predicted probabilities. The `feval` function can take multiple input arguments for new predictor input values, with one input for each predictor variable, which is simpler to use with a model created from a table or dataset array. Note that the `feval` function does not give confidence intervals on its predictions.
• `random` returns predictions with added noise.
Extended Capabilities
Introduced in R2012a | 2,824 | 11,768 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.171875 | 3 | CC-MAIN-2021-39 | latest | en | 0.575429 |
https://mathoverflow.net/questions/348981/measure-of-subsets-of-profinite-groups/349047 | 1,723,456,505,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722641036895.73/warc/CC-MAIN-20240812092946-20240812122946-00194.warc.gz | 316,124,864 | 24,460 | # Measure of subsets of profinite groups
Let $$G$$ be an infinite profinite group, so $$G=\lim_{\longleftarrow}G/N$$ where $$N$$ runs through the open normal subgroups. I have two questions:
1. Is $$G$$ of Haar measure zero in the compact group $$\prod_NG/N$$?
2. What is the relation between the Haar measure of a subset $$E$$ of $$G$$ and the numbers $$\frac{|EN/N|}{|G/N}$$, the size of the image of $$E$$ in $$G/N$$?
• You probably want $G$ connected? Otherwise, just take a finite simple group.
– abx
Commented Dec 23, 2019 at 11:51
• @abx Profinite group is just a totally disconnected compact group. Sure I mean an infinite profinite group as I would edit. Commented Dec 23, 2019 at 11:57
• Yes, I think that 1 has a positive answer (it's a good exercise).
– YCor
Commented Dec 23, 2019 at 14:43
• For 2, there's an obvious inequality (for $E$ measurable, $\lambda(E)\le\inf_N|EN/N|/|G/N|$) which for $E$ dense of measure zero is clearly not an equality.
– YCor
Commented Dec 23, 2019 at 14:47
• @YCor I choose this title because of my first question. Commented Dec 23, 2019 at 15:01
1) The measure of a closed subgroup $$H$$ of a profinite group $$G$$ is $$\frac{1}{\vert G:H \vert}$$. So $$G$$ has measure zero in $$\prod G/N$$ if and only if it has infinite index. This way you should be able to show that $$G$$ always has measure zero in $$\prod G/N$$.
2) As Yves mentioned, you always have the inequality $$\mu(S) \le \inf \frac{\vert NS/N\vert}{\vert G/N \vert}$$ for any measurable subset $$S$$. In fact the right-hand side is the measure of the closure of $$S$$. If $$S$$ is closed it is an equality. If $$G$$ is the profinite completion of an abstract countable group $$\Gamma$$, then $$\mu(\Gamma)=0$$, but the closure of $$\Gamma$$ is $$G$$. | 544 | 1,762 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 26, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.609375 | 4 | CC-MAIN-2024-33 | latest | en | 0.895236 |
https://www.bbc.co.uk/bitesize/guides/zwkn7p3/revision/3 | 1,611,296,220,000,000,000 | text/html | crawl-data/CC-MAIN-2021-04/segments/1610703529128.47/warc/CC-MAIN-20210122051338-20210122081338-00447.warc.gz | 686,290,711 | 226,650 | # Cell models
It's straightforward to model the cells of organisms using cubes. By doing this we can easily see how the surface area to volume ratio changes as organisms increase in size.
We can investigate the effect of increasing size on surface area to volume ratios using models based on cubes:
So, as the volume increases, the surface area does not increase at the same rate.
If a graph is drawn:
Question
What is the surface area to volume ratio of the highlighted mark?
This cube will have a surface area:volume ratio of 1.
The volume = 6 × 6 × 6 = 216
The surface area = 6 × (6 × 6) = 216
A stacked bar chart can be drawn to illustrate the proportions of surface area and volume.
In the below table scientists have estimated the surface area:volume ratios of various organisms.
OrganismSurface area in square metresVolume in cube metresSurface area:volume
Bacterium6 × 10−121 × 10−186,000,000:1
Blow fly6 × 10−41 × 10−6600:1
Whale6 × 1041 × 1060.06
Large organisms have:
• mechanisms to increase surface area proportionately, such as additional absorbing areas or adaptations of shape
• transport systems and keep distances for diffusion to a minimum
Organisms living in harsh environmental conditions may reduce their surface area, for example cacti have adapted to have less surface area to lose water through. | 311 | 1,335 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.484375 | 3 | CC-MAIN-2021-04 | latest | en | 0.907197 |
https://www.sololearn.com/ru/Discuss/2258806/booleans | 1,716,600,850,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058770.49/warc/CC-MAIN-20240525004706-20240525034706-00784.warc.gz | 877,083,255 | 190,661 | Новый курс! Каждый программист должен знать генеративный ИИ!
+ 1
Booleans
x = True y = False z = False x or y and z == True. Why is 'and' treated before 'or' ? Is this a python thing? Help me understand.
25th Apr 2020, 12:51 AM
Tomiwa Joseph
5 ответов
+ 3
'and' is before 'or' in every programming language. There is an order of operations just like in math. https://en.m.wikipedia.org/wiki/Order_of_operations#Programming_languages
25th Apr 2020, 1:01 AM
Henri Evjen
+ 2
arr = [1, True, 'a', 2] print('a' in arr in arr) Output is false. While: print(('a': in arr) in arr) Output is true Schindlabua Henri Evjen Is python right to left without brackets for 'in' operations?
25th Apr 2020, 7:56 AM
Tomiwa Joseph
+ 1
Even cooler: `and` is like multiplication and `or` is like addition in a pretty strict mathematical sense. true behaves a bit like 1, false like 0: true and true == true 1 * 1 == 1 false and true == false 0 * 1 == 0 false or false == false 0 + 0 == 0 false or true == true 0 + 1 == 1 (with 1+1 we have to squint a little to see it.) And of course multiplication goes before addition :)
25th Apr 2020, 3:00 AM
Schindlabua
+ 1
I don't write much python, but looks like it. Right-associativity is not terribly uncommon though. Think of `a ** b ** c`, or in other languages, `a ? b : c ? d : e` or `a = b = c`.
25th Apr 2020, 10:39 AM
Schindlabua
0
Thanks. Cool eye opener. *Not every programming language though. There are exceptions.
25th Apr 2020, 1:12 AM
Tomiwa Joseph
Актуальное сегодня
Help! | 486 | 1,511 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.984375 | 3 | CC-MAIN-2024-22 | latest | en | 0.772738 |
https://community.esri.com/thread/47809 | 1,561,367,652,000,000,000 | text/html | crawl-data/CC-MAIN-2019-26/segments/1560627999298.86/warc/CC-MAIN-20190624084256-20190624110256-00011.warc.gz | 401,168,622 | 23,329 | # Path Distance algorithm
Discussion created by simam on Mar 20, 2012
Latest reply on Apr 10, 2012 by PYadav-esristaff
Hi,
I have two questions regardin the algorithm which is used for calculating the cost_distance values in the 'Path Distance' tool. As it is described in its help page, cost-distance calculation follows the formula:
Cost-Distance = cost-surface*surface-distance*([friction a * Horizontal Factor a + Friction b* Horizontal Factor b]/2)* vertical factor
which is multiplied by 1.4 in case of diagonal movement from cell to cell.
1- what happens if we do not define any Vertical Factor in the tool? what is the default value which would be assigned to it?
when it comes to calculate the accumulated-cost it uses the following formula:
accumulated-cost-distnace= a1+ surface-distance*([friction b* Horizontal Factor b + Friction c* Horizontal Factor c]/2)* vertical factor, and my next question is :
2-why it doesn't use the "Cost-Surface" term in this step anymore? doesn't it really follow the same concep as the previous formula to calculate the cost of movement to proceed to the next cells or is there any other explanation for it?
Thank you in advance for your help,
Regards,
Sima | 280 | 1,208 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.796875 | 3 | CC-MAIN-2019-26 | latest | en | 0.918876 |
www.meshp.org | 1,611,778,386,000,000,000 | text/html | crawl-data/CC-MAIN-2021-04/segments/1610704832583.88/warc/CC-MAIN-20210127183317-20210127213317-00701.warc.gz | 159,740,663 | 7,441 | Category
# Calculate car loan – this is how you should proceed
A new car is associated with considerable costs. The car has to be paid for, usually the strong brand and AU are renewed, and the vehicle has to be insured and taxed. A car salesman once added that one could also not expect to get the car with a full tank. In particular, novice drivers are often surprised at how much the new car will cost them. They often resort to bank funding. But before that, it is important to carefully calculate the necessary car loan.
## Calculate a car loan: What is the term “car loan”?
Car loans are loans that banks grant specifically to be able to buy a used or new car. The borrowed money may not be used for any other purpose. But where do the costs start with the purchase of a vehicle and are therefore covered by the car loan and where do they end? If you want to calculate your necessary car loan, this question is of enormous importance. The puristic variant of a car loan says: It can only be used for the purchase price. However, this variant is extremely rare.
Usually, some scope for the new production of strong brand and AU is also allowed. In fact, the credit line is actually flexible. In theory, you can apply for significantly more money than you actually need. However, when examining the application, the necessary costs for the purchase of the vehicle are added by the financial institution. Any difference between the requested money and the actual costs is deleted from the loan amount. However, some banks also allow (especially so-called car banks), insurance (liability, second car, etc.) and taxes to be repaid via the loan. Before you can calculate your car loan, you first need a precise catalog of benefits, which your own insurance actually offers.
## Calculate a car loan: use aids
If you know the service catalog of a car loan offer and if you agree, you can start to calculate the actual car loan. This happens in two phases, for which the bank provides you with the appropriate tools in the form of an online loan calculator: First you determine the necessary loan amount yourself, which can also be checked by the bank, but then you deal with the question which is much more important for the financial institution: how do you repay the car loan? Banks offer flexible loan terms that start at twelve months and often last up to 84 months.
It should be borne in mind that although the longer the loan term is chosen, the monthly rate decreases, but the interest burden increases, so you actually pay more and more and a long-term financial burden becomes a problem for many people at some point. The reverse conclusion, however, cannot be that the shorter the loan term, the better. The monthly installments would otherwise skyrocket and threaten normal living standards. If the rate is too high (more than ten percent of the monthly net income), the bank will probably not agree to the loan application. | 598 | 2,941 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.5625 | 3 | CC-MAIN-2021-04 | latest | en | 0.978846 |
https://www.math.net/interquartile-range | 1,720,795,817,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514404.71/warc/CC-MAIN-20240712125648-20240712155648-00826.warc.gz | 644,560,511 | 3,527 | # Interquartile range
In statistics, the interquartile range (IQR) is a measure of how spread out the data is. It is equal to the difference between the 75th and 25th percentiles, referred to as the third (Q3) and first quartiles (Q1), respectively. Thus, the IQR is comprised of the middle 50% of the data, and is therefore also referred to as the midspread, or middle 50%.
The IQR is particularly useful when data is contaminated (e.g. has many outliers) because it excludes extreme values. It can also be used to find outliers in a set of data.
## How to find the IQR
Given a set of data ordered from smallest to largest,
{3, 4, 7, 11, 12, 16, 21, 22, 30, 32, 105}
the IQR can be found by subtracting Q1 from Q3, or:
IQR = Q3 - Q1
Refer to the quartile page for more information on how to find each quartile. Q2 is the median of the set of data, Q1 is the median of the data between the first element and Q2, and Q3 is the median of the data between Q3 and the final element of the set.
Thus, in the set above, Q1 is 7, Q2 is 16, and Q3 is 30. The interquartile range is therefore:
IQR = 30 - 7 = 23
Example
Find the IQR for the set:
{1, 4, 6, 9, 15, 21, 22, 27, 35, 40, 41, 56}
The following formulas can be used to determine the position of the quartiles in the set
Q1 = 0.25(n + 1)
Q3 = 0.75(n + 1)
where n is the number of terms in the set. Thus:
Q1 = 0.25(12 + 1) = 3.25
Q3 = 0.75(12 + 1) = 9.75
The decimal values indicate that the quartile lies between the elements closest to the value. Thus, Q1 lies between the 3rd and 4th element in the set, and Q3 lies between the 9th and 10th elements. Averaging the terms in those positions yields Q1 and Q3:
Thus, the interquartile range can be calculated as:
IQR = 37.5 - 7.5 = 30
## Using the IQR to find outliers
The IQR can be used to find outliers (values in the set that lie significantly outside the expected value). Values that lie farther than 1.5 times the IQR away from either end of the IQR (Q1 or Q3) are considered outliers, as shown in the figure below:
Thus, the expected range of values is:
[Q1 - 1.5(IQR), Q3 + 1.5(IQR)]
Anything outside the above range of values is an outlier.
Example
Find any outliers for the following set of data:
{1, 3, 4, 6, 13, 20, 25, 26, 28, 62, 95}
Q2 is the median, and since there are 11 elements in the set, Q2 is the middle value. Q1 is the median of the first 25% of values (first element through Q1) and Q3 is the median of the last 25% of values (Q3 through the final element). Q1 is highlighted in green, Q2 in red, and Q3 in blue:
{1, 3, 4, 6, 13, 20, 25, 26, 28, 62, 95}
IQR is the difference between Q3 and Q1:
IQR = 28 - 4 = 24
Thus any value outside of the range
[4 - 1.5(24), 28 + 1.5(24)] = [-32, 64]
is an outlier. All of the data, except for 95, lies within the above range. Thus, 95 is the only outlier in the set. | 927 | 2,867 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.9375 | 5 | CC-MAIN-2024-30 | latest | en | 0.927416 |
http://mathhelpforum.com/statistics/193446-confidence-interval-print.html | 1,526,818,755,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794863410.22/warc/CC-MAIN-20180520112233-20180520132233-00284.warc.gz | 191,239,103 | 3,015 | # Confidence interval
Printable View
• Dec 4th 2011, 05:10 PM
deezy
Confidence interval
Find a 95% confidence interval for the percentage of cars on a certain highway that have poorly adjusted brakes, using a random sample of 500 cars stopped at a roadblock on that highway, 87 of which had poorly adjusted brakes.
I think I need to find k, with
$\displaystyle k = \frac{cs}{\sqrt n}$
n = 500, $\displaystyle \mu = 87$.
Not sure how to find c or s.
• Dec 4th 2011, 05:21 PM
pickslides
Re: Confidence interval
Try this
$\displaystyle \frac{87}{500}\pm 1.96\times \sqrt{\frac{\frac{87}{500}\left(1-\frac{87}{500}\right)}{500}}$
• Dec 4th 2011, 06:05 PM
deezy
Re: Confidence interval
Quote:
Originally Posted by pickslides
Try this
$\displaystyle \frac{87}{500}\pm 1.96\times \sqrt{\frac{\frac{87}{500}\left(1-\frac{87}{500}\right)}{500}}$
Where did that come from? When I calculated it I got .5063 and -0.1583.
http://edugen.wileyplus.com/edugen/c...25/math086.gif
This is the answer, but I don't know how c or $\displaystyle s^2$ was calculated.
For c, I tried $\displaystyle .5(1+.95) = .975$ and looking up the value for .975 on the t distribution chart with 50 - 1 = 49 degrees of freedom. But 49 was not on the chart.
For $\displaystyle s^2$, I'm not understanding why it is $\displaystyle 87 * 413 / 500$. | 420 | 1,322 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.84375 | 4 | CC-MAIN-2018-22 | latest | en | 0.855151 |
https://math.stackexchange.com/questions/4516678/is-the-number-of-finite-subgroups-of-a-finite-by-cyclic-group-finite | 1,718,426,976,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861583.78/warc/CC-MAIN-20240615031115-20240615061115-00556.warc.gz | 349,099,179 | 36,035 | # Is the number of finite subgroups of a finite-by-cyclic group finite?
Let $$G$$ be a finite-by-cyclic group, i.e. it contains a finite normal subgroup $$N$$ such that $$G/N$$ is cyclic.
My question is that: Does $$G$$ contain only finitely many finite subgroups?
What I've tried: Assume that $$G/N$$ is finite cyclic. Then $$G$$ is a finite group and then it has only a finite number of subgroups.
• "Assume that $G/N$ is finite cyclic. Then $G$ is a finite group" : why ? Commented Aug 22, 2022 at 11:07
• @Arthur It was a question for the OP... Thanks for the answer though. Commented Aug 22, 2022 at 11:13
• @TheSilverDoe Fair enough. I usually address such questions more explicitly to the OP precisely to avoid these misunderstandings. It is impossible for me to know from your original comment whether you want my help or not. Commented Aug 22, 2022 at 11:15
• (Although if you care to look, you will see that just my previous comment on this site is exactly on the opposite side of this issue, speaking of the merits of letting the OP answer the questions in the comments. So I'm not exactly consistent about it.) Commented Aug 22, 2022 at 12:18
Yes, $$G$$ contains only finitely many finite subgroups. As you pointed out, this is trivial if $$G/N$$ is finite, so we will consider the case $$G/N \cong \mathbb{Z}$$.
Let $$K$$ be a finite subgroup of $$G$$, then $$KN$$ is also a finite subgroup of $$G$$ (since $$N$$ is finite and normal) and therefore $$KN/N$$ is a finite subgroup of $$G/N \cong \mathbb{Z}$$. We conclude that $$KN/N = \left\{0\right\}$$, hence $$KN = N$$ and so $$K < N$$. This means that we have only a finite number of choices for $$K$$. | 465 | 1,673 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 20, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.71875 | 4 | CC-MAIN-2024-26 | latest | en | 0.923289 |
https://studylib.net/doc/8506501/file | 1,611,251,353,000,000,000 | text/html | crawl-data/CC-MAIN-2021-04/segments/1610703527224.75/warc/CC-MAIN-20210121163356-20210121193356-00003.warc.gz | 584,816,999 | 13,564 | # file
```1. The _________ is the rate of return a firm must earn on its investments in projects in
order to maintain the market value of its stock.
A. yield to maturity
B. cost of capital
C. internal rate of return
D. gross profit margin
2. The cost of each type of capital depends on the
A. risk-free cost of that type of funds.
B. business risk of the firm.
C. financial risk of the firm.
D. All of the above.
3. When determining the after-tax cost of a bond, the face value of the issue must be
adjusted to the net proceeds amounts by considering
A. the risk.
B. the floatation costs.
C. the approximate returns.
D. the taxes.
4. A firm has a beta of 1.2. The market return equals 14 percent and the risk-free rate of
return equals 6 percent. The estimated cost of common stock equity is
A. 6 percent.
B. 7.2 percent.
C. 14 percent.
D. 15.6 percent.
5. The investment opportunity schedule combined with the weighted marginal cost of
capital indicates
A. those projects that a firm should select.
B. those projects that will result in the highest cash flows.
C. which projects are acceptable given the firm's cost of capital.
D. which combination of projects will fit within the firm's capital budget.
6. _______ costs are a function of time, not sales, and typically contractual.
A. Fixed
B. Semi-variable
C. Variable
D. Operating
7. As fixed operating costs increase and all other factors are held constant, the degree of
operating leverage will
A. increase.
B. decrease.
C. remain unchanged.
D. change in an undetermined direction.
8. A corporation borrows \$1,000,000 at 10 percent annual rate of interest. The firm has a
40 percent tax rate. The yearly, after-tax cost of this debt is
A. \$40,000.
B. \$60,000.
C. \$100,000.
D. \$166,667.
9. A corporation has \$10,000,000 in 10 percent preferred stock outstanding and a 40
percent tax rate. The amount of earnings before interest and taxes (EBIT required to pay
the preferred dividends is
A. \$1,000,000.
B. \$400,000.
C. \$600,000.
D. \$1,666,667.
10. The conflict resulting from a manager's desire to increase the firm's risk without
increasing current borrowing costs and lenders' desire to limit lending is one effect of the
_______ problem.
A. agency
B. leverage
C. capital
D. variable cost
11. In order to enhance the wealth of stockholders and to send positive signals to the
market, corporations generally raise funds using the following order:
A. retained earnings, equity, debt.
B. retained earnings, debt, equity.
C. debt, retained earnings, equity.
D. equity, retained earnings, debt
12. As debt is substituted for equity in the capital structure and the debt ratio increases,
the behavior of the overall cost of capital is partially explained by
A. the tax-deductibility of interest payments.
B. the increase in the number of common shares outstanding.
C. the reduction in risk as perceived by the common shareholders.
D. the decrease in the cost of equity.
13. The major shortcoming of the EBIT-EPS approach to capital structure is that
A. the technique does not promote the maximization of shareholder wealth.
B. the technique does not consider the cost of capital
C. the technique only considers leverage-related risk.
D. the technique does not maximize earnings per share.
14. At a firm's quarterly dividend meeting held April 9, the directors declared a \$.50 per
share cash dividend for the holders of record on Monday, May 1. The firm's stock will
sell ex-dividends on
A. April 9
B. May 5
C. April 25
D. April 27
15. Paying a stock dividend ____________ the retained earnings account.
A. decreases
B. has no effect on
C. increases
D. reorganizes
16. Dividend policy is a form of
A. capital budgeting policy
B. financing policy
C. working capital policy
D. dividend reinvestment policy
17. A firm has current after-tax earnings of \$1,000,000 and has declared a cash dividend
of \$400,000. The firm's dividend payout ration is
A. 2.5%
B. 2.0 percent
C. 4.0 percent
D. 40 percent
18. A stock split has __________ effect on the firm's capital structure.
A. little
B. no
C. a measurable
D. a detrimental
19. The repurchase of stock __________ the earnings per share and __________ the
market price
of stock.
A. increases, increases
B. decreases, decreases
C. increases, decreases
D. decreases, increases.
20. The purpose of a stock split is to | 1,100 | 4,319 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2021-04 | longest | en | 0.906916 |
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<html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8"> <meta name="generator" content="pdf2htmlEX"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <link rel="stylesheet" href="https://static.pudn.com/base/css/base.min.css"> <link rel="stylesheet" href="https://static.pudn.com/base/css/fancy.min.css"> <link rel="stylesheet" href="https://static.pudn.com/prod/directory_preview_static/62684f4e4f8811599eff0481/raw.css"> <script src="https://static.pudn.com/base/js/compatibility.min.js"></script> <script src="https://static.pudn.com/base/js/pdf2htmlEX.min.js"></script> <script> try{ pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({}); }catch(e){} </script> <title></title> </head> <body> <div id="sidebar" style="display: none"> <div id="outline"> </div> </div> <div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/62684f4e4f8811599eff0481/bg1.jpg"><div class="c x0 y1 w2 h2"><div class="t m0 x1 h3 y2 ff1 fs0 fc0 sc0 ls0 ws0">小功率直流随动系统设计</div><div class="t m0 x2 h4 y3 ff1 fs1 fc0 sc0 ls0 ws0">【摘要】：<span class="_ _0"></span><span class="sc1">本文对小功率<span class="_ _0"></span>直流随动系统进行<span class="_ _0"></span>了研究与设计。首<span class="_ _0"></span>先对随动系统<span class="_ _0"></span>进行了实验建模</span></div><div class="t m0 x2 h4 y4 ff1 fs1 fc0 sc1 ls0 ws0">与实验测试<span class="_ _0"></span>，构建了随动<span class="_ _0"></span>系统的系统框图<span class="_ _0"></span><span class="ff2">;</span>然后<span class="_ _0"></span>采用频率法为系统<span class="_ _0"></span>设计超前校正<span class="_ _0"></span>装置<span class="ff2">, <span class="_ _0"></span></span>并使用</div><div class="t m0 x2 h4 y5 ff2 fs1 fc0 sc1 ls0 ws0">Matlab<span class="ff1">计算<span class="_ _0"></span>机仿真<span class="_ _0"></span>软件<span class="_ _0"></span>对<span class="_ _0"></span>系统进<span class="_ _0"></span>行了仿真<span class="_ _1"></span></span>;<span class="ff1">最后对<span class="_ _0"></span>校正装<span class="_ _0"></span>置进行了<span class="_ _0"></span>电路设<span class="_ _0"></span>计与制作<span class="_ _0"></span>，对系统</span></div><div class="t m0 x2 h4 y6 ff1 fs1 fc0 sc1 ls0 ws0">进<span class="_ _0"></span>行校<span class="_ _0"></span>正<span class="_ _0"></span>，<span class="_ _0"></span>使<span class="_ _0"></span>系<span class="_ _0"></span>统<span class="_ _0"></span>满足<span class="_ _0"></span>了<span class="_ _0"></span>性<span class="_ _0"></span>能指<span class="_ _0"></span>标<span class="_ _0"></span>要<span class="_ _0"></span>求<span class="_ _0"></span>。<span class="_ _0"></span>其建<span class="_ _0"></span>模<span class="_ _0"></span>、<span class="_ _0"></span>仿真<span class="_ _0"></span>以<span class="_ _0"></span>及<span class="_ _0"></span>校<span class="_ _0"></span>正<span class="_ _0"></span>网络<span class="_ _0"></span>设<span class="_ _0"></span>计<span class="_ _0"></span>方法<span class="_ _0"></span>简<span class="_ _0"></span>单<span class="_ _0"></span>易<span class="_ _0"></span>行<span class="_ _1"></span><span class="ff2">,<span class="_ _0"></span></span>对</div><div class="t m0 x2 h4 y7 ff1 fs1 fc0 sc1 ls0 ws0">研究其他种类的随动控制系统具有一定的借鉴作用。</div><div class="t m0 x2 h4 y8 ff1 fs1 fc0 sc0 ls0 ws0">【关键词】：<span class="sc1">小功率直流随动系统 建模 仿真 超前校正</span></div><div class="t m0 x3 h4 y9 ff3 fs1 fc0 sc1 ls0 ws0">Abstract<span class="ff1 sc0">：</span></div><div class="t m0 x4 h5 ya ff4 fs2 fc0 sc1 ls0 ws0">In<span class="_ _0"></span> <span class="_ _0"></span>this<span class="_ _1"></span> <span class="_ _0"></span>paper,<span class="_ _0"></span> <span class="_ _0"></span>low-power<span class="_ _1"></span> <span class="_ _0"></span>DC<span class="_ _0"></span> <span class="_ _0"></span>servo<span class="_ _1"></span> <span class="_ _0"></span>system<span class="_ _0"></span> <span class="_ _0"></span>to<span class="_ _1"></span> <span class="_ _0"></span>carry<span class="_ _0"></span> <span class="_ _1"></span>out <span class="_ _1"></span>a<span class="_ _0"></span> <span class="_ _0"></span>study<span class="_ _1"></span> and<span class="_ _1"></span> <span class="_ _0"></span>design.<span class="_ _0"></span> <span class="_ _0"></span>With<span class="_ _1"></span> <span class="_ _0"></span>the<span class="_ _0"></span> <span class="_ _0"></span>fir<span class="_ _0"></span>st<span class="_ _0"></span> <span class="_ _0"></span>experimental</div><div class="t m0 x2 h5 yb ff4 fs2 fc0 sc1 ls0 ws0">dynamic<span class="_ _0"></span> <span class="_ _1"></span>system<span class="_ _0"></span> <span class="_ _0"></span>modeling<span class="_ _1"></span> <span class="_ _0"></span>and<span class="_ _0"></span> <span class="_ _0"></span>e<span class="_ _0"></span>xperimental<span class="_ _0"></span> <span class="_ _0"></span>testing,<span class="_ _1"></span> <span class="_ _0"></span>to<span class="_ _0"></span> <span class="_ _1"></span>build<span class="_ _0"></span> <span class="_ _0"></span>a<span class="_ _1"></span> <span class="_ _0"></span>servo<span class="_ _0"></span> <span class="_ _1"></span>system<span class="_ _0"></span> <span class="_ _0"></span>block<span class="_ _1"></span> <span class="_ _0"></span>diagram<span class="_ _0"></span> <span class="_ _0"></span>o<span class="_ _0"></span>f<span class="_ _0"></span> <span class="_ _0"></span>the<span class="_ _1"></span> <span class="_ _0"></span>system;</div><div class="t m0 x2 h5 yc ff4 fs2 fc0 sc1 ls0 ws0">and<span class="_ _2"> </span> <span class="_ _2"> </span>then<span class="_ _2"> </span> <span class="_ _2"> </span>usin<span class="_ _3"></span>g<span class="_ _2"> </span> <span class="_ _2"> </span>the<span class="_ _2"> </span> <span class="_ _2"> </span>frequenc<span class="_ _3"></span>y<span class="_ _2"> </span> <span class="_ _2"> </span>of<span class="_ _2"> </span> <span class="_ _2"> </span>law<span class="_ _2"> </span> <span class="_ _4"> </span>advanced<span class="_ _4"> </span> <span class="_ _2"> </span>system<span class="_ _2"> </span> <span class="_ _2"> </span>design<span class="_ _2"> </span> <span class="_ _4"> </span>correction<span class="_ _2"> </span> <span class="_ _2"> </span>device,<span class="_ _2"> </span> <span class="_ _2"> </span>and<span class="_ _4"> </span> <span class="_ _2"> </span>use<span class="_ _2"> </span> <span class="_ _2"> </span>computer</div><div class="t m0 x2 h5 yd ff4 fs2 fc0 sc1 ls0 ws0">simulation <span class="_ _0"></span>software<span class="_ _0"></span> <span class="_ _0"></span>Matlab <span class="_ _0"></span>simulation <span class="_ _0"></span>o<span class="_ _0"></span>f <span class="_ _0"></span>the <span class="_ _0"></span>system;<span class="_ _0"></span> <span class="_ _0"></span>the <span class="_ _0"></span>end<span class="_ _0"></span> o<span class="_ _0"></span>f <span class="_ _0"></span>the <span class="_ _0"></span>calibration<span class="_ _0"></span> <span class="_ _0"></span>device<span class="_ _0"></span> a<span class="_ _0"></span> <span class="_ _0"></span>circuit <span class="_ _0"></span>design<span class="_ _0"></span> <span class="_ _0"></span>and</div><div class="t m0 x2 h5 ye ff4 fs2 fc0 sc1 ls0 ws0">production, to <span class="_ _0"></span>correct the sy<span class="_ _0"></span>stem, allowing the <span class="_ _0"></span>system to <span class="_ _0"></span>meet the <span class="_ _0"></span>performance requirements. <span class="_ _0"></span>Its modeling,</div><div class="t m0 x2 h5 yf ff4 fs2 fc0 sc1 ls0 ws0">simulation <span class="_ _1"></span>and <span class="_ _0"></span>calibration<span class="_ _0"></span> <span class="_ _0"></span>network <span class="_ _1"></span>de<span class="_ _3"></span>sign<span class="_ _0"></span> <span class="_ _0"></span>is <span class="_ _0"></span>simple,<span class="_ _0"></span> <span class="_ _0"></span>the<span class="_ _0"></span> <span class="_ _0"></span>study<span class="_ _0"></span> <span class="_ _0"></span>of <span class="_ _1"></span>oth<span class="_ _3"></span>er<span class="_ _0"></span> ty<span class="_ _0"></span>pes <span class="_ _1"></span>of <span class="_ _0"></span>servo <span class="_ _1"></span>control <span class="_ _0"></span>system<span class="_ _0"></span> has<span class="_ _0"></span> <span class="_ _0"></span>a</div><div class="t m0 x2 h5 y10 ff4 fs2 fc0 sc1 ls0 ws0">certain reference.</div><div class="t m0 x2 h4 y11 ff3 fs1 fc0 sc1 ls0 ws0">Keywords<span class="ff1 sc0">：</span><span class="ff4 fs2"> Low-power DC servo system Modeling Simulation Lead correction</span></div></div></div><div class="pi" data-data='{"ctm":[1.611850,0.000000,0.000000,1.611850,0.000000,0.000000]}'></div></div> </body> </html>
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# Answer: In Exercises 49 52, use a graphing utility to
ISBN: 9780618851522 246
## Solution for problem 11.4.51 Chapter 11.4
Precalculus With Limits A Graphing Approach | 5th Edition
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Problem 11.4.51
In Exercises 49 52, use a graphing utility to complete the table and estimate the limit of the sequence as n approaches infinity. Then find the limit algebraically.
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# HW1 - A T B with tr C = n X i =1 c ii c = c ii 1 ≤ i ≤...
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Electrical Engineering and Computer Science Department University of Central Florida COT 6600– Quantum Computing Fall 2010 (dcm) Homework Assignment 1. – Due Wednesday, September 22, 2008. Problem 1 (20 points) Show that: e = cos θ + i sin θ and ( e ) n = cos( ) + i sin( ) . Hint: Review Taylor series expansion of a function before attempting to prove Euler’s and De Moivre’s formulae. Problem 2 (20 points) Show that C m × n the set of all m × n matrices with complex entries is a complex vector space. Hint: Review the properties of a complex vector space and show that V, W C m × n each one of the properties is satisfied. Problem 3 (20 points) Find the transpose, W T , the conjugate, W * , and adjoint, W of the complex matrix W : W = 6 - 3 i 2 + 12 i - 19 i 0 5 + 2 . 1 i 17 1 2 + 5 i 3 - 4 . 5 i Problem 4 (20 points) Given matrices A, B, C R n × n (square matrices with real elements) the inner product h A, B i is defined as:
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Unformatted text preview: ( A T B ) with tr( C ) = n X i =1 c ii , c = [ c ii ] 1 ≤ i ≤ n. Construct the inner product of A = 6 2 4 13 9 11 5 17 1 3 8 10 17 16 15 14 and B = -6 2-4 13 9-11 5-17-1 3-8 10 17-16 15-14 Given matrices V,W ∈ C n × n (square matrices with complex elements) the inner product h V,W i is defined as: h V,W i = tr ( V † W ) . Construct the inner product of V = 6 + i 1 + 2 i 3 + 4 i 3-i 9 1-2 i 2 + 3 i 1-i i 3-2 i 8 5-4 i 1 + i 7 6 + 5 i 9-7 i and W = -6 + i-i 2-i 4 + i i-1-i-1 2 i 8-7 i 10-i 2 + 5 i-7 7 i Problem 5 (20 points) Construct the tensor and the outer products of the matrices A and B in Problem 4....
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http://mathoverflow.net/questions/79921/polynomials-with-prescribed-points-to-match-prescribed-bounds | 1,467,153,025,000,000,000 | text/html | crawl-data/CC-MAIN-2016-26/segments/1466783397213.30/warc/CC-MAIN-20160624154957-00122-ip-10-164-35-72.ec2.internal.warc.gz | 188,339,854 | 16,323 | # Polynomials with prescribed points to match prescribed bounds
Consider real polynomials on the interval $I=[-1,1]$. It is easy to see that the smallest degree for a non-negative polynomial with given zeros $x_1,\dots,x_s\in I^\circ$ is $n=2s$ (e.g. $P(x) = \prod_{i=1}^s (x-x_i)^2$ works).
My question is:
What is the smallest degree for a polynomial such that it is bounded by $\pm 1$ on $I$ and attains the value $1$ on a set $x_1^+,\dots,x_s^+$ and the value $-1$ on a set $x_1^-,\dots,x_r^-$?
Background: I know that the fact about nonnegative polynomials with presribed zeros can be generalized to "generalized polynomials" built from Tchebycheff-systems (due to a theorem by Krein). I would love to see a similar theorem on bounded generalized polynomials which attain the bounds at prescribed points.
Edit: In this question I leanerd from the answer of Gjergji Zaimi that there are bounds on the degree of increasing interpolating polyomials. How does the bounds change for monotone interpolation are described above? Are there (algebraical or numerical) methods to calculate the polynomial?
It seems to me that monotone interpolating polynomials are not treated in the current literature and are not subject of current research. Is that right, and if so is there a special reason for that?
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You understand that the answer depends heavily on the location of the points and, thereby, is by no means as simple and clean as in the non-negative case, right? – fedja Nov 3 '11 at 13:17
That is pretty clear. I am willing to add more assumptions and not hoping for a very clean answer. – Dirk Nov 3 '11 at 13:35
@Dirk and @fedia: why does such a polynomial exist for every choice of $x^+, x^-$? Say, what is the polynomial if $x_1^+=1/5, x_2^+=1/4, x_1^-=1/3, x_2^-=1/2$? – Mark Sapir Nov 3 '11 at 22:13
Such a polynomial always exists: Take a Chebycheff polynomial $P$ of degree high enough (such that is attains the values 1 and -1 in the right order at point $t_i$. Then choose a monotone polynomial $Q$ which maps the points $t_i$ to the prescribed $x_i$ and take $P\circ Q$. Such monotone polynomial interpolants exit by a theorem of Young (ams.org/mathscinet-getitem?mr=0212455). – Dirk Nov 4 '11 at 7:58
Oh, there is a simpler argument: Since there are piecewise monotone interpolating polynomials we can use them directly to interpolate the $x^+$ and $x^-$ (and possibly adding zero interpolation points inbetween). – Dirk Nov 4 '11 at 8:56
Let $D$ be the minimum distance between $x$'s, merging all the $x$'s into one list of length $N$.
Let $k$ be an integer $\ge \max(16\log(8/D^2),10N)\ /\ D^2$.
Then a polynomial of degree of $6(k+1)(N-1)$ suffices.
Proof: Let $p(x) = \frac{1}{2}(3 q(x) - q^3(x))$, where $q(x) = \sum_i \pm \Pi_{j \neq i} r_{ij}(x)$ and
$$r_{ij}(x) = \left(1-\frac{(x-x_i)^2}{4}\right)^k \frac{(x-x_j)^2}{(x_i-x_j)^2}$$
Then $p$ is clearly of the specified degree, and has the specified values at the $x_i$'s.
The key is to show that $p$ is bounded by $\pm1$.
Small terms: When $|x-x_i| > D/2$, $r_{ij}(x) \le (1-D^2/16)^k (4/D^2) < %(1-D^2/16)^{\large(16/D^2)\log(8/D^2)}(4/D^2) < 1/2$, so $\Pi_{j \neq i} r_{ij}(x) < 2^{-(N-1)}$.
Large terms: When $|x-x_i| < D/2$,
$$r_{ij}(x) \le \left(1-\frac{(x-x_i)^2}{4}\right)^k \left(1 - \frac{x-x_i}{x_j-x_i}\right)^2 \le \left(1-\frac{(x-x_i)^2}{4}\right)^k \left(1 + \frac{2|x-x_i|}{kD}\right)^k$$ $$\le \left(1+ \frac{2|x-x_i|}{kD}-\frac{(x-x_i)^2}{4}\right)^k \le \left(1+\frac{4}{k^2D^2}\right)^k \le e^{\large 4/D^2k} \le e^{\large 4/10N} \le (3/2)^{\large 1/N}.$$ So $\Pi_{j \neq i} r_{ij}(x) < 3/2$
Since each $x$ is within $D/2$ of at most one $x_i$, $q(x)$ is the sum of at most one large term bounded by 3/2, and by $N-1$ small terms bounded by $2^{-(N-1)}$. So $|q|\le2$, and $|p|\le1$.
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That looks very interesting! Christmas time prevents that I check the answer in detail right now and since I am not sure, if I am going to come back here in due time I check this answer as correct right now to cash the bounty. Maybe I'll come back with questions later. – Dirk Dec 26 '13 at 17:28
Dirk, happy to answer any questions. For what it's worth, you can check that, e.g. $p$ behaves appropriately in @Mark Sapir's example, with $k=51654$, $q=250000 \left(1-\frac{1}{4} \left(x-\frac{1}{5}\right)^2\right)^{3k} (x-1/2)^2(x-1/3)^2(x-1/4)^2+921600 \left(1-\frac{1}{4} \left(x-\frac{1}{4}\right)^2\right)^{3k} (x-1/2)^2(x-1/3)^2(x-1/5)^2-291600 \left(1-\frac{1}{4} \left(x-\frac{1}{3}\right)^2\right)^{3k}(x-1/2)^2(x-1/4)^2(x-1/5)^2-6400 \left(1-\frac{1}{4} \left(x-\frac{1}{2}\right)^2\right)^{3k} (x-1/3)^2 (x-1/4)^2 (x-1/5)^2$. – Matt F. Dec 29 '13 at 11:06 | 1,664 | 4,663 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.96875 | 3 | CC-MAIN-2016-26 | latest | en | 0.918497 |
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