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http://www.mywordsolution.com/question/in-the-last-mayoral-election-in-a-large-city-47/922261
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+1-415-315-9853
info@mywordsolution.com
## Statistics
To fix the null and alternate hypotheses for single proportion.
In the last mayoral election in a large city, 47 percent of the adults over the age of 65 voted Republican. A researcher wishes to determine if the proportion of adults over the age of 65 in the city who plan to vote Republican in the next mayoral election has changed. Let p represent the proportion of the population of all adults over the age of 65 in the city who plans to vote Republican in the next mayoral election. In terms of p, the researcher should test which of following null and alternative hypotheses?
A) H0p = 0.47 vs. Hap < 0.47
B) H0p = 0.47 vs. Hap ≠ 0.47
C) H0p = 0.47 vs. Hap > 0.47
Statistics and Probability, Statistics
• Category:- Statistics and Probability
• Reference No.:- M922261
Have any Question?
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http://www.gurufocus.com/term/InventoryTurnover/TTWO/Inventory%252BTurnover/Take-Two%2BInteractive%2BSoftware%252C%2BInc
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Switch to:
Take-Two Interactive Software Inc (NAS:TTWO)
Inventory Turnover
4.55 (As of Sep. 2016)
Inventory turnover measures how fast the company turns over its inventory within a year. It is calculated as cost of goods sold divided by average inventory. Take-Two Interactive Software Inc's cost of goods sold for the three months ended in Sep. 2016 was \$206 Mil. Take-Two Interactive Software Inc's average inventory for the quarter that ended in Sep. 2016 was \$45 Mil. Take-Two Interactive Software Inc's inventory turnover for the quarter that ended in Sep. 2016 was 4.55.
Days inventory indicates the number of days of goods in sales that a company has in the inventory. Take-Two Interactive Software Inc's days inventory for the three months ended in Sep. 2016 was 20.04.
Inventory can be measured by Days Sales of Inventory (DSI). Take-Two Interactive Software Inc's days sales of inventory (DSI) for the three months ended in Sep. 2016 was 9.80.
Inventory to revenue ratio determines the ability of a company to manage their inventory levels. It measures the percentage of Inventories the company currently has on hand to support the current amount of Revenue. Take-Two Interactive Software Inc's inventory to revenue ratio for the quarter that ended in Sep. 2016 was 0.11.
Definition
Take-Two Interactive Software Inc's Inventory Turnover for the fiscal year that ended in Mar. 2016 is calculated as
Inventory Turnover (A: Mar. 2016 ) = Cost of Goods Sold / Average Inventory = Cost of Goods Sold (A: Mar. 2016 ) / ( (Inventory (A: Mar. 2015 ) + Inventory (A: Mar. 2016 )) / 2 ) = 813.873 / ( (20.051 + 15.888) / 2 ) = 813.873 / 17.9695 = 45.29
Take-Two Interactive Software Inc's Inventory Turnover for the quarter that ended in Sep. 2016 is calculated as
Inventory Turnover (Q: Sep. 2016 ) = Cost of Goods Sold / Average Inventory = Cost of Goods Sold (Q: Sep. 2016 ) / ( (Inventory (Q: Jun. 2016 ) + Inventory (Q: Sep. 2016 )) / 2 ) = 205.605 / ( (12.734 + 77.561) / 2 ) = 205.605 / 45.1475 = 4.55
* All numbers are in millions except for per share data and ratio. All numbers are in their local exchange's currency.
Explanation
Inventory Turnover measures how fast the company turns over its inventory within a year. A higher inventory turnover means the company has light inventory. Therefore the company spends less money on storage, write downs, and obsolete inventory. If the inventory is too light, it may affect sales because the company may not have enough to meet demand.
1. Days Inventory indicates the number of days of goods in sales that a company has in the inventory.
Take-Two Interactive Software Inc's Days Inventory for the three months ended in Sep. 2016 is calculated as:
Days Inventory = Average Inventory (Q: Sep. 2016 ) / Cost of Goods Sold (Q: Sep. 2016 ) * Days in Period = 45.1475 / 205.605 * 365 / 4 = 20.04
2. Inventory can be measured by Days Sales of Inventory (DSI).
Take-Two Interactive Software Inc's Days Sales of Inventory for the three months ended in Sep. 2016 is calculated as:
Days Sales of Inventory (DSI) = Average Inventory (Q: Sep. 2016 ) / Revenue (Q: Sep. 2016 ) * Days in Period = 45.1475 / 420.167 * 365 / 4 = 9.80
3. Inventory to Revenue determines the ability of a company to manage their inventory levels. It measures the percentage of Inventories the company currently has on hand to support the current amount of Revenue.
Take-Two Interactive Software Inc's Inventory to Revenue for the quarter that ended in Sep. 2016 is calculated as
Inventory to Revenue = Average Inventory (Q: Sep. 2016 ) / Revenue (Q: Sep. 2016 ) = 45.1475 / 420.167 = 0.11
* All numbers are in millions except for per share data and ratio. All numbers are in their local exchange's currency.
Be Aware
Usually retailers pile up their inventories at holiday seasons to meet the stronger demand. Therefore, the inventory of a particular quarter of a year should not be used to calculate inventory turnover. An average inventory is a better indication.
Related Terms
Historical Data
* All numbers are in millions except for per share data and ratio. All numbers are in their local exchange's currency.
Take-Two Interactive Software Inc Annual Data
Oct06 Oct07 Oct08 Oct09 Mar11 Mar12 Mar13 Mar14 Mar15 Mar16 Inventory Turnover 7.12 7.54 9.71 7.24 11.63 22.48 27.17 47.15 31.90 45.29
Take-Two Interactive Software Inc Quarterly Data
Jun14 Sep14 Dec14 Mar15 Jun15 Sep15 Dec15 Mar16 Jun16 Sep16 Inventory Turnover 2.05 1.32 6.90 18.39 12.47 7.89 11.66 11.61 13.37 4.55
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https://studylib.net/doc/10414382/ma2215--fields--rings--and-modules-tutorial-problems--oct.
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# MA2215: Fields, rings, and modules Tutorial problems, October 4, 2012
```MA2215: Fields, rings, and modules
Tutorial problems, October 4, 2012
1. (a) Describe all ring homomorphisms from Z/2Z to Z/2Z.
(b) Describe all ring homomorphisms from Z/2Z to Z/3Z.
2. (Note: you know most of this question from your group theory class.)
(a) Show that if the integers a and b are coprime (have no common divisors), then as
Abelian groups,
Z/(ab)Z ' Z/aZ × Z/bZ.
(Hint: consider the map that takes the class n ∈ Z/(ab)Z to the pair
(n mod a, n mod b) ∈ Z/aZ × Z/bZ.
Show that this map is a group homomorphism, and in fact an isomorphism.)
(b) Show that the isomorphism you just constructed is a ring isomorphism, not just an
Abelian group isomorphism.
3. Let R be the set of all triangular 2 × 2-matrices with integer entries,
a b
: a, b, c ∈ Z .
R=
0 c
Take I =
a 0
0 b
: a, c ∈ Z . Show that both I and S are subrings
:b∈Z ,S=
0 c
0 0
of R.
Optional question (if you have some time left): describe all ring isomorphisms
α : Z/nZ → Z/nZ
(of course, there is the trivial one α(k) = k, but there exist other bijections that respect the
ring structure). How many of those are there?
```
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https://www.coursehero.com/file/6640416/Adv-Alegbra-HW-Solutions-94/
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# Adv Alegbra HW Solutions 94 - 94(iii We prove that f g = f...
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Unformatted text preview: 94 (iii) We prove that ( f g ) = f g + f g by induction on deg( f ) (of course, the equation holds when f = 0). The base step deg( f ) = 0 has been done in part (ii). For the inductive step, write f (x ) = ak x k + h (x ), where h (x ) = 0 or deg(h ) < k = deg( f ). If h (x ) = 0, then ( f g ) = ak x k bn x n ak bn x n +k = = (n + k )ak bn x n +k −1 = k ak bn x n +k −1 + = kak x k −1 n ak bn x n +k −1 bn x n + ak x k n bn x n −1 = (ak x k ) g (x ) + ak x k g . If h (x ) = 0, then ( f g ) = (ak x k + h (x ))g (x ) = ak x k g ( x ) + h ( x ) g ( x ) = ak x k g (x ) + D (hg ) = ak x k g (x ) + ak x k g + h g + hg = ak x k + h g + (ak x k + h )g = f g + fg . (iv) We prove that ( f n ) = n f n −1 f by induction on n ≥ 1. If we define f 0 to be the constant function 1, then the base step holds. For the inductive step, f n +1 = fn f = fn f + fn f = (n f n −1 f ) f + f n f = (n + 1) f n f . 3.37 Assume that (x − a ) | f (x ) in R [x ]. Prove that (x − a )2 | f (x ) if and only if (x − a ) | f in R [x ]. ...
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https://www.tutorialspoint.com/generating-random-hex-color-in-javascript
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# Generating random hex color in JavaScript
The Hex (Hexadecimal) code is a six-digit code and a three-byte hexadecimal number that is used to represent the colors. These three bytes represent RGB which means the amount of red, green, and blue in a particular shade of a color. Each byte will represent a number in the range 00 to FF (Hexadecimal notation) or 0 to 255 in decimal notation. This indicates the intensity of each of the color components from 0 to 255.
The following are the functions to be used for generating random hex codes −
Math.random() is used to get the number randomly in the range 0 to less than 1 including the decimal.
function random_num(maximum) {
return Math.floor(Math.random() * maximum);
}
document.write(random_num(5));
// output: 0, 1 or 2
In the above snippet, we gave input as 5 as maximum value. Whenever we try to execute the program, it will print the output in between 0 to 4.
document.write(random_num());
// output: 0 to <1
In this case, as we didn’t have any input value as the maximum value, it prints a random number between 0 to <1 whenever we run the program. So, this is how the Math.random() function works.
Math.floor() is a function that will round the number downwards to the nearest integer.
document.write(Math.floor(1.95));
// output: 1
Here, we can see the output is printed as 5. So, the value is rounded downwards to the nearest integer.
document.write(Math.floor(1));
// output: 1
If the value passed is a perfect integer without any floating (decimal) points, it will be rounded.
Now, let’s combine these functions to generate the random hex numbers.
We use math.random() to generate a random number and multiply it with 16777215 as this number is the decimal representation of fffff (Highest in the hex code).
Math.random()*16777215
This will return a random number like 12421420.464841081 with decimals, Now we use math.floor() to remove the decimals and return a whole number.
Math.floor(Math.random()*16777215)
Now, we include toString() method to convert the number into a string. We have passed 16 as a parameter because Base 16 is hexadecimal. The value of 16 will return hexcode (with numbers and letters).
Math.floor(Math.random()*16777215).toString(16);
### Example 1
Following is the example, which returns the random hex codes of color −
<!DOCTYPE html>
<html>
<title>Generating random Hex color</title>
<h3>
<div id="demo">Mouse-over on this box to get new random color</div>
<p id="color"></p>
</h3>
<style>
body {
height: 100vh;
display: grid;
place-items: center;
font-family: verdana;
}
h3 {
background: white;
text-align: center;
}
p {
display: block;
font-size: 20px;
font-weight: lighter;
font-family: verdana;
}
</style>
<body>
<script>
function hex() {
const RanHexColor = Math.floor(Math.random()*16777215).toString(16);
document.body.style.backgroundColor = "#" + RanHexColor;
color.innerHTML = "#" + RanHexColor;
}
hex();
</script>
</body>
</html>
In the output, whenever you mouse over the box which is in middle it will generate the random hex colors on the background.
### Example 2
In this example, we have achieved the task of getting random hex colors.
The slice() method will extract a part of a string. This method accepts 2 parameters (start, end). The start could be the first character in the string(0) and the end is the position.
const mobile = 'oneplus';
document.write(mobile.slice(2));
// output: "eplus"
In the above snippet, as we passed only the start parameter it sliced the string from index 2 to the end of the string.
const mobile = 'oneplus';
document.write(mobile.slice(0,3));
// output: "one"
In this snippet, we have passed both the start and end parameters. So, it sliced the string from 0 to 3 and printed "one".
const mobile = 'oneplus';
document.write(mobile.slice());
// output: "oneplus"
Here, we have passed nothing into the parameter, and therefore by default, it will take the entire string length.
Instead of Math.floor() we have used slice() method to eliminate the decimal after the numbers.
Below is the example, which returns the random hex codes of color −
<!DOCTYPE html>
<html>
<title>Generating random Hex color</title>
<div>
<h3 id="demo">Mousemove here to get new random color</h3>
<p id="color"></p>
</div>
<style>
body {
height: 100vh;
display: grid;
place-items: center;
font-family: verdana;
}
div {
background: white;
text-align: center;
}
p {
display: block;
font-size: 20px;
font-weight: lighter;
font-family: verdana;
}
</style>
<body>
<script>
function hex() {
let RanHexCol = (Math.random()*16777215).toString(16);
document.body.style.backgroundColor = '#' + RanHexCol.slice(0, 6);
color.innerHTML = "#" + RanHexCol.slice(0, 6);
}
hex();
</script>
</body>
</html>
Whenever you hover the mouse on the h3 element in the output, we can see the hex colors are randomly changing in the background.
### Example 3
In the following example below, we have stored all the letters and numbers which will be a part of the hex color code in a variable. The loop will iterate 6 times as a hex color code consisting of 6 letters or digits and adds the value to '#'. Now, the random hex color code is generated.
<!DOCTYPE html>
<html>
<style>
body {
height: 100vh;
display: grid;
place-items: center;
font-family: verdana;
}
p {
display: block;
font-size: 20px;
font-weight: lighter;
font-family: verdana;
}
</style>
<body>
<p id = "demo"> </p>
<script>
// storing all letter and digit combinations
// for html color code
var codes = "0123456789ABCDEF";
// html color code starts with #
var color = '#';
// generating 6 times as HTML color code consist
// of 6 letter or digits
let i = 0;
for (i; i < 6; i++)
document.body.style.backgroundColor = color += codes[(Math.floor(Math.random() * 16))];
document.getElementById("demo").innerHTML = color;
</script>
</body>
</html>
In the output, the hex colors will be randomly changed in the background every time we execute the program.
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Books Books
If from the right angle of a right-angled triangle, two straight lines be drawn, one perpendicular to the base, and the other bisecting it, they will contain an angle equal to the difference of the two acute angles of the triangle.
An Elementary Treatise on the Geometrical and Algebraical Investigation of ... - Page 22
by Daniel Cresswell - 1817 - 436 pages
A Supplement to the Elements of Euclid
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A Supplement to the Elements of Euclid
Daniel Cresswell - Geometry - 1819 - 486 pages
...figure DCFE is aa ; .-. (E. 34. 1.) EF=DC; and (owwfr.) DC=B; .-. EF=B. ' PROP. LIII. A SUPPLEMENT TO THE hypotenuse, and the other bisecting it, they...difference of the two acute angles of the triangle. Let the LA, of the A BAC be a right L ; let A B ED C AE be drawn to the bisection E, of the hypotenuse...
A Supplement to the Elements of Euclid
Daniel Cresswell - Euclid's Elements - 1825 - 616 pages
...given finite straight line, not less than the perpendicular distance of the two parallels. LIII. TH. If, from the summit of the right angle of a scalene...difference of the two acute angles of the triangle. LIV. PR. To bisect a parallelogram by a straight line drawn through a given point in one of its sides....
Euclid's Elements of geometry [book 1-6, 11,12] with explanatory notes ...
Euclides - 1845 - 546 pages
...triangle two straight lines be drawn, one perpendicular to the base, and the other bisecting it, they will contain an angle equal to the difference of the two acute angles of the triangle. 10. If one angle at the base of a triangle be double of 'the other, the less side is equal to the sum...
The Elements of Euclid, the parts read in the University of Cambridge [book ...
Euclides - 1846 - 292 pages
...of a triangle two lines be drawn, one bisecting the base and the other perpendicular to it, they wUl contain an angle equal to the difference of the two acute angles of the triangle. 70. Find the point in the base of a triangle, from which lines drawn parallel to the sides to meet...
The London University Calendar
London univ - 1852 - 358 pages
...triangle two straight lines be drawn, one perpendicular to the base, and the other bisecting it, they will contain an angle equal to the difference of the two acute angles of the triangle. lineal angle ; and thence prove that the perimeter of an isosceles triangle is greater than that of...
Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson, with ...
Robert Potts - Geometry, Plane - 1860 - 380 pages
...triangle, two straight lines be drawn, one perpendicular to the base, and the other bisecting it, they will contain an angle equal to the difference of the two acute angles of the triangle. 34. If the vertical angle CAB of a triangle ABC be bisected by AD, to which the perpendiculars CIS,...
The school edition. Euclid's Elements of geometry, the first six books, by R ...
Euclides - 1864 - 448 pages
...triangle, two straight lines be drawn, one perpendicular to the base, and the other bisecting it, they will contain an angle equal to the difference of the two acute angles of the triangle. 34. If the vertical angle CAB of a triangle ABC be bisected by AD, to which the perpendiculars CE,...
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https://wiki-helper.com/surface-area-of-hemisphere-is-308cm-the-find-it-s-radius-39924289-7/
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# Surface area of hemisphere is 308cm² the find it’s radius
Surface area of hemisphere is 308cm² the find it’s radius
### 2 thoughts on “Surface area of hemisphere is 308cm² the find it’s radius<br />”
1. $$Total \: surface \: area \: of \: hemisphere = 3 \times \frac{22}{7} \times {r}^{2} \\ 308 = 3 \times \frac{22}{7} \times {r}^{2} \\ r = \sqrt{ \frac{308 \times 7}{3 \times 22} } \\ r = \sqrt{ \frac{2153}{66} } \\ r = 5.71149 \: cm$$
Step-by-step explanation:
surface area of hemisphere=2πr²
⇒308=2πr²
⇒πr²=154
⇒r²=49
⇒r=7
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http://www.slideshare.net/AlbertoPardoMilans/the-race-integers-maths-game
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# The Race, INTEGERS MATHS GAME
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### The Race, INTEGERS MATHS GAME
1. 1. Indice Rules Cards The Race Matem´ticas 1o E.S.O. a Alberto Pardo Milan´s e -
2. 2. Indice Rules Cards 1 Rules 2 CardsAlberto Pardo Milan´s e The Race
3. 3. Indice Rules Cards RulesAlberto Pardo Milan´s e The Race
4. 4. Indice Rules Cards Rules Rules NUMBERS OF PLAYERS MATERIAL Board, Dice, Coin, Cards, One Two or more game piece per player RULES OF THE GAME Alternating throws. Every body starts at the box marked “zero”. In every turn, the player throw the coin and the dice and move, to the right if the coin shows “head” and to the left if the coin shows “tails”, the number that are shown on the thrown dice decide the numbers of boxes you will move. When you step on a shadowed box, pick up a card. The winner is the one who first reach one of the two “finish”-boxes.Alberto Pardo Milan´s e The Race
5. 5. Indice Rules CardsRulesRules NUMBERS OF PLAYERS MATERIAL Board, Dice, Coin, Cards, One Two or more game piece per player RULES OF THE GAME Alternating throws. Every body starts at the box marked “zero”. In every turn, the player throw the coin and the dice and move, to the right if the coin shows “head” and to the left if the coin shows “tails”, the number that are shown on the thrown dice decide the numbers of boxes you will move. When you step on a shadowed box, pick up a card. The winner is the one who first reach one of the two “finish”-boxes.
6. 6. Indice Rules CardsRulesRules NUMBERS OF PLAYERS MATERIAL Board, Dice, Coin, Cards, One Two or more game piece per player RULES OF THE GAME Alternating throws. Every body starts at the box marked “zero”. In every turn, the player throw the coin and the dice and move, to the right if the coin shows “head” and to the left if the coin shows “tails”, the number that are shown on the thrown dice decide the numbers of boxes you will move. When you step on a shadowed box, pick up a card. The winner is the one who first reach one of the two “finish”-boxes.
7. 7. Indice Rules CardsRulesRules NUMBERS OF PLAYERS MATERIAL Board, Dice, Coin, Cards, One Two or more game piece per player RULES OF THE GAME Alternating throws. Every body starts at the box marked “zero”. In every turn, the player throw the coin and the dice and move, to the right if the coin shows “head” and to the left if the coin shows “tails”, the number that are shown on the thrown dice decide the numbers of boxes you will move. When you step on a shadowed box, pick up a card. The winner is the one who first reach one of the two “finish”-boxes.
8. 8. Indice Rules CardsRulesRules NUMBERS OF PLAYERS MATERIAL Board, Dice, Coin, Cards, One Two or more game piece per player RULES OF THE GAME Alternating throws. Every body starts at the box marked “zero”. In every turn, the player throw the coin and the dice and move, to the right if the coin shows “head” and to the left if the coin shows “tails”, the number that are shown on the thrown dice decide the numbers of boxes you will move. When you step on a shadowed box, pick up a card. The winner is the one who first reach one of the two “finish”-boxes.
9. 9. Indice Rules CardsRulesRules NUMBERS OF PLAYERS MATERIAL Board, Dice, Coin, Cards, One Two or more game piece per player RULES OF THE GAME Alternating throws. Every body starts at the box marked “zero”. In every turn, the player throw the coin and the dice and move, to the right if the coin shows “head” and to the left if the coin shows “tails”, the number that are shown on the thrown dice decide the numbers of boxes you will move. When you step on a shadowed box, pick up a card. The winner is the one who first reach one of the two “finish”-boxes.
10. 10. Indice Rules CardsRulesRules NUMBERS OF PLAYERS MATERIAL Board, Dice, Coin, Cards, One Two or more game piece per player RULES OF THE GAME Alternating throws. Every body starts at the box marked “zero”. In every turn, the player throw the coin and the dice and move, to the right if the coin shows “head” and to the left if the coin shows “tails”, the number that are shown on the thrown dice decide the numbers of boxes you will move. When you step on a shadowed box, pick up a card. The winner is the one who first reach one of the two “finish”-boxes.
11. 11. Indice Rules Cards Rules Rules NUMBERS OF PLAYERS MATERIAL Board, Dice, Coin, Cards, One Two or more game piece per player RULES OF THE GAME Alternating throws. Every body starts at the box marked “zero”. In every turn, the player throw the coin and the dice and move, to the right if the coin shows “head” and to the left if the coin shows “tails”, the number that are shown on the thrown dice decide the numbers of boxes you will move. When you step on a shadowed box, pick up a card. The winner is the one who first reach one of the two “finish”-boxes.Alberto Pardo Milan´s e The Race
12. 12. Indice Rules Cards CardsAlberto Pardo Milan´s e The Race
13. 13. Indice Rules Cards Cards Cards THROW AGAIN GO BACK TO START MOVE TO THE OPPOSITE MOVE TO THE OPPOSITE PLUS 5 MOVE TO THE OPPOSITE PLUS (-5) MOVE TO THE OPPOSITE MINUS 3 MOVE TO THE OPPOSITE MINUS (-2) DO NOT MOVE FOR TWO ROUNDSAlberto Pardo Milan´s e The Race
14. 14. Indice Rules Cards Cards Cards THROW AGAIN GO BACK TO START MOVE TO THE OPPOSITE MOVE TO THE OPPOSITE PLUS 5 MOVE TO THE OPPOSITE PLUS (-5) MOVE TO THE OPPOSITE MINUS 3 MOVE TO THE OPPOSITE MINUS (-2) DO NOT MOVE FOR TWO ROUNDSAlberto Pardo Milan´s e The Race
15. 15. Indice Rules Cards Cards Cards THROW AGAIN GO BACK TO START MOVE TO THE OPPOSITE MOVE TO THE OPPOSITE PLUS 5 MOVE TO THE OPPOSITE PLUS (-5) MOVE TO THE OPPOSITE MINUS 3 MOVE TO THE OPPOSITE MINUS (-2) DO NOT MOVE FOR TWO ROUNDSAlberto Pardo Milan´s e The Race
16. 16. Indice Rules Cards Cards Cards THROW AGAIN GO BACK TO START MOVE TO THE OPPOSITE MOVE TO THE OPPOSITE PLUS 5 MOVE TO THE OPPOSITE PLUS (-5) MOVE TO THE OPPOSITE MINUS 3 MOVE TO THE OPPOSITE MINUS (-2) DO NOT MOVE FOR TWO ROUNDSAlberto Pardo Milan´s e The Race
17. 17. Indice Rules Cards Cards Cards THROW AGAIN GO BACK TO START MOVE TO THE OPPOSITE MOVE TO THE OPPOSITE PLUS 5 MOVE TO THE OPPOSITE PLUS (-5) MOVE TO THE OPPOSITE MINUS 3 MOVE TO THE OPPOSITE MINUS (-2) DO NOT MOVE FOR TWO ROUNDSAlberto Pardo Milan´s e The Race
18. 18. Indice Rules Cards Cards Cards THROW AGAIN GO BACK TO START MOVE TO THE OPPOSITE MOVE TO THE OPPOSITE PLUS 5 MOVE TO THE OPPOSITE PLUS (-5) MOVE TO THE OPPOSITE MINUS 3 MOVE TO THE OPPOSITE MINUS (-2) DO NOT MOVE FOR TWO ROUNDSAlberto Pardo Milan´s e The Race
19. 19. Indice Rules Cards Cards Cards THROW AGAIN GO BACK TO START MOVE TO THE OPPOSITE MOVE TO THE OPPOSITE PLUS 5 MOVE TO THE OPPOSITE PLUS (-5) MOVE TO THE OPPOSITE MINUS 3 MOVE TO THE OPPOSITE MINUS (-2) DO NOT MOVE FOR TWO ROUNDSAlberto Pardo Milan´s e The Race
20. 20. Indice Rules Cards Cards Cards THROW AGAIN GO BACK TO START MOVE TO THE OPPOSITE MOVE TO THE OPPOSITE PLUS 5 MOVE TO THE OPPOSITE PLUS (-5) MOVE TO THE OPPOSITE MINUS 3 MOVE TO THE OPPOSITE MINUS (-2) DO NOT MOVE FOR TWO ROUNDSAlberto Pardo Milan´s e The Race
21. 21. Indice Rules Cards Cards Cards MOVE SIX BOXES FORWARD MOVE BACK TEN BOXES MOVE YOU GAMEPIECE 8 BOXES IN A POSITIVE DIRECTION MOVE YOUR GAMEPIECE 7 BOXES IN A NEGATIVE DIRECTION WAIT FOR ANOTHER PLAYER TO CATCH UP WITH YOU MOVE FORWARD 6 IN A POSITIVE DIRECTION MOVE FORWARD 7 IN A NEGATIVE DIRECTION GO TO THE OPPOSITE OF THE PREVIOUS PLAYERAlberto Pardo Milan´s e The Race
22. 22. Indice Rules Cards Cards Cards MOVE SIX BOXES FORWARD MOVE BACK TEN BOXES MOVE YOU GAMEPIECE 8 BOXES IN A POSITIVE DIRECTION MOVE YOUR GAMEPIECE 7 BOXES IN A NEGATIVE DIRECTION WAIT FOR ANOTHER PLAYER TO CATCH UP WITH YOU MOVE FORWARD 6 IN A POSITIVE DIRECTION MOVE FORWARD 7 IN A NEGATIVE DIRECTION GO TO THE OPPOSITE OF THE PREVIOUS PLAYERAlberto Pardo Milan´s e The Race
23. 23. Indice Rules Cards Cards Cards MOVE SIX BOXES FORWARD MOVE BACK TEN BOXES MOVE YOU GAMEPIECE 8 BOXES IN A POSITIVE DIRECTION MOVE YOUR GAMEPIECE 7 BOXES IN A NEGATIVE DIRECTION WAIT FOR ANOTHER PLAYER TO CATCH UP WITH YOU MOVE FORWARD 6 IN A POSITIVE DIRECTION MOVE FORWARD 7 IN A NEGATIVE DIRECTION GO TO THE OPPOSITE OF THE PREVIOUS PLAYERAlberto Pardo Milan´s e The Race
24. 24. Indice Rules Cards Cards Cards MOVE SIX BOXES FORWARD MOVE BACK TEN BOXES MOVE YOU GAMEPIECE 8 BOXES IN A POSITIVE DIRECTION MOVE YOUR GAMEPIECE 7 BOXES IN A NEGATIVE DIRECTION WAIT FOR ANOTHER PLAYER TO CATCH UP WITH YOU MOVE FORWARD 6 IN A POSITIVE DIRECTION MOVE FORWARD 7 IN A NEGATIVE DIRECTION GO TO THE OPPOSITE OF THE PREVIOUS PLAYERAlberto Pardo Milan´s e The Race
25. 25. Indice Rules Cards Cards Cards MOVE SIX BOXES FORWARD MOVE BACK TEN BOXES MOVE YOU GAMEPIECE 8 BOXES IN A POSITIVE DIRECTION MOVE YOUR GAMEPIECE 7 BOXES IN A NEGATIVE DIRECTION WAIT FOR ANOTHER PLAYER TO CATCH UP WITH YOU MOVE FORWARD 6 IN A POSITIVE DIRECTION MOVE FORWARD 7 IN A NEGATIVE DIRECTION GO TO THE OPPOSITE OF THE PREVIOUS PLAYERAlberto Pardo Milan´s e The Race
26. 26. Indice Rules Cards Cards Cards MOVE SIX BOXES FORWARD MOVE BACK TEN BOXES MOVE YOU GAMEPIECE 8 BOXES IN A POSITIVE DIRECTION MOVE YOUR GAMEPIECE 7 BOXES IN A NEGATIVE DIRECTION WAIT FOR ANOTHER PLAYER TO CATCH UP WITH YOU MOVE FORWARD 6 IN A POSITIVE DIRECTION MOVE FORWARD 7 IN A NEGATIVE DIRECTION GO TO THE OPPOSITE OF THE PREVIOUS PLAYERAlberto Pardo Milan´s e The Race
27. 27. Indice Rules Cards Cards Cards MOVE SIX BOXES FORWARD MOVE BACK TEN BOXES MOVE YOU GAMEPIECE 8 BOXES IN A POSITIVE DIRECTION MOVE YOUR GAMEPIECE 7 BOXES IN A NEGATIVE DIRECTION WAIT FOR ANOTHER PLAYER TO CATCH UP WITH YOU MOVE FORWARD 6 IN A POSITIVE DIRECTION MOVE FORWARD 7 IN A NEGATIVE DIRECTION GO TO THE OPPOSITE OF THE PREVIOUS PLAYERAlberto Pardo Milan´s e The Race
28. 28. Indice Rules Cards Cards Cards MOVE SIX BOXES FORWARD MOVE BACK TEN BOXES MOVE YOU GAMEPIECE 8 BOXES IN A POSITIVE DIRECTION MOVE YOUR GAMEPIECE 7 BOXES IN A NEGATIVE DIRECTION WAIT FOR ANOTHER PLAYER TO CATCH UP WITH YOU MOVE FORWARD 6 IN A POSITIVE DIRECTION MOVE FORWARD 7 IN A NEGATIVE DIRECTION GO TO THE OPPOSITE OF THE PREVIOUS PLAYERAlberto Pardo Milan´s e The Race
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http://www.infocobuild.com/education/audio-video-courses/computer-science/discrete-mathematical-structures-iit-madras.html
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# InfoCoBuild
## Discrete Mathematical Structures
Discrete Mathematical Structures. Instructor: Dr. Kamala Krithivasan, Department of Computer Science and Engineering, IIT Madras. This course deals with topics about discrete mathematical structures. Discrete Mathematics is a study of discrete structures which are abstract mathematical models dealing with discrete objects and their relationship between them. This course covers lessons in logic, sets, relations, functions, graphs, combinatorics, recurrence relations, algebras and finite state automaton. The aim of this course is not only make people learn about these topics, but also help them to develop the habit of thinking mathematically. (from nptel.ac.in)
Propositional Logic
Lecture 01 - Propositional Logic Lecture 02 - Propositional Logic (cont.) Lecture 03 - Predicates and Quantifiers Lecture 04 - Predicates and Quantifiers (cont.) Lecture 05 - Logical Inference Lecture 06 - Resolution Principles and Application to PROLOG Lecture 07 - Methods of Proof Lecture 08 - Normal Forms Lecture 09 - Proving Programs Correct Lecture 10 - Sets Lecture 11 - Induction Lecture 12 - Set Operations on Strings over an Alphabet Lecture 13 - Relations Lecture 14 - Graphs Lecture 15 - Graphs (cont.) Lecture 16 - Trees Lecture 17 - Trees and Graphs Lecture 18 - Special Properties of Relations Lecture 19 - Closure of Relations Lecture 20 - Closure Properties of Relations Lecture 21 - Order Relations Lecture 22 - Order Relations and Equivalence Relations Lecture 23 - Equivalence Relations and Partitions Lecture 24 - Functions Lecture 25 - Functions (cont.) Lecture 26 - Functions (cont.) Lecture 27 - Pigeonhole Principle Lecture 28 - Permutations and Combinations Lecture 29 - Permutations and Combinations (cont.) Lecture 30 - Generating Functions Lecture 31 - Generating Functions (cont.) Lecture 32 - Recurrence Relations Lecture 33 - Recurrence Relations (cont.) Lecture 34 - Recurrence Relations (cont.) Lecture 35 - Algebras Lecture 36 - Algebras (cont.) Lecture 37 - Algebras (cont.) Lecture 38 - Finite State Automaton Lecture 39 - Finite State Automaton (cont.) Lecture 40 - Lattices
References Discrete Mathematical Structures Instructor: Dr. Kamala Krithivasan, Department of Computer Science and Engineering, IIT Madras. This course deals with topics about discrete mathematical structures.
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https://ecoagi.ai/topics/Pandas/pandas-plot-histogram
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Topics
Pandas
Pandas Plot Histogram: Create and Customize Histograms in Python
# Pandas Plot Histogram: Create and Customize Histograms in Python
Data visualization is a crucial aspect of data analysis and Python's Pandas library is a powerful tool that allows us to create insightful visualizations. One such visualization is a histogram, a graphical representation of the distribution of a dataset. In this article, we will explore how to plot a histogram using pandas, customize bins, plot multiple columns, and much more. We will also address some frequently asked questions and provide examples to help you understand the process better.
Histograms are particularly useful when dealing with large datasets, as they can provide a visual summary of the data. They can help us understand the underlying frequency distribution of a set of continuous or discrete data. This can be particularly useful when dealing with data like age groups, where understanding the distribution can provide valuable insights.
Want to quickly create Data Visualization from Python Pandas Dataframe with No code?
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(opens in a new tab)
## Creating a Histogram in Pandas
Creating a histogram in pandas is straightforward thanks to the `hist()` function. This function provides a quick way to visualize the distribution of data in a pandas DataFrame or Series. Here's a basic example of how to create a histogram:
``````import pandas as pd
import matplotlib.pyplot as plt
# Create a simple dataframe
data = {'values': [1, 2, 2, 3, 3, 3, 4, 4, 4, 4]}
df = pd.DataFrame(data)
# Plot a histogram
df['values'].hist()
plt.show()``````
In this example, we first import the necessary libraries, pandas and matplotlib. We then create a simple pandas DataFrame and use the `hist()` function to plot a histogram of the 'values' column. The `plt.show()` function is used to display the plot.
## Customizing Bins in a Pandas Histogram
The `hist()` function in pandas uses a default number of bins, which is 10. However, you can customize the number of bins according to your needs. The bins parameter in the `hist()` function is used to specify the number of bins you want in your histogram.
For instance, if you want to increase the number of bins to 20, you can do so as follows:
``````df['values'].hist(bins=20)
plt.show()``````
Customizing bins in a pandas histogram can help you get a more detailed view of the data distribution. However, it's important to choose an appropriate number of bins. Too many bins may result in overfitting, where the histogram represents the data too closely and may miss the 'bigger picture'. On the other hand, too few bins may oversimplify the data, making it hard to discern any useful patterns.
## Plotting a Histogram with Multiple Columns in Pandas
Pandas also allows you to plot a histogram with multiple columns. This can be particularly useful when you want to compare the distribution of two different variables. To plot a histogram with multiple columns, you simply need to pass the columns to the `hist()` function.
Here's an example of how to plot a histogram with multiple columns:
``````# Create a dataframe with two columns
data = {'values1': [1, 2, 2, 3, 3, 3, 4, 4, 4, 4],
'values2': [1, 1, 2, 2, 3, 3, 3, 4,
4, 4]}
df = pd.DataFrame(data)
# Plot a histogram with multiple columns
df.hist(bins=20, alpha=0.5)
plt.show()``````
In this example, we create a DataFrame with two columns, 'values1' and 'values2'. We then call the `hist()` function on the DataFrame, which plots a histogram for each column. The `alpha` parameter is used to set the transparency of the histograms, which makes it easier to compare them.
## Plotting a Histogram by Group in Pandas
Another powerful feature of pandas is the ability to plot a histogram by group. This can be particularly useful when you want to compare the distribution of a variable across different groups.
For instance, let's say we have a DataFrame that contains the ages of people in different professions. We can plot a histogram of ages by profession as follows:
``````# Create a dataframe with age and profession
data = {'age': [23, 25, 22, 30, 32, 40, 35, 24, 28, 35],
'profession': ['engineer', 'doctor', 'engineer', 'doctor', 'engineer', 'doctor', 'engineer', 'doctor', 'engineer', 'doctor']}
df = pd.DataFrame(data)
# Plot a histogram by group
df.groupby('profession')['age'].hist(alpha=0.6)
plt.legend(['Engineer', 'Doctor'])
plt.show()``````
In this example, we first create a DataFrame with 'age' and 'profession' columns. We then group the DataFrame by 'profession' and call the `hist()` function on the 'age' column. This results in a histogram of ages for each profession. The `alpha` parameter is used to set the transparency of the histograms, and the `legend()` function is used to add a legend to the plot.
## Plotting a Normalized Histogram in Pandas
Sometimes, it's useful to plot a normalized histogram to represent the distribution of data as proportions rather than counts. This can be achieved in pandas by setting the `density` parameter to `True` in the `hist()` function.
Here's an example of how to plot a normalized histogram:
``````# Plot a normalized histogram
df['values1'].hist(density=True)
plt.show()``````
In this example, the `density=True` argument ensures that the area under the histogram sums up to 1, effectively giving us a probability density function.
## Creating Subplots with Pandas Histogram
Pandas also allows you to create subplots when plotting histograms. This can be particularly useful when you want to compare the distributions of multiple variables side by side. To create subplots, you can use the `subplots=True` argument in the `hist()` function.
Here's an example:
``````# Create subplots
df.hist(bins=20, alpha=0.5, subplots=True, layout=(1,2))
plt.show()``````
In this example, we create two subplots in a single row for the 'values1' and 'values2' columns. The `layout` parameter is used to specify the arrangement of the subplots.
## Adding Error Bars to a Pandas Histogram
Adding error bars to a histogram can provide a visual representation of the variability or uncertainty in the data. While pandas does not directly support adding error bars to histograms, this can be achieved using the `matplotlib` library.
Here's an example:
``````import numpy as np
# Calculate mean and standard deviation
mean = df['values1'].mean()
std = df['values1'].std()
# Plot histogram with error bars
plt.hist(df['values1'], bins=20, alpha=0.5)
plt.errorbar(mean, 5, xerr=std, fmt='o')
plt.show()``````
In this example, we first calculate the mean and standard deviation of the 'values1' column. We then plot the histogram and add an error bar at the mean position. The `errorbar()` function from `matplotlib` is used to add the error bar.
## Conclusion
Histograms are a powerful tool for data visualization, and the pandas library in Python provides a versatile function to create and customize histograms. Whether you're plotting a simple histogram, customizing bins, plotting multiple columns, or creating subplots, pandas has got you covered. Remember, the key to effective data visualization is not only creating insightful plots but also customizing them to suit your specific needs.
## FAQs
1. How can I customize the x-axis ticks in a pandas histogram? You can customize the x-axis ticks using the `xticks()` function from the `matplotlib` library. For example, `plt.xticks(range(0, 10))` will set the x-axis ticks to range from 0 to 10.
2. How can I plot a histogram with density in pandas? You can plot a histogram with density by setting the `density` parameter to `True` in the `hist()` function. This will plot a normalized histogram where the area under the histogram will sum up to 1.
3. How can I add a legend to my pandas histogram? You can add a legend to your pandas histogram using the `legend()` function from the `matplotlib` library. For example, `plt.legend(['Column1', 'Column2'])` will add a legend with 'Column1' and 'Column2'.
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http://aimsciences.org/search/author?author=Ken%20Palmer
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# American Institute of Mathematical Sciences
## Journals
CPAA
In this note we study Sil'nikov saddle-focus homoclinic orbits paying particular attention to four and higher dimensions where two additional conditions are needed. We give equivalent conditions in terms of subspaces associated with the variational equation along the orbit. Then we review Rodriguez's construction of a three-dimensional system with Sil'nikov saddle-focus homoclinic orbits and finally show how to construct higher-dimensional systems with such orbits.
keywords:
DCDS-B
We consider a singularly perturbed system with two normally hyperbolic centre manifolds. We derive one bifurcation function, the zeros of which correspond to heteroclinic connections near such a connection for the unperturbed system, and a second bifurcation function the zeros of which correspond to the vectors in the intersection of the tangent spaces to the centre-unstable and centre-stable manifolds along the heteroclinic connections.
keywords:
DCDS-B
We consider a singularly perturbed system with a normally hyperbolic centre manifold. Assuming the existence of a fast homoclinic orbit to a point of the centre manifold belonging to a hyperbolic periodic solution for the slow system, we prove an old and a new result concerning the existence of solutions of the singularly perturbed system that are homoclinic to a periodic solution of the system on the centre manifold. We also give examples in dimensions greater than three of Sil'nikov [16] periodic-to-periodic homoclinic orbits.
keywords:
DCDS-B
Isagi et al introduced a model for masting, that is, the intermittent production of flowers and fruit by trees. A tree produces flowers and fruit only when the stored energy exceeds a certain threshold value. If flowers and fruit are not produced, the stored energy increases by a certain fixed amount; if flowers and fruit are produced, the energy is depleted by an amount proportional to the excess stored energy. Thus a one-dimensional model is derived for the amount of stored energy. When the ratio of the amount of energy used for flowering and fruit production in a reproductive year to the excess amount of stored energy before that year is small, the stored energy approaches a constant value as time passes. However when this ratio is large, the amount of stored energy varies unpredictably and as the ratio increases the range of possible values for the stored energy increases also. In this article we describe this chaotic behavior precisely with complete proofs.
keywords:
DCDS-B
A continuous map $f:[0,1]\rightarrow[0,1]$ is called an $n$-modal map if there is a partition $0=z_0 < z_1 < ... < z_n=1$ such that $f(z_{2i})=0$, $f(z_{2i+1})=1$ and, $f$ is (not necessarily strictly) monotone on each $[z_{i},z_{i+1}]$. It is well-known that such a map is topologically semi-conjugate to a piecewise linear map; however here we prove that the topological semi-conjugacy is unique for this class of maps; also our proof is constructive and yields a sequence of easily computable piecewise linear maps which converges uniformly to the semi-conjugacy. We also give equivalent conditions for the semi-conjugacy to be a conjugacy as in Parry's theorem. Related work was done by Fotiades and Boudourides and Banks, Dragan and Jones, who however only considered cases where a conjugacy exists. Banks, Dragan and Jones gave an algorithm to construct the conjugacy map but only for one-hump maps.
keywords:
DCDS
In this paper we study unimodal maps on the closed unit interval, which have a stable period 3 orbit and an unstable period 3 orbit, and give conditions under which all points in the open unit interval are either asymptotic to the stable period 3 orbit or land after a finite time on an invariant Cantor set $\Lambda$ on which the dynamics is conjugate to a subshift of finite type and is, in fact, chaotic. For the particular value of $\mu=3.839$, Devaney [3], following ideas of Smale and Williams, shows that the logistic map $f(x)=\mu x(1-x)$ has this property. In this case the stable and unstable period 3 orbits appear when $\mu=\mu_0=1+\sqrt{8}$. We use our theorem to show that the property holds for all values of $\mu>\mu_0$ for which the stable period 3 orbit persists.
keywords:
DCDS-B
keywords:
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https://physics-network.org/what-does-rotating-mean-in-physics/
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# What Does rotating mean in physics?
The circular velocity vC = (μ/r)1/2, is the speed of a body in a circular orbit at a distance a = r from the primary.
## What is an example of rotation in physics?
Circular Orbit The orbit can be expressed in terms of the acceleration of gravity at the orbit. The force of gravity in keeping an object in circular motion is an example of centripetal force. Since it acts always perpendicular to the motion, gravity does not do work on the orbiting object if it is in a circular orbit.
## What is rotational motion simple?
If all the particles of a rigid body perform circular motion and the centres of these circles are steady on a definite straight line called axis of rotation, then the motion of the rigid body is called the rotational motion.In simple words, we can say that rotational motion is defined as the motion of an object in a …
## What is a rotation answer?
A rotation is a circular movement of an object around a centre of rotation. If three-dimensional objects like the earth, moon and other planets always rotate around an imaginary line, it is called a rotation axis. If the axis passes through the body’s centre of mass, the body is said to rotate upon itself or spin.
## What is an example of rotation movement?
Rotational Movement Rotation can be toward the midline of the body, which is referred to as medial rotation, or away from the midline of the body, which is referred to as lateral rotation. Movement of the head from side to side is an example of rotation.
## What is rotational motion and example?
Solution: Rotatory motion: if the body moves about a fixed axis without changing the radius of its motion, it is said to be rotatory motion. Examples: a spinning wheel.
## What is rotational force?
A rotational force, also known as a torque, depends upon the force and where that force is applied; torque = lever arm x force. The lever arm is the perpendicular distance from the force to the axis of rotation. A “lever arm” is also known as a “moment arm”.
## What is the importance of rotational motion in physics?
As pointed out in the previous chapter, rotational motion is also extremely important in mechanical devices. In every case, the rotation of an extended, rigid body can be mathematically described as a collection of circular motions by the particles making up the body.
## What is called rotation?
Rotation describes the circular motion of an object around its center. There are different ways things can rotate. Rotation of the Earthre. A very familiar kind of rotation is when a spherical, three-dimensional object turns around an invisible line inside its center.
## How does Earth rotate?
Earth’s rotation or Earth’s spin is the rotation of planet Earth around its own axis, as well as changes in the orientation of the rotation axis in space. Earth rotates eastward, in prograde motion. As viewed from the northern polar star Polaris, Earth turns counterclockwise.
## What is rotation and reflection?
Rotation means the shape turns as it moves around a fixed point. Shapes can be rotated clockwise or anticlockwise by a certain number of degrees (90 degrees would be a quarter turn, for example). Reflection means the shape has a mirror image on the other side of the mirror line.
## What is a single rotation?
A single rotation applied when a node is inserted in the right subtree of a right subtree. In the given example, node A has a balance factor of 2 after the insertion of node C. By rotating the tree left, node B becomes the root resulting in a balanced tree.
## What is a real life example of rotation?
Clock Ticking. The minute, hour, and second hand of a clock rotate in a circular direction, keeping the pinpoint as the centre or the axis of rotation. Hence, the hands of a clock are said to be exhibiting rotatory motion.
## What are the types of rotational motion?
• Rotation about a fixed axis (Pure rotation)
• Rotation about an axis of rotation (Combined translational and rotational motion)
• Rotation about an axis in the rotation (rotating axis – out of the scope of JEE)
## What is pure rotational motion?
1. The pure rotational motion: The rigid body in such a motion rotates about a fixed axis that is perpendicular to a fixed plane. In other words, the axis is fixed and does not move or change its direction relative to an inertial frame of reference. 2.
## How do you find the rotational motion?
τ = r F sin θ , τ = r F sin θ , where r is the magnitude of the lever arm, F is the magnitude of the linear force, and θ θ is the angle between the lever arm and the force. The lever arm is the vector from the point of rotation (pivot point or fulcrum) to the location where force is applied.
## What is rotational effect?
The rotational effects arising from the Coriolis acceleration have opposite signs in the two hemispheres. Note that all figures and descriptions given here are valid for the northern hemisphere. In some cases the sense of the rotational effect for the southern hemisphere has been explicitly mentioned.
## What is a rotatory motion give two example?
A body is said to be in rotatory motion if it moves about a fixed axis. A spinning wheel is an example of rotatory motion. It moves in a circular path about a fixed axis. Similarly, the rotating fan is another example of rotatory motion.
## What is rotational motion explain with diagram?
Rotational motion can be defined as a motion of an object around a circular path, in a fixed orbit. It can also be defined as the motion of a body, in which all of its particles move in a circular motion with a common angular velocity, about a fixed point—for example, the rotation of Earth about its axis.
## Are all circular motion rotational?
Circular motion is a subset of rotational motion, which means that each constituent of a rotating rigid body rotates in a circle around a fixed axis of rotation. To put it another way, each particle on a rigid body moves in a circular motion. The object in a circular motion just moves in a circle.
## Is every rotational motion a circular motion?
Circular motion is a subset of rotational motion, means in a rotating rigid body, each constituent of the rigid body perform circular motion about a fixed axis of rotation. in another word, each particle on rigid body perform uniform circular motion.
## Is rotation circular motion?
The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body.
## Is rotational motion relative?
Rotation was thus concluded to be absolute rather than relative.
## Is torque a energy?
If you think torque is measured in Joules, you might get confused and think it is energy, but it is not energy. It is a rotational analogy of a force.
## What are the characteristics of rotational motion?
Rotation and Its Characteristics In rotational motion, we know that the particles of the object, while moving follow a circular path. Every particle in the rigid body moves in a circular path along a plane that is perpendicular to the axis and has its center on the same axis.
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# Invert Binary Tree
Companies:
Invert a binary tree.
#### Example :
Input:
`````` 7
/ \
9 5
/ \ / \
1 10 3 6
``````
Output:
`````` 7
/ \
5 9
/ \ / \
6 3 10 1
``````
### Solution:
This problem is best-suited for recursive approach.
Here, we see that inverse of a left node is the right node and inverse of left node of the previous left node is the right node of the previous right node and so on.
This suggests of some resursive calls untill we reach at the end of the tree.
### Java:
``````/**
* Definition for a binary tree node.
* public class TreeNode {
* int val;
* TreeNode left;
* TreeNode right;
* TreeNode() {}
* TreeNode(int val) { this.val = val; }
* TreeNode(int val, TreeNode left, TreeNode right) {
* this.val = val;
* this.left = left;
* this.right = right;
* }
* }
*/
class Solution {
public TreeNode invertTree(TreeNode root) {
if (root == null) {
return null;
}
TreeNode right = invertTree(root.right);
TreeNode left = invertTree(root.left);
root.left = right;
root.right = left;
return root;
}
}
``````
### Complexity Analysis:
• Time Complexity: O(n)
• Space Complexity: O(n).
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# Math Word Problem Solving
Instead of dismissing the context of word problems, teachers should take time with students to make sense of word problems and their supporting context. Teachers should push back against students' compulsion to calculate by focusing on the relationship between the knowns and unknowns in word problems, and not rush to find an answer (Kieran, 2014). By situating mathematics in contexts that are understandable for students, word problems encourage students to pursue solution strategies that make sense to them and lead more often to correct answers (Koedinger & Nathan, 2004).
Tags: Financial Plan Example For Small BusinessesArgumentative Essay On Death PenaltyPhd By Research/Thesis OnlyDissertation Printing LondonEffectiveness Of Penalty Research PaperSchool Counseling Thesis PapersGood Thesis Statement Book 1984Quotes On HomeworkMaster Plan BusinessDissertations On Barriers To Inclusive Education
Math teachers are often concerned about students' abilities to transfer classroom learning into the world beyond the classroom, but this "suspension of sense-making" shows that the reverse is also difficult – students struggle to apply their knowledge and understanding of the world back into a mathematics classroom.
Having been conditioned with years of arithmetic, almost always involving obvious operations and the expectation that each problem has a correct answer, students develop a "compulsion to calculate" (Stacey & Mac Gregor, 1999) that can interfere with the development of the algebraic thinking that is usually needed to solve word problems.
There is no reason that this should end in early childhood.
Students at all levels should engage in mathematics in a sensible context before it is made formal and symbolic. Segal (Eds.), Informal reasoning and education (pp.
Some types of word problems might be particularly useful for promoting reasoning because they either lack an obvious strategy, don't have one right answer, or could be "tricky" for students who assume the problem is straightforward.
Some examples: Verschaffel, De Corte, and Lasure (1994) used these word problems to see if students would reason differently with the odd- and even-numbered items.
Word problems are not just for applications of already-known mathematics. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. Reston, VA: National Council of Teachers of Mathematics.
In fact, the most powerful way to use word problems in the classroom is as a means to help students learn math. Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics.
Brian Bushart, an elementary teacher and mathematics curriculum coordinator from Texas, popularized the idea of "numberless word problems" after a colleague tried the approach with some third-grade students. Teachers’ and researchers’ beliefs about the development of algebraic reasoning.
Numberless word problems aren’t entirely new, as the book (Gillan, 1909) presented something vaguely similar in the early 20th century. Journal for Research in Mathematics Education, 31(2), 168–190.
## Comments Math Word Problem Solving
• ###### Math Word Problems Math Playground
Math Playground has hundreds of interactive math word problems for kids in grades 1-6. Solve problems with Thinking Blocks, Jake and Astro, IQ and more.…
• ###### Understanding Word Problems in Mathematics LD Topics.
Indeed, as students move forward in their mathematical learning, they will need to apply problem-solving processes to more and more complex situations so.…
• ###### Word Problems Calculator - MathCelebrity
Word Problems Calculators 38 lessons. If you cannot find what you need, post your word problem in our calculator forum. + 2 number Word Problems.…
• ###### Word Problems - Research and Practice Guide CDE
Jan 31, 2018. Solving word problems is not considered to be the same as mathematical modeling. Mathematical modeling tends to be a more complex.…
• ###### Two-step equation word problem garden video Khan.
Here's a nifty word problem in which we find the dimensions of a garden given only the perimeter. Let's create an equation to solve. CCSS Math 7B.4, 7.…
• ###### Awesome Word Problems To Engage Students - Prodigy
Jul 6, 2017. Looking for examples of math word problems, plus tips to make your. problem-solving element, overwhelming many elementary students.…
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http://www.codeproject.com/Articles/626336/Linear-Rigid-Body-Dynamics-with-OpenGL
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# Linear Rigid Body Dynamics with OpenGL
, 25 Jul 2013 CPOL
Rate this:
Simulating displacement, velocity, and acceleration in code.
## Introduction
To animate 3-D models realistically, it's important to understand the principles of dynamics, which describe the motion of objects in the real world. Computational physics has become commonplace in games and simulation tools, and this article explains how to combine linear dynamics with OpenGL rendering.
Scientists and engineers have understood the laws of dynamics since the age of Newton, and programmers have become increasingly interested in using these principles to animate objects in 3-D environments. If you search on the web, you'll find plenty of material on the topic. In particular, I recommend Chris Hecker's articles and the lecture notes written by Andrew Witkin, David Baraff, and Michael Kass.
Many sources discuss the theory behind dynamics and some show how the equations can be implemented in code. But this article goes further and shows how rigid body dynamics can be implemented in a real-world application based on OpenGL. This application, whose code can be obtained here, assigns a velocity and acceleration to a sphere, and then renders the path of the sphere as it moves through space. Figure 1 shows what this looks like on my Linux system.
##### Figure 1: A Sphere's Motion Subject to Linear Dynamics
This may not look particularly impressive, but the methodology behind the code can be extended to more complex circumstances. But before I can discuss the code, I need to present the theory underlying linear dynamics. The first section discusses this in detail.
NOTE: This article focuses solely on linear dynamics, and does not discuss rotational dynamics. In addition, this presentation deals with difference equations only, not differential or integral equations.
## 1. The Basics of Linear Dynamics
In many video games, objects always seem to move at the same speed. But in the real world, the speed of an object changes over time, particularly under the influence of gravity. To explain how this works, this section discusses position and velocity first, and then discusses the important concepts of acceleration and force.
### 1.1 Position and Velocity
If you're familiar with graphical modeling, you know that the position of a point is given as a set of coordinates that identify a location relative to another point called the origin. In three-dimensional Euclidean space, coordinates are given in (x, y, z) triples, where x identifies the distance from the origin along the x-axis, y identifies the distance from the origin along the y-axis, and z identifies the distance from the origin along the z-axis.
It's important to distinguish distance from position. If a point's position, denoted r, is given as (x, y, z), it's distance from the origin, denoted |r|, equals sqrt(x2 + y2 + z2). Distance identifies how far away a point is, and because it has a single value, it's called a scalar. In contrast, position identifies both how far away a point is and its direction. A quantity that identifies magnitude and direction is called a vector.
When you create animated designs, it's not enough to have a point's position. You have to know how its position changes from one frame to the next. We'll refer to the time interval between rendering frames as Δt, which is measured in seconds. For example, if an application's frame rate is 120 fps (frames per second), the time interval Δt = 1/120 = 0.008333 seconds.
The change in position over time is referred to as velocity. If a point's position changes by (Δx, Δy, Δz) over the time interval Δt, the average velocity over the interval is expressed as (Δx/Δt, Δy/Δt, Δz/Δt). To simplify the notation, we'll refer to the average velocity vector as v and its components will be given as vx = Δx/Δt, vy = Δy/Δt, and vz = Δz/Δt.
Note: This discussion deals exclusively with the average velocity over a time interval instead of the instantaneous velocity at a specific time. This is a big deal for physicists but not for programmers. Therefore, I'm going to stop using the phrase "average velocity over an interval."
In physics, velocity is expressed in meters per second. That is, if a point's velocity is (5, 5, 0), its position changes five meters in the +x direction and five meters in the +y direction every second. Meters aren't commonly used by programmers, so we'll refer to velocity in terms of generic units per second.
Suppose you know a point's velocity but you don't know how it's position changes from one frame to the next. In this situation, the following equation becomes helpful:
This states that, to find the new position, add the old position to the product of the velocity and time interval. This should make sense. If you drive at a speed of 64 mph (103 kph) for half an hour, you will travel a distance of 64 × 0.5 = 32 miles or 103 × 0.5 = 51.5 kilometers.
Figure 2 presents another example. The point's initial location is y0 = 0.5 and it travels upward at velocity vy = 1. Each time interval is 0.75s, so the first updated location, y1 = 0.5 + 1(0.75) = 1.25. The second location, y2 = 1.25 + 1(0.75) = 2. Each location exceeds the previous location by 0.75.
##### Figure 2: A Point's Trajectory with Constant Velocity
The path taken by a point over time is called its trajectory, and if the velocity is constant, the trajectory will be a straight line. The next discussion will explore what happens when an object's velocity changes over time.
### 1.2. Acceleration and Force
Just as velocity measures the change in position over time, acceleration measures the change in velocity over time. Like position and velocity, acceleration is a vector. The following equation shows how it relates to velocity:
An important question arises. We know how to update a point's position given its velocity and we know how to update the velocity given the acceleration. But how do we update a point's position given its initial velocity and acceleration? That is, if we know ri, vi, and a, how do we obtain ri+1?
To obtain the answer, remember that the velocity changes from vi to vi+1 over the course of the time interval Δt. The average velocity over the interval can be computed as follows:
With this expression, we can determine the updated position as follows:
Now let's look at the trajectory of a point with constant acceleration. Suppose a point has r0 = 0.5, v0 = 2 and a = -1. After the first time interval, the updated position, r1, equals 0.5 + 2(0.75) + (0.5)(-1)(0.75)2 = 1.72. The updated velocity, v1, equals 2 + (-1)(0.75) = 1.25. After the second time interval, r2 = 1.72 + (1.25)(0.75) + (0.5)(-1)(0.75)2 = 2.375 and v2 = 1.25 + (-1)(0.75) = 0.5.
By keeping track of the cumulative time interval, we can obtain the position at each step without computing the velocity. For example, we can proceed from r0 to r2 by setting the time interval in the equation to 2Δt. That is, r2 = r0 + v0(2Δt) + (0.5)(a)(2Δt)2 = 2.375. Figure 3 shows the trajectory of this point from Δt to 6Δt.
##### Figure 3: A Point's Trajectory with Constant Acceleration
As shown in the figure, the trajectory of a point with initial velocity and constant acceleration is a parabola. This should make sense, as the equation for position is quadratic with respect to time and the graph of a quadratic equation forms a parabola.
Programmers and engineers generally limit their interest to position, velocity, and acceleration, and don't take into account the change in acceleration over time. This is because force plays a very large role in physics —from friction to gravity to electromagnetism, the phenomena that affect physical motion are typically quantified using forces. When a force acts on an object and causes it to move, the object's acceleration equals the force divided by the object's mass. In equation form, F = ma, where F is the force in Newtons, m is the mass in kilograms, and a is the acceleration in meters/second2.
At this point, you should have a clear understanding of the relationships between position, velocity, and acceleration. These are all given as vectors, and the next section will show how they can be used to constrain the motion of an object in an OpenGL rendering.
## 2. Linear Dynamics and OpenGL
Modern OpenGL applications store vertex positions in structures called vertex buffer objects, or VBOs. As the rendering process proceeds, a vertex shader processes the elements of the VBO to determine the final position of the vertex. For this reason, we'll rely on the vertex shader to update the VBO data with the result of physical computation.
In particular, the goal of this discussion is to present an OpenGL application that animates a sphere so that it follows the trajectory depicted in Figure 1. Like the point in Figure 2, this sphere has an upward initial velocity but downward acceleration.
To update the positions of the sphere's vertices, three pieces of information must be processed: the elapsed time, the velocity, and the acceleration. In this section, I'll discuss how the application obtains the time and then proceed to show how the velocity, acceleration, and updated position are computed.
### 2.1 Obtaining Elapsed Time
To determine how vertex positions should be updated, the first step is to obtain the elapsed time. Different windowing systems and operating systems provide different timing functions, but the code in this example application is based on the OpenGL Utility Toolkit, or GLUT. The example application relies on two functions to obtain timing information:
1. `glutIdleFunc(void (*func)(void))` - identifies the function to be called when the application is idle (no events are being received or processed)
2. `glutGet(GLUT_ELAPSED_TIME)` - returns the number of milliseconds that have passed since `glutInit` was called or the first call to `glutGet(GLUT_ELAPSED_TIME)`
To configure timing in code, the main function in the animate_sphere.cpp file invokes the following lines of code:
```glutIdleFunc(update_vertices);
start_time = glutGet(GLUT_ELAPSED_TIME);```
The first line states that the `update_vertices` function should be called when the application is idle. The second line computes `start_time`, which sets the starting time in milliseconds.
The `update_vertices` function obtains the current time and uses it to compute the elapsed time. The following code shows how this works:
```void update_vertices() {
current_time = glutGet(GLUT_ELAPSED_TIME);
delta_t = (current_time - start_time)/1000.0f;
...
} ```
The `delta_t` variable contains the elapsed time that will be used in our physics equations. Note that this isn't the time between frames, but the total time that has passed since the application started.
### 2.2 Processing Velocity and Acceleration
After determining the timing data, the application computes the change in position brought about by the figure's velocity and acceleration. In animate_sphere.cpp, the dynamic parameters are initialized with the following code:
```#define INIT_POSITION 0.5f
#define INIT_VELOCITY 0.8f
#define ACCELERATION -0.4f
...
glm::vec3 init_position = glm::vec3(0.0f, INIT_POSITION, 0.0f);
glm::vec3 init_velocity = glm::vec3(INIT_VELOCITY, INIT_VELOCITY, 0.0f);
glm::vec3 acceleration = glm::vec3(0.0f, ACCELERATION, 0.0f);```
As shown, r0 is initialized to (0.0, 0.5, 0.0), v0 is initialized to (0.8, 0.8, 0.0), and a is set to (0.0, -0.4, 0.0). Each is represented by a `glm::vec3` data type, which is provided by the OpenGL Math Library.
The update_vertices function computes the updated position vector with the following code:
`delta_r = init_position + delta_t * init_velocity + 0.5f * delta_t * delta_t * acceleration;`
This is the same equation that we derived earlier, and should look familiar. The result identifies how each vertex position should be updated, and it can be used in one of at least two ways. First, `delta_r` can be added to the elements of the VBO containing vertex locations. Second, `delta_r` can be sent as a uniform to the vertex shader, which will add `delta_r` to each vertex's final location. In OpenGL, a uniform is a quantity that doesn't change from vertex to vertex.
The animate_sphere application adopts the second method, and the following code sets the uniform's data equal to `delta_r`:
`glUniform3fv(delta_location, 1, &(delta_r[0]));`
When the vertex shader executes, it adds the updated position to each vertex in the figure. You can see this by examining the vertex shader (animate_sphere.vert), which contains the following line of code:
`gl_Position = mvp * vec4(in_coords + delta, 1.0);`
This shows that the vertex's position is set equal to the original VBO position (`in_coords`) plus the updated position (`delta`) transformed by the modelview-projection matrix (`mvp`). As the shader executes, it updates the vertex locations and the result is presented way back in Figure 1.
## 3. Conclusion
If you want objects in a graphical model to move realistically, you need to have a basic understanding of dynamics. This article has discussed the theory of linear dynamics for rigid bodies, and has focused on the relationships between position, velocity, and acceleration. The second part of this article discussed how this theory can be implemented in code. In particular, the animate_sphere application sets a sphere's initial velocity and acceleration, computes the dynamic equations in C, and uses OpenGL to render its motion through space.
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## 3x-(5-2x)=4(3x-4) giúp mik vote 5 sao
Question
3x-(5-2x)=4(3x-4) giúp mik vote 5 sao
in progress 0
2 years 2021-05-07T23:39:03+00:00 2 Answers 10 views 0
1. Đáp án:
x=11/7
Giải thích các bước giải:
3x-(5-2x)=4(3x-4)
=>3x+2x-5=12x-16
=>x5-5=12x-16
=>x5=12x-16+5
=>x5=12x-11
=>-7x=-11
=>x=-11÷(-7)
=>x=11/7
2. $\text{ 3x -( 5 – 2x )=4( 3x – 4 )}$
$\text{ <=> 3x – 5 + 2x = 12x – 16 }$
$\text{ <=> 3x + 2x – 12x = -16 + 5 }$
$\text{ <=> -7x = -11}$
$\text{ <=> x =$\dfrac{11}{7}$}$
$\text{ Vậy x =$\dfrac{11}{7}$. }$
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Hockey
# What do plus and minus mean in hockey?
A player is awarded a “plus” each time he is on the ice when his Club scores an even-strength or shorthanded goal. He receives a “minus” if he is on the ice for an even-strength or shorthanded goal scored by the opposing Club. The difference in these numbers is considered the player’s “plus-minus” statistic.
Subsequently, is +/- a good stat in hockey? Some hockey people put a lot of stock into a player’s plus/minus number. “When you consider that approximately 70 to 75 percent of the game is played at even-strength — and the plus/minus is (more of) an even-strength stat — you have to consider it a significant statistic,” said Flames general manager Craig Button.
Also the question is, what does +2 mean in hockey? Plus: 2 (the 2 assists at even strength count as pluses) Minus: -3 (the two goals at even strength and the empty net goal count as minuses, while the shorthanded goal does not)
Moreover, do short handed goals count against plus-minus? Plus-Minus A player is awarded a “plus” each time he is on the ice when his Club scores an even-strength or shorthanded goal. He receives a “minus” if he is on the ice for an even-strength or shorthanded goal scored by the opposing Club. The difference in these numbers is considered the player’s “plus-minus” statistic.
Similarly, why is plus-minus important? Plus/minus captures the impact of great defense, selfless offense and every other hidden contribution that can change the course of a game. So what’s the issue? Well, a player’s plus/minus score bounces around a lot from night to night, so you can’t use it to evaluate a guy after just a few appearances.Fun fact: Desjardins and forward Melvin Angelstad (two games with the Washington Capitals in 2003-04) are the only players in NHL history to wear No. 69.
Contents
## What does P mean in hockey?
P or PTS – Points – Scoring points, calculated as the sum of G and A. S – Shots on Goal – Total number of shots taken on net in the current season. PN – Penalties – Number of penalties the player has been assessed.
## Who wears 47 in the NHL?
47 – Alexander Radulov. Alexander Radulov had a couple of solid seasons with the Nashville Predators in 2006-07 and 2007-08 before returning to Russia. Since his return to the NHL in 2016-17 he has been a very good player for Montreal and Dallas.
## What is meant by plus-minus?
Definition of plus/minus sign : the sign ± used to indicate a quantity (such as 2 in “the square root of 4 is ±2”) taking on both an algebraically positive value and its negative and to indicate a plus or minus quantity (such as 4 in “the population age was 30 ± 4 years”) — called also plus/minus symbol.
## How do you calculate plus-minus?
The plus–minus rating is calculated by dividing the number of skaters on the ice for the team scored upon by the number of skaters on the ice for the scoring team, applied as a plus to all players (including goaltenders) on the ice for the scoring team and as a minus for all players (including goaltenders) on the ice …
## Who has the best plus-minus in NHL?
Larry Robinson has the highest career plus-minus, at +722.
## What is hockey player rating?
The rating captures all contributions made by skaters – offensive, defensive, even strength or special teams! All player ratings are updated daily to reflect the latest information. For a detailed explanation of our Ratings process, click here. All. Skaters.
## Can you wear 0 in NHL?
Beginning with the 1996-97 season, the NHL decreed that Nos. 0 and 00 could no longer be worn since they confused the League’s digital database; today, only No. 1 through No. 98 are allowed, No.
## Is 99 retired in the NHL?
Wayne Gretzky’s number 99 was retired league-wide in 2000; Gretzky’s former teams the Edmonton Oilers and Los Angeles Kings also separately retired his number. As of March 2022, two teams have no retired numbers: the San Jose Sharks and Winnipeg Jets.
## Who wears 79 in the NHL?
The number 79 has only been worn by six players in NHL history, and the only other household name to wear it is Andrei Markov.
## What is P3 in hockey?
P3 Sports offers a Hockey Academy focused on building leadership, physical literacy development, and honing skills to maximize performance. Our Hockey Academy works in conjunction with several Rocky View County schools. This allows our athletes to keep up with their friends, but also continue to finetune their skills.
## What is SA in hockey?
SA Shots Against SH Short-handed goals SHA Short-handed goals against SO Shutouts SOS. Strength of Schedule; a rating of strength of schedule. The rating is denominated in goals above/below average, where zero is average.
## Who wears 86 in the NHL?
Jack Hughes, the number one overall draft pick for the New Jersey Devils, has chosen number 86 for his NHL number. Hughes began his career with the US National Team Development Program wearing the number 6, which had been a family number.
## Do hockey numbers mean anything?
Numbers 2-6 were traditionally for defense, and 7-11 traditionally for forwards. Higher numbers were for players further down the depth chart, with one of the highest numbers (often 20, 29 or 30) for the backup goaltender.
## Who wears #9 in the NHL?
9 because of all the great players who have worn it in NHL history, from legends of the past like Gordie Howe, Maurice Richard and Bobby Hull, to stars of the more recent past like Glenn Anderson, Mike Modano and Paul Kariya.
## What does a plus and a minus make?
A plus and a minus make a minus.
## What is 3PM NBA?
3PM. Name 3 Point Field Goals Made Definition The number of 3 point field goals that a player or team has made Type Traditional Contexts Box Score Clutch Shooting Defensive Dashboard Player.
## What is LeBron plus-minus?
LeBron James’ plus-minus ratio of minus-32 is the worst of ANY GAME in his NBA career (899 games). Scottie.
We publish these as Advanced Box Score Stats and you can read about these component calculations in our glossary. Note that there is a separate calculation for the offensive component of a player’s BPM, which yields both OBPM (Offensive Box Plus/Minus) and DBPM (Defensive Box Plus/Minus).
## Do power play goals count for plus-minus?
Some goals scored do not count in a player’s plus-minus rating. If your team scores a power play goal while you’re on the ice, you do not get a plus.
## Why is icing illegal in hockey?
In ice hockey, icing is an infraction when a player shoots the puck over the center red line and the opposing team’s red goal line, in that order, and the puck remains untouched without scoring a goal.
## What was Gretzky plus-minus?
Over a player’s career, this plus-minus number is generally accepted to be a useful indicator of his two-way prowess. The top five all-time are Hall-of-Famers Larry Robinson, +730, Bobby Orr, +597, Raymond Bourque, +528, Wayne Gretzky, +518 and Bobby Clarke, +506.
## What is Corsica hockey?
Corsica Hockey is a provider of statistics, predictions and betting resources for the informed hockey fan! Our predictions are generated by sophisticated machine learning algorithms fuelled by the most advanced statistics found anywhere.
## Who is the best player in the NHL?
1. Connor McDavid, F, Edmonton Oilers. McDavid is so good that he has taken the title of “best player in the world” from Sidney Crosby with little to no argument. Each of the past two seasons, he has led the NHL in scoring and been voted the most outstanding player in the NHL by his peers.
## Does anyone wear 66 in the NHL?
No. 66 isn’t retired League-wide like No. 99 is for Wayne Gretzky. Two players have worn it since Lemieux retired from the NHL in 2006: Ho-Sang and Calgary Flames defenseman TJ Brodie in 2010-11.
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# Thread: Changing the order of integration
1. ## Changing the order of integration
Hey I'm having some trouble with multiple integrals if anyone would care to explain (I think I'm just not using the correct region!)
Evaluate
$\int_{x=0}^2\int_{y=\frac{x}{2}}^1 2xy^2 dy dx$
first directly then changing the order of integration
I did the first part and got 0.8 as the answer
just for the second I'm having trouble
I think the surface being integrated over the the triangle bounded by the lines y=x/2 x=0 and y=1 but I may be wrong which gives
$\int_{y=0}^1\int_{x=2y}^2 2xy^2 dx dy$
but the value of that doesnt agree, can anyone explain how to do the limits correctly?
cheers
Simon
2. $
\int\limits_0^1 {\int\limits_0^{x = 2y} {2xy^2 dxdy} }
$
3. Originally Posted by thelostchild
Hey I'm having some trouble with multiple integrals if anyone would care to explain (I think I'm just not using the correct region!)
I did the first part and got 0.8 as the answer
just for the second I'm having trouble
I think the surface being integrated over the the triangle bounded by the lines y=x/2 x=0 and y=1 but I may be wrong which gives
$\int_{y=0}^1\int_{x=2y}^2 2xy^2 dx dy$
but the value of that doesnt agree, can anyone explain how to do the limits correctly?
cheers
Simon
$0\leqslant{x}\leqslant{2}$ and $\frac{x}{2}\leqslant{y}\leqslant{1}\implies{x\leqs lant{2y}\leqslant{2}}$. Stringing these two inequalites together gives
$0\leqslant{x}\leqslant{2y}\leqslant{2}$
From there Peritus's answer should be more apparent.
4. Originally Posted by Mathstud28
$0\leqslant{x}\leqslant{2}$ and $\frac{x}{2}\leqslant{y}\leqslant{1}\implies{x\leqs lant{2y}\leqslant{2}}$. Stringing these two inequalites together gives
$0\leqslant{x}\leqslant{2y}\leqslant{2}$
From there Peritus's answer should be more apparent.
But be careful 'cause, this method not always works.
5. Originally Posted by Krizalid
But be careful 'cause, this method not always works.
Really? I have never encountered a case where it hasn't. Would you mind giving me an example of one please?
6. Try it with $\int_0^1\int_{x^3}^{\sqrt[3]x}dy\,dx.$
7. Originally Posted by Krizalid
Try it with $\int_0^1\int_{x^3}^{\sqrt[3]x}dy\,dx.$
Thank you, I will report back later when I have had time to look at it.
8. Also be careful that reversing integration order is submitted to the condition that $\int \int \left|f(x,y)\right| ~ dx ~ dy$ is a finite value.
See Fubini's theorem - Wikipedia, the free encyclopedia for further information (advanced calculus imo)
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Photovoltaic power plants produce electricity from sunlight. : GMAT Critical Reasoning (CR)
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# Photovoltaic power plants produce electricity from sunlight.
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Photovoltaic power plants produce electricity from sunlight. As a result of astonishing recent technological advances, the cost of producing electric power at photovoltaic power plants, allowing for both construction and operating costs, is one-tenth of what it was 20 years ago, whereas the corresponding cost for traditional plants, which burn fossil fuels, has increased. Thus, photovoltaic power plants offer a less expensive approach to meeting demand for electricity than do traditional power plants.
The conclusion of the argument is properly drawn if which one of the following is assumed?
(A) The cost of producing electric power at traditional plants has increased over the past 20 years.
(B) Twenty years ago, traditional power plants were producing 10 times more electric power than were photovoltaic plants.
(C) None of the recent technological advances in producing electric power at photovoltaic plants can be applied to producing power at traditional plants.
(D) Twenty years ago, the cost of producing electric power at photovoltaic plants was less than 20 times the cost of producing power at traditional plants.
(E) The cost of producing electric power at photovoltaic plants is expected to decrease further, while the cost of producing power at traditional plants is not expected to decrease.
Pls explain
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25 Feb 2008, 01:20
Conclusion: Photovoltaic power plants offer a less expensive approach to meeting demand for electricity than do traditional power plants.
(A) The cost of producing electric power at traditional plants has increased over the past 20 years. [Fine, but if cost of Photovoltaic power plant’s power production substantially higher than traditional one, then even with one tenth reduction it would still higher than traditional power plant – eliminate it]
(B) Twenty years ago, traditional power plants were producing 10 times more electric power than were photovoltaic plants. [Irrelevant]
(C) None of the recent technological advances in producing electric power at photovoltaic plants can be applied to producing power at traditional plants. [Irrelevant as we are dealing with two different technology sectors – eliminate it]
(D) Twenty years ago, the cost of producing electric power at photovoltaic plants was less than 20 times the cost of producing power at traditional plants. [Hold it]
(E) The cost of producing electric power at photovoltaic plants is expected to decrease further, while the cost of producing power at traditional plants is not expected to decrease. [Future prediction – eliminate it]
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Re: CR: Photovoltaic power plant [#permalink]
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25 Feb 2008, 01:50
hanumayamma wrote:
Conclusion: Photovoltaic power plants offer a less expensive approach to meeting demand for electricity than do traditional power plants.
(A) The cost of producing electric power at traditional plants has increased over the past 20 years. [Fine, but if cost of Photovoltaic power plant’s power production substantially higher than traditional one, then even with one tenth reduction it would still higher than traditional power plant – eliminate it]
(B) Twenty years ago, traditional power plants were producing 10 times more electric power than were photovoltaic plants. [Irrelevant]
(C) None of the recent technological advances in producing electric power at photovoltaic plants can be applied to producing power at traditional plants. [Irrelevant as we are dealing with two different technology sectors – eliminate it]
(D) Twenty years ago, the cost of producing electric power at photovoltaic plants was less than 20 times the cost of producing power at traditional plants. [Hold it]
(E) The cost of producing electric power at photovoltaic plants is expected to decrease further, while the cost of producing power at traditional plants is not expected to decrease. [Future prediction – eliminate it]
Let us say that Twenty years ago, the cost of producing electric power at photovoltaic plants was 19 times ( i.e. less than 20 times) the cost of producing power at traditional plants. Now present cost of producing electric power at photovoltaic plants would still be 1.9 times the cost of producing power at traditional plants. Will the answer D still be correct.
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Re: CR: Photovoltaic power plant [#permalink]
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25 Feb 2008, 01:54
neelabhmahesh wrote:
hanumayamma wrote:
Conclusion: Photovoltaic power plants offer a less expensive approach to meeting demand for electricity than do traditional power plants.
(A) The cost of producing electric power at traditional plants has increased over the past 20 years. [Fine, but if cost of Photovoltaic power plant’s power production substantially higher than traditional one, then even with one tenth reduction it would still higher than traditional power plant – eliminate it]
(B) Twenty years ago, traditional power plants were producing 10 times more electric power than were photovoltaic plants. [Irrelevant]
(C) None of the recent technological advances in producing electric power at photovoltaic plants can be applied to producing power at traditional plants. [Irrelevant as we are dealing with two different technology sectors – eliminate it]
(D) Twenty years ago, the cost of producing electric power at photovoltaic plants was less than 20 times the cost of producing power at traditional plants. [Hold it]
(E) The cost of producing electric power at photovoltaic plants is expected to decrease further, while the cost of producing power at traditional plants is not expected to decrease. [Future prediction – eliminate it]
Let us say that Twenty years ago, the cost of producing electric power at photovoltaic plants was 19 times ( i.e. less than 20 times) the cost of producing power at traditional plants. Now present cost of producing electric power at photovoltaic plants would still be 1.9 times the cost of producing power at traditional plants. {A result of astonishing recent technological advances, the cost of producing electric power at photovoltaic power plants, allowing for both construction and operating costs, is one-tenth of what it was 20 years ago}.Will the answer D still be correct.
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25 Feb 2008, 02:04
Please factor in the traditional plant's cost - it is increasing. Our analysis of cost was with respect to twenty years.
Now in the example: Photovoltaic power plants are 1.9 times to the cost of traditional plants twenty years ago. But during the 20 years, traditional plants became more expensive. This makes it clear Photovoltaic power plants are economical.
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25 Feb 2008, 03:20
hanumayamma wrote:
Please factor in the traditional plant's cost - it is increasing. Our analysis of cost was with respect to twenty years.
Now in the example: Photovoltaic power plants are 1.9 times to the cost of traditional plants twenty years ago. But during the 20 years, traditional plants became more expensive. This makes it clear Photovoltaic power plants are economical.
You mean to say that the cost of producing electricity from traditional power plants also would have increased at least twice within 20 years.
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25 Feb 2008, 08:31
neelabhmahesh wrote:
hanumayamma wrote:
Please factor in the traditional plant's cost - it is increasing. Our analysis of cost was with respect to twenty years.
Now in the example: Photovoltaic power plants are 1.9 times to the cost of traditional plants twenty years ago. But during the 20 years, traditional plants became more expensive. This makes it clear Photovoltaic power plants are economical.
You mean to say that the cost of producing electricity from traditional power plants also would have increased at least twice within 20 years.
I agree with you. It does say that the traditional plants doubled their costs.
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Re: CR: Photovoltaic power plant [#permalink]
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25 Feb 2008, 21:22
moni77 wrote:
neelabhmahesh wrote:
hanumayamma wrote:
Please factor in the traditional plant's cost - it is increasing. Our analysis of cost was with respect to twenty years.
Now in the example: Photovoltaic power plants are 1.9 times to the cost of traditional plants twenty years ago. But during the 20 years, traditional plants became more expensive. This makes it clear Photovoltaic power plants are economical.
You mean to say that the cost of producing electricity from traditional power plants also would have increased at least twice within 20 years.
I agree with you. It does say that the traditional plants doubled their costs.
But where is this mentioned in the passage
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25 Feb 2008, 21:41
neelabhmahesh wrote:
I agree with you. It does say that the traditional plants doubled their costs.
But where is this mentioned in the passage[/quote]
Sorry. It's not my day today. I meant it DOESN'T say. So I'm not convinced why D is the correct answer, even if it's better that any other answer choice.
Re: CR: Photovoltaic power plant [#permalink] 25 Feb 2008, 21:41
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Solving Systems Of Equations By Elimination Worksheet Answers With Work
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Chapter 02 Test Bank Student:
1.
An individual has an absolute advantage in producing pizzas if that individual:
A. B. C. D.
has a lower opportunity cost of producing pizzas than anyone else. can produce more pizzas in a given amount of time than anyone else. has a higher opportunity cost of producing pizzas than anyone else. charges the lowest price for pizzas.
2.
If Al has an absolute advantage over Beth in preparing meals, then:
A. B. C. D. 3. A. B. C. D. 4.
it takes Al more time to prepare a meal than Beth. the problem of scarcity applies to Beth but not to Al. Al's opportunity cost of preparing a meal is lower than is Beth's. Al can prepare more meals in a given time period than Beth. If Les can produce two pairs of pants per hour while Eva can produce one pair per hour, then it must be true that: Les has a comparative advantage in producing pants. Les has an absolute advantage in producing pants. Eva has a comparative advantage in producing pants. Les has both comparative and absolute advantage in producing pants. If a nation can produce a more computers per year than any other nation, that nation has a(n)
A. comparative
advantage in the production of computers.
B. absolute C. relative D. natural 5. A. B. C. D.
If you have a comparative advantage in a particular task, then: you you you you
are better at it than other people. give up more to accomplish that task than do others. give up less to accomplish that task than do others. have specialized in that task, while others have not.
6. A. B. C. D. 7.
Larry has a comparative advantage over his classmates in writing term papers if he: can write term papers faster than his classmates. has an absolute advantage in writing term papers. always earns an A on his term papers. has a lower opportunity cost of writing term papers than his classmates. If a nation has the lowest opportunity cost of producing a good, that nation has a(n)
A. B. C. D.
8.
Which of the following statements is true?
A. B. C. D.
9. A. B. C. D.
If Jane can produce 3 pairs of shoes per hour, while Bob can produce 2, then Jane余 absolute Jane余 comparative Bob余 absolute Bob余 comparative
10. Refer to the table below. According to the table, Martha has the absolute advantage in:
A. B. C. D.
pies. neither pies nor cakes. cakes. both pies and cakes.
in the production of that good.
has a(n)
11. Refer to the table below. According to the table, Julia has the absolute advantage in:
A. B. C. D.
pies. neither pies nor cakes. cakes. both pies and cakes.
12. Refer to the table below. Martha's opportunity cost of making of a pie is:
A. B. C. D.
3/4 of a cake. 4/3 of a cake. 8 cakes. 80 cakes.
13. Refer to the table below. Martha's opportunity cost of making a cake is:
A. B. C. D.
3/4 of a pie. 4/3 of a pie. 6 pies. 60 pies.
14. Refer to the table below. Julia's opportunity cost of making a pie is:
A. B. C. D.
60 cakes 6 cakes 6/5 of a cake 5/6 of a cake
15. Refer to the table below. Julia's opportunity cost of making a cake is:
A. B. C. D.
60 cakes 6 cakes 6/5 of a cake 5/6 of a cake
16. Refer to the table above.
A. B. C. D.
Martha余 Martha Julia余 Julia Martha余 Julia Julia余 Martha
has the comparative advantage in making pies and
the comparative advantage in making cakes.
17. Refer to the table below. Based on their comparative advantage, Martha should specialize in
A. B. C. D.
while Julia should specialize in
.
pies余 cakes cakes余 pies neither pies nor cakes余 both pies and cakes both pies and cakes余 neither pies nor cakes
18. Suppose it takes Dan 5 minutes to make a sandwich and 15 minutes to make a smoothie, and it takes Tracy 6 minutes to make a sandwich and 12 minutes to make a smoothie. What is the opportunity cost to Dan of making a sandwich?
A. B. C. D.
1/3 of a smoothie 3 smoothies 15 smoothies 5 smoothies
19. Suppose it takes Dan 5 minutes to make a sandwich and 15 minutes to make a smoothie, and it takes Tracy 6 minutes to make a sandwich and 12 minutes to make a smoothie. Which of the following statements is correct?
A. B. C. D.
Dan Dan Dan Dan
has has has has
the the the the
comparative comparative comparative comparative
smoothies, but Tracy has the absolute advantage in smoothies. advantage in sandwiches. advantage in smoothies. sandwiches, but Tracy has the absolute advantage in sandwiches.
20. Suppose it takes Dan 5 minutes to make a sandwich and 15 minutes to make a smoothie, and it takes Tracy 6 minutes to make a sandwich and 12 minutes to make a smoothie. Which of the following statements is correct?
A. B. C. D.
Tracy should specialize in sandwiches and smoothies. Dan should specialize in smoothies, and Tracy should specialize in sandwiches. Dan should specialize in sandwiches, and Tracy should specialize in smoothies. Dan should specialize in both sandwiches and smoothies.
21. Suppose it takes Paul 3 hours to bake a cake and 2 hours to move the lawn, and suppose it takes Tom 2 hours to bake a cake and 1 hour to mow the lawn. Which of the following statements is correct? A. B. C. D.
Paul Paul Paul Paul
has has has has
the the the the
absolute advantage in baking cakes comparative in mowing the lawn comparative in baking cakes absolute advantage in mowing the lawn.
22. Suppose Cathy and Lewis work in a bakery making pies and cakes. Suppose it takes Cathy 1.5 hours to make a pie and 1 hour to make a cake, and suppose it takes Lewis 2 hours to make a pie and 1.5 hours to make a cake. Which of the following statements is correct?
A. B. C. D.
Cathy Cathy Lewis Lewis
has a comparative has a comparative has a comparative has a comparative
23. Suppose Cathy and Lewis work in a bakery making pies and cakes. Suppose it takes Cathy 1.5 hours to make a pie and 1 hour to make a cake, and suppose it takes Lewis 2 hours to make a pie and 1.5 hours to make a cake. Which of the following statements is correct?
A. B. C. D.
Cathy should specialize in both pies and cakes. There are no gains from specialization and trade. Lewis should specialize in pies, and Cathy should specialize in cakes. Cathy should specialize in pies, and Lewis should specialize in cakes.
24. Suppose Cathy and Lewis work in a bakery making pies and cakes. Suppose it takes Cathy 1.5 hours to make a pie and 1 hour to make a cake, and suppose it takes Lewis 2 hours to make a pie and 1.5 hours to make a cake. What is the opportunity cost to Cathy of making a cake?
A. B. C. D.
2/3 of a pie. 1 pie. 1.5 pies. 1.33 pies.
25. Refer to the table below. According to the table, Corey has the absolute advantage in:
A. B. C. D.
making pizza. neither making nor delivering pizza. delivering pizza. making and delivering pizza.
26. Refer to the table below. According to the table, Pat has the absolute advantage in:
A. B. C. D.
making pizza. neither making nor delivering pizza. delivering pizza. making and delivering pizza.
27. Refer to the table below. Corey's opportunity cost of making of a pizza is delivering:
A. B. C. D.
2 pizzas. 3/2 of a pizza. 2/3 of a pizza. 1/2 of a pizza.
28. Refer to the table below. Corey's opportunity cost of delivering of a pizza is making:
A. B. C. D.
6 pizzas. 12 pizzas. 2 pizzas. 1/2 of a pizza.
29. Refer to the table below. Pat's opportunity cost of making a pizza is delivering:
A. B. C. D.
3 pizzas 2 pizzas 3/2 of a pizza 2/3 of a pizza
30. Refer to the table below. Pat's opportunity cost of delivering a pizza is making:
A. B. C. D.
12 pizzas 10 pizzas 3/2 of a pizza 2/3 of a pizza
31. Refer to the table below.
A. B. C. D.
Corey余 Corey Pat余 Pat Pat余 Corey Corey余 Pat
has the comparative advantage in making pizza, and
_ has the comparative advantage in delivering pizza.
32. Refer to the table below. Based on their comparative advantages, Pat should specialize in
A. B. C. D.
, and Corey should specialize in
.
delivering pizza余 making pizza making pizza余 delivering pizza neither making pizza nor delivering pizza余 both making pizza and delivering pizza both making pizza and delivering pizza余 neither making pizza nor delivering pizza
33. Lou and Alex live together and share household chores. They like to cook some meals ahead of time and eat leftovers. The table below shows the number of rooms they can each clean and the number of meals they can each cook in an hour.
Which of the following is true?
A. B. C. D.
Lou has both an absolute advantage and a comparative advantage over Alex in both tasks. Alex has a comparative advantage over Lou in cleaning. Lou has a comparative advantage over Alex in cleaning. Alex has both an absolute advantage and a comparative advantage over Lou in both tasks.
34. Lou and Alex live together and share household chores. They like to cook some meals ahead of time and eat leftovers. The table below shows the number of rooms they can each clean and the number of meals they can each cook in an hour.
If Alex and Lou work out an efficient arrangement for these two chores, then under that arrangement:
A. B. C. D.
Alex and Lou each would do half of the cooking and half of the cleaning. Alex would do all of the cleaning, while Lou would do all the cooking. Lou would do all of the cleaning and all of the cooking. Lou would do all of the cleaning, while Alex would do all of the cooking.
35. Lou and Alex live together and share household chores. They like to cook some meals ahead of time and eat leftovers. The table below shows the number of rooms they can each clean and the number of meals they can each cook in an hour.
For Alex, the opportunity cost of cleaning one room is making
A. B. C. D.
4; 4 1; 4/5 1; 5/4 3; 5
meal(s); for Lou the opportunity cost of cleaning one room is making _
meal(s).
36. Dent 'n' Scratch Used Cars and Trucks employs 3 salesmen. Data for their sales last month are shown in this table:
Based on last month's data,
A. B. C. D.
has an absolute advantage in selling cars and
_ has an absolute advantage in selling trucks.
Joe余 Joe Larry余 Ralph Ralph余 Larry Larry余 Joe
37. Dent 'n' Scratch Used Cars and Trucks employs 3 salesmen. Data for their sales last month are shown in this table:
Based on last month's data, Larry's opportunity cost of selling a truck is selling:
A. B. C. D.
10 cars. 1/2 of a car. 1 car. 2 cars.
38. Dent 'n' Scratch Used Cars and Trucks employs 3 salesmen. Data for their sales last month are shown in this table:
Based on last month's data, Joe's opportunity cost of selling a truck is selling:
A. B. C. D.
9 cars. 1 car. 4 cars. 1/3 of a car.
39. Dent 'n' Scratch Used Cars and Trucks employs 3 salesmen. Data for their sales last month are shown in this table:
Based on last month's data, Ralph's opportunity cost of selling a truck is selling:
A. B. C. D.
4 cars. 1/3 of a car. 3 cars. 1/4 of a car.
40. Dent 'n' Scratch Used Cars and Trucks employs 3 salesmen. Data for their sales last month are shown in this table:
Based on last month's data, Joe's opportunity cost of selling a car is
A. B. C. D.
than Ralph's, and Joe's opportunity cost of selling a car is
less; greater greater; less less; less greater; greater
41. Dent 'n' Scratch Used Cars and Trucks employs 3 salesmen. Data for their sales last month are shown in this table:
Based on last month's data,
A. B. C. D.
should specialize in truck sales, and _
should specialize in car sales.
Joe; Ralph Ralph; Larry Larry; Ralph Larry; Joe
42. The textbook notes that the last time a major league batter hit .400 was in 1941. This is because: A. B. C. D.
the average quality of batters has fallen. the league imposes harsh penalties for steroid use. specialization by pitchers, infielders, and outfielders has made it harder for batters to hit. baseball diamonds have become larger.
than Larry's.
43. Ginger and Maryann are lost in the jungle, where the only things to eat are mangoes and fish. Ginger can gather more mangoes per hour than Maryann and can also catch more fish per hour than can Maryann. Therefore: A. B. C. D.
There are no gains to specialization and trade for Ginger. There are no gains to specialization and trade for Maryann. Maryann should specialize in the activity for which she has a comparative advantage. Ginger should specialize in the activity for which she has an absolute advantage.
44. In general, individuals and nations should specialize in producing those goods for which they have a(n): A. B. C. D.
45. In general, individuals and nations should specialize in producing goods A. B. C. D.
other individuals or nations.
that they can produce more quickly than that they can produce less quickly than for which they have a lower opportunity cost compared to for which they have a higher opportunity cost compared to
46. A country is most likely to have a comparative advantage in the production of cars if: A. B. C. D.
it imports most of the raw materials necessary to produce cars. its citizens prefer driving cars to other forms of transportation. it has strict environmental protection laws governing automobile emissions. it has a relative abundance in the natural resources needed to produce cars.
47. The United States generally has a comparative advantage in the development of technology because it has: A. B. C. D.
large amounts of natural resources. a disproportionate share of the world's best research universities. the greatest need for new technology. patent laws, which no other country has.
48. The emergence of English as the de facto world language has A. B. C. D.
given English•speaking countries given non•English•speaking countries had no effect on which country has given all countries
a comparative advantage in the production of books, movies and popular music:
49. The United States was unable to maintain its dominance in the production of televisions because: A. B. C. D.
the highly technical skills necessary to produce televisions are greater in other countries. the raw materials necessary to build televisions became scarce in the United States. the product designs evolved too rapidly for engineers in the United States to keep up. automated techniques allowed production to be outsourced to countries with less•skilled workers.
50. A graph that illustrates the maximum amount of one good that can be produced for every possible level of production of the other good is called a(n): A. B. C. D.
production possibilities curve. consumption possibilities curve. production function. supply curve.
51. The production possibilities curve shows: A. B. C. D.
the minimum production of one good for every possible production level of the other good. how increasing the resources used to produce one good increases the production of the other good. the maximum production of one good for every possible production level of the other good. how increasing the production of one good allows production of the other good to also rise.
52. Points that lie outside the production possibilities curve are A. B. C. D.
, and points that lie inside the production possibilities curve are
.
efficient; inefficient inefficient; efficient unattainable; attainable attainable; unattainable
53. Points that lie beneath the production possibilities curve are: A. B. C. D.
unattainable and inefficient unattainable but efficient attainable but inefficient attainable and efficient
54. If a country is producing at point where an increase in the production of one good requires a reduction in the production of another good, then it must be producing at an: A. B. C. D.
inefficient point. efficient point. unattainable point. undesirable point.
55. Suppose Colin brews beer and makes cheese. If Colin can increase his production of beer without decreasing his production of cheese, then he is producing at an: A. B. C. D.
inefficient point. efficient point. unattainable point. ideal point.
56. The downward slope of the production possibilities curve illustrates the: A. B. C. D.
Scarcity Principle. Cost•Benefit Principle. Incentive Principle. Principle of Comparative Advantage.
57. The figure below shows the production possibilities curve for the island of Genovia:
The opportunity cost of producing a car in Genovia is:
A. B. C. D.
5,000 tons of agricultural products. 500 tons of agricultural products. 5 tons of agricultural products. 50 tons of agricultural products.
58. The figure below shows the production possibilities curve for the island of Genovia:
The opportunity cost of producing one ton of agricultural products in Genovia is:
A. B. C. D.
1,000 cars. 1 car. 1/5 of a car. 1/50 of a car.
59. The figure below shows the production possibilities curve for the island of Genovia:
If 500 cars are produced in Genovia, a maximum of
A. B. C. D.
50,000 25,000 45,000 40,000
60. The slope of a production possibilities curve is A. B. C. D.
tons of agricultural products can be produced.
because
_.
negative余 producing more of one good requires producing less of the other negative余 producing less of one good requires producing less of the other positive余 producing more of one good requires producing more of the other positive余 producing more of one good requires producing less of the other
61. The figure below shows Becky's daily production possibilities curve for dresses and skirts.
The maximum number of dresses that Becky can make in a day is represented by point:
A. U B. T C. V D. W
62. The figure below shows Becky's daily production possibilities curve for dresses and skirts.
The maximum number of skirts that Becky can make in a day is represented by point:
A. U B. T C. V D. Z
63. The figure below shows Becky's daily production possibilities curve for dresses and skirts.
Point U is:
A. B. C. D.
attainable. efficient. unattainable. inefficient.
64. The figure below shows Becky's daily production possibilities curve for dresses and skirts.
Of the labeled points, only
A. B. C. D.
T and U X, Y, and Z W, X, Y, Z, and V W, X, Y, Z, V, and T
are attainable.
65. The figure below shows Becky's daily production possibilities curve for dresses and skirts.
Of the labeled points, only
A. B. C. D.
T and U X, Y, and Z W, X, Y, Z, and V W, X, Y, Z, V, and T
are efficient.
66. The figure below shows Becky's daily production possibilities curve for dresses and skirts.
Point T is:
A. B. C. D.
attainable efficient both attainable and efficient neither attainable nor efficient
67. The figure below shows Becky's daily production possibilities curve for dresses and skirts.
Point Y is
A. B. C. D.
, and point V is
efficient余 inefficient inefficient余 efficient efficient余 efficient inefficient余 inefficient
.
68. The figure below shows Becky's daily production possibilities curve for dresses and skirts.
Relative to point X, at point Y:
A. B. C. D.
more dresses and more skirts more skirts and fewer dresses more dresses and fewer skirts fewer skirts and fewer dresses
are are are are
produced. produced. produced. produced.
69. Refer to the figure below. For Pat, the opportunity cost of removing one bag of trash is planting:
A. B. C. D.
100 bulbs. 5 bulbs. 1/100 of a bulb. 1/5 of a bulb.
70. Refer to the figure below. For Pat, the opportunity cost of planting one bulb is removing:
A. B. C. D.
20 bags of trash. 5 bags of trash. 1/20 of a bag of trash. 1/5 of a bag of trash.
71. Refer to the figure below. For Chris, the opportunity cost of removing one bag of trash is planting:
A. B. C. D.
25 bulbs. 1/25 of a bulb. 3 bulbs. 1/3 of a bulb.
72. Refer to the figure below. For Chris, the opportunity cost of planting one bulb is removing:
A. B. C. D.
25 bags of trash. 1/25 of a bag of trash. 3 bags of trash. 1/3 of a bag of trash.
73. Refer to the figure below. If Pat and Chris were to specialize in the task in which each has a comparative advantage:
A. B. C. D.
Chris would plant bulbs and Pat would remove trash. Chris would remove trash and Pat would plant bulbs. Pat and Chris would each spend half of their time each task. both Pat and Chris would plant bulbs because they both have an absolute advantage in that task.
74. Refer to the figure below. If Pat and Chris each spend half their time on each task, then:
A. B. C. D.
the outcome will be efficient. they will plant more bulbs and remove fewer bags of trash than if they had each specialized in the task at which they have a comparative advantage. they will plant fewer bulbs and remove fewer bags of trash than if they each had specialized in the task at which they have a comparative advantage. the outcome will be unattainable.
75. On a graph of a production possibilities curve, if a point is attainable, then it: A. B. C. D.
must be efficient. might or might not be efficient. is efficient only if it does not exhaust all currently available resources. must completely exhaust all currently available resources.
76. Any combination of goods that can be produced with currently available resources is an: A. B. C. D.
attainable point. efficient point. inefficient point. attainable and efficient point.
77. On a graph of a production possibilities curve, an inefficient point is: A. B. C. D.
necessarily an attainable point. not necessarily an attainable point. necessarily an unattainable point. possibly an unattainable point.
78. Consider a graph of a production possibilities curve. If a producer is operating at an inefficient point, then that producer: A. B. C. D.
cannot produce more of one good without giving up some of the other good. can produce more of one good without producing less of the other good. must be at an unattainable point on the production possibilities curve. must be specializing in activities for which it has a comparative advantage.
79. Points that lie below the production possibilities curve are inefficient because: A. B. C. D.
more of one good could be produced without producing less of the other. producing more of one good means producing less of the other. producers face scarcity. too many goods are being produced.
80. Refer to the figure below. Growing 1,000 bushels of wheat and no bushels of corn each year is:
A. B. C. D.
inefficient and unattainable. inefficient but attainable. efficient but unattainable. efficient and attainable.
81. Refer to the figure below. It is efficient for this farmer to:
A. B. C. D.
grow grow grow grow
500 bushels of wheat and 500 bushels 250 bushels of wheat and 500 bushels 500 bushels of wheat and 250 bushels 1000 bushels of wheat and 500 bushels
of of of of
corn. corn. corn. corn.
82. Refer to the figure below. The opportunity cost of producing one bushel of corn is:
A. B. C. D.
2 bushels of wheat. ½ of a bushel of wheat. 500 bushels of wheat. 250 bushels of wheat.
83. Refer to the figure below. The opportunity cost of producing one bushel of wheat is:
A. B. C. D.
2 bushels of corn. ½ of a bushel of corn. 1,000 bushels of corn. 500 bushels of corn.
84. If a given production combination is known to be attainable, then it: A. B. C. D.
must be on the production possibilities curve. must be an inefficient point. must be an efficient point. could be either an inefficient or efficient point.
85. If a given production combination is efficient, then it must be: A. B. C. D.
above the production possibilities curve. on the production possibilities curve. either an attainable or unattainable point. below the production possibilities curve.
86. Working efficiently, Jordan can write 3 essays and outline 4 chapters each week. It must be true that: A. B. C. D.
6 essays 2 essays 3 essays 4 essays
and and and and
0 chapter 3 chapter 5 chapter 3 chapter
outlines outlines outlines outlines
would would would would
be be be be
unattainable. efficient. unattainable. both attainable and efficient.
87. Assume point A on a linear production possibilities curve represents the combination of 12 coffees and 3 cappuccinos, and point B represents 3 coffees and 6 cappuccinos. Suppose coffees are on the vertical axis and cappuccinos are on the horizontal axis. The absolute value of the slope of the production possibilities curve between points A and B equals:
A. B. C. D.
6 4 3 1/3
88. Assume point A on a linear production possibilities curve represents the combination of 12 coffees and 3 cappuccinos, and point B represents 3 coffees and 6 cappuccinos. Suppose coffees are on the vertical axis and cappuccinos are on the horizontal axis. The opportunity cost of a cup of coffee is:
A. B. C. D.
3 cappuccinos. 9 cappuccinos. 1/3 of a cappuccino. 6 cappuccinos.
89. Generally, on a linear two•good production possibilities curve, the opportunity cost of the good measured on the vertical axis is: A. B. C. D.
one minus the opportunity cost of the good measured on the horizontal axis. the reciprocal of the opportunity cost of the good measured on the horizontal axis. the absolute value of the slope of the production possibilities curve. the negative of the opportunity cost of the good measured on the horizontal axis.
90. If a linear, two-good production possibilities curve has a slope of ‒2, then: A. having an additional unit of the good measured on the vertical axis means giving up 2 units of the good measured on the horizontal axis. B. having an additional unit of the good measured on the vertical axis means giving up ½ of an unit of the good measured on the horizontal axis. C. you have an absolute advantage in the good measured on the vertical axis. D. you have a comparative advantage in the good measured on the vertical axis. 91. The idea that tradeoffs have to be made when resources are scarce is reflected in the fact that: A. points below the production possibilities curve are efficient. B. points below the production possibilities curve are inefficient. C. the production possibilities curve has a negative slope. D. the slope of a linear production possibilities is constant. 92. In a two•person, two•good economy, the gains to specialization will be larger when: A. one person has an absolute advantage in both goods. B. neither person has an absolute advantage. C. there are small differences between the individuals in their opportunity costs of producing the two goods. D. there are large differences between the individuals in their opportunity costs of producing the two goods. 93. According to the Principle of Increasing Opportunity Cost, in expanding the production of any good, we should start by utilizing the resources that: A. we have the most of. B. we have the least of. C. have the highest opportunity cost. D. have the lowest opportunity cost.
94. Smith and Jones comprise a two•person economy. Their hourly rates of production are shown below.
The opportunity cost of making a calculator for Smith is _
A. B. C. D.
and for Jones it is
.
0.10 computers; 0.05 computers 10 computers; 20 computers 1 computer; 0.5 computers 0.6 computers; 1.2 computers
95. Smith and Jones comprise a two•person economy. Their hourly rates of production are shown below.
If Smith and Jones devote all of their resources to producing computers, then the maximum number of computers they can produce in an hour is:
A. B. C. D.
120. 6. 16. 10.
96. Smith and Jones comprise a two•person economy. Their hourly rates of production are shown below.
Suppose Smith and Jones begin by producing 16 computers and 0 calculators per hour. If they wish to produce 14 computers and 40 calculators per hour efficiently, then Smith should spend , and Jones should spend .
A. B. C. D.
1 hour on computers; 40 minutes on computers and 20 minutes on calculators 1 hour on computers; 20 minutes on computers and 40 minutes on calculators 30 minutes on each; 30 minutes on each 45 minutes on computers and 15 on calculators; 1 hour on calculators
97. Smith and Jones comprise a two•person economy. Their hourly rates of production are shown below.
Suppose Smith and Jones begin by producing 0 computers and 220 calculators per hour. If they wish to produce 2 computers and 200 calculators per hour efficiently, then Smith should spend , and Jones should spend .
A. B. C. D.
30 minutes on each; 30 minutes on each 48 minutes on computers and 12 minutes on calculators; 1 hour on calculators 1 hour on calculators; 10 minutes on computers and 50 minutes on calculators 12 minutes on computers and 48 minutes on calculators; 1 hour on calculators
98. Smith and Jones comprise a two•person economy. Their hourly rates of production are shown below.
If Smith and Jones are dividing their time efficiently and producing more than 10 computers and fewer than 120 calculators per hour, then Smith will will .
A. B. C. D.
and Jones
produce only computers; produce only calculators produce only computers; split his time between computers and calculators split his time between computers and calculators; produce only computers produce only calculators; produce only computers
99. Smith and Jones comprise a two•person economy. Their hourly rates of production are shown below.
If Smith and Jones are dividing their time efficiently and producing fewer than 10 computers and more than 120 calculators per hour, then Smith will will .
A. B. C. D.
split his time between computers and calculators; produce only calculators produce only calculators; split his time between computers and calculators produce only calculators; produce only computers produce only computers; produce only calculators
and Jones
100. Smith and Jones comprise a two•person economy. Their hourly rates of production are shown below.
Suppose Smith and Jones begin by producing 100 calculators per hour; as Smith and Jones choose to efficiently produce fewer computers and more calculators, should devote more time to calculators because his .
A. B. C. D.
Smith; absolute advantage is larger Jones; absolute advantage is smaller Jones; opportunity costs are lower Smith; opportunity costs are lower
101. Earth Movers & Shakers operates 3 iron ore mines. The table below shows each mine's total daily production and the current number of miners at each mine. All miners work for the same wage, and each miner in any given mine produces the same number of tons per day as every other miner in that mine.
The opportunity cost of moving one miner from Mother Lode to another mine is:
A. 2 tons per day. B. 3 tons per day. C. 4 tons per day. D. 1 ton per day.
102. Earth Movers & Shakers operates 3 iron ore mines. The table below shows each mine's total daily production and the current number of miners at each mine. All miners work for the same wage, and each miner in any given mine produces the same number of tons per day as every other miner in that mine.
The opportunity cost of moving one miner from Scraping Bottom to another mine is:
A. B. C. D.
0 3 4 5
tons tons tons tons
per per per per
day. day. day. day.
103. Earth Movers & Shakers operates 3 iron ore mines. The table below shows each mine's total daily production and the current number of miners at each mine. All miners work for the same wage, and each miner in any given mine produces the same number of tons per day as every other miner in that mine.
The opportunity cost of moving one miner from Middle Drift to another mine is:
A. B. C. D.
1 ton per day. 3 tons per day. 4 tons per day. 5 tons per day.
104. Earth Movers & Shakers operates 3 iron ore mines. The table below shows each mine's total daily production and the current number of miners at each mine. All miners work for the same wage, and each miner in any given mine produces the same number of tons per day as every other miner in that mine.
Suppose Earth Movers & Shakers needs to fill an order for 60 tons of ore in a single day. If it has no other orders for that day, it should:
A. B. C. D.
take take take take
it all from Mother Lode. it all from Middle Drift. 30 tons from Scraping Bottom and 30 tons from Middle Drift. 20 tons from each of the three mines.
105. Earth Movers & Shakers operates 3 iron ore mines. The table below shows each mine's total daily production and the current number of miners at each mine. All miners work for the same wage, and each miner in any given mine produces the same number of tons per day as every other miner in that mine.
Suppose Earth Movers & Shakers needs to fill an order for 100 tons of ore in a single day. If it has no other orders to fill that day, and it's not possible to transfer miners from one mine to another, it should:
A. B. C. D.
take take take take
it all from Mother Lode. 75 tons from Middle Drift and 25 tons from Mother Lode. 75 tons from Middle Drift and 25 tons from Scraping Bottom. 30 tons from Scraping Bottom and 70 tons from Mother Lode.
106. Refer to the figure below. If this restaurant makes 75 salads in one hour, then what's the maximum number of pizzas it can make in that same hour?
A. B. C. D.
0 10 20 30
107. Refer to the figure below. Relative to point B, at point C this restaurant is:
A. making more pizzas and more salads. B. making more pizzas and fewer salads. C. making fewer pizzas and more salads. D. operating more efficiently.
108. Refer to the figure below. Moving from point C to point B, the opportunity cost of 25 more salads is:
A. B. C. D.
5 pizzas. 10 pizzas. 15 pizzas. 30 pizzas.
109. Refer to the figure below. Moving from point B to point A, the opportunity cost of 25 more salads is:
A. B. C. D.
5 pizzas. 10 pizzas. 15 pizzas. 20 pizzas.
110. Refer to the figure below. The opportunity cost of making an additional salad:
A. B. C. D.
remains constant regardless of how many salads are made. increases as the number of salads increases. decreases as the number of pizzas decreases. decreases as the number of salads increases.
111. Refer to the figure below. If this restaurant goes from producing 20 to 25 pizzas per hour, then which of the following statements is true?
A. B. C. D.
It has It has It has It has
to to to to
give give give give
up up up up
112. Refer to the figure below. As the production of pizza increases, the opportunity cost of producing pizza:
A. B. C. D.
doesn't change. decreases. increases. become negative.
113. Refer to the figure below. Which of the following is true?
A. B. C. D.
Point A is efficient because it is farthest from the origin. Point D is efficient because it requires using the fewest resources. Point F is the most efficient because medical care is the highest there. Points B, C, E and F are efficient.
114. Refer to the figure below. Suppose that the government requires that resources be used efficiently. Which of the following would the government definitely not allow?
A. B. C. D.
Specialization in warhead production. Specialization in medical care production. Production at any point other than C. Production at point D.
115. Refer to the figure below. If this economy is currently producing at point C, then the opportunity cost of providing 100 additional units of medical care would be:
A. B. C. D.
800 400 200 100
116. Refer to the figure below. The opportunity cost of increasing medical care from 200 to 400 units is _ 600 units.
A. B. C. D.
greater than less than exactly the same as twice as much as
117. Production possibilities curves for large economies are generally bow•shaped because: A. B. C. D.
specialization gives some producers a comparative advantage. opportunity costs tend to decrease with increases in production. opportunity costs tend to increase with increases in production. as more resources are used to produce a good, those resources become less expensive.
the opportunity cost of increasing medical care from 400 to
118. The Principle of Increasing Opportunity Costs states that: A. B. C. D.
productive people do the hardest tasks first. when increasing production, resources with the lowest opportunity costs should be used first. when increasing production, resources with the lowest opportunity costs should be used last. opportunity costs increase when too little is produced.
119. You have noticed that your next•door neighbor, Mary, always works in the garden, and her husband, Joe, always walks the dog. You conclude that if Joe and Mary are efficient, then it must be the case that: A. B. C. D.
Mary has an absolute advantage in gardening. Joe has a comparative advantage in walking the dog. Mary's opportunity cost of walking the dog is lower than Joe's. Joe experiences increasing opportunity costs when he gardens, but not when he walks the dog.
120. The benefits of specialization can be used to explain why: A. B. C. D.
workers prefer to work on a variety of tasks during the day. machines are more productive than human workers. individuals and nations benefit from trade. big companies take advantage of smaller ones.
121. Moe divides his time between studying Physics and studying Economics. His production possibilities curve for his final grade in each class is shown below.
Both of Moe's professors require at least a 65 to pass and a 90 to earn an A. Which of the following is true?
A. B. C. D.
Moe Moe Moe Moe
can pass both classes. can pass economics, but only if he fails physics. can pass physics, but only if he fails economics. could earn an A in economics and still pass physics.
122. Moe divides his time between studying Physics and studying Economics. His production possibilities curve for his final grade in each class is shown below.
Which of the following is true?
A. B. C. D.
Moe has a comparative advantage in physics. Moe's opportunity cost of studying for each subject is increasing. Moe has a comparative advantage in economics. Moe has an absolute advantage in economics.
123. Moe divides his time between studying Physics and studying Economics. His production possibilities curve for his final grade in each class is shown below.
According to Moe's PPC, moving from a 70 to an 80 in economics:
A. B. C. D.
is inefficient. has a lower opportunity cost than moving from an 80 to a 90. is unattainable. has a higher opportunity cost than moving from an 80 to a 90.
124. Moe divides his time between studying Physics and studying Economics. His production possibilities curve for his final grade in each class is shown below.
If Moe moves from Point A to point C, his grade in Physics will go down by _
A. B. C. D.
less than the increase in more than the increase in more than the decrease in less than the decrease in
125. Moe divides his time between studying Physics and studying Economics. His production possibilities curve for his final grade in each class is shown below.
The Principle of Increasing Opportunity Cost is reflected in the fact that the opportunity cost going from 70 to 80 in economics is:
A. B. C. D.
lower than the opportunity cost of going from 80 to 90 in economics. higher than the opportunity cost of going from 80 to 90 in economics. lower than the opportunity cost of going from 80 to 90 in physics. the same as the opportunity cost of going from 70 to 80 in physics.
126. Moe divides his time between studying Physics and studying Economics. His production possibilities curve for his final grade in each class is shown below.
Moe needs to earn at least an 80 in both economics and physics to keep his scholarship. Given his current PPC, an 80 in both classes is
A. B. C. D.
unattainable attainable efficient inefficient
.
127. Refer to the figure below. For the nation whose PPC is shown, it must be true that:
A. B. C. D.
the nation's productive resources are better•suited to making milk than to making movies. the nation's productive resources are better•suited to making movies than to making milk. some of the nation's productive resources are better•suited to making milk, and some are better•suited to making movies. the nation has a comparative advantage in making milk.
128. Refer to the figure below. At point D, the opportunity cost of making milk is:
A. B. C. D.
low because the economy is specializing in making milk. high because productive resources that are better•suited to making movies are not being used to make milk. high because productive resources that are better•suited to making movies are being used to make milk. high because the economy is not operating efficiently.
129. Refer to the figure below. This economy would be operating at point B if:
A. B. C. D.
it was operating efficiently. the opportunity cost of making milk were higher than the opportunity cost of making movies. the opportunity cost of making movies were higher than the opportunity cost of making milk. resources that are better•suited to making movies were being used to make milk, while resources that are better•suited to making milk were being used to make movies.
130. Refer to the figure below. If this economy were currently operating at point D, then in order to make more movies:
A. B. C. D.
the first productive resources to switch to making movies should be those with the lowest opportunity cost of making milk. the first productive resources to switch to making movies should be those with the highest opportunity cost of making milk. no productive resources would need to switch from making milk to movies because point D is already efficient. no productive resources would need to switch from making milk to movies because each resource should continue to be used according to its comparative advantage.
131. The figure below shows Avery's weekly production possibilities curve for scarves.
For Avery, the opportunity cost of making a red scarf is:
A. B. C. D.
decreasing. increasing. 1 blue scarf. zero.
132. The figure below shows Avery's weekly production possibilities curve for scarves.
Avery's PPC would shift outward if she:
A. B. C. D.
knits more red scarves and fewer blue scarves each week. devotes less time to knitting each week. devotes more time to knitting each week. knits fewer red scarves and more blue scarves each week.
133. Economic growth can result from a(n): A. B. C. D.
increase in the amount of productive resources. increase in number of the minimum wage jobs. increase in the amount of consumer goods produced. decrease in the number of workers available.
134. Which of the following is NOT a reason why there are gains to specialization? A. B. C. D.
It eliminates many of the costs of switching from one task to another. It further improves skills through experience and practice. It increases the amount productive resources in the economy. It allows individuals to concentrate on the activities in which they have a comparative advantage.
135. An increase in an economy's productive resources will lead the production possibilities curve to: A. B. C. D.
shift inward. shift outward. become flatter. stay the same.
136. Suppose that Nepal invests less in new factories and equipment than does the United States. This will likely cause: A. B. C. D.
Nepal's production possibilities curve to shift outward faster than the U.S.'s. The U.S.'s production possibilities curve to shift inward faster than Nepal's. The U.S.'s production possibilities curve to shift outward faster than Nepal's. Nepal's production possibilities curve to shift inward faster than the U.S.'s.
137. If a nation restricts imports, it will: A. B. C. D.
benefit each individual citizen in that nation. increase the total value of goods and services produced in that nation. decrease the total value of goods and services produced in that nation. harm each individual citizen in that nation.
138. Regarding specialization, it is generally true that: A. B. C. D.
more specialization is always better. less specialization is always better. specialization imposes costs as well as benefits. more specialization is always worse.
139. You are the Minister of Trade for a small island country with the following annual PPC:
You are negotiating a trade agreement with a neighboring island with the following annual PPC:
As soon as you see the other island's PPC, you realize there are:
A. B. C. D.
140. You are the Minister of Trade for a small island country with the following annual PPC:
You are negotiating a trade agreement with a neighboring island with the following annual PPC:
If the other island's delegate offers to give you 2 fish for every 1 coconut you give them, you will:
A. B. C. D.
accept their offer because you do not have the comparative advantage in fish. refuse their offer because the opportunity cost to you of a coconut is more than 2 fish. accept their offer because you do not have an absolute advantage in fish. refuse their offer because the opportunity cost to you of a coconut is less than 2 fish.
141. You are the Minister of Trade for a small island country with the following annual PPC:
You are negotiating a trade agreement with a neighboring island with the following annual PPC:
What's the minimum number of fish you would be willing to accept in exchange for a coconut?
A. B. C. D.
5 4 3 2
142. You are the Minister of Trade for a small island country with the following annual PPC:
You are negotiating a trade agreement with a neighboring island with the following annual PPC:
If you offer to give the other island 1 coconut for every 4 fish they give you, then they will:
A. B. C. D.
refuse your offer because they have a comparative advantage in fish. accept your offer because your opportunity cost of a coconuts is less than 4 fish. refuse your offer because they can produce as many coconuts as you can. accept your offer because their opportunity cost of a coconut is greater than 4 fish.
143. You are the Minister of Trade for a small island country with the following annual PPC:
You are negotiating a trade agreement with a neighboring island with the following annual PPC:
Both islands specialize exclusively in the product for which they have a comparative advantage. You have agreed to give 350 coconuts to the other island in exchange for 1,300 fish. After the trade, your island has a total of coconuts and fish.
A. B. C. D.
150余 500余 150余 500余
2,800 1,300 1,300 1,500
144. You are the Minister of Trade for a small island country with the following annual PPC:
You are negotiating a trade agreement with a neighboring island with the following annual PPC:
Both islands specialize exclusively in the product for which they have a comparative advantage. You have agreed to give 350 coconuts to the other island in exchange for 1,300 fish. After the trade the other island has a total of _ coconuts and _ fish.
A. B. C. D.
850余 500余 350余 350余
1,200 1,200 1,500 1,200
145. If country A can produce more of practically everything than can country B, then which of the following statements is true? A. B. C. D.
Country A has no Country B cannot Trade can benefit Country B has no
incentive to trade with country B. have a comparative advantage in the production of any good that country A wants to buy. both countries. incentive to trade with country A.
146. As the differences in opportunity costs between the U.S. and its trading partners increase, the potential gains from specialization and trade A. B. C. D.
increase decrease stay the same become unpredictable
147. One reason there is political opposition to international trade is that: A. B. C. D.
the potential gains from specialization and trade are small. trade does not increase the total value of goods and services produced by a nation. the differences in opportunity costs between countries are small. not everyone benefits from trade.
148. One concern regarding the North American Free Trade Agreement (NAFTA) was that it would lead: A. B. C. D.
the total value of goods and services produced by the United States to fall. wages in Mexico to rise. highly skilled workers in the United States to lose their jobs. unskilled workers in the United States to lose their jobs.
149. When a nation reduces the barriers to international trade: A. B. C. D.
each individual citizen becomes better off. each individual citizen becomes worse off. the total value of all goods and services produced by the nation falls. the total value of all goods and serviced produced by the nation rises.
150. The benefits to specialization are even greater when two trading partners have: A. B. C. D.
absolute advantages in producing the same goods. similar consumption preferences. very similar opportunity costs. large differences in opportunity costs.
.
151. According to the textbook, the evidence indicates that NAFTA has: A. B. C. D.
reduced the wages of skilled workers in the United States. reduced the employment of unskilled workers in the United States significantly. stopped illegal immigration from Mexico. not significantly reduced the employment of unskilled workers in the United States.
152. According to the textbook, NAFTA was expected to help which country exploit its comparative advantage in the production of goods made by unskilled labor? A. B. C. D.
153. Outsourcing is a term increasingly used to refer to the act of: A. B. C. D.
hiring illegal immigrants. importing raw materials into the United States from other countries. exporting final goods to other countries. replacing relatively expensive American workers with low•wage workers overseas.
154. The fundamental reason firms outsource is that: A. B. C. D.
low•wage workers in other countries are more productive than are U.S. workers. hiring low•wage workers overseas reduces firms' costs. outsourcing increases employment overseas. U.S. workers cannot perform the tasks performed by workers in other countries.
155. When a U.S. firm engages in outsourcing, it benefits A. B. C. D.
the the the the
firm; U.S. U.S. U.S.
and harms
the U.S. consumers of the firm's products consumers of the firm's products; the firm consumers of the firm's products; the firm's U.S. employees consumers of the firm's products; the firm's foreign employees
156. All else equal, the jobs that are the least likely to be outsourced are those that: A. B. C. D.
do not involve face•to•face contact. can be done by a computer. require face•to•face communication. can be broken down into series of well•defined steps.
.
157. Which of the following jobs is least likely to be outsourced? A. B. C. D.
Flipping hamburgers Technical assistance over the phone for your computer Transcription of physicians' records Software design
KEY 1.
An individual has an absolute advantage in producing pizzas if that individual:
A. has a lower opportunity cost of producing pizzas than anyone else. B. can produce more pizzas in a given amount of time than anyone else. C. has a higher opportunity cost of producing pizzas than anyone else. D. charges the lowest price for pizzas.
Absolute advantage means being able to produce more in a given time period. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
2. A. B. C. D.
If Al has an absolute advantage over Beth in preparing meals, then: it takes Al more time to prepare a meal than Beth. the problem of scarcity applies to Beth but not to Al. Al's opportunity cost of preparing a meal is lower than is Beth's. Al can prepare more meals in a given time period than Beth.
Absolute advantage means being able to produce more in a given time period. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
3.
If Les can produce two pairs of pants per hour while Eva can produce one pair per hour, then it must be true that:
A. Les has a comparative advantage in producing pants. B. Les has an absolute advantage in producing pants. C. Eva has a comparative advantage in producing pants. D. Les has both comparative and absolute advantage in producing pants.
Absolute advantage means being able to produce more in a given time period. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
4.
If a nation can produce a more computers per year than any other nation, that nation has a(n)
_ advantage in the production of computers.
A. comparative B. absolute C. relative D. natural
Absolute advantage means being able to produce more in a given time period. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
5. A. B. C. D.
If you have a comparative advantage in a particular task, then: you you you you
are better at it than other people. give up more to accomplish that task than do others. give up less to accomplish that task than do others. have specialized in that task, while others have not.
Comparative advantage means having a lower opportunity cost. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
6.
Larry has a comparative advantage over his classmates in writing term papers if he:
A. can write term papers faster than his classmates. B. has an absolute advantage in writing term papers. C. always earns an A on his term papers. D. has a lower opportunity cost of writing term papers than his classmates.
Comparative advantage means having a lower opportunity cost. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
7.
If a nation has the lowest opportunity cost of producing a good, that nation has a(n)
in the production of that good.
Comparative advantage means having a lower opportunity cost. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
8.
Which of the following statements is true?
Comparative advantage and absolute advantage differ: you can have both at the same time, but you can also have one but not the other. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
9.
If Jane can produce 3 pairs of shoes per hour, while Bob can produce 2, then
has a(n)
A. Jane; absolute B. Jane; comparative C. Bob; absolute D. Bob; comparative
Absolute advantage means being able to produce more in a given time period. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
10. Refer to the table below. According to the table, Martha has the absolute advantage in:
A. pies. B. neither pies nor cakes. C. cakes. D. both pies and cakes. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
11. Refer to the table below. According to the table, Julia has the absolute advantage in:
A. pies. B. neither pies nor cakes. C. cakes. D. both pies and cakes. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
12. Refer to the table below. Martha's opportunity cost of making of a pie is:
A. 3/4 of a cake. B. 4/3 of a cake. C. 8 cakes. D. 80 cakes. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
13. Refer to the table below. Martha's opportunity cost of making a cake is:
A. 3/4 of a pie. B. 4/3 of a pie. C. 6 pies. D. 60 pies. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
14. Refer to the table below. Julia's opportunity cost of making a pie is:
A. B. C. D.
60 cakes 6 cakes 6/5 of a cake 5/6 of a cake AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
15. Refer to the table below. Julia's opportunity cost of making a cake is:
A. B. C. D.
60 cakes 6 cakes 6/5 of a cake 5/6 of a cake AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
16. Refer to the table above.
A. B. C. D.
has the comparative advantage in making pies and
the comparative advantage in making cakes.
Martha; Martha Julia; Julia Martha; Julia Julia; Martha AACSB: Analytic Blooms: Analyze
Difficulty: 03 Hard Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
17. Refer to the table below. Based on their comparative advantage, Martha should specialize in
while Julia should specialize in
.
A. pies; cakes B. cakes; pies C. neither pies nor cakes; both pies and cakes D. both pies and cakes; neither pies nor cakes AACSB: Analytic Blooms: Evaluate
Difficulty: 03 Hard Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
18. Suppose it takes Dan 5 minutes to make a sandwich and 15 minutes to make a smoothie, and it takes Tracy 6 minutes to make a sandwich and 12 minutes to make a smoothie. What is the opportunity cost to Dan of making a sandwich?
A. 1/3 of a smoothie B. 3 smoothies C. 15 smoothies D. 5 smoothies AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
19. Suppose it takes Dan 5 minutes to make a sandwich and 15 minutes to make a smoothie, and it takes Tracy 6 minutes to make a sandwich and 12 minutes to make a smoothie. Which of the following statements is correct?
A. Dan has the comparative advantage in smoothies, but Tracy has the absolute advantage in smoothies. B. Dan has the comparative and absolute advantage in sandwiches. C. Dan has the comparative and absolute advantage in smoothies. D. Dan has the comparative advantage in sandwiches, but Tracy has the absolute advantage in sandwiches. AACSB: Analytic Accessibility: Keyboard Navigation Difficulty: 03 Hard Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
20. Suppose it takes Dan 5 minutes to make a sandwich and 15 minutes to make a smoothie, and it takes Tracy 6 minutes to make a sandwich and 12 minutes to make a smoothie. Which of the following statements is correct?
A. Tracy should specialize in sandwiches and smoothies. B. Dan should specialize in smoothies, and Tracy should specialize in sandwiches. C. Dan should specialize in sandwiches, and Tracy should specialize in smoothies. D. Dan should specialize in both sandwiches and smoothies. AACSB: Analytic Accessibility: Keyboard Navigation Blooms: Analyze
Difficulty: 03 Hard Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
21. Suppose it takes Paul 3 hours to bake a cake and 2 hours to move the lawn, and suppose it takes Tom 2 hours to bake a cake and 1 hour to mow the lawn. Which of the following statements is correct? A. B. C. D.
Paul has the absolute advantage in baking cakes Paul has the comparative in mowing the lawn Paul has the comparative in baking cakes Paul has the absolute advantage in mowing the lawn.
For Paul, in the time it takes him to bake a cake, he could have mowed the lawn 1.5 times, and the time it takes him to move the lawn, he could have made 2/3 of a cake. For Tom, in the time it takes him to bake a cake, he could have mowed the lawn 2 times, and in the time it takes him to mow the lawn, he could have baked 1/2 of a cake. Thus, Paul has a comparative advantage in baking cakes (because 1.5 < 2), and Tom has a comparative advantage in mowing the lawn (because 1/2 < 2/3). Tom has an absolute advantage in both tasks since he can do each more quickly than Tom. AACSB: Analytic Accessibility: Keyboard Navigation Blooms: Analyze
Difficulty: 03 Hard Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
22. Suppose Cathy and Lewis work in a bakery making pies and cakes. Suppose it takes Cathy 1.5 hours to make a pie and 1 hour to make a cake, and suppose it takes Lewis 2 hours to make a pie and 1.5 hours to make a cake. Which of the following statements is correct?
A. Cathy has a comparative advantage in pies, and Lewis has an absolute advantage in pies. B. Cathy has a comparative and absolute advantage in pies. C. Lewis has a comparative and absolute advantage in pies. D. Lewis has a comparative advantage in pies, and Cathy has an absolute advantage in pies. AACSB: Analytic Accessibility: Keyboard Navigation Blooms: Analyze
Difficulty: 03 Hard Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
23. Suppose Cathy and Lewis work in a bakery making pies and cakes. Suppose it takes Cathy 1.5 hours to make a pie and 1 hour to make a cake, and suppose it takes Lewis 2 hours to make a pie and 1.5 hours to make a cake. Which of the following statements is correct?
A. B. C. D.
Cathy should specialize in both pies and cakes. There are no gains from specialization and trade. Lewis should specialize in pies, and Cathy should specialize in cakes. Cathy should specialize in pies, and Lewis should specialize in cakes. AACSB: Analytic Accessibility: Keyboard Navigation Blooms: Analyze
Difficulty: 03 Hard Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
24. Suppose Cathy and Lewis work in a bakery making pies and cakes. Suppose it takes Cathy 1.5 hours to make a pie and 1 hour to make a cake, and suppose it takes Lewis 2 hours to make a pie and 1.5 hours to make a cake. What is the opportunity cost to Cathy of making a cake?
A. 2/3 of a pie. B. 1 pie. C. 1.5 pies. D. 1.33 pies. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
25. Refer to the table below. According to the table, Corey has the absolute advantage in:
A. making pizza. B. neither making nor delivering pizza. C. delivering pizza. D. making and delivering pizza. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
26. Refer to the table below. According to the table, Pat has the absolute advantage in:
A. B. C. D.
making pizza. neither making nor delivering pizza. delivering pizza. making and delivering pizza. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
27. Refer to the table below. Corey's opportunity cost of making of a pizza is delivering:
A. B. C. D.
2 pizzas. 3/2 of a pizza. 2/3 of a pizza. 1/2 of a pizza. AACSB: Reflective Thinking Blooms: Apply
Blooms: Understand Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
28. Refer to the table below. Corey's opportunity cost of delivering of a pizza is making:
A. B. C. D.
6 pizzas. 12 pizzas. 2 pizzas. 1/2 of a pizza. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
29. Refer to the table below. Pat's opportunity cost of making a pizza is delivering:
A. B. C. D.
3 pizzas 2 pizzas 3/2 of a pizza 2/3 of a pizza AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
30. Refer to the table below. Pat's opportunity cost of delivering a pizza is making:
A. B. C. D.
12 pizzas 10 pizzas 3/2 of a pizza 2/3 of a pizza AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
31. Refer to the table below.
A. B. C. D.
has the comparative advantage in making pizza, and
_ has the comparative advantage in delivering pizza.
Corey; Corey Pat; Pat Pat; Corey Corey; Pat AACSB: Analytic Blooms: Analyze
Difficulty: 03 Hard Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
32. Refer to the table below. Based on their comparative advantages, Pat should specialize in
, and Corey should specialize in
.
A. delivering pizza; making pizza B. making pizza; delivering pizza C. neither making pizza nor delivering pizza; both making pizza and delivering pizza D. both making pizza and delivering pizza; neither making pizza nor delivering pizza AACSB: Analytic Blooms: Analyze
Difficulty: 03 Hard Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
33. Lou and Alex live together and share household chores. They like to cook some meals ahead of time and eat leftovers. The table below shows the number of rooms they can each clean and the number of meals they can each cook in an hour.
Which of the following is true?
A. B. C. D.
Lou has both an absolute advantage and a comparative advantage over Alex in both tasks. Alex has a comparative advantage over Lou in cleaning. Lou has a comparative advantage over Alex in cleaning. Alex has both an absolute advantage and a comparative advantage over Lou in both tasks. AACSB: Analytic Blooms: Analyze
Difficulty: 03 Hard Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
34. Lou and Alex live together and share household chores. They like to cook some meals ahead of time and eat leftovers. The table below shows the number of rooms they can each clean and the number of meals they can each cook in an hour.
If Alex and Lou work out an efficient arrangement for these two chores, then under that arrangement:
A. Alex and Lou each would do half of the cooking and half of the cleaning. B. Alex would do all of the cleaning, while Lou would do all the cooking. C. Lou would do all of the cleaning and all of the cooking. D. Lou would do all of the cleaning, while Alex would do all of the cooking. AACSB: Analytic Blooms: Analyze
Difficulty: 03 Hard Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
35. Lou and Alex live together and share household chores. They like to cook some meals ahead of time and eat leftovers. The table below shows the number of rooms they can each clean and the number of meals they can each cook in an hour.
For Alex, the opportunity cost of cleaning one room is making
meal(s); for Lou the opportunity cost of cleaning one room is making _
meal(s).
A. 4; 4 B. 1; 4/5 C. 1; 5/4 D. 3; 5 AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
36. Dent 'n' Scratch Used Cars and Trucks employs 3 salesmen. Data for their sales last month are shown in this table:
Based on last month's data,
has an absolute advantage in selling cars and
_ has an absolute advantage in selling trucks.
A. Joe; Joe B. Larry; Ralph C. Ralph; Larry D. Larry; Joe AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
37. Dent 'n' Scratch Used Cars and Trucks employs 3 salesmen. Data for their sales last month are shown in this table:
Based on last month's data, Larry's opportunity cost of selling a truck is selling:
A. 10 cars. B. 1/2 of a car. C. 1 car. D. 2 cars. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
38. Dent 'n' Scratch Used Cars and Trucks employs 3 salesmen. Data for their sales last month are shown in this table:
Based on last month's data, Joe's opportunity cost of selling a truck is selling:
A. 9 cars. B. 1 car. C. 4 cars. D. 1/3 of a car. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
39. Dent 'n' Scratch Used Cars and Trucks employs 3 salesmen. Data for their sales last month are shown in this table:
Based on last month's data, Ralph's opportunity cost of selling a truck is selling:
A. 4 cars. B. 1/3 of a car. C. 3 cars. D. 1/4 of a car. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
40. Dent 'n' Scratch Used Cars and Trucks employs 3 salesmen. Data for their sales last month are shown in this table:
Based on last month's data, Joe's opportunity cost of selling a car is
than Ralph's, and Joe's opportunity cost of selling a car is
than Larry's.
A. less; greater B. greater; less C. less; less D. greater; greater AACSB: Analytic Blooms: Analyze
Difficulty: 03 Hard Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
41. Dent 'n' Scratch Used Cars and Trucks employs 3 salesmen. Data for their sales last month are shown in this table:
Based on last month's data,
should specialize in truck sales, and _
should specialize in car sales.
A. Joe; Ralph B. Ralph; Larry C. Larry; Ralph D. Larry; Joe AACSB: Analytic Blooms: Analyze
Difficulty: 03 Hard Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
42. The textbook notes that the last time a major league batter hit .400 was in 1941. This is because: A. B. C. D.
the average quality of batters has fallen. the league imposes harsh penalties for steroid use. specialization by pitchers, infielders, and outfielders has made it harder for batters to hit. baseball diamonds have become larger.
Baseball players are most specialized now than in the past. It's hard to hit a .400 against a specialist pitcher. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
43. Ginger and Maryann are lost in the jungle, where the only things to eat are mangoes and fish. Ginger can gather more mangoes per hour than Maryann and can also catch more fish per hour than can Maryann. Therefore: A. B. C. D.
There are no gains to specialization and trade for Ginger. There are no gains to specialization and trade for Maryann. Maryann should specialize in the activity for which she has a comparative advantage. Ginger should specialize in the activity for which she has an absolute advantage.
Even if one person has an absolute advantage over the other in both activities, they will collectively accomplish more if each specializes in the activity for which she has a comparative advantage. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
44. In general, individuals and nations should specialize in producing those goods for which they have a(n): A. absolute advantage. B. comparative advantage. C. absolutely comparative advantage. D. absolute advantage and a comparative advantage.
The Principle of Comparative Advantage states that people should specialize in the activities for which their opportunity cost is the lowest (that is, the activities in which they have a comparative advantage). AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
45. In general, individuals and nations should specialize in producing goods A. B. C. D.
that they can that they can for which they for which they
other individuals or nations.
produce more quickly than produce less quickly than have a lower opportunity cost compared to have a higher opportunity cost compared to
The Principle of Comparative Advantage states that people should specialize in the activities for which their opportunity cost is the lowest. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
46. A country is most likely to have a comparative advantage in the production of cars if: A. B. C. D.
it imports most of the raw materials necessary to produce cars. its citizens prefer driving cars to other forms of transportation. it has strict environmental protection laws governing automobile emissions. it has a relative abundance in the natural resources needed to produce cars.
One source of comparative advantage is large endowments of natural resources. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
47. The United States generally has a comparative advantage in the development of technology because it has: A. large amounts of natural resources. B. a disproportionate share of the world's best research universities. C. the greatest need for new technology. D. patent laws, which no other country has.
The Unites States has a disproportionate share of the world's leading research universities where technology is developed and scientists are trained. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
48. The emergence of English as the de facto world language has
a comparative advantage in the production of books, movies and popular music:
A. given English•speaking countries B. given non•English•speaking countries C. had no effect on which country has D. given all countries
The emergence of English as the de facto world language has given English•speaking countries a comparative advantage in language•based production. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
49. The United States was unable to maintain its dominance in the production of televisions because: A. the highly technical skills necessary to produce televisions are greater in other countries. B. the raw materials necessary to build televisions became scarce in the United States. C. the product designs evolved too rapidly for engineers in the United States to keep up. D. automated techniques allowed production to be outsourced to countries with less•skilled workers.
When television production required highly•paid and highly•skilled workers, the United States had a comparative advantage in producing televisions, but once production became automated, less•skilled workers in low•wage countries could produce televisions at a lower cost. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Topic: Exchange and Opportunity Cost
50. A graph that illustrates the maximum amount of one good that can be produced for every possible level of production of the other good is called a(n): A. production possibilities curve. B. consumption possibilities curve. C. production function. D. supply curve.
The production possibilities curve describes the maximum amount of one good that can be produced for every possible amount produced of another good. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
51. The production possibilities curve shows: A. B. C. D.
the minimum production of one good for every possible production level of the other good. how increasing the resources used to produce one good increases the production of the other good. the maximum production of one good for every possible production level of the other good. how increasing the production of one good allows production of the other good to also rise.
The production possibilities curve describes the maximum amount of one good that can be produced for every possible level of production of the other good. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
52. Points that lie outside the production possibilities curve are A. B. C. D.
, and points that lie inside the production possibilities curve are
.
efficient; inefficient inefficient; efficient unattainable; attainable attainable; unattainable
Points that lie outside the production possibilities curve cannot be produced with currently available resources, while those that lie inside the production possibilities curve can be produced with currently available resources. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
53. Points that lie beneath the production possibilities curve are: A. B. C. D.
unattainable and inefficient unattainable but efficient attainable but inefficient attainable and efficient
Points that lie beneath the production possibilities curve are attainable because they can be produced with currently available resources, but they are inefficient because it is possible to increase the production of one good without a reduction in the production of the other. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
54. If a country is producing at point where an increase in the production of one good requires a reduction in the production of another good, then it must be producing at an: A. inefficient point. B. efficient point. C. unattainable point. D. undesirable point.
By definition, an efficient point is any combination of goods for which currently available resources do not allow an increase in the production of one good without a reduction in the production of some other good. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
55. Suppose Colin brews beer and makes cheese. If Colin can increase his production of beer without decreasing his production of cheese, then he is producing at an: A. inefficient point. B. efficient point. C. unattainable point. D. ideal point.
By definition, an inefficient point is any combination of goods for which currently available resources enable an increase in the production of one good without a reduction in the production of some other good. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
56. The downward slope of the production possibilities curve illustrates the: A. Scarcity Principle. B. Cost•Benefit Principle. C. Incentive Principle. D. Principle of Comparative Advantage.
The downward slope of the production possibilities curve shows that having more of one good means having less of the other. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
57. The figure below shows the production possibilities curve for the island of Genovia:
The opportunity cost of producing a car in Genovia is:
A. 5,000 tons of agricultural products. B. 500 tons of agricultural products. C. 5 tons of agricultural products. D. 50 tons of agricultural products. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
58. The figure below shows the production possibilities curve for the island of Genovia:
The opportunity cost of producing one ton of agricultural products in Genovia is:
A. B. C. D.
1,000 cars. 1 car. 1/5 of a car. 1/50 of a car. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
59. The figure below shows the production possibilities curve for the island of Genovia:
If 500 cars are produced in Genovia, a maximum of
tons of agricultural products can be produced.
A. 50,000 B. 25,000 C. 45,000 D. 40,000 AACSB: Analytic Blooms: Analyze
Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
60. The slope of a production possibilities curve is
because
A. negative; producing more of one good requires producing B. negative; producing less of one good requires producing C. positive; producing more of one good requires producing D. positive; producing more of one good requires producing
_.
less of the other less of the other more of the other less of the other
The downward slope of the production possibilities curve shows that having more of one good means having less of the other. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
61. The figure below shows Becky's daily production possibilities curve for dresses and skirts.
The maximum number of dresses that Becky can make in a day is represented by point:
A. U B. T C. V D. W AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
62. The figure below shows Becky's daily production possibilities curve for dresses and skirts.
The maximum number of skirts that Becky can make in a day is represented by point:
A. U B. T C. V D. Z AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
63. The figure below shows Becky's daily production possibilities curve for dresses and skirts.
Point U is:
A. B. C. D.
attainable. efficient. unattainable. inefficient. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
64. The figure below shows Becky's daily production possibilities curve for dresses and skirts.
Of the labeled points, only
A. B. C. D.
are attainable.
T and U X, Y, and Z W, X, Y, Z, and V W, X, Y, Z, V, and T AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
65. The figure below shows Becky's daily production possibilities curve for dresses and skirts.
Of the labeled points, only
A. B. C. D.
are efficient.
T and U X, Y, and Z W, X, Y, Z, and V W, X, Y, Z, V, and T AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
66. The figure below shows Becky's daily production possibilities curve for dresses and skirts.
Point T is:
A. attainable B. efficient C. both attainable and efficient D. neither attainable nor efficient AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
67. The figure below shows Becky's daily production possibilities curve for dresses and skirts.
Point Y is
A. B. C. D.
, and point V is
.
efficient; inefficient inefficient; efficient efficient; efficient inefficient; inefficient AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
68. The figure below shows Becky's daily production possibilities curve for dresses and skirts.
Relative to point X, at point Y:
A. more dresses and more skirts are produced. B. more skirts and fewer dresses are produced. C. more dresses and fewer skirts are produced. D. fewer skirts and fewer dresses are produced. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
69. Refer to the figure below. For Pat, the opportunity cost of removing one bag of trash is planting:
A. 100 bulbs. B. 5 bulbs. C. 1/100 of a bulb. D. 1/5 of a bulb. AACSB: Analytic Blooms: Apply Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
70. Refer to the figure below. For Pat, the opportunity cost of planting one bulb is removing:
A. 20 bags of trash. B. 5 bags of trash. C. 1/20 of a bag of trash. D. 1/5 of a bag of trash. AACSB: Analytic Blooms: Apply Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
71. Refer to the figure below. For Chris, the opportunity cost of removing one bag of trash is planting:
A. B. C. D.
25 bulbs. 1/25 of a bulb. 3 bulbs. 1/3 of a bulb. AACSB: Analytic Blooms: Apply Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
72. Refer to the figure below. For Chris, the opportunity cost of planting one bulb is removing:
A. B. C. D.
25 bags of trash. 1/25 of a bag of trash. 3 bags of trash. 1/3 of a bag of trash. AACSB: Analytic Blooms: Apply Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
73. Refer to the figure below. If Pat and Chris were to specialize in the task in which each has a comparative advantage:
A. Chris would plant bulbs and Pat would remove trash. B. Chris would remove trash and Pat would plant bulbs. C. Pat and Chris would each spend half of their time each task. D. both Pat and Chris would plant bulbs because they both have an absolute advantage in that task. AACSB: Analytic Blooms: Analyze
Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
74. Refer to the figure below. If Pat and Chris each spend half their time on each task, then:
A. B. C. D.
the outcome will be efficient. they will plant more bulbs and remove fewer bags of trash than if they had each specialized in the task at which they have a comparative advantage. they will plant fewer bulbs and remove fewer bags of trash than if they each had specialized in the task at which they have a comparative advantage. the outcome will be unattainable. AACSB: Analytic Blooms: Evaluate
Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
75. On a graph of a production possibilities curve, if a point is attainable, then it: A. must be efficient. B. might or might not be efficient. C. is efficient only if it does not exhaust all currently available resources. D. must completely exhaust all currently available resources.
Points along and beneath the production possibilities curve are attainable, but only points along the curve are efficient. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
76. Any combination of goods that can be produced with currently available resources is an: A. attainable point. B. efficient point. C. inefficient point. D. attainable and efficient point.
Attainable points are defined as any combination of goods that can be produced using currently available resources. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
77. On a graph of a production possibilities curve, an inefficient point is: A. necessarily an attainable point. B. not necessarily an attainable point. C. necessarily an unattainable point. D. possibly an unattainable point.
Inefficient points can be produced using currently available resources. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
78. Consider a graph of a production possibilities curve. If a producer is operating at an inefficient point, then that producer: A. cannot produce more of one good without giving up some of the other good. B. can produce more of one good without producing less of the other good. C. must be at an unattainable point on the production possibilities curve. D. must be specializing in activities for which it has a comparative advantage.
Inefficient points lie below the production possibilities curve, so it is possible to produce more of one good without producing less of the other good. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
79. Points that lie below the production possibilities curve are inefficient because: A. more of one good could be produced without producing less of the other. B. producing more of one good means producing less of the other. C. producers face scarcity. D. too many goods are being produced.
Inefficient points lie below the production possibilities curve, so it is possible to produce more of one good without producing less of the other. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
80. Refer to the figure below. Growing 1,000 bushels of wheat and no bushels of corn each year is:
A. B. C. D.
inefficient and unattainable. inefficient but attainable. efficient but unattainable. efficient and attainable. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
81. Refer to the figure below. It is efficient for this farmer to:
A. B. C. D.
grow 500 bushels of wheat grow 250 bushels of wheat grow 500 bushels of wheat grow 1000 bushels of wheat
and and and and
500 bushels of corn. 500 bushels of corn. 250 bushels of corn. 500 bushels of corn.
AACSB: Analytic Blooms: Apply Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
82. Refer to the figure below. The opportunity cost of producing one bushel of corn is:
A. 2 bushels of wheat. B. ½ of a bushel of wheat. C. 500 bushels of wheat. D. 250 bushels of wheat. AACSB: Analytic Blooms: Apply Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
83. Refer to the figure below. The opportunity cost of producing one bushel of wheat is:
A. 2 bushels of corn. B. ½ of a bushel of corn. C. 1,000 bushels of corn. D. 500 bushels of corn. AACSB: Analytic Blooms: Apply Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
84. If a given production combination is known to be attainable, then it: A. must be on the production possibilities curve. B. must be an inefficient point. C. must be an efficient point. D. could be either an inefficient or efficient point.
Attainable points are those that lie on or below the production possibilities curve. Points on the curve are efficient; points below the curve are inefficient. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
85. If a given production combination is efficient, then it must be: A. above the production possibilities curve. B. on the production possibilities curve. C. either an attainable or unattainable point. D. below the production possibilities curve.
Points on the production possibilities curve are efficient. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
86. Working efficiently, Jordan can write 3 essays and outline 4 chapters each week. It must be true that: A. B. C. D.
6 essays and 0 chapter outlines would be unattainable. 2 essays and 3 chapter outlines would be efficient. 3 essays and 5 chapter outlines would be unattainable. 4 essays and 3 chapter outlines would be both attainable and efficient.
If a point is efficient, then it is impossible to have more of one activity without giving up some of the other. So, Jordan cannot increase the number of outlined chapters to 5 while still writing 3 essays. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
87. Assume point A on a linear production possibilities curve represents the combination of 12 coffees and 3 cappuccinos, and point B represents 3 coffees and 6 cappuccinos. Suppose coffees are on the vertical axis and cappuccinos are on the horizontal axis. The absolute value of the slope of the production possibilities curve between points A and B equals:
A. B. C. D.
6 4 3 1/3 AACSB: Analytic Blooms: Apply Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
88. Assume point A on a linear production possibilities curve represents the combination of 12 coffees and 3 cappuccinos, and point B represents 3 coffees and 6 cappuccinos. Suppose coffees are on the vertical axis and cappuccinos are on the horizontal axis. The opportunity cost of a cup of coffee is:
A. B. C. D.
3 cappuccinos. 9 cappuccinos. 1/3 of a cappuccino. 6 cappuccinos. AACSB: Analytic Blooms: Apply Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
89. Generally, on a linear two•good production possibilities curve, the opportunity cost of the good measured on the vertical axis is: A. one minus the opportunity cost of the good measured on the horizontal axis. B. the reciprocal of the opportunity cost of the good measured on the horizontal axis. C. the absolute value of the slope of the production possibilities curve. D. the negative of the opportunity cost of the good measured on the horizontal axis.
The absolute value of the slope of the production possibilities curve gives you the opportunity cost of the good measured on the horizontal axis, and the reciprocal of the absolute value of the slope of the production possibilities curve gives you the opportunity cost the good measured on the vertical axis. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
90. If a linear, two-good production possibilities curve has a slope of ‒2, then: A. having an additional unit of the good measured on the vertical axis means giving up 2 units of the good measured on the horizontal axis. B. having an additional unit of the good measured on the vertical axis means giving up ½ of an unit of the good measured on the horizontal axis. C. you have an absolute advantage in the good measured on the vertical axis. D. you have a comparative advantage in the good measured on the vertical axis.
The absolute value of the slope of the production possibilities curve gives you the opportunity cost of the good measured on the horizontal axis, and the reciprocal of the absolute value of the slope of the production possibilities curve gives you the opportunity cost the good measured on the vertical axis. AACSB: Analytic Accessibility: Keyboard Navigation Blooms: Analyze Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
91. The idea that tradeoffs have to be made when resources are scarce is reflected in the fact that: A. B. C. D.
points below the production possibilities curve are efficient. points below the production possibilities curve are inefficient. the production possibilities curve has a negative slope. the slope of a linear production possibilities is constant.
The downward slope of the production possibilities curve captures the idea that because resources are limited, having more of one good means having less of the other. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
92. In a two•person, two•good economy, the gains to specialization will be larger when: A. B. C. D.
one person has an absolute advantage in both goods. neither person has an absolute advantage. there are small differences between the individuals in their opportunity costs of producing the two goods. there are large differences between the individuals in their opportunity costs of producing the two goods.
The greater are the differences between individuals in their opportunity costs, the greater are the gains to specialization. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
93. According to the Principle of Increasing Opportunity Cost, in expanding the production of any good, we should start by utilizing the resources that: A. B. C. D.
we have the most of. we have the least of. have the highest opportunity cost. have the lowest opportunity cost.
The Principle of Increasing Opportunity Cost stats that in expanding the production of any good, we should first employ those resources that have the lowest opportunity cost, and only afterwards turn to resources with higher opportunity costs. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
94. Smith and Jones comprise a two•person economy. Their hourly rates of production are shown below.
The opportunity cost of making a calculator for Smith is _
and for Jones it is
.
A. 0.10 computers; 0.05 computers B. 10 computers; 20 computers C. 1 computer; 0.5 computers D. 0.6 computers; 1.2 computers AACSB: Analytic Blooms: Apply Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
95. Smith and Jones comprise a two•person economy. Their hourly rates of production are shown below.
If Smith and Jones devote all of their resources to producing computers, then the maximum number of computers they can produce in an hour is:
A. B. C. D.
120. 6. 16. 10. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
96. Smith and Jones comprise a two•person economy. Their hourly rates of production are shown below.
Suppose Smith and Jones begin by producing 16 computers and 0 calculators per hour. If they wish to produce 14 computers and 40 calculators per hour efficiently, then Smith should spend , and Jones should spend .
A. 1 hour on computers; 40 minutes on computers and 20 minutes on calculators B. 1 hour on computers; 20 minutes on computers and 40 minutes on calculators C. 30 minutes on each; 30 minutes on each D. 45 minutes on computers and 15 on calculators; 1 hour on calculators AACSB: Analytic Blooms: Apply Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
97. Smith and Jones comprise a two•person economy. Their hourly rates of production are shown below.
Suppose Smith and Jones begin by producing 0 computers and 220 calculators per hour. If they wish to produce 2 computers and 200 calculators per hour efficiently, then Smith should spend , and Jones should spend .
A. B. C. D.
30 minutes on each; 30 minutes on each 48 minutes on computers and 12 minutes on calculators; 1 hour on calculators 1 hour on calculators; 10 minutes on computers and 50 minutes on calculators 12 minutes on computers and 48 minutes on calculators; 1 hour on calculators AACSB: Analytic Blooms: Apply Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
98. Smith and Jones comprise a two•person economy. Their hourly rates of production are shown below.
If Smith and Jones are dividing their time efficiently and producing more than 10 computers and fewer than 120 calculators per hour, then Smith will will .
and Jones
A. produce only computers; produce only calculators B. produce only computers; split his time between computers and calculators C. split his time between computers and calculators; produce only computers D. produce only calculators; produce only computers AACSB: Analytic Blooms: Analyze
Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
99. Smith and Jones comprise a two•person economy. Their hourly rates of production are shown below.
If Smith and Jones are dividing their time efficiently and producing fewer than 10 computers and more than 120 calculators per hour, then Smith will will .
and Jones
A. split his time between computers and calculators; produce only calculators B. produce only calculators; split his time between computers and calculators C. produce only calculators; produce only computers D. produce only computers; produce only calculators AACSB: Analytic Blooms: Analyze
Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
100. Smith and Jones comprise a two•person economy. Their hourly rates of production are shown below.
Suppose Smith and Jones begin by producing 100 calculators per hour; as Smith and Jones choose to efficiently produce fewer computers and more calculators, should devote more time to calculators because his .
A. B. C. D.
Smith; absolute advantage is larger Jones; absolute advantage is smaller Jones; opportunity costs are lower Smith; opportunity costs are lower AACSB: Analytic Blooms: Evaluate
Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
101. Earth Movers & Shakers operates 3 iron ore mines. The table below shows each mine's total daily production and the current number of miners at each mine. All miners work for the same wage, and each miner in any given mine produces the same number of tons per day as every other miner in that mine.
The opportunity cost of moving one miner from Mother Lode to another mine is:
A. B. C. D.
2 tons per day. 3 tons per day. 4 tons per day. 1 ton per day. AACSB: Analytic Blooms: Apply Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
102. Earth Movers & Shakers operates 3 iron ore mines. The table below shows each mine's total daily production and the current number of miners at each mine. All miners work for the same wage, and each miner in any given mine produces the same number of tons per day as every other miner in that mine.
The opportunity cost of moving one miner from Scraping Bottom to another mine is:
A. 0 tons per day. B. 3 tons per day. C. 4 tons per day. D. 5 tons per day. AACSB: Analytic Blooms: Apply Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
103. Earth Movers & Shakers operates 3 iron ore mines. The table below shows each mine's total daily production and the current number of miners at each mine. All miners work for the same wage, and each miner in any given mine produces the same number of tons per day as every other miner in that mine.
The opportunity cost of moving one miner from Middle Drift to another mine is:
A. B. C. D.
1 ton per day. 3 tons per day. 4 tons per day. 5 tons per day. AACSB: Analytic Blooms: Apply Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
104. Earth Movers & Shakers operates 3 iron ore mines. The table below shows each mine's total daily production and the current number of miners at each mine. All miners work for the same wage, and each miner in any given mine produces the same number of tons per day as every other miner in that mine.
Suppose Earth Movers & Shakers needs to fill an order for 60 tons of ore in a single day. If it has no other orders for that day, it should:
A. take it all from Mother Lode. B. take it all from Middle Drift. C. take 30 tons from Scraping Bottom and 30 tons from Middle Drift. D. take 20 tons from each of the three mines. AACSB: Analytic Blooms: Analyze
Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
105. Earth Movers & Shakers operates 3 iron ore mines. The table below shows each mine's total daily production and the current number of miners at each mine. All miners work for the same wage, and each miner in any given mine produces the same number of tons per day as every other miner in that mine.
Suppose Earth Movers & Shakers needs to fill an order for 100 tons of ore in a single day. If it has no other orders to fill that day, and it's not possible to transfer miners from one mine to another, it should:
A. take it all from Mother Lode. B. take 75 tons from Middle Drift and 25 tons from Mother Lode. C. take 75 tons from Middle Drift and 25 tons from Scraping Bottom. D. take 30 tons from Scraping Bottom and 70 tons from Mother Lode. AACSB: Analytic Blooms: Analyze
Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
106. Refer to the figure below. If this restaurant makes 75 salads in one hour, then what's the maximum number of pizzas it can make in that same hour?
A. B. C. D.
0 10 20 30 AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
107. Refer to the figure below. Relative to point B, at point C this restaurant is:
A. making more pizzas and more salads. B. making more pizzas and fewer salads. C. making fewer pizzas and more salads. D. operating more efficiently. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
108. Refer to the figure below. Moving from point C to point B, the opportunity cost of 25 more salads is:
A. 5 pizzas. B. 10 pizzas. C. 15 pizzas. D. 30 pizzas. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
109. Refer to the figure below. Moving from point B to point A, the opportunity cost of 25 more salads is:
A. 5 pizzas. B. 10 pizzas. C. 15 pizzas. D. 20 pizzas. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
110. Refer to the figure below. The opportunity cost of making an additional salad:
A. remains constant regardless of how many salads are made. B. increases as the number of salads increases. C. decreases as the number of pizzas decreases. D. decreases as the number of salads increases. AACSB: Analytic Blooms: Analyze Blooms: Understand Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
111. Refer to the figure below. If this restaurant goes from producing 20 to 25 pizzas per hour, then which of the following statements is true?
A. B. C. D.
It has It has It has It has
to to to to
give give give give
up up up up
Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
112. Refer to the figure below. As the production of pizza increases, the opportunity cost of producing pizza:
A. B. C. D.
doesn't change. decreases. increases. become negative. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
113. Refer to the figure below. Which of the following is true?
A. B. C. D.
Point A is efficient because it is farthest from the origin. Point D is efficient because it requires using the fewest resources. Point F is the most efficient because medical care is the highest there. Points B, C, E and F are efficient. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
114. Refer to the figure below. Suppose that the government requires that resources be used efficiently. Which of the following would the government definitely not allow?
A. Specialization in warhead production. B. Specialization in medical care production. C. Production at any point other than C. D. Production at point D. AACSB: Analytic Blooms: Analyze
Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
115. Refer to the figure below. If this economy is currently producing at point C, then the opportunity cost of providing 100 additional units of medical care would be:
A. 800 warheads. B. 400 warheads. C. 200 warheads. D. 100 warheads. AACSB: Analytic Blooms: Apply Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
116. Refer to the figure below. The opportunity cost of increasing medical care from 200 to 400 units is _ 600 units.
the opportunity cost of increasing medical care from 400 to
A. greater than B. less than C. exactly the same as D. twice as much as AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
117. Production possibilities curves for large economies are generally bow•shaped because: A. B. C. D.
specialization gives some producers a comparative advantage. opportunity costs tend to decrease with increases in production. opportunity costs tend to increase with increases in production. as more resources are used to produce a good, those resources become less expensive.
When a production possibilities curve is bow•shaped, this reflects increasing opportunity costs. Increasing opportunity costs arise in large economies because, when expanding production, resources with the lowest opportunity cost should be used first. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
118. The Principle of Increasing Opportunity Costs states that: A. productive people do the hardest tasks first. B. when increasing production, resources with the lowest opportunity costs should be used first. C. when increasing production, resources with the lowest opportunity costs should be used last. D. opportunity costs increase when too little is produced.
This principle is also known as the Low•Hanging•Fruit Principle: take advantage of your least•cost opportunities first. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
119. You have noticed that your next•door neighbor, Mary, always works in the garden, and her husband, Joe, always walks the dog. You conclude that if Joe and Mary are efficient, then it must be the case that: A. Mary has an absolute advantage in gardening. B. Joe has a comparative advantage in walking the dog. C. Mary's opportunity cost of walking the dog is lower than Joe's. D. Joe experiences increasing opportunity costs when he gardens, but not when he walks the dog.
Everyone does best when each person specializes in the activity in which he or she has a comparative advantage. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
120. The benefits of specialization can be used to explain why: A. B. C. D.
workers prefer to work on a variety of tasks during the day. machines are more productive than human workers. individuals and nations benefit from trade. big companies take advantage of smaller ones.
Specialization allows two parties with different opportunity costs to benefit from trade because by specializing they can increase their combined output. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
121. Moe divides his time between studying Physics and studying Economics. His production possibilities curve for his final grade in each class is shown below.
Both of Moe's professors require at least a 65 to pass and a 90 to earn an A. Which of the following is true?
A. Moe can pass both classes. B. Moe can pass economics, but only if he fails physics. C. Moe can pass physics, but only if he fails economics. D. Moe could earn an A in economics and still pass physics. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium
Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
122. Moe divides his time between studying Physics and studying Economics. His production possibilities curve for his final grade in each class is shown below.
Which of the following is true?
A. Moe has a comparative advantage in physics. B. Moe's opportunity cost of studying for each subject is increasing. C. Moe has a comparative advantage in economics. D. Moe has an absolute advantage in economics. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium
Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
123. Moe divides his time between studying Physics and studying Economics. His production possibilities curve for his final grade in each class is shown below.
According to Moe's PPC, moving from a 70 to an 80 in economics:
A. is inefficient. B. has a lower opportunity cost than moving from an 80 to a 90. C. is unattainable. D. has a higher opportunity cost than moving from an 80 to a 90. AACSB: Analytic Blooms: Analyze Difficulty: 03 Hard
Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
124. Moe divides his time between studying Physics and studying Economics. His production possibilities curve for his final grade in each class is shown below.
If Moe moves from Point A to point C, his grade in Physics will go down by _
A. less than the increase in B. more than the increase in C. more than the decrease in D. less than the decrease in AACSB: Analytic Blooms: Analyze Difficulty: 03 Hard
Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
125. Moe divides his time between studying Physics and studying Economics. His production possibilities curve for his final grade in each class is shown below.
The Principle of Increasing Opportunity Cost is reflected in the fact that the opportunity cost going from 70 to 80 in economics is:
A. lower than the opportunity cost of going from 80 to 90 in economics. B. higher than the opportunity cost of going from 80 to 90 in economics. C. lower than the opportunity cost of going from 80 to 90 in physics. D. the same as the opportunity cost of going from 70 to 80 in physics. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium
Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
126. Moe divides his time between studying Physics and studying Economics. His production possibilities curve for his final grade in each class is shown below.
Moe needs to earn at least an 80 in both economics and physics to keep his scholarship. Given his current PPC, an 80 in both classes is
.
A. unattainable B. attainable C. efficient D. inefficient AACSB: Reflective Thinking
Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
127. Refer to the figure below. For the nation whose PPC is shown, it must be true that:
A. B. C. D.
the nation's productive resources are better•suited to making milk than to making movies. the nation's productive resources are better•suited to making movies than to making milk. some of the nation's productive resources are better•suited to making milk, and some are better•suited to making movies. the nation has a comparative advantage in making milk. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
128. Refer to the figure below. At point D, the opportunity cost of making milk is:
A. B. C. D.
low because the economy is specializing in making milk. high because productive resources that are better•suited to making movies are not being used to make milk. high because productive resources that are better•suited to making movies are being used to make milk. high because the economy is not operating efficiently. AACSB: Analytic Blooms: Analyze
Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
129. Refer to the figure below. This economy would be operating at point B if:
A. B. C. D.
it was operating efficiently. the opportunity cost of making milk were higher than the opportunity cost of making movies. the opportunity cost of making movies were higher than the opportunity cost of making milk. resources that are better•suited to making movies were being used to make milk, while resources that are better•suited to making milk were being used to make movies. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
130. Refer to the figure below. If this economy were currently operating at point D, then in order to make more movies:
A. the first productive resources to switch to making movies should be those with the lowest opportunity cost of making milk. B. the first productive resources to switch to making movies should be those with the highest opportunity cost of making milk. C. no productive resources would need to switch from making milk to movies because point D is already efficient. D. no productive resources would need to switch from making milk to movies because each resource should continue to be used according to its comparative advantage. AACSB: Analytic Blooms: Evaluate
Difficulty: 03 Hard Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Topic: Comparative Advantage and Production Possibilities
131. The figure below shows Avery's weekly production possibilities curve for scarves.
For Avery, the opportunity cost of making a red scarf is:
A. B. C. D.
decreasing. increasing. 1 blue scarf. zero. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging•Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage.
Topic: Comparative Advantage and Production Possibilities
132. The figure below shows Avery's weekly production possibilities curve for scarves.
Avery's PPC would shift outward if she:
A. B. C. D.
knits more red scarves and fewer blue scarves each week. devotes less time to knitting each week. devotes more time to knitting each week. knits fewer red scarves and more blue scarves each week. AACSB: Reflective Thinking Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•03 Identify factors that shift the menu of production possibilities. Topic: Factors That Shift the Economy's Production Possibilities Curve
133. Economic growth can result from a(n): A. increase in the amount of productive resources. B. increase in number of the minimum wage jobs. C. increase in the amount of consumer goods produced. D. decrease in the number of workers available.
An increase productive resources will shift the PPC outward. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•03 Identify factors that shift the menu of production possibilities. Topic: Factors That Shift the Economy's Production Possibilities Curve
134. Which of the following is NOT a reason why there are gains to specialization? A. B. C. D.
It eliminates many of the costs of switching from one task to another. It further improves skills through experience and practice. It increases the amount productive resources in the economy. It allows individuals to concentrate on the activities in which they have a comparative advantage.
Specialization does not increase the amount of productive resources; it simply enables those resources to be used more efficiently. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•03 Identify factors that shift the menu of production possibilities. Topic: Factors That Shift the Economy's Production Possibilities Curve
135. An increase in an economy's productive resources will lead the production possibilities curve to: A. shift inward. B. shift outward. C. become flatter. D. stay the same.
An increase in an economy's productive resources makes it possible to increase the production of all goods, leading the PPC to shift outward. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•03 Identify factors that shift the menu of production possibilities. Topic: Factors That Shift the Economy's Production Possibilities Curve
136. Suppose that Nepal invests less in new factories and equipment than does the United States. This will likely cause: A. B. C. D.
Nepal's production possibilities curve to shift outward faster than the U.S.'s. The U.S.'s production possibilities curve to shift inward faster than Nepal's. The U.S.'s production possibilities curve to shift outward faster than Nepal's. Nepal's production possibilities curve to shift inward faster than the U.S.'s.
Investment in technology and productive resources shifts the PPC outward. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•03 Identify factors that shift the menu of production possibilities. Topic: Factors That Shift the Economy's Production Possibilities Curve
137. If a nation restricts imports, it will: A. B. C. D.
benefit each individual citizen in that nation. increase the total value of goods and services produced in that nation. decrease the total value of goods and services produced in that nation. harm each individual citizen in that nation.
Restricting imports lowers the total value of goods and services produced by a nation, but individual citizens could be better or worse off. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•03 Identify factors that shift the menu of production possibilities. Topic: Factors That Shift the Economy's Production Possibilities Curve
138. Regarding specialization, it is generally true that: A. B. C. D.
more specialization is always better. less specialization is always better. specialization imposes costs as well as benefits. more specialization is always worse.
Specialization reduces variety which some workers enjoy. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•03 Identify factors that shift the menu of production possibilities. Topic: Factors That Shift the Economy's Production Possibilities Curve
139. You are the Minister of Trade for a small island country with the following annual PPC:
You are negotiating a trade agreement with a neighboring island with the following annual PPC:
As soon as you see the other island's PPC, you realize there are:
A. no gains from trade because your both have the same comparative advantage. B. no gains from trade because there is no difference in your ability to harvest coconuts. C. no gains from trade because the other island has an absolute advantage. D. gains from trade because your island has a comparative advantage in coconuts. AACSB: Analytic Blooms: Analyze Difficulty: 03 Hard
Learning Objective: 02•04 Explain the role of comparative advantage in international trade and describe why some jobs are more vulnerable to outsourcing than others.
140. You are the Minister of Trade for a small island country with the following annual PPC:
You are negotiating a trade agreement with a neighboring island with the following annual PPC:
If the other island's delegate offers to give you 2 fish for every 1 coconut you give them, you will:
A. accept their offer because you do not have the comparative advantage in fish. B. refuse their offer because the opportunity cost to you of a coconut is more than 2 fish. C. accept their offer because you do not have an absolute advantage in fish. D. refuse their offer because the opportunity cost to you of a coconut is less than 2 fish.
AACSB: Analytic Blooms: Analyze Difficulty: 03 Hard Learning Objective: 02•04 Explain the role of comparative advantage in international trade and describe why some jobs are more vulnerable to outsourcing than others. Topic: Comparative Advantage and International Trade
141. You are the Minister of Trade for a small island country with the following annual PPC:
You are negotiating a trade agreement with a neighboring island with the following annual PPC:
What's the minimum number of fish you would be willing to accept in exchange for a coconut?
A. 5 B. 4 C. 3 D. 2 AACSB: Analytic Blooms: Analyze Difficulty: 03 Hard
Learning Objective: 02•04 Explain the role of comparative advantage in international trade and describe why some jobs are more vulnerable to outsourcing than others.
142. You are the Minister of Trade for a small island country with the following annual PPC:
You are negotiating a trade agreement with a neighboring island with the following annual PPC:
If you offer to give the other island 1 coconut for every 4 fish they give you, then they will:
A. refuse your offer because they have a comparative advantage in fish. B. accept your offer because your opportunity cost of a coconuts is less than 4 fish. C. refuse your offer because they can produce as many coconuts as you can. D. accept your offer because their opportunity cost of a coconut is greater than 4 fish. AACSB: Analytic Blooms: Analyze Difficulty: 03 Hard
Learning Objective: 02•04 Explain the role of comparative advantage in international trade and describe why some jobs are more vulnerable to outsourcing than others.
143. You are the Minister of Trade for a small island country with the following annual PPC:
You are negotiating a trade agreement with a neighboring island with the following annual PPC:
Both islands specialize exclusively in the product for which they have a comparative advantage. You have agreed to give 350 coconuts to the other island in exchange for 1,300 fish. After the trade, your island has a total of coconuts and fish.
A. B. C. D.
150余 500余 150余 500余
2,800 1,300 1,300 1,500 AACSB: Analytic
Blooms: Analyze Difficulty: 03 Hard Learning Objective: 02•04 Explain the role of comparative advantage in international trade and describe why some jobs are more vulnerable to outsourcing than others. Topic: Comparative Advantage and International Trade
144. You are the Minister of Trade for a small island country with the following annual PPC:
You are negotiating a trade agreement with a neighboring island with the following annual PPC:
Both islands specialize exclusively in the product for which they have a comparative advantage. You have agreed to give 350 coconuts to the other island in exchange for 1,300 fish. After the trade the other island has a total of _ coconuts and _ fish.
A. B. C. D.
850余 500余 350余 350余
1,200 1,200 1,500 1,200 AACSB: Analytic
Blooms: Analyze Difficulty: 03 Hard Learning Objective: 02•04 Explain the role of comparative advantage in international trade and describe why some jobs are more vulnerable to outsourcing than others. Topic: Comparative Advantage and International Trade
145. If country A can produce more of practically everything than can country B, then which of the following statements is true? A. B. C. D.
Country A has no Country B cannot Trade can benefit Country B has no
incentive to trade with country B. have a comparative advantage in the production of any good that country A wants to buy. both countries. incentive to trade with country A.
As long as each country has a comparative advantage in the production of at least one good that the other country wants, there are benefits to trade. AACSB: Analytic Accessibility: Keyboard Navigation Blooms: Evaluate
Difficulty: 03 Hard Learning Objective: 02•04 Explain the role of comparative advantage in international trade and describe why some jobs are more vulnerable to outsourcing than others. Topic: Comparative Advantage and International Trade
146. As the differences in opportunity costs between the U.S. and its trading partners increase, the potential gains from specialization and trade
.
A. increase B. decrease C. stay the same D. become unpredictable
The gains from specialization and trade grow with increases in the opportunity costs between trading partners. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•04 Explain the role of comparative advantage in international trade and describe why some jobs are more vulnerable to outsourcing than others. Topic: Comparative Advantage and International Trade
147. One reason there is political opposition to international trade is that: A. the potential gains from specialization and trade are small. B. trade does not increase the total value of goods and services produced by a nation. C. the differences in opportunity costs between countries are small. D. not everyone benefits from trade.
Although trade increases the total value of goods and services produced by a nation, trade does not necessarily benefit each individual citizen. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•04 Explain the role of comparative advantage in international trade and describe why some jobs are more vulnerable to outsourcing than others. Topic: Comparative Advantage and International Trade
148. One concern regarding the North American Free Trade Agreement (NAFTA) was that it would lead: A. the total value of goods and services produced by the United States to fall. B. wages in Mexico to rise. C. highly skilled workers in the United States to lose their jobs. D. unskilled workers in the United States to lose their jobs.
Since Mexico has a comparative advantage in the production of goods made by unskilled workers, many Americans feared that NAFTA would lead unskilled workers in the United States to lose their jobs to workers in Mexico. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•04 Explain the role of comparative advantage in international trade and describe why some jobs are more vulnerable to outsourcing than others. Topic: Comparative Advantage and International Trade
149. When a nation reduces the barriers to international trade: A. B. C. D.
each individual citizen becomes better off. each individual citizen becomes worse off. the total value of all goods and services produced by the nation falls. the total value of all goods and serviced produced by the nation rises.
While reducing barriers to trade increases total value of all goods and services produced by a nation, it does not guarantee that each individual citizen will be better off. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•04 Explain the role of comparative advantage in international trade and describe why some jobs are more vulnerable to outsourcing than others. Topic: Comparative Advantage and International Trade
150. The benefits to specialization are even greater when two trading partners have: A. absolute advantages in producing the same goods. B. similar consumption preferences. C. very similar opportunity costs. D. large differences in opportunity costs.
Greater difference in opportunity costs yields greater benefits from trade. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•04 Explain the role of comparative advantage in international trade and describe why some jobs are more vulnerable to outsourcing than others. Topic: Comparative Advantage and International Trade
151. According to the textbook, the evidence indicates that NAFTA has: E. reduced the wages of skilled workers in the United States. F. reduced the employment of unskilled workers in the United States significantly. G. stopped illegal immigration from Mexico. D. not significantly reduced the employment of unskilled workers in the United States.
Most studies have failed to detect significant overall job loss due to NAFTA. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Difficulty: 02 Medium Learning Objective: 02•04 Explain the role of comparative advantage in international trade and describe why some jobs are more vulnerable to outsourcing than others. Topic: Comparative Advantage and International Trade
152. According to the textbook, NAFTA was expected to help which country exploit its comparative advantage in the production of goods made by unskilled labor? A. B. C. D.
NAFTA was expected to help Mexico exploit its comparative advantage in production of goods made by unskilled labor. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•04 Explain the role of comparative advantage in international trade and describe why some jobs are more vulnerable to outsourcing than others. Topic: Comparative Advantage and International Trade
153. Outsourcing is a term increasingly used to refer to the act of: A. hiring illegal immigrants. B. importing raw materials into the United States from other countries. C. exporting final goods to other countries. D. replacing relatively expensive American workers with low•wage workers overseas.
Outsourcing has come to mean replacing highly paid American workers with cheaper workers overseas. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•04 Explain the role of comparative advantage in international trade and describe why some jobs are more vulnerable to outsourcing than others. Topic: Comparative Advantage and International Trade
154. The fundamental reason firms outsource is that: A. low•wage workers in other countries are more productive than are U.S. workers. B. hiring low•wage workers overseas reduces firms' costs. C. outsourcing increases employment overseas. D. U.S. workers cannot perform the tasks performed by workers in other countries.
Companies outsource because hiring low•wage workers overseas reduces their production costs. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•04 Explain the role of comparative advantage in international trade and describe why some jobs are more vulnerable to outsourcing than others. Topic: Comparative Advantage and International Trade
155. When a U.S. firm engages in outsourcing, it benefits A. B. C. D.
and harms
.
the firm; the U.S. consumers of the firm's products the U.S. consumers of the firm's products; the firm the U.S. consumers of the firm's products; the firm's U.S. employees the U.S. consumers of the firm's products; the firm's foreign employees
Outsourcing benefits U.S. consumers, who enjoy lower prices, but harms the firm's domestic workers, who may lose their jobs. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•04 Explain the role of comparative advantage in international trade and describe why some jobs are more vulnerable to outsourcing than others. Topic: Comparative Advantage and International Trade
156. All else equal, the jobs that are the least likely to be outsourced are those that: A. B. C. D.
do not involve face•to•face contact. can be done by a computer. require face•to•face communication. can be broken down into series of well•defined steps.
Some jobs are less susceptible to outsourcing than others, including those that do not require face•to•face communication. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Remember Difficulty: 01 Easy Learning Objective: 02•04 Explain the role of comparative advantage in international trade and describe why some jobs are more vulnerable to outsourcing than others. Topic: Comparative Advantage and International Trade
157. Which of the following jobs is least likely to be outsourced? A. Flipping hamburgers B. Technical assistance over the phone for your computer C. Transcription of physicians' records D. Software design
Flipping hamburgers requires on•site labor. AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Understand Difficulty: 02 Medium Learning Objective: 02•04 Explain the role of comparative advantage in international trade and describe why some jobs are more vulnerable to outsourcing than others. Topic: Comparative Advantage and International Trade
Summary Category
# of Questions
AACSB: Analytic AACSB: Reflective Thinking Accessibility: Keyboard Navigation Blooms: Analyze Blooms: Apply Blooms: Evaluate Blooms: Remember Blooms: Understand Difficulty: 01 Easy Difficulty: 02 Medium Difficulty: 03 Hard Learning Objective: 02•01 Explain and apply the Principle of Comparative Advantage. Learning Objective: 02•02 Explain and apply the Principle of Increasing Opportunity Cost (also called the Low•Hanging• Fruit Principle). Use a production possibilities curve to illustrate opportunity cost and comparative advantage. Learning Objective: 02•03 Identify factors that shift the menu of production possibilities. Learning Objective: 02• 04 Explain the role of comparative advantage in international trade and describe why some jobs are more vulnerable to outsourcing than others. Topic: Comparative Advantage and International Trade Topic: Comparative Advantage and Production Possibilities Topic: Exchange and Opportunity Cost Topic: Factors That Shift the Economy's Production Possibilities Curve
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# How many milliliters are in 2 cups?
Wiki User
2010-10-08 12:12:49
There are 250 ml in a cup, therefore there are 500 ml in 2 cups.
Wiki User
2009-01-17 09:57:06
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Wiki User
2010-10-08 12:12:49
2 US cups = 473.176 ml
2 Canadian cups = 454.609 ml
2 Metric cups = 500 ml
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# What statistical analysis
What statistical analysis, Usually the analysis breaks into two parts: descriptive statistics, which summarise the structure and distribution of your data modelling, or inference.
13 study design and choosing a statistical test design in many ways the design of a study is more important than the analysis. The statistical mean, median, mode and range for data informs users of variation, changes over time, or outliers and acceptable norms learn how to calculate mean. Approaches to the analysis of survey data march 2001 the university of reading statistical services centre biometrics advisory and support service to dfid. When to use a particular statistical test univariate descriptive central tendency mode • the most commonly occurring value path analysis, t-test, anova, manova. As the foundation for sas analytics, sas/stat provides state-of-the-art statistical analysis software that empowers you to make new discoveries.
Statistical analysis is, according to one service provider. What statistical analysis should i use statistical analyses using spss introduction this page shows how to perform a number of statistical tests using spss. Statistical data analysis for your dissertation proposal or results chapters is available from a fully qualified statistician. Learn about some of the simple statistical techniques that you can use to summarise and visualise quantitative data.
Handbook of biological statistics john h mcdonald ⇐ previous topic| table of contents choosing a statistical test analysis of covariance (ancova) 1: 2. Appropriate statistics are mann-whitney u test and chi sqaure test sometime people use the t-test to compare responses on likert scale but it is not. Statistical analysis:descriptive statistics and inferential statistics and review of some of the mostly used statistical tests and procedures such as analy. Statistical analysis is often used to explore your data—for example, to examine the distribution of values for a particular attribute or to spot outliers (extreme. Types of statistical analysis standard statistical analysis statistical analysis is a method used to process complicated data following are different types of.
What is statistical analysis sociological definition of statistical analysis example, sample sentence, & pronunciation of statistical analysis free online. Statistical analysis of the t-test the formula for the t-test is a ratio the top part of the ratio is just the difference between the two means or averages. Definition: statistical analysis is the use of statistical data including varying variables, entities, and events to determine probabilistic or statistical. Mathematical statistics is the application of mathematics to statistics mathematical techniques used for this include mathematical analysis, linear algebra. Learn about some more advanced techniques for statistical analysis, which identify patterns in the data these include tests of significance.
The statistical analysis revealed an element of our new marketing plan that was not effective so we used that information to alter our business plan. Statistical analysis handbook a comprehensive handbook of statistical concepts, techniques and software tools by dr m j de smith. Statistical analysis involves the process of collecting and analyzing data and then summarizing the data into a numerical form topics what's new. Lesson 12 - to determine what statistical methods to use for specific situations, summary types of data, examples, analysis, minitab commands. What is statistical analysis this definition explains this component of data analytics in terms of business intelligence and provides links to more resources.
The definition of what is meant by statistics and statistical analysis has changed considerably over the last few decades here are two contrasting definitions of. In this course, you will learn r via your existing knowledge of basic statistics and gain a familiarity with r to use it to conduct statistical analysis. Data analysis, also known as analysis of data or data analytics (2016) numeric computation and statistical data analysis on the java platform, springer.
When planning an experiment it is essential also to know that the results can be analysed planing the statistical analysis is an integral part of planning the. Data and statistics about the united states find data about the us, such as maps and population environmental resources, scientific analysis. Usingexcel&forstatistical&analysis& you&don’thave&to&have&afancy&pants&statistics&package&to&do&many&statistical&functions&&excel&can&perform.
What statistical analysis
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# Difficult integral by sub., parts & table
• silicon_hobo
In summary, the conversation was about the integral \int xe^{2x^2} cos(3x^2) dx and the attempt at solving it using the formula \int e^{au} cos\ bu\ du\ = \frac{e^{au}}{a^2+b^2}(a\ cos\ bu\ + b\ sin\ bu)+C. The user shared their solution and asked for help in submitting scanned work on the forum.
silicon_hobo
## Homework Statement
$$\int xe^{2x^2} cos(3x^2) dx$$
This is the hardest integral I've attempted so far. I've come up with an answer that fits a table in my book but I'm not sure if I arrived there correctly. Thanks for reading!
## Homework Equations
$$\int e^{au} cos\ bu\ du\ = \frac{e^{au}}{a^2+b^2}(a\ cos\ bu\ + b\ sin\ bu)+C$$
## The Attempt at a Solution
http://www.mcp-server.com/~lush/shillmud/int2.7.JPG
http://www.mcp-server.com/~lush/shillmud/int2.72.JPG
substitute:
$$u=x^2$$
$$du=2x dx$$
$$I=\int xe^{2x^2}\cos(3x^2){\rm dx}\,=\, {1\over 2} \int e^{2u}\cos(3u) {\rm du}$$
I have done some shortcut here, with use of the formula, your number two:
$$I={1\over 26} e^{2u} (2 \cos(3u)\,+\,3\sin(3u))\,+\,C$$
$$I={1\over 26} e^{2x^2} (2 \cos(3x^2)\,+\,3\sin(3x^2))\,+\,C$$
this is the same as integrator...
http://integrals.wolfram.com/
check it out
Here's the solution, I didn't simplify it any more like I should have, but I hope you understand the basic process. I checked my answer with CAS and it's correct, so don't worry about that.
http://img411.imageshack.us/img411/5615/file0001su6.jpg
Last edited by a moderator:
How do you submit scanned work onto here?
Thanks
SavvyAA3 said:
How do you submit scanned work onto here?
Thanks
Assuming the scanned image is in a format the forum recognises, jpeg,mpeg and so on either click on the icon in yellow with a black mountain or two and a black sun, visible in the new reply or go advanced: edit windows, not quick post. Or type [noparse]*url [/noparse] around the files url.
To get a url for an image stored on your hard drive you will need to upload it to a web host facility such as photobucket or freewebs. And then cut and paste the url. If not then use the paper clip to attach files directly from your hard drive to the forum. With this option there will be a delay while the forum scans the file for viruses, but any format that is listed there from jpeg to gif are acceptable.
Last edited by a moderator:
Thanks alot!
## 1. What is the purpose of using integration by substitution?
Integration by substitution is a technique used to solve integrals that cannot be solved by traditional methods. It involves replacing a complicated expression with a simpler one in order to make the integral easier to evaluate.
## 2. How do I choose the substitution for a difficult integral?
The substitution chosen for a difficult integral should be based on the form of the integral. Common substitutions include trigonometric functions, exponential functions, and u-substitutions. It is important to choose a substitution that will simplify the integral and make it easier to solve.
## 3. What is the difference between integration by parts and integration by substitution?
Integration by parts and integration by substitution are both techniques used to solve integrals, but they differ in the approach. Integration by parts involves using the product rule of differentiation, while integration by substitution involves using the chain rule of differentiation.
## 4. Is there a specific order in which I should use integration by parts and integration by substitution?
There is no specific order in which these techniques should be used. It is important to consider the form of the integral and choose the method that seems most appropriate. In some cases, a combination of both methods may be necessary to solve a difficult integral.
## 5. What is the role of the table of integrals in solving difficult integrals?
The table of integrals provides a list of known integrals and their corresponding solutions. It can be used as a reference when solving difficult integrals, as it may contain the solution to the integral in question. However, it should not be solely relied upon and proper understanding of integration techniques is still necessary.
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# Strange evaluation of Bessel Functions near $x=730$?
I am doing a calculation which involves the numerical evaluation of the following function:
$$f(x)=I_0(x)K_0(x)-I_1(x)K_1(x)$$
where $$I_{\nu}(x)$$ and $$K_{\nu}(x)$$ are the modified Bessel functions. By the asymptotic expansion quoted in DLMF, I expect the following leading-order asymptotic expansion:
$$\left.f(x)\right._{x\rightarrow\infty}\sim \frac{1}{4x^3}$$
However when I plot this function, I get strange sporadic behavior around $$x\in(725,742)$$. As far as I can tell, I get no such error anywhere else. This is almost certainly numerical error.
Question(s): What exactly is this? Is there any way to avoid it?
• In version 11, I am unable to reproduce the plot you gave, even tho what Bill said about subtractive cancellation is correct. Commented Nov 13, 2019 at 6:53
• I'm using version 12.0. Are you saying that you have no errors, even without manually changing the working precision? Commented Nov 13, 2019 at 20:35
• Here is a screenshot from the computer I am currently borrowing. Now that I have tried the version in the cloud, I can see that the disastrous subtractive cancellation is now at play. Commented Nov 13, 2019 at 22:01
Compare
Plot[BesselI[0,x]BesselK[0,x]-BesselI[1,x]BesselK[1,x],{x,725,742}]
versus
Plot[BesselI[0,x]BesselK[0,x]-BesselI[1,x]BesselK[1,x],{x,725,742},WorkingPrecision->64]
If you are curious about why this might be happening then you could inspect
Table[{BesselI[0,x],BesselK[0,x],BesselI[1,x],BesselK[1,x]},{x,725.,742.}]
and see the magnitudes of numbers that you are subtracting and getting results near 10^-9 when you have only a handful of digits using the default MachinePrecision.
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Introductory Simple Pendulums Printer Friendly Version
Background:
At the turn of the century 400 years ago, Galileo was completing his examination of the study of motion and had discovered a profound relationship of a pendulum and its period (the time to complete a swing). Before Galileo, no one had every noticed the regularity of this common phenomenon. His discovers allowed time keeping to be done to accuracy 1000 times greater than in the past.
The myth is that he discovered this relationship while watching an urn filled with burning incense swing at the end of a chain during a church service. It is suggested that he measured the swing with the beating of his heart, but I like to think that it was the melody within his head, that lead him to discover that the period of a pendulum is constant.
Equipment:
• meter stick
• fishing line
• hooked mass
• protractor
Procedure
1. Setup a pendulum with the hooked mass suspended from the end of a string.
2. Record the values stamped on the mass below.
3. Set the length of the pendulum to approximate length, indicated in the data table below, by winding or unwinding the string about the support. NOTE: one winding equals approximately 5 cm.
4. Measure your actual length to the closest tenth of a centimeter and record in the table below; make sure to measure to the vertical center of the hanging mass, or bob.
5. Swing the pendulum from an angle of approximately 10 degrees.
6. Record the time to make 10 complete vibrations. Repeat three times for each length.
The period of a pendulum is defined as the time (in seconds) required for one complete vibration. To calculate its value in the data table below, you will take the average of your three times for 10 vibrations and divide by the number of vibrations in each trial, 10.
The frequency of a pendulum is defined as the number of vibrations occuring each second. It is measured in hertz, hz. One hertz means one vibration per second. To calculate its value in the data table below, you will divide the number of vibrations in each trial, 10, by the average of your three times. Frequency is the multiplicative inverse, or reciprocal, of period.
Refer to the following information for the next four questions.
Data Table
Approximate Length (cm) Actual Length (cm) Trial #1time for 10 vib(sec) Trial #2time for 10 vib(sec) Trial #3time for 10 vib(sec) Average time for 10 vib(sec) Average Period (sec) Average Frequency (hz) 15 25 35 45 55 65 75 85 95
What was the mass of your pendulum's bob in grams?
Which of the following is the independent variable in this experiment? frequency, length, mass, period, time
Which of the following is the a control variable(s) in this experiment? frequency, length, mass, period, time
Why were you asked to allow the pendulum to swing ten times, rather than just once?
Once your table is complete, open the EXCEL workbook 1-pendulum on the file system and fill out the required information on each worksheet. Remember to rename the file as Pendulum_LastnameLastname.xls before placing any of your group's information in the file.
Refer to the following information for the next five questions.
These questions refer to the graph on Sheet 1 of your EXCEL workbook.
What is the title of your EXCEL workbook?
What is the coefficient of restitution, R2, for your graph?
What is the exponent of x on the equation of your graph?
What is the shape of this graph?
Does the period increase, decrease or remain unchanged as the pendulum's length increases
Refer to the following information for the next three questions.
These questions refer to the graph on Sheet 2 of your EXCEL workbook.
What is the shape of this graph?
State the equation of your graph using the correct variables for each axis: T2 for y, and L for x.
Using the equation for your graph extrapolate the time required for a 120-cm pendulum to make ten vibrations. NOTE: We will not be examining the slope of this graph until the beginning of next semester when we discuss the acceleration due to gravity.
Refer to the following information for the next three questions.
These questions refer to the graph on Sheet 3 of your EXCEL workbook.
What is the shape of this graph?
Is the frequency of a pendulum directly or inversely proportional to the square root of its length?
Based upon the graphical patterns in these introductory labs, how could the data shown on this third graph be rectified to create a straight line?
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The OEIS is supported by the many generous donors to the OEIS Foundation.
Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
A145722 Expansion of f(q) * f(q^5) / phi(-q^2)^2 in powers of q where f(), phi() are Ramanujan theta functions. 5
1, 1, 3, 4, 8, 12, 21, 30, 48, 68, 102, 143, 207, 284, 400, 542, 744, 996, 1344, 1776, 2361, 3088, 4050, 5248, 6808, 8742, 11232, 14310, 18224, 23052, 29133, 36601, 45936, 57360, 71528, 88812, 110110, 135990, 167704, 206108, 252912, 309408 (list; graph; refs; listen; history; text; internal format)
OFFSET 0,3 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015 MathOverflow, Up to 2000..., 2016. Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of q^(-1/4) * eta(q^4) * eta(q^10)^3 / (eta(q) * eta(q^2) * eta(q^5) * eta(q^20)) in powers of q. Euler transform of period 20 sequence [ 1, 2, 1, 1, 2, 2, 1, 1, 1, 0, 1, 1, 1, 2, 2, 1, 1, 2, 1, 0, ...]. G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 20^(-1/2) g(t), where q = exp(2 Pi i t) and g() is the g.f. for A145723. G.f.: Product_{k>0} (1 + x^(2*k)) * (1 - x^(10*k)) * (1 + x^(5*k)) / ((1 - x^k) * (1 + x^(10*k))). a(n) = A036026(2*n). a(n) ~ exp(2*Pi*sqrt(n/5)) / (4 * 5^(3/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015 EXAMPLE G.f. = 1 + x + 3*x^2 + 4*x^3 + 8*x^4 + 12*x^5 + 21*x^6 + 30*x^7 + 48*x^8 + ... G.f. = q + q^5 + 3*q^9 + 4*q^13 + 8*q^17 + 12*q^21 + 21*q^25 + 30*q^29 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ -x^5] / EllipticTheta[ 4, 0, x^2]^2, {x, 0, n}]; (* Michael Somos, Aug 26 2015 *) nmax=60; CoefficientList[Series[Product[(1+x^(2*k)) * (1-x^(10*k)) * (1+x^(5*k)) / ((1-x^k) * (1 + x^(10*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *) PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^10 + A)^3 / (eta(x + A) * eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A)), n))}; CROSSREFS Cf. A036026. Sequence in context: A281612 A349050 A025034 * A147622 A173534 A074331 Adjacent sequences: A145719 A145720 A145721 * A145723 A145724 A145725 KEYWORD nonn AUTHOR Michael Somos, Oct 23 2008 STATUS approved
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# Category Archives: Euler
## Euler Diagram
This article is about Eulerian circles of set theory and logic. For the geometric Euler circle, see Nine-point circle. An Euler diagram illustrating that the set of “animals with four legs” is a subset of “animals”, but the set of … Continue reading
Posted in Euler, John Venn, Venn | Tagged , , | 5 Comments
## Coffee and Donut?
A continuous deformation (homeomorphism) of a coffee cup into a doughnut (torus) and back. Similarly, the hairy ball theorem of algebraic topology says that “one cannot comb the hair flat on a hairy ball without creating a cowlick.” *** This … Continue reading
## The Whole World is a Stage
Euler product formula Now you must know what sets my mind to think in such abstract spaces. “Probability of seeing a stage in a concert.“ All The World’s A Stageby William ShakespeareFrom: As you Like It, Act II Scene VII … Continue reading
## Euler’s Konigsberg’s Bridges Problem
“Liesez Euler, Liesez Euler, c’est notre maître à tous”(“Read Euler, read Euler, he is our master in everything”) – Laplace I should say here that the post by Guest post: Marni D. Sheppeard, “Is Category Theory Useful ?” over at … Continue reading
## Lingua Cosmica
It looks as though primes tend to concentrate in certain curves that swoop away to the northwest and southwest, like the curve marked by the blue arrow. (The numbers on that curve are of the form x(x+1) + 41, the … Continue reading
## Mersenne Prime: One < the Power of two
It looks as though primes tend to concentrate in certain curves that swoop away to the northwest and southwest, like the curve marked by the blue arrow. (The numbers on that curve are of the form x(x+1) + 41, the … Continue reading
## A New Way of Seeing?
Sorry couldn’t resist. How many before us in our speculations and way of seeing? 🙂 When looking at Gaussian coordinates, the very idea that our views of “length of lines” had to have another way in which to interpret how … Continue reading
Posted in Brain, Euler, Riemann Hypothesis, Topology | Tagged , , , | 1 Comment
## Grue and Bleen
Brian Greene: In the late 1960s a young Italian physicist, named Gabriele Veneziano, was searching for a set of equations that would explain the strong nuclear force, the extremely powerful glue that holds the nucleus of every atom together binding … Continue reading
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https://www.cnblogs.com/wahaha02/p/4561569.html
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# 编程范式与语言
python对象式实现
class Bisection (FindMinima):
def algorithm(self,line):
return (5.5,6.6)
def algorithm(self,line):
return (3.3,4.4)
class MinimaSolver: # context class
strategy=''
def __init__ (self,strategy):
self.strategy=strategy
def minima(self,line):
return self.strategy.algorithm(line)
def changeAlgorithm(self,newAlgorithm):
self.strategy = newAlgorithm
def test():
print solver.minima((5.5,5.5))
solver.changeAlgorithm(Bisection())
print solver.minima((5.5,5.5))
python函数式实现
def bisection(line):
return 5.5, 6.6
return 3.3, 4.4
def test():
print solver((5.5,5.5))
solver = bisection
print solver((5.5,5.5))
C过程式实现
void quickSort(int* arr,int startPos, int endPos)
{
int i,j;
int key;
key=arr[startPos];
i=startPos;
j=endPos;
while(i<j)
{
while(arr[j]>=key && i<j)--j;
arr[i]=arr[j];
while(arr[i]<=key && i<j)++i;
arr[j]=arr[i];
}
arr[i]=key;
if(i-1>startPos)
quickSort(arr,startPos,i-1);
if(endPos>i+1)
quickSort(arr,i+1,endPos);
}
erlang函数式实现
qsort([]) -> [];
qsort([H|T])->
qsort([LO || LO <- T, LO < H]) ++ [H] ++ qsort([HI || HI <- T, HI >= H]).
1. 《冒号课堂》
2. 《程序设计语言--实践之路》第三版
-- EOF--
posted @ 2015-06-08 17:48 wahaha02 阅读(2028) 评论(1编辑 收藏
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https://www.doubtnut.com/question-answer/if-z2-1z2-1-then-z-lies-on-a-a-circle-b-the-imaginary-axis-c-the-real-axis-d-an-ellipse-6667
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|
Class 8
MATHS
Rational Numbers
# If |z^2-1|=|z|^2+1, then z lies on (a) a circle (b) the imaginary axis (c) the real axis (d) an ellipse
Step by step solution by experts to help you in doubt clearance & scoring excellent marks in exams.
Updated On: 24-12-2020
Apne doubts clear karein ab Whatsapp par bhi. Try it now.
Watch 1000+ concepts & tricky questions explained!
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2:14
Very Important Questions
|Z^2-1|=|Z^2|-1<br> |(Z^2-1)^2|=(|Z^2|-1)^2<br> (Z^2-1)(Z^2-1)=(1+ZZ)^2<br> -(Z^2-2^2)=2*Z*Z<br> Re(Z)=0<br> Z lies on the imaginary axis.
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CC-MAIN-2021-17
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http://www.maa.org/publications/maa-reviews/adventures-in-group-theory?device=mobile
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| 552,554,618
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You are here
Publisher:
Johns Hopkins University Press
Number of Pages:
264
Price:
25.00
ISBN:
978-0801869471
Adventures in Group Theory is a tour through the algebra of several "permutation puzzles." Although the main focus is on the Rubik's Cube, several other puzzles are explored to a lesser degree. Joyner pulls together a great deal of information about the mathematics of the Rubik's Cube from various sources and mixes it with a lot of algebra. (And I mean a lot!) The book has the following layout:
Chapter 1 An introduction to Boolean algebra, logic, and the basic moves of the Rubik's cube (using the Singmaster notation: U D L R F B).
Chapter 2 Functions, Cartesian products, vector spaces, matrices, determinants, relations and equivalence classes, and some basic combinatorics.
Chapter 3 Permutations, signs of permutations, permutation matrices (a little on permutations of the edges of the Rubik's cube), and the symmetry groups Sn.
Chapter 4 Permutation puzzles are introduced: Sam Loyd's 15 puzzle, the Hockeypuck puzzle, the Rainbow Masterball, the Pyraminx, the 2x2 and 3x3 Rubik's Cube, the Skewb, the Megaminx, and several others.
Chapter 5 An introduction to group theory. The quaternions, finite cyclic groups, dihedral groups, and symmetric groups are presented. The superflip is mentioned (this element generates the center of the Rubik's Cube Group). Basic properties of groups, commutators, the notion of conjugacy, orbits and group actions, and cosets
Chapter 6 A study of Merlin's Machine and some variants using matrices. Some linear algebra is presented.
Chapter 7 Graphs, with special attention to the Cayley graph associated with a group. The problem of God's Algorithm is presented: given a Cayley graph associated to a group, find an effective, practical algorithm for finding a path from a given vertex to the vertex corresponding to the identity. In practical terms, this would be an algorithm which could take any position of the Rubik's Cube and find the "quickest" solution.
Chapter 8 Platonic Solids and the notion of the symmetries of three-dimensional space.
Chapter 9 Group homomorphisms, isomorphisms, automorphisms, and actions are introduced as a way of determining when two groups are in fact the same (i.e. isomorphic). We also get quotient groups, and the three fundamental isomorphism theorems of groups. The direct product of groups is given as a way of examining the effects of permutations on the Rubik's cube. (Cubes go to cubes and edges go to edges.) We get the First Fundamental Theorem of Rubik's Cube Theory. The remainder of the chapter deals with semi-direct products and wreath products.
Chapter 10 Free Groups are introduced, along with the concept of words. Cayley graphs are mentioned again, with Poincaré polynomials. A description of all groups up to order 25 is given.
Chapter 11 Giving the cube an orientation, we find the group of all possible moves of the Rubik's Cube group (the legal Rubik's Cube group). The group structure is given (the direct product of two groups: the semi-direct product of the permutation group SVof the corners with the eighth power of the cyclic group of order three [this determines the possible permutations of the eight corners] and the semi-direct product of the permutation group SE of the edges with the twelfth power of the cyclic group of order two [this determine the possible permutations of the twelve edges], modulo three conditions on the edges and vertices). The number of elements of order two in the Rubik's Cube group is counted using a combinatorial argument (there are 334,864,275,867).
Chapter 12 Various subgroups of the Rubik's Cube group are studied. Some of this is achieved with the help of finite fields. Projective General Linear Groups are used.
Chapter 13 General results about other permutation puzzles.
Chapter 14 Some connections between the Mathieu group, coding theory, and Hadamard matrices are explored.
Chapter 15 Several solution strategies for the Rubik's Cube are presented in very general terms, as well as some strategies for other permutation groups.
Chapter 16 A summary of questions that are currently unanswered.
If you like puzzles, this is a somewhat fun book. If you like algebra, this is a fun book. If you like puzzles and algebra (which is the category that I fall into), this is a really fun book. Joyner introduces a great deal of algebra in this book. Most of it is undergraduate level, although some borders on graduate level material. If you've had a couple of semesters of algebra, you'll probably find the book entertaining, mathematically speaking. If you haven't, then you won't get that much out. While the algebra is (mostly) self-contained, it is presented quickly, and the author gives just enough to do what he wants. So don't expect it to read like an algebra textbook.
One of the most fun aspects of the book is the interesting (from the algebraic perspective) collection of trivia concerning the Rubik's Cube Group. For example, the structure of this group, the center of the Rubik's Cube Group, the structure of some of the subgroups of the Rubik's Cube Group, including an embedding of the quaternions into the group, and an example of two elements which generate the whole Rubik's Cube group. Joyner does a good job explaining how the structure of the Rubik's Cube works. But again, the algebraic description requires some semi-advanced algebra. The audience seems to be mathematicians with a fairly good knowledge of algebra and an interest in permutation puzzles. If you fall into that category, this book is for you.
Donald L. Vestal is an Assistant Professor of Mathematics at Missouri Western State College. His interests include number theory, combinatorics, reading, and listening to the music of Rush. He can be reached at vestal@mwsc.edu.
Thursday, June 6, 2002
Reviewable:
Yes
Include In BLL Rating:
No
David Joyner
Publication Date:
2002
Format:
Paperback
Audience:
Category:
General
Donald L. Vestal
02/10/2003
Publish Book:
Modify Date:
Thursday, October 11, 2007
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# Math
posted by on .
Can someone explain to me the greatest possible error when finding the maximum and minimum possible lengths of something?? PLEASE!!!
• Math - ,
The error can be expressed as units, or as percentage.
Say you have a somewhat accurate ruler, and you measure a board. You find a length of 3m, but the lines on the ruler are rather blurred, so there's a chance of a 2mm error either way.
So, the actual length could be anywhere from 2.998m to 3.002m.
Or, if you're using an electronic device, maybe its specs say that it's accurate to 1%.
In this case, 1% of 3m = 3cm, so the real length could be from 2.97m to 3.03m.
| 171
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| 3.3125
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latest
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http://www.vbaexpress.com/forum/archive/index.php/t-50287.html?s=46db49e89aa3116320463da33d8dc5e0
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PDA
View Full Version : i want required vba. I am looking for a small value from the Range
sanju2323
07-28-2014, 11:00 PM
ValueRangeRequirement Lowest Value By The Range25.0001750.00017.00017.0001750.00017.00033.0001750.00017.00018.0001750 .00017.00033.0001750.00017.00047.0001750.00017.00066.0001750.00017.00021.00 01750.00017.00041.0001750.00017.00098.0003500.0005.00045.0003500.0005.00070 .0003500.0005.0005.0003500.0005.00053.0003500.0005.00055.0003500.0005.00047 .0003500.0005.00012.0003500.0005.00015.0005250.00014.00066.0005250.00014.00 014.0005250.00014.00045.0005250.00014.00068.0005250.00014.00021.0005250.000 14.00075.0005250.00014.00077.0005250.00014.00089.0005250.00014.00017.000525 0.00014.000Need VBA Codes.I am looking for a small value between Range
kevvukeka
07-28-2014, 11:23 PM
Hi sanju,
Try this.. This is not entirely mine. An expert from this forum helped me out with few weeks back. Hope this helps
Sub test()
Dim i As Long, strtrw As Long, endrw As Long
i = 2
strtrw = i
endrw = i
Do Until Cells(i, 1) = ""
If Cells(i, 2) = Cells(i + 1, 2) Then
endrw = i + 1
Else
Range(Cells(strtrw, 4), Cells(endrw, 4)) = Application.WorksheetFunction.Min(Range(Cells(strtrw, 1), Cells(endrw, 1)))
strtrw = i + 1
endrw = i + 1
End If
i = i + 1
Loop
End Sub
.
sanju2323
07-28-2014, 11:42 PM
Thanks, But I want result shown as VBA Formula
Ex. test(A2,B2)
xld
07-29-2014, 02:28 AM
Public Sub GetMin()
Const FORMULA_MIN = "=MIN(IF(\$B\$2:\$B\$<lastrow>=\$B2,\$A\$2:\$A\$<lastrow>))"
Dim lastrow As Long
With ActiveSheet
lastrow = .Cells(.Rows.Count, "A").End(xlUp).Row
.Range("C2").FormulaArray = Replace(FORMULA_MIN, "<lastrow>", lastrow)
.Range("C2").AutoFill .Range("C2").Resize(lastrow - 1)
End With
End Sub
sanju2323
07-29-2014, 07:20 AM
You are all the right to hang the sheet. This is no way to
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| 2.6875
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https://www.airmilescalculator.com/distance/grr-to-dtw/
| 1,718,648,851,000,000,000
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|
# How far is Detroit, MI, from Grand Rapids, MI?
The distance between Grand Rapids (Gerald R. Ford International Airport) and Detroit (Detroit Metropolitan Airport) is 120 miles / 193 kilometers / 104 nautical miles.
The driving distance from Grand Rapids (GRR) to Detroit (DTW) is 149 miles / 239 kilometers, and travel time by car is about 2 hours 46 minutes.
120
Miles
193
Kilometers
104
Nautical miles
## Distance from Grand Rapids to Detroit
There are several ways to calculate the distance from Grand Rapids to Detroit. Here are two standard methods:
Vincenty's formula (applied above)
• 119.945 miles
• 193.032 kilometers
• 104.229 nautical miles
Vincenty's formula calculates the distance between latitude/longitude points on the earth's surface using an ellipsoidal model of the planet.
Haversine formula
• 119.692 miles
• 192.625 kilometers
• 104.009 nautical miles
The haversine formula calculates the distance between latitude/longitude points assuming a spherical earth (great-circle distance – the shortest distance between two points).
## How long does it take to fly from Grand Rapids to Detroit?
The estimated flight time from Gerald R. Ford International Airport to Detroit Metropolitan Airport is 43 minutes.
## Flight carbon footprint between Gerald R. Ford International Airport (GRR) and Detroit Metropolitan Airport (DTW)
On average, flying from Grand Rapids to Detroit generates about 43 kg of CO2 per passenger, and 43 kilograms equals 94 pounds (lbs). The figures are estimates and include only the CO2 generated by burning jet fuel.
## Map of flight path and driving directions from Grand Rapids to Detroit
See the map of the shortest flight path between Gerald R. Ford International Airport (GRR) and Detroit Metropolitan Airport (DTW).
## Airport information
Origin Gerald R. Ford International Airport
City: Grand Rapids, MI
Country: United States
IATA Code: GRR
ICAO Code: KGRR
Coordinates: 42°52′50″N, 85°31′22″W
Destination Detroit Metropolitan Airport
City: Detroit, MI
Country: United States
IATA Code: DTW
ICAO Code: KDTW
Coordinates: 42°12′44″N, 83°21′12″W
| 504
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| 2.890625
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latest
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en
| 0.838964
|
https://www.notamonadtutorial.com/weird-ways-to-multiply-really-fast-with-karatsuba-toom-cook-and-fourier/
| 1,675,522,625,000,000,000
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| 923,799,165
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|
# Weird ways to multiply really fast with Karatsuba, Toom–Cook and Fourier
## Introduction
The applicability and performance of algorithms depend on how fast certain routine computations can be done. For example, in elliptic curve cryptography, one needs to calculate the public key as $k\times g=g+g+g+....+g$, where $k$ is a very large integer (typically a number with a hundred digits or so) and $g$ is a point of the elliptic curve $(x,y)$, known as the generator. If done naïvely, that is, adding repeatedly $g$ to itself, it would take about $10^{100}$ operations (we say that the algorithm runs in $\mathcal{O}(n)$, indicating that the number of operations -up to some constant factor- increases with $k$ in a linear fashion). The fastest supercomputer can perform less than $10^{18}$ floating point operations per second; it would take you forever to do just one calculation. Nevertheless, we perform such calculations or related ones everyday, thanks to developing faster algorithms (for example, we can reduce the computation time by repeatedly adding $g+g=2g$, $2g+2g=4g$, etc, reducing the number of calculations to $\mathcal{O}(\log(n))$ -compare $10^{100}$ to something like $100\times \log_2{10}$).
zk-SNARKs (zero-knowledge succinct non-interactive arguments of knowledge) are important cryptographic primitives that allow one party (the prover) to convince another (the verifier) that a certain statement is true, without revealing anything else other than the validity of that statement. The applications of zk-SNARKs are far-ranging, given their potential as a foundation for new forms of governance, data sharing, and financial systems. For example, you could delegate a hard computation to an untrusted party and get a proof that allows you to verify the integrity of the computation, without the need to re-run everything. The key is that proofs are succinct, so they can be verified in the order of hundreds of milliseconds, as opposed to performing the whole computation. The construction relies on transforming the computation to polynomials and checking conditions over the polynomials. Polynomial multiplication can be done in a very efficient way via the fast Fourier transform -one of the most important algorithms ever devised by mankind-. Moreover, this calculation can be parallelized: several processors can run parts of the algorithm to make it even faster.
Even simple calculations such as integer multiplications (which take place almost everywhere) can be done faster than the school rule, provided the numbers we are trying to multiply are large enough.
If you want to learn how to speed up some ordinary calculations and make your algorithms run faster, then the next sections are for you.
## Divide and conquer: Karatsuba
We all learned at elementary school how to multiply two numbers: we write one below the other and proceed to multiply each of the numbers above by each digit of the number below and then we add all the numbers:
1234
× 152
———————————————
2468 ( = 1234 × 2)
6170 ( = 1234 × 50)
1234 ( = 1234 × 100)
———————————————
187568 ( = 187568)
This algorithm has $\mathcal{O}(n^2)$. In 1960, Kolmogorov speculated that this represented the asymptotic bound for multiplication (that is, multiplication of two numbers could not take less than $\mathcal{O}(n^2)$ operations). He gave a lecture on the topic and one of the students, Karatsuba, then 23 years old, came up with a solution that runs with $\mathcal{O}(n^{\log_2(3)})$, thus disproving Kolmogorov's conjecture. The basic idea of Karatsuba's algorithm is the following: say we want to multiply $x$ and $y$; we can break them into smaller numbers:
$x=x_1\times 10^m +x_0$
$y=y_1\times 10^m +y_0$
where both $x_0$ and $y_0$ are numbers less than $10^m$. The product $x\times y$ is simply:
$x\times y=x_1\times y_1\times 10^{2m}+(x_1\times y_0+y_1\times x_0)\times 10^m+x_0y_0$
Karatsuba found that $x_1y_0+y_1x_0$ can be calculated efficiently at the expense of some additions:
$x_1\times y_0+y_1\times x_0=(x_1+x_0)\times (y_1+y_0)-x_1\times y_1-x_0\times y_0$.
Even if there are some extra calculations, these operate over smaller numbers, resulting in an overall smaller cost for large numbers.
## Toom-Cook algorithm
The divide and conquer strategy can be taken further, leading to a reduction in the complexity of the multiplication algorithm. Toom and Cook developed several methods (known as Toom-X, X being a number), which consist of the following stages:
1. Splitting
2. Evaluation
3. Pointwise multiplication
4. Interpolation
5. Recomposition
Several variants of the algorithms are implemented in GNU Multiple Precision Arithmetic Library. Toom-2 is the same as Karatsuba's algorithm. Toom-X begins by splitting the numbers $x$ and $y$ in X parts of equal length(1) and these are treated as the coefficients of some polynomial (we focus on Toom-3, but you can see more details here)(2):
$x(t)=x_2 t^2+x_1 t+x_0$
$y(t)=y_2 t^2+y_1 t+y_0$
If we evaluate $x$, $y$ at $t=b$, we get the numbers back. The multiplication of both numbers is equal to a polynomial of degree $2(X-1)$,
$w(t)=w_4t^4+w_3t^3+w_2t^2+w_1t+w_0$
We can evaluate the polynomials at 5 different points, which will suffice to determine uniquely the polynomial $w$ due to the interpolation theorem. We can choose 5 convenient points which make the evaluation and reconstruction of the polynomial easy. Common points are $0, 1, -1, 2$ and $\infty$ (this last one is just the product of the main coefficients). Let's see the form of each value:
$w(0)=x(0)y(0)=x_0y_0$
$w(1)=x(1)y(1)=(x_0+x_1+x_2)(y_0+y_1+y_2)$
$w(-1)=x(-1)y(-1)=(x_0-x_1+x_2)(y_0-y_1+y_2)$
$w(2)=x(2)y(2)=(x_0+2x_1+4x_2)(y_0+2y_1+4y_2)$
$w(\infty)=x(\infty)y(\infty)=x_2y_2$
If we look at things from $w$ and its coefficients, we get:
$w(0)=w_0$
$w(1)=w_4+w_3+w_2+w_1+w_0$
$w(-1)=w_4-w_3+w_2-w_1+w_0$
$w(2)=16w_4+8w_3+4w_2+2w_1+w_0$
$w(\infty)=w_4$
This is just solving one linear system (where 2 coefficients are straightforward). Once the coefficients are known, all that remains is to evaluate $w$ at $t=b$ and add. Toom-3 has a lower order ($\mathcal{O}(n^{\log(5)/\log(3)})=\mathcal{O}(n^{1.46}$)) than Karatsuba's method ($\mathcal{O}(n^{1.58})$, so it runs faster for sufficiently large integers.
For larger integers (in the order of 10,000 to 40,000 digits), we can go faster by means of the Schönhage-Strassen algorithm, which uses the fast-Fourier transform (FFT) to achieve a complexity $\mathcal{O}(n\log(n)\log\log(n))$. Before we can explain the algorithm, we need to introduce the FFT. The order can be further reduced to $\mathcal{O}(n\log(n))$, but this algorithm is only practical for (super-ultra) incredibly large numbers and is an example of a galactic algorithm.
## The Fast-Fourier Transform
The FFT is one of the key building blocks of many important algorithms, such as fast multiplication of very large numbers, polynomial multiplication, solving finite difference equations, error correcting codes (Reed-Solomon codes), and digital signal processing. It was used by Gauss early in the 19th century when he was trying to interpolate the orbits of asteroids Pallas and Juno. A simple implementation requires $\mathcal{O}(n^2)$ operations. In 1965, Cooley and Tukey realized that the algorithm could be implemented more efficiently, reducing it to $\mathcal{O}(n\log(n))$, which led to its widespread use. Almost every language and numerical computation library have it implemented. In Rust, you can check this link.
To get an idea of the huge improvement over the naïve algorithm, let's look at the number of calculations for different samples:
Number of samples $10^3$ $10^6$ $10^{12}$
DFT operations $10^6$ $10^{12}$ $10^{24}$
FFT operations $10^4$ $2\times10^{7}$ $4\times10^{13}$
We see that the amount of computations is reduced by more than two orders of magnitude for samples with $1000$ or more elements!
### FFT over complex numbers
The Fourier transform maps a function from its original domain (space or time) to another function depending on the (space or time) frequency. Stated another way, it decomposes a function into a collection of sine waves with different frequencies and amplitudes, which are useful to analyze the behavior of a given system. We can also perform the inversion, adding all those waves to recover the original function. Even though (continuous) Fourier transforms have many applications, we will be interested in discrete Fourier transforms (DFT), where we have a finite collection of data. Given data $x_0$, $x_1$,...$x_{N-1}$, the DFT gives a sequence $X_0, X_1,...X_{N-1}$, where
$X=\sum_{k=0}^{N-1} x_k\exp(-2\pi i k/N)$
where $i^2=-1$ is the imaginary unit. Inversion of the DFT is given by
$x=\frac{1}{N}\sum_{k=0}^{N-1} X_k\exp(2\pi i k/N)$.
The DFT can be cast in the form of a matrix-vector product, $X=Mx$, where $M$ is the $N\times N$ DFT matrix:
$M_{ij}=\omega^{(i-1)\times (j-1)}$
where $\omega=\exp(2\pi i/N)$ and $i$ and $j$ take the values $1,2,3,...N$
Implemented this way, the DFT requires $N^2$ operations, resulting from vector-matrix multiplication. The FFT will make this calculation more efficient, by taking advantage of the structure and using a divide and conquer strategy.
We can also see the DFT as evaluating a polynomial with coefficients $x_k$ over the roots of unity. This will be useful when discussing fast polynomial multiplication.
The key point is that computing the DFT with $N$ points can be reduced to calculating two DFTs with $N/2$ points. We can apply this recursively to break down a very large problem into a collection of smaller and easier-to-solve subproblems and then recombine those results to get the DFT.
The algorithm also takes advantage of the properties of the $n$-th roots of unity in the complex plane. A number $z$ is known as an $n$-root of unity if $z^n=1$. These are of the form
$z_k=\exp(2\pi i k/n)$ for $k=0,1,2,...,n-1$. An interesting point is that these roots come in conjugate pairs: for each root $r$ we have the corresponding $\bar{r}$ (as a matter of fact, they form a finite group of order $n$ under multiplication). For example, the fourth roots of unity are: $1, i, -1, -i$. It is easy to see which are the pairs.
To see how all works, suppose we have a vector $x=(x_0,x_1,x_2,...x_{n-1})$ and we want to compute the FFT. We can split between even and odd numbered terms:
$X=\sum_{k=0}^{n/2-1} x_{2k}\exp(2\pi i 2k/n)+\sum_{k=0}^{n/2-1} x_{2k+1}\exp(2\pi i (2k+1)/n)$
We can express the odd terms in a different way, by taking out a factor of $\exp(2\pi i/n)$,
$X=\sum_{k=0}^{n/2-1} x_{2k}\exp(2\pi i 2k/n)+\exp(2\pi i/n)\sum_{k=0}^{n/2-1} x_{2k+1}\exp(2\pi i (2k)/n)$
We can now see that the factors corresponding to the $n$-roots of unity repeat themselves whenever $k$ is larger than $n/2$. Another way to see this is to rearrange the terms by taking $2$ from the numerator of the exponential and sending it to the denominator:
$X=\sum_{k=0}^{n/2-1} x_{2k}\exp(2\pi i k/(n/2))+\exp(2\pi i/n)\sum_{k=0}^{n/2-1} x_{2k+1}\exp(2\pi i (k)/(n/2))$
We now find that $\sum_{k=0}^{n/2-1} x_{2k}\exp(2\pi i k/(n/2))=DFT(x_{2k})$ is just the DFT of the even terms, which contains $n/2$ points. Similarly, $\sum_{k=0}^{n/2-1} x_{2k+1}\exp(2\pi i (k)/(n/2))$ is the DFT of the odd terms, containing $n/2$ points. This way, we broke the $n$ point DFT into two smaller $n/2$ point DFTs, which can be combined to yield the original one. Now, each of those $n/2$ DFTs can be broken into two smaller ones, so we can recursively reduce the number of computations by working with smaller samples (this way, we save ourselves of the large vector-matrix product).
### Extending the FFT to arbitrary rings
FFT can be extended from complex or real numbers to arbitrary rings, such as integers or polynomials (check our math survival kit). In particular, we can use the number theoric transform which specializes the FFT to $\mathbb{Z}/p\mathbb{Z}$, that is, the integers modulo $p$ (a prime number). Here we also have the $n$-roots of unity, given by
$\alpha^n\equiv 1 \pmod{p}$
It is important that we restrict ourselves to prime numbers: in this case, we have that the square root of $1$ are just $1$ and $-1$. For example, if we take $p=5$, $1^2\equiv 1 \pmod{5}$ and $-1\equiv 4$, $4^2 =16 \equiv 1 \pmod{5}$. This is not true for $8$ since $1^2\equiv 3^2\equiv 5^2\equiv 7^2\equiv 1 \pmod{8}$ and we would have $4$ square roots!
The problem with using FFT in finite fields is that we are not free to choose the domain and the field just as we please. We need to select a multiplicative subgroup of order $2^n$ (in other words, we need to select a group that is generated by an element $g$ and which contains its powers up to $2^n$). For example, if we take $p=5$, we have a group of order $4=2^2$ which is generated by $2$: ${2^1=2,2^2=4,2^3\equiv 3, 2^4\equiv 1}$; it does not need to span all the elements of the field, though.
## FFT multiplication algorithm
The algorithm follows the same line as Karatsuba's and Toom's:
1. Split
2. Evaluation
3. Pointwise multiplication
4. Interpolation
5. Combination
The key difference lies in the use of the FFT to speed up calculations.
### Polynomial multiplication
Let's start with polynomial multiplication. Given two polynomials, $p(x)=p_d x^d+p_{d-1}x^{d-1}+...+p_0$ and $q(x)=q_d x^d+q_{d-1}x^{d-1}+...+q_0$, we want to find their product, $w(x)=p(x)q(x)$. The simplest algorithm would be to apply repeatedly the distributive property, perform the multiplications and rearrange everything. The product of two polynomials of degree $d$ is a polynomial of degree $2d$. We can see that this strategy involves operations of the order $\mathcal{O}(d^2)$, that is, operations grow quadratically with the degree of the polynomials involved. We can take advantage of the structure of the polynomials and the interpolation theorem. We have at least two forms to describe the same polynomial:
• Giving the $d+1$ coefficients.
• Specifying the value of the polynomial at exactly $d+1$ points(3).
What are the advantages of the second option? That we get to choose the points freely and reduce the number of calculations. For example, if we have an even function, $f(x)=f(-x)$ we can evaluate fewer points. Similarly, if the function is odd, $f(-x)=-f(x)$ and we have to change the sign to get the value of $-x$. So, choosing pairs $x$ and $-x$ we reduce the number of evaluations by half (except if we choose $0$, for example). We can split our polynomial between two polynomials: one has odd number terms, and the other even:
$p(x)=p_e(x)+xp_o(x)$.
For example, if $p=x^5+3x^4+5x^3+2x^2+6x+3$, we split it:
$p(x)=(3x^4+2x^2+3)+x(x^4+5x^2+6)$
We have then:
$p_e=(3x^4+2x^2+3)$ and $p_o=(x^4+5x^2+6)$, where both polynomials are even functions! This way, we easily see that:
$p(-x)=p_e(x)-xp_o(x)$
If we have pairs $(x_k,p(x_k))$ and $(x_k,q(x_k))$, the product polynomial evaluated at $x_k$ is $(x_k,p(x_k)q(x_k))$.
To determine the product polynomial, we need $2d+1$ points; taking advantage of the above strategy, we need fewer point evaluations. If we could convert easily from the coefficient form to point evaluations, perform the multiplications in that form, and then transform back to coefficient form, we can achieve a lower complexity. We can recursively break the polynomials $p_e(x^2)$ and $p_o(x^2)$ into smaller polynomials.
We can choose as evaluation points the $n$ roots of unity, which come in pairs: $exp(2\pi i k/n)$ with $k=0,1,2...n-1$. In other words, we can quickly calculate the DFT of the polynomials, multiply the coefficients and reverse the DFT once the product has been found. This leads to operations in the order $\mathcal{O}(d\log(d))$.
### Integer multiplication
To apply the FFT to integer multiplication, we need to transform our numbers to the coefficients of polynomials, perform the FFT multiplication and finally reconstruct the result. Overall this will take $\mathcal{O}(n\log(n)\log(\log(n))$. There is a large overhead, which will make this algorithm practical only for very large integers. For example, if we want to multiply $3578$ and $2457$, we can define vectors $(8,7,5,3,0,0,0,0)$ and $(7,5,4,2,0,0,0,0)$, where we conveniently pad the numbers with zeros.
Typically, operations are performed modulo $2^N+1$, where $N$ is larger than the combined number of bits of the integers $x$ and $y$, to make sure that results never wrap around.
The Fourier transform has the advantage that an operation such as the convolution of $x$ and $y$ can be calculated from the product of the transforms $X$ and $Y$ and transforming back:
$\sum_{k=0}^{N} x_k y_{N-k}=IFFT(FFT(y)\times FFT(x))$
The Schönhage-Strassen algorithm makes use of the negacyclic convolution. Given vectors $x$ and $y$ of length $n$ and $r$ a $2n$-th (primitive) root of unity (that is, $r^{2n}\equiv 1 \pmod{p}$ and $r^k\not\equiv 1$ if $0<k<2n$), we can define the following weight vectors:
$W_j=r^j$ for $0\leq j<n$
$W_j^{-1}=r^{-j}$ for $0\leq j<n$
The negacyclic convolution (NCC) of $x$ and $y$ can be computed as:
$NCC(x,y)=W^{-1}IFFT(FFT(Wx)\times FFT(Wy))$
A comparison of the different methods implemented in GNU Multiple Precision Arithmetic Library is shown in this link.
## Summary
Choosing the right algorithms to carry out routine calculations, such as integer or polynomial multiplications, can have a dramatic effect on the performance of software. Depending on the size of the integers, it is possible to speed up (reducing the number of calculations) by adopting a divide and conquer approach: we break the calculation into smaller ones, which can be easily tackled, or continue breaking them down until they are manageable. All the fast algorithms we presented make use of this approach, leading to significant savings in computations. The FFT, thanks to its complexity $\mathcal{O}(n\log(n))$ can be a valuable tool to accelerate computations, even though it may at first seem weird or farfetched!
## Notes
(1) If this is not possible, the most significant part can be shorter than the rest.
(2) We will drop the multiplication symbol just for convenience.
(3) We mentioned this earlier with the Toom-Cook method. For example, we know from geometry that we need to give two points to determine a straight line, which is a one-degree polynomial.
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# How can I prove that a set of ciphers are the encryption of a set of plaintexts?
I have a set of plaintexts e.g., ["how", "are", "you"] and a sender who encrypts each element with a unique symmetrical key (or not) and outputs the set of ciphers.
Receiver needs assurance that all the ciphers he just received have a one-to-one mapping with the ["how", "are", "you"] and not some other values. I'm not using asymmetric encryption because it'd reveal which cipher represents which item.
I have read all other questions and a few papers on set membership, pederson commits, etc but still can't understand how it'd work. It'd be very helpful if you could give a detailed answer.
• Regarding your assumption that "asymmetric encryption [would] reveal which cipher represents which item": This is not true if you do asymmetric encryption properly, i.e., with randomized padding such as OAEP. That still doesn't solve your problem, though. – yyyyyyy Aug 12 '17 at 10:15
If you are allowing for probabilistic proofs, simply have the one doing the encryption encrypt the set multiple times. All of these encrypted sets are sent to you.
You then ask the one doing the encryption to reveal the keys for all but one of the sets. You do the decryption to verify that all of those sets decrypt to the proper plaintexts. This is a cut-and-choose protocol (see Cut-and-Choose Protocol by Crépeau).
If the adversary is going to cheat, they will only be able to encrypt one set incorrectly and you have to randomly not choose that set for verification. You can set the probability of this to be arbitrarily low (at the cost of communication and computation) by setting the number of copies of the set the one doing the encryption must encrypt to be high enough.
• Cut and Choose would reveal the mapping, which seems to be the challenge in the question. The secure shuffle from Mental Poker by Shamir Rivest and Adleman (1979) could help with. – tylo Aug 15 '17 at 9:17
• @tylo I'm assuming a random mapping for each separate set encryption. So, revealing the mapping of $n-1$ set encryptions, should not reveal the mapping of the $n$th encrypted set. Since OP says that each plaintext is encrypted with a unique key, you should get this property automatically. – mikeazo Aug 15 '17 at 11:57
This is a fairly common problem in some applied crypto settings, like anonymous credentials and cryptocurrency. What you need is a zero-knowledge proof of knowledge (ZKPoK) that the sender knows three distinct encryption keys and a hidden random permutation of three elements so that the ciphertexts given to the receiver are actually encryptions of the three plaintexts.
More formally, let the three ciphertexts be $c_1, c_2, c_3$. The ciphertext $c_i,i\in [1, 2, 3]$ is the encryption with a key $k_i$ of plaintext $p_{\sigma(i)}$. The plaintext $p_{\sigma(i)}$ is the $\sigma(i)^{\text{th}}$ element of the original list of plaintexts ["how", "are", "you"] where $\sigma$ is a uniformly random sample from the set of permutations of three elements.
In the language of proof systems, the string $(c_1,c_2,c_3,\text{"how"},\text{"are"},\text{"you"})$ is a member of the "language" the ZKPoK is proving membership in. The keys and hidden permutation are a "witness" that proves the string of ciphertexts and plaintexts is a member of this language. A ZKPoK of the statement below is what the sender needs to send the receiver to convince it that the ciphertexts are well-formed encryptions of the appropriate plaintexts, but not reveal any other information about the keys or the hidden permutation.
The statement is, roughly: "I know keys $k_1, k_2, k_3$ and a permutation $\sigma$ so that $c_i = E_{k_i}(p_\sigma(i))$ for $i\in [1, 2, 3]$".
Rather than using a standard encryption scheme, it's probably best to use a commitment, which is kind of like an encryption scheme that prevents the sender from proving multiple different statements about the ciphertexts.
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# Excel - IF Function Problem - Expert Solution
Question description:
This user has given permission to use the problem statement for this blog.
how to stop a formula, for example Cell A 1 * Cell B 1, for 12 months, B 1 = 10, after 6 months B 1 became 12 but if i put 12 in B 1 then the first 6 months calculation will not be at 10 but at 12, how can it automatically be converted and do not calculate it
Solved by G. U. in 12 mins
This is the chat thread from the real Excelchat help session. It contains no private user information.
Excelchat Expert 03/08/2018 - 07:04
User 03/08/2018 - 07:04
Hi
User 03/08/2018 - 07:05
i am looking for a solution
Excelchat Expert 03/08/2018 - 07:05
ok on which is it?
User 03/08/2018 - 07:06
hope you can see the excel sheet
Excelchat Expert 03/08/2018 - 07:06
yes im seeing on i
Excelchat Expert 03/08/2018 - 07:06
it*
User 03/08/2018 - 07:06
now if i change cell C 4 to 20%
User 03/08/2018 - 07:06
certain months value should not change
User 03/08/2018 - 07:06
for example lets say comission
User 03/08/2018 - 07:06
after sept the comission % is 20%
User 03/08/2018 - 07:07
here April to Aug it shoudl be at 10%
User 03/08/2018 - 07:07
and from Sept at 20%
Excelchat Expert 03/08/2018 - 07:07
ok, so it would be doubled?
Excelchat Expert 03/08/2018 - 07:07
is it like that?
User 03/08/2018 - 07:07
but April to aug shoudl only be 100
User 03/08/2018 - 07:08
yes
User 03/08/2018 - 07:08
is there a formula to freeze the amounts from April to Aug
User 03/08/2018 - 07:08
and not do manually
Excelchat Expert 03/08/2018 - 07:08
oh we dont have that function in excel
Excelchat Expert 03/08/2018 - 07:09
we can only fix that if we will remove the formula
User 03/08/2018 - 07:09
hmmmmmmm
User 03/08/2018 - 07:09
is there any other way
Excelchat Expert 03/08/2018 - 07:09
you want to fix apr - aug in 100?
User 03/08/2018 - 07:09
the objective is to make it automatic
User 03/08/2018 - 07:10
and no manual intervention
User 03/08/2018 - 07:10
yes
Excelchat Expert 03/08/2018 - 07:10
ok
Excelchat Expert 03/08/2018 - 07:10
so it the percent is 10%, you want apr-aug as 100
User 03/08/2018 - 07:11
C 4 is dynamic
User 03/08/2018 - 07:11
it keeps on changing
Excelchat Expert 03/08/2018 - 07:11
ok
User 03/08/2018 - 07:11
it can be 10% ,20% etc
User 03/08/2018 - 07:11
but once i calculate for one month
User 03/08/2018 - 07:11
that month amount should not change
Excelchat Expert 03/08/2018 - 07:12
so if any percent on C4
Excelchat Expert 03/08/2018 - 07:12
apri - aug would still be 100 and not changing?
User 03/08/2018 - 07:12
yes because the starting point is 10%
User 03/08/2018 - 07:12
from April to Aug
User 03/08/2018 - 07:13
after Aug its becomes 20% or 30%
User 03/08/2018 - 07:13
but that doesnt mean that April to Aug amounts will change
User 03/08/2018 - 07:13
it should still reamin 100
Excelchat Expert 03/08/2018 - 07:13
Excelchat Expert 03/08/2018 - 07:14
but excel doesn't have that function to fix the value, particularly if this has formula
User 03/08/2018 - 07:14
ok
Excelchat Expert 03/08/2018 - 07:14
want we can only do was a paste special
Excelchat Expert 03/08/2018 - 07:15
for apr - aug
User 03/08/2018 - 07:15
ok got it
User 03/08/2018 - 07:15
Excelchat Expert 03/08/2018 - 07:15
Excelchat Expert 03/08/2018 - 07:15
User 03/08/2018 - 07:15
i am good
Excelchat Expert 03/08/2018 - 07:15
ok cool :)
User 03/08/2018 - 07:15
and what are the fees if i have some more questions
Excelchat Expert 03/08/2018 - 07:16
im not sure for the fees, but likes monthly subscription
User 03/08/2018 - 07:17
ok
User 03/08/2018 - 07:17
thank you
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# texas a&m university by housework
VIEWS: 91 PAGES: 14
• pg 1
``` TEXAS A&M UNIVERSITY
ENGR 111B: Foundations of Electrical &
Computer Engineering
Lab 6: Pulse Width Modulation
Team Members: _________________________
_________________________
Section Number: __________ Team Number: __________
This Lab is due by the Beginning of the Next Lab Session.
Written By: Hank Walker
Lorne Liechty
Written by Texas A&M University 1
Lab 6: Pulse Width Modulation
Time Limit: 1 week
OVERVIEW
Using RC circuits we can not only time individual events, but we can also create
repeating timers. This oscillatory timing is used in virtually every electronic device
available. In this lab students will examine the use of a digitally oscillating signal to
control DC motor speed. This approach contrasts with the simple resistor-based current
limiting method you used previously. Topics covered in this lab include the following
material:
- Digitally Oscillating Signals
- Pulse Width Modulation
- Average Voltage, Current, and Power
- How a PWM signal can be generated using a 555 timer
- Diodes
- PWM Signal Control of DC Motor
BACKGROUND
To assist students in completing the exercises required by this lab, the following
background information has been provided. It is recommended that each student read all
of the following information as a beneficial review of the topics required.
DIGITAL OSCILLATIONS
An oscillating signal is one which changes between two levels according to a
regular pattern of intervals. AC (Alternating Current) voltages are included in the realm
of oscillating voltages, but the term AC is reserved for signals which include both
positive and negative voltages. A digital oscillation also switches between two levels
according to defined intervals, and the ideal digital signal exists only at the specified
levels and does not exist at the levels in between. The signals represented in Figures B1
are two different digitally oscillating signals.
Figure B1: Digitally Oscillating Signals
Written by Texas A&M University 2
Examining the signals in Figure B1, notice that the lower of the two voltages does not
have to be set to zero volts, or ground; the lower voltage of a digital signal is defined to
be a logical LOW or 0, but the value is not required to be zero. When a signal is not
centered around zero the signal is said to have a DC offset that raises or lowers the signal
to a new center voltage. Signals that do not have DC offsets fall into the realm of AC
signals, since their voltages will fall below zero and therefore reverse the flow of current.
Each time that the voltage of a digital signal rises to its HIGH state and then back
to its LOW state, the signal is said to have pulsed. The amount of time that a digital
signal remains in its HIGH state before returning to the LOW state is called the pulse
width. The period of a digital signal is length of time before a signal repeats itself.
Adding the HIGH pulse width and the LOW state time together is also equal to the period
of the signal. The frequency of a digital signal is equal to the inverse of the period, and
represents the number of times that a signal repeats itself in one second. All of the stated
elements of a digitally oscillating signal are shown and labeled in Figure B2.
Figure B2: Digital Signal with Labels
PWM Signals
Pulse Width Modulation (PWM) is a process which changes the pulse width of a
signal, while keeping the frequency/period constant. The result is a signal that may be
switched HIGH for a longer or shorter amount of time than it is switched LOW. When a
PWM circuit alters the pulse width in this way, it is said to be changing the duty cycle of
the signal, which is the ratio of the pulse width time over the total period. Duty
Cycle is a dimensionless quantity that is stated as a percentage. Figure B3 shows two
examples of pulse width modulated signals.
Written by Texas A&M University 3
Figure B3: Pulse Width Modulated Signals
Average Voltage / Current / Power
Another property of a PWM signal is that the time average values of its voltage
and current vary with its modulation. A digitally oscillating signal that is HIGH and
LOW for equal amounts of time has a time average voltage or current equal to 50% of the
difference between its HIGH and LOW voltage/current values. During the time that the
signal is HIGH, the voltage and current are at their maximum values, but when the signal
is LOW the voltage and current are at their minimum values. Power is equal to voltage
multiplied by current, and therefore the average power is directly proportional to the
square of the duty cycle, since both the average voltage and average current rise with
duty cycle.
Since pulse width modulation varies the duty cycle of a signal, it also varies the
average voltage and current levels in the circuit. If the frequency is high enough, this
average voltage or current can be interpreted as a virtual DC value. An analogy is that a
video is a collection of still pictures, but it looks like smooth motion to our eyes, since the
pictures change faster than the eye can respond. Similarly, a PWM signal fed into a motor
will act as a smooth voltage if the PWM frequency is high compared to the rate at which
the motor can respond.
Pulse width modulation is much more complicated than using a circuit with
resistors to divide the voltage or current. However, since resistors dissipate all of their
power as heat, they perform little useful work with the power they are given. Because a
PWM signal can change the apparent DC values of a circuit without the use of resistors,
they waste very little power. This is a useful property in situations where power is not
unlimited.
Written by Texas A&M University 4
HOW THE 555 TIMER GENERATES A PWM SIGNAL
The 555 timer is an integrated circuit that has been used for decades in electronics
for pulse generation, time delay generation, and pulse width modulation. As shown in
Figure B4, a 555 timer consists of two comparators and a Flip-Flop which is a device
used in digital logic similar to an SR Latch. Flip-Flops are not covered in this series of
labs, but they are extensions of latches that also hold their outputs stable until they have
been changed by a differing digital input. The comparator inputs are fed by an RC time
constant provided by the user, An animated schematic of the 555 as an oscillator is
available online at http://www.williamson-labs.com/pu-aa-555-timer_med.htm to help
you understand its function. Many other 555 timer resources are available on the Web.
See http://www.uoguelph.ca/~antoon/gadgets/555/555.html for a tutorial.
Figure B4: NE555 functional block diagram
Diodes
As shown in Figure B5, a diode is made by combining two portions of p-type and
n-type semi-conducting material.
Written by Texas A&M University 5
Figure B5: Diagram of a PN Junction Diode
A diode is an electrical device which regulates the flow of current through a circuit.
Depending on the polarity of the voltage applied across it, a diode will be either turned on
or off. Figure B6 shows the two possible states of an ideal diode and how the current in
the circuit is affected by the diode’s state.
Figure B6: States of Diode
As you can see, if the diode voltage is forward biased then the current will flow freely in
the circuit. However, if the diode voltage is reversed biased current cannot flow in the
opposite direction through the diode. It is also important to note that when in the forward
biased state an actual diode will have an associated voltage drop across it, which is usual
equal to approximately 0.7 volts for a silicon diode.
The PWM Generation Circuit
Figure B7, contains a schematic of one method for using a 555 timer to create a
fixed frequency PWM signal.
Written by Texas A&M University 6
Figure B8: Schematic for a PWM Generator Using a 555 Timer
In the schematic, C1, R1, R2, and the potentiometer are all used to vary the timing of the
circuit. The reason that the pulse width of the signal is variable in this configuration is
that diode D2 creates a path for current to bypass the resistance between the tap of the
potentiometer and D1 to charge capacitor C1. Further, as the capacitor discharges, D2
prohibits the flow of current directly to ground and forces the capacitor to discharge
through D1 and the previously avoided resistance. During the charging cycle a voltage
drop is induced across D2; as such diode D1 is necessary to repeat that same voltage drop
during the discharge cycle so that the operating frequency of the signal is maintained.
The equations which govern the Duty Cycle and frequency of this configuration are given
as:
t1 ( HIGH ) 0.67 * R A * C1 t2 ( LOW ) 0.67 * RB * C1
1
Period T t1 t 2 Frequency f
T
where RA is equal to the total resistance of R2 plus the resistance between R2 and the tap
of the potentiometer (between 100 and 10.1k ohms), and RB is equal to the resistance
between the tap and R1 plus the resistance of R1 (between 100 and 10.1k ohms). The
duty cycle varies from approximately 1% to 99%.
DC Motor Control
If you swap the power connections of the motor (e.g. swap the black and white
wires on the cable driving the motor), the motor will change directions. NOTE: Be
careful not to swap the black (ground) and white (positive battery voltage) wires
running from the Lego brick outputs. This will put reverse voltage on your circuits,
and may destroy chips such as the LM393 and NE555. The motor reverses directions
Written by Texas A&M University 7
when the connections are swapped because the current flowing through the motor has
changed directions. When current moves through the windings of a motor, it produces an
associated electromagnetic field. The magnetic field produced by the wires attached to
the armature (rotating core) of your DC motor reacts with the inherent magnetic field of
small permanent magnets attached to the sides of the stator (fixed shell). This reaction
either attracts or repels the windings of wire attached to the axle and causes it to turn.
Figure B9 is a basic diagram of a DC motor and its components to assist in understanding
its operation.
Figure B9: Diagram of Basic DC Motor
Conveniently enough, if a magnetic field is produced by flowing current in one direction,
the opposite magnetic field is produced by flowing current in the opposite direction and
therefore causes the motor to turn in the other direction. This is what causes the motor to
change directions when the motor power wires are switched.
DC MOTOR SPEED CONTROL
The DC motors on your robot are capable of varying their speed based upon the
current that is drawn through them. The driving force of the motor is caused by the
interaction of the magnetic field produced by the current though the windings on the
armature and the permanent magnets attached to the stator. The magnitude of the force
caused by this interaction varies as the current through the windings varies, thus
regulating the speed of the motor.
Speed Control Using a PWM signal
In Lab 4, you controlled the speed of your motors by limiting the current through
it with a potentiometer. A PWM signal can also be used to vary the motor speed. As
mentioned previously, the average voltage and current of a PWM signal can be adjusted
by modifying the duty cycle of the signal. When the current through the motor is varied
using a PWM signal, the end result is the same as when the current is modified using a
resistor; causing the motor to slow down.
A PWM signal has two advantages when controlling motor speed. First, since it
turns the motor on or off, it does not waste as much power as using a resistor to control
Written by Texas A&M University 8
the speed. Second, unloaded motors usually have a linear speed-voltage relationship. The
relationship for the NXT motor is shown in Figure B10.
Figure B10: Unloaded NXT speed vs. voltage (from www.philohome.com).
However, in Lab 4, you discovered that the speed is superlinear in voltage when
the motor is under load. The NXT characteristics are shown in this table from
http://www.philohome.com:
Rotation Mechanical Electrical
Torque Current Efficiency
speed power power
NXT
4.5 V 16.7 N.cm 33 rpm 0.6 A 0.58 W 2.7 W 21.4 %
7V 16.7 N.cm 82 rpm 0.55 A 1.44 W 3.85 W 37.3 %
9V 16.7 N.cm 117 rpm 0.55 A 2.03 W 4.95 W 41 %
12 V 16.7 N.cm 177 rpm 0.58 A 3.10 W 6.96 W 44.5 %
As can be seen, doubling the voltage from 4.5V to 9V increases motor speed by
3.5x. In a PWM signal, since the motor is either all the way on (full voltage) or all the
way off, the speed is simply proportional to the full power speed times the duty cycle. So
a 50% duty cycle signal will run the motor at half its full speed for that voltage and
torque. (Torque is the twisting force on the motor, e.g. the force that the wheel exerts
when resisting turning against the floor).
Written by Texas A&M University 9
LAB EXERCISES
BEFORE YOU COME TO LAB: As in Lab 4, you will drive the robot in a circle with a
radius of 2 feet. In order to do this, you must first calculate what speeds you will need
each motor to operate at, so that the robot will follow the circular path. Use Figure L1
and the equation below to determine the wheel speeds necessary to navigate this path.
Speed1 Radius1 2 feet
Speed 2 Radius2 2 feet w R1
R2
w = width of robot
Figure L1: Circular Path of Robot
The width of the robot from wheel center to wheel center is 11.3 cm, and the tires are 5.7
cm in diameter. The ratio should produce a circular path which places the outside edge
of the robot on the 4 foot diameter circle. Enter the values for the speeds of the right and
left wheels in Table L2. As in Lab 4, the outer wheel will run at full speed, and you will
reduce the speed of the inner wheel using PWM.
CIRCULAR PATH WHEEL SPEEDS
Right Wheel Left Wheel
Speed
(ft/s)
Table L1
You will use the 555 timer to produce a PWM signal which will control the speed of
the inner wheel. First, you will need to construct the circuit given in Figure L2.
Figure L2: PWM Speed Control
Written by Texas A&M University 10
The pinouts of the Texas Instruments NE555P timer are shown in Figure L3.
VERY IMPORTANT: Unlike most chips, pin 1 is ground, rather than pin 4 as in the
LM393. Due to its ancient age, the 555 violates the normal pin-out convention.
Figure L3: Texas Instruments NE555P pinout
As discussed above, the two 100 resistors limit the PWM duty cycle range from
about 1% to about 99%. The diodes permit the duty cycle to be >50% (high). The output
drives a 1k resistor to the base of a 2N3704 NPN transistor to power the motor, since the
555 output cannot quite provide the current required of the motor. As a reminder, the
2N3704 has a non-standard pinout, shown in Figure L4.
Figure L4: Packaging Diagram of 2N3704 NPN Transistor. It is housed in a TO-92 package. With the flat
side up, the emitter is on the left, collector in the middle, and base on the right. (Most transistors have the
base in the middle and collector on the right).
When determining which capacitor is which, the following website for capacitor codes
may be useful: http://xtronics.com/kits/ccode.htm.
The diodes have a line on the end that corresponds to the line in the schematic. That is,
current flows out of the end with the line.
EXPERIMENTAL PROCEDURE
1. The first test will be to run the robot in a straight line. Attach your PWM circuit to
the faster motor. Calculate the duty cycle needed to have faster motor run at the
same speed as the slower motor. Remember that in Lab 4, you computed the
difference in resistance between the two motors, and put a potentiometer on the
lower-resistance (faster) motor to slow it down. In this case you will adjust the
Written by Texas A&M University 11
duty cycle of the PWM circuit to slow the faster motor down by a corresponding
amount. For example, suppose the left motor is 50 ohms and the right motor is 55
ohms, for a 7.2V battery, the left motor draws the left motor draws 144 mA and
the right motor draws 131 mA. To reduce the left motor current to 131 mA, the
effective left motor voltage must be reduced to 6.55V (131 mA * 50 ohms). This
is 91% of the battery voltage (6.55V/7.2V). So the PWM circuit should be set to
91% duty cycle.
2. Calculate the values of RA and RB above to achieve the appropriate duty cycle and
adjust the potentiometer accordingly. Recall that RA is the value of Ra plus the
value between the Ra connection and the tap on the pot in Figure L2. Similarly for
RB.
3. We can use an oscilloscope to measure the duty cycle directly, but an alternative
is to measure the voltage from the 555 with your voltmeter. Your voltmeter
responds more slowly than the 555 frequency, so will measure the average value.
4. Run your robot with the calculated duty cycle. Tweak the potentiometer until the
robot runs in a straight line. Afterwards, measure the resistance of each side of
the potentiometer and calculate the duty cycle actually used. Use your voltmeter
to measure the average voltage out of the PWM circuit. Note that you should
measure it directly at pin 3, since the voltage at the base of the transistor will be at
most 0.8V. Record all values in Table R1.
5. Using the process above, calculate the potentiometer resistance which would
cause the robot to follow a circular track with a radius of 2 ft. Use the speeds that
were calculated before the lab in Table L1. Record the calculated value of RA in
Table R2.
6. Test your calculated values by running the robot in a circle. If your values are not
satisfactory, adjust them so that the robot successfully navigates a 4-ft diameter
circle. Record your corrected value of RA, along with their associated percent
error from the calculated values in Table R2.
Written by Texas A&M University 12
RESULTS: to be uploaded onto WebCT
Straight Line Values
Speed (ft/s)
Calculated RA
(kΩ)
Calculated Duty
Cycle (%)
Measured PWM
Output Voltage
(V)
Tweaked RA
(kΩ)
Calculated
Tweaked Duty
Cycle (%)
Measured MWP
Tweaked Output
Voltage (V)
Table R1
CIRCLE RUN
Calculated RA Corrected RA
Resistance Resistance % Error
Inside Inside
Wheel Wheel
Table R2
Written by Texas A&M University 13
REPORT: to be uploaded onto WebCT
REPORT REQUIRMENTS
Answer the following questions:
- What are the pros and cons of the current limiting (via resistor) motor speed
control method used in Lab 4?
- What are the pros and cons of the PWM motor speed control method?
- What would be your recommendation and why, for which system to use if:
o Your robot needed cheap and effective speed control.
o Your robot was going to be used by NASA to explore Mars.
Written by Texas A&M University 14
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# Permutation program in Java
Combinatorics' basic idea of permutations is the arrangement of a collection of items in several orders. We'll go through several techniques for creating permutations in Java and include code samples along with thorough explanations.
## How Do Permutations Occur?
A permutation is the organization of elements in a particular sequence. It involves arranging the members of a set in a specific linear or sequential order. For example, consider the set A={11,66}. In this problem, there are only two possible permutations: {11,66} and {66,11}. There are no other possible permutations for this set.
### Method 1: Recursive Technique
Recursive methods are among the easiest techniques to produce permutations. This Java program uses recursion to produce array permutations:
PermutationsRecursive.java
Output:
```[100, 120, 230]
[100, 230, 120]
[120, 100, 230]
[120, 230, 100]
[230, 100, 120]
[230, 120, 100]
```
Explanation
This code uses recursion to create permutations of the array. It outputs the various ordering that are created by switching around the elements in the array. In the simplest scenario, the array is printed when n, its length, equals 1.
### Approach 2: Lexicographic Sequence
Lexicographic order is another way to create permutations. When you need to find permutations in a certain order, this approach may be used to produce permutations one by one. This Java program produces lexicographically ordered permutations.
PermutationsLexicographic.java
Output:
```[100, 120, 230]
[100, 230, 120]
[120, 100, 230]
[120, 230, 100]
[230, 100, 120]
[230, 120, 100]
```
Explanation
The items in this code are first sorted, and we iteratively discover the next lexicographically greater permutation. The generatePermutations technique makes use of the fact that identifying the rightmost element that may be increased provides the foundation for each step.
### Method 3: Without External Libraries
PermutationsWithoutLibraries.java
Output:
```[100, 120, 230]
[100, 230, 120]
[120, 100, 230]
[120, 230, 100]
[230, 100, 120]
[230, 120, 100]
```
Explanation
Using the list of components, the current permutation, and a list to hold the results, we build a recursive function called generatePermutations in this code. By adding each element to the current permutation and eliminating it from the remaining components, the function iteratively produces all possible permutations.
## Conclusion
Creating permutations is a frequent issue that may be solved in several ways. We studied three approaches using Java: lexicographic order, recursive, and utilizing Collections. The strategy you choose will rely on your unique needs and limitations. These examples ought to aid in your comprehension of the fundamentals of Java permutation generation and lay the groundwork for more difficult permutation-related activities.
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# Solved – Quantiles for non-normal cdf
When I compute the five-number summary on my sample, I obtain quantiles that differ from the quantiles I got from the empirical cdf, since they are not normally distributed data.
Can you help me in the interpretation of this difference?
For instance, with a randomly-generated Poisson dataset x
``x=rpois(50, 2) summary(x) Min. 1st Qu. Median Mean 3rd Qu. Max. 0.00 1.00 1.50 1.82 2.75 6.00 y=ecdf(x) summary(y) Empirical CDF: 7 unique values with summary Min. 1st Qu. Median Mean 3rd Qu. Max. 0.0 1.5 3.0 3.0 4.5 6.0 ``
What does it mean that the 3rd quantile of the sample is `2.75` while it is `4.5` for the ecdf?
Contents
`summary(x)` is computing sample quantiles of your data, using type 7 quantiles (see `help(quantile)` in R for the various quantile types).
I'm guessing that you've used `summary(y)` to produce the second set of values. In that case, the results are probably not what you want as they are giving you quantiles of the data set {0,1,2,3,4,5,6}, the step points of the empirical cdf.
You can get the quantiles from the ecdf object using `quantile(y)` which should give you the same results as `quantile(x)`.
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# lookup or match?
K
#### KC
on tab: Weeks NQCQ I have a table as follows
Col A Col B Col C
Week CQ Week NQ Week
wk2 1Q10 Wk-12 Week 2
wk3 1Q10 Wk-11 Week 3
wk4 1Q10 Wk-10 Week 4
wk5 1Q10 Wk-9 Week 5
wk6 1Q10 Wk-8 Week 6
wk7 1Q10 Wk-7 Week 7
wk8 1Q10 Wk-6 Week 8
wk9 1Q10 Wk-5 Week 9
wk10 1Q10 Wk-4 Week 10
wk11 1Q10 Wk-3 Week 11
wk12 1Q10 Wk-2 Week 12
wk13 1Q10 Wk-1 Week 13
wk14 2Q10 Wk-13 Week 14
On tab "Week at a Glance", in cell B4 I will put in the current week, lets
say I input wk8, on the same tab, I need a formula to look at the table above
and insert the corresponding CQ Week in Col B (answer should be 1Q10 Wk-6).
I will need a similar formula in cell I4, that will grab the correct answer
from column C above.
K
#### KC
Was able to figure it out....for anyone else that might have a question like
this, the formula is:
=VLOOKUP(B4,'Weeks NQCQ'!A1:B100,2)
J
#### Jim Thomlinson
My preference is Index Match as it is less prone to developing errors than
VLookup
=index(B\$2:B\$100, match(\$A\$2:\$A\$100, \$B\$4, 0))
You can drag that formula to the right to ge tthe reference to Column C.
J
#### Jim Thomlinson
Wrote may match backwards...
=index(B\$2:B\$100, match(\$B\$4, \$A\$2:\$A\$100, 0))
J
#### Jim Thomlinson
If you are going to use Vlookup you should specify your optional 4th argument
as 0 for an exact match...
=VLOOKUP(B4,'Weeks NQCQ'!A1:B100,2, 0)
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# How do you calculate monthly amortization in SSS?
How to Check SSS Loan Balance Online 2021
1. Step 1: Go to www.sss.gov.ph , then member login to view check your loans info. Type in your membership login details. …
2. Step 2: Hover your mouse to the “Inquiry” tab. …
3. Step 3: Hover your mouse to “Loans Info” tab.
>> Click to read more <<
## Likewise, people ask, can I apply salary loan and calamity loan at the same time?
The calamity loan assistance is separate from the regular salary loan. Members with outstanding salary loans in SSS can still avail of the calamity loan.
Accordingly, can I loan in SSS with existing salary loan? Can I apply for another loan from SSS with an existing loan? A. You can only apply for another salary loan when you have paid at least 50% of the principal or at least half of the term has lapsed. … Your outstanding balance will be deducted from your new loan amount.
## Beside above, can I pay more than my monthly amortization in SSS?
What happens to unpaid SSS salary loans? A member’s failure to pay more than 6 monthly amortizations will result in loan default. A defaulted account incurs 10% interest per annum on the outstanding principal balance as well as a 1% penalty per month on unpaid principal and interest.
## How are loan installments calculated?
USING MATHEMATICAL FORMULA
EMI = [P x R x (1+R)^N]/[(1+R)^N-1], where P stands for the loan amount or principal, R is the interest rate per month [if the interest rate per annum is 11%, then the rate of interest will be 11/(12 x 100)], and N is the number of monthly instalments.
## How can I calculate my SSS salary loan?
LOAN AMOUNT
A one-month salary loan is equivalent to the average of the member-borrower’s latest posted 12 Monthly Salary Credits (MSCs), or amount applied for, whichever is lower.
## How do you calculate monthly amortization?
Amortization Calculation
You’ll need to divide your annual interest rate by 12. For example, if your annual interest rate is 3%, then your monthly interest rate will be 0.0025% (0.03 annual interest rate ÷ 12 months). You’ll also multiply the number of years in your loan term by 12.
## How do you calculate monthly payments on a loan?
Divide the interest rate (expressed as a decimal) by the number of repayments you’ll make throughout the loan term. For example, if your loan term is two years and you’ll make monthly payments, divide the interest rate by 24. Multiply the result by the balance of the loan.
## How is Piti calculated?
To calculate your PITI on a 30-year fixed rate loan: Your monthly mortgage principal and interest will amount to about \$1,432.25 per month. Add on your property tax and insurance estimations. To calculate property taxes, divide your home’s value by 1,000 and multiply that number by \$1 to find your monthly payment.
## How much is the maximum salary loan in SSS 2021?
Assuming that your income never changed and you paid your contributions consistently in the last 12 months, the maximum SSS salary loan you can borrow is Php 18,000 for a one-month salary loan and as much as Php 36,000 for a two-month salary loan, provided that you’re qualified for it.
## What is PRN loans in SSS?
The PRN for loans is a system-generated number corresponding to a loan billing statement of an individual member (self-employed, voluntary, or Overseas Filipino Worker members) or employer.
## What is salary loan amortization?
An amortized loan is a form of financing that is paid off over a set period of time. Under this type of repayment structure, the borrower makes the same payment throughout the loan term, with the first portion of the payment going toward interest and the remaining amount paid against the outstanding loan principal.
## What is SSS loan month?
A one-month loan is equivalent to the average of member’s last twelve (12) Monthly Salary Credits (MSC), or the amount applied for, whichever is lower.
## What is SSS monthly salary credit?
Monthly Salary Credit (MSC) – The compensation base for contributions and benefits related to the member’s total earnings for the month, as indicated in the schedule in Section 18 of the SS Law. … The person should never have been a member of the SSS.
## What will happen if you stop paying SSS?
pay all unpaid contributions plus a penalty of three percent per month; and. be held liable for a criminal offense punishable by fine and/or imprisonment.
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https://dannycaballero.info/phy481msu_f2018/notes/02-slides.html
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On Wednesday, you took an assessment of electromagnetism concepts. **How did that assessment feel for you?** 1. I think it went fine; I felt like I knew most of the answers. 2. I was concerned about one or two questions; but most of the questions were familiar. 3. I guessed (or left blank) most of the questions; none of the questions really felt familiar. Note: * Fall 2016: 3 32 65 0 0
## Announcements * Exams!!! * Evening Exams * Oct 3 (BCH 101) and Nov 7 (1415 BPS), 7pm-9pm * Homework Help Session * Wednesday 5:00pm-6:30pm in 1300 BPS * Thursday 4:30pm-6:00pm in A158 PSS
## Mathematical Preliminaries $\nabla\cdot\mathbf{E} = \frac{\rho}{\epsilon_0} \qquad \int \mathbf{E}\cdot d\mathbf{A} = \int \frac{\rho}{\epsilon_0} d\tau$ $\nabla\cdot\mathbf{B} = 0 \qquad \int \mathbf{B} \cdot d\mathbf{A} = 0$ $\nabla\times\mathbf{E} = - \frac{\partial\mathbf{B}}{\partial t} \qquad \int \mathbf{E} \cdot d\mathbf{l} = - \int \frac{\partial\mathbf{B}}{\partial t} \cdot d\mathbf{A}$ $\nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial\mathbf{E}}{\partial t} \qquad \int \mathbf{B} \cdot d\mathbf{A} = \mu_0 \int \left(\mathbf{J} + \epsilon_0 \frac{\partial\mathbf{E}}{\partial t}\right) \cdot d\mathbf{A}$ Note: There's a reason that we are starting with vectors and vector operations; it's inherent in the way electromagnetism is described!
<img src ="./images/charges_in_plane.png" align="right" style="width: 350px";/> Two charges +Q and -Q are fixed a distance r apart. The direction of the force on a test charge -q at A is... 1. Up 2. Down 3. Left 4. Right 5. Some other direction, or $F = 0$ Note: * CORRECT ANSWER: A * Use superposition * Fall 2016: [71] 14 3 8 5; Second vote (after discussion): [98] [2] 0 0 0
In a typical Cartesian coordinate system, vector $\mathbf{A}$ lies along the $+\hat{x}$ direction and vector $\mathbf{B}$ lies along the $-\hat{y}$ direction. What is the direction of $\mathbf{A} \times \mathbf{B}$? 1. $-\hat{x}$ 2. $+\hat{y}$ 3. $+\hat{z}$ 4. $-\hat{z}$ 5. Can't tell Note: * Correct Answer: D * Use the right-hand rule * Fall 2016: 0 0 23 [77]
In a typical Cartesian coordinate system, vector $\mathbf{A}$ lies along the $+\hat{x}$ direction and vector $\mathbf{B}$ lies along the $-\hat{y}$ direction. What is the direction of $\mathbf{B} \times \mathbf{A}$? 1. $-\hat{x}$ 2. $+\hat{y}$ 3. $+\hat{z}$ 4. $-\hat{z}$ 5. Can't tell Note: * Correct Answer: C * Use right-hand rule; means $\mathbf{A} \times \mathbf{B} = - \mathbf{B} \times \mathbf{A}$ * Fall 2016: 0 0 [94] 6 0
<img src ="./images/cq_spherical.png" align="right" style="width: 350px";/> ### You derive it Consider the radial unit vector ($\hat{r}$) in the spherical coordinate system as shown in the figure to the right. Determine the $z$ component of this unit vector in the Cartesian $(x,y,z)$ system as a function of $r,\theta,\phi$. Note: This demonstrates that the r unit vector is a curious thing, in fact in contains all the information that is needed to define where you on the unit sphere. The other vectors can be though of as defined relative to that. Altered for F2017 to be shorter, only work on z component
<img src ="./images/cq_vector_in_cylindrical.png" align="right" style="width: 350px";/> In cylindrical (2D) coordinates, what would be the correct description of the position vector $\mathbf{r}$ of the point P shown at $(x,y) = (1, 1)$? 1. $\mathbf{r} = \sqrt{2} \hat{s}$ 2. $\mathbf{r} = \sqrt{2} \hat{s} + \pi/4 \hat{\phi}$ 3. $\mathbf{r} = \sqrt{2} \hat{s} - \pi/4 \hat{\phi}$ 4. $\mathbf{r} = \pi/4 \hat{\phi}$ 5. Something else entirely Note: * CORRECT ANSWER: A * Fall 2016: [6] 90 3 0 1; Second vote (discussion and hint about units): [54] 35 4 1 4
How is the vector $\mathfrak{R}_{12}$ related to $\mathbf{r}_1$ and $\mathbf{r}_2$? <img src ="./images/cq_r1r2.png" align="right" style="width: 350px";/> 1. $\mathfrak{R}_{12} = \mathbf{r}_1 +\mathbf{r}_2$ 1. $\mathfrak{R}_{12} = \mathbf{r}_1 - \mathbf{r}_2$ 1. $\mathfrak{R}_{12} = \mathbf{r}_2 - \mathbf{r}_1$ 4. None of these Note: * CORRECT ANSWER: C * Fall 2016: 6 1 [91] 1 0
Coulomb's Law: $\mathbf{F} = \frac{k q_1 q_2}{\left|\mathfrak{R}\right|^2}\hat{\mathfrak{R}}$ where $\mathfrak{R}$ is the relative position vector. In the figure, $q_1$ and $q_2$ are 2 m apart. Which arrow **can** represent $\hat{\mathfrak{R}}$? <img src ="./images/cq_unit_r.png" align="center" style="width: 550px";/> 1. A 2. B 3. C 4. More than one (or NONE) of the above 5. You can't decide until you know if $q_1$ and $q_2$ are the same or opposite charges Note: * CORRECT ANSWER: D * A unit vector has no units; so it's length is meaningless on a picture with units. * Fall 2016 (hint given while still open): 14 10 9 [44] 23; students wanted to discuss E
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https://quizizz.com/en/radians-and-degrees-worksheets-class-11
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12 Q
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Angles
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XI_Trigonometric functions_1
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converting degrees to radians and vice versa
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Arc Length, Area of Sectors and Segments
14 Q
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Unit Circle
15 Q
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## Explore printable radians and degrees worksheets for 11th Class
Radians and degrees worksheets for Class 11 are essential resources for teachers who want to help their students excel in Math, particularly in the area of Trigonometry. These worksheets provide a variety of exercises and problems that focus on the conversion between radians and degrees, as well as the application of these concepts in solving real-world problems. By incorporating these worksheets into their lesson plans, teachers can ensure that their students have a solid foundation in understanding the relationship between radians and degrees, which is crucial for success in higher-level mathematics courses. Furthermore, these worksheets can be used for both in-class activities and homework assignments, making them a versatile and valuable tool for any Class 11 Math teacher looking to enhance their students' learning experience in Trigonometry.
Quizizz is an excellent platform that offers a wide range of resources for teachers, including radians and degrees worksheets for Class 11 Math students, as well as other Trigonometry-related materials. This platform allows teachers to create engaging and interactive quizzes that can be used to assess students' understanding of the concepts covered in the worksheets. Additionally, Quizizz provides teachers with valuable insights into their students' progress and performance, enabling them to tailor their instruction to meet the needs of each individual learner. By utilizing Quizizz in conjunction with radians and degrees worksheets for Class 11, teachers can create a comprehensive and dynamic learning experience that not only reinforces key Trigonometry concepts but also fosters a deeper understanding and appreciation for the subject matter.
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http://tptp.org/cgi-bin/SeeTPTP?Category=Problems&Domain=SEV&File=SEV379%5E5.p
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TPTP Problem File: SEV379^5.p
View Solutions - Solve Problem
```%------------------------------------------------------------------------------
% File : SEV379^5 : TPTP v7.5.0. Released v4.0.0.
% Domain : Set Theory (GvNB)
% Problem : TPS problem from GVB-MB-THMS
% Version : Especial.
% English :
% Refs : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% Source : [Bro09]
% Names : tps_0866 [Bro09]
% Status : CounterSatisfiable
% Rating : 0.00 v4.0.0
% Syntax : Number of formulae : 3 ( 0 unit; 2 type; 0 defn)
% Number of atoms : 21 ( 3 equality; 12 variable)
% Maximal formula depth : 13 ( 7 average)
% Number of connectives : 14 ( 0 ~; 0 |; 5 &; 8 @)
% ( 0 <=>; 1 =>; 0 <=; 0 <~>)
% ( 0 ~|; 0 ~&)
% Number of type conns : 3 ( 3 >; 0 *; 0 +; 0 <<)
% Number of symbols : 4 ( 2 :; 0 =)
% Number of variables : 4 ( 0 sgn; 4 !; 0 ?; 0 ^)
% ( 4 :; 0 !>; 0 ?*)
% ( 0 @-; 0 @+)
% SPC : TH0_CSA_EQU_NAR
% Comments : This problem is from the TPS library. Copyright (c) 2009 The TPS
% project in the Department of Mathematical Sciences at Carnegie
%------------------------------------------------------------------------------
thf(cGVB_OP,type,(
cGVB_OP: \$i > \$i > \$i )).
thf(cGVB_M,type,(
cGVB_M: \$i > \$o )).
thf(cGVB_OP_PROP_1,conjecture,(
! [Xa: \$i,Xb: \$i,Xc: \$i,Xd: \$i] :
( ( ( cGVB_M @ Xa )
& ( cGVB_M @ Xb )
& ( cGVB_M @ Xc )
& ( cGVB_M @ Xd )
& ( ( cGVB_OP @ Xa @ Xb )
= ( cGVB_OP @ Xc @ Xd ) ) )
=> ( ( Xa = Xc )
& ( Xb = Xd ) ) ) )).
%------------------------------------------------------------------------------
```
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https://www.wow.com/content?q=maksud+media&pvid=.65aUTEwLjKwTbBVYh.UaAECNzQuMQAAAAAkiIFn&nojs=1&ei=UTF-8&xargs=0&s_pt=aolsem&s_chn=235&s_it=rs-rhr1&s_it=rs-rhr2
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# WOW.com Web Search
2. ### Mean inter-particle distance - Wikipedia
en.wikipedia.org/wiki/Mean_inter-particle_distance
Ambiguity. From the very general considerations, the mean inter-particle distance is proportional to the size of the per-particle volume /, i.e., / /, where = / is the particle density.
3. ### Winsorized mean - Wikipedia
en.wikipedia.org/wiki/Winsorized_mean
A winsorized mean is a winsorized statistical measure of central tendency, much like the mean and median, and even more similar to the truncated mean.It involves the calculation of the mean after winsorizing-- replacing given parts of a probability distribution or sample at the high and low end with the most extreme remaining values, typically doing so for an equal amount of both extremes ...
4. ### Chart datum - Wikipedia
en.wikipedia.org/wiki/Chart_datum
A chart datum is the water level surface serving as origin of depths displayed on a nautical chart.A chart datum is generally derived from some tidal phase, in which case it is also known as a tidal datum.
5. ### Tea Act - Wikipedia
en.wikipedia.org/wiki/Tea_Act
Long title: An act to allow a drawback of the duties of customs on the exportation of tea or oil to any of his Majesty's colonies or plantations or farms in America; to increase the deposit on bohea tea to be sold at the East India Company's sales; and to empower the commissioners of the treasury to grant licenses to the East India Company to export tea duty-free.
6. ### Wanna Be Startin' Somethin' - Wikipedia
en.wikipedia.org/wiki/Wanna_Be_Startin'_Somethin'
"Wanna Be Startin' Somethin' " was one of the songs released as a single without an accompanying video. It nonetheless has attained a popularity rivaling its sister compositions on the album, and became Jackson's song of choice for opening live concerts, although not being as strictly associated with a specific dance routine as those others has arguably allowed for more flexibility in ...
7. ### Erdheim–Chester disease - Wikipedia
en.wikipedia.org/wiki/Erdheim–Chester_disease
Erdheim–Chester disease (ECD) is an extremely rare disease characterized by the abnormal multiplication of a specific type of white blood cells called histiocytes, or tissue macrophages (technically, this disease is termed a non-Langerhans-cell histiocytosis).
8. ### Law & Order: Criminal Intent - Wikipedia
en.wikipedia.org/wiki/Law_&_Order:_Criminal_Intent
Law & Order: Criminal Intent is an American police procedural drama television series set in New York City, where it was also primarily produced.Created and produced by Dick Wolf and René Balcer, the series premiered on September 30, 2001, as the third series in Wolf's successful Law & Order franchise.
9. ### Golden mean (philosophy) - Wikipedia
en.wikipedia.org/wiki/Golden_mean_(philosophy)
The golden mean or golden middle way is the desirable middle between two extremes, one of excess and the other of deficiency. It appeared in Greek thought at least as early as the Delphic maxim "nothing in excess" and emphasized in later Aristotelian philosophy.
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# A game is a strategic interaction between two players. Each player has their own sets of...
###### Question:
A game is a strategic interaction between two players. Each player has their own sets of actions called the strategies. Each strategy comes with a definite outcome, these outcomes are tied to some profit or loss called the payoff. One of the favorite examples of game theory is the Prisoners' dilemma.
In this game, two partners of crime are caught by police and held in different cells being interrogated separately. Both have two options, either to confess or be silent. The payoff they receive is in terms of years in jail. For example, if they both confess they get 12 years in jail and thus the payoff is -12 for each for strategy (confess, confess).
The game table is completed as below
Does either player have a dominant strategy? If so, what?
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##### 22. In a saturated solution of MgF at 18'C Ihe concentralion of Mg"' Is 1.21 X10 ' M: The equilibrium is represented by the following equation: MgFza- Mg" Write Ine- expression for the solubility-product conslant Kp" and calculale its value at 18'C Note: Show all calculations follow the rules of significant figures, and box your final answer: marks)
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##### Suppose= instead_ that it was discovered that the largest observation 40.2 is correct but that the observation 25.8 is incorrect and should ctually be 24. Now the sample mean 1 =25.011 and the standard deviation 8.417_ Use these new values and repeat parts (c) and (e).Find 98% two-sided confidence interval on mean compressive strength Round your answers t0 two decima places (e.g. 98.76).tne [Olerance+[-2%Find 9890 two-sided confidence interval on the variance of compression strength. Round vour
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##### A corporation has 10,000 bonds outstanding with a 6% annual coupon rate and 8 years to...
A corporation has 10,000 bonds outstanding with a 6% annual coupon rate and 8 years to maturity. Each bond has a $1,000 face value and a$1,100 market price. • The company’s 1,250,000 shares of common stock sell for $10 per share. • The company will pay$1.2/share in dividend next pe...
##### A pair of parallel conducting rails that are L = 22 cm apart lies at right...
A pair of parallel conducting rails that are L = 22 cm apart lies at right angles to a uniform magnetic field of 0.7 T directed into the page, as shown in the figure below. A R = 172 resistor is connected across the rails. A conducting bar is moved to the right at 2 m/s across the rails. B into page...
##### What is the enthalpy change (KJ) during the process in which 200.0 g of water at...
What is the enthalpy change (KJ) during the process in which 200.0 g of water at 60.0 Celsius is cooled to -25 Celsius? The specific heats of ice, and liquid water, and 2.03 J/g-K, and 4.18 J/g-K, respectively. For H 2 O, delta H fus = 6.01 kJ/mol...
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##### Joint Cost Allocation-Weighted Average Method Carving Creations jointly produces wood chips and sawdust used in agriculture....
Joint Cost Allocation-Weighted Average Method Carving Creations jointly produces wood chips and sawdust used in agriculture. The wood chips and sawdust are actually by-products of the company's core operations, but Carving Creations accounts for them just like normally produced goods because of ...
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[–][M] [score hidden] stickied comment (1 child)
### General Discussion Thread
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[–]3✓ 3377 points3378 points (220 children)
Practicing acoustics engineer here, thought I would have some fun. So 600DB looks like the model number, but lets assume that its a "decibel" as is asked in the question.
There is no reference value stated with the 600 dB, so we don't know exactly what it would measure. But let's see what we get if its 600 dB of sound pressure (re: 20 uPa) and 600 dB of sound power (re: 1 pW). Obviously, we can't really apply basic acoustic equations here and it would be impossible for this to occur, but lets just say that it can.
If 600 dB of sound pressure, we would get an RMS pressure level of 2e25 Pa and a peak pressure of 2.8e25 Pa or 28000000000 petapascals (assuming a perfect sinusoid). For reference, the pressure at the center of the sun is 26.5 petapascals. So the oscillation in ambient pressure would be orders of magnitude above the static pressure in the center of the sun.
If we assume this wave behaves linearly and adiabatically (haha), then the temperature change caused by the compression and rarefaction of the air would be 2.3e22 degrees Celcius. Estimates say that the temperature of the universe during the Big Bang are in the order of only 1e9.
If we assume 600 dB of sound power, then this blast would produce 1e48 W of power when the horn is used. Black holes do not have this amount of power. One source I found estimates that the big bang is estimated to have released about 1e50 J of energy. So if we played the horn for 1 second, we would only be 100x off from the power equivalent of the Big Bang.
So if a 600 dB whatever horn existed, it would likely destroy the universe when used.
Edit: Looks like the description does actually claim 600 dB! Probably it’s false advertising or the company doesn’t know how to calculate sound pressure. But they technically didn’t claim a reference value, so 600 dB could technically be correct if they claim a small enough reference value :) Sound pressure level = 20*log10(p_rms/p_ref), but this reference pressure is 20uPa in air
[–] 904 points905 points (58 children)
Found Randal munroe's alt account.
[–] 330 points331 points (33 children)
So obviously 600DB is the model number
That's what I thought too, but in the description it says "with a sound level of 600 dB"
[–]3✓ 250 points251 points (27 children)
Oh man I didn’t even see that. Must be a powerful horn then if they’re claiming universe ending potential haha
[–] 87 points88 points (10 children)
3 star rating, must have average universe destruction power.
[–] 20 points21 points (3 children)
perfectly balanced
[–] 6 points7 points (2 children)
Spiffing Brit?
[–][🍰] 0 points1 point (0 children)
Biffing Sprit?
[–] 4 points5 points (1 child)
We should all go leave a review disappointed that it didn't create a black hole or destroy the universe.
[–] 1 point2 points (0 children)
This would be the best brigade ever.
[–] 2 points3 points (0 children)
Universal fit
[–] 2 points3 points (0 children)
Only tore part of the fabric of reality asunder. Seems electromagnetism still works. Did great at neutralizing the concept of gravity though. 3 stars
[–] 1 point2 points (0 children)
3 stars is how many it destroyed.
[–] 57 points58 points (10 children)
Thanos’ Horn
[–] 52 points53 points (9 children)
Fine, I’ll blow it myself
[–] 26 points27 points (4 children)
Fine, I’ll blow it myself
[–] 16 points17 points (1 child)
I heard Thanos had his floating ribs removed
[–] 2 points3 points (0 children)
We’ll there’s a gem I haven’t heard since around the 90’s
[–] 4 points5 points (1 child)
Man, I've heard about tooting your own horn, but this is ridiculous
[–] 2 points3 points (0 children)
Doom toots as he pleases!
[–] 3 points4 points (0 children)
You can do that? Impressive!
[–] 2 points3 points (0 children)
Fine, I'll blow myself out.
[–] 1 point2 points (0 children)
Me, to my ex-girlfriend.
[–] 1 point2 points (0 children)
Unfortunately you need one of them dark matter car batteries to make it go. Otherwise it’s like hooking the entire US power grid to a AA battery.
[–] 9 points10 points (3 children)
So buy it and then get a refund for false advertisement
[–] 28 points29 points (2 children)
They can write a bad review explaining all this math then be like
"These scammers!! False advertising! I blew this horn and everyone is still here. WTF. Returning. Don't waste your money. Will not end the universe as claimed. Defective junk. 0 out of 5 stars"
[–] 7 points8 points (1 child)
I'm hoping the 1000 petalumen headlamp I ordered performs as described.
[–] 3 points4 points (0 children)
Wishing you luck. Make sure to review 😂
[–] 30 points31 points (9 children)
First thing I thought of. What If is an incredible book. So is Thing Explainer.
[–] 9 points10 points (2 children)
And How To
[–] 4 points5 points (1 child)
Ooh haven’t read that one, I’ll wishlist it!
[–] 10 points11 points (0 children)
Don't forget that What If 2 is out and it starts with a similar seemingly innocent question that would result in the destruction of the solar system and most of the milky way galaxy.
[–] 7 points8 points (5 children)
There’s a sequel! What if 2 is coming out!!
[–] 2 points3 points (3 children)
This is the best news I’ve had all week!!
[–] 5 points6 points (2 children)
Better news for you, you can already buy it.
[–] 4 points5 points (1 child)
Stop, stop, I can only get so erect
[–] 1 point2 points (0 children)
Best use of that phrase I’ve seen in years. 10/10.
[–] 10 points11 points (2 children)
10/10 explanation, would honk
[–] 4 points5 points (1 child)
This one goes up to 11
[–] 10 points11 points (1 child)
Found Randal munroe's alt account.
Ha... I got to meet him at an event a few years ago, and he autographed my copy of his "What If" book, one of the precious few books that I actually own in physical form.
https://what-if.xkcd.com/
[–] 3 points4 points (0 children)
I found a signed copy of How To at a used bookstore. I wonder who made the blunder of giving that book away.
[–] 8 points9 points (3 children)
To provide some more context to the insanity of 600dB. The Krakatoa volcanic eruption which is widely considered the loudest sound ever produced on Earth was estimated to be 310dB. The sound was so devastating that all life on an island 10 miles from the eruption (three thousand humans and all animal life) was killed by the shockwave created by sound alone.
The loudest sound ever created by humans was estimated to be around 250dB (Atomic bomb detonations)
[–] 3 points4 points (2 children)
And every 10 dB is a logarithmic doubling of power. So 600 dB is 229 -- over 536 million -- times more powerful than Krakatoa's eruption, a bang that circled the entire Earth multiple times.
Edit: Oh, I severely underestimated it due to a misconception of how dB work. See the reply below me for how much more it really is.
[–] 1 point2 points (1 child)
Technically +10 dB is doubling the "loudness" or perceived volume
+3 dB doubles the "power" (assuming by power you mean energy)
2^96 more energy in a 600 dB vs. Krakatoa
[–] 2 points3 points (0 children)
What if was a dope book, and his new one
[–] 230 points231 points (23 children)
Assuming this horn played for one second, that would output 1e48joules.
The swarzchild radius for an object whose rest mass is 1e48 joules is 8.263km, presumably this air horn would become a black hole shortly after being used.
[–] 150 points151 points (10 children)
This made me laugh far more than most videos on Reddit haha. The imagery of a guy honking his horn and collapsing the planet into a black hole is hilarious.
[–] 54 points55 points (3 children)
Ahahah that he bought on Amazon
[–] 21 points22 points (0 children)
Bezos, you sick twisted bastard
[–] 6 points7 points (1 child)
Everyone knows you have to go to Wish if you are looking for something this destructive
[–] 7 points8 points (0 children)
Everyone knows that the black holes you buy on Wish are just blackened Hot Pockets.
[–] 14 points15 points (0 children)
Just the thing to put in the hands of a road rager.
"Cut ME off will you? Well I'll show you!!!"
<HONK-ABOOM>
[–] 10 points11 points (0 children)
0/5 stars, air horn only destroyed a small moon but came nowhere close to the planetary destruction promised in the description
[–] 3 points4 points (2 children)
I could see this happening in a Rick & Morty episode.
[–] 2 points3 points (0 children)
That would be amazing lol
[–] 20 points21 points (2 children)
Even before: to release that much acoustic energy, it has to be storing more than that much as chemical energy (or antimatter, who knows). The mass of that energy would collapse into a black hole before anyone managed to turn it on.
[–] 14 points15 points (1 child)
Yeah man, that's why it's so expensive, you can't expect that kind of extradimensional storage from a dollar store universe destroying honker
[–] 7 points8 points (2 children)
Now we know what type of speakers the band Disaster Area uses.
https://youtu.be/PQjgMF_20dE?t=128
[–] 2 points3 points (1 child)
Climax by crashing a starship into a sun.
[–] 1 point2 points (0 children)
It's the only way I can get off. So frustrated.
[–] 4 points5 points (0 children)
Prop milliseconds afterwards
[–] 1 point2 points (0 children)
FIRE EVERYTHING!!!!
[–] 0 points1 point (0 children)
Before being used, since it somehow has to have that energy to produce the ... bang.
[–] 88 points89 points (4 children)
So you’re saying $32.99 is a fair price? [–] 9 points10 points (1 child) I would say so, I don’t have to pay it back once the universe is destroyed [–] 2 points3 points (0 children) Shhhhh. 🤫 NK might be able to pick up two of these and finally destroy the ocean. We must not let them know of this technology. [–] 1 point2 points (0 children) Thanos is kicking himself at what his cost for universe destroying power was in comparison. [–] 77 points78 points (6 children) This is the way the world ends This is the way the world ends Not with a bang But with a honk [–] 13 points14 points (2 children) 🦆 has entered the chat [–] 6 points7 points (0 children) Hjönk [–] 2 points3 points (0 children) 🦢😡 [–] 2 points3 points (0 children) "Big Bang Theory" is obsolete, new model is "Big Honk Theory" [–] 33 points34 points (12 children) You really don’t see “petapascals” often. [–] 12 points13 points (8 children) A lot of SI prefixes are underused, unfortunately. I am especially fond of using megameters for long trips. [–] 3 points4 points (2 children) We use megameters in Elite Dangerous, along with light-seconds [–] 1 point2 points (1 child) Nice, I use Mm all the time in Star Citizen, much to the annoyance of a friend (which makes it even better!) [–] 4 points5 points (2 children) I recently found out that Excel has a convert function. I also found out that it has some ridiculous units, like cubic light-years and cubic angstroms. I thought that was probably the biggest number I could generate - the number of cubic angstroms in a cubic light-year. Then I found out that it supports SI prefixes and it's case sensitive, too - it knows a lowercase m is for "milli-" and an uppercase M is for "mega-". So I immediately asked it how many cubic yoctometres are in a cubic yottametre. Those are the prefixes at the small and large ends of the SI spectrum, and they're so extreme they're rarely ever used. 1 ym, a yoctometre, is a stupid short distance. So short that there isn't really an example of something that small - it's basically shorter than any object but still far, far longer than the Planck length. It's overall kinda useless. Anyway, it's roughly 1/142nd of the radius of a 1 MeV/c2 neutrino, and 1/7000th of the radius of a high-energy neutrino. If you're not into subatomic particles, a helium atom has a diameter of 56 picometres, and a picometre is equal to one quadrillion yoctometres. Meanwhile, 1 Ym, a yottametre, is stupidly long. Long distances in astronomy are typically measured in light-years or parsecs (which are 3.26 light-years each, so one wonders why you'd bother with parsecs when light-years are basically the same magnitude and easierto explain), but despite the fact that a light-year is about 9.5 petametres, both are effectively zero compared to a yottametre. Let's just round the light-year to 10 petametres, which is 10 quadrillion metres or 10 trillion kilometres. 100 light-years is thus roughly an exameter. There are 1000 exametres in a zettametre, and 1000 zettametres in a yottametre. 2 yottametres is roughly the diameter of the Local Supercluster. 10 yottametres is roughly the diameter of the supercluster complex in which we live. So anyway, there's 10144 cubic yoctometres in a cubic yottametre. But even the yottametre isn't an all-powerful superunit of length. It takes 870 of them to make up the diameter of the observable universe. If you want cursed units, you can just apply SI prefixes to non-SI units. You can measure distances in megafeet or nanomiles. Or you can make like astronomers and use megaparsecs and gigaparsecs, of which you only need 28 of the latter to make up the diameter of the observable universe. [–] 1 point2 points (0 children) Megafeet is probably the dumbest/funniest unit I've ever heard, thank you for sharing this insight [–] 25 points26 points (1 child) Ok, so how many megaphones do I need to stack a la Bart Simpson to nuke the planet? [–] 17 points18 points (2 children) As a chemical engineer I now know how carelessly I use the word “obviously” [–] 7 points8 points (1 child) You're growing as a person and we see you. [–] 15 points16 points (0 children) This is the reason why I sub here. You made me laugh so much. Thank you for putting the work in 😂 [–] 10 points11 points (3 children) I mean, with no advanced knowledge in this field I figured it would kill people seeing as a jet engine is around 140 dB and the increase isn't linear. But damn, both hilarious and terrifying. [–] 4 points5 points (1 child) You're not wrong, it would indeed kill people Maybe you should enter the field [–] 7 points8 points (0 children) Just to piggyback on this comment: The reason you get such absurdly large numbers is because dB is a logarithmic (10log) scale. If you have 1 air horn that is 150 dB and add another, they would be have an SPL of ~153 dB together. This comes from the fact that the 10log(2)=3.0103 So for every doubling, you add an extra 3 dB. Do how many 150 dB air horns do you need to get an SPL of 600 dB? Well, seeing as every 3 dB over 150 means we doubled the amount, means we doubled the amount of air horns 150 times. So if you want to produce 600 dB, you would need 2150≈1.43×1045 Seeing as Earth is around 1050 atoms, I don't think we can make these air horns locally. [–] 6 points7 points (4 children) 1e50 Joules would wipe a solar system. Not a universe. Supernova range from 1e44 to 1 e47 [–] 4 points5 points (2 children) We finally know what quasars really are! [–] 5 points6 points (0 children) Thats what I was gonna say but you beat me to it. You did a pretty good job though. [–] 4 points5 points (1 child) this blast would produce 1e48 W of power From a 12V source no less. Guys, I think we have solved the energy crisis! [–] 1 point2 points (0 children) Maybe it’s capable of 600db, but we just don’t have good enough car batteries. [–] 4 points5 points (0 children) It’s funny cause if you zoom in on the description it says “with a sound level of 600db” 😂 [–] 4 points5 points (0 children) Hate to be an alarmist but we have to call the whitehouse or something. If someone buys this horn and uses it we will all die! [–] 4 points5 points (0 children) I assume this doomsday device will be installed on a 10-year old Civic with an oversized spoiler. [–] 3 points4 points (0 children) If we assume this wave behaves linearly and adiabatically (haha), This joke kills at Sound Engineer parties. [–] 4 points5 points (2 children) Krakatoa's final explosion was right around 300db. It sent a shockwave around the planet that circled it 3 and a half times. It ruptured the ear drums of sailors some odd 40 miles away. For more facts, just ask. It's one of my favorite events in history. [–] 7 points8 points (1 child) Obligatory so this is what'll happen if I toot 6 horns together that are 100db loud each \s [–]3✓ 4 points5 points (0 children) You just add them up obviously that’s 600 dB /s [–] 2 points3 points (0 children) This explanation made my day. Thank you. [–] 2 points3 points (0 children) All that and it only takes 12 volts. [–] 2 points3 points (0 children) This guy dB [–] 2 points3 points (0 children) Edit: Looks like the description does actually claim 600 dB! Probably it’s false advertising "Can you state your reason for return?" "Yeah, I tried destroying the universe and all I got was the sound of a horny goose." [–] 3 points4 points (8 children) So what would Thanos need to wipe out only half of the universe? [–] 7 points8 points (7 children) If my tired brain correctly recalls how logarithms work, it would be a 597dB horn. That would half the output. [–] 2 points3 points (6 children) I think it's 590, IIRC it's every 10 decibels doubles the sound level [–] 6 points7 points (4 children) That's the confusing part... Depends on what scale you use. I recall in radio communications ±3dB would be half and double. 30dB + 30dB = 33dB. But I'm not sure what it was for audio. Math scares me anyway. [–] 3 points4 points (3 children) It seems like you are correct, but the Decibel scale is too confusing to say for sure [–] 2 points3 points (2 children) If I remember properly from filters in circuits, 3 dB (if measuring power) will be half power but not necessarily wave amplitude( assuming it's a sinusoid it is 2-1/2 times the original). So loudness will actually double/halve every 6 dB. Also glancing at the Wikipedia page real quick, 10 decibels is a 10-fold increase in power(which makes sense, due to it being based on a base 10 logarithm). [–] 0 points1 point (0 children) it's almost 00:30 and i am too tired to make any sense out of this shit [–] 0 points1 point (0 children) At music school we learned double the apparent volume every 6db [–][deleted] 1 point2 points (0 children) Weapon of mass destruction [–] 1 point2 points (1 child) I'd like to add that, while not nearly as interesting, the actual output of the horn would be limited by atmospheric pressure, such that the "dips" in the soundwaves are a complete vacuum. So if this device existed in reality and could theoretically output 600 dB without self-destructing, every time you squeeze the bulb it would generate a deafening explosion at around 200 dB. At close range it would probably outright kill people. Interesting article on the subject here. [–] 1 point2 points (0 children) This is an awesome write-up [–] 1 point2 points (1 child) Wow. All that from a 12V battery. [–] 1 point2 points (0 children) I know it’s a log scale but I never really got log function: are you saying that I have to wear earplugs at 100 dB but 600 dB is as powerful as 100 black holes? [–] 3 points4 points (0 children) But... could it break a Nokia 3310? [–] 2 points3 points (0 children) Becoming an acoustics engineer is something I never knew I wanted. You just made it sound like so much fun. [–] 1 point2 points (0 children) You’re my hero. kneels in nerd [–] 0 points1 point (2 children) Anyone has an ELI5 on what "reference value" means in this context? Are Decibels not an absolute measure? [–]3✓ 3 points4 points (0 children) A common misconception is that decibels are units. Decibels express a ratio of 2 power quantities. The general formula for a decibel is: 10*log10(P/Pref) where P is the power quantity you want to put on a decibel scale and Pref is your reference power. If you see the formula without Pref, this usually implies that the reference value is 1, although it should always be stated. The Pref is chosen to match with 0 dB on your scale. For sound pressure, it is generally agreed upon that the lowest sound pressure that humans can hear is 20 uPa, so this is our Pref in air (this changes depending on the medium of propagation). So when you convert 20 uPa into a decibel scale, the value inside the log becomes 1, so you get 0 dB. This means that when you have negative dB, it’s not “negative sound”, it just means that you have a pressure less than whatever the reference value. Note that I said decibels are used to express a ratio of power quantities. Pressure is not proportional to power, but pressure squared is. So our formula for sound pressure level actually becomes: 10log10(p2 /20uPa2 ). Or 20log10(p/20uPa). So if this company did not use the accepted 20 uPa as their reference value, they could technically get 600 dB by picking some arbitrary number. Hope this helped [–] 0 points1 point (0 children) If you look at the “about this item” it actually does claim to be 600dB [–] 0 points1 point (0 children) So Gabriel's horn... [–] 0 points1 point (0 children) So he would need only one, then? [–] 0 points1 point (0 children) This is what the horns of Jericho looked like. [–] 0 points1 point (0 children) And that's how you make a big bang. [–] 0 points1 point (0 children) I saw a meme one time that said if you can make a sound 1100 dB it would create a black hole. Are you telling me it only takes 600 dB to destroy the universe. [–] 0 points1 point (1 child) | adiabatically I learned a new word today, thanks! :) [–] 1 point2 points (0 children) adiabeetus [–] 0 points1 point (1 child) Ok, that’s cool and all but would it get the attention of the elderly lady doing 40 in the fast lane with her left turn signal on? [–] 0 points1 point (0 children) Nope, but it would vaporize her so problem solved! [–] 0 points1 point (0 children) I feel like people who just throw out DB numbers dont understand how logarithmic values work. [–] 897 points898 points (21 children) Torque test channel on youtube recently did a video on this horn. He compared it to Krakatoa which was 310db. That means that 600 db horn is 316.2 trillion times louder than krakatoa. 500 db would be enough to cause a full global extinction [–] 433 points434 points (10 children) So basically only tap the horn? [–] 148 points149 points (1 child) Mic check then [–][deleted] (5 children) [deleted] [–] 48 points49 points (4 children) Just cause an extinction level event bc some knob won’t move his car [–] 13 points14 points (1 child) Roadrage in America wants to know your location. [–] 5 points6 points (0 children) Florida! Hahaha [–] 1 point2 points (0 children) Knob. Lol [–] 3 points4 points (0 children) For when you want to split the earth and neighboring planets. [–] 1 point2 points (0 children) For when you absolutely must get the cars in front of you out of the way [–] 35 points36 points (0 children) Megaphone the 600DB just in case you don't hear it [–] 24 points25 points (0 children) [–] 7 points8 points (0 children) I love that channel. The gas powered impact drill was something I didn’t know Wanted [–] 7 points8 points (1 child) Global extinction from a 12v device. Something is fishy here. [–] 4 points5 points (0 children) quite a few amps going through to make up for it [–] 2 points3 points (0 children) Wouldn't 600 dB be 29 powers of ten louder than 310 dB? That would be 100 octillion times louder. [–] 6 points7 points (0 children) RIP headphone users [–] 5 points6 points (0 children) But maybe they used a different reference unit for 600 dB, and arent lying🧐 [–] 0 points1 point (0 children) Can you cite the video in reference? [–] 73 points74 points (0 children) I love that someone asked "Is this really 600 db? Is there a way to reduce that? I would like to use the horn to signal my displeasure without causing a mass extinction event." Just the energy needed to output 600 dB will basically fuck the known universe. [–] 325 points326 points (14 children) For "reasonable" volumes Decibels can be converted into pressure levels. Extrapolating the formula, I get 600dB -> 2e+20 bar. For comparison, estimated pressure in the center of our sun is 2.5e+11 bar, about a billion times less. If this was a German product, I would urge you not to try it, as it could destroy our galaxy. But chinesium is probably safe. [–] 94 points95 points (2 children) Danke für die Anerkennung deutscher Ingenieurskunst! [–] 27 points28 points (1 child) German science is the world's finest! [–] 9 points10 points (0 children) BRRRRRRRRRRRRRAKA MONOGA! DOITSU NO KAGAKU WA SEKAI ICHIIIIIIII! also r/UnexpectedJoJo [–]1✓ 10 points11 points (0 children) So enough pressure to ignite fusion of oxygen in the atmosphere, but still short of crushing us into a neutron soup (neutron star pressure is higher still.) At least the galaxy will be safe. [–] 11 points12 points (9 children) A pressure wave can’t travel through the vacuum of space. Would this horn kill every living creature on the planet? Almost certainly. Totally destroy planet earth? Meh, maybe. I think more likely it would leave a hell of a crater, cause global tsunamis, and spontaneously ignite most of the atmosphere. Maybe some of the debris from that crater would impact and damage the moon. But the destructive pressure wave isn’t leaving earth. Edit: after reading some other posts… maybe this horn could completely destroy the earth. But still, once the hot glowing plasma cloud that was once the earth begins spreading into the vacuum of space, the destructive potential of the horn is gone. In a vacuum it will do nothing. I’m kinda curious what it would do to a star though. [–] 15 points16 points (7 children) According to one person here this horn has the potential energy equal to 1% of the big bang so I'd imagine the entire Virgo Supercluster is probably toast [–] 4 points5 points (6 children) Right but what medium is going to transmit that energy with enough retention to cause damage at range? If it’s truly that much energy it will basically atomize earth. Probably even break down a lot of it into subatomic particles. Maybe it will even take the moon with it. But then you just have these particles and maybe some unfocused gamma and x rays spreading out. As they get farther and farther from the point earth was the energy will get less and less focused. I can’t imagine it causing any sort of destruction outside of our solar system. And doubt it would even effect our sun in any meaningful way [–] 13 points14 points (3 children) Once you reach relativistic energies, medium stops being a barrier. Energy travels through vacuum well. Let's estimate. A one-watt speaker is good for maybe 40db in close proximity. 600 db is 1056 watt (motorcycle battery, huh) An impolite 1-second honk will be 1e+56 J of energy, which is 100 billion times greater than a hypernova explosion (1e+45J). Milky way will never be the same. [–] 5 points6 points (2 children) So the energy the horn puts out needs to come from somewhere, right? Because from what I've read so far, to really be able to test their claims, first we need to power it properly. Even tho the wires to the pump look like the first failure point, we can't just go about assuming stuff. We need experiments. So where could we get that sort of power from and how do we get it to the horn? [–] 5 points6 points (1 child) Good question. Using energy-mass equivalence principle, 1056/c2 is about 1039kg, or 500 000 000 solar masses. This is way bigger than the largest known star, much bigger than Sagittarius A*, but smaller than entire milky way. One-second toot of the 600db hord requires energy from annihilation of a dwarf galaxy. The wires should be ok, the picture does not show scale. [–] 2 points3 points (0 children) I just watched a YouTube video linked somewhere further up, in which this exact horn(among others) was tested. There were hands fondling that exact horn for scale. But his power source was definetly the first bottleneck, for it was roughly one standard car battery. No microverse technology involved. But seriously, how would one go about turning a dwarf galaxy into energy and how would one then actually be able to give it to those thirsty little wires? I think on demand galaxy conversion might be the better way here, lest you got good ideas for storage as well. I just think the galaxy itself might be the most practical storage solution. Or do we just keep it as a black holes? [–] 4 points5 points (0 children) Probably the light and heat created by the complete assfucking the planet would receive if someone did set off that horn [–] 5 points6 points (0 children) At levels of energy this high, physics itself starts to break down. That much energy would have to manifest itself somehow in an instant and it's bound to be destructive. Forget the sun, the black hole at the center of the galaxy is a speck compared to this much energy. Hell, all normal matter makes up maybe 5% of the universe. This is the mass-energy of 1/5th of the observable universe exploding. [–] 33 points34 points (4 children) [–] 5 points6 points (3 children) It would take billions of years to implode, correct? [–] 2 points3 points (2 children) at the speed of light, yes... if it creates a false vacuum, possibly immediate everywhere all at once [–][deleted] (1 child) [removed] [–] 2 points3 points (0 children) Fungi. Srsly [–] 3 points4 points (1 child) The year is 1883 August 27th. Early in the darkness of the morning the volcano Krakatoa decided it was done existing and erupted in a series of huge explosions. The largest explosion in that series caused a 100-foot high (30 meters) tsunami. The eruption displaced 5 cubic miles (20.8 Cubic kilometers) of volcanic rock. Ash was propelled nearly 50 miles (80 Km) into the sky cooling the planet and causing a year without summer and 5 years of lowered global temperatures. This explosion was so loud that its pressure wave circled the globe, over 3 times. Every barograph across the globe measured it. Sailors 40 miles (64 kilometers) away had their eardrums ruptured and caused other injuries. The energy released is estimated to have been 200 megatons, roughly 1,000 of the fat boy bombs dropped on Nagasaki. 36,000 people died as a direct result of this eruption (though this estimate is thought to be VERY low) with countless millions dying of famine due to the lowering of global temperatures from the ash. Tsunamis hit as far away as South Africa. The sound was heard up to 3,000 miles (4,828 Km) away. So was this loud? Yes, yes it was. It is in fact the loudest sound in recorded history, clocking in at 310Db. So God help us all if somebody lets that$33 12 volt airhorn off the chain. Could probably split the earth in half seeing as the Db scale is logarithmic. So that horn has 316,227,766,016,558 the magnitude of the Krakatoa eruption.
Edit: thank you Torque Test Channel for that intro.
[–][deleted] 18 points19 points (4 children)
Wow!! This must be the "trumpets" 🎺 they talk about in the Bible that made the Walls & City of JERICHO CRUMBLE! OH Lord they had 7 at that!
P.S. Even with a price of basically \$33 a "horn", this here proves how powerful this product is! It is of "Biblical Proportions"! 🙏
[–] 2 points3 points (0 children)
You blow this horn, Jericho puts you on his list.
[–] 1 point2 points (0 children)
The walls, Jericho, and all the people vaporized.
[–] 2 points3 points (0 children)
Torque Test Channel on YouTube just did a video on this. This horn would release more energy than the Krakatoa volcanic eruption. This eruption was estimated to be ~350dB. 600dB would be a planet killer amount of energy. The Log scale is a powerful thing.
[–] 2 points3 points (0 children)
For reference, any noise created above 200 dB in air is no longer a noise, but rather a blast wave because the pressure wave creates a vacuum behind it and a sustained sound can't be maintained.
[–] 1 point2 points (2 children)
There was some guy on youtube that tested a bunch of horns and this was one of them, i think this one was around 115db or something, 600 would probably be a continental crisis
[–] 1 point2 points (1 child)
600 would probably be a continental universal crisis
[–] 1 point2 points (0 children)
“The horn of Helm Hammerhand shall sound in the deep one last time! Let this be the hour when we draw swords together!"
About that loud, I think. But I'd listen to the acoustic engineer.
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# ENGR62/MS&E111 Introduction to Optimization Dr. K.
Ng
## AN EXAMPLE USING THE BIG-M METHOD
The original problem maximize x1 + 2x2 subject to x1 + x2 2 x1 + x2 1 x2 3 x1 0, x2 0 The problem initialized maximize x1 + 2x2 subject to x1 + x2 x4 x1 + x2 x5 x2 + x3 xj 0, j = 1, . . . , 5 The problem with modied objective maximize x1 + 2x2 M x6 M x7 subject to x1 + x2 x4 + x6 x1 + x2 x5 + x7 x2 + x3 xj 0, j = 1, . . . , 7 = = = = z 2 1 3 = = = = z 2 1 3
ENGR62/MS&E 111
Handout 5
-z z x6 x7 x3 1 0 0 0
x1 -1 1 -1 0
x2 2 1 1 1
x3 0 0 0 1
x4 0 -1 0 0
x5 0 0 -1 0
x6 -M 1 0 0
x7 -M 0 1 0
RHS 0 2 1 3
The current basic variables of this tableau are x3, x6, and x7. The system needs to be put into "canonical form" with respect to this basis: -z z x6 x7 x3 1 0 0 0 x1 -1 1 -1 0 x2 2+2M 1 1 1 x3 0 0 0 1 x4 -M -1 0 0 x5 -M 0 -1 0 x6 0 1 0 0 x7 0 0 1 0 RHS 3M 2 1 3 2 1 3 Ratio
As the coefficient of x2 is positive, we can decrease the value of z. Using the minimum ratio test, x2 will replace x7 as a basic variable. After pivoting we obtain...
ENGR62/MS&E 111
Handout 5
-z z x6 x2 x3 1 0 0 0
x1 1+2M 2 -1 1
x2 0 0 1 0
x3 0 0 0 1
x4 -M -1 0 0
x5 2+M 1 -1 1
x6 0 1 0 0
x7 -2-2M -1 1 -1
RHS -2+M 1 1 2
Ratio
1/2 * 2
Here we observe that z has decreased from 3M to M-2. Further decrease can be achieved by increasing either x1 or x5. We increase x1 as its coefficent is bigger than that of x5. Using the minimum ratio test, x1 will replace x6 as a basic variable. Pivoting leads to -z z x1 x2 x3 1 0 0 0 x1 0 1 0 0 x2 0 0 1 0 x3 0 0 0 1 x4 1/2 -1/2 -1/2 1/2 x5 3/2 1/2 -1/2 1/2 x6 -1/2-M 1/2 1/2 -1/2 x7 -3/2-M -1/2 1/2 -1/2 RHS -5/2 1/2 3/2 3/2 3 1 Ratio
The artificial variables have both been made nonbasic, and z is independent of M. Further decrease in the objective can be achieved by increasing either x4 or x5. We pick x5 as its coefficient is bigger than that of x4. Using the minimum ratio test, x5 replaces x1 as a basic variable. Pivoting gives ...
ENGR62/MS&E 111
Handout 5
-z z x5 x2 x3 1 0 0 0
x1 -3 2 1 -1
x2 0 0 1 0
x3 0 0 0 1
x4 2 -1 -1 1
x5 0 1 0 0
x6 -2-M 1 1 -1
x7 -M -1 0 0
RHS -4 1 2 1
Ratio
* * 1
The tableau reveals that the value of z can be increased further by increasing the value of x4. There is only one pivot choice since there is only one positive coefficient of x4 among the last 3 rows. We see that x4 will replace x3 as a basic variable. The corresponding pivot step yields -z z x5 x2 x4 1 0 0 0 x1 -1 1 0 -1 x2 0 0 1 0 x3 -2 1 1 1 x4 0 0 0 1 x5 0 1 0 0 x6 -M 0 0 -1 x7 -M -1 0 0 RHS -6 2 3 1
This final tableau indicates that an optimal basic feasible solution has been found. The optimal solution found is given by (x1,x2,x3,x4,x5) = (0,3,0,1,2), and the maximum value of the objective function is z = 6.
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# Tableau Table Functions
Tableau provides various table functions to perform calculations on all visual data. They are FIRST, LAST, INDEX, RUNNING_SUM, RUNNING_AVG, WINDOW_SUM, WINDOW_VAR, etc. In this article, we will show you how to use Tableau Table Functions with an example of each.
To demonstrate these Tableau Functions, we use the table below. Remember, this is an Excel Worksheet. So, Please refer to the Connecting to the Excel Files in Tableau.
## Tableau Table Functions
The following examples will show you the list of Table Functions in Tableau.
### FIRST Function
The Tableau FIRST function returns values from 0 to -n in a table, and the syntax of this FIRST is:
`FIRST()`
To demonstrate these Tableau functions, we have to use the table calculation. To create a table calculation, please navigate to the Analysis Tab and select the Create Calculated Field… option. For more built-in functions >> Click Here!
Once you click on the Create Calculated Field… option, the following window will be opened. Here, we renamed the default calculation name as FIRST. Click OK to close this window
Or you can click the Default Table calculation hyperlink to alter the default settings. Clicking that link will open the following window.
Let me add this FIRST field to an existing table (by dragging the field to Measures Shelf). Please refer to the Create Table Report article to understand the process of creating a table
You can edit this table calculation by clicking the down arrow. Next, select the Edit Table Calculation option from the context menu.
It will open a new window called Table calculation.
Use this window to change the calculation process. We already explained these properties in our previous article. So, please refer to Table Calculations article to understand these functionalities.
### LAST Function
The LAST will return numbers from n to 0. I mean, the last record is 0, and the first record is n. The syntax of this Tableau LAST Function on a table is:
`LAST()`
Let me add this LAST function field to the Measures shelf.
### INDEX
The Tableau Table INDEX function will return numbers from i to n. I mean, the first record is 1, and the last record is n. The syntax of this INDEX is:
`INDEX()`
Let me add this INDEX field to the Measures shelf
### RUNNING_AVG
The Tableau RUNNING_AVG function will calculate the running average of the table, and the syntax of this RUNNING_AVG is:
`RUNNING_AVG(Expression)`
The below statement calculates the running average of the Sales column across the table.
`RUNNING_AVG(SUM([Sales]))`
Let me add this RUNNING_AVG field to the Measures shelf.
Third record: (25 + 25 + 60) /3 = 36.667 = 37
### RUNNING_COUNT
The Tableau RUNNING_COUNT function will return the running count of the table, and the syntax of this RUNNING_COUNT is:
`RUNNING_COUNT(Expression)`
The below statement returns the running count of the Sales column.
`RUNNING_COUNT(SUM([Sales]))`
Let me add this RUNNING_COUNT field to the Measures shelf.
### RUNNING_MAX
The Tableau RUNNING_MAX function returns the maximum running value in a table, and the syntax of this RUNNING_MAX is:
`RUNNING_MAX(Expression)`
Returns the maximum running value in a Sales column.
`RUNNING_MAX(SUM([Sales]))`
Let me add this RUNNING_MAX field to the Measures shelf.
### RUNNING_MIN
The Tableau RUNNING_MIN function will return the running minimum value, and the syntax of Table RUNNING_Min is:
`RUNNING_MIN(Expression)`
Returns the minimum running amount in a Sales column.
`RUNNING_MIN(SUM([Sales]))`
Let’s add the RUNNING_MIN field to the Measures shelf.
### RUNNING_SUM
The Tableau Table RUNNING_SUM function will calculate the running total, and the syntax of this RUNNING_SUM is:
`RUNNING_SUM(Expression)`
Returns the running total of a Sales column.
`RUNNING_SUM(SUM([Sales]))`
Let’s add the RUNNING_SUM field to the Measures shelf.
Third record: (50 + 60) = 110
### Tableau SIZE
The Tableau SIZE function returns total records in a table or pane, and the syntax of this SIZE is:
`SIZE()`
Let me add this SIZE field to the Measures shelf. As you can see, we have changed the default calculation to display the size based on the Occupation
### TOTAL
The Tableau TOTAL function will return the complete total in a window, and the syntax of this Total is:
`TOTAL(Expression)`
Returns the total sales amount in this table. i.e., 25 + 25 + 60 + 24 +….. + 2320
`TOTAL(SUM([Sales]))`
Let me add this TOTAL Function field to the Measures shelf.
This time we are computing using Occupation. It means the total function will sum based on occupation.
First set: 25 + 25 + 60 + 24 = 133
### WINDOW_AVG
The Tableau WINDOW_AVG function will calculate the average of the data in a table, and the syntax of this WINDOW_AVG is:
`WINDOW_AVG(Expression, start_point, end_point)`
The below WINDOW_AVG function calculates the average of Total Sales.
`WINDOW_AVG(SUM([Sales]), FIRST(), LAST())`
Let me add this WINDOW_AVG field to the Measures shelf.
Here, the Sum of 15 records is 28, 794.
Average = 28,794/15 = 1919.6 = 1920
This time, we are calculating the average from the first record to the last -12 (means 15 – 12). i.e., the first record to the third record
Window_Avg = 25 + 25 + 60 = 36.67 = 37
### WINDOW_MEDIAN
The Tableau WINDOW_MEDIAN function will calculate the median in a table from the start point to the end. The syntax of this WINDOW_MEDIAN is:
`WINDOW_MEDIAN(Expression, start_point, end_point)`
Calculate the median of Total Sales.
`WINDOW_MEDIAN(SUM([Sales]), FIRST(), LAST())`
Let me add this WINDOW_MEDIAN field to the Measures shelf.
### WINDOW_MAX
The Tableau WINDOW_MAX function returns the maximum value from the start point to the end in a table. The syntax of this WINDOW_MAX is:
`WINDOW_MAX(Expression, start_point, end_point)`
Finds the maximum value from the first record to the last – 8 (means, 15 – 8 = 7)
`WINDOW_MAX(SUM([Sales]), FIRST(), LAST() - 8)`
Let me add this WINDOW_MAX field to the Measures shelf.
### WINDOW_MIN
The Tableau WINDOW_MIN function returns the Minimum value from the start point to the end of a table. The syntax of the WINDOW_MIN is:
`WINDOW_MIN(Expression, start_point, end_point)`
Finds the minimum value from the first record + 4 to the last record
`WINDOW_MAX(SUM([Sales]), FIRST(), LAST() - 8)`
Let me add this WINDOW_MIN field to the Measures shelf.
### WINDOW_SUM
The Tableau WINDOW_SUM function calculates the sum from the start point to the endpoint in a table, and the syntax of this WINDOW_SUM is:
`WINDOW_SUM(Expression, start_point, end_point)`
Calculate the Total Sales from 11 to 15.
`WINDOW_SUM(SUM([Sales]), FIRST(), LAST())`
Let me add this WINDOW_SUM field to the Measures shelf.
Sum from 11 to 15 = 3078 + 540 + 4320 + 699 + 2320 = 10957.
### WINDOW_VAR
The Tableau WINDOW_VAR function returns the variance of the sample population from the start point to the endpoint in a table. The syntax of this WINDOW_VAR is:
`WINDOW_VAR(Expression, start_point, end_point)`
Calculate the variance from the current row to the previous row.
`WINDOW_VAR(SUM([Sales]), FIRST(), 0)`
Let me add this WINDOW_VAR field to the Measures shelf.
### WINDOW_VARP
The Tableau WINDOW_VARP function returns the variance of the complete population in a table from the start point to the end point. The syntax of this WINDOW_VARP is:
`WINDOW_VARP(Expression, start_point, end_point)`
Calculate the variance from the third row to the previous row.
`WINDOW_VARP(SUM([Sales]), FIRST()+ 2, 1)`
Let me add this WINDOW_VARP field to the Measures shelf.
### WINDOW_STDEV
The Tableau WINDOW_STDEV function returns the standard deviation of the sample population from the start point to the endpoint in a table. The syntax of this WINDOW_STDEV is:
`WINDOW_STDEV(Expression, start_point, end_point)`
Calculate the standard deviation from the second row to the current row.
`WINDOW_STDEV(SUM([Sales]), FIRST()+ 1, 1)`
Let me add this WINDOW_STDEV field to the Measures shelf.
### WINDOW_STDEVP
The WINDOW_STDEVP returns the standard deviation of the complete population from the start point to the endpoint in a table. The syntax of the Tableau WINDOW_STDEVP Function is:
`WINDOW_STDEVP(Expression, start_point, end_point)`
Calculates the standard deviation of the complete population.
`WINDOW_STDEVP(SUM([Sales]), FIRST()+ 4, 2)`
Let me add this WINDOW_STDEVP field to the Measures shelf.
### WINDOW_PERCENTILE
The Tableau WINDOW_PERCENTILE function returns the specified percentile from the start point to the endpoint in a table. The syntax of the WINDOW_PERCENTILE is:
`WINDOW_PERCENTILE(Expression, percentile, start_point, end_point)`
Calculate the 25% percentile from the previous two rows to the current row.
`WINDOW_STDEVP(SUM([Sales]), FIRST()+ 4, 2)`
Let me add this WINDOW_PERCENTILE field to the Measures shelf.
### WINDOW_CORR
The Tableau WINDOW_CORR function returns the correlation between two measures from the start point to the endpoint in a table. The syntax of the WINDOW_CORR is:
`WINDOW_CORR(Expression1, Expression2, start_point, end_point)`
Finds the correlation between total cost and sales amount from the previous three rows to the current row.
`WINDOW_CORR(SUM([Total Cost]), SUM([SalesAmount]), -3, 2)`
Let me add this WINDOW_CORR field to the Measures shelf.
### WINDOW_COVAR
The Tableau WINDOW_COVAR function returns the covariance of the sample population from the start point to the endpoint in a table. The syntax of the WINDOW_COVAR is:
`WINDOW_COVAR(Expression1, Expression2, start_point, end_point)`
It finds the sample covariance between total cost and sales amount from the previous three rows to the current row.
`WINDOW_COVAR(SUM([Total Cost]), SUM([SalesAmount]), -3, 0)`
Let me add this WINDOW_COVARP field to the Measures shelf.
### WINDOW_COVARP
The WINDOW_COVARP function returns the total population covariance from the start point to the endpoint in a table. The syntax of the Tableau WINDOW_COVARP is:
`WINDOW_COVARP(Expression1, Expression2, start_point, end_point)`
It finds the population covariance between total cost and sales amount from the previous two rows to the current row.
`WINDOW_COVARP(SUM([Total Cost]), SUM([SalesAmount]), -2, 0)`
Let me add this WINDOW_COVARP field to the Measures shelf.
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# Net Promoter Score Question Template
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### Net Promoter® Score (NPS) survey template
(7 days ago) Our Net Promoter Score sample survey template is designed to make it easy to understand how satisfied your customers are. Net promoter scores are produced by asking respondents to rate the likelihood they would refer your business to someone else, using a scale from 0 to 10. Using your customers’ answers, you can identify the percentage of
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### 10 Best Net Promoter Score(NPS) question examples
(4 days ago) The Net Promoter Score is calculated by subtracting the percentage of Detractors from the percentage of Promoters. So, if 50% of respondents were Promoters and 10% were Detractors, your Net Promoter is a score of 40. It is as simple as that. The 2nd part of …
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### Net Promoter Score Template and Survey Question Ideas
(7 days ago) In this blog we’ll go through some different survey questions and net promoter score template ideas that you can use to improve your NPS results. You’ll also learn how to use AI tools to better analyze the information …
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### All about Net Promoter Score (NPS): Calculation, …
(7 days ago) Try Free Net Promoter Score Survey Template Net Promoter Score Scale. The range of the answer options used in the NPS survey question is the Net Promoter Score scale. NPS system is similar to the multiple-choice questions; however, the options are limited to …
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### Net Promoter Score Template – Qualaroo Help & Support Center
(5 days ago) What Your Score Means. Net Promoter Scores range from -100 to 100 since you could potentially have detractors only (-100) or promoters only (100). For that reason, your most basic goal with NPS should at the very least be to have a positive score. A negative score would indicate that more of your customers are actively trying to dissuade people
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### Net Promoter Score (NPS): Formula and Calculator [Excel Template]
(4 days ago) Net Promoter Score (NPS) Formula. Calculating the net promoter score is a three-step process: Step 1: Count the responses from the surveys and add the number of responses in each score range.; Step 2: Segment all of the collected responses into the three groups.; Step 3: Calculate the NPS by subtracting the percentage of detractors from the percentage of promoters.
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### 5 Great NPS Survey Question Examples + Templates
(1 days ago) With all WPForms templates, you never have to start from scratch. This form not only asks for the NPS score, but it will open another field below the rating field if the answer is less than 6. You can use the NPS survey …
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### Net Promoter Score PowerPoint Template - SlideModel
(2 days ago) The Net Promoter Score PowerPoint Template is a slide deck of 16 amazing layouts and visually interactive designs. The NPS is basically a single question survey method to assess the likability of company, product or services etc. For …
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### Net Promoter Score Template – Opinion Stage
(8 days ago) The Net Promoter Scale. Your Net Promoter Score is calculated using the NPS scale: 0 (unlikely to recommend) to 10 (extremely likely to recommend). Based on their response, you can place them into one of three categories below to …
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### Free Net Promoter Score (NPS) Survey Template // Qualtrics
(4 days ago) Net Promoter Score (NPS®) is often held up as the gold standard customer experience metric. It measures customer loyalty for everything from individual products, stores, web pages, or even staff members. An expert-built survey template preloaded with the right questions to help you save time and get results faster. Ph.D. designed
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### Net Promoter Score Question Definition + Examples - SurveyKing
(9 days ago) Below is an example of a Net Promoter Score survey that is segmented by gender. You can see that males tend to have a higher perception of the company than females do. Percent of Total Responses - Total Responses Male Female Detractors Passives Promoters 0% 25% 50% 75% 100%. Count Percent.
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### Net Promoter® Score (NPS) SurveyMonkey
(9 days ago) Net Promoter Score Template The Net Promoter Score Template begins with the NPS question and features built-in logic that directs your customers to an open-ended follow-up question based on their rating. The last page of the template includes a series of demographic questions, giving you further insight into who comprises your customer base.
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### How to Calculate Net Promoter Score (NPS): Formula
(9 days ago) Add up the promoters - those who scored 9 and 10. Add up the detractors - those with responses 0 to 6 (included) To calculate the percentage, divide the number of promoters by the total number of responses. Repeat this …
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### 12 Best NPS Survey Questions and Response Templates - 2022
(9 days ago) Jun 15, 2022. Net Promoter Score (NPS) is a global and one of the most popular Customer Satisfaction Metrics that help you understand your business through the eyes of the customers. This popular loyalty metric gauges your business relationship with your customer by measuring their willingness to recommend your company, products, and services
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### Net Promoter Score Guide + NPS Survey Template
(1 days ago) NPS is measured using a single-question survey, where the customers rate their experience on a scale of 0 to 10, while the results are reported with a score from -100 to +100. Don’t worry if those scores confuse you because I’m going to explain everything in as much detail as needed down below. So, now that we’ve defined the Net Promoter
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### Net Promoter Score® (NPS) survey template - Survicate
(Just Now) Because Survicate is easy to use yet powerful,with everything you need in one place. Enable any use case for Marketing, Product, CS and Leadership with one tool. Start tracking NPS, CSAT and CES in minutes. Drag and drop 15 question types to make any custom survey. Run email, link, web and mobile surveys from one tool.
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### Net Promoter Score survey template NPS survey template and …
(5 days ago) Use our free Net Promoter Score survey questions template to know if your customers are likely to refer your company to their friends or colleagues. Classify your customers into promoters, passives and detractors and take actions before you lose them. Collect customer satisfaction data and gain insights into customer experience.
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### Net Promoter Score Survey Questions and Templates (2021)
(4 days ago) Choose the right NPS Survey Question! There are two components to an ideal NPS survey. One is the NPS question which asks customers to rate the business/product/service on a scale of 0 to 10. Second is the open-ended question that seeks to find out why the customer has given the specific score.
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### 3 Net Promoter Score Questions To Ask - SurveyTown
(1 days ago) With this NPS question, you’ll be able to do your Net Promoter Score calculation. #2: The Reason Why Question. Your next question is really a follow up to the Net Promoter Score question. In addition to the main NPS question, you should invite respondents to let you know why they feel the way they do. Provide a text area where your customers
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### Net Promoter Score (NPS) Survey Templates - ProProfs Survey …
(Just Now) Create Net Promoter Score Survey to Measure Customer Loyalty & Growth. Run your NPS tests as and when you require using professionally made NPS survey templates. Use the standard NPS template or make it industry and audience specific using net promoter score sample questions we provide. Customize our net promoter score templates based on your
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### Net Promoter Score ® NPS ® Survey SurveyMonkey
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### NPS survey Net Promoter Score Survey Template - AidaForm
(3 days ago) 1. Calculating the NPS result with AidaForm is easy! Go to the Results > Response Inbox section for your NPS survey and download all responses as a CSV document. Find the column Customer Group and filter the results to see how many respondents fall into Promoters, Passives and Detractors. Use the NPS Calculator to get your NPS score right away. 2.
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### Net Promoter Score® (NPS) Survey Template - Typeform
(7 days ago) Net Promoter Score surveys are commonly used, but not all are built the same. Our forms are completely customizable and intuitive, so while the questions are standard, the experience won’t be. We designed our surveys to be like a conversation, asking questions one-by-one, so your customer will feel listened to.
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### Free Net Promoter Score Survey Template Maze
(4 days ago) Get fast NPS feedback. Product. Marketing. Feedback Survey. With this survey, you’ll gather valuable, qualitative insights that will add context to your net promoter score and help you understand customer satisfaction and brand perception.
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Quantum Logic Explorer < Previous Next > Nearby theorems Mirrors > Home > QLE Home > Th. List > testmod1 GIF version
Theorem testmod1 1214
Description: A modular law experiment. (Contributed by NM, 21-Apr-2012.)
Assertion
Ref Expression
testmod1 (((ca) ∪ ((bc) ∩ (da))) ∩ (a ∪ (b ∩ (d ∪ ((ac) ∩ (bd)))))) = (a ∪ (b ∩ (((ac) ∩ (bd)) ∪ (d ∩ ((ac) ∪ ((bc) ∩ (da)))))))
Proof of Theorem testmod1
StepHypRef Expression
1 testmod 1213 . 2 (((ca) ∪ ((bc) ∩ (da))) ∩ (a ∪ (b ∩ (d ∪ ((ac) ∩ (bd)))))) = ((b ∩ ((((ac) ∪ ((bc) ∩ (da))) ∩ d) ∪ ((ac) ∩ (bd)))) ∪ a)
2 orcom 73 . . 3 ((b ∩ ((((ac) ∪ ((bc) ∩ (da))) ∩ d) ∪ ((ac) ∩ (bd)))) ∪ a) = (a ∪ (b ∩ ((((ac) ∪ ((bc) ∩ (da))) ∩ d) ∪ ((ac) ∩ (bd)))))
3 orcom 73 . . . . . 6 ((((ac) ∪ ((bc) ∩ (da))) ∩ d) ∪ ((ac) ∩ (bd))) = (((ac) ∩ (bd)) ∪ (((ac) ∪ ((bc) ∩ (da))) ∩ d))
4 ancom 74 . . . . . . 7 (((ac) ∪ ((bc) ∩ (da))) ∩ d) = (d ∩ ((ac) ∪ ((bc) ∩ (da))))
54lor 70 . . . . . 6 (((ac) ∩ (bd)) ∪ (((ac) ∪ ((bc) ∩ (da))) ∩ d)) = (((ac) ∩ (bd)) ∪ (d ∩ ((ac) ∪ ((bc) ∩ (da)))))
63, 5tr 62 . . . . 5 ((((ac) ∪ ((bc) ∩ (da))) ∩ d) ∪ ((ac) ∩ (bd))) = (((ac) ∩ (bd)) ∪ (d ∩ ((ac) ∪ ((bc) ∩ (da)))))
76lan 77 . . . 4 (b ∩ ((((ac) ∪ ((bc) ∩ (da))) ∩ d) ∪ ((ac) ∩ (bd)))) = (b ∩ (((ac) ∩ (bd)) ∪ (d ∩ ((ac) ∪ ((bc) ∩ (da))))))
87lor 70 . . 3 (a ∪ (b ∩ ((((ac) ∪ ((bc) ∩ (da))) ∩ d) ∪ ((ac) ∩ (bd))))) = (a ∪ (b ∩ (((ac) ∩ (bd)) ∪ (d ∩ ((ac) ∪ ((bc) ∩ (da)))))))
92, 8tr 62 . 2 ((b ∩ ((((ac) ∪ ((bc) ∩ (da))) ∩ d) ∪ ((ac) ∩ (bd)))) ∪ a) = (a ∪ (b ∩ (((ac) ∩ (bd)) ∪ (d ∩ ((ac) ∪ ((bc) ∩ (da)))))))
101, 9tr 62 1 (((ca) ∪ ((bc) ∩ (da))) ∩ (a ∪ (b ∩ (d ∪ ((ac) ∩ (bd)))))) = (a ∪ (b ∩ (((ac) ∩ (bd)) ∪ (d ∩ ((ac) ∪ ((bc) ∩ (da)))))))
Colors of variables: term Syntax hints: = wb 1 ∪ wo 6 ∩ wa 7 This theorem was proved from axioms: ax-a1 30 ax-a2 31 ax-a3 32 ax-a5 34 ax-r1 35 ax-r2 36 ax-r4 37 ax-r5 38 ax-ml 1122 This theorem depends on definitions: df-a 40 df-t 41 df-f 42 df-le1 130 df-le2 131 This theorem is referenced by: (None)
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## Units of Rate constant
Marvin Lu 1E
Posts: 28
Joined: Fri Sep 25, 2015 3:00 am
### Units of Rate constant
How do we determine the units of a rate constant? The given rate constants in the book are L/mol*s but in the answers the rate constant is just s^-1.
Jake Ney lecture 1 discussion 1F
Posts: 26
Joined: Fri Sep 25, 2015 3:00 am
### Re: Units of Rate constant
The units of rate constants vary by 1st, 2nd and Zero Order reactions. For First Order reactions the units are s-1 because when used in k[A] we get the correct units for d[A]/dt which is mol/L*s. For Second order reactions the units of k are in M-1s-1 because when used in the equation k[A]^2 we get the correct units (mol/L*s) for d[A]/dt.
Chem_Mod
Posts: 18880
Joined: Thu Aug 04, 2011 1:53 pm
Has upvoted: 714 times
### Re: Units of Rate constant
The rate of a reaction is always in $\frac{M}{s}$ so we want a rate constant, which if we have certain inputs (concentrations), we get an answer in $\frac{M}{s}$. So in the case of a first order reaction where the reaction is dependent on the concentration of a single reactant, we have the units of M. Therefore, we want a rate constant that converts M to $\frac{M}{s}$. So our solution to that is to have the rate constant divide our concentration by seconds, represented as s-1.
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F1. Alice and Recoloring 1
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output
The difference between the versions is in the costs of operations. Solution for one version won't work for another!
Alice has a grid of size $n \times m$, initially all its cells are colored white. The cell on the intersection of $i$-th row and $j$-th column is denoted as $(i, j)$. Alice can do the following operations with this grid:
• Choose any subrectangle containing cell $(1, 1)$, and flip the colors of all its cells. (Flipping means changing its color from white to black or from black to white).
This operation costs $1$ coin.
• Choose any subrectangle containing cell $(n, 1)$, and flip the colors of all its cells.
This operation costs $2$ coins.
• Choose any subrectangle containing cell $(1, m)$, and flip the colors of all its cells.
This operation costs $4$ coins.
• Choose any subrectangle containing cell $(n, m)$, and flip the colors of all its cells.
This operation costs $3$ coins.
As a reminder, subrectangle is a set of all cells $(x, y)$ with $x_1 \le x \le x_2$, $y_1 \le y \le y_2$ for some $1 \le x_1 \le x_2 \le n$, $1 \le y_1 \le y_2 \le m$.
Alice wants to obtain her favorite coloring with these operations. What's the smallest number of coins that she would have to spend? It can be shown that it's always possible to transform the initial grid into any other.
Input
The first line of the input contains $2$ integers $n, m$ ($1 \le n, m \le 500$) — the dimensions of the grid.
The $i$-th of the next $n$ lines contains a string $s_i$ of length $m$, consisting of letters W and B. The $j$-th character of string $s_i$ is W if the cell $(i, j)$ is colored white in the favorite coloring of Alice, and B if it's colored black.
Output
Output the smallest number of coins Alice would have to spend to achieve her favorite coloring.
Examples
Input
3 3
WWW
WBB
WBB
Output
3
Input
10 15
WWWBBBWBBBBBWWW
BBBBWWWBBWWWBBB
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74
Note
In the first sample, it's optimal to just apply the fourth operation once to the rectangle containing cells $(2, 2), (2, 3), (3, 2), (3, 3)$. This would cost $3$ coins.
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https://scicomp.stackexchange.com/questions/29575/getting-started-with-fem-ill-conditioned-matrix-when-evaluating-flux-terms-in-c
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# Getting started with FEM: Ill-conditioned matrix when evaluating flux terms in conservation law?
I have a system of conservation laws of the form
$$\frac{\partial \mathbf{q}}{\partial t} + \nabla \cdot \mathbf{F}\!\left(\mathbf{q}\right) = 0$$
I want to use finite elements to solve this system. As an initial choice, I am using Legendre polynomials $\phi$ for my function basis. Let $j$ be a polynomial index. If I substitute the decomposition of my unknown function $\mathbf{q}$ and the flux function $\mathbf{F}$,
$$\mathbf{q}\!\left(t, x\right) = \sum\limits_j \hat{\mathbf{q}}_j\!\left(t\right) \phi_j\!\left(x\right)$$
$$\mathbf{F}\!\left(\mathbf{q}\right) = \sum\limits_j \hat{\mathbf{F}}\!\left(t\right) \phi_j\!\left(x\right)$$
and finally write out the weak form (given some test volume $\Omega$ with boundary $\partial \Omega$)
$$\sum\limits_j \left(\frac{\partial \hat{\mathbf{q}}_j}{\partial t} \int\limits_\Omega \phi_i \phi_j + \hat{\mathbf{h}}_j \oint\limits_{\partial \Omega} \phi_i \phi_j - \hat{\mathbf{F}}_j \cdot \int\limits_\Omega \phi_j \nabla \phi_i\right) = 0$$
Here, I replaced $\mathbf{F} \cdot \hat{\mathbf{n}} = \mathbf{h}$.
My problem occurs when I actually try to evaluate the surface second and third terms inside the parentheses. Because my problem is 1-dimensional, the second term amounts to just evaluating $\phi_i \phi_j$ at 1 and -1 and taking the difference ($\hat{\mathbf{n}} = \pm \hat{\mathbf{x}}$ is the outward-facing normal). In analogy with the stiffness matrix, we could write this as a matrix $\mathbb{B} = \{b_{ij}\}$. Any given element $b_{ij}$ will be either 0 or 2 because the Legendre polynomials take the value of 1 or -1 at the edges of the domain. This matrix is sparse but singular. The third term exhibits a similar problem if we try to write it as a matrix $\mathbb{C} = \{c_{ij}\}$.
The only way I know to solve for the $\hat{\mathbf{F}}_j$ and $\hat{\mathbf{h}}_j$ coefficients is by solving a matrix equation, but all my attempts to solve them (using PETSc) converge slowly and may not even be correct. I would like to use a preconditioner, but I get errors any time I try (PCSETUP_FAILED due to SUBPC_ERROR), even though I have specified the null spaces of these matrices.
Have I approached this problem incorrectly altogether? Or is there a common way of overcoming this challenge, in PETSc or otherwise?
• You mention matrix is singular. It hints you may need a regularizer. May 21, 2018 at 19:06
• You have mix formulation and have two problems here. 1) missing physical equation 2) q has to be in L2 space and F in h-div space to have stability of discrete system. May 24, 2018 at 17:08
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https://allthingsstatistics.com/distributions/hypergeometric-distribution/
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No menu items!
# Hypergeometric Distribution
-
The hypergeometric distribution is a discrete probability distribution that arises when we try to draw a random sample without replacement from a given population. For example, suppose there are N balls in a bag out of which M are white and the remaining N-M are black. Suppose we choose a sample of size n from the bag. Then the probability that the sample has k white balls can be calculated using the pdf of the hypergeometric distribution.
Example: Suppose that a bag contains 10 balls out of which 6 are white and 4 are black. If we choose 4 balls randomly from the bad then calculate the probability that two of these balls are white.
Solution: Let X denote the number of white balls in our sample.
Given N=10, M=6, n=4, and k=2.
Substituting all this in the above formula we get,
P(X=2) = (62)*(42)/(104) = 15*6/210 = 90/210 = 0.429
Mean and Variance of Hypergeometric Distribution:
Example: In the above example, calculate the mean number of white balls selected and the variance.
Solution: Given N=10, M=6, n=4 and k=2.
Substituting all this in the above formula we get,
Mean = 4*6/10 = 2.4 ,that is, 2.4 white balls will be chosen on average.
Variance = (10*6*4*6)/(100*9) = 1.6
Use of hypergeometric distribution:
1. It is used in the theory of Quality control to calculate the number of lots to be selected for inspection of quality of goods.
2. It is used in sampling theory to calculate the probabilities of selecting particular kinds of samples without replacement.
Binomial approximation to Hypergeometric Distribution:
As the value of N tends toward infinity and the value M/N approaches a finite number then the hypergeometric distribution can be approximated using the pdf of the binomial distribution with parameters n and p=M/N.
Hey 👋
I have always been passionate about statistics and mathematics education.
I created this website to explain mathematical and statistical concepts in the simplest possible manner.
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# Mutually exlcusive or not?
• Mar 4th 2008, 02:53 AM
wikji
Mutually exlcusive or not?
Mutually exclusive events are always independent. Is this
true or false? Why?
• Mar 4th 2008, 03:13 AM
mr fantastic
Quote:
Originally Posted by wikji
Mutually exclusive events are always independent. Is this
true or false? Why?
If A and B are mutually exclusive events, then $\Pr(A \cap B) = 0$.
If A and B are independent events, then $\Pr(A \cap B) = \Pr(A) \cdot \Pr(B)$.
So what do you think and why?
Hint: The statement is false.
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http://www.studymode.com/essays/Derivatives-Futures-Contract-And-Inc-Common-701532.html
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# Derivatives: Futures Contract and Inc. Common Stock
Topics: Futures contract, Short, Hedge Pages: 6 (1032 words) Published: May 18, 2011
1. (Problem 1.12) Suppose bank’s loan officer tells you that if you take out a mortgage (i.e., you borrow money to buy a house) you will be permitted to borrow no more than 80% of the value of the house. Describe this transaction using the terminology of short-sales.
We are interested in borrowing the asset “money” to buy a house. Therefore, we go to an owner of the asset, called Bank. The Bank provides the dollar amount, say \$250,000, in digital form in our mortgage account. As \$250,000 is a large amount of money, the bank is subject to substantial credit risk (e.g., we may lose our job) and demands a collateral. Although the money itself is not subject to large variations in price (besides inflation risk, it is difficult to imagine a reason for money to vary in value), the Bank knows that we want to buy a house, and real estate prices vary substantially. Therefore, the Bank wants more collateral than the \$250,000 they are lending. In fact, as the Bank is only lending up to 80% of the value of the house, we could get a mortgage of \$250,000 for a house that is worth \$250,000 ÷ 0.8 = \$312,500. We see that the bank factored in a haircut of \$312,500 - \$250,000 = \$62,500 to protect itself from credit risk and adverse fluctuations in property prices. We buy back the asset money over a long horizon of time by reducing our mortgage through annuity payments.
2. What do hedge funds do:
(a) Hedge?
(b) Speculate?
(c) Arbitrage?
(d) None of the above
(a), (b), (c)
3. During the growing season a corn farmer sells short corn futures contracts in an amount equal to her crop. If after harvesting and selling her crop she maintains the contracts, she is then considered a:
(a) Hedger
(b) Speculator
(c) Arbitrager
(d) None of the above
(b)
4. A firm provides a service that benefits from decreasing employment. This firm has a risk exposure to macro event. All other variables being equal, which of the following derivative securities is the firm most likely use to hedge its exposure?
(a) Short position in an economic futures
(b) Long position in an economic futures
(c) Short position in an interest rate futures
(d) Long position in an interest rate futures
(b)
5. Assume that you purchase 100 shares of Jiffy, Inc. common stock at the bid-ask prices of \$32.00 – \$32.50. When you sell the bid-ask prices are \$32.50 – \$33.00. If you pay a commission rate of 0.5%, what is your profit or loss?
(a) \$0
(b) \$16.25 loss
(c) \$32.50 gain
(d) \$32.50 loss
(d) –(100*32.5)*1.005 + (100*32.5)*0.995 = –3,266.25 + 3,233.75 = –32.5
6. Assume that you open a 100 share short position in Jiffy, Inc. common stock at the bid-ask prices of \$32.00 – \$32.50. When you close your position the bid-ask prices are \$32.50 – \$33.00. You pay a commission rate of 0.5 %. The market interest rate is 5.0 % and the short rebate rate is 3.0 %. What is your additional gain or loss due to leasing the asset?
(a) \$64.00 loss
(b) \$160.00 loss
(c) \$96.00 gain
(d) \$0
(a) 100*32*(0.03–0.05) = –64
7. Catastrophe bonds are bonds that the issuer need not repay if there is a specific event, e.g. earthquake, causing large insurance claims.
a) Who do you think issue catastrophe bonds?
b) Are catastrophe bonds long-term or short-term?
c) Are catastrophe bonds risky or safe investment?
d) Who are the buyers of catastrophe bonds?
e) Why would investors invest into catastrophe bonds?
f) Can catastrophe bonds replace insurance?
a) Usually insurance and reinsurance companies. Sometimes private companies. b) Long-term.
c) Risky.
d) Hedge funds, mutual funds.
e) For...
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http://iceeonline.org/lib/extreme-eigen-values-of-toeplitz-operators
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Download Extreme Eigen Values of Toeplitz Operators by I.I.Jr. Hirschman, D.E. Hughes PDF
By I.I.Jr. Hirschman, D.E. Hughes
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Extra resources for Extreme Eigen Values of Toeplitz Operators
Example text
D~ o R --n Here ~"~ = ~i~ I + Let ... + ~n~n . f] ~ ~ ~ f(~) ~ etc. - v. 0
By Lemma 3a L(t-llxl)L(t-l) -I -~ i as t -+ ~ . Therefore t lira gm~) -- gin(x) 47 Theorem 3d. for all i. x E R Given and all ~n ¢ > 0 there exists given ¢ > 0 such that t > I ~ ( x ) <__A(e)Itl ~ + ~ , ii. , p . Note: Proof. We have ~) for _x ~ Rtm " = t~L(t-l)-If(t-lx+~_m)_ is bounded we clearly have i. , _x E Rtm " Since f By assumption eL-~mI)l~_'~-ml ~) f (~--) = ~m (['~-m)L (I~_-~__mI )l ~-~_mI~ + o (Ll as ~ ~ ~_m " It follows from this that we can find a constant a neighborhood, Nm, of ~-m ' Nm = {~:]~'~--mI < ~} 0 _< f(~) _< ~<1~-~t)1~-~t ~' for ~ E N Thus if _x E a(tjm)N m c Rtm we have 0 ~ f(='l~+~m) ~ ML(t-IIml)t-~Iml~ Therefore t 0 _< gin(x) < ML(t'l)'IL(t-llx l)Ixl ~ Applying Lemma 3b we get such that M and 48 t 0 _< gin(x) <_ M'M(,)[J~[ e for x E ~ ( t , m ) Nm .
P} > 0 We are given a positive constant function L(t)t -~ ~) defined L(t) We have p and a positive con- 0 < t < ~ w h i c h is slowly is slowly oscillating at ~(e) > 0 is decreasing for for w such that 0 < t < ~(e) L(t)t ¢ 0 if for every is increasing and • non-negative m e a s u r a b l e rO if some trans- 6 > 0 . iii. > 0 its m i n i m u m [~-~m ] ~8 inf {f^(~): tinuous the closure of is star-shaped. _ for each R is star--11 is star-shaped (the closure of We have a real function ~m in is star-shaped with respect to the origin.
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# Problem: (Hint: If you dont know where to start, try assuming you have 100g total of solution.) Consider two liquids A and B. The vapor pressure of pure A (molecular weight = 50 g/mol) is 225 torr at 25°C and the vapor pressure of pure B (molecular weight = 75 g/mol) is 90 torr at the same temperature. What is the total vapor pressure at 25°C of a solution that is 25% A amd 75% B by weight? 1. 76 torr 2. 225 torr 3. 335 torr 4. 203 torr 5. 135 torr 6. 191 torr 7. 108 torr 8. 115 torr 9. 124 torr
###### Problem Details
(Hint: If you dont know where to start, try assuming you have 100g total of solution.) Consider two liquids A and B. The vapor pressure of pure A (molecular weight = 50 g/mol) is 225 torr at 25°C and the vapor pressure of pure B (molecular weight = 75 g/mol) is 90 torr at the same temperature. What is the total vapor pressure at 25°C of a solution that is 25% A amd 75% B by weight?
1. 76 torr
2. 225 torr
3. 335 torr
4. 203 torr
5. 135 torr
6. 191 torr
7. 108 torr
8. 115 torr
9. 124 torr
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# Horsepower to watts conversion
## Horsepower to watts conversion calculator
Enter the power in horsepower and press the Convert button:
Select hp unit type: Mechanic horsepower Electrical horsepower Metric horsepower Enter horsepower: hp Result in watts: W
Watts to horsepower conversion ►
#### Horsepower to Kilowatts
The term horsepower was adopted by Scottish inventor James Watt to measure the power output of his steam engine and to compare that power with that of horses horses. Horsepower were later used to measure the power output of piston and turbine engines and electrical motors.
There are now several different types of horsepower, including mechanical, electrical, metric, boiler, brake, and indicated, and they all have different applications in industry. Mechanical and electric are the most commonly used types.
One kilowatt is the power equal to 1,000 watts, or the energy consumption at a rate of 1,000 joules per second.
The kilowatt is a multiple of the watt, which is the SI derived unit for power. In the metric system, "kilo" is the prefix for 103. Kilowatts can be abbreviated as kW; for example, 1 kilowatt can be written as 1 kW.
#### Mechanic / Hydraulic horsepower to kilowatts
One mechanic or hydraulic horsepower is equal to 0.745699872 kilowatts:
1 hp(I) = 745.699872 W = 0.745699872 kW
So the power conversion of horsepower to kilowatts is given by:
P(kW) = 0.745699872 ⋅ P(hp)
#### Example
Convert 10 hp to kW:
P(kW) = 0.745699872 ⋅ 10hp = 7.45699872 kW
#### Electrical horsepower to kilowatts
One electrical horsepower is equal to 0.746 kilowatts:
1 hp(E) = 746 W = 0.746 kW
So the power conversion of horsepower to kilowatts is given by:
P(kW) = 0.746 ⋅ P(hp)
#### Example
Convert 10 hp to kW:
P(kW) = 0.746 ⋅ 10hp = 7.460 kW
#### Metric horsepower to kilowatts
One metric horsepower is equal to 0.73549875 kilowatts:
1 hp(M) = 735.49875 W = 0.73549875 kW
bSo the power conversion of horsepower to kilowatts is given by:
P(kW) = 0.73549875 ⋅ P(hp)
#### Example
Convert 10 hp to kW:
P(kW) = 0.73549875 ⋅ 10hp = 7.3549875 kW
Watts to horsepower conversion ►
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Let’s modify the inquiry slightly by replacing the letter by your alphabetic location : “what is the next element in the serie :2,3,5,4,5,9,6,8?
Of food 14 is the most obvious answer (as declared by all other answers, 2+3=5, 4+5=9, 6+8=14), which provides the letter N as most obvious answer for the initial question.
You are watching: What is the next value? 2 3 e 4 5 i 6 8
But friend could also consider this together three collection alternated without a relation in between them. The very first one 2,4,6 would give 8 together most most likely next element, the second 3,5,8 would provide possibly 12 if the 3rd 5,9 can give united state 13 аs а possible logicаl аnswer, which coincides to the letter M.
Furthermore, if we insteаd think about the letters by your rаnk in the perform of voyels insteаd of their rаnk in the аlphаbet, we obtain 2,3 because that E аnd ns аnd we could for instаnce decision the many logicаl next facet is 4, offering us the voyel O аs the аnswer.
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## Related questions
Ganapati Shinde
0
April 25, 2020 at 7:58 pm
2, 3, E, 4, 5, I, 6, 8, ?2 + 3 = 5 = E (E is in 5th position in аlphаbets)4 + 5 = 9 = ns (I is in nine position in аlphаbets)6 + 8 = 14 = N (n is in 14th place in аlphаbets)2, 3, E, 4, 5, I, 6, 8, N
Himanshu Ajmera
0
April 25, 2020 at 7:58 pm
The аnswer would certainly be letter N.How?Simple A is аssigned no. 1, B no. 2, C no. 3 аnd so no till Z no. 26 because we hаve 26 аlphаbets in English.Now, consider the numbers before аlphаbets. In the sequence 2, 3. Adding these 2 numbers gives us result of 5. Walk by the аbove reasonable we finish up through аlphаbet E. Sаme holds true for next two numbers in sequence 4,5.Continuing utilizing the sаme logic 6,8 outcomes in 14 аnd 14th аlphаbet is N аnd thus the аnswer is N.
See more: How Many Yards In 200 Meters To Yards, Convert 200 Meters To Yards
Kanishk Rao
0
April 25, 2020 at 7:58 pm
The аnswer is NThe series goes choose аdd the no аnd equivalent to thаt discover the аlphаbet.A B C D E F G H i J K l M N O P…1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16..3+2=5 аnd 5 represent E ,similаrly 4+5=9 9 represent I 6+8=14 which represent N
Claude Catudal
0
April 25, 2020 at 7:58 pm
N аs is the 14th letter of thr аlphаbet.2+3=E , 4+5=I, 6+8=N
King Anny Youngdean Umanah
0
April 25, 2020 in ~ 7:58 pm
8, M, 10, 12, Q
Suganya Selvaraj
0
April 25, 2020 in ~ 7:58 pm
Consider just the аlphаbeticаl letters аnd your positions.A,B,C, D, E, F, G, H, I, J, K ,L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z.Not 2,3. That 2+32+3=5 (E- 5th plаce )Not 4, 5. Its 4+54+5=9 ( I- ninth plаce)Not 6,8. That is 6+86+8=14 ( N- 14th plаce)Ans N
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REN Zhong-jun, PENG Xiang-he, HU Ning, YANG Chun-he. Micromechanical Damage Model for Rocks and Concretes Under Triaxial Compression[J]. Applied Mathematics and Mechanics, 2009, 30(3): 309-317.
Citation: REN Zhong-jun, PENG Xiang-he, HU Ning, YANG Chun-he. Micromechanical Damage Model for Rocks and Concretes Under Triaxial Compression[J]. Applied Mathematics and Mechanics, 2009, 30(3): 309-317.
# Micromechanical Damage Model for Rocks and Concretes Under Triaxial Compression
• Rev Recd Date: 2008-12-25
• Publish Date: 2009-03-15
• Based on the analysis of the deformation in an infinite isotropic elastic matrix containing an embedded elliptic crack, subject to far field triaxial compressive stress, the energy release rate and a mixed fracture criterion were obtained by using an energy balance approach. The additional compliance tensor induced by a single closed elliptic microcrack in a representative volume element and its in-plane growth was derived. The additional compliance tensor induced by the kinked growth of the elliptic microcrack was also obtained. The effect of the microcracks, randomly distributed both in geometric characteristics and orientations, was analyzed with the Taylor's scheme by introducing an appropriate probability density function. A micromechanical damage model for rocks and concretes under triaxial compression was obtained and experimentally verified.
• [1] Horii H, Nemat-Nasser S. Overall moduli of solids with microcracks: load-induced anisotropy[J].Journal of the Mechanics and Physics of Solids,1983,31(2):155-171. [2] Li C L, Nordlund E. Deformation of brittle rocks under compression-with particular reference to microcracks[J].Mechanics of Materials,1993,15(3):223-239. [3] Fanella D, Krajcinovic D. A micromechanical damage model for concrete in compression[J]. Engineering Fracture Mechanics,1988,29(1):49-66. [4] Ju J W. On two-dimensional self-consistent micromechanical damage models for brittle solids[J].International Journal of Solids and Structures,1991,27(2):227-258. [5] Feng X Q, Yu S W. A new damage model for microcrack-weakened brittle solids[J].Acta Mechanica Sinica,1993,9(3):251-260. [6] Yu S W, Feng X Q. A micromechanics-based damage model for microcrack-weakened brittle solids[J].Mechanics of Materials,1995,20(1):59-76. [7] Krajcinovic D. Damage mechanics[J].Mechanics of Materials,1989,8(2):117-197. [8] Ju J W, Chen T M. Effective elastic moduli of two-dimensional brittle solids with interacting microcracks—part I: basic formulations[J].Journal of Applied Mechanics,1994,61(2):349-357. [9] Kachanov M. Effective elastic properties of crack solids, critical review of some basic concepts[J].Applied Mechanics Review,1992,45(7):304-335. [10] Kassir M K, Sih G C. Three dimensional stress distribution around an elliptical crack under arbitrary loading[J].Journal of Applied Mechanics,1966,33(3): 601-611. [11] Kassir M K, Sih G C. Three dimensional crack problems[A]. In:Sih G C,Ed.Mechanics of Fracture[C].Leyden: Noordhoff International Publishing, 1975, 382-409. [12] Budiansky B, O'connell R J. Elastic moduli of a cracked solid[J].International Journal of Solids and Structures,1976,12(1): 81-95. [13] 丁遂栋, 孙利民. 断裂力学[M]. 北京：机械工业出版社, 1997. [14] Ashby M F, Hallam S D. The failure of brittle solids containing small cracks under compressive stress state[J].Acta Metallurgica,1986,34(3): 497-510. [15] Kachanov M. A microcrack model of rock inelasticity[J].Mechanics of Materials,1982,1(1):19-41. [16] Horii H, Nemat-Nasser S. Brittle failure in compression: splitting, faulting, and brittle-ductile transition[J].Philosophical Transactions of the Royal Society of London Series A,1986,319(1549):337-374. [17] Li H B, Zhao J, Li T J. Micromechanical modelling of the mechanical properties of a granite under dynamic uniaxial compressive loads[J].International Journal of Rock Mechanics and Mining Sciences,2000,37(6):923-935. [18] X Lee, Ju J W. On three-dimensional self-consistent micromechanical damage models for brittle solids—part Ⅱ: compressive loadings[J].Journal of Engineering Mechanics,1991,117(7): 1515-1536.
### Catalog
###### 通讯作者: 陈斌, bchen63@163.com
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沈阳化工大学材料科学与工程学院 沈阳 110142
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Finding Average Duration Time
✭✭✭✭✭
Hello,
I am trying to get an average duration for a Sheet Summary field but getting "0". My formula is as follows:
=SUMIFS([Total Onsite Time]:[Total Onsite Time], [Store Status]:[Store Status], >"" / COUNTIF([Store Status]:[Store Status], "Store Fully Open"))
What am I missing?
Sharon C
«1
• ✭✭✭✭✭✭
How is the data entered into [Total Onsite Time]?
• ✭✭✭✭✭
The Total onsite time is calculated. The store status "Store Fully Open" is from a drop down list.
• ✭✭✭✭✭✭
How is it calculated? Can you copy/paste the exact formula?
• ✭✭✭✭✭
Here you go,
=IF([Tech Name]@row > "", IF([Tech Arrival Time]@row > "", IF([Store Open Time]@row > "", INT([Install Duration]@row) + "." + ([Install Duration]@row - INT([Install Duration]@row)) * 60)))
• ✭✭✭✭✭✭
Ok. The problem is with the output. When you use
+ "."
or anything else within quotes, you convert the output to a text string.
The most simple fix would be to use a VALUE function to convert it back to a numerical value:
=IF([Tech Name]@row > "", IF([Tech Arrival Time]@row > "", IF([Store Open Time]@row > "", VALUE(INT([Install Duration]@row) + "." + ([Install Duration]@row - INT([Install Duration]@row)) * 60))))
But out of curiosity... Why are you pulling the number like that? What exactly is in the [Install Duration] column?
• ✭✭✭✭✭
The Install Duration column is the calculation hh:mm from a 24 hr time calculation.
• ✭✭✭✭✭✭
What is the formula in [Install Duration]?
• ✭✭✭✭✭
Here is the formula for Install Calculation. I also included the calculations for Open Time and Tech Arrival Time.
Install Calculation: =[Store Open Time-Calculation]2 - [Tech Arrival Time-Calculation]2
Store Open Time Calculation: =((VALUE(LEFT([Store Open Time]@row, FIND(":", [Store Open Time]@row) - 1)) + IF(CONTAINS("p", [Store Open Time]@row), IF(VALUE(LEFT([Store Open Time]@row, FIND(":", [Store Open Time]@row) - 1)) <> 12, 12), IF(VALUE(LEFT([Store Open Time]@row, FIND(":", [Store Open Time]@row) - 1)) = 12, -12))) + (VALUE(MID([Store Open Time]@row, FIND(":", [Store Open Time]@row) + 1, 2)) / 60)) + (([Install End Date]@row - [Install Date]@row) * 24)
Tech Arrival Time Calculation: =(VALUE(LEFT([Tech Arrival Time]@row, FIND(":", [Tech Arrival Time]@row) - 1)) + IF(CONTAINS("p", [Tech Arrival Time]@row), IF(VALUE(LEFT([Tech Arrival Time]@row, FIND(":", [Tech Arrival Time]@row) - 1)) <> 12, 12), IF(VALUE(LEFT([Tech Arrival Time]@row, FIND(":", [Tech Arrival Time]@row) - 1)) = 12, -12))) + (VALUE(MID([Tech Arrival Time]@row, FIND(":", [Tech Arrival Time]@row) + 1, 2)) / 60)
• ✭✭✭✭✭✭
Try this in your [Total Onsite Time] column:
=IF(AND([Tech Name]@row <> "", [Tech Arrival Time]@row <> "", [Store Open Time]@row <> ""), [Install Calculation]@row)
These column names and formulas are really familiar. Did we work together getting this sheet set up?
• ✭✭✭✭✭
Thank you for the fix. It worked like a charm. My Summary Field formula is:
=SUM([Total Onsite Time]:[Total Onsite Time]) / COUNTIF([Store Status]:[Store Status],
Another Smartsheet Hero saves the day👏
• ✭✭✭✭✭
We did. I thought your name was familiar. Voted your solution UP. I guess it would be the SAME Smartsheet Hero saves the day.
Thank you so much again.
• ✭✭✭✭✭✭
Hahaha. Well then in that case... I should have built in a column that has the duration calculated in a numerical value. It is probably called [Calc Duration] or [Calculation Duration] or something like that.
That column was left in place specifically for scenarios like this where you want to use a SUMIFS. Reference THAT column in your SUMIFS, and you shouldn't need an additional column to pull the number and put in the decimal with the INT functions and whatnot.
• ✭✭✭✭✭✭
I just found the thread. Try this (changes in bold - note the removal of a closing parenthesis from the end as well)...
=SUMIFS([Install Duration]:[Install Duration], [Store Status]:[Store Status], <> "") / COUNTIF([Store Status]:[Store Status], "Store Fully Open")
• ✭✭✭✭✭✭
And Summary Field formula:
=SUM([Install Duration]:[Install Duration]) / COUNTIF([Store Status]:[Store Status], <> "")
• ✭✭✭✭✭
I tried that and got a #INVALID VALUE error. This is what ended up working:
=SUM([Total Onsite Time]:[Total Onsite Time]) / COUNTIF([Store Status]:[Store Status], >""
Thank you again.
Help Article Resources
Want to practice working with formulas directly in Smartsheet?
Check out the Formula Handbook template!
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+0
0
54
1
Find the total surface area of the figure.
Apr 12, 2020
#1
+20961
0
Since the two bases are regular hexagons, they can be divided into 6 equilateral triangles.
If the height of an equilateral triangle is 3·sqrt(3), its base is 6 and its area = ½·6·3·sqrt(3) = 9·sqrt(3).
Therefore the area of the regular hexagon = 6·9·sqrt(3) = 54·sqrt(3).
Doubling this to get the area of both bases = 108·sqrt(3).
Each side is a rectangle whose area is 6·8 = 48.
For all 6 sides: 6·48 = 288.
Total surface area = 108·sqrt(3) + 288.
Apr 12, 2020
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# Document 25403445
```MGT201 Mathematical Models
Salkind, Statistics for People Who (Think They) Hate Statistics 6th Edition, SAGE Inc. © 2017
What is Statistics?...and Why Statistics?
Why You Need To Study Statistics:
Ks7aS7YI
What is Statistics?
So, what is statistics?
What is Statistics?
• Statistics is a way of reasoning, along with a
collection of tools and methods, designed to help
us understand the world.
• A set of tools and techniques used for describing,
organizing, and interpreting information or data.
• Statisticians use tools and methods in statistics to:
Analyze small and large amounts of data
Analyze data real time
Interpret the results and help organizations in their
decision making
How is Statistics used in Management?
How is statistical analysis used in
management?
How is Statistics used in Management?
Statistical analysis is used in almost every area in
• Accounting – to audit a company’s accounts
(“statistical audit” is conducted in which a
representative sample of invoices is audited)
• Finance – to decide on which investment
alternatives to invest in (statistics can provide
many ways of measuring risk and expected
returns on investments)
How is Statistics used in Management?
• Marketing– to understand consumer purchasing
patterns, marketers use statistical analysis to learn
about the relationship between various variables
including age group, income level, gender, postal
code, etc. to design promotional campaigns focused
on the appropriate target audience.
• Human Resources Planning – to investigate the
patterns of promotion, recruitment, retirement,
transfers and resignation, an organization analyzes
employee past data to forecast the number of
employees at different levels of the management
pyramid in the future.
How Can I Learn Statistics?
You need to be very proactive in doing the
learning by putting into practice the concepts
and methods the book teaches. Statistics is like
most useful things in life: You must practise it to
really learn it.
Branches of Statistics
What are the two major branches of statistics?
Branches of Statistics
Two major branches of statistics:
a) descriptive statistics
b) inferential statistics
Descriptive Statistics
Used to organize and describe the
characteristics of a particular data set
This collection of data is sometimes called a
data set or just data.
Descriptive Statistics Examples
Average age of everyone in this class (i.e.,
mean)
The most popular college major (i.e., mode)
Practice (find the mean and mode)
Inferential Statistics
Used to make inferences based on a smaller
group of data
A sample is a subset of a population.
A population is all of the occurrences with a
certain characteristic.
Inferential Statistics Examples
Take 100 people with depression, and split
them up into two groups randomly.
With the first group, give them a sugar pill,
and give the second group a new depression
drug.
If you find a statistically significant difference
in both groups, you infer that you would find
similar results in the entire population as well.
Inferential Statistics Example
Based on an analysis of students in MGT 201,
would you infer that most business students at
Stanford College take the bus to school every
day?
- Raise your hand if you do
Inferential Statistics
Why does statistics use samples to make
presents itself in a population?
Inferential Statistics
It is often necessary to take a sample instead
of studying every member of a population due
to one or more of the following reasons:
1. The prohibitive cost of surveying the whole
population.
2. The destructive nature of some tests.
3. The physical impossibility of capturing the
population.
Cases and Variables
What are cases?
What are variables?
Cases
A row of a data table corresponds to an individual
people) we record some characteristics.
Purchase
Order Number
Cases
Name
Ship to
Province
Price
Area
Code
ASIN
10675489
Katherine
H.
Alberta
10.99
403
N
B0000015Y6
10783489
Samuel P.
Nova
Scotia
16.99
902
Y
B000002BK9
12837593
Chris G.
Quebec
15.98
819
N
B000068ZVQ
15783947
Monique D.
Ontario
11.99
905
N
B000001OAA
Variables
The characteristics recorded about each individual or case
are called variables. These are usually shown as the columns of
a data table and identify What has been measured.
Variables
Purchase
Order
Number
Name
Ship to
Province
Price
Area
Code
ASIN
10675489
Katherine
H.
Alberta
10.99
403
N
B0000015Y6
10783489
Samuel P.
Nova
Scotia
16.99
902
Y
B000002BK9
12837593
Chris G.
Quebec
15.98
819
N
B000068ZVQ
15783947
Monique D.
Ontario
11.99
905
N
B000001OAA
Variable Types
What are the types of variables?
Variable Types
When a variable names categories and
those categories, it is called a categorical
variable (also called qualitative variable).
When a variable has measured numerical
values with units and the variable tells us about
the quantity of what is measured, it is called a
quantitative variable.
Variable Types
Categorical variables …
• arise from descriptive responses to questions like
“What kind of advertising do you use?” or “Do
you invest in stock market?”
• may only have two possible values (like “Yes” or
“No”)
• may be a number like a telephone area code
Variable Types (Categorical)
Question
Categories or Responses
Do you invest in the stock market?
__Yes__No
What kind of advertising do you use?
__Magazines__Internet__Direct Mailings
I would recommend this course to
another student.
__Strongly Disagree__Slightly Disagree__Slightly
Agree__Strongly Agree
How satisfied are you with this
product?
__Very Unsatisfied__Unsatisfied__Satisfied__Very
Satisfied
Variable Types (Quantitative)
Some quantitative variables have units. The units
indicate …
• how each value has been measured
• the corresponding scale of measurement
• how much of something we have
• how far apart two values are
Other quantitative variables have no units, such as …
• Number of visits to a web site
• Number of shares of a company traded in Toronto Stock
Exchange
Variable Types
• Some variables can be both categorical and
quantitative
• How data are classified depends on Why we are
collecting the data
For example, variable Age is obviously the quantitative
value, measured in years, that may be used for finding
the average age of customers.
Age categories such as Child, Teen, Adult, or Senior can
be the categorical value used to decide in which music
to offer in a special deal – folk, jazz, hip hop or reggae.
May take on any value
within a given range of
real numbers and
usually arises from a
measurement (not a
counting) process. Eg.
Height, weight, time,
etc.
May have a finite
number. Mostly comes
from a counting
process. Their values
are mostly whole
numbers (counts). Eg.
Numbers of students,
number of university
credits, etc.
Indicates the rank
ordering of items.
Eg. Customer
product review.
Where, 1: very
satisfied, 2:
Satisfied, 3:
Neutral, 4=
Unsatisfied, 5=
Very unsatisfied.
Words that
describe the
categories or
classes of
response. Eg.
1=Male
2=Female.
Numerical variables
Nominal vs. ordinal categorical
variables
A nominal variable is one that has two or more categories, but there is no
intrinsic ordering to the categories. For example, gender is a categorical
variable having two categories (male and female) and there is no intrinsic
ordering to the categories. Hair color is also a categorical variable having
a number of categories (blonde, brown, brunette, red, etc.) and again,
there is no agreed way to order these from highest to lowest. A purely
categorical variable is one that simply allows you to assign categories but
you cannot clearly order the variables. If the variable has a clear
ordering, then that variable would be an ordinal variable.
An ordinal variable is similar to a categorical variable. The difference
between the two is that there is a clear ordering of the variables. For
example, suppose you have a variable, economic status, with three
categories (low, medium and high). In addition to being able to classify
people into these three categories, you can order the categories as low,
medium and high. Now consider a variable like educational experience
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0
# How many people can run under a 12 minute 2 mile?
Updated: 9/18/2023
Wiki User
β 13y ago
not very many
Wiki User
β 13y ago
Earn +20 pts
Q: How many people can run under a 12 minute 2 mile?
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mmm
### How many minute mile running a four hour marathon?
9.154 minutes per mile (rounded)
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https://www.life2coding.com/uri-online-judge-solution-1168-led-intermediate-problem/
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## Problem Summary:
Problem Number: 1168
Problem Name: LED
Author’s Name: Unknown Author
Timelimit: 1
Problem Source: https://www.urionlinejudge.com.br/judge/en/problems/view/1168
## Some Talks about Contest Programming:
An incredible method to enhance your abilities when figuring out how to code is by solving coding problems. Solving different kinds of challenges and riddles can enable you to improve as a problem solver, take in the complexities of a programming dialect, get ready for prospective job interviews, learn new algorithms and more.
An online judge is an online platform to test programs in focused programming challenges. They are likewise used to practice for such challenges. A considerable amount of these platforms also arrange their own programming contests.
## 10 Steps to Solve Any Problems:
1. Read the problem completely at least two or three times (or however many makes you feel comfortable)
2. Identify the subject, the problem belongs to. Is it a sorting or pattern matching problem? Can I use graph theory? Is it related to number theory? etc.
3. Try to solve the problem manually by considering 3 or 4 sample data sets.
4. After concentrate on optimizing the manual steps. Try to make it as simple as possible.
5. Write to write pseudo-code and comments besides the code from the manual steps. One thing you can do is to check after every function is written. Use a good IDE with a debugger, if possible. Don’t need to think much about the syntax. Just focus on the logic and steps.
6. Replace the comments or pseudo-code with real code. Always check if the values and code are behaving as expected before moving to the new line of pseudo-code.
7. Then optimize the real code.
8. Take care of boundary conditions as well.
9. Get feedback from your teammates, professors, and other developers and also ask your question on Stack Overflow if possible. Try to learn from others’ guidelines and what they are handling those problems. A problem may be solved in several ways. So, don’t get disappointed if you can’t think like an expert. You need to stick to the problem and you will gradually become better and quicker in solving problems like others.
10. Practice, Practice, and Practice.
N.B: Try to follow the above steps always. If you still can’t get the problem solved, take a look at the solution below. Don’t just copy paste the code. It will kill your creativity. Try to enjoy contest programming and develop your skills.
## Solution:
```#include <stdio.h>
int main(){
int n, j;
char num[101];
long long leds;
scanf("%d", &n);
for(int i = 0; i < n; i++){
scanf("%s", &num);
j = 0;
leds = 0;
while(true){
if(num[j] == '') break;
if(num[j] == '1') leds += 2;
if(num[j] == '2') leds += 5;
if(num[j] == '3') leds += 5;
if(num[j] == '4') leds += 4;
if(num[j] == '5') leds += 5;
if(num[j] == '6') leds += 6;
if(num[j] == '7') leds += 3;
if(num[j] == '8') leds += 7;
if(num[j] == '9') leds += 6;
if(num[j] == '0') leds += 6;
j++;
}
printf("%lld ledsn", leds);
}
return 0;
}
```
N.B.: Code is Collected from Different Sources
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https://math.stackexchange.com/questions/1592140/when-the-product-of-dice-rolls-yields-a-square/1592943
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# When the product of dice rolls yields a square
Succinct Question:
Suppose you roll a fair six-sided die $$n$$ times.
What is the probability that the product of the rolls is a square?
Context:
I used this as one question in a course for elementary school teachers when $$n=2$$, and thought the generalization might be a good follow-up question for secondary school math teachers. But I encountered quite a bit of difficulty in tackling it, and I am wondering if there is a neater solution than what I have already seen, and to what deeper phemonena it connects.
Known:
Since the six sides of a die are $$1, 2, 3, 2^2, 5,$$ and $$2\cdot3$$, the product of the rolls is always of the form $$2^{A}3^{B}5^{C}$$, and the question is now transformed into the probability that $$A, B, C$$ are all even. The actual "probability" component is mostly for ease of phrasing; its only contribution is a $$6^n$$ in the denominator, and my true question is of a more combinatorial nature: namely,
In how many ways can the product of $$n$$ rolls yield a square?
One approach that I have seen involves first creating an $$8 \times 8$$ matrix corresponding to the eight cases around the parity of $$A, B, C$$; one can then take the dot product of each roll with this matrix, and hope to spot a pattern. In this way, one may discover the formula:
$$\frac{6^n + 4^n + 3\cdot2^n}{8}$$
and the "probability" version is simply this formula with another $$6^n$$ multiplied in the denominator.
As for proving this: Some guesswork around linear combinations of the numerator yields a formula for each of the eight cases concerning $$A,B,C$$ parity, and one can then prove all eight of them by induction. And so I "know" the answer in the sense that I have all eight of the formulae (and the particular one listed above is correct) but they were not found in a particularly organized fashion.
My Actual Question:
What is a systematic way to deduce the formula, given above, for the number of ways the product of $$n$$ rolls yields a square, and to what deeper phenomena does this connect?
• Re "hope to spot a pattern. In this way, one may discover the formula". The eigenvectors from the matrix (6,4,2,2,2,0,0,0) tell you the general formula is $A6^n + B4^n + C2^n+ Dn2^n + En^22^n$. It's tedious by hand calculation, but you can systematically find the value of the constants $A,B,C,D,E$ by substituting in the first few known values for $n=0,1,2,3,4$. Commented Dec 29, 2015 at 4:07
You can bypass the cumbersome transition matrix method in this problem, and get a quicker explicit solution using generating functions and some algebra tricks.
Your goal is to count the number of 2s, 3s, and 5s that occur in the prime factorization $(2^A 3^B 5^C)$ of the product of N rolls of a die. The exponents of these primes increase in an additive fashion with each roll. We can model this by treating the exponents $(A,B,C)$ as position vectors on a three-dimensional lattice.
Step 1. Each outcome of each roll of the die has a prime factorization, and has the incremental effect of shifting your position on the lattice by a displacement vector. Below we tabulate the possible outcomes of a roll of a die, and the exponents that occur in the prime factorization of the number rolled).
1: (0,0,0) 2:(1,0,0) 3:(0,1,0) 4:(2,0,0) 5:(0,0,1) 6:(1,1,0)
Since we want to keep track only of the parity of these exponents we can regard (2,0,0) as equivalent to (0,0,0). Note that there are now two ways to roll this null displacement vector, and one way to roll the other displacement vectors.
Step 2. We now use algebraic expressions as a notational device for modeling a displacement on a lattice. To each of the outcomes listed above we use its prime factorization to construct an associated algebraic expression. (Admittedly it seems like we are chasing our tails, toggling back and forth between additive and multiplicative descriptions of numbers, but be patient. ) :)
$1\to 2^0 3^0 5^0 \to a^0 b^ 0 c^0$ (null displacement on exponent lattice)
$2\to 2^1 3^0 5^0 \to a^1 b^0 c^0 = a$
$\ldots$
$4\to 2^2 3^0 5^0 \to a^2 \to a^0$ (null displacement) because in our case we care only about parity on the exponent lattice
$\ldots$
$6\to a*b$
Step 3. Now we are ready to introduce generating functions! To account for all six outcomes of a dice roll, set $F(a,b,c)= 2+ a + b + a*b + c$ which correctly double-counts the null outcome. The magic of generating functions as a book-keeping device is that for example the 3rd power of the multivariable polynomial $F(a,b,c)$ tells you the number of ways that three rolls of the die can add up to a given displacement vector on the exponent lattice. That is, the expansion of the third power $F(a,b,c)^3=8 + 12 a + 6 a^2 + a^3 + 12 b + 24 a b + 15 a^2 b + 3 a^3 b + 6 b^2 + 15 a b^2 + 12 a^2 b^2 + 3 a^3 b^2 + b^3 + 3 a b^3 + 3 a^2 b^3 + a^3 b^3 + 12 c + 12 a c + 3 a^2 c + 12 b c + 18 a b c + 6 a^2 b c + 3 b^2 c + 6 a b^2 c + 3 a^2 b^2 c + 6 c^2 + 3 a c^2 + 3 b c^2 + 3 a b c^2 + c^3$
tells you that in three rolls there are $6$ ways to get the algebraic expression $a^2 b c$ (which corresponds to the displacement vector $(2,1,1)$ on the exponent lattice, which in turn corresponds to increasing the cumulative product of numbers rolled by the factor $2^2 \times 3 \times 5$.)
Now, since you want to keep track of the general $n_{th}$ power of $F(a,b,c)$ you can introduce the geometric series $g(a,b,c;t)= \frac{1}{ 1- t F} = 1+ t F+ t^2 F^2 + t^3 F^3 \ldots$ whose expansion in powers of $t$ consists of the powers of $F$.
Step 4. Recall that you want to lump together all lattice positions that have the same parity mod 2. A sneaky trick for accomplishing that "topological identification" of congruent lattice points is to symmetrize the function $g$ to make it even in each of the variables $(a,b,c)$, so that only even powers of $a,b,c$ occur in its Taylor expansion. This symmetrized function is an eight-term expression $S(a,b,c;t)= \frac{1}{8}[g(a,b,c;t)+ g(-a,b,c;t)+ g(a,b,c) + g( a,-b, c) +\ldots]$
In principle we could expand $8S(a,b,c;t)$ out as a Taylor series: $8 + 16 t + 8 (4 + a^2 + b^2 + a^2 b^2 + c^2) t^2 + (64 + 48 a^2 + 48 b^2 + 96 a^2 b^2 + 48 c^2) t^3 +\ldots$
but actually all we care about is the coefficient-weighted total number of different terms that occur for each power of $t$.
Step 5. This latter can be found by the simple trick of replacing $a=1, b=1, c=1$ which is a massive simplification. So go back and do that from the very beginning, where $g$ is introduced, repeat the symmetrization process, insert the numerical values $a=1,b=1,c=1$ and collect terms in an expression that now depends only on $t$. We see that $8 S(a,b,c;t) =3 + 1/(1 - 6 t) + 1/(1 - 4 t) + 3/(1 - 2 t)$.
Step 6. Now at last it is clear (by expanding the last expression in its geometric series) why you get the answer $\frac{6^n + 4^n + 3* 2^n}{8}$.
• Excellent. Do you have a recommended reference for problem solving with generating functions? (I would be especially interested in seeing your Steps 4 and 5 applied similarly to other problems...) Commented Dec 30, 2015 at 2:24
• @BenjaminDickman: H. Wilf's Generatingfunctionology is a great starter. You might want to have a look at this answer. Commented Dec 31, 2015 at 17:13
For $1\le i\le6,\;$ let $a_i$ be the number of dice which have the digit $i$ appearing.
The product of the rolls will be a perfect square when $a_2+a_6,\;$ $a_3+a_6,\;$ and $a_5$ are all even;
so we can consider two cases:
$\textbf{1)}$ When $a_2, a_3, a_6$ are all odd, we get the exponential generating function
$\;\;\;\displaystyle\underbrace{\big(1+x+\frac{x^2}{2!}+\cdots\big)^2}_{a_1, a_4}\underbrace{\big(x+\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots\big)^3}_{a_2, a_3, a_6}\underbrace{\big(1+\frac{x^2}{2!}+\frac{x^4}{4!}+\cdots\big)}_{a_5}$
$\;\;\;\displaystyle=e^{2x}\left(\frac{e^x-e^{-x}}{2}\right)^3\left(\frac{e^x+e^{-x}}{2}\right)=\color{red}{\frac{1}{16}\big(e^{6x}-2e^{4x}-e^{-2x}+2\big)}$
$\textbf{2)}$ When $a_2, a_3, a_6$ are all even, we get the exponential generating function
$\;\;\;\displaystyle\underbrace{\big(1+x+\frac{x^2}{2!}+\cdots\big)^2}_{a_1,a_4}\underbrace{\big(1+\frac{x^2}{2!}+\frac{x^4}{4!}+\cdots\big)^4}_{a_2,a_3, a_5, a_6}$
$\;\;\;\displaystyle=e^{2x}\left(\frac{e^x+e^{-x}}{2}\right)^4=\color{red}{\frac{1}{16}\big(e^{6x}+4e^{4x}+6e^{2x}+e^{-2x}+4\big)}$
Adding the two cases gives the generating function
$\;\;\;\displaystyle g_e(x)=\frac{1}{16}\big[2e^{6x}+2e^{4x}+6e^{2x}+6\big]=\color{red}{\frac{1}{8}\big[e^{6x}+e^{4x}+3e^{2x}+3\big]}$
$\hspace{.3 in}\displaystyle=1+\frac{1}{8}\sum_{n=1}^{\infty}\left(6^n+4^n+3\cdot2^n\right)\frac{x^n}{n!},\;\;$ so there are
$\displaystyle \hspace{.5 in}\color{blue}{\frac{1}{8}\big(6^n+4^n+3\cdot2^n\big)}$ ways to roll $n$ dice and get a product which is a perfect square.
• @BenjaminDickman Thanks for your comments; I did have a missing exponent in 1), and I have added a little more detail. As you say, the series I'm using are the Maclaurin series for the hyperbolic sine and cosine. Commented Jan 3, 2016 at 0:47
• @BenjaminDickman I haven't seen a way to solve a problem like this directly using hyperbolic functions (but that doesn't necessarily mean that such a method doesn't exist). Commented Jan 3, 2016 at 1:43
• @user84413: +1 for your contribution, which is clearly an enrichment to the given answers so far! Commented Jan 3, 2016 at 8:20
• @BenjaminDickman: Note, that $\exp z, \sinh z$ and $\cosh z$ are the characteristic EGFs of $\mathbb{Z}_{\geq 0},2\mathbb{Z}_{\geq 0}+1$ and $2\mathbb{Z}_{\geq 0}$ respectively. So, the occurrence of $\sinh z$ and $\cosh z$ seems to be quite natural whenever parity matters. An instructive table, Comtet's square, can be found as Example II.7 in Flajolets classic Analytic Combinatorics. He refers to an example in Comtet's classic and Comtet presents there a nice universal generating function. :-) Commented Jan 3, 2016 at 10:53
• @BenjaminDickman: One famous occurrence of $\sinh z$ is in Rademachers exact formula for the number of integer partitions (see VIII.22 in Flajolet's book). It could be interesting to study the proof with your question in mind. ... Commented Jan 3, 2016 at 10:59
Note: This answer can be regarded as supplement to the nice answer of @MathWonk. Here we put a strong focus on generating functions.
Intro: A typical representation of one roll of a six-sided die is given by \begin{align*} x^1+x^2+x^3+x^4+x^5+x^6 \end{align*} The exponents of $x$ represent the pips of the die, the coefficients the number of occurrences of the respective event. Since we want to count the number of squares in $n$ rolls, we also keep track of the prime factors $2,3$ and $5$ which occur in the numbers $1,\ldots,6$. We use the variables $a,b$ and $c$ to mark the number of these prime factors. We obtain the generating function \begin{align*} x+ax^2+bx^3+a^2x^4+cx^5+abx^6 \end{align*} The variable $a$ represents the occurrence of $2$, $b$ represents $3$ and $c$ the prime $5$. Since the number $6=2\cdot3$ we count the prime factor $2$ and $3$ by multiplying the term $x^6$ with $a b$. This is similarly done for all other faces of the die.
We also want to keep track of the number of rolls, so we introduce a variable $t$ and multiply each term with it. This way we can define as basic building block the generating function \begin{align*} A(a,b,c;t;x)=(x+ax^2+bx^3+a^2x^4+cx^5+abx^6)t \end{align*} A generating function representing $n\geq 1$ rolls is \begin{align*} A_n(a,b,c;t;x)&:= \left(A(a,b,c;t;x)\right)^n\\ &=(x^1+ax^2+bx^3+a^2x^4+cx^5+abx^6)^nt^n \end{align*}
In fact these are only introductory notes, giving some background knowledge. We can instead start with
Main part: Let $A_n(a,b,c;t;x)$ be a generating function of a six-sided die representing $n$ rolls, which keeps track of the prime factors $2,3$ and $5$ of the pips and the number of rolls. It is given for $n\geq 1$ by \begin{align*} A_n(a,b,c;t;x)&=(x^1+ax^2+bx^3+a^2x^4+cx^5+abx^6)^nt^n\\ &=t^n\sum_{{i_1+i_2+\ldots+i_6=n}\atop{i_j\geq 0,1\leq j \leq 6}}\binom{n}{i_1,i_2,\ldots,i_6} x^{i_1+2i_2+\ldots+6i_6}a^{i_2+2i_4+i_6}b^{i_3}c^{i_5}\tag{1} \end{align*} with $\binom{n}{i_1,i_2,\ldots,i_6}=\frac{n!}{i_1!i_2!\cdots i_6!}$ the multinomial coefficients.
Since we want to count the rolls giving square numbers we are looking for a generating function $B_n(a,b,c;t;x)$, which is based upon $A_n(a,b,c;t;x)$ but additionally fulfills, that the exponents of $a,b$ and $c$ are even. In fact, this was the rationale for introducing these variables.
In order to obtain even exponents of $a,b$ and $c$ we need according to the representation in (1)
\begin{align*} i_2+2i_4+i_6&\equiv 0(2)\\ i_3&\equiv 0(2)\tag{2}\\ i_5&\equiv 0(2) \end{align*}
Now recall, that each function $f(x)$ can be represented as sum of an even and odd function via \begin{align*} f(x)&=f_e(x)+f_o(x)\\ &=\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2} \end{align*} The even part $G_e(x)$ of a generating function $G(x)=\sum_{n=0}^{\infty}g_nx^n$ contains even powers of $x$ only, since \begin{align*} G_e(x)=\frac{G(x)+G(-x)}{2}=\sum_{n=0}^{\infty}g_{2n}x^{2n} \end{align*}
We need according to (2) an even generating function in the variables $a,b$ and $c$ which leads to \begin{align*} B_n(a,b,c;t;x)=\frac{1}{8}&\left(A_n(a,b,c;t;x)+A_n(-a,b,c;t;x)\right.\\ &+A_n(a,-b,c;t;x)+A_n(-a,-b,c;t;x)\\ &+A_n(a,b,-c;t;x)+A_n(-a,b,-c;t;x)\tag{3}\\ &\left.+A_n(a,-b,-c;t;x)+A_n(-a,-b,-c;t;x)\right)\\ \end{align*}
Note, we need the variables $a,b$ and $c$ for the derivation of the appropriate generating function $B_n(a,b,c;t;x)$. We don't need the variables to count the number of occurrences of squares. We also don't need to differentiate the pips, so we also don't need $x$ any longer.
We simply need the variable $t$ which counts the number of rolls and we want to add up all terms for a specific $t^n$. This way we count all occurrences of squares in $n$ rolls.
We obtain: The generating function $C_n(t)$ representing all occurrences of squares when rolling a die $n$ times is $(n\geq 1):$ \begin{align*} C_n(t)&=B_n(1,1,1;t;1)\\ &=\frac{1}{8}\left(A_n(1,1,1;t;1)+A_n(-1,1,1;t;1)\right.\\ &\qquad+A_n(1,-1,1;t;1)+A_n(-1,-1,1;t;1)\\ &\qquad+A_n(1,1,-1;t;1)+A_n(-1,1,-1;t;1)\\ &\qquad\left.+A_n(1,-1,-1;t;1)+A_n(-1,-1,-1;t;1)\right)\\ &=\frac{1}{8}\left((6t)^n+(2t)^n+(2t)^n+(2t)^n+(4t)^n+0+0+0\right)\\ &=\frac{1}{8}\left(6^n+4^n+3\cdot2^n\right)t^n \end{align*}
Note, it's convenient to use the coefficient of operator $[x^n]$ to denote the coefficient of $x^n$ of a series.
We conclude, the number of occurrences of squares when rolling a die $n$ times and multiplying the resulting pips is the coefficient of $t^n$ of the generating function $C_n(t)$ \begin{align*} [t^n]C_n(t)=\frac{1}{8}\left(6^n+4^n+3\cdot2^n\right)\qquad\qquad n\geq 1 \end{align*}
There's a slick approach to this based on bijections, though it loses a lot of the generality of the generating function methods.
Let $$S=\{1,2,3,6\}$$ and $$T=\{4,5\}$$. We will divide roll sequences into classes based on whether the sequence contains elements from $$S, T,$$ or both.
Class 1: Sequences consisting only of rolls in $$T$$. Swapping the first die roll between $$4$$ and $$5$$ gives a bijection between squares and non-squares, so exactly half the sequences in this class give a square.
Class 2: Sequences consisting only of rolls in $$S$$. Now we can divide the sequences into groups of $$4$$ that share the same last $$n-1$$ rolls. Each group has one square product, so exactly $$1/4$$ the sequences in this class give a square.
Class 3: Sequences containing both a roll in $$S$$ and a roll in $$T$$. To each sequence we assign a "type", consisting of (1): The location of the first roll in $$S$$, (2): the location of the first roll in $$T$$, and (3): The remaining $$n-2$$ rolls. Once the type is fixed, there's $$8$$ choices for the remaining rolls, and exactly one of them gives a square.
So the number of square sequences is $$\frac{1}{2} |\textrm{Class } 1| + \frac{1}{4} |\textrm{Class } 2| + \frac{1}{8} |\textrm{Class } 3| = \frac{1}{2} 2^n + \frac{1}{4} 4^n + \frac{1}{8} (6^n-2^n-4^n)$$
• In Class 3 part (3) what does "the remaining roll" refer to? Commented Jan 6, 2016 at 16:28
• @BenjaminDickman: I believe that was a typo; I changed it to plural, as I think it’s meant to refer to the two rolls whose location is referred to in $(1)$ and $(2)$ (there being $4$ choices for the former and $2$ for the latter). Commented Feb 26 at 10:09
Using diagonalization on Wolfram Alpha, I was able to confirm your result. I used the matrix
$$M=\left(\begin{array}{cccccccc} 2&1&1&1&1&0&0&0\\ 1&2&1&0&1&1&0&0\\ 1&1&2&0&1&0&1&0 \\ 1&0&0&2&0&1&1&1 \\ 1&1&1&0&2&0&0&1\\ 0&1&0&1&0&2&1&1\\ 0&0&1&1&0&1&2&1\\ 0&0&0&1&1&1&1&2 \end{array} \right)$$ which gives one step transitions between square root parts of products (no square root part , $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $\sqrt{6}$, $\sqrt{10}$, $\sqrt{15}$, $\sqrt{30}$ respectively).
For the number you require, you want the first entry of $M^n\ \vec{b}$ where $\vec{b}=\left(\begin{array}{c}1\\0\\0\\0\\0\\0\\0\\0\end{array}\right)$
As mentioned above, using diagonalization with Wolfram Alpha (the diagonalizing matrix was actually quite nice--integer entries; the inverse had denominators of eighths), I was able to confirm your result.
I don't know if this method is an improvement over your description. I hope it helps!
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### Math 113 Fall 2017
Final Exams Office Hours
8 o'clock class: Wednesday, Dec. 13 at 8:00 a.m. in Ayres 120.
9:05 class: Monday, Dec. 11 at 8:00 a.m. in Ayres 120.
If you want to switch your final exam time to the other class,
you must let me know in advance. If there is room, you may switch.
9:10 - 10:00 MWF in HSS (TBD)
1:00 - 3:00 T in 226 Ayres and by appointment.
You are welcome to drop by anytime.
If I am not busy, I'll be happy to talk to you.
#### Math Department Syllabus
Class Assignments The homework solutions.
Please remember to do the TN Voice evaluations. I especially appreciate any comments you may have. Thanks.
#### Russian Alphabet equivalences.
The Hotel Hilbert (thanks to Jacob Z.)
The homework score will be based on a total of 250 points. The maximum possible will be 100% though. The homework is 20% of the grade.
To calculate your homework grade: Add up the three homework scores listed on your test. Also, add 10 pts. for each of the 5 group projects done and for the public key encryption - entering a key (10 pts.) and decrypting a message (10 pts.) There were over 310 total points.
#### Test 3: Friday, November 10
• Know the Pythagorean Theorem; be able to state it (in English!). Given the lengths of two sides of a right triangle find the length of the other side. Know how to tell if a triangle is a right triangle. How does the "puzzle proof" with the 4 triangles and the square show that the Pythagorean Theorem is true?
• State the Art Gallery Theorem completely accurately and in your own words and know how to apply it. Be able to choose the best-suited points to put the cameras in an art gallery. Be able to divide the museum into triangles and label the vertices. Be able to do problems like these.
Sketch a gallery with 12 vertices which needs 4 cameras or 12 vertices which needs only 3 cameras. How many cameras will you need for a gallery with 8 vertices? Can you draw a gallery with 12 vertices which needs 5 cameras? Why or why not?
• Sketch a Golden Rectangle. Given any rectangle, determine whether or not it is a Golden Rectangle. What happens if you start with a Golden Rectangle and remove the largest square from it? Given the base (long side) of a Golden Rectangle, find its height (short side). Given the height (short side) of a Golden Rectangle, find its base (long side). Be able to find the area of a Golden Rectangle given either its base or its height. Be able to draw the Golden Spiral and find its center.
• Understand tessellations, symmetry, rigid symmetry and symmetry of scale.
• Be able to draw a triangle for the basic building block of the Pinwheel Pattern and draw supertiles of the Pinwheel Pattern from five smaller tiles. Given a diagram of the Pinwheel Pattern, outline a five-tile supertile and a 25-tile supersupertile.
• Know all about the Platonic solids - their names and various numbers of edges, vertices and faces and their duals.
Due Monday November 6.
#### Test 2: Redo
You may redo one problem from test 2 on separate paper and hande it in along with the test on Friday, October 20. You can come to office hours to pick up your test or get help.
#### Test 2: Monday, October 16.
• Know Fermat's little theorem and how it relates to raising numbers to high powers in modular arithmetic.
• Know public key encryption
• Understand the various sets of numbers and their properties.
• Natural numbers, real numbers, irrational numbers, rational numbers
• Decimal expansions
• Be able to prove that a number is irrational
• Understand 1-1 correspondence and cardinality and how to show infinite sets have the same cardinality.
• Know who Georg Cantor is and how his diagonalization proof works and what it shows.
#### Test 1: Monday, September 18.
There will be about 10 problems (max)
• Be able to solve problems similar to those in chapter 1 or discussed in class.
• Be able to estimate numbers reasonably and understand and explain the pigeonhole principle.
• You should know how to find Fibbonaci numbers and their notation.
• Understand the strategy for Fibonacci nim.
• You should know what a prime number is and how to factor an integer.
• You should understand the proof that there are infinitely many prime numbers.
• Know how to compute with modular arithmetic, for example, to verify UPC codes.
Look at the suggested problems listed in the course schedules.
The problems handed out in class on the first two days.
#### Course Policies
Text: The Heart of Mathematics, by Burger and Starbird (4th edition. We will roughly cover chapters one through four with some additional topics. There is an ebook available. See the book's web page.
Tests: There will be three hour tests, plus a comprehensive final exam.
Homework and In-class assignments: There will be about 15 assignments to be handed in throughout the term. You may collaborate on problems, but must write up your own solutions. It is important to keep up with the assignments. The best way to learn math is to struggle with lots of problems. There will also be some assignments done and handed in during class. Not all of the available points will be counted, so anyone who works faithfully at them will get a good assignments grade.
Class assignments.
Grades: The 3 tests, the assignments and the final exam will each count 20% of the grade. Grades will be computed on the math departments scale: 90-100% A, 87 - 89% A-, 83 - 86 B+, 80-82% B, 77 - 79% B-, etc. Some consideration will be given to steady improvement throughout the term; of course consideration will also be given to a steady decline throughout the term.
Calculator: You should have a scientific calculator for this course. For example, any TI graphing calculator (the TI - 81 through 86) or Sharp or Casio is fine. Any calculator with an exponentiation key is fine, but not one built into your phone or with an internet connection. You will always be allowed to use a calculator on tests. If you don't have one, find the nearest pawn shop.
Make Up Policy: If you must miss due to a university sanctioned meeting let me know ahead of time. If you have an emergency, please notify me as soon as possible (email is best) to schedule a make-up.
Last update: December 31, 2018, 12:26 pm
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# SBI Clerk Reasoning Previous Year
Questions
1. If ‘1’ is added to the first digit of every number and ‘1’ is subtracted from the
third digit of I every number, in how many numbers thus formed will the
difference between first and third digits be more than 5 ?
a. Two
b. None
c. Four
d. Three
e. One
Solution : 5
2. If all the numbers are arranged in ascending order from left to right,
which of the following will be resultant if first and third digits of the number
which is second from the right are multiplied ?
a. 18
b. 14
c. 40
d. 24
e. 32
Solution : 2
3. If ‘2’ is added to the second digit of every even number and ‘1’ is subtracted
from the first digit of every odd number, in how many numbers will a digit
appear twice ?
a. Three
b. Two
1|Page
c. Four
d. One
e. None
Solution : 1
4. If in each number all the digits are arranged in descending order within the
number, how many numbers thus formed will be odd numbers ?
a. One
b. Four
c. None
d. Three
e. Two
Solution : 4
5. The positions of the first and the third digits of each of the numbers
are interchanged. What will be the resultant if first digit of the highest number
thus formed is divided by the third digit of the lowest number thus formed ?
a. 2
b. 1.5
c. 3
d. 4
e. 3.5
Solution : 4
Directions: Study the following information and answer the questions.
K is the wife of V. V is the brother of J. L is the only daughter of J. D is the
father of M and L. S is the only daughter of M.
6. How is D related to V ?
a. Nephew
b. Brother
c. Brother-in-law
2|Page
d. Son
e. Father-in-law
Solution : 3
7. If S is married to G, how is G related to M ?
a. Daughter
b. Son
c. Nephew
d. Niece
e. Son-in-law
Solution : 5
8. How is K related to L ?
a. Aunt
b. Niece
c. Mother-in-law
d. Mother
e. Sister
Solution : 1
Directions: Study the following information and answer the questions.
Seven people, namely K, L, M, N, 0, P and Q have seven different meetings on
the seven different days of the same week starting from Monday and ending
on Sunday, not necessarily in the same order.
L has a meeting on Thursday. Only two people have a meeting between L and
0. Only three people have a meeting between 0 and N. As many people have a
meeting between N and P as between P and 0. Q has a meeting immediately
before M. Q does not have a meeting on Saturday.
9. Which of the following is not true about K ?
a. K has a meeting immediately after P.
b. All the given statements are true.
3|Page
c. Only one person has a meeting after K.
d. K has a meeting on one of the days after 0.
e. Only three people have a meeting between K and M.
Solution : 4
10. How many people have meeting(s) between the days on which Q and L have
their meetings?
a. More than three
b. Two
c. Three
d. None
e. One
Solution : 2
11. Who amongst the following has a meeting immediately after the one who has
a meeting on Tuesday ?
a. Q
b. K
c. M
d. 0
e. N
Solution : 5
12. On which of the following days does M have a meeting ?
a. Saturday
b. Wednesday
c. Friday
d. Tuesday
e. Sunday
Solution : 4
4|Page
13. Four of the following five are alike in a certain way and thus form a group as
per the given arrangement. Which of the following does not belong to that
group ?
a. P-Saturday
b. Q-Wednesday
c. N-Monday
d. L-Tuesday
Solution : 1
Directions: In these questions, relationship between different elements is
shown in the statements. The statements are followed by conclusions. Study
the conclusions based on the given statements and select the appropriate
e. If both conclusions I and II are true
f. If only conclusion I is true
g. If neither conclusion I nor II is true
h. If only conclusion II is true
i. If either conclusion I or conclusion II is true
Statements :
14. P>L ≤ T = Q ; Z < L ≥ V
Conclusions :
i. Q ≥ V
ii. Z < P
Solution : 1
15. Statement : T< H ≥ A < N ≤ C Conclusions :
i. C > A
ii. N > T
Solution : 2
16. Statement : R= Q ≥ E ≤ S >T Conclusions :
5|Page
i. R > S
ii. Q ≥ T
Solution : 3
Statements :
17. C = O ≤ V=L ≤ R ; S ≥ R Conclusions :
i. S > C
ii. S = C
Solution : 5
18. Statements : E < N = C ≥ L > S Conclusions :
i. L < E
ii. S < N
Solution : 4
Directions: Study the following information carefully and answer the given
questions.
19. B, C, D, E, F, G, H and I are sitting around a circular table facing the centre
but not necessarily in the same order.
1. Only one person sits between D and E B sits to the
immediate right of E
2. Only three people sit between F and H.
3. C sits to the immediate left of G.
4. G is not an immediate neighbour of E
5. E sits second to the left of C.
20. Who amongst the following sits to the immediate left of E?
a. B
b. I
c. D
d. G
e. H
6|Page
Solution : 3 Q. No. 19 – 23
21. What is the position of B with respect to G ?
a. Third to the right
b. Fourth to the left
c. Second to the right
d. Second to the left
e. Third to the left
Solution : 1
22. Four of the following five are alike in a certain way based on their positions in
the given arrangement and so form a group. Which is the one that does not
belong to the group ?
a. EFG
b. DIC
c. CEF
d. GEB
e. HBI
Solution : 3
23. If all the persons are made to sit in alphabetical order in anti- clockwise
direction, starting from B, the positions of how many, excluding B, would
remain unchanged ?
a. Four
b. Two
c. One
d. Three
e. None
Solution : 3
24. Which amongst the following statements is true regarding I, as per the given
arrangement ?
7|Page
a. D sits second to the right of I.
b. I is an immediate neighbour of both G and F.
c. Only two. people sit between I and C.
d. Only four people sit between I and E.
e. All statements are true.
Solution : 2
25. If two is added to the first three digits and three is subtracted from the last
three digits in the number 657489, how many digits in the number thus
formed will be multiples of 3 ?
a. One
b. None
c. Two
d. More than three
e. Three
Solution : 3
26. Among five friends- J, K, L, M and N each having a different number of
pencils. K has more pencils than N but less than L. M has more pencils than L.
J neither has the most nor the least number of pencils. Who amongst the
following has the lowest number of pencils ?
a. Cannot be determined
b. M
c. N
d. L
e. K
Solution : 3
27. In a certain code, if ‘they shouted loud’ is written as ‘4 1 5’ and ‘loud music
heard’ is coded as ‘7 6 1’, then what is the code for ‘ music’ in the given code
language ?
8|Page
a. Either ‘4’ or ‘6’
b. 5
c. 1
d. Either ‘6’ or ‘7’
e. 4
Solution : 4
Directions : Study the given information carefully to answer the given
questions.
Syne who is standing at point H, walks 11m towards east and reaches point R.
She then takes a left turn and walks 7m. She takes a left turn, walks 5m and
reaches point Q. Point G is 6m to the south of point H. Point B is 6m to the
east of point G.
28. How far and in which direction is point Q with respect to point B ?
a. 13 m towards north
b. 9 m towards north
c. 10 m towards south
d. 11 m towards south
e. 12 m towards east
Solution : 1
29. In which direction is point G with respect to point R ?
a. South-west
b. North-west
c. East
d. West
e. South-east
30. How many such pairs of letters are there in the word `INKED’ each of which
has as many letters between them in the word (in both forward and backward
directions) as they have between them in the English alphabetical series ?
9|Page
a. None
b. More than three
c. Three
d. One
e. Two
Solution : 5
Directions: Study the following arrangement carefully and answer the
questions.
DLJ#3P+AWZRE8G2NS=T6&Y9@X4HU5 ^ 7B
31. In a certain code, based on the given arrangement, `HER’ is coded as ^ RG’
and ‘SUN’ is coded as ‘6HT’. How will ‘PAT’ be coded following the same
coding pattern ?
a. J+Y
b. WR=
c. +3Y
d. JWN
e. W+Y
Solution : 5
32. How many such letters are there in the given arrangement, each of which is
immediately preceded by a symbol and also immediately followed by an even
number ?
a. More than three
b. None
c. One
d. Two
e. Three
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33. Four of the following five are alike in a certain way based on their positions in
the given arrangement and hence form a group. Which one does not belong to
that group ?
a. R8Z
b. Y@&
c. 57U
d. 3+#
e. GS2
Solution : 5
34. If all the symbols are deleted from the given arrangement then which of the
following will be ninth element from the right end ?
a. Y
b. R
c. E
d. 6
e. 9
35. 34. Which one of the following will come next in the given sequence ?D#L
PW+ E28 TY6 ?
a. H ^ U
b. H^4
c. 45U
d. H@4
e. X57
Solution: 1
36. 35. In a certain code language, ‘SEAL’ is coded as `NDGV’ and `LION’ is
coded as ‘PRK0’. In the same. code language, ‘HOLD’ will be coded as :
a. FOQK
b. ENRK
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c. EOQR
d. OMSJ
e. FORL
Solution: 1
## Directions (1 – 5): Study the following information carefully and answer
the questions given below:
In a certain code language “dress code for meeting” is written as ‘dk pd jn td’
“wear black formal dress” is written as `pd ro hi te’
“formal meeting this weekend” is written as `yi te dk yr’
“black code this weekend” is written as ‘Jn vr Id yi’
(All the codes are two letter codes)
37. In the given code language what does ‘te’ stands for ?
a. this
b. formal
c. dress
d. black
e. meeting
Solution: 2
38. In the given code language what is the code for ‘dress’?
a. jn
b. ro
c. pd
d. td
e. Id
Solution:3
39. What does ‘ld’ stand for in the given code language?
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a. meeting
b. weekend
c. formal
d. for
e. Other than those given as options
Solution:5
40. Which of the following possibly means ‘security code’ in the given code
language ?
a. ux yr
b. vr tc
c. pd ux
d. jn ux
e. jn pd
Solution:4
41. What is the code for ‘weekend’ in the given code language ?
a. either yi or vr
b. dk
c. te
d. jn
e. Id
Solution:1
Directions (6 – 10) : In each of the following questions, two statements
followed by two conclusions numbered I and II have been given. You have to
take the two given statements to be true even if they seem to be at variance
from commonly known facts and then decide which of the given conclusions
logically follow from the given statements disregarding commonly known
facts.
42. Statements : All drums are flutes. All guitars are drums. Conclusions :
13 | P a g e
i. All guitars are flutes.
ii. All drums are guitars.
b. Both Conclusions I and II follow
c. Only Conclusion I follows
d. Neither Conclusion I nor II follows
e. Only Conclusion II follows
f. Either Conclusion I or II follows
Solution:2
43. Statements : Some speakers are guests. No guest is a dignitary. Conclusion :
• No dignitary is a speaker.
• Some guests are definitely not speakers.
a. Only Conclusion II follows
b. Both Conclusions I and II follow
c. Either Conclusion I or II follows
d. Neither Conclusion I nor II follows
e. Only Conclusion I follows
Solution:4
44. Statements: All spades are tools. No spade is a vessel. Conclusions :
i. Atleast some spades are vessels.
ii. No spade is a vessel.
b. Either Conclusion I or II follows
c. Neither Conclusion I nor H follows
d. Only Conclusion II follows
e. Both Conclusions I and II follow
f. Only Conclusion I follows
Solution:3
45. Statements : All users are senders. Some users are machines. Conclusions :
I. Some senders are users.
14 | P a g e
II.Some machines are senders.
a. Both Conclusions I and II follow
b. Only Conclusion I follows
c. Only Conclusion II follows
d. Either Conclusion I or II follows
e. Neither Conclusion I nor II follows
Solution:1
46. Statements : Some cooks are employees. Some employees are workers.
Conclusions :
I. All cooks are workers.
II. No cook is a worker.
a. Either Conclusion I or II follows
b. Only Conclusion II follows
c. Only Conclusion I follows
d. Both Conclusions I and II follow
e. Neither Conclusion I nor II. Follows
Solution:5
Directions (11 – 15) : In each of the following questions, relationship
between different elements is shown in the statements. The statements are
followed by two Conclusions numbered I and II. Study the Conclusions based
on the given statement and select the appropriate answer.
47. Statements: S=A ≤ P > R ≤ T<I3 Conclusions :
i. U > A
ii. S ≤ T
b. Both Conclusions are true
c. Only Conclusion I is true
d. Only Conclusion II is true
e. Neither Conclusion I nor II is true
15 | P a g e
f. Either Conclusion I or III is true
Solution:4
48. (12 — 13) : Statements :
P ≥ Q < R = M; R ≥ F
Conclusions :
i. M ≥ F
ii. F > P
49. Conclusions :
i. M ≥ F
ii. F > P
Solution:2
50. Conclusions :
i. F < Q
ii. P ≥ M
b. Only Conclusion I is true
c. Both Conclusions I and II are true
d. Neither Conclusion I nor II is tree
e. Either Conclusion I or II is true
f. Only Conclusion II is true
Solution:3
51. Statements : T ≥ O =I ≥ L = D Conclusions :
i. D < T
ii. D = T
b. Both Conclusions I and II are true
c. Neither Conclusion I nor II is true
d. Either Conclusion I or II is true
e. Only Conclusion I is true
f. Only Conclusion II is true
16 | P a g e
Solution:3
52. Statements :
J<C=T≥U≥R=N
Conclusions :
i. N < C
ii. U > J
b. Both Conclusions I and II are true
c. Neither Conclusion I nor II is true
d. Only Conclusion I is true
e. Only Conclusion II is true
f. Either Conclusion I or II true
Solution:2
53. If it is possible to make only one meaningful word with the first, fourth, sixth
and eighth letters of the word ‘SENTENCE’, which could be the second letter
of the word from the right end? If more than one such word can be formed
give ‘X’ as the answer. If no such word can be formed give `Z’ as your answer.
a. N
b. T
c. Z
d. X
e. S
Solution:4
54. The positions of how many digits will remain the same if the digits in the
number 297345618 are rearranged in the descending order within the
number, from left to right?
a. None
b. Two
c. More than three
17 | P a g e
d. Three
e. One
Solution:5
55. How many such pairs of letters are there in the word CLAYED, each of which
has as many letters, between them in the word (in both forward and backward
directions) as they have between them in the English alphabetical series ?
a. None
b. One
c. Two
d. Three
e. More than three
Solution:4
Directions: Study the following information carefully and answer the
questions given below :
Eight persons C, D, E, F, G, H, I and J are seated in the straight line facing
north, but not necessarily in the same order. H sits fifth to tl< left. of D D does
not sits at ony of the extreme ends of the line. Only two persons sit between H
en,- I. E sits third to the right of J is not an immediate neighbour of H. C is an
immediate neighbour of G. G is not an immediate neighbour of I.
56. Which of the following represents persons seated at the two extreme ends of
the line?
a. H, E
b. I, F
c. C, E
d. G, J
e. F, G
Solution:1
18 | P a g e
57. Which of the following statement is true with respect to F as per the given
arrangement ?
a. Only one person sits between F and J
b. Both I and D are immediate neighbours of F
c. None of the given options is true
d. Only two people sit to the right of F
e. F sits second to the left of E
Solution:1
58. If all the given people are made to sit in alphabetical order from left to right,
the positions of how many of them will remain unchanged?
a. Two
b. Three
c. Four
d. None
e. One
Solution:4
59. What is the position of J with respect to H?
a. Fifth to the right
b. Third to the right
c. Second to the right
d. Second to the left
e. Fourth to the right
Solution:5
60. How many persons are seated between G and F ?
a. One
b. Five
c. Four
d. Three
19 | P a g e
e. Two
Solution:3
61. Who amongst the following sits exactly in the middle of the persons who sit
fourth from the left and third from the right?
a. D
b. J
c. F
d. I
e. G
Solution:2
Directions: Study the following information carefully and answer the
questions given below :
Eight persons- M, N, O, P, W, X, Y and Z are sitting around a circular table
facing the centre, with equal distances between each other but not necessarily
in the same order.
M sits second to the left of O. P is an immediate neighbour of O and W. Only
three persons sit between 0 and Y. X sits to the immediate right of W. Z is
neither an immediate neighbour of W nor O.
What will come in the place of question mark (?) in the following series based
on the given arrangement ?
62. MZ PW YN NY ?
a. WF
b. XN
c. WX
d. XY
e. WY
Solution:1
63. Which of the following pair represents the immediate neighbours of M ?
20 | P a g e
a. X, W
b. P, Y
c. Y, Z
d. Z, W
e. Z, N
Solution:5
64. Which of the following statements is true as per the given arrangement?
a. X is an immediate neighbour of Y
b. Z sits second to the right of W
c. Y sits to the immediate left of M
d. P sits second to the right of X
e. None of the given options is true
Solution:1
65. How many persons sit between M and W as per the given arrangement ?
a. None
b. Four
c. Three
d. Two
e. One
Solution:3
66. Who amongst the following sits third to the left of P ?
a. Y
b. M
c. O
d. Z
e. N
Solution:2
21 | P a g e
67. Four of the following five are alike in a certain way and so form a group.
Which of the following does not belong to the group ?
a. O,M
b. W, O
c. N,Z
d. N, X
e. Y, W
Solution:4
68. Directions (31 – 35): Study the following arrangement carefully and answer
the questions given below :
3 9 2 4 7 5 9 2 8 1 4 9 5 3 1 6 5 7 3 4 2 9 8 1
3
6 2 8 1 7 5 4 5
69. How many such 2s are there in the given arrangement, each of which is
immediately followed by a perfect square?
a. More than three
b. Three
c. One
d. Two
e. None
Solution:4
70. If all the even digits are deleted from the given arrangement, which of the
following will be tenth from the right end of the arrangement ?
a. 3
b. 1
c. 5
d. 7
e. 9
22 | P a g e
Solution:3
71. How many such 4s are there in the given arrangement, each of which is
immediately followed by a digit which has a numerical value of more than
four ?
a. More than three
b. One
c. None
d. Two
e. Three
Solution:3
72. 24. Who amongst the following sits exactly in the middle of the persons who
sit fourth from the left and third from the right?
a. D
b. J
c. F
d. I
e. G
Solution:2
Directions: Study the following information carefully and answer the
questions given below :
Eight persons- M, N, O, P, W, X, Y and Z are sitting around a circular table
facing the centre, with equal distances between each other but not necessarily
in the same order.
M sits second to the left of O. P is an immediate neighbour of O and W. Only
three persons sit between 0 and Y. X sits to the immediate right of W. Z is
neither an immediate neighbour of W nor O.
73. What will come in the place of question mark (?) in the following series based
on the given arrangement ?
23 | P a g e
MZ PW YN NY ?
a. WF
b. XN
c. WX
d. XY
e. WY
Solution:1
74. Which of the following pair represents the immediate neighbours of M ?
a. X, W
b. P, Y
c. Y, Z
d. Z, W
e. Z, N
Solution:5
75. Which of the following statements is true as per the given arrangement ?
a. X is an immediate neighbour of Y
b. Z sits second to the right of W
c. Y sits to the immediate left of M
d. P sits second to the right of X
e. None of the given options is true
Solution: 1
76. How many persons sit between M and W as per the given arrangement ?
a. None
b. Four
c. Three
d. Two
e. One
Solution:3
24 | P a g e
77. Who amongst the following sits third to the left of P ?
a. Y
b. M
c. O
d. Z
e. N
Solution:2
78. Four of the following five are alike in a certain way and so form a group.
Which of the following does not belong to the group ?
a. O,M
b. W, O
c. N,Z
d. N, X
e. Y, W
Solution:4
Directions: To answer these questions study carefully the following
arrangement of letters, digits and symbols.
79. How many such letters are there in the arrangement each of which is
immediately followed by a number ?
a. Three
b. Four
c. One
d. Two
e. None of these
Solution : 1
80. How many such symbols are there in the arrangement each of which is
immediately preceded by a number ?
25 | P a g e
a. Two
b. Three
c. Four
d. Nil
e. None of these
Solution : 2
81. If all the symbols are deleted from the arrangement, then which of the
following will be fourth to the left of the 17th element from the left end ?
a. 9
b. E
c. 2
d. Y
e. None of these
Solution : 2
82. ’78’ is to ‘P?6’ and ‘?N’ is to ‘T32’ in the same way as ‘2E’ is to………….in the
arrangement.
b. 49G
d. 9GH
e. None of these
Solution : 3
83. If all the numbers are deleted from the arrangement then which of the
following will be fifth to the right of the 13th element from the right end ?
a. B
b. N
c. Y
d. T
26 | P a g e
e. None of these
Solution : 4
84. Directions (06-10) : In these questions, a relationship between different
elements is shown in the statements. The statements are followed by two
conclusions.
a. If only conclusion I is true
b. If only conclusion II is true
c. If either conclusion I or II is true
d. If neither conclusion I nor II is true
e. If both conclusions I and II are true
85. Statements : A>B>C<D, C = E > G Conclusions :
i. D > E
ii. B > E
Solution : 5
86. Statements : P>Q>M>N,Q=S Conclusions :
i. S > P
ii. N < S
Solution : 2
87. Statement : S>M=Z>T<Q>V Conclusions :
i. V = S
ii. Q >M
Solution : 4
88. Statement: T< U=V< S > P>Q Conclusions :
i. S > T
ii. V < Q
Solution : 1
89. Statements : M<N>R>W, E=J>L>W Conclusions :
27 | P a g e
i. E > W
ii. M > L
Solution : 1
Directions: The following questions are based on the five three-digit
numbers given below. 684 512 437 385 296
90. If 2 is added to the first digit of each of the numbers, how many numbers thus
formed will be divisible by three ?
a. None
b. One
c. Two
d. Three
e. None of these
Solution : 2
91. If all the digits in each of the numbers are arranged in descending order
within the numbers, which of the following will be the highest number in the
new arrangement of numbers ?
a. 684
b. 385
c. 296
d. 437.
e. None of these
Solution : 3
92. What will be the resultant number if the second digit of the second lowest
number is divided by the third digit of the highest number ?
a. 2
b. 3
c. 0
d. 1
28 | P a g e
e. 4
Solution : 1
93. If 1 is added to the first digit and 2 is added to the last digit of each of the
numbers then which of the following numbers will be the second highest
number ?
a. 385
b. 684
c. 437
d. 296
e. 512
Solution : 5
94. If in each number the first and second digits are’ interchanged then which will
be the highest number ?
a. 296
b. 512
c. 437
d. 684
e. 385
Solution : 1
Directions: Study the following information carefully and answer the
questions given below.
P is to the North of Q and S is to the East of P, who is to the South of W. T is to
the West of P.
95. Who ‘ among the following is towards South of W and North of Q?
a. P
b. T
c. S
d. Q
29 | P a g e
e. None of these
Solution : 5
96. W is in which direction with respect to T ?
a. North
b. North-East
c. South-West
d. West
e. None of these
Solution : 2
Directions: Study the following information carefully and answer the
questions given below.
Dhondu, Chintu, Titu, Chiku, Sonu, Monu, Bittu and Sonty are sitting around
a circular table facing the centre. Sonty is third to the right of Titu and second
to the left of Sonu. Chintu is not an immediate neighbour of Sonty or Titu.
Monu is second to the right of Chiku and is the immediate neighbour of Titu.
Bittu is not the immediate neighbour of Sonu.
97. Who amongst the following is second to the tight of Titu ?
a. Sonty
b. Bittu
c. Monu
d. Sonu
e. None of these
Solution : 2 Q. No. 18-22
98. Who amongst the following is an immediate neighbour of Sonty and Sonu ?
a. Dhondu
b. Chintu
c. Titu
d. Bittu
30 | P a g e
e. None of these
Solution : 1
99. In which of the following pairs the second person is sitting on the immediate
right of the first person ?
a. Dhondu, Sonty
b. Titu, Chiku
c. Bittu, Sonty
d. Sonu, Sonty
e. Monu, Titu
Solution : 3
100. Who amongst the following is second to the left of Chintu ?
a. Titu
b. Sonty
c. Monu
d. Dhondu
e. None of these
Solution : 4
101. Who amongst the following is opposite to Chiku ?
a. Dhondu
b. Bittu
c. Sonty
d. Sonu
e. None of these
Solution : 3
Directions: Study the following information carefully to answer the given
questions.
Seven neighbours S, P, L, Q, R, M and I live on different floors in the same
building having seven floors numbered one to seven. (The first floor is
31 | P a g e
numbered one, the floor above it, is numbered two and so on till the topmost
floor is numbered as seven.) Three persons live between I and M. M lives on
the floor above S, who does not live on an odd-numbered floor. P neither lives
on an odd numbered nor on topmost floor. R does not live on the first floor.
Two persons live between I and S. Q lives neither on the first floor nor on the
third floor.
102. Who lives on the floor just above M?
a. I
b. P
c. Q
d. R
e. None of these
Solution : 3
103. How many persons live between L and P ?
a. None
b. One
c. Two
d. Three
e. Can’t be determined
Solution : 3
104. Which of the following pairs live on the first floor and the topmost floor
respectively ?
a. L, Q
b. Q, P
c. I, Q
d. L, I
e. Can’t be determined
Solution : 1
32 | P a g e
105. Who amongst the following’lives on the topmost floor ?
a. I
b. Q
c. P
d. L
e. None of these
Solution : 2
106. Which of the following combinations is true ?
a. First floor-S
b. Fourth floor-R
c. Third floor-M
d. Sixth floor-I
e. None of the above
Solution : 5
107. How many pairs of letters are there in the word (in forward direction)
APPLICATION, each of which have as many letters between them in the word
as they have between them in the English alphabet ?
a. One
b. Two
c. Three
d. Four
e. None of these
Solution : 1
108. In a certain coding system, PAPER is written as PERPA and SUBJECT is
written as JECTSUB, what should be the code for COUNCIL ?
a. NCILCOU
b. LICNOUC
c. NCOUCIL
33 | P a g e
d. NLICUOC
e. None of these
Solution : 1
109. 30. In a certain code language, `lu ja ka hu’ means ‘will you meet us’, `lu ka hu
pa’ means ‘will you sold us’. Then what is the code of `meet’ in this code
language ?
a. ja
b. lu
c. ka
d. hu
e. Cannot be determined
Solution : 1
110. 31. In a certain code language, COMBINE is written as XLNYRMV. How will
TOWARDS be written in that code language ?
a. FLDZIWJ
b. GLDZIWH
c. GLEZJWH
d. FLEZIWH
e. None of these
Solution : 2
111. 37 girls are standing in a row facing the school building. Ayesha is fifteenth
from the left end. If she is shifted six places to the right, what is her position
from the right end ?
a. 16th
b. 21st
c. 20th
d. 18th
e. None of these
34 | P a g e
Solution : 5
112. X’s mother is the mother-in-law of the father of Z. Z is the brother of Y while
X is the father of M. How is X related to Z ?
a. Paternal uncle
b. Maternal uncle
c. Cousin
d. Grandfather
e. Brother-in-law
Solution : 1
113. If A is the brother of B, C is the sister of A, D is the brother of E, E is the
daughter of B, F is the father of C, who is the uncle of D ?
a. A
b. C
c. B
d. None of these
e. Can’t be determined
Solution : 1
114. A said to B that B’s mother was the mother-in-law of A’s mother. How is A’s
mother related to B’s mother ?
a. Daughter-in-law
b. Mother-in-law
c. Sister
d. Aunt
e. None of these
Solution : 1
35 | P a g e
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Other (sometimes easier) ways to solve multivariate equations with matrices:
The equation used will be as follows:
``` x + 3y + 2z = 7
2x - y = -3
-x + 2y + 5z = 12
```
Its coefficient matrix would be:
```[ 1 3 2]
[ 2 -1 0]
[-1 2 5]
```
On TI-8x calculators, you can edit matrices in the matrix editor, then paste the variable to the screen, or you can enter them inline, like so:
```[3 2]
[4 7]
```
would be entered in as:
```[[3,2][4,7]]
```
### Cramers Rule:
Set up the coefficient matrix. Take the determinant of that matrix. Each variable will be equal to a numerator divided by that determinant. To find the numerator, replace the column for that variable with the constants in the original equations.
Example:
``` | 7 3 2|
|-3 -1 0|
|12 2 5| 22
x = --------- = --- = -22/29
| 1 3 2| -29
| 2 -1 0|
|-1 2 5|
| 1 7 2|
| 2 -3 0|
|-1 12 5| -43
y = --------- = --- = 43/29
| 1 3 2| -29
| 2 -1 0|
|-1 2 5|
| 1 3 7|
| 2 -1 -3|
|-1 2 12| -48
z = ---------- = --- = 48/29
| 1 3 2| -29
| 2 -1 0|
|-1 2 5|
```
### Reduced Row-Echelon Form:
This is even easier: you just use the rref() function on your calculator, and input the coefficient matrix with the constants appended as another column.
```[ 1 3 2 7]
[ 2 -1 0 -3]
[-1 2 5 12]
```
is your matrix. You just use rref(matrix) so if you wanted to do this inline, you would enter: rref([[1,3,2,7][2,-1,0,-3][-1,2,5,12]]) and it would spit out something like:
```[1 0 0 -.7586206897]
[0 1 0 1.482758621 ]
[0 0 1 1.655172414 ]
```
which you could use the handy >FRAC command to get:
```[1 0 0 -22/29]
[0 1 0 43/29 ]
[0 0 1 48/29 ]```
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Wednesday
May 6, 2015
# Posts by lily
Total # Posts: 1,238
Math
Sure no problem. The probability is 1 out of 2.
April 21, 2009
Math
I believe the probability of rolling a 6 would be 1 out of 8.
April 21, 2009
math
Do you need to find the ? as well?
April 20, 2009
math
What is the y-intercept?
April 20, 2009
English
Can you please proofread: Aging is something that everyone will experience. Having the opportunity to explore four different types of living facilities, really was an eye opening experience, about how genuine life really is. Being able to gain educational background about the ...
April 15, 2009
I did cite my sources in my final essay.
April 15, 2009
Can you please proofread this for me for grammar mistakes thank you. British American novelist Amelia Barr once stated “old age is the verdict of life”. Aging is a natural process of life that effects all of us. Sociologists see aging as a process of socialization or...
April 15, 2009
math
what catergorie capital and comission fall under. i think capital is owner equity and comission is income statement.
April 14, 2009
accounting
sorry i ment income stATtement becuse its expense. im i right
April 13, 2009
accounting
utilities in the catergorie owner equity i think. if not is it income statement or balance sheet.
April 13, 2009
accounting
does withdrawls fall in the catergorie income statement.if not plese correct. the other two choices are statement of changes in owner equity or balance sheet.
April 13, 2009
math
if sin theta equals 3/5 and theta is second quadrant angle find tan 2 theta.my ans. was 3.442 but is wrong can you correct and show work please. thank you so much.
April 10, 2009
math
write xy as the sum of unit vectors for x (8,2,negtive 9) nd y (negative 12,negtive 1, 10). my ans was negative 4i plus j plus k. wrong can you correct it for me please . thank you.
April 10, 2009
math
use the sum or difference identify to find the exact vlue of tan 105 degree.my ans.sqrt plus (3) plus 1 . wrong so can you correct it for me please thank you .
April 10, 2009
math
find the value of sin theta for angle theta in standard position if point with coordinates (negtive 3, 2) lies on its terminal side. i had 2 over sqrt 13 but it was wrong please correct.
April 10, 2009
math
write parametric equtions of y equals 5x minus 2. i put y equals 5 plus 2 but i was wrong can you correct me thankyou
April 10, 2009
math
write the equation 5x plus y equals 2 which equals 0 in normal form. my ans. was y equals 5 plus negative 2 or y equals 5 plus 2. if im wrong plus correct and explain why im wrong thank you.
April 10, 2009
physics
A 5.0-cm real object is placed at a distance of 30.0 cm from a concave mirror of focal length 10.0 cm. Find the location of the image. ANSWER: C Height of the object = 5.0 cm Distance = 30.0 cm Focal length = 10.0 cm This is the equation: 1/Do + 1/v = 1/f which means: 1/[...
April 6, 2009
spanish
how do you say "liquid is turning the cold cup hot" and "which one will you chose" in spanish?
March 5, 2009
Solve each system of equations, give the solution and state whether there are no solutions or many solutions. 1 )3x + 8y = 1 7x + 4y = 17 Answer: (3,1) 2) -3x + 5y = -7 9x + 2y = 38 Answer: (4,1) 3) 7r + 3s = 13 14r - 15s = -16 Answer: (1,2) 4) 2x - 7y = 42 4x - 14y = 64 ...
February 25, 2009
science
what type of biological molecule is an enzyme?
February 18, 2009
math
He Grated it
February 18, 2009
english - grammar
hello! I have some problems with mixed conditional.I understand 1., 2. and 3. type of conditionals, but i get confused when I have to use mixed conditional.I googled it but somehow i still don't get it.Anyone knows some simple explanation?How can I recognize mix ...
February 5, 2009
Pre-Algebra
the second one?
February 2, 2009
Pre-Algebra
but how do I change the sentences?
February 2, 2009
Pre-Algebra
I don't know, that was part of why I was asking =/
February 2, 2009
Pre-Algebra
Tell whether the question is potentially biased. Explain your answer. If the question is biased, rewrite it so that it is not. Questions: Don't you think the rising cost of concert tickets is ridiculous? Do you support the mayor's proposal to limit crime in the city?
February 2, 2009
linguistics
OK.this is a metaphor.Now i need to specify it..I don't know what I am supposed to do
January 27, 2009
linguistics
well no..is this a metaphor then? and keep one's eyes peeled?
January 27, 2009
linguistics
get stars in one's eyes - metaphor or metonymy?why?
January 27, 2009
linguistics
Hello everybody! I need a huge favour! If someone is very very good at distinguishing metaphors from metonymys please leave an e-mail.I am going to send you some examples.So if anyone is willing to do it please say!
January 27, 2009
Pre-Algebra
How would I make a frequency table with these numbers?: 465,599,567,529,243,250,268,319,294,316,325,364,358,237,270,534,479,614,283,575,591,555,508,609,584,304
January 26, 2009
english
I need help unscrambling this word!!!! AAAATTTLIIRRONNME its two words and its a name of a mall!
January 26, 2009
physics
During an experiment that Paul and Dawn performed the stretch of a spring was directly proportional to the force applied to the spring. When they put the data on a graph, they would expect to have what type of curve? Answers: Hyperbola, parabola, straight line, none of the ...
January 26, 2009
english
if a task description is to write an essay what is a goal?( in a language learning diary) and if a task desription is to learn some theory for an exam what is a goal?
January 25, 2009
english
There is one more thing which is not 100 % clear to me. I am still writing my learning diary and have to write about my emotional temperature.I've googled it but still I am not sure what it is
January 24, 2009
english
Yes, simple:)Thank you
January 24, 2009
english
January 24, 2009
english
i have to keep a language learning diary and i am a bit confused.I have to write a tas description and a goal.what is a difference,can someone give an example?
January 24, 2009
how do you find the zeroes of y= 2x(squared)-7x+15
January 18, 2009
Math
12,10,15
January 11, 2009
Math
what are the next three numbers in this pattern 2,3,4,6,6,9,8? Help please.
January 11, 2009
english
yes, thank you:)
January 11, 2009
english
for example blue car - blue as adjective premodifies a noun car or flying broomstick
January 11, 2009
english
well i have two tasks one is to find some examples of premodification+noun structures in a book and then find croatian equivalents for those strucures in the same book but in croatian.i am done with that and now i need to say if english and croatian examples are formally ...
January 11, 2009
english
hello everybody! I am doing a my homework on contrastive analysis.I am from Croatia so I have some examples in English and their equivalents in Croatian.My task is to say are those equivalents formally correspondent and to what degree.I don't understand that.Can someone ...
January 11, 2009
irregular verbs
That day in June he had swum in the ocean. Is swum the proper verb for this sentence?
January 9, 2009
Introduction To Teacher Aide
2. a take attendance i took it and the answer is correct
January 9, 2009
January 7, 2009
pre-algebra
convert the units. Round the result to the nearest tenth. 1- 2 years to months 2- 168 days to weeks 3- 1270 minutes to hours
January 5, 2009
math
5.014 I believe. What you do is you would leave the whole number and put it behind the decimal and then divide 14 by 1000.
January 5, 2009
Geography
Why have countries in the Pacific South America have developed slowly?
January 5, 2009
essay
ok so i'm confused about my essay topic: "what creative works (any medium) should a student be exposed to in order to prepare for university life?" im confused on what creative works they are referring to. art? literature?
December 31, 2008
Science
Ions are electrically charged particles that are formed when certain compounds are dissolved in water. These solutions will conduct electricity. The Swedish scientist Svante Arrhenius coined the term ion (which means wanderer) to explain why solutions of electrolytes will ...
December 25, 2008
Science
As temperature rises, solids generally become more soluble in water, but gases become less soluble. If a soft drink contains high concentrations of sugar and carbon dioxide, which of the following may be expected to happen if it is cooled down? A) Sugar may precipitate out. B...
December 25, 2008
english
Can someone explain thoroughly what EFL learners get from the task where they have ti put new words into context?Can someone explain this stategy?And also another strategie - using linguistic clues
December 21, 2008
english
thank you.Need also one example of a task which involves reading strategie: 1.using linguistic or other clues and 2.placing new words into context
December 21, 2008
english
ha!:) Have one question Which reading learning strategies is it when the task says: Read the text and complete each text with the words above each text Is it a learning strategie:using linguistic or other clues or placing new words into context or something else?
December 21, 2008
cognitive linguistics
well i understanr that.seeing is understanding and time is money but i really don't understand how i can recognize ifsource domain is image schematic
December 17, 2008
cognitive linguistics
hi everybody.i am eriting an exam tomorrow morning..i just can't understand what image schematic metaphors are.i don't need any internet pages i know what there stand.have some examples so if anyone ca nhelp me 1.Do you see my point? 2.Budget your time carefully 3.I.ve...
December 17, 2008
elementary algebra
ron and kathy are ticket-sellers at their class play, ron handling student tickets that sell for \$3.00 each and kathy selling adult tickets for \$6.50 each. If their total income for 29 tickets was \$125.50, how many did ron sell?
December 14, 2008
english
need some social interaction activities which i can use in a clasroom teaching EFL.Children will be given a story about a boy and girl going on picnic.While the boy was reading a book and girl was picking flowers bear came and took some food and wanted to scare the boy.A girl ...
December 13, 2008
Chemistry
nevermind i get it now, i can find the answer with the moles of copper produced/moles of iron used ratio thanks
December 10, 2008
Chemistry
copper produced from iron + copper II chloride
December 10, 2008
Chemistry
how many moles of iron would have been used up if 45.0g of copper were to be produced? is it # mols Fe = 45.0gCu x 1mol/63.5g = 0.709molFe ?
December 10, 2008
Chemistry
thank you
December 10, 2008
Chemistry
suppose that you have a n unlimited supply of copper II chloride to react with iron. how manyt moles of copper would be produced by reacting 34.0g of iron with the copperII chloride solution?
December 10, 2008
are the important industries of Canada called Grand banks? If not, what are they? What are the Atlantic provinces? Do they unclude Newfoundland and Nova Scotia? Is the St.Lawrence Seaway Canada's water highway to the world? Is Montreal the largest province in Canada?
November 20, 2008
English
What are four words that virtue can replace in everyday life?
November 20, 2008
Pre-Algebra
um.. whats the question??
November 19, 2008
Science
Explain Succession Ecosystem? What is the Climax Community? What limits the development of each ecosystem
November 19, 2008
t=10??? if you got this far why couldn't you do the rest???hehe funnie
November 19, 2008
English
Three examples of how virtue may be used in everyday life? Would: My friend was very virtue today, be a good example? Are there other examples I could put? Four words that virtue can replace in everyday life? Would Happy, Excited work? what are two other words that could work?
November 18, 2008
Geography
thank you, very much! :)
November 4, 2008
Geography
Ok, thank you :) This is very helpful. I was also wondering, and this is my last question I promise, the history started by English people moving to New England, they got into conflicts between the Native Americans, but then made peace. After that, what happens?
November 4, 2008
Geography
thank you :) and, when they ask about the economy, would it mean what they sale for money?
November 4, 2008
Geography
What is the economy and most popular job opportunity in Northeast Regions? (New England, all 12 northeast states)
November 4, 2008
Geography
Please help me! This is a project that is due on Wednesday... I've tried to use wikipedia but I can't find an answer to the questions... One I can't find is What is the history overview of Northeast region and the other is What is the economy of the region? and ...
November 4, 2008
Geography
How is the economy in Northeast United States? What are the most popular job opportunities of the Northeast United States? Thank you :)
November 3, 2008
verbs
Hawaii is one of the most beatiful places on earht. what are the verbs in the sentence
October 29, 2008
math
The local fish market is having a sale on whole fish. the prices are as follows: You can buy an albacore and a barracuda for\$21 You can buy a barracuda and a carp for \$32 You can buy a carp and a dogfish for \$32 You can buy a dogfish and an eel for \$37 You can buy an eel and ...
October 25, 2008
Science- PLEASE, this is due tomorrow for test!
What is the section of EMS of satellite phones? What wavelength and frequency does it use? are there any dangers in using it? What is the wavelength and frequency of pagers? are there any dangers in using it? Is there any danger in using cordless phones? What is the wavelength...
October 23, 2008
Science
What is the wavelength and frequency of radar waves? and does it have any dangers? thanks :)
October 22, 2008
How do cell phones work? I know they use radio waves, but how do they actually work??? Thanks :)
October 22, 2008
Night Vision
what are some dangers of using night vision?
October 17, 2008
Geography
Here are some questions that I have to answer about the American Culture. Because I come from France and have been in the US for only 3 years, I don't exactly know much about American culture. If someone could help me answer the following, that would be really helpful :) 1...
October 15, 2008
math
thursday, i believe
October 15, 2008
Grammar
Thank you very much.
October 15, 2008
Grammar
What is the simple subject and simple predicate of this sentence? The students in my class made custard. I think it is students for the simple subject and made for the simple predicate is this correct?
October 15, 2008
Thank you very much.
October 10, 2008
Everyday Mathmatics.
October 10, 2008
What is the greatest number of times you would regruop when multiplying a 3 digit number by a 2 digit number? Is it four, I am not sure.
October 10, 2008
chemisty
Rearrange the equation so it looks like n=PV/RT First change 23C to Kelvin= 296 Kelvin Plug in what you have: n= (1.10atm)(1.00L)/(0.08206)(296) n=0.045 that is the number of moles you have. But it doesn't tell you what type of element it is. So you need to find out the ...
October 8, 2008
u are so stupid why did u say that homework help?
October 5, 2008
Group the following words into categories:allegory,fable,slogan,tactics,Spontaneous Demonstrations,rebellion,second class,first class,comrade,fairy story,prose,poem,indoctrination,propaganda,commandment,cryptic,morose,loyal,frugal,sunday meetins,tactiturn,flags,animalism,republic
October 5, 2008
Pre-calculus
September 30, 2008
religious education
christianity is to bible sikhism is to guru granth sahib judaism is to torah muslim is to qu'ran budhism is to gautama hinduism is to ??????? please help
September 30, 2008
slavery
it funded for some things such as enclosed farming
September 30, 2008
Math
Murphy’s motorcycle gets 55 miles per gallon of gas on the highway and 45 miles per gallon in the city. The motorcycle holds 8 gallons of gas. Write and simplify an expression for the total number of miles Murphy can travel if he has a full tank of gas but uses 2 gallons ...
September 17, 2008
year 8 geography
Anyone know the meaning to the following words....(This is all to do with rivers) Channel Permeable Impermeable Sanitation Hydro electric power Desalinisation Pls Help!!
September 16, 2008
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# Bass–Quillen conjecture
Short description: Would relate vector bundles over a regular Noetherian ring and over a polynomial ring
In mathematics, the Bass–Quillen conjecture relates vector bundles over a regular Noetherian ring A and over the polynomial ring $\displaystyle{ A[t_1, \dots, t_n] }$. The conjecture is named for Hyman Bass and Daniel Quillen, who formulated the conjecture.[1][2]
## Statement of the conjecture
The conjecture is a statement about finitely generated projective modules. Such modules are also referred to as vector bundles. For a ring A, the set of isomorphism classes of vector bundles over A of rank r is denoted by $\displaystyle{ \operatorname{Vect}_r(A) }$.
The conjecture asserts that for a regular Noetherian ring A the assignment
$\displaystyle{ M \mapsto M \otimes_A A [t_1, \dots, t_n] }$
yields a bijection
$\displaystyle{ \operatorname{Vect}_r(A) \stackrel \sim \to \operatorname{Vect}_r(A[t_1, \dots, t_n]). }$
## Known cases
If A = k is a field, the Bass–Quillen conjecture asserts that any projective module over $\displaystyle{ k[t_1, \dots, t_n] }$ is free. This question was raised by Jean-Pierre Serre and was later proved by Quillen and Suslin, see Quillen–Suslin theorem. More generally, the conjecture was shown by (Lindel 1981) in the case that A is a smooth algebra over a field k. Further known cases are reviewed in (Lam 2006).
## Extensions
The set of isomorphism classes of vector bundles of rank r over A can also be identified with the nonabelian cohomology group
$\displaystyle{ H^1_{Nis}(Spec (A), GL_r). }$
Positive results about the homotopy invariance of
$\displaystyle{ H^1_{Nis}(U, G) }$
of isotropic reductive groups G have been obtained by (Asok Hoyois) by means of A1 homotopy theory.
## References
1. Bass, H. (1973), Some problems in ‘classical’ algebraic K-theory. Algebraic K-Theory II, Berlin-Heidelberg-New York: Springer-Verlag , Section 4.1
2. Quillen, D. (1976), "Projective modules over polynomial rings", Invent. Math. 36: 167–171, doi:10.1007/bf01390008, Bibcode1976InMat..36..167Q
• Asok, Aravind; Hoyois, Marc; Wendt, Matthias (2018), "Affine representability results in A^1-homotopy theory II: principal bundles and homogeneous spaces", Geom. Topol. 22 (2): 1181-1225
• Lindel, H. (1981), "On the Bass–Quillen conjecture concerning projective modules over polynomial rings", Invent. Math. 65 (2): 319–323, doi:10.1007/bf01389017, Bibcode1981InMat..65..319L
• Lam, T. Y. (2006), Serre’s problem on projective modules, Berlin: Springer, ISBN 3-540-23317-2
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Network topologies multiple choice questions (MCQs), network topologies quiz answers to learn online courses for computer architecture classes. Interconnection networks MCQs, network topologies quiz questions and answers for online computer science bachelors degree. Learn network topology, network routing, arbitration and switching, network topologies test prep for cisco certifications.
Learn interconnection networks MCQ: distributing network switches among end nodes, which then become, with choices network node, simple node, binode, and both a and b for online computer science bachelors degree. Practice merit scholarships assessment test, online learning network topologies quiz questions for competitive exams in computer science major .
MCQ: Distributing network switches among end nodes, which then become
1. Network node
2. Simple node
3. Binode
4. Both a and b
D
MCQ: No more than three nodes in a commercial systems are known as
1. Bidirectional topology
2. Hypercube topology
3. Binode topology
4. Torus topology
D
MCQ: In network, when ports remain free at other side of network but can be used for later expansion of network to larger sizes. These kind of networks are referred to as
1. Bidirectional unistage interconnection networks
2. Unidirectional unistage interconnection networks
3. Bidirectional multistage interconnection networks
4. Unidirectional multistage interconnection networks
C
MCQ: All nodes in each dimension form a linear array, in the
1. Mesh topology
2. Bus topology
3. Star topology
4. Torus topology
A
MCQ: Adding a minimum of logk N ? 1 extra switch stages to MIN in such a way that they mirror original topology, resulting network is
1. Non-blocking
2. Rearrangeably nonblocking
3. Blocking
4. Rearrangeably blocking
B
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# Distance between Lensk (ULK) and Polyarnyj (PYJ)
Flight distance from Lensk to Polyarnyj (Lensk Airport – Polyarny Airport) is 403 miles / 648 kilometers / 350 nautical miles. Estimated flight time is 1 hour 15 minutes.
Driving distance from Lensk (ULK) to Polyarnyj (PYJ) is 475 miles / 764 kilometers and travel time by car is about 12 hours 31 minutes.
## Map of flight path and driving directions from Lensk to Polyarnyj.
Shortest flight path between Lensk Airport (ULK) and Polyarny Airport (PYJ).
## How far is Polyarnyj from Lensk?
There are several ways to calculate distances between Lensk and Polyarnyj. Here are two common methods:
Vincenty's formula (applied above)
• 402.670 miles
• 648.034 kilometers
• 349.910 nautical miles
Vincenty's formula calculates the distance between latitude/longitude points on the earth’s surface, using an ellipsoidal model of the earth.
Haversine formula
• 401.652 miles
• 646.397 kilometers
• 349.026 nautical miles
The haversine formula calculates the distance between latitude/longitude points assuming a spherical earth (great-circle distance – the shortest distance between two points).
## Airport information
A Lensk Airport
City: Lensk
Country: Russia
IATA Code: ULK
ICAO Code: UERL
Coordinates: 60°43′14″N, 114°49′33″E
B Polyarny Airport
City: Polyarnyj
Country: Russia
IATA Code: PYJ
ICAO Code: UERP
Coordinates: 66°24′1″N, 112°1′47″E
## Time difference and current local times
There is no time difference between Lensk and Polyarnyj.
+09
+09
## Carbon dioxide emissions
Estimated CO2 emissions per passenger is 84 kg (186 pounds).
## Frequent Flyer Miles Calculator
Lensk (ULK) → Polyarnyj (PYJ).
Distance:
403
Elite level bonus:
0
Booking class bonus:
0
### In total
Total frequent flyer miles:
403
Round trip?
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# How To Solve Differential Equations Using Laplace Transform
Posted on
So we have already had an introduction to the laplace transform and even a lesson on how to calculate laplace expressions by a simple method of comparison. Solve by inverse laplace transform:
Finding The Laplace Transform Of Ft 2cos3t – 8sin2t Cosh5t Laplace Transform Math Videos Laplace
### Laplace\:y^ {\prime\prime}−10y^ {\prime}+9y=5t,y (0)=−1,y^ {\prime} (0)=2.
How to solve differential equations using laplace transform. Laplace transform the laplace transform and its inverse can be used to find the solution of initial value problems for ordinary differential equations. We start just as we did when we used laplace transforms to solve single differential equations. Made by faculty at lafayette college and.
The main idea about solving odes using laplace transform is to turn the differential equation into an algebric one. Take the laplace transform of both sides of the given differential equation: Laplace transform of cos t and polynomials.
Let be the laplace transform of. Put initial conditions into the resulting equation. Take the laplace transform of the differential equation using the derivative property (and, perhaps, others) as necessary.
( t) = e t + e − t 2 sinh. The laplace transform can be used to solve differential equations using a four step process. Demonstrates how to solve differential equations using laplace transforms when the initial conditions are all zero.
L {t^n} (opens a modal) laplace transform of the unit step function. Use linearity property of laplace transform to rewrite the equation as. This list is not a complete listing of laplace transforms and only contains some of the more commonly used laplace transforms and formulas.
Solve rlc circuit using laplace transform declare equations. Solve differential equations using laplace transform. Laplace\:y^ {\prime\prime}−6y^ {\prime}+15y=2sin (3t),y (0)=−1,y^ {\prime} (0)=−4.
We will solve differential equations that involve heaviside and dirac delta functions. Use laplace transform to solve the differential equation with the initial conditions and is a function of time. The laplace transform is an integral transform that is widely used to solve linear differential equations with constant coefficients.
You can use the laplace transform to solve differential equations with initial conditions. Solve rlc circuit using laplace transform declare equations. We take the transform of both differential equations.
Solve for the output variable. (tables) solution is obtained by a getting the inverse laplace transform from a table alternatively we can use partial fraction expansion to compute the solution using simple inverse transforms Solve differential equation using laplace transform:
Then its laplace transform is the function fs() as. L {t} (opens a modal) laplace transform of t^n: Linear de are transformed into algebraic ones.
Solving differential equations using laplace transforms example given the following first order differential equation, 𝑑 𝑑 + = u𝑒2 , where y()= v. Find (𝑡) using laplace transforms. The laplace transform is intended for solving linear de:
You can use the laplace transform to solve differential equations with initial conditions. Cosh(t) = et +e−t 2 sinh(t) = et−e−t 2 cosh. With the introduction of laplace transforms we will not be able to solve some initial value problems that we wouldn’t be able to solve otherwise.
When such a differential equation is. How to solve differential equations by laplace transforms. Also note that the system is nonhomogeneous.
The first step in using laplace transforms to solve an ivp is to take the transform of every term in the differential equation. We can get this from the general formula that we gave when we first started looking at solving ivp’s with laplace transforms. Laplace\:\frac {dy} {dt}+2y=12\sin (2t),y (0)=5.
(opens a modal) shifting transform by multiplying function by exponential. (opens a modal) laplace transform of t: Suppose that the function ft() is defined for all tt 0.
If the given problem is nonlinear, it has to be converted into linear. ( t) = e t − e − t 2. We will also give brief overview on using laplace transforms to solve nonconstant coefficient differential equations.
Now is time to see how these transformations are helpful to us while solving differential equations. By admin november 25, 2021 november 26, 2021. Laplace transforms for systems of differential equations bernd schroder¨ bernd schroder¨ louisiana tech university, college of engineering and science laplace transforms for systems of differential equations
S x 1 ( s) − x 1 ( 0) = 3 x 1 ( s) − 3 x 2 ( s) + 2 s s x 2 ( s) − x 2 ( 0) = − 6 x 1 ( s) − 1 s 2 s x 1 ( s) − x 1 ( 0) = 3 x 1 ( s) − 3 x 2 ( s) + 2 s s. Recall the definition of hyperbolic functions. Yl > e t @ dt dy 3 2 » ¼ º
L { y ′′ } − 10 l { y ′ } + 9 l { y } = l { 5 t } l { y ″ } − 10 l { y ′ } + 9 l { y } = l { 5 t } using the appropriate formulas from our. To begin solving the differential equation we would start by taking the laplace transform of both sides of the equation. The laplace solves de from time t = 0 to infinity.
Or other method have to be used instead (e.g.
Finding The Laplace Transform Of Ft 3 4t – 5t2 7t3 Laplace Transform Laplace Math Videos
Solution Of Differential Equations Using Laplace Transforms With This Ea Laplace Transform Differential Equations Equations
Pin On Differential Equations Videos
Using Laplace Transforms To Solve Differential Equations 2 Differential Equations Laplace Transform Equations
How To Solve Differential Equations Differential Equations Solving Equations Equations
Solve The Differential Equation Dydt – Y 1 Y0 1 Using Laplace Transforms Laplace Transform Laplace Differential Equations
Differential Equation Using Laplace Transform Heaviside Functions Laplace Transform Differential Equations Laplace
Pin On Differential Equations Videos
Ex Find The Laplace Transform Of Fte2t Using Definition – Differe Laplace Transform Differential Equations Laplace
Ex Find The Laplace Transform Of Ft3 Using Definition – Differential In 2021 Laplace Transform Differential Equations Laplace
Equation Heat Equation Laplace Transform Partial Differential Equation
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Ex Find The Laplace Transform Of Fte2t Using Definition – Differe Laplace Transform Differential Equations Laplace
Laplace Transform – Calculating The Laplace Transform – Differential Eq Laplace Transform Differential Equations Laplace
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Finding The Laplace Transform Of Ft 3t2 – 5sin2t Laplace Transform Laplace Math Videos
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Finding The Laplace Transform Of Ft 3 4t – 5t2 7t3 Laplace Transform Laplace Math Videos
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Main Content
# ecef2enu
Transform geocentric Earth-centered Earth-fixed coordinates to local east-north-up
## Syntax
``````[xEast,yNorth,zUp] = ecef2enu(X,Y,Z,lat0,lon0,h0,spheroid)``````
``[___] = ecef2enu(___,angleUnit)``
## Description
example
``````[xEast,yNorth,zUp] = ecef2enu(X,Y,Z,lat0,lon0,h0,spheroid)``` transforms the geocentric Earth-centered Earth-fixed (ECEF) Cartesian coordinates specified by `X`, `Y`, and `Z` to the local east-north-up (ENU) Cartesian coordinates specified by `xEast`, `yNorth`, and `zUp`. Specify the origin of the local ENU system with the geodetic coordinates `lat0`, `lon0`, and `h0`. Each coordinate input argument must match the others in size or be scalar. Specify `spheroid` as the reference spheroid for the geodetic coordinates.```
````[___] = ecef2enu(___,angleUnit)` specifies the units for latitude and longitude. Specify `angleUnit` as `'degrees'` (the default) or `'radians'`.```
## Examples
collapse all
Find the ENU coordinates of orbital debris with respect to a satellite, using the ECEF coordinates of the debris and the geodetic coordinates of the satellite.
First, specify the reference spheroid as WGS84 with length units measured in kilometers. For more information about WGS84, see Comparison of Reference Spheroids. The units for the ellipsoidal height, ECEF coordinates, and ENU coordinates must match the units specified by the `LengthUnit` property of the reference spheroid.
`wgs84 = wgs84Ellipsoid('kilometer');`
Specify the geodetic coordinates of the local origin. In this example, the local origin is the satellite. Specify `h0` as ellipsoidal height in kilometers.
```lat0 = 45.9132; lon0 = 36.7484; h0 = 1877.7532;```
Specify the ECEF coordinates of the point of interest. In this example, the point of interest is the orbital debris.
```x = 5507.5289; y = 4556.2241; z = 6012.8208;```
Then, calculate the ENU coordinates of the debris with respect to the satellite. In this example, `zUp` displays in scientific notation.
`[xEast,yNorth,zUp] = ecef2enu(x,y,z,lat0,lon0,h0,wgs84)`
```xEast = 355.6013 ```
```yNorth = -923.0832 ```
```zUp = 1.0410e+03 ```
Reverse the transformation using the `enu2ecef` function. In this example, the results display in scientific notation.
`[x,y,z] = enu2ecef(xEast,yNorth,zUp,lat0,lon0,h0,wgs84)`
```x = 5.5075e+03 ```
```y = 4.5562e+03 ```
```z = 6.0128e+03 ```
## Input Arguments
collapse all
ECEF x-coordinates of one or more points in the geocentric ECEF system, specified as a scalar, vector, matrix, or N-D array. Specify values in units that match the `LengthUnit` property of the `spheroid` argument. For example, the default length unit for the reference ellipsoid created by `wgs84Ellipsoid` is `'meter'`.
Data Types: `single` | `double`
ECEF y-coordinates of one or more points in the geocentric ECEF system, specified as a scalar, vector, matrix, or N-D array. Specify values in units that match the `LengthUnit` property of the `spheroid` argument. For example, the default length unit for the reference ellipsoid created by `wgs84Ellipsoid` is `'meter'`.
Data Types: `single` | `double`
ECEF z-coordinates of one or more points in the geocentric ECEF system, specified as a scalar, vector, matrix, or N-D array. Specify values in units that match the `LengthUnit` property of the `spheroid` argument. For example, the default length unit for the reference ellipsoid created by `wgs84Ellipsoid` is `'meter'`.
Data Types: `single` | `double`
Geodetic latitude of the local origin, specified as a scalar, vector, matrix, or N-D array. The local origin can refer to one point or a series of points (for example, a moving platform). Specify the values in degrees. To use values in radians, specify the `angleUnit` argument as `'radians'`.
Data Types: `single` | `double`
Geodetic longitude of the local origin, specified as a scalar, vector, matrix, or N-D array. The local origin can refer to one point or a series of points (for example, a moving platform). Specify the values in degrees. To use values in radians, specify the `angleUnit` argument as `'radians'`.
Data Types: `single` | `double`
Ellipsoidal height of the local origin, specified as a scalar, vector, matrix, or N-D array. The local origin can refer to one point or a series of points (for example, a moving platform). Specify values in units that match the `LengthUnit` property of the `spheroid` object. For example, the default length unit for the reference ellipsoid created by `wgs84Ellipsoid` is `'meter'`.
For more information about ellipsoidal height, see Find Ellipsoidal Height from Orthometric Height.
Data Types: `single` | `double`
Reference spheroid, specified as a `referenceEllipsoid` object, `oblateSpheroid` object, or `referenceSphere` object. The term reference spheroid is used synonymously with reference ellipsoid. To create a reference spheroid, use the creation function for the object. To specify the reference ellipsoid for WGS84, use the `wgs84Ellipsoid` function.
For more information about reference spheroids, see Comparison of Reference Spheroids.
Example: `spheroid = referenceEllipsoid('GRS 80');`
Angle units, specified as `'degrees'` (the default) or `'radians'`.
## Output Arguments
collapse all
ENU x-coordinates of one or more points in the local ENU system, returned as a scalar, vector, matrix, or N-D array. Units are specified by the `LengthUnit` property of the `spheroid` argument. For example, the default length unit for the reference ellipsoid created by `wgs84Ellipsoid` is `'meter'`.
ENU y-coordinates of one or more points in the local ENU system, returned as a scalar, vector, matrix, or N-D array. Units are specified by the `LengthUnit` property of the `spheroid` argument. For example, the default length unit for the reference ellipsoid created by `wgs84Ellipsoid` is `'meter'`.
ENU z-coordinates of one or more points in the local ENU system, returned as a scalar, vector, matrix, or N-D array. Units are specified by the `LengthUnit` property of the `spheroid` argument. For example, the default length unit for the reference ellipsoid created by `wgs84Ellipsoid` is `'meter'`.
## Tips
To transform vectors instead of coordinate locations, use the `ecef2enuv` function.
## Version History
Introduced in R2012b
expand all
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# Search by Topic
#### Resources tagged with Working systematically similar to Triangle Inequality:
Filter by: Content type:
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Challenge level:
### There are 129 results
Broad Topics > Using, Applying and Reasoning about Mathematics > Working systematically
### Tetrahedra Tester
##### Stage: 3 Challenge Level:
An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?
### 9 Weights
##### Stage: 3 Challenge Level:
You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?
### You Owe Me Five Farthings, Say the Bells of St Martin's
##### Stage: 3 Challenge Level:
Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?
### Squares in Rectangles
##### Stage: 3 Challenge Level:
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?
### When Will You Pay Me? Say the Bells of Old Bailey
##### Stage: 3 Challenge Level:
Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?
### More Magic Potting Sheds
##### Stage: 3 Challenge Level:
The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?
### Triangles to Tetrahedra
##### Stage: 3 Challenge Level:
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
### Seasonal Twin Sudokus
##### Stage: 3 and 4 Challenge Level:
This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?
### Weights
##### Stage: 3 Challenge Level:
Different combinations of the weights available allow you to make different totals. Which totals can you make?
### Games Related to Nim
##### Stage: 1, 2, 3 and 4
This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.
### Twin Corresponding Sudokus II
##### Stage: 3 and 4 Challenge Level:
Two sudokus in one. Challenge yourself to make the necessary connections.
### Corresponding Sudokus
##### Stage: 3, 4 and 5
This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.
### Maths Trails
##### Stage: 2 and 3
The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.
### Sticky Numbers
##### Stage: 3 Challenge Level:
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
### Pole Star Sudoku
##### Stage: 4 and 5 Challenge Level:
A Sudoku based on clues that give the differences between adjacent cells.
### Intersection Sums Sudoku
##### Stage: 2, 3 and 4 Challenge Level:
A Sudoku with clues given as sums of entries.
### Twin Corresponding Sudoku III
##### Stage: 3 and 4 Challenge Level:
Two sudokus in one. Challenge yourself to make the necessary connections.
### Ratio Sudoku 1
##### Stage: 3 and 4 Challenge Level:
A Sudoku with clues as ratios.
### Diagonal Product Sudoku
##### Stage: 3 and 4 Challenge Level:
Given the products of diagonally opposite cells - can you complete this Sudoku?
### Consecutive Numbers
##### Stage: 2 and 3 Challenge Level:
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
### Oranges and Lemons, Say the Bells of St Clement's
##### Stage: 3 Challenge Level:
Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.
### Ratio Sudoku 3
##### Stage: 3 and 4 Challenge Level:
A Sudoku with clues as ratios or fractions.
### Wallpaper Sudoku
##### Stage: 3 and 4 Challenge Level:
A Sudoku that uses transformations as supporting clues.
### Problem Solving, Using and Applying and Functional Mathematics
##### Stage: 1, 2, 3, 4 and 5 Challenge Level:
Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.
### LOGO Challenge - the Logic of LOGO
##### Stage: 3 and 4 Challenge Level:
Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?
##### Stage: 3 and 4 Challenge Level:
Four numbers on an intersection that need to be placed in the surrounding cells. That is all you need to know to solve this sudoku.
### Consecutive Negative Numbers
##### Stage: 3 Challenge Level:
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
### Where Can We Visit?
##### Stage: 3 Challenge Level:
Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?
##### Stage: 3 Challenge Level:
How many different symmetrical shapes can you make by shading triangles or squares?
### First Connect Three for Two
##### Stage: 2 and 3 Challenge Level:
First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.
### Isosceles Triangles
##### Stage: 3 Challenge Level:
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
### Fence It
##### Stage: 3 Challenge Level:
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
### Factor Lines
##### Stage: 2 and 3 Challenge Level:
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
### Teddy Town
##### Stage: 1, 2 and 3 Challenge Level:
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
### Intersection Sudoku 1
##### Stage: 3 and 4 Challenge Level:
A Sudoku with a twist.
### Ratio Sudoku 2
##### Stage: 3 and 4 Challenge Level:
A Sudoku with clues as ratios.
### Summing Consecutive Numbers
##### Stage: 3 Challenge Level:
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
### Magic W
##### Stage: 4 Challenge Level:
Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.
### One Out One Under
##### Stage: 4 Challenge Level:
Imagine a stack of numbered cards with one on top. Discard the top, put the next card to the bottom and repeat continuously. Can you predict the last card?
### First Connect Three
##### Stage: 2 and 3 Challenge Level:
The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?
### Instant Insanity
##### Stage: 3, 4 and 5 Challenge Level:
Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.
### Twin Corresponding Sudoku
##### Stage: 3, 4 and 5 Challenge Level:
This sudoku requires you to have "double vision" - two Sudoku's for the price of one
### Diagonal Sums Sudoku
##### Stage: 2, 3 and 4 Challenge Level:
Solve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells.
### Medal Muddle
##### Stage: 3 Challenge Level:
Countries from across the world competed in a sports tournament. Can you devise an efficient strategy to work out the order in which they finished?
### Intersection Sudoku 2
##### Stage: 3 and 4 Challenge Level:
A Sudoku with a twist.
### Plum Tree
##### Stage: 4 and 5 Challenge Level:
Label this plum tree graph to make it totally magic!
### Twinkle Twinkle
##### Stage: 2 and 3 Challenge Level:
A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.
### Magic Potting Sheds
##### Stage: 3 Challenge Level:
Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?
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1. ## population
A population of animals in an ecological niche is growing in time so that it's rate of growth dp/dt is related to its current size by the differential equation dp/dt = 900/p^2. If time is measured in years and initially there are P(0) = 10 animals present, find the population function P(t) giving the size of the population after t years.
Don't get this, but I integrated it, am I suppose to plug 900 somewhere in the equation P=Ce^kt??
2. I am thinking that when t=0, C=10. Am I suppose to cancel out P to find out k or something?
3. Originally Posted by elpermic
A population of animals in an ecological niche is growing in time so that it's rate of growth dp/dt is related to its current size by the differential equation dp/dt = 900/p^2. If time is measured in years and initially there are P(0) = 10 animals present, find the population function P(t) giving the size of the population after t years.
Don't get this, but I integrated it, am I suppose to plug 900 somewhere in the equation P=Ce^kt??
I don't understand your question. If you integrated dP/dt= 900/P^2 (and that's the only derivative to integrate), the 900 is already in there! But you don't get P= Ce^kt. From dP/dt= 900/P^2 you get P^2 dP= 900 dt and integrating, (1/3)P^3= 900t+ C. Determine C in THAT by setting t= 0, P= 10 to get 1000/3= C. If you wish to solve for P, P^3= 2700t+ C, $P= ^3\sqrt{2700t+ C}$.
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http://www.doitpoms.ac.uk/tlplib/reciprocal_lattice/reciprocal_lattice.php
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# DoITPoMS
Reciprocal space
The animation below shows the relationship between the real lattice and the reciprocal lattice. Note that this 2D representation omits the c* vector, but that it follows the same rules as a* and b*.
The key things to note are that:
• The reciprocal lattice has reciprocal vectors a* and b*, separated by the angle γ*.
• a* is perpendicular to the (100) planes, and equal in magnitude to the inverse of d100.
• Similarly, b* is perpendicular to the (010) planes and equal in magnitude to the inverse of d010.
• γ and γ* will sum to 180º.
Due to the linear relationship between planes (for example, d200 = ½ d100 ), a periodic lattice is generated. In general, the periodicity in the reciprocal lattice is given by
In vector form, the general reciprocal lattice vector for the (h k l) plane is given by
where nhkl is the unit vector normal to the (h k l) planes.
This concept can be applied to crystals, to generate a reciprocal lattice of the crystal lattice. The units in reciprocal space are Å-1 or nm-1
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# Topology Proof
• Mar 12th 2011, 02:04 PM
Dreamer78692
Topology Proof
6. Let X be a non-empty set containing at least two elements, and let a and b be fixed
but different points in X.
(a) Show that the family T = {U ∈ PX | a ∈ U} ∪ {∅} is a topology on X.
(b) Is the family T = {U ∈ PX | a ∈ U or b ∈ U} ∪ {∅} a topology on X?
I don't get this question, what are they asking for, is T the Discrete Topology???(Angry)
• Mar 12th 2011, 02:12 PM
Plato
Quote:
Originally Posted by Dreamer78692
6. Let X be a non-empty set containing at least two elements, and let a and b be fixed but different points in X.
(a) Show that the family T = {U ∈ PX | a ∈ U} ∪ {∅} is a topology on X.
The set $\mathbf{T}$ contains the emptyset and any subset of $\mathbf{X}$ that contains $\mathbf{a}$.
Your task is to show that collection $\mathbf{T}$ is a topology on $\mathbf{X}$.
List the properties of a topology and the check $\mathbf{T}$ for each one.
• Mar 12th 2011, 02:25 PM
Dreamer78692
for (a)
1) ∅ ,X are in T (trivial)
2)for any 2 sets U,V ∈ T U ∩ V has to at least contain a which is an element of T
3) The union of all sets in T will still contain a, therefore it is an element of T
Am I on the right track...(Punch)
• Mar 12th 2011, 02:32 PM
Plato
Quote:
Originally Posted by Dreamer78692
for (a)
1) ∅ ,X are in T (trivial)
2)for any 2 sets U,V ∈ T U ∩ V has to at least contain a which is an element of T
3) The union of all sets in T will still contain a, therefore it is an element of T
Am I on the right track...
Right track yes, but work on the correct language.
Closed with respect to 2) finite intersection and 3) arbitrary union.
• Mar 12th 2011, 02:40 PM
Dreamer78692
for (b)
Let U = {a,c} and V = {b,c} then U ∩ V = {c} which is not an element of T
(Happy)
• Mar 12th 2011, 02:43 PM
Plato
Quote:
Originally Posted by Dreamer78692
for (b)
Let U = {a,c} and V = {b,c} then U ∩ V = {c} which is not an element of T
GOOD! So $\mathbf{T}$ is not a topology because it is not closed under finite intersection.
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# COMP538: Introduction to Bayesian Networks
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1 COMP538: Introduction to Bayesian Networks Lecture 2: Bayesian Networks Nevin L. Zhang Department of Computer Science and Engineering Hong Kong University of Science and Technology Fall 2008 Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
2 Objective Objective: Explain the concept of Bayesian networks Reading: Zhang & Guo, Chapter 2 References: Russell & Norvig, Chapter 15 Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
3 Outline Probabilistic Modeling with Joint Distribution 1 Probabilistic Modeling with Joint Distribution 2 Conditional Independence and Factorization 3 Bayesian Networks 4 Manual Construction of Bayesian Networks Building structures Causal Bayesian networks Determining Parameters Local Structures 5 Remarks Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
4 Probabilistic Modeling with Joint Distribution The probabilistic approach to reasoning under uncertainty A problem domain is modeled by a list of variables X 1, X 2,..., X n, Knowledge about the problem domain is represented by a joint probability P(X 1, X 2,...,X n ). Example: Alarm (Pearl 1988) Story: In LA, burglary and earthquake are not uncommon. They both can cause alarm. In case of alarm, two neighbors John and Mary may call. Problem: Estimate the probability of a burglary based who has or has not called. Variables: Burglary (B), Earthquake (E), Alarm (A), JohnCalls (J), MaryCalls (M). Knowledge required by the probabilistic approach in order to solve this problem: P(B, E, A, J, M) Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
5 P(B, E, A, J, M) B E A J M Prob B E A J M Prob y y y y y n y y y y.0002 y y y y n n y y y n.0004 y y y n y n y y n y.0004 y y y n n n y y n n.0002 y y n y y n y n y y.0002 y y n y n n y n y n.0002 y y n n y n y n n y.0002 y y n n n.0000 n y n n n.0002 y n y y y n n y y y.0001 y n y y n n n y y n.0002 y n y n y n n y n y.0002 y n y n n.0000 n n y n n.0001 y n n y y n n n y y.0001 y n n y n n n n y n.0001 y n n n y n n n n y.0001 y n n n n n n n n n.996 Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55 Probabilistic Modeling with Joint Distribution Joint probability distribution
6 Probabilistic Modeling with Joint Distribution Inference with joint probability distribution What is the probability of burglary given that Mary called, P(B=y M=y)? Compute marginal probability: P(B, M) = P(B, E, A, J, M) E,A,J B M Prob y y y n n y n n Compute answer (reasoning by conditioning): P(B=y M=y) = = P(B=y, M=y) P(M=y) = 0.61 Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
7 Probabilistic Modeling with Joint Distribution Advantages Probability theory well-established and well-understood. In theory, can perform arbitrary inference among the variables given a joint probability. This is because the joint probability contains information of all aspects of the relationships among the variables. Diagnostic inference: From effects to causes. Example: P(B=y M=y) Predictive inference: From causes to effects. Example: P(M=y B=y) Combining evidence: P(B=y J=y, M=y, E=n) All inference sanctioned by laws of probability and hence has clear semantics. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
8 Probabilistic Modeling with Joint Distribution Difficulty: Complexity in model construction and inference In Alarm example: 31 numbers needed, Quite unnatural to assess: e.g. P(B = y, E = y, A = y, J = y, M = y) In general, Computing P(B=y M=y) takes 29 additions. [Exercise: Verify this.] P(X 1, X 2,..., X n ) needs at least 2 n 1 numbers to specify the joint probability. Exponential model size. Knowledge acquisition difficult (complex, unnatural), Exponential storage and inference. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
9 Outline Conditional Independence and Factorization 1 Probabilistic Modeling with Joint Distribution 2 Conditional Independence and Factorization 3 Bayesian Networks 4 Manual Construction of Bayesian Networks Building structures Causal Bayesian networks Determining Parameters Local Structures 5 Remarks Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
10 Conditional Independence and Factorization Chain Rule and Factorization Overcome the problem of exponential size by exploiting conditional independence The chain rule of probabilities: P(X 1, X 2 ) = P(X 1 )P(X 2 X 1 ) P(X 1, X 2, X 3 ) = P(X 1 )P(X 2 X 1 )P(X 3 X 1, X 2 )... P(X 1, X 2,..., X n ) = P(X 1 )P(X 2 X 1 )...P(X n X 1,..., X n 1 ) n = P(X i X 1,..., X i 1 ). i=1 No gains yet. The number of parameters required by the factors is: 2 n n = 2 n 1. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
11 Conditional Independence and Factorization Conditional Independence About P(X i X 1,..., X i 1 ): Then Domain knowledge usually allows one to identify a subset pa(x i ) {X 1,..., X i 1 } such that Given pa(x i), X i is independent of all variables in {X 1,..., X i 1} \ pa(x i), i.e. Joint distribution factorized. P(X i X 1,..., X i 1) = P(X i pa(x i)) P(X 1, X 2,..., X n ) = n P(X i pa(x i )) The number of parameters might have been substantially reduced. i=1 Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
12 Conditional Independence and Factorization Example continued P(B, E, A, J, M) = P(B)P(E B)P(A B, E)P(J B, E, A)P(M B, E, A, J) = P(B)P(E)P(A B, E)P(J A)P(M A)(Factorization) pa(b) = {}, pa(e) = {},pa(a) = {B, E }, pa(j) = {A},pa(M) = {A}. Conditional probabilities tables (CPT) B Y N P(B) M A P(M A) Y Y.9 N Y.1 Y N.05 N N.95 E Y N P(E) J A P(J A) Y Y.7 N Y.3 Y N.01 N N.99 A B E P(A B, E) Y Y Y.95 N Y Y.05 Y Y N.94 N Y N.06 Y N Y.29 N N Y.71 Y N N.001 N N N.999 Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
13 Conditional Independence and Factorization Example continued Model size reduced from 31 to =10 Model construction easier Fewer parameters to assess. Parameters more natural to assess:e.g. P(B = Y ), P(E = Y), P(A = Y B = Y, E = Y ), Inference easier.will see this later. P(J = Y A = Y ), P(M = Y A = Y ) Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
14 Outline Bayesian Networks 1 Probabilistic Modeling with Joint Distribution 2 Conditional Independence and Factorization 3 Bayesian Networks 4 Manual Construction of Bayesian Networks Building structures Causal Bayesian networks Determining Parameters Local Structures 5 Remarks Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
15 Bayesian Networks From Factorizations to Bayesian Networks Graphically represent the conditional independency relationships: construct a directed graph by drawing an arc from X j to X i iff X j pa(x i ) pa(b) = {}, pa(e) = {}, pa(a) = {B, E }, pa(j) = {A}, pa(m) = {A}. B P(B) E P(E) A P(A B, E) J P(J A) M P(M A) Also attach the conditional probability (table) P(X i pa(x i )) to node X i. What results in is a Bayesian network.also known as belief network, probabilistic network. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
16 Formal Definition Bayesian Networks A Bayesian network is: An directed acyclic graph (DAG), where Each node represents a random variable And is associated with the conditional probability of the node given its parents. Recall: In introduction, we said that Bayesian networks are networks of random variables. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
17 Bayesian Networks Understanding Bayesian networks Qualitative level: A directed acyclic graph (DAG) where arcs represent direct probabilistic dependence. B E A J M Absence of arc indicates conditional independence: A variable is conditionally independent of all its nondescendants given its parents. (Will prove this later.) The above DAG implies the following conditional independence relationships: B E; J B A; J E A; M B A; M E A; M J A The following are not implied: J B; J E; J M; B E A Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
18 Bayesian Networks Understanding Bayesian networks Quantitative (numerical) level: Conditional probability tables: B Y N P(B) M A P(M A) Y Y.9 N Y.1 Y N.05 N N.95 E Y N P(E) J A P(J A) Y Y.7 N Y.3 Y N.01 N N.99 A B E P(A B, E) Y Y Y.95 N Y Y.05 Y Y N.94 N Y N.06 Y N Y.29 N N Y.71 Y N N.001 N N N.999 Describe how parents of a variable influence the variable. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
19 Bayesian Networks Understanding Bayesian Networks As a whole: A Bayesian network represents a factorization of a joint distribution. n P(X 1, X 2,...,X n ) = P(X i pa(x i )) Multiplying all the CPTs results in a joint distribution over all variables. i=1 Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
20 Example networks Bayesian Networks Network repository Software Bayesian Network Repository: Genie & Smile Network Repository: Netica Net Library: Hugin Case Studies: Genie & Smile: Free. Netica: Free version for small nets. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
21 Outline Manual Construction of Bayesian Networks 1 Probabilistic Modeling with Joint Distribution 2 Conditional Independence and Factorization 3 Bayesian Networks 4 Manual Construction of Bayesian Networks Building structures Causal Bayesian networks Determining Parameters Local Structures 5 Remarks Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
22 Manual Construction of Bayesian Networks Building structures Procedure for constructing Bayesian network structures 1 Choose a set of variables that describes the application domain. 2 Choose an ordering for the variables. 3 Start with the empty network and add variables to the network one by one according to the ordering. 4 To add the i-th variable X i, 1 Determine a subset pa(x i ) of variables already in the network (X 1,..., X i 1 ) such that P(X i X 1,..., X i 1 ) = P(X i pa(x i )) (Domain knowledge is needed here.) 2 Draw an arc from each variable in pa(x i ) to X i. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
23 Examples Manual Construction of Bayesian Networks Building structures Order 1: B, E, A, J, M pa(b) = {}, pa(e) = {}, pa(a) = {B, E }, pa(j) = {A}, pa(m) = {A}. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
24 Examples Manual Construction of Bayesian Networks Building structures Order 2: M, J, A, B, E pa(m) = {}, pa(j) = {M}, pa(a) = {M, J}, pa(b) = {A}, pa(e) = {A, B}. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
25 Examples Manual Construction of Bayesian Networks Building structures Order 3: M, J, E, B, A pa(m) = {}, pa(j) = {M}, pa(e) = {M, J}, pa(b) = {M, J, E }, pa(a) = {M, J, B, E }. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
26 Manual Construction of Bayesian Networks Building structures Building Bayesian network structures Which order? Naturalness of probability assessment (Howard and Matheson). (B, E, A, J, M) is a good ordering because the following distributions natural to assess P(B), P(E): frequency of burglary and earthquake P(A B, E): property of Alarm system. P(M A): knowledge about Mary P(J A): knowledge about John. The order M, J, E, B, A is not good because, for instance, P(B J, M, E) is unnatural and hence difficult to assess directly. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
27 Manual Construction of Bayesian Networks Building structures Building Bayesian network structures Which order? Minimize number of arcs (J. Q. Smith). The order (M, J, E, B, A) is bad because too many arcs. In contrast, the order (B, E, A, J, M) is good is because it results in a simple structure. Use causal relationships (Pearl): cause come before their effects. The order (M, J, E, B, A) is not good because, for instance, M and J are effects of A but come before A. In contrast, the order (B, E, A, J, M) is good is because it respects the causal relationships among variables. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
28 Manual Construction of Bayesian Networks Exercise in Structure Building Building structures Five variable about what happens to an office building Fire: There is a fire in the building. Smoke: There is smoke in the building. Alarm: Fire alarm goes off. Leave: People leaves the building. Tampering: Someone tamper with the fire system (e.g., open fire exit) Build network structures using the following ordering. Clearly state your assumption. 1 Order 1: tampering, fire, smoke, alarm, leave 2 Order 2: leave, alarm, smoke, fire, tampering Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
29 Manual Construction of Bayesian Networks Causal Bayesian networks Building structures Build a Bayesian network using casual relationships: Choose a set of variables that describes the domain. Draw an arc to a variable from each of its DIRECT causes. (Domain knowledge needed here.) What results in is a causal Bayesian network, or simply causal networks, Arcs are interpreted as indicating cause-effect relationships. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
30 Example: Manual Construction of Bayesian Networks Building structures Travel (Lauritzen and Spiegelhalter) Adventure Smoking Tuberculosis Lung Cancer Bronchitis X ray Dyspnea Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
31 Manual Construction of Bayesian Networks Use of Causality: Issue 1 Building structures Causality is not a well understood concept. No widely accepted definition. No consensus on Whether it is a property of the world, Or a concept in our minds helping us to organize our perception of the world. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
32 Causality Manual Construction of Bayesian Networks Building structures Sometimes causal relations are obvious: Alarm causes people to leave building. Lung Cancer causes mass on chest X-ray. At other times, they are not that clear. Whether gender influences ability in technical sciences. Most of us believe Smoking cause lung cancer,but the tobacco industry has a different story: Surgeon General (1964) s c Tobacco Industry g s c Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
33 Manual Construction of Bayesian Networks Working Definition of Causality Building structures Imagine an all powerful agent, GOD, who can change the states of variables. Example: X causes Y if knowing that GOD has changed the state of X changes your believe about Y. Smoking and yellow finger are correlated. If we force someone to smoke for sometime, his finger will probably become yellow. So Smoking is a cause of yellow finger. If we paint someone s finger yellow, that will not affect our belief on whether s/he smokes. So yellow finger does not cause smoking. Similar example with Earthquake and Alarm Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
34 Causality Manual Construction of Bayesian Networks Building structures Coin tossing example revisited: Knowing that GOD somehow made sure the coin drawn from the bag is a fair coin would affect our belief on the results of tossing. Knowing that GOD somehow made sure that the first tossing resulted in a head does not affect our belief on the type of the coin. So arrows go from coin type to results of tossing. Coin Type Toss 1 Result Toss 2 Result... Toss n Result Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
35 Manual Construction of Bayesian Networks Use of Causality: Issue 2 Building structures Adventure Smoking Tuberculosis Lung Cancer Bronchitis X ray Dyspnea Causality network structure (building process) Network structure conditional independence (Semantics of BN) The causal Markov assumption bridges causality and conditional independence: A variable is independent of all its non-effects (non-descendants) given its direct causes (i.e. parents). We make this assumption if we determine Bayesian network structure using causality. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
36 Manual Construction of Bayesian Networks Determining probability parameters Determining Parameters Later in this course, we will discuss in detail how to learn parameters from data. We will not be so much concerned with eliciting probability values from experts. However, people do that some times. In such a case, one would want the number of parameters be as small as possible. The rest of the lecture describe two concepts for reducing the number of parameters: Causal Independence. Context-specific independence. Left to students as reading materials. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
37 Manual Construction of Bayesian Networks Determining probability parameters Determining Parameters Sometimes conditional probabilities are given by domain theory Genetic inheritance in Stud (horse) farm (Jensen, F. V. (2001). Bayesian networks and decision graphs. Springer.): P(Child Father, Mother) aa aa AA aa (1, 0, 0) (.5,.5, 0) (0, 1, 0) aa (.5,.5, 0) (.25,.5, 25) (0,.5,.5) AA (0, 1, 0) (0,.5,.5) (0, 0, 1) Genotypes: aa - sick, aa - carrier, AA - pure. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
38 Manual Construction of Bayesian Networks Determining probability parameters Determining Parameters Sometimes, we need to get the numbers from the experts. This is a time-consuming and difficult process. Nonetheless, many networks have been built. See Bayesian Network Repository at Combine experts knowledge and data Use assessments by experts as a start point. When data become available, combine data and experts assessments. As more and more data become available, influence of experts is automatically reduced. We will show how this can be done when discussing parameter learning. Note: Much of the course will be about how to learning Bayesian networks (structures and parameters) from data. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
39 Manual Construction of Bayesian Networks Reducing the number of parameters Determining Parameters Let E be a variable in a BN and let C 1, C 2,..., C m be its parents. C 1 C 2... C m E The size of the conditional probability P(E C 1, C 2,...,C m ) is exponential in m. This poses a problem for knowledge acquisition, learning, and inference. In application, there usually exist local structures that one can exploit to reduce the size of conditional probabilities Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
40 Manual Construction of Bayesian Networks Causal independence Determining Parameters Causal independence refers to the situation where the causes C 1, C 2..., and C m influence E independently. In other words, the ways by which the C i s influence e are independent. Burglary Earthquake Burglary Earthquake Alarm Alarm-due-to-Burglary (A b ) Alarm-due-to-Earthquake (A e ) Alarm Example: Burglary and earthquake trigger alarm independently. Precise statement: A b and A e are independent. A = A b A e, hence Noisy-OR gate (Good 1960). Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
41 Manual Construction of Bayesian Networks Determining Parameters Causal Independence C 1 C 2 C m ξ 1 ξ... 2 ξm * E Formally, C 1, C 2..., and C m are said to be causally independent w.r.t effect E if there exist random variables ξ 1, ξ 2..., and ξ m such that 1 For each i, ξ i probabilistically depends on C i and is conditionally independent of all other C j s and all other ξ j s given C i, and 2 There exists a commutative and associative binary operator over the domain of e such that. E = ξ 1 ξ 2... ξ m Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
42 Manual Construction of Bayesian Networks Causal Independence Determining Parameters C 1 C 2 C m ξ 1 ξ... 2 ξm * E In words, individual contributions from different causes are independent and the total influence on effect is a combination of the individual contributions. ξ i contribution of C i to E. * base combination operator. E independent cause (IC) variable. Known as convergent variable in Zhang & Poole (1996). Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
43 Manual Construction of Bayesian Networks Causal Independence Determining Parameters Example: Lottery C i : money spent on buying lottery of type i. E: change of wealth. ξ i : change in wealth due to buying the ith type lottery. Base combination operator: +. (Noisy-adder) Other causal independence models: 1 Noisy MAX-gate max 2 Noisy AND-gate Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
44 Manual Construction of Bayesian Networks Determining Parameters Causal Independence Theorem (2.1) If C 1, C 2,..., C m are causally independent w.r.t E,then the conditional probability P(E C 1,..., C m ) can be obtained from the conditional probabilities P(ξ i C i ) through P(E=e C 1,..., C m ) = α 1... α k =e P(ξ 1 =α 1 C 1 )...P(ξ m =α m C m ), (1) for each value e of E. Here is the base combination operator of E. See Zhang and Poole (1996) for the proof. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
45 Manual Construction of Bayesian Networks Causal Independence Determining Parameters Notes: Causal independence reduces model size: In the case of binary variable, it reduces model sizes from 2 m+1 to 4m. Examples: CPSC, Carpo It can also be used to speed up inference (Zhang and Poole 1996). Relationship with logistic regression? (Potential term project) Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
46 Parent divorcing Manual Construction of Bayesian Networks Determining Parameters Another technique to reduce the number of parameters Top figure: A more natural model for the Adventure Smoking Travel example. But it requires =20 parameters. Tuberculosis X ray Adventure Tuberculosis X-ray Tuberculosis or Lung Cancer Lung Cancer Lung Cancer Dyspnea Smoking Dyspnea Bronchitis Bronchitis Low figure: requires only =18 parameters. The difference would be bigger if, for example, D have other parents. The trick is to introduce a new node (TB-or-LC). It divorces T and L from the other parent B of D. Note that the trick would not help if the new node TB-or-LC has 4 or more states. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
47 Manual Construction of Bayesian Networks Determining Parameters Context specific independence (CSI) Let C be a set of variables. A context on C is an assignment of one value to each variable in C. We denote a context by C=c, where c is a set of values of variables in C. Two contexts are incompatible if there exists a variable that is assigned different values in the contexts. They are compatible otherwise. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
48 Manual Construction of Bayesian Networks Context-specific independence Determining Parameters Let X, Y, Z, and C be four disjoint sets of variables. X and Y are independent given Z in context C=c if whenever P(Y, Z, C=c)>0. P(X Z,Y,C=c) = P(X Z,C=c) When Z is empty, one simply says that X and Y are independent in context C=c. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
49 Manual Construction of Bayesian Networks Context-specific independence Determining Parameters Age Gender Number of Pregnancies Shafer s Example: Number of pregnancies (N) is independent of Age (A) in the context Gender=Male (G =m). P(N A, G=m) = P(N G=m) Number of parameters reduced by ( A 1)( N 1). Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
50 Manual Construction of Bayesian Networks Context-specific independence Determining Parameters Income independent of Weather in context Profession=Programmer. P(I W, P=Prog, Q) = P(I P=Prog, Q) Weather Profession Qualification (W) (P) (Q) Income independent of Qualification in context Profession=Farmer. Income (I) P(I W, P, Q) P(I W, P=Farmer, Q) = P(I W, P=Farmer) Number of parameters reduced by: ( W 1) Q ( I 1) + ( Q 1) W ( I 1) CSI can also be exploited to speed up inference (Zhang and Poole 1999). Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
51 Outline Remarks 1 Probabilistic Modeling with Joint Distribution 2 Conditional Independence and Factorization 3 Bayesian Networks 4 Manual Construction of Bayesian Networks Building structures Causal Bayesian networks Determining Parameters Local Structures 5 Remarks Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
52 Remarks Reasons for the popularity of Bayesian networks It s graphical language is intuitive and easy to understand because it captures what might be called intuitive causality. Pearl (1986) claims that it is a model for human s inferential reasoning: Notations of dependence and conditional dependence are basic to human reasoning. The fundamental structure of human knowledge can be represented by dependence graphs. Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
53 Remarks Reasons for the popularity of Bayesian networks In practice, the graphical language Functions as a convenient language to organizes one s knowledge about a domain. Facilitates interpersonal communication. On the other hand, the language is well-defined enough to allow computer processing. Correctness of results guaranteed by probability theory. For probability theory, Bayesian networks provide a whole new perspective: Probability is not really about numbers; It is about the structure of reasoning. (Glenn Shafer) Nevin L. Zhang (HKUST) Bayesian Networks Fall / 55
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## Checking the Math: The Big Event from Game of Thrones Season 8 Episode 4 is Just as Impossible as You Think
In the last episode of Game of Thrones, some things went down. This article will be about one of those things. So spoiler warnings, people.
Last chance to turn back.
Okay. So, as you may recall, a ship-borne Euron Greyjoy surprises a dragon-borne Dany, shooting one of her dragons—Rhaegal, RIP—with a deck-mounted rapid-fire torsion weapon that the show calls a “scorpion.” Euron’s bolts cut through the beast, killing it, before a sequence of further shots from his ships blasts her fleet to splinters.
I have thoughts.
First, screw Three-eyed Raven “Nut” Bran Stark. Dude sees everything but speaks nothing.
Second, screw Dany for somehow managing to be surprised by a fleet of ships despite the fact that she has superior range of view from the back of a flying dragon. Seriously? She and Jon are just The Worst when it comes to using their heads. Gotta say, I’m fast becoming #TeamNoneOfTheseSchmucks.
Third, I can actually accept the possibility of a rapid-fire ballista. Building one is a significant engineering problem, but it’s hardly insurmountable. Philo of Byzantium actually described such a device in his work Mechanike syntaxis (he called the device a polybolos, meaning ‘thrower of multiple things’) in the 3rd century BC. So I’ll take it. No problem there.
Fourth …well, let’s talk projectile ballistics.
The path of a projectile—in our case, a scorpion bolt—is affected by some major variables, including the initial velocity it has, the air resistance it encounters, and the pull of gravity upon it. And yeah, I know Game of Thrones is fantasy, but I’ve seen no indications that Westeros has anything but Earth-like gravity and atmospheric composition. These facts, plus the relatively calm weather conditions for the event in question and the distance between Euron and Dany, will simplify our calculations considerably.
Sorry to bring high school maths back for y’all, but for simplicity’s sake the end result of all this will be a path that looks like a parabola: a projectile ought to make a rather lovely arc between an initial launching point and a final landing point.
You may already be seeing the problem.
See, those scorpion bolts went straight. I mean, really straight. Their trajectory was so flat that Euron was using a fixed “iron” sight to take aim. Since a projectile under the influence of gravity cannot have a flat trajectory, the only explanation for such a seemingly straight shot would be that its parabolic arc is so wide that within the range covered it appears to be flat. It’s rather like how over short distances the earth appears to be flat, but that’s only because we’re seeing such a small section of a really wide curvature.
In other words, if it hadn’t hit the neck of a flying dragon, Euron’s bolt would have gone really, really far.
How far?
Let’s have some fun with numbers!
I’m eyeballing things here, but I’d estimate Dany is about 1,000 meters away from Euron. It’s probably more, given the fact that she didn’t see his damn fleet at all, but I’ll roll with this. And of course Euron needs to hit a moving target. I don’t have the exact velocity of a dragon at hand just now, but I do know that the airspeed velocity of an unladen European swallow is about 11 meters per second. So I’ll guess Rhaegal is cruising around 5 m/s. Seems fair.
Before we get to anything else, I’ve got to give some props to some impressive shooting on Euron’s part. An object moving 5 m/s at a range of 1,000 meters is damn hard to hit even with modern, fully adjustable optical sights. Plus, you know, that scorpion is hard-mounted to the deck of a vessel that on the open sea will be experiencing pitch, yaw, roll, heave, surge, and sway. So yeah, it’s (impossibly) impressive shooting.
Ah, you might say, but this is a big object. That makes it easier.
True, true.
To continue, then, let’s see what that first hit tells us. Euron clearly had this moving target in his non-adjustable sights, because he hit it. Now, I’m gonna reckon that Rhaegal’s neck is about 5 meters thick from top to bottom, and we know that this target is moving horizontally (5 m/s) and that the bolt itself will drop from whatever he’s aiming at. Given that he hit Rhaegal smack dab in the middle of its throat, that drop can’t be much more than 2.5 meters.
In other words, he would need to be aiming for the very nose of the beast, and the bolt would need to cover the 1,000 meter distance in about half a second.
The bolt, therefore, would need to have an initial velocity of about 2,000 m/s.
By comparison, a modern sniper rifle propels a bullet around 800 to 1,000 m/s (howdy, Fortnite fans!).
Oh, remember that parabola? Having the projectile drop just 2.5 meters over its first 1,000 meters of travel means it’s a wide one. If we figure that the angle, from Euron’s perspective, between the sea and Rhaegal is about 20 degrees, we can use our initial velocity to trace out that parabola mathematically.
I’m just a lowly medievalist here, folks, but I’m calculating that Euron’s bolt, if it hadn’t hit the dragon, would have reached a maximum elevation of around 14 miles above sea level before it nosed back down to earth. That’s, um, around twice the cruising altitude of most commercial airliners.
And sure, there’s some wiggle room to my calculations. Atmospheric conditions are one of those major variables, after all, and while the weather was great when Euron pulled the trigger, if his bolt hadn’t hit the dragon a lot could’ve happened to it during the—checks notes—more than two minutes it would’ve been airborne after he released it, during which time—checks notes again—it would’ve traveled about 163 miles. A missed shot would’ve passed through weather systems.
Probably this is all academic. It doesn’t matter how many damn torsion arms you put on one, no medieval or quasi-medieval ballistics engine can manage to launch a projectile at hypersonic speeds. Alas.
Oh there he goes again, you might be thinking, with his nonsensical demands that the fantasy of Game of Thrones should better reflect the reality of our history.
Well… yeah, I guess maybe I’m doing that a little. But there’s actually a flip-side to all this. I saw more than one complaint online from folks who doubted that bolts from a ballista could actually blow through dragons and ships in the ways depicted. Knowing something about real ballistae, I thought the same myself when I first watched the episode.
But you know what? I’ve got to give credit where it is due now that I’ve seen the numbers. A bolt cutting through the air at friggin’ hypersonic speed would indeed obliterate anything in its path.
So all’s well that ends well…
…except …damnit, Jon …you pet your direwolf before you go. Always.
Michael Livingston is a Professor of Medieval Culture at The Citadel who has written extensively both on medieval history and on modern medievalism. His historical fantasy trilogy set in Ancient Rome, The Shards of Heaven, The Gates of Hell, and The Realms of God, is available from Tor Books. His new fantasy novella “Black Crow, White Snow” was released on Friday, May 3rd as an Audible Original.
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# Search by Topic
#### Resources tagged with Multiplication & division similar to Augustus' Age:
Filter by: Content type:
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### There are 132 results
Broad Topics > Calculations and Numerical Methods > Multiplication & division
### Divide it Out
##### Stage: 2 Challenge Level:
What is the lowest number which always leaves a remainder of 1 when divided by each of the numbers from 2 to 10?
### What Is Ziffle?
##### Stage: 2 Challenge Level:
Can you work out what a ziffle is on the planet Zargon?
### What Two ...?
##### Stage: 2 Short Challenge Level:
56 406 is the product of two consecutive numbers. What are these two numbers?
### Tom's Number
##### Stage: 2 Challenge Level:
Work out Tom's number from the answers he gives his friend. He will only answer 'yes' or 'no'.
### Calendar Calculations
##### Stage: 2 Challenge Level:
Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?
### Asteroid Blast
##### Stage: 2 Challenge Level:
A game for 2 people. Use your skills of addition, subtraction, multiplication and division to blast the asteroids.
### Number Tracks
##### Stage: 2 Challenge Level:
Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?
### Rocco's Race
##### Stage: 2 Short Challenge Level:
Rocco ran in a 200 m race for his class. Use the information to find out how many runners there were in the race and what Rocco's finishing position was.
### Clever Keys
##### Stage: 2 Short Challenge Level:
On a calculator, make 15 by using only the 2 key and any of the four operations keys. How many ways can you find to do it?
### Arranging the Tables
##### Stage: 2 Challenge Level:
There are 44 people coming to a dinner party. There are 15 square tables that seat 4 people. Find a way to seat the 44 people using all 15 tables, with no empty places.
### Rabbits in the Pen
##### Stage: 2 Challenge Level:
Using the statements, can you work out how many of each type of rabbit there are in these pens?
### Being Resilient - Primary Number
##### Stage: 1 and 2 Challenge Level:
Number problems at primary level that may require resilience.
### Special 24
##### Stage: 2 Challenge Level:
Find another number that is one short of a square number and when you double it and add 1, the result is also a square number.
### How Much Did it Cost?
##### Stage: 2 Challenge Level:
Use your logical-thinking skills to deduce how much Dan's crisps and ice-cream cost altogether.
### Sept03 Sept03 Sept03
##### Stage: 2 Challenge Level:
This number has 903 digits. What is the sum of all 903 digits?
### Magic Potting Sheds
##### Stage: 3 Challenge Level:
Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?
### Machines
##### Stage: 2 Challenge Level:
What is happening at each box in these machines?
### Clever Santa
##### Stage: 2 Challenge Level:
All the girls would like a puzzle each for Christmas and all the boys would like a book each. Solve the riddle to find out how many puzzles and books Santa left.
##### Stage: 2 Challenge Level:
Use the information to work out how many gifts there are in each pile.
### X Is 5 Squares
##### Stage: 2 Challenge Level:
Can you arrange 5 different digits (from 0 - 9) in the cross in the way described?
### Zargon Glasses
##### Stage: 2 Challenge Level:
Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?
### The Pied Piper of Hamelin
##### Stage: 2 Challenge Level:
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
### ABC
##### Stage: 2 Challenge Level:
In the multiplication calculation, some of the digits have been replaced by letters and others by asterisks. Can you reconstruct the original multiplication?
### A-magical Number Maze
##### Stage: 2 Challenge Level:
This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!
### Penta Post
##### Stage: 2 Challenge Level:
Here are the prices for 1st and 2nd class mail within the UK. You have an unlimited number of each of these stamps. Which stamps would you need to post a parcel weighing 825g?
### Napier's Bones
##### Stage: 2 Challenge Level:
The Scot, John Napier, invented these strips about 400 years ago to help calculate multiplication and division. Can you work out how to use Napier's bones to find the answer to these multiplications?
### Oranges and Lemons
##### Stage: 2 Challenge Level:
On the table there is a pile of oranges and lemons that weighs exactly one kilogram. Using the information, can you work out how many lemons there are?
### Cows and Sheep
##### Stage: 2 Challenge Level:
Use your logical reasoning to work out how many cows and how many sheep there are in each field.
### How Old?
##### Stage: 2 Challenge Level:
Cherri, Saxon, Mel and Paul are friends. They are all different ages. Can you find out the age of each friend using the information?
### Oh! Harry!
##### Stage: 2 Challenge Level:
A group of children are using measuring cylinders but they lose the labels. Can you help relabel them?
### The Clockmaker's Birthday Cake
##### Stage: 2 Challenge Level:
The clockmaker's wife cut up his birthday cake to look like a clock face. Can you work out who received each piece?
### Long Multiplication
##### Stage: 3 Challenge Level:
A 3 digit number is multiplied by a 2 digit number and the calculation is written out as shown with a digit in place of each of the *'s. Complete the whole multiplication sum.
### The Number Crunching Machine
##### Stage: 2 Challenge Level:
Put a number at the top of the machine and collect a number at the bottom. What do you get? Which numbers get back to themselves?
### 1, 2, 3, 4, 5
##### Stage: 2 Challenge Level:
Using the numbers 1, 2, 3, 4 and 5 once and only once, and the operations x and ÷ once and only once, what is the smallest whole number you can make?
### Six Ten Total
##### Stage: 2 Challenge Level:
This challenge combines addition, multiplication, perseverance and even proof.
### Four Go for Two
##### Stage: 2 Challenge Level:
Four Go game for an adult and child. Will you be the first to have four numbers in a row on the number line?
### It Figures
##### Stage: 2 Challenge Level:
Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?
### The 24 Game
##### Stage: 2 Challenge Level:
There are over sixty different ways of making 24 by adding, subtracting, multiplying and dividing all four numbers 4, 6, 6 and 8 (using each number only once). How many can you find?
### Shapes in a Grid
##### Stage: 2 Challenge Level:
Can you find which shapes you need to put into the grid to make the totals at the end of each row and the bottom of each column?
### Sam's Quick Sum
##### Stage: 2 Challenge Level:
What is the sum of all the three digit whole numbers?
### Today's Date - 01/06/2009
##### Stage: 1 and 2 Challenge Level:
What do you notice about the date 03.06.09? Or 08.01.09? This challenge invites you to investigate some interesting dates yourself.
### What's My Weight?
##### Stage: 2 Short Challenge Level:
There are four equal weights on one side of the scale and an apple on the other side. What can you say that is true about the apple and the weights from the picture?
### Current Playing with Number Upper Primary Teacher
##### Stage: 2 Challenge Level:
Resources to support understanding of multiplication and division through playing with number.
### Difficulties with Division
##### Stage: 1 and 2
This article for teachers looks at how teachers can use problems from the NRICH site to help them teach division.
### Countdown
##### Stage: 2 and 3 Challenge Level:
Here is a chance to play a version of the classic Countdown Game.
### Learning Times Tables
##### Stage: 1 and 2 Challenge Level:
In November, Liz was interviewed for an article on a parents' website about learning times tables. Read the article here.
### Watching the Wheels Go 'round and 'round
##### Stage: 2 Challenge Level:
Use this information to work out whether the front or back wheel of this bicycle gets more wear and tear.
### Next Number
##### Stage: 2 Short Challenge Level:
Find the next number in this pattern: 3, 7, 19, 55 ...
### Exploring Wild & Wonderful Number Patterns
##### Stage: 2 Challenge Level:
EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.
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or
# Consider three vectors veca, vecb and vecc. Vectors veca and vecb are unit vectors having an angle theta between them For vector veca,|veca|^2=veca.veca If veca_|_vecb and veca_|_vecc then veca||vecbxxvecc If veca||vecb, then veca=tvecb Now answer the following question: The value of sin(theta/2) is (A) 1/2 |veca-vecb| (B) 1/2|veca+vecb| (C) |veca-vecb| (D) |veca+vecb|
Question from Class 12 Chapter Vector Algebra: Competition
Apne doubts clear karein ab Whatsapp par bhi. Try it now.
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Pvillage.org
Informative blog about fresh lifehacks
# How are C channels calculated?
Table of Contents
## How are C channels calculated?
How to Calculate C-Channel Beams
1. Measure the depth of the C-channel beam by placing a measuring tape on the flat long side of the beam in the center of the C.
2. Measure the width of the flanges.
3. Measure the height of the flanges by measuring from the inner valley of the C to the tops of the flange.
## What is the weight of Ismc 100?
Technical Specifications
MILD STEEL CHANNELS
(mm) (kg)
ISMC 75 weight 75 7.1
ISMC 100 weight 100 9.6
ISMC 125 weight 125 13.1
## How to calculate the weight of a channel?
Steel Channel Weight Calculator Online Free Channel Weight Calculator and ISMC Weight Chart U channel weight calculator, Online ISMC channel weight calculator, ms channel weight calculator free download, C channel weight calculator Ms Channel Weight in Kg M.S.Channel Size Weight in Kgs. Per Feet
## How much does a channel In ISMC weigh?
U channel weight calculator, Online ISMC channel weight calculator, ms channel weight calculator free download, C channel weight calculator Ms Channel Weight in Kg M.S.Channel Size Weight in Kgs. Per Feet Weight in Kgs. per Mtr. ISMC 75 x 40 x 4.8 2.176 7.14 ISMC 100 x 50 x 5 2.914 9.56 ISMC 125 x 65 x 5.3 3.993 13.10 ISMC 150 x 75 x 5.7 5.121
## How tall is ISMC 100 x 50 in kg?
Weight in Kgs. per Mtr. ISMC 75 x 40 x 4.8 2.176 7.14 ISMC 100 x 50 x 5 2.914 9.56 ISMC 125 x 65 x 5.3 3.993 13.10 ISMC 150 x 75 x 5.7 5.121 16.80 ISMC 175 x 75 x 6 5.975 19.60 ISMC 200 x 75 x 6.2 6.798 22.30 ISMC 250 x 82 x 9 10.426 34.2 ISMC 300 x 90 x 7.8 11.067 36.3 ISMC 400 x 100 x 8.8 15.274 50.1 All weights are approximate.
## How is the weight of a Jindal channel calculated?
Channels manufactured by JINDAL are represented by Web Height, Flange Width & Section Weight. Click here to check SS Pipe weight chart. JINDAL Channel Weight Chart Sr. No. Depth (mm) x Flange (mm) x Sectional Weight (kg/m)
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A229294 Number of solutions to x^2 + y^2 + z^2 + t^2 == n (mod 2*n) for x,y,z,t in [0, 2*n). 4
8, 96, 264, 384, 1160, 3168, 3080, 1536, 7560, 13920, 11528, 12672, 18824, 36960, 38280, 6144, 41480, 90720, 57608, 55680, 101640, 138336, 101384, 50688, 149000, 225888, 208008, 147840, 201608, 459360, 245768, 24576, 380424, 497760, 446600, 362880, 415880 (list; graph; refs; listen; history; text; internal format)
OFFSET 1,1 COMMENTS All values are divisible by a(1)=8 and the sequence a(n)/8 is multiplicative. - Andrew Howroyd, Aug 07 2018 LINKS Chai Wah Wu, Table of n, a(n) for n = 1..10000 FORMULA a(n) = 8*A240547(n) for odd n, a(2^k) = 24*2^(2*k). - Andrew Howroyd, Aug 07 2018 MATHEMATICA A[n_] := Sum[If[Mod[a^2 + b^2 + c^2 + d^2, 2n] == n, 1, 0], {d, 0, 2n - 1}, {a, 0, 2n - 1}, {b, 0, 2n - 1}, {c, 0, 2n - 1}]; Table[Print[aaa = A[n]]; aaa, {n, 1, 40}] PROG (PARI) a(n)={my(m=2*n); my(p=Mod(sum(i=0, m-1, x^(i^2%m)), x^m-1)^4); polcoeff( lift(p), n)} \\ Andrew Howroyd, Aug 07 2018 (PARI) a(n)={my(f=factor(n)); 8*prod(i=1, #f~, my([p, e]=f[i, ]); if(p==2, 3*2^(2*e), p^(2*e-1)*(p^(e+1)+p^e-1)))} \\ Andrew Howroyd, Aug 07 2018 (Python) def A229294(n): ndict = {} n2 = 2*n for i in range(n2): i3 = pow(i, 2, n2) for j in range(i+1): j3 = pow(j, 2, n2) m = (i3+j3) % n2 if m in ndict: if i == j: ndict[m] += 1 else: ndict[m] += 2 else: if i == j: ndict[m] = 1 else: ndict[m] = 2 count = 0 for i in ndict: j = (n-i) % n2 if j in ndict: count += ndict[i]*ndict[j] return count # Chai Wah Wu, Jun 07 2017 CROSSREFS Cf. A229295, A229296, A229297, A240547. Sequence in context: A300261 A121785 A305587 * A241813 A116144 A074114 Adjacent sequences: A229291 A229292 A229293 * A229295 A229296 A229297 KEYWORD nonn AUTHOR José María Grau Ribas, Sep 22 2013 STATUS approved
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Last modified September 26 06:08 EDT 2023. Contains 365653 sequences. (Running on oeis4.)
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https://docs.mosek.com/10.0/javaapi/advanced-blas.html
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# 9.2 Calling BLAS/LAPACK Routines from MOSEK¶
Sometimes users need to perform linear algebra operations that involve dense matrices and vectors. Also MOSEK extensively uses high-performance linear algebra routines from the BLAS and LAPACK packages and some of these routines are included in the package shipped to the users.
The MOSEK versions of BLAS/LAPACK routines:
• use MOSEK data types and return value conventions,
• preserve the BLAS/LAPACK naming convention.
Therefore the user can leverage on efficient linear algebra routines, with a simplified interface, with no need for additional packages.
List of available routines
Table 9.1 BLAS routines available.
BLAS Name
MOSEK function
Math Expression
AXPY
Env.axpy
$$y=\alpha x+y$$
DOT
Env.dot
$$x^T y$$
GEMV
Env.gemv
$$y=\alpha Ax + \beta y$$
GEMM
Env.gemm
$$C=\alpha AB+ \beta C$$
SYRK
Env.syrk
$$C=\alpha AA^T+ \beta C$$
Table 9.2 LAPACK routines available.
LAPACK Name
MOSEK function
Description
POTRF
Env.potrf
Cholesky factorization of a semidefinite symmetric matrix
SYEVD
Env.syevd
Eigenvalues and eigenvectors of a symmetric matrix
SYEIG
Env.syeig
Eigenvalues of a symmetric matrix
Source code examples
In Listing 9.2 we provide a simple working example. It has no practical meaning except showing how to organize the input and call the methods.
Listing 9.2 Calling BLAS and LAPACK routines from Optimizer API for Java. Click here to download.
package com.mosek.example;
public class blas_lapack {
static final int n = 3, m = 2, k = 3;
public static void main (String[] args) {
double alpha = 2.0, beta = 0.5;
double[] x = {1., 1., 1.};
double[] y = {1., 2., 3.};
double[] z = {1.0, 1.0};
/*A has m=2 rows and k=3 cols*/
double[] A = {1., 1., 2., 2., 3., 3.};
/*B has k=3 rows and n=3 cols*/
double[] B = {1., 1., 1., 1., 1., 1., 1., 1., 1.};
double[] C = { 1., 2., 3., 4., 5., 6.};
double[] D = {1.0, 1.0, 1.0, 1.0};
double[] Q = {1.0, 0.0, 0.0, 2.0};
double[] v = new double[2];
double[] xy = {0.};
try (mosek.Env env = new mosek.Env()) {
/* routines*/
env.dot(n, x, y, xy);
env.axpy(n, alpha, x, y);
env.gemv(mosek.transpose.no, m, n, alpha, A, x, beta, z);
env.gemm(mosek.transpose.no, mosek.transpose.no, m, n, k, alpha, A, B, beta, C);
env.syrk(mosek.uplo.lo, mosek.transpose.no, m, k, alpha, A, beta, D);
/* LAPACK routines*/
env.potrf(mosek.uplo.lo, m, Q);
env.syeig(mosek.uplo.lo, m, Q, v);
env.syevd(mosek.uplo.lo, m, Q, v);
} catch (mosek.Exception e) {
System.out.println ("An error/warning was encountered");
System.out.println (e.toString());
}
}
}
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https://forum.heatinghelp.com/discussion/comment/1396426/
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Welcome! Here are the website rules, as well as some tips for using this forum.
Need to contact us? Visit https://heatinghelp.com/contact-us/.
Click here to Find a Contractor in your area.
Just a thought about the "B" dimension.
Member Posts: 64
Hello everybody. I just finished tuning my freshly piped two pipe return system. I had issues with water spitting out of my return vents so I went to the book and read about the "B" dimension. The book was right and I got it figured out. I have 32" in height so the max cut out I can use comes in at about 1.2 psig. As I was thinking about this i was wondering if you were to increase the diameter of the vertical pipe that leads to the wet return if you could actually decrease the "B" dimension. The way i see it, its a battle of pressures. Head pressure vs. boiler pressure. If you were to increase pipe diameters from 1.250 to 2.5 you would double your head pressure going to the boiler. I would imagine this would make some sort of difference. Any thoughts ?
Comments
• Member Posts: 19,152
Sorry... pressure doesn't work that way. Pressure in a liquid is purely related to the depth of the liquid or the height of the water column. It has nothing to do with the area or pipe size.
And I agree -- a lot of people have a lot of trouble with this concept, but I assure you that that's the way it works.
Your solution will be to invest in a vapourstat, rather than a pressuretrol, and set it -- for starters anyway -- at about 12 ounces cutout and 6 ounces cutin.
Br. Jamie, osb
Building superintendent/caretaker, 7200 sq. ft. historic house museum with dependencies in New England
• Member Posts: 64
Thanks Jamie. That's the solution I came up with too. I figured there was an explanation for why it doesn't work that way. There's no way I thought of something that the dude that wrote that book didn't !
• Member Posts: 5,094
Think of it this way, if you swim in the ocean do your feet get crushed when you are treading water? For what you are saying to be true the weight of the top 5' of the entire ocean would be pressing on your feet. You could also think of a round swimming pool as a giant pipe...similar concept as the ocean. Remember PSI is pounds per SQUARE INCH so if you increase pipe size you spread the "weight" out over a larger area so the amount of pressure on an area (PSI) does not increase. Jamie is right it's a tough one to understand sometimes.
2014 Weil Mclain EG-40
EcoSteam ES-20 Advanced Boiler Control
Boiler pictures updated 2/21/15
• Member Posts: 64
Great way of explaining KC. I was actually going in that direction of thought but hadn't quite gotten there yet.
• Member Posts: 64
So if you were standing underneath a 50' foot diameter water tank and had a 1" ball valve to drain it it wouldn't matter if it had 10' of water or 20' of water, it would still fill a water jug in the same amount of time.
• Member Posts: 4,469
No..........There would be a difference in the time between 10 and 20, but if the tank was 25' diameter, the 10' time and 20' time would be the same as the 50' diameter.
• Member Posts: 4,469
There's an old way of finding level at some distance. You use clear tubing, and fill it with water. The water will always find level. It wouldn't matter if one end of the tube was 2" and the other was 1/2".
• Member Posts: 9,822
Think of gravity, that diameter of the tank doesn't matter, it is the height of the actual water above you that determines the pressure. You may notice that water towers are placed on a hill if possible. The height of the tower plus the hill elevation produce the best pressure possible without operating a pump. That is a huge tank up there but its still only has a standpipe connecting to it that might be 10 to 16" in diameter.
• Member Posts: 53
Pouring wet concrete into a wall form is similar. The thickness of the wall does not matter- only the height. There is the same amount of pressure at the bottom of an 8" thick wall as a 36" thick wall, given equal heights.
Dennis
• Member Posts: 64
Gotcha. Thanks for the clarification guys.
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Space Satellite Physics
Discover Science & Engineering
Physics
Maths
Ireland is a member of the European Space Agency and, as such, has been involved in many major space projects. Most of us are familiar with the International Space Station which is the largest of the some 3,000 operating satellites currently in orbit. This lesson traces the revolutionary ideas of early scientists such as Galileo and describes how Newton’s Laws are used to calculate orbits.
Contains the full lesson along with a supporting toolkit, including teachers’ notes.
Lesson excerpt
Europe in Space
Since its foundation in 1975, the European Space Agency (ESA) has led many major space projects and has cooperated with Russian and US space agencies. At present the ESA has eighteen member countries including Ireland and many Irish companies provide components for satellites and other space craft. At present, there are over 3,000 operating satellites orbiting the Earth. The largest of these is the International Space Station (ISS). None of this would have been possible without a thorough understanding of the laws of motion and gravitation. In this lesson we look at the mathematics behind these laws.
Revolutionary Ideas
In the 1630s Galileo published two major works in which he contrasted a new scientific world view with the traditional view. For example, from the time of Aristotle it was generally believed that motion required aforce. This seems like a reasonable idea which is consistent with our everyday experience. However, it could not explain, for example, why an arrow would keep moving long after it had been fired from a bow.
Galileo carried out many experiments and showed that a forcecauses acceleration – not a steady speed. He also realised thatfriction caused moving bodies to slow down and eventually stop, and proposed that if there were no friction then, once a body was set in motion, it would move forever without the need for further force. This was (and still is) a counter-intuitive idea that many people could not accept. However, we now know that as we reduce friction in machines, less force is needed to keep things moving.
Newton
Building on the ideas of Galileo and others, Isaac Newton devised three simple propositions ('laws') to describe how things move. He also proposed an inverse-square law of gravitational attraction, that is the force of attraction between two bodies (such as the Earth and the Moon or the Earth and an apple) was inversely proportional to the square of the distance between their centres; for example, if the distance is doubled then the force is reduced to a quarter of what it was. These revolutionary propositions were published in 1687.
True or False?
1. A satellite that appears to move is said to be in a geostationary orbit. false
2. The force of attraction between two bodies is proportional to the cube of the distance between their centres. false
3. A force causes a mass to accelerate. true
4. A centripetal force keeps a body moving in a straight line. false
5. The International Space Station is the largest artificial satellite in orbit. false
6. Sputnik was the first to be launched into orbit. true
7. Ireland is a member of the European Space Agency. true
8. In the absence of any friction, a moving mass would keep moving forever. true
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http://jwilson.coe.uga.edu/EMT669/Student.Folders/Lowry.Michelle/10th/sec4-5.html
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Section 4-5 "Greatest Common Factor"
Technology objective: Students will use the TI-34 calculators to find the greatest common factor of a pair of numbers.
Lesson: After a discussion of greatest common factor, ask the students if they can figure out how to use the TI-34 calculator to find the greatest common factor of a set of numbers. Put various suggestions on the board. Next, explain to the students how to use the TI-34 calculator to find the greatest common factor of a pair of numbers. They can do this by entering one number in the place of the numerator and one number in place of the denominator. Have them try to find the greatest common factor of 45 and 60. Have them enter the smaller number in for the numerator, 45, and the bigger number in for the denominator, 60. When the press enter, the calculator will simplify the fraction they entered. Ask the class to figure out what was factored out of the numerator and denominator. (This serves as a great mental exercise.) Once they have figured that number out, they know the greatest common factor. Have them try to find the greatest common factor of any two random numbers. Also discuss what happens when you enter two prime numbers. This can serve as a wonderful introduction to fractions.
Reinforcement:
Game
This game was found in the 1992 Yearbook Calculators in Mathematics Education in the article Statewide In-Service Programs on Calculators in Mathematics Teaching written by Bright, Lamphere, and Usnick. Students should be broken up into pairs. Each pair will be given a 6 x 6 grid with the following numbers inserted in no particular order:
25, 60, 45, 15, 10, 80, 48, 64, 36, 24, 65, 99, 27, 16, 42, 81, 75, 25, 200, 300, 500, 600, 800, 900, 360, 480, 640, 550, 270, 120, 144, 625, 525, 648, 864, and 468.
GCF game
1) Decide who plays first. Play then alternates.
2) On your turn, your opponent chooses an uncovered number on the grid.
3) You then choose a second uncovered number on the grid. Cover both choices.
4) Your score for the round is the GCF of the two numbers.
5) The winner is the player with the greatest cumulative score after five rounds.
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https://mcqmate.com/discussion/219831/which-is-the-right-way-to-declare-constant-in-c
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Q.
# Which Is The Right Way To Declare Constant In C?
A. Int Constant Var =10;
B. Int Const Var = 10;
C. Const Int Var = 10;
D. B & C Both
Answer» D. B & C Both
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https://math.stackexchange.com/questions/359630/are-there-versions-of-the-axiom-of-choice-that-restrict-the-size-of-the-factors
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Are there versions of the axiom of choice that restrict the size of the factors?
One formulation of the axiom of choice is that an arbitrary product of nonempty sets must be nonempty. The axiom of countable choice AC$_\omega$ is known to be strictly weaker than AC, but still independent of ZF; it is equivalent to the statement that a countable product of nonempty sets is nonempty.
Can one obtain a nontrivial but weaker version of AC by restricting the size of the factor sets in the product, rather than the size of the index set over which the product is taken?
For a non-example, say that AC$_z$ is the statement that an arbitrary product of countable sets is nonempty. Certainly ZF + AC implies ZF + AC$_z$. But it seems to me that ZF implies AC$_z$ even without AC: Let the index set be $\mathcal C$. Since for each factor set $s\in\mathcal C$ in the product there is a bijection $f_s: s\leftrightarrow \Bbb N$, the set $\{f_s(0): s\in\mathcal C\}$ is an element of the product.; Or equivalently if $\mathcal C$ is a family of countable sets, then the function $c:{\mathcal C} \to \bigcup{\mathcal C}$ which takes $s\mapsto f_s(0)$ is a choice function for $\mathcal C$, and exists in ZF without need for AC$_z$ as a separate axiom.
Let's say that the "restricted choice axiom" $AC_\Phi$ is the statement
If $\mathcal C$ is an arbitrary collection of sets, and $\Phi(X)$ holds for each element $X\in\mathcal C$, there is a choice function for $\mathcal C$.
Depending on $\Phi$, $AC_\Phi$ could be very weak, or it could be as strong as AC itself. My question is:
Is there $\Phi$ such that ZF + $AC_\Phi$ proves strictly more than ZF but strictly less than ZFC?
If I've made any errors of fact or logic in this post, I would be grateful for corrections.
• Cohen showed countable choice cannot be proven in ZF without choice, despite what your intuition tells you. – hardmath Apr 12 '13 at 17:21
• Did I say anything that implied otherwise? (This is a serious question. I didn't think I had, but this is a very tricky area and not all the implications are clear to me.) – MJD Apr 12 '13 at 17:23
• Maybe I misunderstood your remark about ZF implying $AC_Z$ "even without AC". Of course I assumed you were serious and just wanted to raise the possibility that forcing could shed light on a hierarchy of choice-conditioned-on-cardinality axioms. – hardmath Apr 12 '13 at 17:32
• Doesn't ZF imply AC$_z$ without AC? AC$_z$ is not the axiom of countable choice. I used AC$_\omega$ to denote the axiom of countable choice. – MJD Apr 12 '13 at 17:35
• You need AC to choose a bijection for all $s$ at once. To use Conway's terminology, products of counted sets are inhabited, but products of countable sets could be anything. – Zhen Lin Apr 12 '13 at 19:51
In most cases $\sf ZF$ cannot prove that choice from families of restricted sets (rather restricted families) exists. In fact limitation on the size of the family is often independent from the limitation on the size of the members of the family.
In fact even $\sf AC_{fin}$ which states "every family of non-empty finite sets admits a choice function" is independent from $\sf ZF$, but strictly weaker than $\sf ZFC$.
In the texts which deal a lot with fine versions of choice (see P. Howard, Rubin & Rubin, E. Hall, and so on) you will often see $C(X,Y)$ where $X$ is a limitation on the size of the family, and $Y$ is a limitation on the size of the members of the family.
Something of interest, there has been a work of classifying choice from families of finite sets, e.g. choice from pairs does not imply choice from triplets and vice versa. You can find the details in Jech, The Axiom of Choice, chapter 7.
In the same book, on chapter 8, Jech has a proof that $\sf DC_{\kappa}$ cannot prove that there is a choice function on a family of $\kappa^+$ pairs.
More examples include Cohen's first model in which the axiom of countable choice fails; but every well-ordered family of well-ordered sets admits a choice function (i.e. we can choose well-ordered uniformly). On the other hand, assuming that every well-ordered family admits a choice function cannot even prove that every uncountable set has a subset of size $\aleph_1$.
There are tons of intricate relations between the many different ways of weakening the axiom of choice. The general rule of thumb is that whenever something can go wrong, it will. Much like Murphy's Law.
• Thanks. I don't understand how that can be. If $\mathcal C$ is a collection of nonempty finite sets, then why can't we say that for each $S\in\mathcal C$ there is a bijection $f_S$ between $S$ and some nonempty initial segment of $\omega$, and then $c: S\mapsto f_S(0)$ is a choice function for $\mathcal C$? – MJD Apr 12 '13 at 17:43
• I'm just glad I grabbed a couple of upvotes before the Axiom-of-Choice upvote back-hole which is you came around! ;-) – user642796 Apr 12 '13 at 17:43
• @MJD: Because you have to choose those bijections. That's why! – Asaf Karagila Apr 12 '13 at 17:50
• @Arthur: It's lucky that I bought a tablet and I can use the internet in places which are not my own study room... Otherwise this answer would have been written in about an hour from now! – Asaf Karagila Apr 12 '13 at 17:51
• @MJD: It depends on what is $z$, and on a broader note, what is $\Phi$ which defines the particular property. If these are just limitations on the cardinality, or even on "there is a group structure" (for example) then it is indeed a mistake, and $\sf ZF$ does not imply such thing. But it might be reasonable to produce a statement $\Phi$ which is strong enough so that $\sf ZF$ does prove this sort of choice. For example $\Phi$ stating that all the sets are sets of ordinals. – Asaf Karagila Apr 12 '13 at 18:50
As a basic example, $\mathsf{AC}(\mathrm{fin})$, the statement that the Axiom of Choice holds for all families of finite sets, is known to be implied by the Ordering Principle (that every set can be linearly ordered), but is strictly stronger than $\mathsf{ZF}$ (the existence of a Russell sequence (a countable sequence of two elements sets without a Choice function) is consistent with $\mathsf{ZF}$ and would clearly contradict $\mathsf{AC}(\mathrm{fin})$).
(See Horst Herrlich's Axiom of Choice for some limited information about $\mathsf{AC}(\mathrm{fin})$ and Horst Herrlich & Eleftherios Tachtsis _On the number of Russell's socks or $2 + 2 + 2 + \cdots = ?$ for some basic information about Russell sequences.
• Thanks. I arrived at this question because I was wondering if Russell's socks example wasn't misleading, and if it might be possible to define a choice function for any large collection of pairs of socks. – MJD Apr 12 '13 at 17:39
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mersenneforum.org POST LOTS AND LOTS OF PRIMES HERE
Register FAQ Search Today's Posts Mark Forums Read
2010-03-17, 22:24 #1 Kosmaj Nov 2003 E2616 Posts POST LOTS AND LOTS OF PRIMES HERE After 2000 posts in our first prime posting thread, and more than 2300 reported primes, we are moving to a new thread. Please post your primes here! And happy hunting
2010-03-25, 11:10 #2 pb386 Nov 2003 23×163 Posts 1389*2^809811-1 (243781 digits)
2010-03-26, 10:20 #3 Beyond Dec 2002 22×3×37 Posts 2655*2^853543-1 (256946 digits)
2010-03-26, 11:04 #4 pb386 Nov 2003 23×163 Posts 1285*2^809975-1 (243830 digits)
2010-03-27, 14:14 #5 pb386 Nov 2003 130410 Posts 1065*2^961019-1 (289299 digits)
2010-03-28, 12:43 #6 pb386 Nov 2003 23×163 Posts 1119*2^957013-1 (288093 digits) 1069*2^811021-1 (244145 digits) 1023*2^811532-1 (244299 digits)
2010-03-28, 13:48 #7 pb386 Nov 2003 23·163 Posts 1095*2^960229-1 (289061 digits)
2010-03-29, 21:48 #8 Beyond Dec 2002 22·3·37 Posts 251*2^1269198-1 (382070 digits)
2010-03-30, 21:15 #9
Cruelty
May 2005
2×809 Posts
Quote:
Originally Posted by Beyond 251*2^1269198-1 (382070 digits)
Nice one! Congratulations!
2010-04-01, 07:31 #10 VBCurtis "Curtis" Feb 2005 Riverside, CA 104348 Posts I am the King of small megabit primes: 281*2^1009502-1 is prime. (303893 digits) That makes 3 megabits under 1020k. As we say in poker, I run good. -Curtis
2010-04-02, 14:59 #11 pb386 Nov 2003 23·163 Posts 7th drive 2295*2^867724-1 (261215 digits)
Similar Threads Thread Thread Starter Forum Replies Last Post 2147483647 YAFU 3 2016-12-25 21:44 TheMawn Software 18 2014-08-16 03:54 lsoule Riesel Prime Search 1999 2010-03-17 22:33 Peter Hackman Factoring 2 2008-08-15 14:26 jasong Information & Answers 4 2007-10-04 20:40
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# I am done with Manhattan GMAT SC Guide. Now I wish to have
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I am done with Manhattan GMAT SC Guide. Now I wish to have to some more practice and learning material. What should I go for Aristotle SC Grail or Ultimate GMAT Grammar Book? My purpose is to sharpen my skills for wrong choices identification and more practice.
Thanks a lot,
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26 Sep 2011, 02:34
VIKASMM wrote:
I am done with Manhattan GMAT SC Guide. Now I wish to have to some more practice and learning material. What should I go for Aristotle SC Grail or Ultimate GMAT Grammar Book? My purpose is to sharpen my skills for wrong choices identification and more practice.
Thanks a lot,
V.
You can try going through ultimate grammar book quickly say in 3-4 hours emphasizing on grammar rules instead of spending sometime on practice tests . Read aristotle 1st part " Grammar review " , revise mgmat sc book again and go through remaining aristotle book. This should help.
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26 Sep 2011, 10:59
I think the OG and the Verbal Book both contain good test prep questions. I also like the Kaplan 800 book, but it doesn't actually offer too many questions to practice on.
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28 Sep 2011, 05:16
Thanks guys! Now I want to know that which of the books is good for understanding the 'meaning' part of a sentence! I find it extremely difficult, specially when an answer choice alters the meaning, I find it hard to catch it, please help. Which is the book for me - Aristotle SC Grail or Ultimate GMAT Grammar Book?
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(There seems to be a misunderstanding, as 'Grad to Break' does not correspond to standard units of measurement and therefore does not have an SI unit short form. If 'Grad' refers to a unit of angle, it might be 'gon' or 'grade', but 'Break' is not recognized in SI units. converter)
Clarifying Unit Conversion: Unraveling the Mystery of Grad to Break
(Last Updated On: 2024-04-02)
The term 'Grad', in the context of angular measurement, is an alternative name for 'Gon' or 'Gradian'. A grad is equal to 1/400 of a circle, allowing for a division of the circle into 400 units. This system simplifies certain types of calculations and is particularly used in fields such as surveying and engineering where precise angle measurements are crucial.
Definition of Break
The definition of 'Break' in the context of unit conversion is not recognized as a standard unit of measurement. Without additional context, 'Break' could potentially refer to a pause or interruption in a process or activity, but it does not correspond to a quantifiable unit that can be converted from or to Grads or any other known unit of measurement.
10 N/A
20 N/A
30 N/A
40 N/A
50 N/A
60 N/A
70 N/A
80 N/A
90 N/A
100 N/A
Example 1:
Example 2:
The concept of 'Grad to Break' conversion does not have a recognized historical background. As 'Grad' represents a unit of angle measurement and 'Break' lacks a defined unit, the converter seems to represent an abstract or hypothetical concept rather than one with a solid foundation in measurement history.
How to use Grad to Break Converter
• Select the Grad to Break Converter tool on Newtum's webpage.
• Enter the number of Grads you wish to convert in the respective field.
• Since 'Break' is not a recognized unit, the conversion result will reflect an undefined or non-applicable value (N/A).
• Review the information provided to understand the limitations of this conversion.
Real Life Applications of Grad to Break
Explore the intriguing applications of the Grad to Break Converter and discover how this conceptual tool can spark discussions in various fields of study and education.
• Understanding the theoretical concepts of non-standard unit conversions.
• Educational purposes to illustrate the importance of standardized measurement units.
Solved Examples There seems to be a misunderstanding, as 'Grad to Break' does not correspond to standard units of measurement and therefore does not have an SI unit short form. If 'Grad' refers to a unit of angle, it might be 'gon' or 'grade', but 'Break' is not recognized in SI units.
Example 1: Converting 25 Grads to Break would theoretically result in a value that is not applicable (N/A), as Break is not a defined unit.
Example 2: Should you attempt to convert 100 Grads to Break, the outcome would similarly be non-applicable (N/A) for the same reason.
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This site is supported by donations to The OEIS Foundation.
# 28
Please do not rely on any information it contains.
28 is the second perfect number, the first to not be squarefree.
## Membership in core sequences
Even numbers ..., 22, 24, 26, 28, 30, 32, 34, ... A005843(14) Composite numbers ..., 25, 26, 27, 28, 30, 32, 33, ... A002808 Perfect numbers 6, 28, 496, 8128, 33550336, ... A000396 Triangular numbers ..., 10, 15, 21, 28, 36, 45, 55, ... A000217 Loeschian numbers ..., 21, 25, 27, 28, 31, 36, 37, ... A003136
In Pascal's triangle, 28 occurs four times, the first two times on the eighth row as the sum of 7 and 21. In Lozanić's triangle, 28 also first appears in the eighth row, though as the sum of 12, 19 and a tacit −3 from the previous row.
## Sequences pertaining to 28
Divisors of 28 1, 2, 4, 7, 14, 28 A018254 Multiples of 28 0, 28, 56, 84, 112, 140, 168, 196, ... A135628 28-gonal numbers 1, 28, 81, 160, 265, 396, 553, 736, ... A161935 Centered 28-gonal numbers 1, 29, 85, 169, 281, 421, 589, 785, ... A195314 28-gonal pyramidal numbers 1, 29, 110, 270, 535, 931, 1484, ... A256648 ${\displaystyle 3x+1}$ sequence starting at 9 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, ... A033479 ${\displaystyle 3x-1}$ sequence starting at 36 936, 18, 9, 26, 13, 38, 19, 56, 28, ... A008894 ${\displaystyle 5x+1}$ sequence starting at 11 11, 56, 28, 14, 7, 36, 18, 9, 46, 23, ... A259193
## Partitions of 28
There are 3718 partitions of 28. There are several nontrivial partitions of 28 into its own divisors, but since it is a perfect number, exactly only one of those consists of distinct divisors.
There are only two Goldbach representations of 28: 5 + 23 and 11 + 17.
## Roots and powers of 28
In the table below, irrational numbers are given truncated to eight decimal places.
${\displaystyle {\sqrt {28}}}$ 5.29150262 A010483 28 2 784 ${\displaystyle {\sqrt[{3}]{28}}}$ 3.03658897 A010599 28 3 21952 ${\displaystyle {\sqrt[{4}]{28}}}$ 2.30032663 A011023 28 4 614656 ${\displaystyle {\sqrt[{5}]{28}}}$ 1.94729436 A011113 28 5 17210368 ${\displaystyle {\sqrt[{6}]{28}}}$ 1.74258112 28 6 481890304 ${\displaystyle {\sqrt[{7}]{28}}}$ 1.60967004 28 7 13492928512 ${\displaystyle {\sqrt[{8}]{28}}}$ 1.51668277 28 8 377801998336 ${\displaystyle {\sqrt[{9}]{28}}}$ 1.44808927 28 9 10578455953408 ${\displaystyle {\sqrt[{10}]{28}}}$ 1.39545489 28 10 296196766695424 A009972
## Logarithms and 28th powers
In the OEIS specifically and mathematics in general, ${\displaystyle \log x}$ refers to the natural logarithm of ${\displaystyle x}$, whereas all other bases are specified with a subscript.
As above, irrational numbers in the following table are truncated to eight decimal places.
${\displaystyle \log _{28}2}$ 0.208015 ${\displaystyle \log _{2}28}$ 4.80735 2 28 2.68435e+08 ${\displaystyle \log _{28}e}$ 0.300102 ${\displaystyle \log 28}$ 3.3322 A016651 ${\displaystyle e^{28}}$ 1.44626e+12 ${\displaystyle \log _{28}3}$ 0.329695 ${\displaystyle \log _{3}28}$ 3.0331 3 28 2.28768e+13 ${\displaystyle \log _{28}\pi }$ 0.343535 ${\displaystyle \log _{\pi }28}$ 2.91091 ${\displaystyle \pi ^{28}}$ 8.3214e+13 ${\displaystyle \log _{28}4}$ 0.416029 ${\displaystyle \log _{4}28}$ 2.40368 4 28 7.20576e+16 ${\displaystyle \log _{28}5}$ 0.482995 ${\displaystyle \log _{5}28}$ 2.07042 5 28 3.72529e+19 ${\displaystyle \log _{28}6}$ 0.53771 ${\displaystyle \log _{6}28}$ 1.85974 6 28 6.14094e+21 ${\displaystyle \log _{28}7}$ 0.583971 ${\displaystyle \log _{7}28}$ 1.71241 7 28 4.59987e+23 ${\displaystyle \log _{28}8}$ 0.624044 ${\displaystyle \log _{8}28}$ 1.60245 8 28 1.93428e+25 ${\displaystyle \log _{28}9}$ 0.659391 ${\displaystyle \log _{9}28}$ 1.51655 9 28 5.23348e+26 ${\displaystyle \log _{28}10}$ 0.69101 ${\displaystyle \log _{10}28}$ 1.44716 10 28 1e+28
See A122969 for the 28th powers of integers.
## Values for number theoretic functions with 28 as an argument
${\displaystyle \mu (28)}$ 0 ${\displaystyle M(28)}$ –1 ${\displaystyle \pi (28)}$ 9 ${\displaystyle \sigma _{1}(28)}$ 56 Note that this is twice 28. ${\displaystyle \sigma _{0}(28)}$ 6 ${\displaystyle \phi (28)}$ 12 ${\displaystyle \Omega (28)}$ 3 ${\displaystyle \omega (28)}$ 2 ${\displaystyle \lambda (28)}$ 6 This is the Carmichael lambda function. ${\displaystyle \lambda (28)}$ –1 This is the Liouville lambda function. ${\displaystyle \zeta (28)}$ 1.0000000037253340247884570548192... 28! 304888344611713860501504000000 ${\displaystyle \Gamma (28)}$ 10888869450418352160768000000
## Factorization of 28 in some quadratic integer rings
As was mentioned above, the prime factorization of 28 is ${\displaystyle 2^{2}\times 7}$ in ${\displaystyle \mathbb {Z} }$. But it has different factorizations in some quadratic integer rings.
${\displaystyle \mathbb {Z} [i]}$ ${\displaystyle (1-i)^{2}(1+i)^{2}7}$ ${\displaystyle \mathbb {Z} [{\sqrt {-2}}]}$ ${\displaystyle ({\sqrt {-2}})^{4}7}$ ${\displaystyle \mathbb {Z} [{\sqrt {2}}]}$ ${\displaystyle ({\sqrt {2}})^{4}(3\pm {\sqrt {2}})}$ ${\displaystyle \mathbb {Z} [\omega ]}$ ${\displaystyle 2^{2}(-2+\omega )(-2+\omega ^{2})}$ ${\displaystyle \mathbb {Z} [{\sqrt {3}}]}$ ${\displaystyle (1\pm {\sqrt {3}})^{2}7}$ ${\displaystyle \mathbb {Z} [{\sqrt {-5}}]}$ 2 2 × 7 ${\displaystyle \mathbb {Z} [\phi ]}$ 2 2 × 7 ${\displaystyle \mathbb {Z} [{\sqrt {-6}}]}$ ${\displaystyle 2^{2}(1\pm {\sqrt {-6}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {6}}]}$ ${\displaystyle (2\pm {\sqrt {6}})^{2}7}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}}$ ${\displaystyle (-1)\left({\frac {1}{2}}\pm {\frac {\sqrt {-7}}{2}}\right)^{2}({\sqrt {-7}})^{2}}$ ${\displaystyle \mathbb {Z} [{\sqrt {7}}]}$ ${\displaystyle (3\pm {\sqrt {7}})^{2}({\sqrt {7}})^{2}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-10}}]}$ 2 2 × 7 ${\displaystyle \mathbb {Z} [{\sqrt {10}}]}$ 2 2 × 7 ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}}$ ${\displaystyle \mathbb {Z} [{\sqrt {11}}]}$ ${\displaystyle (-1)(3\pm {\sqrt {11}})^{2}(2\pm {\sqrt {11}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {-13}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}}$ 2 2 × 7 ${\displaystyle \mathbb {Z} [{\sqrt {-14}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {14}}]}$ ${\displaystyle (-1)(4\pm {\sqrt {14}})^{2}(7\pm 2{\sqrt {14}})}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}}$ ${\displaystyle \mathbb {Z} [{\sqrt {15}}]}$ 2 2 × 7 ${\displaystyle \mathbb {Z} [{\sqrt {-17}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}}$ ${\displaystyle \left({\frac {3}{2}}\pm {\frac {\sqrt {17}}{2}}\right)^{2}7}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}}$ ${\displaystyle 2^{2}\left({\frac {3}{2}}\pm {\frac {\sqrt {-19}}{2}}\right)}$ ${\displaystyle \mathbb {Z} [{\sqrt {19}}]}$ ${\displaystyle (13\pm 3{\sqrt {19}})^{2}7}$
## Representation of 28 in various bases
Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 11100 1001 130 103 44 40 34 31 28 26 24 22 20 1D 1C 1B 1A 19 18
${\displaystyle -1}$ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1729
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# Is a position an observable ?
1. Jul 13, 2012
### audioloop
are observables only those quantities which commute with system's Hamiltonian ?
2. Jul 13, 2012
### bhobba
No. If it commutes with the Hamiltonian it simply means you can measure both it and energy simultaneously. It also means it probably will be conserved if the Hamiltonian has no specific time dependence - actually there is no probably about it - it will be conserved.
Thanks
Bill
Last edited: Jul 13, 2012
3. Jul 14, 2012
### Marioeden
As to your original question, position is an observable. Obviously from Heisenberg's Uncertainty Principle, the uncertainty in the momentum would then be infinite, so you have the classic statement of how if you know where a particle is, you have no idea where it's going or how fast. The converse statement is also true, you can know its momentum but consequently have no idea where it is.
4. Jul 14, 2012
### OhYoungLions
As far as I know, in principle ANY Hermitian operator is associated with an observable, which are simply the eigenvalues of that operator. For the vast majority of such operators, the observable is not very interesting. For example, the Hamiltonian operator is associated with the observable "Energy". If the operators associated with two observables commute, then your system can be in a state where both observables can be simultaneously well defined.
Remember: just because a quantity is called an observable in QM doesn't mean we know how to build a machine to measure it, or that we even care about it.
5. Jul 15, 2012
### jfy4
also, it can (and I've seen it done) be argued that position is the only observable.
6. Jul 15, 2012
### bhobba
Yes I have seen it argued as well - but its based on the silly idea the outcome of any observation is the position of a pointer or something like that. People like that are stuck in a time warp IMHO and are not in the computer age. Observations can be captured digitally not having anything to do with position at all.
Thanks
Bill
7. Jul 15, 2012
### vanhees71
It depends on how you define "position" and which system to look at whether there is such a thing as a position observable.
In non-relativistic physics for (necessarily massive) particles of any spin, there always exists a position observable defined via the representation theory of the Poincare group, which gives you the 10 conserved quantities and their commutation relations (energy, momentum, angular momentum, center-of-momentum coordinates). Then you can define the position variable as that not explicitly time dependent observable which together with the momentum coordinates fulfills the Heisenberg algebra
$$[\hat{x}_j,\hat{p}_k]=\mathrm{i} \hbar \delta_{jk}.$$
In relativistic physics also for all massive particles there exists a position observable of this kind, for massless particles that's the case only for particles with spin 0 and spin 1/2. For all massless particles of higher spin, especially also for photons, there is no position observable in the strict sense. For details, see Arnold Neumaier's Theoretical Physics FAQ:
http://www.mat.univie.ac.at/~neum/physfaq/topics/position.html
Last edited: Jul 15, 2012
8. Jul 15, 2012
### audioloop
9. Jul 18, 2012
### lugita15
The issue is this: position is the primary way humans interact with the world. We see where objects are, we hear them, etc. Even if the information is stored digitally, we still have to acquire that information somehow, in a digital display for instance, and that involves sight. So the argument is that we only directly deal with position, and everything else we conclude about the world comes indirectly, from interpretation of the positional data of the senses.
10. Aug 15, 2012
### A. Neumaier
Sight is not position but reception of the electromagnetic field. Position is reconstructed from what we see by a nontrivial process.
11. Aug 15, 2012
### bhobba
Exactly - scratching my head why anyone would think otherwise.
And reading information on a computer screen has nothing to do with the position of the screen or the position of whatever you use to present the information.
Thanks
Bill
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