url stringlengths 6 1.61k | fetch_time int64 1,368,856,904B 1,726,893,854B | content_mime_type stringclasses 3 values | warc_filename stringlengths 108 138 | warc_record_offset int32 9.6k 1.74B | warc_record_length int32 664 793k | text stringlengths 45 1.04M | token_count int32 22 711k | char_count int32 45 1.04M | metadata stringlengths 439 443 | score float64 2.52 5.09 | int_score int64 3 5 | crawl stringclasses 93 values | snapshot_type stringclasses 2 values | language stringclasses 1 value | language_score float64 0.06 1 |
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http://stackoverflow.com/questions/7536465/create-a-2d-array-with-a-nested-loop | 1,369,418,458,000,000,000 | text/html | crawl-data/CC-MAIN-2013-20/segments/1368704933573/warc/CC-MAIN-20130516114853-00079-ip-10-60-113-184.ec2.internal.warc.gz | 234,538,390 | 11,815 | # Create a 2D array with a nested loop
The following code
``````n = 3
matrix = [[0] * n] * n
for i in range(n):
for j in range(n):
matrix[i][j] = i * n + j
print(matrix)
``````
prints
``````[[6, 7, 8], [6, 7, 8], [6, 7, 8]]
``````
but what I expect is
``````[[0, 1, 2], [3, 4, 5], [6, 7, 8]]
``````
Why?
-
## 3 Answers
Note this:
``````>>> matrix = [[0] * 3] * 3
>>> [x for x in matrix]
[[0, 0, 0], [0, 0, 0], [0, 0, 0]]
>>> [id(x) for x in matrix]
[32484168, 32484168, 32484168]
>>>
``````
Three rows but only one object.
See the docs especially Note 2 about the `s * n` operation.
Fix:
``````>>> m2= [[0] * 3 for i in xrange(5)]
>>> [id(x) for x in m2]
[32498152, 32484808, 32498192, 32499952, 32499872]
>>>
``````
Update: Here are some samples of code that gets an answer simply (i.e. without `iter()`):
``````>>> nrows = 2; ncols = 4
>>> zeroes = [[0 for j in xrange(ncols)] for i in xrange(nrows)]
>>> zeroes
[[0, 0, 0, 0], [0, 0, 0, 0]]
>>> ap = [[ncols * i + j for j in xrange(ncols)] for i in xrange(nrows)]
>>> ap
[[0, 1, 2, 3], [4, 5, 6, 7]]
>>>
``````
-
The docs says "...This often haunts new Python programmers...". True! But why this feature can be useful? – user949952 Sep 24 '11 at 2:20
Try running `matrix[0][0] = 0` after that. Notice it now becomes:
``````[[0, 7, 8], [0, 7, 8], [0, 7, 8]]
``````
So it's changing all three at the same time.
That seems to explain it.
-
Thanks Mk12 for the link. And I have the same question: whether this is a "feature" of Python or a "bug."? – user949952 Sep 24 '11 at 2:08
It isn't something I expected, that's for sure. But it makes sense. – Mk12 Sep 24 '11 at 2:13
@user949952: The docs (see my answer) say that it takes a shallow copy of the sequence. So it's a feature. – John Machin Sep 24 '11 at 2:14
``````>>> it = iter(range(9))
>>> [[next(it) for i in range(3)] for i in range(3)]
[[0, 1, 2], [3, 4, 5], [6, 7, 8]]
``````
just replace `3` with `n` and `9` with `n**2`
Also (just answer the "why?"), you're making copies of the same list with multiplication and, therefore, modifying one of them will modify all of them.
- | 811 | 2,110 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.09375 | 3 | CC-MAIN-2013-20 | latest | en | 0.844482 |
https://au.mathworks.com/matlabcentral/answers/516676-write-a-function-called-tri_area-returns-the-area-of-a-triangle-with-base-b-and-height-h | 1,620,358,919,000,000,000 | text/html | crawl-data/CC-MAIN-2021-21/segments/1620243988774.96/warc/CC-MAIN-20210507025943-20210507055943-00067.warc.gz | 111,853,355 | 34,409 | # write a function called tri_area returns the area of a triangle with base b and height h
497 views (last 30 days)
Andrew Ayman on 9 Apr 2020
Commented: Walter Roberson on 4 Apr 2021
hello this is my function code and command window code and there is a message of invalid expression at line 2 and i dont know what is the wrong can anyone help me
function [area] = tri_area([b,h]);
tri_area([b,h])=(0.5)*(b)*(h)
area=tri_area([b,h])
end
%command window
area = tri_area[3,2])
Christine Mizzi on 27 Aug 2020
What is the purpose for writing two output arguments in the code? i.e. [area, tri_area]
If the user is calling the area of a triangle wouldn't that be only one output argument?
Torsten on 9 Apr 2020
Edited: darova on 9 Apr 2020
function area = tri_area(b,h)
area = 0.5*b*h;
end
From the command window
A = tri_area(3,2)
Walter Roberson on 10 Aug 2020
Have you considered adding disp statements so you can see what parameters are being passed for the random input case?
Imane Tahar on 19 Nov 2020
function area = tri_area(b,h)
area = (b*h)/2
end
Ramakant Gupta on 15 May 2020
Edited: Walter Roberson on 2 Jun 2020
function area = tri_area(b,h)
area = 0.5*b*h;
end
##### 2 CommentsShowHide 1 older comment
madhan ravi on 2 Jun 2020
Maybe he wanted to test his first answer xD in the forum.
Eshan Pansare on 28 Aug 2020
How to solve the random inputs part?
prudhvi gandham on 6 Nov 2020
function area = tri_area(b,h)
area = 0.5*b*h;
end
Walter Roberson on 4 Apr 2021
Siya Desai on 4 Apr 2021
Edited: Walter Roberson on 4 Apr 2021
function
function [area] = tri_area (b,h)
tri_area = (0.5)*(b)*(h)
tri_area(2,3) %any random input
Walter Roberson on 4 Apr 2021
result = tri_area(2,3) %any random input
tri_area = 3
Output argument "area" (and maybe others) not assigned during call to "solution>tri_area".
function [area] = tri_area (b,h)
tri_area = (0.5)*(b)*(h)
end | 596 | 1,875 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.859375 | 3 | CC-MAIN-2021-21 | latest | en | 0.743651 |
https://onlinerecnik.com/recnik/engleski/srpski/hyperbola | 1,624,160,848,000,000,000 | text/html | crawl-data/CC-MAIN-2021-25/segments/1623487655418.58/warc/CC-MAIN-20210620024206-20210620054206-00606.warc.gz | 386,044,333 | 9,439 | hyperbola | englesko - srpski prevod
## hyperbola
imenicamatematika
ETYM Greek, prop., an overshooting, excess, i.e., of the angle which the cutting plane makes with the base. Related to Hyperbole.
An open curve formed by a plane that cuts the base of a right circular cone.
Curve formed when cone is cut by plane which makes angle with the base greater than that made by side of cone.
In geometry, a curve formed by cutting a right circular cone with a plane so that the angle between the plane and the base is greater than the angle between the base and the side of the cone. All hyperbolae are bounded by two asymptotes (straight lines which the hyperbola moves closer and closer to but never reaches).
A hyperbola is a member of the family of curves known as conic sections.
A hyperbola can also be defined as a path traced by a point that moves such that the ratio of its distance from a fixed point (focus) and a fixed straight line (directrix) is a constant and greater than 1; that is, it has an eccentricity greater than 1.
### 1.hiperbola
ženski rodmatematika
1. geom. Kriva u ravni koja ima svojstvo da je za svaku njenu tačku razlika otstojanja od dve stalne tačke uvek ista (konstantna); up. elipsa, parabola;
2. ret. Preterivanje, preterano uvećavanje onoga što treba kazati ili preterano umanjivanje onoga o čemu se govori, u nameri da se na to obrati što veća pažnja, npr.: "LJuto cvili do neba se čuje", "Pravi se manji od makova zrna".
hyperbole
## Naši partneri
Škole stranih jezika | Sudski tumači/prevodioci | 435 | 1,537 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.109375 | 3 | CC-MAIN-2021-25 | latest | en | 0.756387 |
https://www.jiskha.com/search/index.cgi?query=highway | 1,502,979,539,000,000,000 | text/html | crawl-data/CC-MAIN-2017-34/segments/1502886103316.46/warc/CC-MAIN-20170817131910-20170817151910-00576.warc.gz | 924,631,590 | 10,767 | # highway
737 results
### physics
A highway is made of concrete slabs that are 14.0 m long at 20° C. (a) If the temperature range at the location of the highway is from -17.8° C to +40.8° C, what size expansion gap should be left (at 20° C) to prevent buckling of the highway? (b) How large are the gaps at...
### word problem
Millwood city is constructing a new highway through town. The construction crew can complete 5 3/6 miles of road each month. The total new highway will be 54 miles long. Will the highway be finished in 8 months? I know that 36 2/5 miles of highway can be completed in 6 1/2 ...
### calculus
a road that runs perpendicular to a highway leads to a farmhouse located one mile off the highway an automobile travels down the highway past the road leading to the farmhouse at a speed of 60 mph how fast is the distance between the farmhouse an automobile increasing when the...
### Math
Suppose that the amount of time it takes to build a highway varies directly with the length of the highway and inversely with the number of workers. Suppose also that it takes 100 workers 6 weeks to build 4 miles of highway. How long will it take 160 workers to build 16 miles ...
### physics
A salesperson leaves the office and drives 26 km north along a straight highway. A turn is made onto a highway that leads in a direction of 60 degrees. The driver continues on the highway for a distance of 62 km and then stops. What is the total displacement of the salesperson...
### PHYSICS
A salesperson leaves the office and drives 26 km north along a straight highway. A turn is made onto a highway that leads in a direction of 60 degrees. The driver continues on the highway for a distance of 62 km and then stops. What is the total displacement of the salesperson...
### math
A rancher with 7000 yds of fencing wants to enclose a rectangular field that borders a straight highway and then wants to devide it into two plots with a fence parellel to the highway. If no fence is needed along the highway, what is the largest area that the farmer can enclose?
### algebra
One paving machine can surface 1 km of highway in 6 hours. Another can surface 1 km of highway in 8 hours. how long would it take to surface 1 km of highway with the two machines working together?
### MATH
A truck enters a highway driving 60mph. A car enter the highway at the same place 14 minutes later and drives 66mph in the dame direction. From the time the car enters the highway, how long will it take the car to pass th truck?
A highway is to be built between two towns, one of which lies 44.0 km south and 74.0 km west of the other. What is the shortest length of highway that can be built between the two towns, and at what angle would this highway be directed?
### physics
A highway is to be built between two towns, one of which lies 24.0 km south and 76.0 km west of the other. What is the shortest length of highway that can be built between the two towns, and at what angle would this highway be directed?
### math
the new highway will be a total of 54 miles long . will the highway be finish in 8 months
### Math
The new highway will be a total of 54 mils long. Will the highway be finished in 8 months?
### math
the new highway will be a total of 54 miles long. will the highway be finishe in 8 months
### math
using the example above, the new highway will be a total of 54 miles long. will the highway be finished in 8 mouths
### Calculus
A road perpendicular to a highway leads to a farmhouse located 6 mile away. An automobile traveling on the highway passes through this intersection at a speed of 75mph. How fast is the distance between the automobile and the farmhouse increasing when the automobile is 10 miles...
### physics
A highway is being constructed to accommodate traffic for speeds of 22.8 m/s. If the angle of the bank is 6.28° and there is no friction force, what is the highway's curve radius?
### Math, Geometry
The percent grade of a highway is the amount that the highway raises or falls in a given horizontal distance. For example, a highway with a four percent grade rises 0.04 mile for every 1 mile of horizontal distance. a. How many feet does a highway with a six percent grade rise...
### Math
The percent grade of a highway is the amount that the highway raises or falls in a given horizontal distance. For example, a highway with a four percent grade rises 0.04 mile for every 1 mile of horizontal distance. a. How many feet does a highway with a six percent grade rise...
### calc
The department of transportation has proposed a new 7-mile stretch of highway. The elevation of the highway x miles along the path is given by E(x)=.004 (x^4-8x^3)+2. Where is the steepest point
### Math
Millwood City is constructing a new highway through town. The construction crew can complete 5 3/5 miles of road each month. 6Can the highway be finished in 8 months?
### Science
A highway is to be built between two towns, one of which lies 98.0 km south and 24.0 km west of the other. (a) What is the shortest length of highway that can be built between the two towns, and (b) at what angle would this highway be directed, as a positive angle with respect...
### physics
A highway is to be built between two towns, one of which lies 49.0 km south and 95.0 km west of the other. (a) What is the shortest length of highway that can be built between the two towns, and (b) at what angle would this highway be directed, as a positive angle with respect...
### physics
A highway is to be built between two towns, one of which lies 47.0 km south and 69.0 km west of the other. (a) What is the shortest length of highway that can be built between the two towns, and (b) at what angle would this highway be directed, as a positive angle with respect...
### physics
a highway is to be built between two towns, one of which lies 35.0 km south and 72.0 km west of the other. what is the shortest length of highway that can be built between the two towns, and at what angle would this highway be with respect due to west? can some one solve this ...
### intermediate algebra
A farmer with 3000 feet wants to enclose a rectangular plot that borders on a straight highway. If the farmer does not fence the side along the highway, what is the largest area that can be enclosed?
### math
A farmer with 2000 meters of fencing wants to enclose a rectangular plot that borders on a straight highway. If the farmer does not fence the side along the highway, what is the largest area that can be enclosed?
### math
Mr. Mitchells office is on a winding side road next to a highway exit.He is driving on the highway at 60mph, and at this rate he expected to arrive at his office at 8:45a.m.However at 8:30 he was forced to leave the highway and take the side road, driving at a steady speed of ...
### Calculus
A road perpendicular to a highway leads to a farmhouse located 6 mile away. An automobile traveling on the highway passes through this intersection at a speed of 55mph. How fast is the distance between the automobile and the farmhouse increasing when the automobile is 2 miles ...
### Math
According to a rating agency, a car's MPG (miles per gallon) ratings are: 25 MPG for city and 30 MPG for highway driving. A driver spent \$20.00 (the gasoline costs \$1.25/gallon) on a combined city and highway trip of 450 miles. How many miles "C" in city and "H" on highway was...
### intermediate algebra
A farmer with 3000 feet of fencing wants to enclose a rectangular plot that borders on a straight highway. If the farmer does not fence the side along the highway, what is the largest area that can be enclosed?
### Math
According to a rating agency, a car's MPG (miles per gallon) ratings are: 25 MPG for city and 30 MPG for highway driving. A driver spent \$20.00 (the gasoline costs \$1.25/gallon) on a combined city and highway trip of 450 miles. How many miles "C" in city and "H" on highway was...
### Math
According to a rating agency, a car's MPG (miles per gallon) ratings are: 25 MPG for city and 30 MPG for highway driving. A driver spent \$20.00 (the gasoline costs \$1.25/gallon) on a combined city and highway trip of 450 miles. How many miles "C" in city and "H" on highway was...
### Math
Officer Weihe is in a patrol car sitting 25ft from the highway and observes Jacob in his car approaching. At a particular instant, t seconds, Jacob is x feet down the highway. The line of sight to Jacob makes an angle of theta radians to a perpendicular to the highway. Find (...
### English
Choose the sentence that is written in active voice. A. A serious accident was prevented on the highway by Roger's quick thinking. B. Roger's quick thinking prevented a serious accident on the highway. C. A serious accident was prevented on the highway. D. A serious accident ...
### physics
A 1000 kg car is travelling on the highway at a constant speed of 110 km/h. The highway makes a turn that is banked at an angle of 10o and has a radius of 160 m. What is the coefficient of friction between the car tires and the pavement in the curve?
### Math
Rochelle's car average 32.5 miles per gallon the highway.At that rate,how many gallons of gas will her car use for driving on the highway for 260 miles?
### algebra
at the airport the new run way will be parallel to a highway. on the scale drawing of the airport the equation that represents the highway is 6y=8x-11. what is the equation for the runway
### physics
Driving down Highway 401 in Ontario, it had rained and stopped raining by the time I came to the freezing rain on the highway. I knew there was a problem when the cars in front of me started to fly off the highway at the curve in the road. Assume the curve in the road is ...
### Trigonometry
Points A and B 1000 meter apart are plotted on a straight highway running east and west. from A, the bearing of tower C is N32*W and from be, the bearing of C is N64*E. approximate the shortest distance of the tower from the highway. please explain........
### economics
To save on gasoline expenses, Edith and Mathew agreed to carpool together for traveling to and from work. Edith preferred to travel on I-20 highway as it was usually the fastest, taking 25 minutes in the absence of traffic delays. Mathew pointed out that traffic jams on the ...
### Calculus - Related Rates
A road perpendicular to a highway leads to a farmhouse located 5 miles away. An automobile traveling on the highway passes through this intersection at a speed of 65mph. How fast is the distance between the automobile and the farmhouse increasing when the automobile is 1 miles...
### trigonometry
An airplane is flying at an elevation of 5150 ft, directly above a straight highway. Two motorists are driving cars on the highway on opposite sides of the plane, and the angle of depression to one car is 35° and to the other is 55°. How far apart are the cars? (Round your ...
### Social Studies
What were the effects of the interstate highway system? 1.it caused decline in air 2.it connected surban areas to metropolitan areas 3.highway constructions projects created jobs 4.it enabled people to drive longer distances at faster speeds my best answer is 1 and 3 can u ...
### math
Mark's house is 90 miles away from New York. You travel 15 miles on city streets, and then 75 miles on the highway. a. your average speed is 25 miles per hour in the city and 60 miles per hour on the highway. Use the formula time=distance/rate. How much time will you spend ...
### math
two cars are approaching each other on the same highway one from san bernardino at 80 mph and the other from las vegas at 60 mph how far apart are they on the highway one hour before they meet?
### physics
a car travels along a highway with a velocity of 24m/s, west. then exits the highway and 4s later, its instantaneous velocity is 16m/s, 45degrees north of west. Calculate the magnitude of the average accelaration of the car during the 5s interval?
### economics
To save on gasoline expenses, Edith and Mathew agreed to carpool together for traveling to and from work. Edith preferred to travel on I-20 highway as it was usually the fastest, taking 25 minutes in the absence of traffic delays. Mathew pointed out that traffic jams on the ...
### math
The transmitter site of radio station WGGW and the transmitter site of another radio station, WGWB, are on the same highway 100 miles apart. The radio signal from the transmitter site of WGWB can be received only within a radius of 60 miles in all directions from the WGWB ...
### algebra 2
The fuel efficiency rating for a certain car is 28 miles per gallon on the highway and 21 miles per gallon in the city. During one week the car traveled a total of 10.5 gallons of fuel. a) please set up an equation(s) that would model the situation.(be sure to identify ...
### trigonometry
An airplane is flying at an elevation of 5150 ft, directly above a straight highway. Two motorists are driving cars on the highway, both on one side of the plane. If the angle of depression to one car is 34° and to the other is 51°, how far apart are the cars? (Round your ...
### Calculus
a rectangular lot adjacent to a highway is to be enclosed by a fencing cost \$ 2.50 per foot.along the highway \$1.50 per foot on the other sides find the dimension of the largest lot that can be fenced off for \$270. Thank you God bless. Badly need it
### Algebra 2
The shape of a park can be modeled by a circle with the equation x^2 + y^2 = 1600. A stretch of highway near the park is modeled by the equation y = 1/40(x - 40)^2. At what points does a car on the highway enter or exit the park? I am soooooo lost!
### physics
A car travels along a highway with a velocity of 24 m/s, west. The car exits the highway and 4.0 s later, its instantaneous velocity is 16 m/s, 45o north of west. What is the magnitude of the average acceleration of the car during the five-second interval?
### math
a cb radio station c is located 3 mi from the interstate highway h. the station has a range of 6.1 mi in all directions from the station. if the interstate is along a straight line, how many miles of highway are in the range of this staion?
### Math
Could someone answer this question so I understand it. The answer is 28.0, but i'd like to know how to do it. Thanks Randomly selected cars were weighed, and the highway fuel consumption amounts (in miles/gal) were determined. For 20 cars, the linear correlation coefficient is...
### Geometry Part 2
A highway makes an angle of 6 with the horizontal.This angle is maintained for a horizontal distance of 5 miles.To the nearest hundredth of a mile, how high does the highway rise in this 5- mile section?Show the steps you use to find the distance.
### physical
A highway of concrete slabs is to be built in the Libyan desert, where the highest air temperature recorded is 57.8 °C. The temperature is 23.1 °C during construction of the highway. The slabs are measured to be 11.7 m long at this temperature. How wide should the expansion ...
### math
It takes 4 workers five days to clean 180 miles of highway. How many workers are needed to clean 250 miles of highway in just 6 days?
### maths
A straight highway leads to the foot of a tower of height 50m. From the top of the tower, the angles of depression of two cars standing on the highway are 30 degree and 60 degree. What is the distance between the two cars and how far is each car from the tower ?
### maths
A straight highway leads to the foot of a tower of height 50m. From the top of the tower, the angles of depression of two cars standing on the highway are 30 degree and 60 degree. What is the distance between the two cars and how far is each car from the tower ?
### English
What does diverge mean in the following sentence? After traveling approximately 3.5 miles, the old highway will diverge. Stay in the left lane and follow until you see the new mall. a. the highway will end. b. the highway will divide into two lanes that go in different ...
### PHYSICS
A banked circular highway curve is designed for traffic moving at 60 km/h. The radius of the curve is 205 m. Traffic is moving along the highway at 40 km/h on a rainy day. What is the minimum coefficient of friction between tires and road that will allow cars to negotiate the ...
### physics
A banked circular highway curve is designed for traffic moving at 84 km/h. The radius of the curve is 152 m. Traffic is moving along the highway at 43 km/h on a rainy day. What is the minimum coefficient of friction between tires and road that will allow cars to negotiate the ...
### Physics Mechanics
A banked circular highway curve is designed for traffic moving at 57 km/h. The radius of the curve is 211 m. Traffic is moving along the highway at 48 km/h on a rainy day. What is the minimum coefficient of friction between tires and road that will allow cars to negotiate the ...
### Math
Millwood City is constructing a new highway through town. The construction crew can complete 5 3/5 miles of road each month. Can the highway be finished in 8 months? 6 1/2 months have already passed by and so far they have 36 2/5 miles complete. Thanks! My friend and I are ...
### Ap calc
A traffic engineer monitors the rate at which cars enter a freeway onramp during rush hour. From her data, she estimates that between 4:30pm and 5:30pm the rate R(t) at which cars enter is given by: R(t)= 100(1-0.00012t^2) car per min., where t is time in min. since 4:30. a. ...
To save on gasoline expenses, Edith and Mathew agreed to carpool together for traveling to and from work. Edith preferred to travel on I-20 highway as it was usually the fastest, taking 25 minutes in the absence of traffic delays. Mathew pointed out that traffic jams on the ...
### math
suppose that it takes 100 workers 6 weeks to build 4 miles of highway. how many miles of highway could 160 workers build in 15 weeks
### Math
Fifteen smart cars were randomly selected and the highway mileage of each was noted. The analysis yielded a mean of 47 miles per gallon and a standard deviation of 5 miles per galon. Which of the following would represent a 90% confidence interval for the average highway ...
### Math - Linear Equations
A car gets 21 mi/gal in city driving and 28 mi/gal highway driving. If 18 gal of gas are used in traveling 448 mi, how many miles were driven in the city, how many driven on the highway (assuming that only the given rates of usage were actually used)?
### AP Calculus AB
A traffic engineer monitors the rate at which cars enter a freeway onramp during rush hour. From her data, she estimates that between 4:30 pm and 5:30 pm, the rate R(t) at which cars enter is given by: R(t)=100(1-0.00012t^2) cars per minute, where t is time in minutes since 4:...
### trigonometry
a surveyor 1standing on a highway observed the angle of elevation of the airplain as 22.5°. at the same time, a surveyor 2 observed the angle of elevation of the same airplane to be 30°. if the airplane was 850m directly above a point on the highway, find the distance ...
### math
A driver is headed north on a long, straight highway and sees this sign: Nearville 150 miles Farville 160 miles Then, surprisingly, an hour later she sees this apparently inconsistent sign on the same highway: Nearville 100 miles Farville 109 miles How can this be possible?
### Physics
A banked circular highway curve is designed for traffic moving at 60 km/h. The radius of the curve is 207 m. Traffic is moving along the highway at 42 km/h on a rainy day. What is the minimum coefficient of friction between tires and road that will allow cars to take the turn ...
### physics
A banked circular highway curve is designed for traffic moving at 55 km/h. The radius of the curve is 220 m. Traffic is moving along the highway at 38 km/h on a rainy day. What is the minimum coefficient of friction between tires and road that will allow cars to take the turn ...
### physics
A banked circular highway curve is designed for traffic moving at 57 km/h. The radius of the curve is 216 m. Traffic is moving along the highway at 42 km/h on a rainy day. What is the minimum coefficient of friction between tires and road that will allow cars to take the turn ...
### Math
The city highway department bought 3/4 tons of salt to put on the road when it snows. They have used 2/7 of salt. How many tons has the city highway department used?
### physics
A car is traveling at a constant speed of 27.1 m/s on a highway. At the instant this car passes an entrance ramp, a second car enters the highway from the ramp. The second car starts from rest and has a constant acceleration. What acceleration must it maintain, so that the two...
### phyics
A car is traveling at a constant speed of 31.8 m/s on a highway. At the instant this car passes an entrance ramp, a second car enters the highway from the ramp. The second car starts from rest and has a constant acceleration. What acceleration must it maintain, so that the two...
### physics
A car is traveling at a constant speed of 25.5 m/s on a highway. At the instant this car passes an entrance ramp, a second car enters the highway from the ramp. The second car starts from rest and has a constant acceleration. What acceleration must it maintain, so that the two...
### Physics
A banked circular highway curve is designed for traffic moving at 65 km/h. The radius of the curve is 207 m. Traffic is moving along the highway at 43 km/h on a rainy day. What is the minimum coefficient of friction between tires and road that will allow cars to negotiate the ...
### Calculus
A straight driveway perpendicular to a straight highway leads to a house located one mile away. A truck travels past the house traveling 60 miles per hour. How fast is the distance between the truck and the house increasing when the truck is three miles past the intersection ...
### Algebra 2
Your car gets 25 mi/gal around town and 30 mi/gal on the highway. a. If 50% of the miles you drive are on the highway and 50% are around town, what is your overall average miles per gallon? b. If 60% of the miles you drive are on the highway and 40% are around town, what is ...
### Civics
The Emergency Highway Energy Conservation Act was passed in 1974. Under this act, states would only get federal highway funding if they set a speed limit of 55 mph on all four-lane highways. Why would the federal government deny funding the states instead of just directly ...
### math
John said that he had been working for the highway department for about 9.75 years. Which of the following could be the actual amount of time, in years, that John had been working for the highway department. f) 9.25 g) 9.35 h) 9.65 i) 9.95
### Math
John said that he had been working for the highway department for about 9.75 years. Which of the following could be the actual amount of time, in years, that John had been working for the highway department. f) 9.25 g) 9.35 h) 9.65 i) 9.95
### math
monica keeps a supply of dimes and quarters in her car for highway tolls.A week's supply of toll coins contains 10 more dimes than quarters and totals \$11.50. How many quarters does monica spend on highway tolls in one week? F. 26 G. 30 H. 36 I. 40
### trig
a state trooper is hidden 30ft. from a highway. One second after a truck passes, the angle beta between the highway and the line of ovservation from the patrol car to the the truck is measured. if the speed limit is 55mph and a speeding ticket is issued for speeds of 5mph or ...
### calculus
You work for the silver there is a home with a driveway 2 miles long extending to the house from a nearby highway. the nearest connection box is along the highway but 5 miles from the driveway. it costs the company \$100 per mile to install cable along the highway and \$140 per ...
### social studies
The Emergency Highway Energy Conservation Act was passed in 1974. Under this act, states would only get federal highway funding if they set a speed limit of 55 mph on all four-lane highways. Why would the federal government deny funding the states instead of just directly ...
### Physics 12
A sports car is travelling along a highway at a constant velocity of 36 m/s [N] exceeding the speed limit. It passes an unmarked police car parked on the shoulder of the highway. The police car accelerates from rest, 3.0 s after the sports car passed it, at the rate of 6.0 m/...
### physics
meerany A flat (unbanked) curve on a highway has a radius of 220.0 Ill. A car rounds the curve at a speed of 25.0 m/s. (a) What is the minimum coefficient of friction that will prevent sliding? (b) Suppose the highway is icy and the coefficient of friction between the tires ...
### Math
Hey can anybody please help me with this? Im stuck? I got the first part right, but I don't really know. Ok here's the problem: Annie kept track of the cars she saw on the highway? She saw twice as many Toyotas as Skyalks. The number of Pintos was eight less than the number of...
### math
A traveler travels on a highway 80 miles at 55 miles per hour, he then travels back on the same highway at 45 miles per hour. What was his average rate of speed. (Hint: it's not 50 miles per hour) If it's not 50, i'm so confused!!! Please help!!! (and please show me how you ...
### Calculus
A section of highway connecting two hillsides with grades of 6% and 4% is to be build between two points that are separated by a horizontal distance of 2000 feet. At the point where the two hillsides come together, there is a 50-foot difference in elevation. a) Design a ...
### Calculus
A section of highway connecting two hillsides with grades of 6% and 4% is to be build between two points that are separated by a horizontal distance of 2000 feet. At the point where the two hillsides come together, there is a 50-foot difference in elevation. a) Design a ...
### math
Extra Credit opportunity: A driver is headed north on a long, straight highway and sees this sign: Nearville 150 miles Farville 160 miles Then, surprisingly, an hour later she sees this apparently inconsistent sign on the same highway: Nearville 100 miles Farville 109 miles ...
### Calculus
1)A north south highway, intersects an east west highway, at point P. A vehicle process point P @ 1pm, traveling east at a constant speed of 60km/h. At the same instant another vehiclist 5km north of P, travelling south at 80km/h, find the time, when the 2 vehicles are closest...
### calculus
this is a optimization problem A construction company has been offered a contract for \$7.8 million to construct and operate a trucking route for five years to transport ore from a mine site to a smelter. The smelter is located on a major highway, and the mine is 3 km into a ... | 6,355 | 26,816 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.578125 | 3 | CC-MAIN-2017-34 | latest | en | 0.951715 |
https://strangeherring.com/what-are-the-factors-of-game-theory/ | 1,696,450,906,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233511406.34/warc/CC-MAIN-20231004184208-20231004214208-00789.warc.gz | 595,120,677 | 16,827 | # What Are the Factors of Game Theory?
//
Diego Sanchez
Game theory is a branch of mathematics that deals with the study of decision-making in strategic situations, where the outcome of an individual’s choice depends on the choices made by others. It is widely used in various fields such as economics, political science, psychology, and biology. In this article, we will explore the different factors that make up game theory.
## Players
The first factor in game theory is the players involved. Players can be individuals or groups that make decisions based on their own interests. The number of players can range from two to many, and each player has a set of possible choices or strategies.
### Strategies
A strategy is a plan of action that a player can take in a given situation. A player’s strategy depends on what they believe other players will do. In game theory, there are two types of strategies – pure and mixed.
A pure strategy is when a player chooses one specific action regardless of what others do. For example, in a game of rock-paper-scissors, if a player always chooses rock as their move, it is considered a pure strategy.
On the other hand, a mixed strategy involves choosing multiple actions with certain probabilities. For example, if there are two possible moves – A and B – and a player chooses A 60% of the time and B 40% of the time based on what they think other players will do.
### Payoffs
The payoff in game theory refers to what each player gains or loses depending on the outcome of the game. It can be represented as numerical values assigned to each possible outcome.
For example, consider a simple game where two players have two possible strategies each – cooperate (C) or defect (D). If both cooperate, they both receive 3 points each (CC).
If one cooperates while the other defects (CD), then the defector gets 5 points and the cooperator gets 1 point. If both defect, then they both receive 2 points each (DD). The numerical values assigned to these outcomes are the payoffs.
### Information
In game theory, information refers to what each player knows about the other players’ strategies and payoffs. There are two types of information – complete and incomplete.
Complete information means that all players know all the possible strategies and payoffs of the game. On the other hand, incomplete information means that some players do not know certain aspects of the game.
### Equilibrium
Equilibrium is a state where no player can improve their outcome by changing their strategy, given what they know about other players’ strategies. In game theory, there are different types of equilibria such as Nash equilibrium and dominant strategy equilibrium.
Nash equilibrium is a state where no player can improve their outcome by changing their strategy, assuming that all other players stick to their current strategies. Dominant strategy equilibrium is a state where one strategy dominates over all others for a player, regardless of what other players do.
## Conclusion
Game theory has many practical applications in various fields such as economics, political science, psychology, and biology. Understanding its different factors such as players, strategies, payoffs, information, and equilibrium can help us make better decisions in strategic situations. By using these elements effectively in our analysis of games we can make better decisions based on logic rather than chance or impulse. | 677 | 3,452 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4 | 4 | CC-MAIN-2023-40 | latest | en | 0.95101 |
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### MEAP Preparation - Grade 4 Mathematics2.17 Convert Units of Weight - US Customary System
Example: Convert 5.5 tons to lb 1ton = 2000 lbs Hence 5.5 tons = 2000 x 5.5 lbs Answer: 11,000 lbs Directions: Answer the following. Also write at least 10 examples of your own.
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• Unlimited practice with auto-generated 'WIZ MATH' quizzes | 390 | 1,414 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.15625 | 3 | CC-MAIN-2020-24 | latest | en | 0.770002 |
https://www.practicalclinicalskills.com/course-contents-ekg-module/315/ventricular-rhythms | 1,726,525,283,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651714.51/warc/CC-MAIN-20240916212424-20240917002424-00750.warc.gz | 879,515,274 | 14,505 | Sign-in, or Join a paid plan. Join Restore
# Ventricular Rhythms Module
## Ventricular Rhythms
Thomas E. O'Brien
AS CCT CRAT RMA
## Learning Objectives
At the conclusion of this training module the reader will be able to:
• Recall and apply the 5-steps of heart rhythm interpretation
• Recognize the difference between regular and irregular rhythms
• Recall the normal range for PR interval and QRS complex
• Recognize the features and qualifying criteria for the following complexes and rhythms:
• Premature Ventricular Complexes
• Agonal Rhythm
• Idioventricular Rhythm
• Accelerated Idioventricular Rhythm
• Ventricular Tachycardia
• Ventricular Fibrillation
• Asystole
Lessons
## Intro
Rhythm analysis using a 5 step method will be practiced in this section. Analyze tracings in the following order.
• Rhythm Regularity
• Heart Rate
• P wave morphology
• P R interval or PRi
• QRS complex duration and morphology
## Step 1
### Rhythm Regularity
• Carefully measure from the tip of one R wave to the next, from the beginning to the end of the tracing.
• A rhythm is considered “regular or constant” when the distance apart is either the same or varies by 1 ½ small boxes or less from one R wave to the next R wave.
## Step 2
### Heart Rate Regular (Constant) Rhythms
• The heart rate determination technique used will be the 1500 technique.
• Starting at the beginning of the tracing through the end, measure from one R wave to the next R wave (ventricular assessment), then P wave to P wave (atrial assessment), then count the number of small boxes between each and divide that number into 1500. This technique will give you the most accurate heart rate when analyzing regular heart rhythms. You may include ½ of a small box i.e. 1500/37.5 = 40 bpm (don’t forget to round up or down if a portion of a beat is included in the answer).
## Step 2-2
### Heart Rate - Irregular Rhythms
• If the rhythm varies by two small boxes or more, the rhythm is considered “irregular”.
• The heart rate determination technique used for irregular rhythms will be the “six-second technique”.
• Simply count the number of cardiac complexes in six seconds and multiply by ten.
## Step 3
### P wave Morphology (shape)
• Lead II is most commonly referenced in cardiac monitoring
• In this training module, lead two will specifically be referenced unless otherwise specified.
## Step 4
### PR interval (PRi)
Constant PR Interval
Variable PR Interval
• Measurement of the PR interval reflects the amount of time from the beginning of atrial depolarization to the beginning of ventricular depolarization.
• Plainly stated, this measurement is from the beginning of the P wave to the beginning of the QRS complex.
• The normal range for PR interval is: 0.12 – 0.20 seconds (3 to 5 small boxes)
• It is important that you measure each PR interval on the rhythm strip.
• Some tracings do not have the same PRi measurement from one cardiac complex to the next. Sometimes there is a prolonging pattern, sometimes not.
• If the PR intervals are variable, report them as variable, but note if a pattern is present or not.
## Step 5
### QRS complex
• QRS represents ventricular depolarization.
• It is very important to analyze each QRS complex on the tracing and report the duration measurement and describe the shape (including any changes in shape).
• As discussed in step 3, when referring to P waves, remember changes in the shape of the waveform can indicate the locus of stimulation has changed or a different conduction pathway was followed. It is no different when analyzing the QRS complex. The difference is that in step 3, we were looking at atrial activity. Now we are looking at ventricular activity.
• Measure from the beginning to the end of ventricular depolarization.
• The normal duration of the QRS complex is: 0.06 – 0.10 second
## Close
• The previous slides presented the five-steps of rhythm analysis. These five steps must be followed regardless of how simple of complex the tracing is you are reviewing.
• The information gathered in these steps are telling a story.
• The title of that story is the interpretation.
## Introduction Part 1
• Rhythms are often named according to the origin of the electrical activity in the heart or the structure where the problem is occurring.
• Ventricular Rhythms are aptly named due to the locus of stimulation being the ventricles (Purkinje network).
• Dysrhythmias in this category occur as a result of either a failure of the higher (faster) pacemakers within the heart or an abnormal locus of stimulation within the ventricles is occurring at a faster rate than the other pacemaker sites and thus takes over as the pacemaker of the heart.
• Remember, the fastest electricity in the heart (regardless of location) will dictate the heart rate.
## Introduction Part 2
• Each rhythm in this category will share unique morphologic features which separate them from other rhythms.
• Other than Asystole and Ventricular Fibrillation which are unique even within this category, the remaining ventricular rhythms typically present without P waves and will display a wide, bizarre QRS complex (measuring 0.12 seconds or greater).
• After learning the unique features just described, it is simply a matter of recalling the heart rate range associated with the dysrhythmia.
## Chart of Types
### Ventricular Rhythms
(no P wave, wide - bizarre QRS if present)
Rhythm Rate Asystole 0 Agonal less than 20 bpm Idioventricular 20-40 bpm Accelerated Idioventricular 40-100 bpm *Ventricular Tachycardia > 100 bpm *Ventricular Fibrillation Electrical Chaos
*Only two rhythms that are treated with defibrillation.
## Part 1
Regardless of rhythm category, morphology of waveforms and pattern of occurrence are important aspects to include in an accurate interpretation.
Unifocal – abnormal complexes are of the same shape
Multifocal – abnormal complexes are of two or more different shapes. This indicates the impulse causing the PVC’s are coming from different locations.
Bigeminy – abnormal complexes occur every second complex
Trigeminy– abnormal complexes occur every third complex
## Part 2
Regardless of rhythm category, morphology of waveforms and pattern of occurrence are important aspects to include in an accurate interpretation.
Quadrigeminy– abnormal complexes occur every fourth complex
Couplet – Two PVC’s together
Run of Ventricular Tachycardia
(V Tach) – Three or more PVC’s in a row at a rate of 100 bpm or greater. Also known as Triplet PVC’s or Salvo PVC’s
## Description 1
• PVC rhythms may occur for a number of different reasons i.e., diet, fatigue, stress, disease, ischemia to name a few.
• Premature complexes frequently occur in bradycardic rhythms, but may occur almost any time.
• PVC’s occur when an early electrical impulse occurs from a location in either ventricle.
## Description 2
• This early impulse causes an early cardiac complex which disrupts the underlying rhythm.
• The locus of stimulation being different, results in a change in the morphology of the cardiac complex.
• Note the absence of P wave and the wide, bizarre QRS complex.
• PVC’s can occur occasionally or frequently.
• PVC’s can be observed with or without a pattern
## PVC EKG Strip
Analyze this tracing using the five steps of rhythm analysis.
• Rhythm: Irregular
• Rate: 60
• P Wave: upright and uniform, absent on early complex
• PR interval: 0.16 sec
• QRS: 0.08 sec, early complex wide and bizarre 0.16 sec
• Interpretation: Sinus Rhythm (NSR) with PVC
## Description
• This is a life-threatening dysrhythmia. Agonal rhythm is often the last ordered semblance of organized electrical activity in the heart prior to death.
• Heart rate is less than 20 bpm, without P waves and a wide, bizarre QRS complex.
• The rate is often so slow, that on a singular six-second rhythm strip it will be impossible to determine whether the rhythm is regular or irregular. There must be at lest three complexes on the tracing to make this call. Many times there will only be one or two complexes captured on the ecg strip.
## Practice Strip
Analyze this tracing using the five steps of rhythm analysis.
• Rhythm: Cannot determine (only two complexes)
• Rate: 20 (using the six-second technique)
• Rate: 14 (using the 1500 technique – 110 small boxes/1500)
• P Wave: absent
• PR interval: n/a
• QRS: Wide and bizarre 0.20 sec
• Interpretation: Idioventricular Rhythm
## Description
• The morphologic features continue with the dysrhythmia. No P wave, wide and bizarre QRS.
• The heart rate is between 20 – 40 bpm.
## Practice Strip
Analyze this tracing using the five steps of rhythm analysis.
• Rhythm: Regular
• Rate: 34
• P Wave: absent
• PR interval: n/a
• QRS: Wide and bizarre 0.24 sec
• Interpretation: Idioventricular Rhythm ECG
## Description
• The morphologic features continue with the dysrhythmia. No P wave, wide and bizarre QRS.
• The heart rate is between 40 – 100 bpm.
• Note the abnormal shape and width of the QRS complexes.
• Patients with this dysrhythmia may actually be hemodynamically stable when the heart rate is within the “normal” range. This rhythm must always be reported whether the patient can tolerate it or not.
## Practice Strip
Analyze this tracing using the five steps of rhythm analysis.
• Rhythm: Regular
• Rate: 41
• P Wave: absent
• PR interval: n/a
• QRS: Wide and bizarre 0.16 sec
• Interpretation: Accelerated Idioventricular Rhythm
## Description
• The morphologic features continue with the dysrhythmia. No P wave, wide and bizarre QRS.
• Ventricular Tachycardia occurs when the rate exceeds 100 bpm.
• Approximately 50% of patients become unconscious at the onset of ventricular tachycardia.
• Although patients in V Tach may be treated with a defibrillator, not all patients in Ventricular Tachycardia require this level of treatment.
• Depending upon their level of consciousness and blood pressure. The patient may be treated with medications, synchronized cardioversion or in the worst case scenario a defibrillator and BLS/ACLS response.
• This rhythm must always be reported whether the patient appears stable or not. (Follow your local reporting and treatment protocols)
## Practice Strip
Analyze this tracing using the five steps of rhythm analysis.
• Rhythm: Regular
• Rate: 150
• P Wave: absent
• PR interval: n/a
• QRS: Wide and bizarre
• Interpretation: Ventricular Tachycardia
## Description
• The morphologic features are different with this dysrhythmia. No P wave and no QRS complexes. This rhythm presents with a chaotic waveform which reflects the electrical chaos occurring within the heart.
• The heart is not actually beating as we know it. The chaos occurs as a result of small regions of tissue which are independently depolarizing.
• This rapid disorganized electrical activity actually makes the heart appear to quiver in response to this activity. Some have described it as shaking like Jello.
• Fibrillatory waves may be coarse or very fine. This is based upon their size. The longer V Fib occurs, the smaller the waveforms are likely to be.
• (Description continues on next slide)
## Description (continued)
• Coarse Ventricular Fibrillation (coarse V Fib) is when a majority of the waveforms measure 3 mm or greater
• Fine V Fib (fine vfib) is when a majority of the waveforms measure less than 3 mm
• This is absolutely a life-threatening dysrhythmia which requires, immediate, effective, and aggressive care.
• If your patient is talking to you when you see this on the monitor, then your patient is not in V Fib. Always, check your patient first, but there will likely be a loose or disconnected lead wire or electrode.
## VFib Practice Strip
Analyze this vfib rhythm strip using the five steps of rhythm analysis.
• Rhythm: Irregular
• Rate: Unable to determine
• P Wave: absent
• PR interval: absent
• QRS: absent
• Interpretation: Ventricular Fibrillation (Fine)
## Description of Asystole ECG
• Asystole dysrhythmia occurs when there is a total absence of electrical activity in the heart.
• The patient is clinically dead.
• Sometimes this tracing is referred to as straight line or flat line.
• There will be an absence of P waves and QRS complexes.
• True Asystole is an absolute medical emergency.
• If your patient is speaking to you, they are NOT in Asystole. Check your attachments and equipment.
## Practice Strip
Analyze this tracing using the five steps of rhythm analysis.
• Rhythm: Absent
• Rate: 0
• P Wave: absent
• PR interval: n/a
• QRS: absent
• Interpretation: Asystole
## Ventricular Asystole
• Another form of Asystole you may encounter is called Ventricular Asystole.
• The features are the same as traditional Asystole, with one exception.
• There will be P waves present in this tracing.
• The patient is clinically dead. The patient will not survive with just atrial depolarization.
## Practice Strip
Analyze this tracing using the five steps of rhythm analysis.
• Rhythm: Regular
• Rate: Atria – 52, Ventricles - 0
• P Wave: Upright and uniform
• PR interval: absent
• QRS: absent
• Interpretation: Ventricular Asystole
## Question #1
When analyzing a rhythm strip, it qualifies as being regular when
A. the QT intervals are the same
B. the PR interval measures the same
C. the QRS complexes measures the same
D. the R - R intervals measure the same
## Question #2
Which of the following steps is not one of the five-steps of rhythm analysis?
A. PR interval measurement
B. Rhythm regularity
C. QT Interval
D. QRS complex measurement
## Question #3
Which of the following is considered normal range of the QRS complex?
A. 0.12 - 0.20 minutes
B. 0.06 - 0.10 minutes
C. 0.12 - 0.20 seconds
D. 0.06 - 0.10 seconds
## Question #4
Which of the following is considered normal range of the PR interval?
A. 0.12 - 0.20 minutes
B. 0.06 - 0.10 minutes
C. 0.12 - 0.20 seconds
D. 0.06 - 0.10 seconds
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https://answerofmath.com/solved-fat-shattering-dimension/ | 1,696,230,646,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233510983.45/warc/CC-MAIN-20231002064957-20231002094957-00508.warc.gz | 115,917,836 | 18,433 | Solved – Fat-shattering dimension
A set of points \$X = {x}\$ is \$gamma\$-shattered by a set of functions \$mathcal{F}\$ if there are real numbers \$r_x\$ indexed by \$x\$ such that for any binary vector \$b\$ defining labeling of points from \$X\$ we can find a function \$f in mathcal{F}\$ such that \$f(x) geq r_x + gamma\$ if \$x\$ has label 1 and \$f(x) leq r_x – gamma\$ if \$x\$ has label -1.
In this definition, what is the role of numbers \$r_x\$? If we don't use \$r_x\$ in the definition, what will change?
Contents | 159 | 530 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.640625 | 3 | CC-MAIN-2023-40 | latest | en | 0.778021 |
https://www.mql5.com/en/docs/standardlibrary/mathematics/stat/noncentralchisquare/mathcumulativedistributionnoncentralchisquare | 1,519,073,163,000,000,000 | text/html | crawl-data/CC-MAIN-2018-09/segments/1518891812788.42/warc/CC-MAIN-20180219191343-20180219211343-00521.warc.gz | 906,001,972 | 7,810 | # MathCumulativeDistributionNoncentralChiSquare
Calculates the probability distribution function of noncentral chi-squared distribution with the nu and sigma parameters for a random variable x. In case of error it returns NaN.
double MathCumulativeDistributionNoncentralChiSquare( const double x, // value of random variable const double nu, // parameter of distribution (number of degrees of freedom) const double sigma, // noncentrality parameter const bool tail, // flag of calculation, if lower_tail=true, then the probability of random variable not exceeding x is calculated const bool log_mode, // calculate the logarithm of the value, if log_mode=true, then the natural logarithm of the probability is returned int& error_code // variable to store the error code );
Calculates the probability distribution function of noncentral chi-squared distribution with the nu and sigma parameters for a random variable x. In case of error it returns NaN.
double MathCumulativeDistributionNoncentralChiSquare( const double x, // value of random variable const double nu, // parameter of distribution (number of degrees of freedom) const double sigma, // noncentrality parameter int& error_code // variable to store the error code );
Calculates the probability distribution function of noncentral chi-squared distribution with the nu and sigma parameters for an array of random variables x[]. In case of error it returns false. Analog of the pchisq() in R.
bool MathCumulativeDistributionNoncentralChiSquare( const double& x[], // array with the values of random variable const double nu, // parameter of distribution (number of degrees of freedom) const double sigma, // noncentrality parameter const bool tail, // flag of calculation, if lower_tail=true, then the probability of random variable not exceeding x is calculated const bool log_mode, // flag to calculate the logarithm of the value, if log_mode=true, then the natural logarithm of the probability is calculated double& result[] // array for values of the probability function );
Calculates the probability distribution function of noncentral chi-squared distribution with the nu and sigma parameters for an array of random variables x[]. In case of error it returns false.
bool MathCumulativeDistributionNoncentralChiSquare( const double& x[], // array with the values of random variable const double nu, // parameter of distribution (number of degrees of freedom) const double sigma, // noncentrality parameter double& result[] // array for values of the probability function );
Parameters
x
[in] Value of random variable.
x[]
[in] Array with the values of random variable.
nu
[in] Parameter of distribution (number of degrees of freedom).
sigma
[in] Noncentrality parameter.
tail
[in] Flag of calculation. If true, then the probability of random variable not exceeding x is calculated.
log_mode
[in] Flag to calculate the logarithm of the value. If log_mode=true, then the natural logarithm of the probability is calculated.
error_code
[out] Variable to store the error code.
result[]
[out] Array for values of the probability function. | 969 | 3,440 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3 | 3 | CC-MAIN-2018-09 | latest | en | 0.10353 |
http://oeis.org/A174051/internal | 1,591,308,899,000,000,000 | text/html | crawl-data/CC-MAIN-2020-24/segments/1590347458095.68/warc/CC-MAIN-20200604192256-20200604222256-00293.warc.gz | 85,492,372 | 3,096 | The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.
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A174051 Composite numbers of the form x^2+y^2, gcd(x,y) = 1 1
%I
%S 10,25,26,34,50,58,65,74,82,85,106,122,125,130,145,146,169,170,178,
%T 185,194,202,205,218,221,226,250,265,274,289,290,298,305,314,325,338,
%U 346,362,365,370,377,386,394,410,425,442,445,458,466,481,482,485,493,505
%N Composite numbers of the form x^2+y^2, gcd(x,y) = 1
%C Composite numbers in A008784. - _R. J. Mathar_, Jul 08 2012
%H Charles R Greathouse IV, <a href="/A174051/b174051.txt">Table of n, a(n) for n = 1..10000</a>
%e 10 is in the sequence because 10 = 1^2 + 3^2 = 2*5;
%e 25 is in the sequence because 25 = 3^2 + 4^2 = 5*5;
%e 65 is in the sequence because 65 = 1 + 8^2 = 4^2 + 7^2 = 5*13.
%p with(numtheory):T:=array(0..50000000):U=array(0..50000000 ):k:=1:for x from 1 to 1000 do:for y from x to 1000 do:if type(x^2+y^2,prime)=false and gcd(x,y)=1 then T[k]:=x^2+y^2:k:=k+1:else fi: od :od:mini:=T[1]:ii:=1:for p from 1 to k-1 do:or n from 1 to k-1 do:if T[n] < mini then mini:= T[n]:ii:=n: indice:=U[n]: else fi:od:print(mini):T[ii]:= 99999999: ii:=1:mini:=T[1] :od:
%o (PARI) list(lim)=my(v=List(), x2, t); lim\=1; for(x=3, sqrtint(lim-1), x2=x^2; for(y=1, min(x-1, sqrtint(lim-x2)), if(gcd(x, y)==1 && !isprime(t=x2+y^2), listput(v, t)))); Set(v) \\ _Charles R Greathouse IV_, Jan 27 2018
%Y Cf. A002808, A002313, A037942.
%K nonn
%O 1,1
%A _Michel Lagneau_, Mar 06 2010
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Last modified June 4 17:50 EDT 2020. Contains 334828 sequences. (Running on oeis4.) | 766 | 1,888 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.53125 | 4 | CC-MAIN-2020-24 | latest | en | 0.509015 |
https://ao.ms/how-to-calculate-dominant-primes-in-golang/ | 1,675,162,488,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764499857.57/warc/CC-MAIN-20230131091122-20230131121122-00592.warc.gz | 117,309,895 | 9,632 | ## The challenge#
The prime number sequence starts with: `2,3,5,7,11,13,17,19...`. Notice that `2` is in position `one`.
`3` occupies position `two`, which is a prime-numbered position. Similarly, `5``11` and `17` also occupy prime-numbered positions. We shall call primes such as `3,5,11,17` dominant primes because they occupy prime-numbered positions in the prime number sequence. Let’s call this `listA`.
As you can see from listA, for the prime range `range(0,10)`, there are `only two` dominant primes (`3` and `5`) and the sum of these primes is: `3 + 5 = 8`.
Similarly, as shown in listA, in the `range (6,20)`, the dominant primes in this range are `11` and `17`, with a sum of `28`.
Given a `range (a,b)`, what is the sum of dominant primes within that range? Note that `a <= range <= b` and `b` will not exceed `500000`.
## The solution in Golang#
Option 1:
``````package solution
func Solve(a, b int) int {
sum := 0
sv := make([]bool, b+1)
pos := 1
if a <= 3 && b >= 3 {
sum += 3
}
for i := 3; i <= b; i += 2 {
if sv[i] == false {
pos++
if i >= a && pos%2 == 1 && sv[pos] == false {
sum += i
}
for j := i + i; j <= b; j += i {
sv[j] = true
}
}
}
return sum
}
``````
Option 2:
``````package solution
func Solve(a, b int) (sum int) {
primes := make([]int, 0, 5000)
primes = append(primes, 2, 3, 5, 7)
prime := func(n int) int {
for n > len(primes) {
next := primes[len(primes) - 1] + 1
for i := 0; primes[i]*primes[i] <= next; i++ {
if next%primes[i] == 0 { next += 1 ; i = -1 }
}
primes = append(primes, next)
}
return primes[n-1]
}
for i := 1 ;; i++ {
dominantPrime := prime(prime(i))
if dominantPrime > b { break }
if dominantPrime < a { continue }
sum += dominantPrime
}
return
}
``````
Option 3:
``````package solution
func isPrime(n int) bool {
for i := 2; i*i < n+i; i++ {
if n%i == 0 {
return false
}
}
return n > 1
}
func Solve(a, b int) int {
c, pos := 0, 0
for i := 0; i <= b; i++ {
if isPrime(i) {
pos++
if i >= a && isPrime(pos) {
c += i
}
}
}
return c
}
``````
## Test cases to validate our solution#
``````package solution_test
import (
. "github.com/onsi/ginkgo"
. "github.com/onsi/gomega"
)
func dotest(s, g, exp int) {
var ans = Solve(s,g)
Expect(ans).To(Equal(exp))
}
var _ = Describe("Example tests", func() {
It("It should work for basic tests", func() {
dotest(0,10, 8)
dotest(2,200, 1080)
dotest(1000,100000,52114889)
dotest(4000,500000,972664400)
})
})
`````` | 867 | 2,407 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.1875 | 4 | CC-MAIN-2023-06 | latest | en | 0.404389 |
https://www.reddit.com/r/WatchPeopleDieInside/comments/10gcv06/when_you_dont_know_how_to_pronounce/j53cr2n/ | 1,686,151,470,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224653930.47/warc/CC-MAIN-20230607143116-20230607173116-00728.warc.gz | 1,029,221,385 | 34,326 | ×
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[–] 201 points202 points (63 children)
You have to solve that puzzle [edit: without getting a bankrupt after getting the wedge], then have the highest total at the end of the show to win the game, then you go to the bonus round where you spin a smaller wheel that has sparkly hotdog-style folded-up cards on it. If you won the million dollar wedge, one of those blind cards will have one million dollars written in it. The rest are normal bonus prizes, usually something like \$20,000 - 40,000 or a car.
The host holds onto the card you land on, not revealing the bonus prize yet, and then you play the bonus round. Regardless of the results, the card is opened up after the bonus puzzle. If you successfully solved the bonus puzzle, you add the amount on the card to your total winnings. If not, you still get to keep the money you won during the regular part of the show. The guest(s) you brought will walk onto the stage and show varying levels of affection and excitement, including but not limited to hugs.
So he did not necessarily miss out on \$1 million, just a small random chance to win it if everything else went right.
[–] 0 points1 point (0 children)
Did Games Workshop write the rules for this gameshow?
[–] 2 points3 points (0 children)
This game show starting to sound like that one from friends
[–] 131 points132 points (60 children)
Dumb question, but why does it matter if you pronounce it wrong?
[–] 0 points1 point (0 children)
Especially on a name
[–] 0 points1 point (1 child)
I think it’s so you can’t “wing it” if there are some missing letters but you don’t know the answer. It’s to stop people kinda mumbling their way through half filled words. Not really fair in this case as he clearly knew the answer.
[–] 1 point2 points (0 children)
Makes sense!
[–] 129 points130 points (26 children)
Agreed. The letters are worth money in this game.
He spelled it right. He won.
[–] 2 points3 points (1 child)
But those aren’t the rules of the game that the contestant agreed to. The host even goes on to explain that it was ruled incorrect because proper nouns have to be pronounced correctly. I would guess this is because a proper name’s pronunciation is more subjective and usually dictated by the person even if it is not a technically correct pronunciation.
For example, in the US the family name “Lafayette” is likely pronounced differently depending on where the person is from. Where I grew up in LA it’s “laffy-ET” but in other parts of the south they say “la-FAY-et” and if it were my last name I would personally only recognize the first pronunciation as correct.
[–] 1 point2 points (0 children)
And both pronunciations of Lafayette would be fine on this show.
This guy wasn't even remotely close. The woman who answered next was also off (a-keel-eez) but was marked as correct enough.
[–] 24 points25 points (23 children)
because imagine if the c was missing and he tried to solve the puzzle saying mythologikal hero akillies
it means they would have to allow any mispronounced word
[–] 0 points1 point (0 children)
Yes but the C wasn't missing.
[–] 1 point2 points (3 children)
You have to say the names of each letter out loud, so there’s no way to misspell on the show.
Most of the time they’re just guessing at letters anyway.
[–] 1 point2 points (2 children)
you dont have to uncover all letters to solve
[–] 2 points3 points (1 child)
Well, duh.
Then you can guess the puzzle. You have to say the words, but they don’t have to SOUND a certain way.
In this case, no guessing was required, AND his mispronunciation still properly acknowledged the right letters.
[–] 0 points1 point (0 children)
The evidence says the only thing that matters is the SOUND so i don't know how you come to that conclusion.
[–] 33 points34 points (17 children)
[–] 4 points5 points (16 children)
in this instance ,yes. imagine he thinks the missing letter is a "c" or a "k" and hes unsure.
so in that case he should solve and not pick a letter since both pronunciations lead to a win (thats how you are saying it should be ) but it is not to protect against these issues .
[–] 13 points14 points (15 children)
You only win if you've spelled it correctly. The pronunciation part is b.s.
[–] 2 points3 points (1 child)
That’s not true. You don’t have to spell anything at all to win.
[–] 0 points1 point (0 children)
[–] 4 points5 points (12 children)
you dont understand that u can solve without uncovering all leters?
[–] 0 points1 point (0 children)
No I get that; just tripped up on why how you pronounce it matters.
[–] 3 points4 points (1 child)
But he DID uncover all letters. I think they are arguing only for that specific case.
I think that would be fair tbh. Don't now how to pronounce smth? K, just go ahead an uncover all letters to show that you know what it shoud say. If you know how to pronounce it, easy, just solve early.
[–] 1 point2 points (0 children)
so make it a special case? i guess they could but its considered an intelligence based game so they are ok with keeping the rules ridged which adds a slight bit more difficulty
[–] 1 point2 points (8 children)
But can be pronounced correctly without knowing the spelling?
[–] 1 point2 points (3 children)
yeah like if hes unsure its a c or k just pronounce it either way without uncovering the letter since he may be wrong and if they allow mispronounced words they have to accept it or only allow mispronounced words when all letters are uncovered .
note: jeopardy does not accept mispronounced words or misspelled words in final jeopardy
[–] -2 points-1 points (3 children)
Or mispronounced because have only read, and lack western civ historical context to know the th is pronounced as a "k" welcome to that crt shit y'all hate! There is an inherent bias in that bulshit... hope you don,t live in florida or your kids won't learn, either!?
[–] 0 points1 point (4 children)
Pat mentioned it was a proper name at the end of the clip, partly cut off.
[–] 0 points1 point (3 children)
And yeah? He spelled it correctly. This is a spelling game, not a pronunciation game.
[–] 0 points1 point (0 children)
lol I didn't make the rules and neither did you. You want to argue about them, give Pat or the producers of wheel of fortune a call and tell them how you feel.
[–] 2 points3 points (1 child)
It's actually a "Guess the Riddle" game. Say the wrong answer then you're wrong. Simple as that. The letters are there as hunts.
[–] 0 points1 point (0 children)
Hints? OK, makes sense. Still feels unfair to this guy...
[–] -4 points-3 points (0 children)
Because it was wrong. It's not tomato, tomato in this case.
[–] 5 points6 points (12 children)
[–] 18 points19 points (11 children)
That's stupid though, If you misprounce the word you're still saying it though.
[–] 0 points1 point (0 children)
That's the thing though - that's not always the case. Say for example the clue was "popular online game" and the letters were "HEAR__S_ONE" and you put those letters together and made the guess "Heartstone", it would be obvious that you don't actually know the answer and made a guess that is close but not the correct. The show could decide to give you the benefit of the doubt and assume you meant to say "Hearthstone", but that would be unfair to those that actually know the correct answer (and know how to count / spell).
[–] 0 points1 point (9 children)
Small mispronunciations are probably overlooked but this was so mispronounced it was obviously the guy had no idea what this was. That pause before the buzzer sounded was the judges deciding the answer didn't pass muster.
I don't even know how you'd guess "Aychelis" from those letters unless you've never seen any Greek words before. It's also a common name for a part of your body not some esoteric character from a little known corner of Greek mythology.
[–] 7 points8 points (8 children)
It's a name, do we even know how is it actually supposed to be pronounced.
I'm not a native speaker, I Def know what the word is, but I already forgot how it's pronounced in the west. So are you implying that anyone that has an accent shouldn't be able to participate?
[–] 2 points3 points (4 children)
Doesn't really matter if you have an accent or not, you still learn the right way to pronounce things in whatever language, especially names.
"Hello, my name's Steve.
"Hello, Stevvay, my name is Paul.
"Nice to meet you, Pa-ool."
[–] 0 points1 point (3 children)
How do you pronounce the word iran?
[–] -1 points0 points (2 children)
I pronounce 'Iran', with the 'I' as in, 'internal' or 'bit'.
I'm English, if it's relevant.
[–] 0 points1 point (1 child)
I ran or e run? My point being that the typically accepted I ran pronunciation in the west is not how Iranians accept it. But regardless of your pronunciation, we both know what you're talking about.
[–] -3 points-2 points (2 children)
I didn't hear an accent
[–] 6 points7 points (1 child)
Everyone has an accent
[–] 0 points1 point (0 children)
They each, including the host, had generally the same American accent. Which is the same accent I have. So I did not hear an accent. I did not say he had no accent an his accent has no bearing on how he pronounced the puzzle here.
[–] 42 points43 points (10 children)
I don't know, but is it "Phil A she Oh," or "Fell A Tee Oh?" I don't care as long as I get a great BJ, but it seems the game show really wants "Phil A she Oh!"
[–] 1 point2 points (0 children)
When I was young, a friend pronounced it flee- toe. I tried to look it up and it was many more years, after I had seen the word fellatio and pronounced it correctly for years that my mind flashed back to my early teens and realized that flee-toe was his attempt at sounding out fellatio and I almost wanted to cry.
[–] 9 points10 points (0 children)
Me reading it as the other way to read A and not understanding wtf that meant lol
[–] 11 points12 points (6 children)
Rhymes with Horatio.
[–] 1 point2 points (0 children)
For-ay-shee-oh
[–] 3 points4 points (1 child)
Ho•ray•tee•oh, got it.
[–] 4 points5 points (0 children)
Everyone's a comedian!
[–] 3 points4 points (0 children)
Lmao | 2,673 | 10,335 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.59375 | 3 | CC-MAIN-2023-23 | latest | en | 0.938706 |
http://www.jiskha.com/display.cgi?id=1349386147 | 1,462,049,559,000,000,000 | text/html | crawl-data/CC-MAIN-2016-18/segments/1461860112727.96/warc/CC-MAIN-20160428161512-00206-ip-10-239-7-51.ec2.internal.warc.gz | 620,648,589 | 3,534 | Saturday
April 30, 2016
# Homework Help: math
Posted by Taylor on Thursday, October 4, 2012 at 5:29pm.
the figure represents a building in the shap of a regular hexagon. using th e scale 1cm=177m, what is hte perimeter of hte building 1.1cm? | 78 | 244 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.640625 | 3 | CC-MAIN-2016-18 | longest | en | 0.948636 |
https://www.studypool.com/discuss/370027/an-object-is-accelerating-at-175-m-s2?free | 1,481,207,262,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698541529.0/warc/CC-MAIN-20161202170901-00011-ip-10-31-129-80.ec2.internal.warc.gz | 983,581,770 | 14,147 | ##### An object is accelerating at 175 m/s2.
Physics Tutor: None Selected Time limit: 1 Day
An object is accelerating at 175 m/s2. If the net force is tripled and the mass of the object is doubled, what is the new acceleration?
Feb 2nd, 2015
Please note that I can write this:
a1 = F1/M1, where a = 175 m/sec^2, F1 is the magnitude of the force which is acting on the particle, whose mass is M1.
Now I consider a force acting on the particle whose magnitude is F2= 3 F1, and the mass of that particle is M2 = 2 M1, so I can write, for the new acceleration a2:
a2 = F2/M2= (3 F1)/ (2* M1) = (3/2) * (F1/M1) = (3/2) * a1 = (3/2)* 175
soo a2 = (3/2) a1 = (3/2) *175 = 262.5 m/sec^2
Feb 2nd, 2015
...
Feb 2nd, 2015
...
Feb 2nd, 2015
Dec 8th, 2016
check_circle | 281 | 766 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.90625 | 4 | CC-MAIN-2016-50 | longest | en | 0.917322 |
https://matsci.org/t/an-interesting-observation/28515 | 1,716,936,325,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971059160.88/warc/CC-MAIN-20240528220007-20240529010007-00446.warc.gz | 324,668,066 | 4,563 | # An interesting observation
Dear all,
I was trying to do uniaxial tensile loading test of a nanostructure of a very well known metal, copper, by using the popular script as provided at https://icme.hpc.msstate.edu/mediawiki/index.php/Uniaxial_Tension
As a part of this calculation I was using
fix 1 all npt temp 300 300 1 y 0 0 1 z 0 0 1 drag 1
For the value of P_start and P_final as “Zero”, I’m observing no fracture in the sample even after a tensile strain of 250%!!!
But once I make those values as non-zero, e.g., 0.01, the same sample breaks/fractures at around a strain of ~40-45%.
Could anybody please enlighten this issue.
Best,
R
Which direction you are pulling on it? If you make the pressure zero in the direction of pulling then there might be no fracture.
Thanks for your response. I’m maintaining ~0 pressure in Y and Z, while the tensile direction is X.
No fracture in Cu(100), any explanation ?
Best,
R | 249 | 929 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.515625 | 3 | CC-MAIN-2024-22 | latest | en | 0.884108 |
https://getpractice.com/subjects/maths/exponents-and-powers | 1,660,021,046,000,000,000 | text/html | crawl-data/CC-MAIN-2022-33/segments/1659882570901.18/warc/CC-MAIN-20220809033952-20220809063952-00702.warc.gz | 277,716,244 | 3,544 | ### Exponents and Powers
goals
If $3^a + 3^b=756, 7^a + 2^c=375, 5^a + 3=128$, then the value of $a+b+c$ is
Simplify and express each of the following in exponential form:$25^{ 4 }\div { 5 }^{ 3 }$
Write the number of power of $5$.
Find
$\sqrt[3]{{ - 125}} \times \sqrt[3]{{ 3375}}$
Numbers can be classified into two categories depending on their divisible conditions.
They are (i) Even numbers $(2p) \ \forall p \epsilon N$ (ii) odd numbers $(2p + 1) \ \forall p \epsilon N$ | 169 | 477 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.0625 | 4 | CC-MAIN-2022-33 | latest | en | 0.658464 |
https://discuss.codingblocks.com/t/the-running-median-of-stream-of-integers-problem/37068 | 1,702,018,304,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100724.48/warc/CC-MAIN-20231208045320-20231208075320-00809.warc.gz | 236,703,227 | 4,061 | # The running median of stream of integers problem
please tell me how to implement this code as Im not able to do so
Hey @aslesha.j7
Create two heaps. One max heap to maintain elements of lower half and one min heap to maintain elements of higher half at any point of time…
Take initial value of median as 0.
For the first two elements add smaller one to the maxHeap on the left, and bigger one to the minHeap on the right. Then process stream data one by one,
Step 1: Add next item to one of the heaps
if next item is smaller than maxHeap root add it to maxHeap,
else add it to minHeap
Step 2: Balance the heaps (after this step heaps will be either balanced or
one of them will contain 1 more item)
if number of elements in one of the heaps is greater than the other by
more than 1, remove the root element from the one containing more elements and
add to the other one
Then at any given time you can calculate median like this:
If the heaps contain equal amount of elements;
median = (root of maxHeap + root of minHeap)/2
Else
median = root of the heap with more elements
I hope I’ve cleared your doubt. I ask you to please rate your experience here
Your feedback is very important. It helps us improve our platform and hence provide you
the learning experience you deserve.
On the off chance, you still have some questions or not find the answers satisfactory, you may reopen
the doubt. | 323 | 1,399 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.859375 | 3 | CC-MAIN-2023-50 | latest | en | 0.879114 |
https://uk.answers.yahoo.com/activity/questions?show=K33XGEPBBXCO6QS7K6NB46RCVM&t=g | 1,495,927,541,000,000,000 | text/html | crawl-data/CC-MAIN-2017-22/segments/1495463609305.10/warc/CC-MAIN-20170527225634-20170528005634-00362.warc.gz | 1,025,927,438 | 25,993 | • ### What anime is this from?
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3 answers · Comics & Animation · 6 years ago | 3,689 | 15,729 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.421875 | 3 | CC-MAIN-2017-22 | longest | en | 0.922451 |
http://oeis.org/A191325 | 1,566,055,008,000,000,000 | text/html | crawl-data/CC-MAIN-2019-35/segments/1566027313428.28/warc/CC-MAIN-20190817143039-20190817165039-00246.warc.gz | 139,060,110 | 3,611 | This site is supported by donations to The OEIS Foundation.
Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
A191325 Increasing sequence generated by these rules: a(1)=1, and if x is in a then [3x/2] and [5x/2] are in a, where [ ]=floor. 2
1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 15, 17, 18, 19, 22, 25, 27, 28, 30, 32, 33, 37, 40, 42, 45, 47, 48, 49, 55, 60, 62, 63, 67, 70, 72, 73, 75, 80, 82, 90, 92, 93, 94, 100, 105, 108, 109, 112, 117, 120, 122, 123, 135, 137, 138, 139, 141, 150, 155, 157, 162, 163, 167, 168, 175, 180, 182, 183, 184, 187, 200, 202, 205, 207, 208, 211, 225 (list; graph; refs; listen; history; text; internal format)
OFFSET 1,2 COMMENTS See A191323. LINKS Ivan Neretin, Table of n, a(n) for n = 1..10000 EXAMPLE 1 -> 2 -> 3,5 -> 4,7,12 -> 6,10,17,18,30 -> MATHEMATICA h = 3; i = 0; j = 5; k = 0; f = 1; g = 15; Union[Flatten[NestList[{Floor[h #/2] + i, Floor[j #/2] + k} &, f, g]]] (* A191325 *) CROSSREFS Cf. A191323. Sequence in context: A144769 A165737 A033104 * A190209 A048856 A067175 Adjacent sequences: A191322 A191323 A191324 * A191326 A191327 A191328 KEYWORD nonn AUTHOR Clark Kimberling, May 30 2011 STATUS approved
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Last modified August 17 11:14 EDT 2019. Contains 326057 sequences. (Running on oeis4.) | 623 | 1,505 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.796875 | 3 | CC-MAIN-2019-35 | latest | en | 0.674941 |
https://playtaptales.com/numbers/quinseptuagintacentillion/ | 1,670,118,578,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710953.78/warc/CC-MAIN-20221204004054-20221204034054-00477.warc.gz | 491,513,131 | 4,233 | ## Quinseptuagintacentillion
A Quinseptuagintacentillion (1 Quinseptuagintacentillion) is 10 to the power of 528 (10^528). This is a grandiosely astronomical number!
## How many zeros in a Quinseptuagintacentillion?
There are 528 zeros in a Quinseptuagintacentillion.
## What's before Quinseptuagintacentillion?
A Quattuorseptuagintacentillion is smaller than a Quinseptuagintacentillion.
## What's after Quinseptuagintacentillion?
A Seseptuagintacentillion is larger than a Quinseptuagintacentillion.
## Quinseptuagintacentillionaire
A Quinseptuagintacentillionaire is someone whos assets, net worth or wealth is 1 or more Quinseptuagintacentillion. It is unlikely anyone will ever be a true Quinseptuagintacentillionaire. If you want to be a Quinseptuagintacentillionaire, play Tap Tales!
## Is Quinseptuagintacentillion the largest number?
Quinseptuagintacentillion is not the largest number. Infinity best describes the largest possible number - if there even is one! We cannot comprehend what the largest number actually is.
## Quinseptuagintacentillion written out
Quinseptuagintacentillion is written out as:
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
## Big Numbers
This is just one of many really big numbers! | 704 | 1,897 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.78125 | 3 | CC-MAIN-2022-49 | latest | en | 0.688688 |
https://math.answers.com/other-math/What_is_one_third_of_450.00 | 1,669,695,055,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710685.0/warc/CC-MAIN-20221129031912-20221129061912-00751.warc.gz | 425,362,943 | 48,484 | 0
# What is one third of 450.00?
Wiki User
2012-01-14 01:02:57
1/3 x 450 = 150. If the number refers to a unit of currency, don't forget to round to include the appropriate currency symbol and round to the correct number of decimal places if necessary.
Wiki User
2012-01-14 01:02:57
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## A number a power of a variable or a product of the two is a monomial while a polynomial is the of monomials
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2020-08-14 07:00:36
150.00 | 162 | 505 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.671875 | 3 | CC-MAIN-2022-49 | latest | en | 0.849347 |
https://cracku.in/51-two-lighthouses-located-at-points-a-and-b-on-the-e-x-xat-2020 | 1,716,705,892,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058868.12/warc/CC-MAIN-20240526043700-20240526073700-00886.warc.gz | 149,705,381 | 24,992 | Question 51
# Two lighthouses, located at points A and B on the earth, are 60 feet and 40 feet tall respectively. Each lighthouse is perfectly vertical and the land connecting A and B is perfectly flat. The topmost point of the lighthouse at A is A’ and of the lighthouse at B is B’. Draw line segments A’B and B’A, and let them intersect at point C’. Drop a perpendicular from C’ to touch the earth at point C. What is the length of CC’ in feet?
Solution
Triangle ACC' is similar to triangle ABB'
Considering AC = a, BC = b, CC' = h, AA' is given as 60, BB' is given to be 40.
AC/AB = CC'/BB' = h/40.
$$\left(\frac{a}{a+b}\right)\ =\ \frac{h}{40}$$ (1)
Similarly triangle BCC' is similar to BAA'.
BC/AB = CC'/AA' = h/60.
$$\left(\frac{b}{a+b}\right)\ =\ \frac{h}{60}$$ (2)
Adding (1) and (2).
$$\frac{h}{40}+\frac{h}{60\ }=\ 1$$
1/h = $$\left(\frac{1}{40}+\frac{1}{60}\right)$$
h = 24
Using crossed ladder theorem
$$\frac{1}{CC'}=\frac{1}{AA'}+\frac{1}{BB'}$$=1/60 +1/40 =5/120 =24.
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Boost your Prep! | 405 | 1,238 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.15625 | 4 | CC-MAIN-2024-22 | latest | en | 0.828867 |
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When a beam of light waves hits a transparent material such as glass the speed of the waves slows down. If the waves meet the interface at an angle the beam bends - this bending is called refraction.
The speed of light in air is 3x108 ms-1 (ca) and its speed in glass is about 2x108 ms-1 (cg).
In the animation the angle of incidence of the waves in air is 40o, the refractive index of the glass (ca/cg) = 1.5 and this gives an angle of refraction of the wave in glass of 25.4o.
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kellybell
Posts: 473
Joined: Mon Jun 28, 2004 2:40 pm
### Math Topics - Double digit addition and subtraction
Renai wrote:I'm at a loss of how else to present this concept. Funny thing is, she gets the subtraction with borrowing. It's the addition with carrying that's causing her problems. I know she "gets" the subtraction because Friday she wanted to play a computer game that required both. I quickly showed her how to work the problems and every time I checked, her work was correct on the whiteboard.
I had initially skipped these chapters to firm up her math facts, and am coming back to them.
We've used base 10 blocks (even though she doesn't really like to use manipulatives) with a place value chart. She works stuff on the whiteboard. She still likes to split the numbers into units and tens. She'll add the right side, put that answer in a circle in the side, then put those numbers in the right columns. This afternoon she thought she'd get done quicker by adding the left side first. Of course, she had to erase all the answers. How frustrating is that!?
So I don't know what else to do. Any ideas? skip again until the end of the year? I'm up for it, lol! There's still plenty else to do in this book.
I know you said she doesn't like manipulatives, but pennies and dimes might work. Trade 10 pennies for a dime, etc. So, if you add 7 pennies and 8 more pennies, you make a group of 10 and trade it for a dime and have five pennies left over.
Since she already naturally understands borrowing, show her that she can borrow by trading the dime for 10 pennies. Maybe since she already gets this concept, when you add with the coins, maybe it will click in that direction too.
Play with those pennies and dimes!
I'd start with facts that she just might know (or easily figure out). For example, she probably knows that 9 + 2 = 11. You could easily model this with pennies. Make a group of 10, trade for a dime, and have one left over. After she's fine with demonstrating facts she knows, try problems she's not memorized (say, 32+88). Same concepts.
Another thing that I was reminded of when you mentioned adding the left first is that we had some silly motivational speaker at work (long before my mommy days) who asked us (a group of several dozen adults) how to add and everyone agreed "right to left." He showed us that you can just as easily add "left to right;" it's just not what we were taught. I had forgotten about it and don't really remember the concept (the speaker wasn't teaching math, just showing us to think out of the box or something -- I guess he didn't make much of an impact if all I remember is the math thing).
Anyway, if you add these numbers "left to right":
8 9
+ 3 5
You could add the 8 and 3 and put the 11 in the "tens column." Then when you add the 9 and 5, you put the 4 in the "ones column" and add the extra 1 to the tens column and you get 124. Same as if you went right to left. However, that's not how any curriculum that I know of teaches addition. I just remember this speaker saying it's just as easy. You're carrying all the same, but in a different way.
It's just figuring out what makes a kid understand, what makes that light go on. My youngest dd does fine in math ... once she gets a concept. Some concepts are like second nature to her and others are totally alien.
Kelly, wife to Jim since 1988, mom to Jamie (a girl, 1994), Mary (1996), Brian (1998) and Stephanie (2001).
cbollin
kellybell wrote: However, that's not how any curriculum that I know of teaches addition. I just remember this speaker saying it's just as easy. You're carrying all the same, but in a different way.
It's just figuring out what makes a kid understand, what makes that light go on.
Singapore math teaches something similar to this, doesn't it? Or maybe the dust from cleaning my basement is got to my head and I just think that Singapore introduces this in the end of 1B? Except that Singapore doesn't teach it this way vertically, it does it horizontally. When Singapore gets around to the vertical stacking, it does more traditional carrying style.
-crystal
Renai
Posts: 35
Joined: Fri Dec 02, 2005 12:48 pm
Location: New Mexico
Contact:
I'd forgotten about money, silly me. She likes money :). I'll try that today. Does it seem strange that she can borrow, but forgets regrouping?? Or is it just me?
It's funny you mention adding from left to right. I actually do that now that I'm older. Don't know when I figured it out, but it's not something I was taught. The only reason I wanted dd to do right-left is because I thought it would help her to see for example with 15, put 5 in the ones total and the 1 in the tens column. She kept forgetting to carry the one. Hmmm, you know, yesterday she did start writing the number correctly- from left-right. I'll see how it goes today.
(I'm actually using Singapore as a supplement, but we only have 1A so far.)
Renai
Wife to Enrique
Mom of two dd- 9/99 & 1/11
Bilingual homeschool
2004-05 SL
2005-2012 MFW
2012-2013 K12
2013-2014 dual-language charter
2014-2015 MFW Ancients/young'un- MFW preschool, reading books
http://creativeplaybilingual.blogspot.com/
donutmom
Posts: 67
Joined: Tue Nov 21, 2006 5:41 am
Pennies and dimes are what clicked for our son.
My nephew got the borrowing right away, too, but not the carrying. So you're not alone!
Another idea. . . use graph paper with larger squares (large enough for your child to manage to write a number in each square). Then have the problems on the paper--whether you or she write it. Tell her you can only have one number in each square. So when they add the 7 + 5 in the ones column, they can't write 12 in the square below it. So show how to write the 2 and carry the one. Can work well with the money on hand as well.
Did I write that in a way that makes sense?? Maybe not. I'm coming off a not so nice case of the flu, and hubby's out of town, besides the house being in a state of remodeling. I don't really know if I'm coming or going right now! I guess I'll be going now--to bed.
Dee
Renai
Posts: 35
Joined: Fri Dec 02, 2005 12:48 pm
Location: New Mexico
Contact:
Yes, Dee, that makes a lot of sense! I'm glad we're not alone in the borrowing/carrying thing too. I actually do have that graph paper in the house (somewhere- been put away for a couple of years now).
We took a break from math yesterday... we'll be back at it today with money and graph paper.
Thanks all for the assistance!
Renai
Wife to Enrique
Mom of two dd- 9/99 & 1/11
Bilingual homeschool
2004-05 SL
2005-2012 MFW
2012-2013 K12
2013-2014 dual-language charter
2014-2015 MFW Ancients/young'un- MFW preschool, reading books
http://creativeplaybilingual.blogspot.com/ | 1,719 | 6,768 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.03125 | 4 | CC-MAIN-2020-24 | latest | en | 0.973952 |
https://astronomy.stackexchange.com/questions/40252/how-to-manually-locate-a-certain-declination-in-the-sky-for-a-given-location-n | 1,716,952,502,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971059167.30/warc/CC-MAIN-20240529010927-20240529040927-00739.warc.gz | 84,889,839 | 36,294 | # How to manually locate a certain declination in the sky for a given location? (no sextant available)
Consider this example. I am standing in a flat surface, at latitude X, looking to the equator, in a random day of the year. (For simplicity, let's assume X > 23.5º)
I want to find the position of the sun at noon on the winter solstice (that is, the highest point in the sky of the sun, the day which this reaches the lowest altitude), location which has a declination of 23.5º.
We know the sextant does exactly that. But say I don't have a sextant. I want to find an alternative method to do the same.
If my latitude is X, then such position of the sun is X-23.5º above the horizon (looking North). I could then divide this result by 90, and get a proportion of the sky between the horizon and my zenit where that point is. But this is too rough.
Perhaps one way would be to have a long stick and a measuring tape. For a given value of X-23.5º there must be some relationship between my eye, the vertical height of the stick and my horizontal distance to the stick which allows me to find exactly the position in the sky.
Or perhaps another method?
• Are you able to manufacture a tool to help with this? Dec 6, 2020 at 19:41
• @JamesK Absolutely. As long as it is rudimentary Dec 6, 2020 at 21:20
• Seems like pretty much anything that works is effectively a sextant. If you can get something with a joint with enough friction to stay wherever you leave it, but still lets you move it easily, you can move it so it matches the declenation. Dec 7, 2020 at 4:34
$$\tan(\theta/2)= c/2l$$
Where $$\theta$$ is the angle above the horizon, $$l$$ is the position of the crossbar from the eye and $$c$$ is the length of the crossbar. | 446 | 1,737 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 4, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.609375 | 4 | CC-MAIN-2024-22 | latest | en | 0.945323 |
https://www.nottingham.ac.uk/xpert/scoreresults.php?keywords=Why%20study%20the%20Didache?start=4620&end&start=22260&end=22280 | 1,560,849,467,000,000,000 | text/html | crawl-data/CC-MAIN-2019-26/segments/1560627998708.41/warc/CC-MAIN-20190618083336-20190618105336-00360.warc.gz | 863,225,148 | 31,094 |  
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Author(s): No creator set | 1,557 | 6,897 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.921875 | 3 | CC-MAIN-2019-26 | latest | en | 0.880535 |
https://www.homeworkminutes.com/subject/topic/379/General-Statistics/ | 1,571,580,497,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570986710773.68/warc/CC-MAIN-20191020132840-20191020160340-00182.warc.gz | 922,685,222 | 40,151 | • General Statistics
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Statistics Statistics / General Statistics 2019-08-09 Answer Now 0 Yes \$1.00 | 338 | 1,327 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.65625 | 3 | CC-MAIN-2019-43 | latest | en | 0.70068 |
https://philosophy-question.com/library/lecture/read/42883-is-the-union-of-rational-and-irrational-numbers | 1,675,302,848,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764499954.21/warc/CC-MAIN-20230202003408-20230202033408-00174.warc.gz | 470,225,759 | 7,780 | # Is the union of rational and irrational numbers?
## Is the union of rational and irrational numbers?
Numbers that are not rational are called irrational numbers. The real line consists of the union of the rational and irrational numbers. ... Any rational number is trivially also an algebraic number.
## Is a rational times an irrational rational?
If you multiply any irrational number by the rational number zero, the result will be zero, which is rational. Any other situation, however, of a rational times an irrational will be irrational. A better statement would be: "The product of a non-zero rational number and an irrational number is irrational."
## Are some rational numbers irrational?
Rational numbers can never be irrational. ... They cannot be both rational and irrational.
Irrational numbers include the square root, cube root, fourth root, and nth root of many numbers. Whenever a number is preceded with a radical sign, the number is called a radical. Not all radicals are irrational. For example, √4 is not an irrational number.
## How do you tell if a root is rational or irrational?
Real numbers have two categories: rational and irrational. If a square root is not a perfect square, then it is considered an irrational number. These numbers cannot be written as a fraction because the decimal does not end (non-terminating) and does not repeat a pattern (non-repeating).
## Is 125 rational or irrational?
125 is a rational number because it can be expressed as the quotient of two integers: 125 ÷ 1.
## Is the square root of 90 rational or irrational?
Since sqrt(90) is an irrational number, by definition it cannot be represented as a ratio of two integers.
## What are factors of 90?
We know that the factors of the number 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90. Now, to find the sum of the factors, add all the numbers, you will get a total of 234.
## How do you distribute radicals?
Distribute (or FOIL) to remove the parenthesis. Remember that you can multiply numbers outside the radical with numbers outside the radical and numbers inside the radical with numbers inside the radical, assuming the radicals have the same index.
## How do you divide radicals in order?
Dividing Radicals: When dividing radicals (with the same index), divide under the radical, and then divide in front of the radical (divide any values multiplied times the radicals). Divide out front and divide under the radicals. Then simplify the result. You have just "rationalized" the denominator!
## What is the product rule for radicals?
The Product Rule states that the product of two or more numbers raised to a power is equal to the product of each number raised to the same power. The same is true of roots: x√ab=x√a⋅x√b a b x = a x ⋅ b x . ... Combining radicals is possible when the index and the radicand of two or more radicals are the same.
## What is the quotient rule for radicals?
We also have a second rule called the quotient rule for radicals, which can help you simplify a radical expression containing a fraction. This rule states that given the nth root of a fraction a over b, that this will equal the nth root of a over the nth root of b. ... Notice we can simplify both radicals.
## Can you divide by a radical?
To divide two radicals, you can first rewrite the problem as one radical. The two numbers inside the square roots can be combined as a fraction inside just one square root. Once you do this, you can simplify the fraction inside and then take the square root.
## How are radicals used in everyday life?
Radical expressions are utilized in financial industries to calculate formulas for depreciation, home inflation and interest. Electrical engineers also use radical expressions for measurements and calculations. Biologists compare animal surface areas with radical exponents for size comparisons in scientific research.
## Can a radical be rational?
Radicals can be rational numbers, but they are not always rational numbers.
## How do you explain rational exponents?
A rational exponent is an exponent that is a fraction. For example, can be written as . Can't imagine raising a number to a rational exponent? They may be hard to get used to, but rational exponents can actually help simplify some problems.
## What is a rational equation?
A rational equation is an equation containing at least one fraction whose numerator and denominator are polynomials, ... A common way to solve these equations is to reduce the fractions to a common denominator and then solve the equality of the numerators. | 963 | 4,576 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.46875 | 4 | CC-MAIN-2023-06 | latest | en | 0.931787 |
http://www.purplemath.com/learning/viewtopic.php?p=3219 | 1,524,415,001,000,000,000 | text/html | crawl-data/CC-MAIN-2018-17/segments/1524125945624.76/warc/CC-MAIN-20180422154522-20180422174522-00096.warc.gz | 502,045,376 | 6,536 | ## what are the 3 integers in arithmetic seq. w/ prime product?
Fractions, ratios, percentages, exponents, number patterns, word problems without variables, etc.
marty.frmn
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### what are the 3 integers in arithmetic seq. w/ prime product?
what are the 3 integers in arithmetic sequence whose product is prime????
stapel_eliz
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Since primality is a property of whole numbers, and since 1 and 0 are not primes, then 0 cannot be one of the numbers. Also, the product must be positive and greater than 1. In addition, you cannot have an even number in the sequence, since then 2 would be a factor, and the product would not be prime.
If the three integers (call them r, s, and t) are non-trivial, then their product, rst, will generally be non-trivial. Clearly none of them can be zero or even. The only factors which can safely be ignored are 1 and -1, as they "collapse" into the product.
Can you think of a way to work with these two values, and possibly one or two others, to create an arithmetical sequence (a string of three equidistant values) whose product rst > 1 is prime?
marty.frmn
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### Re: what are the 3 integers in arithmetic seq. w/ prime product?
yep.... got that one....
with -1 and 1.... comes along -3 which is neither an even number nor a 0....
so they satisfy the product being a prime condition and are in arithmetic sequence....
so the numbers have to be -3,-1 and 1....
stapel_eliz
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That's what I came up with, too. Good work! | 460 | 1,664 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.53125 | 4 | CC-MAIN-2018-17 | latest | en | 0.936382 |
https://www.12000.org/my_notes/kamek/mma_12_1_maple_2020/KERNELsubsection316.htm | 1,720,827,851,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514459.28/warc/CC-MAIN-20240712224556-20240713014556-00872.warc.gz | 522,214,997 | 2,883 | #### 2.316 ODE No. 316
$\left (2 x y(x)^3+y(x)\right ) y'(x)+2 y(x)^2=0$ Mathematica : cpu = 0.130987 (sec), leaf count = 48
$\left \{\{y(x)\to 0\},\text {Solve}\left [x=-\frac {1}{4} e^{-\frac {1}{2} y(x)^2} \text {Ei}\left (\frac {y(x)^2}{2}\right )+c_1 e^{-\frac {1}{2} y(x)^2},y(x)\right ]\right \}$ Maple : cpu = 0.043 (sec), leaf count = 53
$\left \{ y \left ( x \right ) =0,y \left ( x \right ) =\sqrt {-2\,{\it RootOf} \left ( {{\rm e}^{{\it \_Z}}}{\it Ei} \left ( 1,{\it \_Z} \right ) +4\,{{\rm e}^{{\it \_Z}}}{\it \_C1}-4\,x \right ) },y \left ( x \right ) =-\sqrt {-2\,{\it RootOf} \left ( {{\rm e}^{{\it \_Z}}}{\it Ei} \left ( 1,{\it \_Z} \right ) +4\,{{\rm e}^{{\it \_Z}}}{\it \_C1}-4\,x \right ) } \right \}$ | 364 | 727 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.15625 | 3 | CC-MAIN-2024-30 | latest | en | 0.246366 |
http://www.netlib.org/lapack/explore-html-3.6.1/de/ddd/lapacke_8h_ad5f14b1b1e18eabfe65eec156467ef35.html | 1,537,777,022,000,000,000 | text/html | crawl-data/CC-MAIN-2018-39/segments/1537267160233.82/warc/CC-MAIN-20180924070508-20180924090908-00540.warc.gz | 370,024,937 | 4,099 | LAPACK 3.6.1 LAPACK: Linear Algebra PACKage
lapack_int LAPACKE_zlarft ( int matrix_layout, char direct, char storev, lapack_int n, lapack_int k, const lapack_complex_double * v, lapack_int ldv, const lapack_complex_double * tau, lapack_complex_double * t, lapack_int ldt )
Definition at line 36 of file lapacke_zlarft.c.
41 {
42 lapack_int ncols_v, nrows_v;
43 if( matrix_layout != LAPACK_COL_MAJOR && matrix_layout != LAPACK_ROW_MAJOR ) {
44 LAPACKE_xerbla( "LAPACKE_zlarft", -1 );
45 return -1;
46 }
47 #ifndef LAPACK_DISABLE_NAN_CHECK
48 /* Optionally check input matrices for NaNs */
49 ncols_v = LAPACKE_lsame( storev, 'c' ) ? k :
50 ( LAPACKE_lsame( storev, 'r' ) ? n : 1);
51 nrows_v = LAPACKE_lsame( storev, 'c' ) ? n :
52 ( LAPACKE_lsame( storev, 'r' ) ? k : 1);
53 if( LAPACKE_z_nancheck( k, tau, 1 ) ) {
54 return -8;
55 }
56 if( LAPACKE_zge_nancheck( matrix_layout, nrows_v, ncols_v, v, ldv ) ) {
57 return -6;
58 }
59 #endif
60 return LAPACKE_zlarft_work( matrix_layout, direct, storev, n, k, v, ldv, tau,
61 t, ldt );
62 }
lapack_logical LAPACKE_z_nancheck(lapack_int n, const lapack_complex_double *x, lapack_int incx)
#define LAPACK_ROW_MAJOR
Definition: lapacke.h:119
lapack_int LAPACKE_zlarft_work(int matrix_layout, char direct, char storev, lapack_int n, lapack_int k, const lapack_complex_double *v, lapack_int ldv, const lapack_complex_double *tau, lapack_complex_double *t, lapack_int ldt)
lapack_logical LAPACKE_lsame(char ca, char cb)
Definition: lapacke_lsame.c:36
#define LAPACK_COL_MAJOR
Definition: lapacke.h:120
void LAPACKE_xerbla(const char *name, lapack_int info)
#define lapack_int
Definition: lapacke.h:47
lapack_logical LAPACKE_zge_nancheck(int matrix_layout, lapack_int m, lapack_int n, const lapack_complex_double *a, lapack_int lda)
Here is the call graph for this function: | 609 | 1,836 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.578125 | 3 | CC-MAIN-2018-39 | latest | en | 0.22794 |
https://brainmass.com/business/accounting/wealth-conservation-reducing-wealth-and-state-tax-through-annual-gifts-to-children-222084 | 1,643,075,936,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320304749.63/warc/CC-MAIN-20220125005757-20220125035757-00207.warc.gz | 213,781,932 | 75,842 | Explore BrainMass
# Wealth Conservation: Reducing Wealth and State Tax Through Annual Gifts to Children
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This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!
TAX STRATEGY PROBLEM
Pedro Bourbone is the founder and owner of a highly successful small business and, over the past several years, has accumulated a significant amount of personal wealth. His portfolio of stocks and bonds is worth nearly \$5,000,000 and generates income from dividends and interest of nearly \$250,000 per year. With his salary from the business and his dividends and interest, Pedro has taxable income of approximately \$600,000 per year and is clearly in the top individual marginal tax bracket. Pedro is married and has three children, ages 16, 14, and 12. Neither his wife nor his children are employed and have no income. Pedro has come to you as his CPA to discuss ways to reduce his individual tax liability as well as to discuss the potential estate tax upon his death. You mention the possibility of making gifts each year to his children. Explain how annual gifts to his children will reduce both his income during lifetime and his estate tax at death.
#### Solution Preview
Pay attention to the fact that the earning assets are consistently being drawn down.
The annual gift tax exclusion with gift-splitting is \$22,000. Pedro and his wife together can give to each child up to \$22,000. Employing the strategy enables Pedro to decrease his assets earning money by \$66,000 per year and to shift some of the income he is earning on his assets to his minor children. The top tax bracket is 35%. The first year that he and his ...
#### Solution Summary
This solution consists of a word problem and the professional response to the inquiry. The subject matter is how to systematically use the Gift and Estate Tax Laws appropriately to reduce the estate tax at death.
\$2.49 | 413 | 2,008 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2022-05 | latest | en | 0.98633 |
https://jonathanlewis.wordpress.com/2019/06/09/cpu-percent/ | 1,632,606,588,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780057775.50/warc/CC-MAIN-20210925202717-20210925232717-00195.warc.gz | 374,197,034 | 23,676 | ## June 9, 2019
### CPU percent
Filed under: 12c,AWR,Oracle — Jonathan Lewis @ 2:31 pm BST Jun 9,2019
A recent post on the ODC General Database forum asked for an explanation of the AWR report values “%Total CPU” and “%Busy CPU” under the “Instance CPU” label, and how the “%Busy CPU “ could be greater than 100%. Here’s a text reproduction of the relevant sample supplied:
## Host CPU
CPUs Cores Sockets Load Average Begin Load Average End %User %System %WIO %Idle 2 2 1 0.30 1.23 10.7 5.6 5.3 77.7
## Instance CPU
%Total CPU %Busy CPU %DB Time waiting for CPU (Resource Manager) 29.8 133.8 0.0
The answer is probably “It’s 12.1 and it’s a programmer error”.
• Note that the Host CPU %Idle is not consistent with the three usage figures: 10.7 + 5.6 + 5.3 = 21.6 whereas 100 – 77.7 = 22.3.
• So let’s run with 22.3% and see what else we can notice: 29.8 / 22.3 = 1.3363 – that’s pretty close (when expressed as a percentage) to 133.8%
### Hypothesis:
Someone did the division the wrong way round when trying to work out the percentage of the host’s non-idle CPU that could be attributed to the instance. In this example the “%Busy CPU” should actually report 100 * 22.3 / 29.8 = 74.8%
Note – the difference between 133.8 and 133.63 can be attributed to the fact that the various figures reported in this bit of the AWR are rounded to the nearest 1 decimal place.
Note 2 – I don’t think this error is present in 11.2.0.4 or 12.2.0.1
1. There are many problems with AWR reports in 12.1
For instance, AWR reports do not work properly with PDBs in 12.1 or 12.2, as a number metrics are missing.
AWR does appear to work properly in 19c PDBs. I have not tested 18c.
Comment by jkstill — June 9, 2019 @ 4:12 pm BST Jun 9,2019
2. If we’re “trying to work out the percentage of the host’s non-idle CPU that could be attributed to the instance”, it makes sense to divide the instance CPU / host non-idle, so arguably the division done is the right way – what’s weird is that “% total cpu” of the instance is higher than that of the host…
Comment by Nickolay — June 10, 2019 @ 12:44 pm BST Jun 10,2019
This site uses Akismet to reduce spam. Learn how your comment data is processed. | 671 | 2,193 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2021-39 | latest | en | 0.896564 |
https://www.physicsoverflow.org/19246/obtaining-the-%24-frac-1-2-pi-%24-factor-in-the-fourier-transform | 1,721,471,912,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763515079.90/warc/CC-MAIN-20240720083242-20240720113242-00621.warc.gz | 826,881,089 | 27,960 | # Obtaining the $\frac{1}{2\pi}$ factor in the Fourier transform
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This MathWorld page gives this definition of a Fourier transform: $$F(k) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i k x}dx.$$ But, I wish to speak in terms of linear frequency $\nu$ and time $t$ rather than in terms of wavenumber $k$ and position $x$, so I will use the substitutions $k \rightarrow \nu$ and $x \rightarrow t$ rewrite this as: $$F(\nu) = \int_{-\infty}^{\infty} f(t) e^{-i2\pi \nu t}dt \; \; \; \text{(eq. 1).}$$
Is this substitution valid, or did I miss a factor?
Now, angular frequency $\omega$ and linear frequency $\nu$ are related by $\omega = 2 \pi \nu$ so I can rewrite in terms of the angular frequency $\omega$: $$F({\omega\over2\pi}) = F(\nu) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t}dt \; \; \; \text{(eq. 2).}$$ However, my physics professor's distributed notes give this definition of the Fourier transform $F(\omega)$ of $f(t)$: $$F(\omega) = \frac{1}{2\pi} \int_{-\infty}^{\infty} f(t) e^{-i\omega t}dt \; \; \; \text{(eq. 3)}$$
How can I convert equation (2) to equation (3), to obtain the $\frac{1}{2\pi}$ factor?
This post imported from StackExchange Mathematics at 2014-06-16 11:26 (UCT), posted by SE-user Andrew
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The answer is, as always, in wikipedia. In your derivation, the factor that is not coming up in the direct transformation, will show up in the inverse one. In your professor's, the factor that is in the direct transform will not be present in the inverse one. As long as you define them matching each other, you will be ok. It is also pretty common to split the difference and have a $1/\sqrt{2\pi}$ in front of each.
This post imported from StackExchange Mathematics at 2014-06-16 11:26 (UCT), posted by SE-user Jaime
answered Dec 15, 2012 by (90 points)
One detail is slightly incorrect. The factor doesn't show up in the inversion formula either since Wikipedia, following Laurent Schwartz, has chosen a unitary and symmetrical convention, explained in my long answer. Perhaps you could edit your answer slightly.
This post imported from StackExchange Mathematics at 2014-06-16 11:26 (UCT), posted by SE-user joseph f. johnson
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I have recently been thinking a lot about this. You can say it is a question of taste, but why not think about taste, eh?
First, in a nutshell, you cannot convert one equation to the other because the $F$ in one is numerically different from the other. Wikipedia is using the most natural, the most symmetrical, the most «unitary» conventions, as I will explain later. But to transform to what your professor has, first do what you did, which is fine. Then you have to realise the prof is using a different definition of F which is not equal to other people's. What Jaime says is also correct.
Now for the in depth discussion.
Physically, the Fourier expansion (for example, of a periodic function) should not have any numerical factors in it since it is supposed to be an expansion in terms of a basis. That is,
$$\cos (t) = 0 + 0\sin t + 0\sin 2t + \dots + 1\cos t + 0\cos 2t + \dots .$$
Laurent Schwartz, for me, is the one who influenced a whole generation of mathematicians to say, well, the most basic period is period 1. So the basis functions must be $\cos 2\pi t$ etc. Okay, really he, like most scientists and engineers after the war, used complex exponentials. So the basis for complex-valued functions on the real line with unit periodicity are $e^{i 2\pi nt}$ with $n$ an integer and $i$ a fixed choice of a square root of unity. Don't use the unit circle since we want to allow $t$ to be time.
This will be an orthonormal basis if you choose the measure to be Lebesgue measure on the unit interval, again, the most natural choice. (Forget the unit circle, the unit interval with total measure one is the most natural. If you are going to use the unit circle, okay, as long as you use a measure with total mass unity.)
The other advantage here is that $n$ is the natural frequency $f$ rather than the angular frequency. This is good for statisticians or anyone else who uses Hz or cps. Angular frequency really is less natural. I have even seen an author who changes the definition of $e$ to get rid of the factor of $2\pi$ in order to make the appearance of these formulae as natural as their reality. One could alter $i$ by a scalar, too...
So, my point is, firstly, the inversion formula or expansion formula should not have any constants. Secondly, it is possible to pick a very natural, «unitary» set of basis functions and frequencies. This forces where the two and the pi go.
Now the Fourier transform is forced on you, i.e., the Fourier transform is how you get the coefficients from the given function, and the inverse transform is how you get the function from the coefficients, and these do not live on the same space. It is good to make this distinction as clear as possible, since it crops up again for Lie Groups. (It gets a little blurred for the real line.)
The Fourier transform really takes the easiest possible form, since the basis is orthonormal,
$$\hat x (n) = \int_0^1 x(t) e^{-i2\pi nt} dt$$
as we merely have to take the Hermitian inner product of $f$ with each basis element in order to get the expansion coefficient. (The negative $i$ is because it is the Hermitian inner product, not a real symmetric inner product.)
For the case of a non-periodic function $x(t)$, the analogues look the same. The transform to get the coefficients for each frequency $f$ is
$$\hat x (f) = \int_{-\infty}^\infty x(t) e^{-i2\pi ft} dt$$
and the inverse transform, that gives the expansion of $x$ in terms of each frequency, is
$$x(t) = \int_{-\infty}^\infty \hat x(f) e^{i2\pi ft} df$$
and everything is easy to memorise: no coefficients in the expansion, unit periodicity, unit measure, reverse the sign of $i$ in the transform formula since it is a Hermitian form.
If for any reason you have to change units, like, days to years, since it's not periodic daily but only yearly, just change variables $t$ goes to $t'$ consistently in every formula but realise that now the frequency means in years, so of course what you now call $\hat x(f)$ is no longer equal to what you earlier called $\hat x(f)$ because $f$ means something quite different due to the change in units. $x(t)$ is the same, and the Fourier transform still means the same thing as before but its numerical value has changed since you changed the units. Now that is exactly what your professor did: the change was from $dt$ to $\frac1{2\pi}dt$. Unnatural, in a way, but there it is. People who prefer angular frequency tend to do this. It is analogous to a change of time units and has a similar effect on the formulas.
If you change the definition of the transform, you have to change the definition of the inverse transform. In order to be completely clear, let's make this rival definition of the Fourier transform use a squiggly instead of a hat:
$$\tilde x(f) = \int_{-\infty}^\infty x(t) e^{-i ft} {dt\over 2\pi}.$$
What is the relationship of $\tilde x$ to $\hat x$?. First, you must change the $e^{-i ft}$, which is appropriate when $f$ means angular frequency, to match the $e^{-i 2\pi ft}$ we use in our definition, which we use because it is normal frequency.
BTW, many people use the convention of capitalising the name of the function, $x$, to get its Fourier transform, $X$. But statisticians can't do that since we already have the convention that lower case is a sample signal, and upper case is for the random variable or stochastic process. So, like mathematicians but for the opposite reason, we prefer the hat or squiggle notation.
So change $f$ in the integral: write $\omega= {2\pi}f$. (You could, instead, change $t$. As I say, changing the $f$ to denote angular frequency is a lot like changing the time units.) Then
$$\hat x({\omega\over2\pi}) = \int_{-\infty}^\infty x(t) e^{-i \omega t} dt.$$ But the right hand side is, in turn, equal to $2\pi\tilde x(\omega)$. In short, $$\hat x(f) = 2\pi\tilde x(\omega),$$ where the relation between angular frequency and normal frequency has to be kept in mind, $$\omega= {2\pi}f,$$ i.e., $${\omega\over 2\pi}=f$$ and so $df = {d\omega\over2\pi}$.
Therefore, substituting into the inversion formula we would use for $\hat x$, we get the appropriate one to use for the professor's definition:
$$x(t) = \int_{-\infty}^\infty 2\pi\tilde x(\omega) e^{i2\pi ft} df$$ $$= 2\pi\int_{-\infty}^\infty \tilde x(\omega) e^{i\omega t} {d\omega\over2\pi}= \int_{-\infty}^\infty \tilde x(\omega) e^{i\omega t} d\omega.$$
What has caused all the trouble is your professor's using $f$ for angular frequency. Consistenly keeping track of the physical meaning of the different units actually helps one understand the numerical factors occurring in the Fourier transform and expansion formulas. So now we, reluctantly, agree to write $f$ instead of $\omega$ for angular frequency in order to agree with your professor. And we get...no, no, I won't do it...
This post imported from StackExchange Mathematics at 2014-06-16 11:26 (UCT), posted by SE-user joseph f. johnson
answered Feb 12, 2013 by (500 points)
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https://ebezpieczni.org/and-pdf/1271-half-wave-full-wave-and-bridge-rectifier-pdf-492-748.php | 1,643,360,903,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320305423.58/warc/CC-MAIN-20220128074016-20220128104016-00070.warc.gz | 274,179,878 | 7,277 | Thursday, March 18, 2021 8:57:49 AM
# Half Wave Full Wave And Bridge Rectifier Pdf
File Name: half wave full wave and bridge rectifier .zip
Size: 24680Kb
Published: 18.03.2021
If you know what is a rectifier , then you may know the ways to reduce the ripple or voltage variations on a direct DC voltage by connecting capacitors across the load resistance. This method may be suitable for low power applications , but not for applications that need a steady and smooth DC supply. One method to improve on this is to use every half-cycle of the input voltage instead of every other half-cycle waveform.
## Full Wave Rectifier Circuit
Related documents. Advantage of using full wave bridge is more efficiency, reduce more ripple and higher average voltage, current. Design the circuit diagram. Half Wave Rectifier Theory. The only dissimilarity is half wave rectifier has just one-half cycles positive or negative whereas in full wave rectifier has two cycles positive and negative.
It is within the power rectification arena that the bridge rectifier is the most common form of rectifier. The full wave rectifier is more complicated than the half wave version, but the full wave rectifier offers some significant advantages, and as a result it is almost exclusively used in this area. The concept of the full wave rectifier is that it utilises both halves of the waveform to provide an output and this greatly improves its efficiency. A further advantage when used in a power supply is that the resulting output is much easier to smooth. When using a smoothing capacitor, the time between the peaks is much greater for a half wave rectifier than for a full wave rectifier. It can be seen from the circuit diagram, that the fundamental frequency within the rectified waveform is twice that of the source waveform - there are twice as many peaks in the rectified waveform. This can often be heard when there is a small amount of background hum on an audio circuit.
A Full wave rectifier is a circuit arrangement which makes use of both half cycles of input alternating current AC and converts them to direct current DC. In our tutorial on Half wave rectifiers , we have seen that a half wave rectifier makes use of only one-half cycle of the input alternating current. This process of converting both half cycles of the input supply alternating current to direct current DC is termed full wave rectification. Full wave rectifier can be constructed in 2 ways. The first method makes use of a centre tapped transformer and 2 diodes.
## Full wave bridge rectifier
It utilizes only half of AC cycle for the conversion process. The Half-Wave Rectifier is unidirectional; it means it will allow the conduction in one direction only. This is the reason that it is called Half Wave Rectifier. While Full-wave Rectifier, is bi-directional, it conducts for positive half as well as negative half of the cycle. Thus, it is termed as full wave rectifier. Half Wave Rectifier circuit consists of a single diode and a step-down transformer, the high voltage AC will be converted into low voltage AC with the help of step-down transformer. After this, a diode connected in the circuit will be forward biased for positive half of AC cycle and will be reversed biased during negative half.
That means the full wave rectifier converts AC to DC more efficiently than the half wave rectifier. Manufacturing of the center-tapped transformer is quite expensive and so Full wave rectifier with center-tapped transformer is costly. Consider the half-wave rectifier circuit with resistive load. Besides the general rules above, specific guidelines for this lab: 1. Rectifier broadly divided into two categories: Half wave rectifier and full wave rectifier. This DC is not constant and varies with time.
In both cases, rectification is performed by utilizing the characteristic that current flows only in the positive direction in a diode. Full-wave rectification rectifies the negative component of the input voltage to a positive voltage, then converts it into DC pulse current utilizing a diode bridge configuration. In contrast, half-wave rectification removes just the negative voltage component using a single diode before converting to DC. From this, it can be said that full-wave rectification is a more efficient method than half-wave rectification since the entire waveform is used. Also, a ripple voltage that appears after smoothing will vary depending on the capacitance of this capacitor and the load.
Diode D1 conducts during the positive half-wave of the voltage. Diode D2 7: Waveforms of the single-phase, double-way, full-wave bridge rectifier. Looking at.
## full wave rectifier derivation pdf
Diode as a rectifiera rectifier is a device that converts alternating current to direct current. Amir Azmi. A bridge rectifier with an efficient filter is ideal for any type of general power supply applications like charging a battery, powering a dc device like a motor, led etc etc. Similarly, the current that flows acr oss the diode is the same as flows in the load. Substituting these values in 2 we get.
Peak inverse voltage PIV Peak inverse voltage or peak reverse voltage is the maximum voltage a diode can withstand in the reverse bias condition. The rectifier efficiency of a full-wave rectifier is A full wave rectifier produces positive half cycles at the output for both half cycles of the input.
Mierda! - вскипел Халохот. Беккеру удалось увернуться в последнее мгновение. Убийца шагнул к .
Но Цифровая крепость никогда не устареет: благодаря функции меняющегося открытого текста она выдержит людскую атаку и не выдаст ключа. Новый стандарт шифрования. Отныне и навсегда.
Две минуты спустя Джабба мчался вниз к главному банку данных. ГЛАВА 85 Грег Хейл, распластавшись, лежал на полу помещения Третьего узла.
У нас нет времени, чтобы… - Никакая служба здесь не появится, Сьюзан. У нас столько времени, сколько. Сьюзан отказывалась понимать. Не появится.
- Стратмор хмыкнул, раздумывая, как поступить, потом, по-видимому, также решил не раскачивать лодку и произнес: - Мисс Флетчер, можно поговорить с вами минутку. За дверью. - Да, конечно… сэр.
- Salida. Выпустите. Кардинал Хуэрра послушно кивнул. | 1,393 | 6,217 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.796875 | 3 | CC-MAIN-2022-05 | latest | en | 0.92009 |
http://metamath.tirix.org/mpeascii/bitr3i | 1,721,052,970,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514707.52/warc/CC-MAIN-20240715132531-20240715162531-00386.warc.gz | 21,851,896 | 1,607 | # Metamath Proof Explorer
## Theorem bitr3i
Description: An inference from transitive law for logical equivalence. (Contributed by NM, 2-Jun-1993)
Ref Expression
Hypotheses bitr3i.1
`|- ( ps <-> ph )`
bitr3i.2
`|- ( ps <-> ch )`
Assertion bitr3i
`|- ( ph <-> ch )`
### Proof
Step Hyp Ref Expression
1 bitr3i.1
` |- ( ps <-> ph )`
2 bitr3i.2
` |- ( ps <-> ch )`
3 1 bicomi
` |- ( ph <-> ps )`
4 3 2 bitri
` |- ( ph <-> ch )` | 165 | 432 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.703125 | 3 | CC-MAIN-2024-30 | latest | en | 0.645307 |
http://www.aspmessageboard.com/showthread.php?163684-Total-figure-not-accurate | 1,477,120,596,000,000,000 | text/html | crawl-data/CC-MAIN-2016-44/segments/1476988718840.18/warc/CC-MAIN-20161020183838-00404-ip-10-171-6-4.ec2.internal.warc.gz | 321,297,203 | 13,086 | Total figure not accurate?
# Thread: Total figure not accurate?
1. ncw
Senior Member
Join Date
Dec 1969
Posts
614
## Total figure not accurate?
<%<BR>Do While Not rs.EOF<BR>strTotal = rs("qty_issued") * rs("stock_price")<BR>response.write "<tr><td>" & rs("stock") & "</td>"<BR>response.write "<td>" & rs("qty") & "</td>"<BR>response.write "<td>" & formatnumber(strTotal,2) & "</td></tr>"<BR>strGrandTotal = clng(strGrandTotal) + clng(strTotal)<BR>rs.MoveNext<BR>Loop<BR>response. write "<tr><td><b>Gran d Total = " & formatnumber(strGrandTotal,2) & "</b></td></tr>"<BR>%><BR><BR>I wonder why my grand total figure not accurate esp the total of cents. Sample<BR>Stock Qty \$<BR>Flyers 600 162.36<BR>chair 1 16.50<BR>Grand total = 178.00<BR><BR>How do I fix this? Thank you.
2. Senior Member
Join Date
Dec 1969
Posts
16,931
## RE: Total figure not accurate?
Clng:<BR>http://msdn.microsoft.com/library/default.asp?url=/library/en-us/script56/html/vsfctclng.asp<BR>"For example, use CInt or CLng to force integer arithmetic in cases where currency, single-precision, or double-precision arithmetic normally would occur."<BR><BR>CDbl:<BR>http://msdn.microsoft.com/library/en-us/script56/html/vsfctcdbl.asp<BR><BR>Craig.<BR><BR>
#### Posting Permissions
• You may not post new threads
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• | 482 | 1,527 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.765625 | 3 | CC-MAIN-2016-44 | latest | en | 0.650604 |
https://economics.stackexchange.com/questions/55055/how-to-calculate-the-long-run-equilibrium-of-firms-in-a-cournot-competition | 1,702,202,838,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679101779.95/warc/CC-MAIN-20231210092457-20231210122457-00756.warc.gz | 252,847,926 | 41,697 | # How to calculate the long-run equilibrium of firms in a Cournot competition?
I am trying to solve the following question:
'In a scenario in which there exist multiple identical firms with a large supply of products available, each firm must decide how much to provide to the market. Each firm has the same cost function, given by C(q) = 10q and market demand is given by Q = 150 - P.
Entering the market entails a fixed cost F to be incurred by each firm in the entry stage. Suppose there is one period of Cournot competition after entry.
Determine the long-run equilibrium number of firms n^0.'
How can this be determined without knowing the exact initial fixed cost? Are long-run profits under Cournot competition 0 as well (like they are under perfect competition)? Thank you!
As Giskard’s answer to the linked question says, since the firms have symmetric costs, in the long run, they all make $$0$$ profit.
The inverse demand function is
$$Q = 150 - P \implies P(Q) = 150 - Q$$
The profit function is
$$\Pi_i = P(Q) q_i - TC_i(q_i)$$
Substituting all the given functions,
$$\Pi_i = 150 q_i - Q q_i - 10 q_i - F$$
Now we find the optimal production for each firm:
By the product rule on $$Q q_i$$ and noting $$\frac{\partial Q}{\partial q_i} = 1$$,
$$\frac{\partial \Pi_i}{\partial q_i} = 140 - Q - q_i = 0$$
Since the firms have symmetric costs, they have the same optimal productions, which implies $$Q = n q_i$$
$$140 - (n+1) q_i = 0$$
Isolating $$q_i$$
$$q_i = \frac{140}{n+1}$$
$$Q = \frac{140 n}{n+1}$$
From the inverse demand function we get
$$P = \frac{10 (n+15)}{n+1}$$
Substituting these into the $$0$$ profit condition,
$$10 \cdot \frac{n+15}{n+1} \cdot \frac{140}{n+1} - 10 \cdot 35 - F = 0$$
After doing some algebra we get
$$\frac{F + 350}{1400} (n+1)^2 = n + 15$$
$$\frac{F + 350}{1400} (n+1)^2 = (n+1) + 14$$
$$\frac{F + 350}{1400} (n+1)^2 - (n+1) - 14 = 0$$
$$(F + 350)(n+1)^2 - 1400 (n+1) - 19600 = 0$$
This is a quadratic equation in $$n+1$$ with coefficients
$$a = F + 350, b = -1400, c = -19600$$
Using the quadratic formula we get
$$n + 1 = \frac{140(5 \pm \sqrt{F + 375})}{F+350}$$
$$n = \frac{140(5 \pm \sqrt{F + 375})}{F+350} - 1$$
Since $$\sqrt{F + 375} \geq \sqrt{375} > \sqrt{25} = 5$$, the solution with the minus sign is negative so we discard it.
Therefore, the number of firms is
$$n = \frac{140(5 + \sqrt{F + 375})}{F+350} - 1$$
We got the number of firms as a function of the fixed cost $$F$$. To get an actual number, you would need to know the fixed cost value.
EDIT: Maybe you have to use the fact that in the long run, the fixed costs are $$0$$, so you would follow the same procedure and just get rid of $$F$$.
In that case, the number of firms would be:
$$n = \frac{140(5+\sqrt{375})}{350} - 1 \approx 8.75$$
Since there can’t be fractional firms, here I’d say the number of firms would be $$8$$ since it’s the highest integer value in which the firms don’t make negative economic profits. | 950 | 2,977 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 30, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.96875 | 4 | CC-MAIN-2023-50 | longest | en | 0.861457 |
https://socratic.org/questions/how-do-you-graph-y-ln-abs-x-3-1 | 1,597,215,326,000,000,000 | text/html | crawl-data/CC-MAIN-2020-34/segments/1596439738878.11/warc/CC-MAIN-20200812053726-20200812083726-00087.warc.gz | 494,234,931 | 6,179 | # How do you graph y=ln abs(x+3)?
Nov 11, 2016
#### Explanation:
Note that $\ln \left(x + 3\right)$ is not defined if $x + 3 < 0$ i.e. $x < - 3$.
Further if $x = - 2$, $g \left(- 2\right) = \ln \left(- 2 + 3\right) = \ln 1 = 0$ and therefore in the interval $\left(- 2 , \infty\right)$ $\ln \left(x + 3\right)$ is above $x$-axis and in interval $\left(- 3 , - 2\right)$, it is below $x$-axis.
Also $g ' \left(x\right) = \frac{1}{x + 3}$ and hence in the interval $\left(- 3 , \infty\right)$, $g ' \left(x\right)$ is always positive and hence $g \left(x\right)$ is a continuously increasing function and its graph looks like
graph{ln(x+3) [-10, 10, -5, 5]}
Hence, for $y = | \ln \left(x + 3\right) |$,
While in the interval $\left(- 2 , \infty\right)$, it is exactly same as for $\ln \left(x + 3\right)$, in interval $\left(- 3 , - 2\right)$ it will be positive too and a reflection of curve above. Obviously, we have a discontinuity at $x = - 2$
graph{|ln(x+3)| [-10, 10, -5, 5]} | 373 | 986 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 19, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.53125 | 5 | CC-MAIN-2020-34 | latest | en | 0.644687 |
https://en.youscribe.com/catalogue/documents/education/university/the-university-of-new-south-wales-1723481 | 1,606,206,958,000,000,000 | text/html | crawl-data/CC-MAIN-2020-50/segments/1606141176049.8/warc/CC-MAIN-20201124082900-20201124112900-00148.warc.gz | 274,642,573 | 52,799 | 3 Pages
English
# THE UNIVERSITY OF NEW SOUTH WALES
Description
Niveau: Supérieur, Master
THE UNIVERSITY OF NEW SOUTH WALES DEPARTMENT OF STATISTICS Exercises for MATH3811, Statistical Inference One Sample Problems. Two Sample Problems 1. In a study of drug abuse in a suburb A it was found that the median I.Q. of arrested abusers who were 16 years of age or older was 107. In order to test whether the median I.Q. of arrested abusers aged 16 or more in another suburb B is also 107, the I.Q.'s of 12 such persons from the suburb B were determined and are given below: 108, 111, 99, 127, 109, 100, 90, 94, 135, 104, 119, 117 a) Carry out an appropriate sign test. b) Give a non-parametric point estimate and find 95% confidence limits for the median I.Q. of drug abusers aged 16 or more in the suburb B. 2. The following table gives the level of the chemical serotonin in the blood of a patient before and after injecting 2.5mg of the drug reserpine. Patient 1 2 3 4 5 6 7 8 9 10 Level before 0.61 0.59 0.68 0.67 0.70 0.56 0.64 0.59 0.64 0.64 Level after 0.56 0.46 0.59 0.53 0.79 0.50 0.54 0.52 0.72 0.60 a) Carry out a sign test of the hypothesis that there is no difference in the serotonin level of a patient before and after injecting reserpine.
• another suburb
• alcohol treatment
• wilcoxon test
• who lived
• pure alcohol
• boy
• patient before
• exact wilcoxon
• test whether
• random variable
Subjects
Informations | 439 | 1,440 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.890625 | 3 | CC-MAIN-2020-50 | latest | en | 0.834549 |
https://www.euclideanspace.com/physics/relativity/lorentz/lorentzRotation/index.htm | 1,590,899,510,000,000,000 | text/html | crawl-data/CC-MAIN-2020-24/segments/1590347410745.37/warc/CC-MAIN-20200531023023-20200531053023-00428.warc.gz | 715,477,988 | 4,726 | # Physics - Velocity as a Rotation in Space-Time
## Overview
This page investigates velocities in moving frames of reference. We investigate whether ideas from relativity can be used in classical Euclidean space.
## Velocity as a Rotation in Space-Time
In relativity space and time is treated as a combined 4 dimensional space instead of separate 3 and 1 dimensional spaces. Here we stay with classical physics but try to use the space-time idea. Just to be clear: we are using conventional Euclidean space and are not using and concepts like the constant speed of light.
So imagine that Betty and Charles start walking at different speeds from the same point in time and space. Their positions can be plotted in space and time by a stationary person (Adam) and the plot may look like this:
The equations for these lines are:
x = vbt for Betty and
x = vct for Charles
where:
• vb= velocity of Betty
• vc= velocity of Charles
What if we want to measure these things in the frame of reference of say Charles? then we would get:
Since Charles does not move relative to himself then his line will be vertical (along the time axis). We get these graphs by subtracting vbt from the distance axis, which gives:
x = vbt - vct = (vb- vc)t for Betty and
x = vct - vct = 0 for Charles
x = -vct for absolute values such as the start point
So the axes are skewed relative to the absolute frame of reference.
For completeness here is the space-time diagram relative to Betty:
giving the equations:
x = vbt - vbt = 0 for Betty and
x = vct - vbt = (vc- vb)t for Charles
x = -vbt for absolute values such as the start point.
## Representing as a Rotation
Is it possible, instead of skewing the distance axis, to skew the time axis or rotate the axes?
tan(θ)=x/t ?
## Transform due to Relative Velocity
If we want to transform an event measured in two reference frames one of which is traveling at velocity v compared with the other we could use:
x -> x0 + vt
where:
• v = relative velocity of the two reference frames.
which would look like this:
—»
in other words point:
t x
is transformed to:
t x + vt
## Galilean Transforms
Galilean Transforms give a transform from on inertial frame to another in classical Newtonian physics.
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2005-03-04, 00:01 #1 Dougy Aug 2004 Melbourne, Australia 23·19 Posts Mersenne composites (with prime exponent) G'day, Just wondering if anyone has got a collection of the largest known Mersenne numbers with prime exponents. According to Mathworld p = 7068555.(2^121301)-1 and q = 2p+1 = 7068555.(2^121302)-1 is the largest known Sophie Germain prime. Since p is of the form 4n-1 (ie. a Lucasian prime) we use a theorem of Euler's to show that q divides 2^p-1, and hence 2^p-1 is composite. If my calculations are correct, the number of digits of 2^p-1 is roughly 10^36523. Does anyone know of any larger Mersenne composites with prime exponents? - Dougy
2005-03-04, 19:07 #2 philmoore "Phil" Sep 2002 Tracktown, U.S.A. 3×373 Posts This is an interesting question. Note that large as your 2^p-1 is, it is still miniscule compared to the largest known composite Fermat number F(2478782)! I see that the largest known Sophie Germain prime entry has roughly doubled in number of digits in the past 5 years, so I wouldn't expect that looking for larger S.G. primes is going to lead to any dramatic breakthroughs. I have been searching recently for factors of large iterated Mersenne numbers M(M(p)) where M(p) is a Mersenne prime. Will Edgington keeps a webpage of progress on these numbers at: http://www.garlic.com/~wedgingt/MMPstats.txt I am currently testing 16*(2^20996011-1)+1 to see if it is a factor of M(M(20996011)), but I estimate that my chances of success are roughly 1 in 2.5 million. There are now 42 of these iterated Mersenne numbers known, but no factors are known for any larger than M(M(31)), so I don't expect any sudden progress in the near future. But there must be an easier way to increase the upper bound of largest known composite Mersenne number (with prime number exponent). Ideas, anyone?
2005-03-04, 19:33 #3
ewmayer
2ω=0
Sep 2002
República de California
2D7116 Posts
Quote:
Originally Posted by philmoore I am currently testing 16*(2^20996011-1)+1 to see if it is a factor of M(M(20996011)), but I estimate that my chances of success are roughly 1 in 2.5 million.
Hi, Phil:
Why so low? On average, factor candidates 2*k*p + 1 with small k have a relatively much larger chance of being factors than those with large k, that's why even if one tries just the smallest-possible candidates k = 1 for a bunch of known-composite Mersennes, one will generally find some nontrivial fraction to be factors. Could you explain the heuristics you used to arrive at your probabilistic estimate?
-Ernst
2005-03-04, 23:09 #4 philmoore "Phil" Sep 2002 Tracktown, U.S.A. 3·373 Posts Those odds are really just a guess. I arrived at it by analogy with Fermat numbers. I believe it was a conjecture of Keller that if k*2^m+1 is prime, the probability of it dividing some Fermat number seem to be about 1/k. So I guessed by analogy that if p is prime and 2*k*p+1 is also prime, then the odds of 2*k*p+1 being a divisor of 2^p-1 are roughly 1/k. Of course this can't be exactly right, since if k = 1, 2p+1 is a divisor of 2^p-1 only when p is = 3 mod 4. (Otherwise, it divides 2^p+1.) Still, for the sake of argument, take my case of k=8. I checked first that 16*(2^20996011-1)+1 has no divisors less than 2^32. Then I used the new version of Prime95 to run P-1 and ECM on this number, which is also 2^20996015-15. I ran P-1 to bounds B1=250,000 and B2=10,000,000 and then I ran 15 ECM curves to bounds B1=11,000 and B2=1,100,000. (In retrospect, I wish I had run 33 curves with B1=5000 and B2=500,000 instead.) Now, I am estimating that I certainly would have found any factors of at least up to 40 bits, this raises the probability that my number is prime to about 1 in 300,000. But if it then has only a 1 in 8 chance of dividing M(M(20996011)), that puts my overall chances as 1 in 2,400,000. That may be an underestimate, but I still don't expect the odds to be too much better than 1 in 2 million. I would enjoy hearing any comments anyone may have. After this one, I plan to test 2*48*(2^24036583-1)+1 and 2*45*(2^25964951-1)+1 as well, but because of the larger k values, I really think that my current test presents the best probability of finding a factor of a large iterated Mersenne number.
2005-03-11, 12:14 #5 Dougy Aug 2004 Melbourne, Australia 15210 Posts Okay, I hope my calculations are correct, but the number of digits of F2478785 = 2^(2^2478785)+1 is greater than 10^746187. So this is phenomonally bigger then the largest known mersenne number with prime exponent. P. Ribenboim, The Book of Prime Number Records (1988): The largest known composite Mersenne number is Mq with q = 16695*2^3002-1. 2q+1 is prime. (or at least probably) Obviously this out out of date, but this used Euler's theorem to find the largest known Mersenne composite (with prime exponent), and I haven't heard of any other theorems that will give other Mersenne composites. So perhaps the same method is still valid at this stage.
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https://www.jiskha.com/questions/8670/Drbob222-even-if-you-just-tell-me-that-there-is-an-answer-to-the-following-problem | 1,566,144,922,000,000,000 | text/html | crawl-data/CC-MAIN-2019-35/segments/1566027313936.42/warc/CC-MAIN-20190818145013-20190818171013-00406.warc.gz | 851,026,940 | 5,472 | # Question,drbob222,math
Drbob222, even if you just tell me that there is an answer to the following problem then i know that there is an answer because i've tried many ways and i can't get x alone.
-4(2x-3) = -8x +5
Is there an actual answer for this one?
No. There is no answer, the question is flawed, impossible.
No, it isnt. the initial step of -4 times 2 is -8, not -1 as you posted. Also, you added eight x to both sides in the second line, but it became a negative on the left. Cant do that.
-4x+12=8x+5
-4x-8x=5-12
-12x=-7
x=12/7
i think this mught be right
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More Similar Questions | 979 | 3,203 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.21875 | 3 | CC-MAIN-2019-35 | latest | en | 0.957308 |
http://www.enotes.com/homework-help/express-following-partial-fractions-3-x-1-x-2-x-3-442891 | 1,461,931,487,000,000,000 | text/html | crawl-data/CC-MAIN-2016-18/segments/1461860111324.43/warc/CC-MAIN-20160428161511-00006-ip-10-239-7-51.ec2.internal.warc.gz | 496,487,967 | 12,065 | # Express the following as partial fractions: `(3)/((x+1)(x+2)(x+3))`
### 1 Answer |Add Yours
Posted on
`3/((x+1)(x+2)(x+3)) = A/(x+1)+B/(x+2)+C/(x+3)`
`3 = A((x+2)(x+3)+B(x+1)(x+3)+C(x+1)(x+2)`
When x = -1
`3 = A(-1+2)(-1+3)+B(-1+1)(-1+3)+C(-1+1)(-1+2)`
`3 = 2A+0+0`
`A = 3/2`
When x = -2
`3 = B(-2+1)(-2+3)`
`3 = -B`
`B = -3`
When x = -3
`3 = C(-3+1)(-3+2)`
`3 = 2C`
`C = 3/2`
So by partial fractions;
`3/((x+1)(x+2)(x+3)) = (3/2)/(x+1)-3/(x+2)+(3/2)/(x+3)`
Sources:
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## Jess is running her first ultramarathon (100 miles) with her
This topic has 2 expert replies and 1 member reply
gmattesttaker2 Legendary Member
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#### Jess is running her first ultramarathon (100 miles) with her
Sat Jun 21, 2014 6:29 pm
Hello,
Can you please assist with this?
Thanks a lot,
Sri
OA: Miles Kerry is ahead: 8
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Hi pyaramosam,
GMAT assassins aren't born, they're made,
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pyaramosam Newbie | Next Rank: 10 Posts
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Tue Feb 17, 2015 10:16 pm
We're told that the race is 100 miles and that it's broken in HALVES (50 miles each). The question gives us the rates for both Jess and Kerri during the FIRST HALF of the race (5mph and 6mph, respectively) and during the SECOND HALF of the race (8mph and 6mph, respectively). ???
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Hi pyaramosam,
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Rich
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what is the square root of 74
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Use a calculator...? There's no point in asking this question and you aren't required to remember it....
Report as
Rounded to two decimal places, the square root of 74 is equal to 8.60.
Report as
Report as
To the nearest thousandth, 8.602, but if you're looking for simplest rational form, that's as simplified as it's gonna get.
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8.602325267042627 is the square root of 74
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Pages: 12
Allow me:
Well, take the function f(x) = x, (-pi, pi), f(x+2(pi)n) = f(x), n E (here E is being used as "is an element of," not the letter) Z (all integers), (-infinity, infinity). Essentially, it is the line f(x) = x, from -pi to pi. At pi, it goes from pi to negative pi, creating a saw wave (think of it being |/|/|/|/|/ from negative infinity to infinity, where the bars are at pi*n).
Now, there's an interesting relationship between the coefficients of a fourier series (which is just using sine and cosine to represent a periodic function that isn't normally represented with sine and cosine, like our earlier saw wave) and the function itself:
sum(n=1, n -> infinity) (|cn(sum of the coefficients)|2) = int(-pi to pi)(f(x)2dx)/(2pi). In other words, the sum where n goes from 1 to infinity of the absolute value of the fourier coefficients squared is equal to the integral from -pi to pi of the function squared over 2pi. As for the coefficients:
bn = int(-pi to pi)(f(x)cos(nx)dx)/pi, n >= 0, which equals 0 (if f(x) = x, then it's zero, because the integral of an odd function from -a to a is zero)
an = int(pi to pi)(f(x)sin(nx)dx)/pi, n>=1, which equals (if you want, I'll solve the integral in a subsequent post) (2(-1)n+1)/n, n >= 1.
With that, we know that cn = an, since bn = 0. Therefore, the sums absolute value of (2(-1)n+1)/n squared from 1 to infinity should be equal to the integral from -pi to pi of x2 over 2 pi.
Well, the absolute value of an ends up just being 1/n (we pull out the two in front of the sum of the coefficients beforehand). Thus, the sum from one to infinity of 1/n2 is equal to the integral of x2 from -pi to pi, all over 4 pi. This, of course, ends up being pi2/6.
Does anyone know where you can find both a complete definition of a converging series, and the lowest level methods which are used to prove wether a series converges or diverges?
Topic archived. No new replies allowed.
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In this video, I discuss the homework for this lesson.
#### 3 Dimensional Analysis 15:50
In this lesson we discuss dimensional analysis (sometimes called unit analysis) and work through some examples.
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#### 14 Two Dimensional Vector Wrap Up 07:55
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#### 15 Projectile Motion 11:21
Projectile motion takes what we've learned about 2 dimensional vectors and applies the concepts of velocity, distance, freefall, and acceleration to describe the movement of projectiles.
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#### 17 Newton's First Law of Motion 12:57
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#### 18 Newton's First Law Wrap Up 05:30
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#### 26 Momentum and Impulse Wrap Up 08:45
In this video, I discuss the homework for this lesson.
#### 27 Wave Fundamentals 15:42
In this introduction we cover a lot of the basic topics and formulas which will serve as the foundation for following lessons.
#### 28 Wave Fundamentals Wrap Up 02:28
In this video, I discuss the homework for this lesson.
#### 29 Sound Waves 11:31
In this lesson we take a conceptual look at many of the fundamental ideas behind sound waves as well as apply some of the wave formulas we learned in the previous lesson to sound.
#### 30 Sound Wave Wrap Up 02:10
In this video, I discuss the homework for this lesson.
#### 31 Light 14:20
In this lesson we look into the nature of the electromagnetic spectrum, light, and the speed of light.
#### 32 Light Wrap Up 01:52
In this video, I discuss the homework for this lesson.
#### 33 What Is A Light Year? 07:55
In this short lesson we explain what a light year really is learn how many kilometers and miles light travels in a year.
#### 34 Lightyear Wrap Up 01:52
In this video, I discuss the homework for this lesson.
#### 35 Colors 16:23
In this lesson we discuss the color spectrum.
#### 36 Color Wrap Up 04:48
In this video, I discuss the homework for this lesson.
#### 37 Reflection and Refraction 15:23
In our last unit on waves we focus on how they can be reflected and refracted. We discuss how this relates to light waves but we also look briefly at how this can relate to sound waves as well.
#### 38 Reflection and Refraction Wrap Up 01:29
In this video, I discuss the homework for this lesson.
#### 39 Lenses 05:31
This short lesson takes a very conceptual look at lenses.
#### 40 Lens Wrap Up 06:06
This short lesson takes a very conceptual look at lenses.
#### 41 Energy 12:21
In this lesson we cover work, power, potential energy, and kinetic energy.
#### 42 Energy Wrap Up 06:12
In this video, I discuss the homework for this lesson.
#### 43 An Introduction To Gravity 11:22
In this video we talk about gravity as a concept as well as look at the mathematical formulas used to calculate gravitational attraction between two masses and find the gravity of a single mass.
#### 44 Gravity Wrap Up 02:37
In this video, I discuss the homework for this lesson.
#### 45 Gravitational Interactions 11:49
Now that we know some of the basics of gravity, let's look more at how masses interact with each other gravitationally.
#### 46 Gravitational Interaction Wrap Up 07:22
In this video, I discuss the homework for this lesson.
#### 47 Atoms Fundamentals 11:38
In this lesson we discuss that particles that make up atoms and the different phases of matter.
#### 48 Atom Fundamentals Wrap Up 03:10
In this video, I discuss the homework for this lesson.
#### 49 Solids 14:32
In this lesson we look at the atomic nature of solids and learn how to calculate the density of an object.
#### 50 Solids Wrap Up 03:09
In this video, I discuss the homework for this lesson.
#### 51 Liquids 13:13
Here we take a closer look at liquids and concepts such as buoyancy, the Archimedes Principle, and Pascal's Principle.
#### 52 Liquids Wrap Up 01:31
In this video, I discuss the homework for this lesson.
#### 53 Gases 11:46
In this lesson we discuss the last phase of matter we will cover in this course, which is gas.
#### 54 Gas Wrap Up 02:40
In this video, I discuss the homework for this lesson.
#### 55 Electrostatics 17:08
In this lesson we introduce Coulomb's Law and discuss concepts behind electrostatics.
#### 56 Electrostatic Wrap Up 03:26
In this video, I discuss the homework for this lesson. At the beginning of the video I state that I'm about to talk about the lens homework and it even says "lenses" on the board but, rest assured, I cover the electrostatic worksheet. Sorry about any confusion!
#### 57 Electric Fields 10:25
In this lesson we build off of what we've learned about electrostatics in order to learn about electric fields.
#### 58 Electric Fields Wrap Up 04:04
In this video, I discuss the homework for this lesson.
#### 59 Electric Potential 06:52
In this short lesson we discuss what electric potential is and how to calculate it.
#### 60 Electric Potential Wrap Up 03:05
In this video, I discuss the homework for this lesson.
#### 61 Electric Current 12:55
In this lesson we discuss potential difference and current.
#### 62 Electric Current Wrap Up 05:10
In this video, I discuss the homework for this lesson.
#### 63 Electric Circuits 13:21
In our last lesson we look at circuit diagrams, the differences between series and parallel circuits, as well as how to determine the resistance of a circuit when resistors are in series and/or in parallel.
#### 64 Electric Circuit Wrap Up 08:09
In this video, I discuss the homework for this lesson.
Course Description:
This course covers topics that would be taught in most Physics classrooms across North America. This course contains 9 units and 33 interesting lessons. Each lesson includes a worksheet and a wrap up video which contains detailed explainations for the solutions to the problems in the lesson.
Course Goals:
Upon completion of this course, each student will have a foundational understanding of the core concepts of physics and will be ready to move onward to more advanced phyics classes at the high school or college levels.
Course Includes:
• Over 10 hours of video content
• 33 video lessons
• 33 practice worksheets
• 33 lesson wrap up videos (I review each problem and provide the answer step by step)
• 33 short quizes
Target Audience:
This course for for students who are interested in physics because...
• They want to develop stronger problem solving skills
• They want to major in a field that requires an understanding of physics
• They enjoy mathmatics
• They enjoy science
• They want to understand more about how the world works from a scientific perspective
• They want to sharper their critical thinking skills
• They enjoy challenging themselves with new concepts
Course Requirements:
This course is for students which have taken introductory algebra classes or are currently enrolled in one. A knowledge of basic geometry will also be advantageous.
Course Topics:
• Metric Unit Conversion
• Dimensional Analysis
• Scientific Notation
• One Dimensional Vectors
• Speed and Velocity
• Freefall and Acceleration
• Two Dimensional Vectors
• Projectile Motion
• Newton's First Law of Motion
• Newton's Second Law of Motion
• Friction
• Newton's Third Law of Motion
• Momentum and Impulse
• Wave Fundamentals
• Sound
• Light
• Light years
• Colors
• Reflection and refraction
• Lenses
• Energy
• Gravity
• Gravitational Interactions
• Atoms
• Solids
• Liquids
• Gases
• Electrostatics
• Electric Fields
• Electric Potential
• Electric Current
• Electric Circuits
• Teacher: Justin
• Areas of expertise: Physics, Biology
• Education: Masters, Science Education
• Interests: Physics, astronomy, geology, computer science, programming, gaming, spending time outdoors, spending time with my family.
• Skills:
• Associations: Certified teacher in the state of Texas, General Science 8-12. I'm also a certified administrator in the state of Texas.
• Issues I care about: Environmental issues, safety, quality education, scientific literacy among youth
#### Metric Unit Conversion Practice Key
These answers given are approximate, meaning the student may or may not have chosen to round up their answers. If the students rounded up logically, or chose not to round up at all then I would still give them credit if I was grading the paper, so long as they are still close to the answer given. | 2,542 | 10,934 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.796875 | 4 | CC-MAIN-2022-21 | latest | en | 0.810541 |
https://geeksprep.com/write-a-program-to-find-the-wave-array/ | 1,643,443,427,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320300573.3/warc/CC-MAIN-20220129062503-20220129092503-00057.warc.gz | 318,234,896 | 21,624 | # Write a program to find the wave array.
Given an integer array we have to find the wave of array and print the result on screen.
so what is wave of array?
when we have to find print the array in arr[0] >= arr[1] <= arr[2] >= arr[3] form then that array is wave array.
Example:
``````Input:7 6 4 5 0 1
output:1 0 5 4 7 6 ``````
Logic:
``````step 1- sort the array
step 3-print the resultant array.``````
code:
```#include<bits/stdc++.h>
using namespace std;
int main()
{
int n;
cin>>n;
int a[n];
for(int i=0;i<n;i++)
{
cin>>a[i];
}
sort(a, a+n);
for (int i=0; i<n-1; i+= 2)
{
swap(a[i],a[i+1]);
}
for (int i=0; i<n; i++)
{
cout<<a[i]<<" ";
}
cout<<endl;
}
```
Output:
``````6
7 6 4 5 0 1
1 0 5 4 7 6 ``````
Time Complexity:O(Nlogn)
Space Complexity:O(1)
Scroll to Top | 292 | 783 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.5625 | 3 | CC-MAIN-2022-05 | longest | en | 0.637519 |
https://drmf-beta.wmflabs.org/index.php?title=Formula:KLS:01.06:07&oldid=2059 | 1,660,724,811,000,000,000 | text/html | crawl-data/CC-MAIN-2022-33/segments/1659882572870.85/warc/CC-MAIN-20220817062258-20220817092258-00390.warc.gz | 231,707,452 | 11,734 | # Formula:KLS:01.06:07
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
${\displaystyle{\displaystyle{\displaystyle\frac{1}{2\cpi\iunit}\int_{-\iunit% \infty}^{\iunit\infty}\frac{\Gamma\left(a+s\right)\Gamma\left(b+s\right)\Gamma% \left(c+s\right)\Gamma\left(d+s\right)\Gamma\left(a-s\right)\Gamma\left(b-s% \right)\Gamma\left(c-s\right)\Gamma\left(d-s\right)}{\Gamma\left(2s\right)% \Gamma\left(-2s\right)}\,ds{}=\frac{2\,\Gamma\left(a+b\right)\Gamma\left(a+c% \right)\Gamma\left(a+d\right)\Gamma\left(b+c\right)\Gamma\left(b+d\right)% \Gamma\left(c+d\right)}{\Gamma\left(a+b+c+d\right)}}}}$
## Proof
We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.
## Symbols List
: ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
: integral : http://dlmf.nist.gov/1.4#iv
: Euler's gamma function : http://dlmf.nist.gov/5.2#E1 | 333 | 944 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 4, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.578125 | 3 | CC-MAIN-2022-33 | latest | en | 0.499268 |
http://wizardofodds.com/ask-the-wizard/probability/puzzles/ | 1,427,552,211,000,000,000 | text/html | crawl-data/CC-MAIN-2015-14/segments/1427131297587.67/warc/CC-MAIN-20150323172137-00003-ip-10-168-14-71.ec2.internal.warc.gz | 301,219,932 | 39,033 | # Probability - Puzzles
## Wizard Recommends
• \$3000 Welcome Bonus
• \$11000 Welcome Bonus
• \$11000 Welcome Bonus
• 400% Welcome Bonus
I am interested in finding out some specific information on the odds of rolling dice. If you have 6 dice and roll them all at once, the odds of rolling all ones are 1 in 46,656. My question is what are the odds of rolling one to five ones. I am really interested in finding out the formula that should be used to calculate this type of problem.
Ken
The probability of rolling x ones out of y dice is combin(y,x)*(1/6)x*(5/6)y-x. See my section on probabilities in poker for an explanation of the combin(x,y) function. For example the probability of rolling 4 ones is combin(6,4)*(1/6)4*(5/6)2 = 0.803755%.
### Number of Ones in Six Dice
Ones Probability
0 0.3348980
1 0.4018776
2 0.2009388
3 0.0535837
4 0.0080376
5 0.0006430
6 0.0000214
Total 1.0000000
Eight golfers went to a new course. The caddy master put 8 bags on four carts at random. The golfers put 8 marked golf balls in a hat. The balls were thrown in the air. The 2 closes balls to each other were partners. In every case the partners’ golf bags were already on the same cart. What is the probability of that the golf bags were paired up correctly before the throw?
Anonymous
The formulaic answer for the number of combinations would be combin(8,2)*combin(6,2)*combin(4,2)/fact(4) = 25*15*6/24 = 105. Another way to solve the number of combinations would be to take one golfer at random. There are 7 possible people to pair him with. Then pick another golfer at random from the six left. There are 5 possible people to pair him with. Then pick another golfer at random from the four left. There are 3 possible people to pair him with. So the number of combination is 7*5*3 = 105. Thus the answer is 1 in 105.
A friend sent me this, I was wondering if there was a formula as to how this works.
Anonymous
Often these mind reading number puzzles work because an interesting mathematical oddity. If the sum of digits of a number is divisible by 9 then the number it self is divisible by 9. Let’s try it on the phone number of the Las Vegas Tropicana (702-739-2222). The sum of digit is 7+0+2+7+3+9+2+2+2+2 = 36. 36 divides evenly by 9, so 702739222 must also be divisible by 9. Here is a proof of this.
1. Let n bet any integer. Express n as d0*1 + d1*10 + d2*100+ d3*1000+ ... + dn*10n, where dn is the first digit, dn-1 is the second, and so on.
2. n = [d0 + d1 + d2 + ... + dn ] + [d1*9 + d2*99+ d3*999+ ...+ dn*999...9 ( a number with n nines)]
3. n = [d0 + d1 + d2 + ... + dn ] + 9*[d1*1 + d2*11+ d3*111+ ... dn*111...1 (a number with n ones)]
4. 9*any integer is evenly divisible by 9. So if d0 + d2 + d2 + ... + dn , or the sum of digits, is divisible by 9, then the entire number must be divisible by 9.
Now that we have that proof out of the way we can look at this magic trick. The problem asks you to pick any number. Then rearrange the digits to make a second number. Then subtract the smaller number from the larger number.
The answer is always going to have a sum of digits divisible by 9. Why? For every digit in the original number it appears somewhere else in the other number. Going one set of digits at a time, changing all the other numbers to zero, we could boil down each set as +/- n*[10x - 10y] (where x>=y and n is the digit) = +/-n *10y * (10x-y - 1) = 10y * (a number composed of only nines) = a number divisible by 9.
Let’s look at an example. Let the original number be 1965. Scramble it up to get 6951. 6951 - 1965 = 6*(1000-10) + 9*(100-100) + 5*(10-1) + 1*(1-1000) = 6*990 + 9*0 + 5*9 + 6*-999. Note that each part is divisible by 9, thus the number you get after subtracting must also be divisible by 9, and finally the sum of digits is also divisible by 9.
The trick then asks you to circle a number except 0 and enter the sum of all the other digits. The program then only needs to add a number to the number you entered so that the sum is divisible by 9. For example if you said the sum of your digits was 13 then you must have circled a 5, because 13+5 = a number divisible by 9.
The reason you can’t circle a zero is because if you did and then entered a number already divisible by 9 then the program wouldn’t know whether you circled a 0 or a 9.
Great site. I refer to it often as a gambler with an interest in probability and statistics, but this question actually pertains to my work. My HR Department insists that I rate my small staff (5 people) on a bell curve--one in the top 5% of all employees, one in the next 20%, one in the next 50%, one in the next 20%, and one in the bottom 5%. The company has approximately 5000 employees. What is the probability of such a small sample size fitting this distribution?
Anonymous
Thanks for the compliment. This is a good problem. The probability that exactly one employee will be in the bottom 5% is 5*(.05)*(.95)4 = 0.203627. Given that one employee is in the lowest 5% the probability of exactly one in the next 20% is 4*(.2/.95)*(.75/.95)3 = 0.414361. Given these two underachievers the probability of exactly one in the next 50% out of the remaining 75% is 3*(.5/.75)*(.25/.75)2 = 0.222222. The probability that one of the remaining two falls in the lower 20% of 25% is 2*(.2/.25)*(.05/.25) = 0.32. Taking the product of all these probabilities we get 0.006, or 3/5 of 1%.
To the fellow who asked the question about order statistics (column #100), I have two quibbles: one small and one large. Your method failed to make a finite population correction, which I grant is trivial with 5000 employees, but it certainly wouldn’t have been had there been 20 employees!
More importantly, however, you implicitly assume that managers have no effect on their employees. Suppose good managers, through judicious hiring and firing, or through above-average motivational skills, raise the average level of their employees. Without accounting for this effect we will either upward- or downward-bias the resulting probabilities. I’m sure you knew this, but I am sensitized to it because I do a lot of calculations like this in discrimination cases and failure to adjust for things we can adjust for (in this case a group-specific effect) can often lead people astray.
Anonymous
Thank you for those good points. However the alternative to no control over the distribution of job performance ratings is rating inflation. The manager will be put in a position of giving out bloated ratings to keep his staff happy. As a government worker for ten years I speak with some experience on this. When I taught at UNLV there was no average class GPA standard but there were certain expectations about what a grading curve should look like at the end of the semester. At least in a college setting I thought that made for a reasonable policy. Perhaps in a business environment some sort of common sense medium would also be best.
Do there exist any famous unsolved problems facing gaming mathematicians? Like a Fermat's Last Theorem in the gambling world. If so, would you please share an example.
Anonymous
Good question. I can't think of any.
How does this work: www.1800gotjunk.com/genie/?
Anonymous
Let’s express your number as 10t+u. You are asked to subtract each digit, leaving you with 10t+u-t-u = 9t, a number divisible by 9. Note how all the numbers divisible by 9 have the same item, which is the one the genie predicts.
I am about to take a professional licensure examination. The regulations provide that:
1. The examination shall consist of 7 subjects.
2. For each subject, 60 multiple choice questions shall be asked.
3. Each multiple-choice question shall have four possible answers, but only one correct answer.
4. In order to pass, an examinee must obtain a general average of at least 75% and must not have a grade lower than 65% in any subject.
My question is, if an examinee merely guesses all his answers, what is his chance of passing the exam? Stated differently, what is the probability of passing the exam by sheer luck?
Anonymous
To satisfy the 75% requirement the student must get at least 315 out of the 420 questions right. The expected number of correct answers from guessing is 420*0.25=105. The standard deviation is (420*0.25*0.75)^0.5 = 8.87412. So the candidate must exceed expectations by 210 questions, or 210/8.87412=23.66432 standard deviations. The probability of doing this is way off the charts. If every living thing on earth took this test, answering randomly, I doubt anyone or anything would pass. I won’t even get into the other requirement.
If a university’s football team has a 10% chance of winning game 1 and a 30% chance of winning game 2, and a 65% chance of losing both games, what are their chances of winning exactly once?
Anonymous
If we assumed the games were independent then the probability of losing both would be 90%*70%=63%. But since you say the probability of losing both is actually 65% (which is more than the 63%), that means the two events are correlated. If the probability of losing both is 65% and just losing game 2 is 70%, then the probability of winning game 1 and losing game 2 must be 5%. Using the same logic the probability of losing game 1 and winning game 2 must be 25%. That only leaves 5% for winning both games. So the probability of winning exactly once is 25%+5% = 30%.
On the game show Let’s Make a Deal, there are three doors. For the sake of example, let’s say that two doors reveal a goat, and one reveals a new car. The host, Monty Hall, picks two contestants to pick a door. Every time Monty opens a door first that reveals a goat. Let’s say this time it belonged to the first contestant. Although Monty never actually did this, what if Monty offered the other contestant a chance to switch doors at this point, to the other unopened door. Should he switch?
Anonymous
Yes! The key to this problem is that the host is predestined to open a door with a goat. He knows which door has the car, so regardless of which doors the players pick, he always can reveal a goat first. The question is known as the "Monty Hall Paradox." Much of the confussion about it is because often when the question is framed, it is incorrectly not made clear the host knows where the car is, and always reveals a goat first. I think put some of the blame on Marilyn Vos Savant, who framed the question badly in her column. Let’s assume that the prize is behind door 1. Following are what would happen if the player (the second contestant) had a strategy of not switching.
• Player picks door 1 --> player wins
• Player picks door 2 --> player loses
• Player picks door 3 --> player loses
Following are what would happen if the player had a strategy of switching.
• Player picks door 1 --> Host reveals goat behind door 2 or 3 --> player switches to other door --> player loses
• Player picks door 2 --> Host reveals goat behind door 3 --> player switches to door 1 --> player wins
• Player picks door 3 --> Host reveals goat behind door 2 --> player switches to door 1 --> player wins
So by not switching the player has 1/3 chance of winning. By switching the player has a 2/3 chance of winning. So the player should definitely switch.
For further reading on the Monty Hall paradox, I recommend the article at Wikipedia.
I disagree with your answer to the Monty Hall question in the November 19, 2004 column. Assuming the car is behind door one there are actually four possibilities as follows, where the prize is behind door 1.
• Player picks door 1 --> shown 2 --> switch to 3, lose
• Player picks door 1 --> shown 3 --> switch to 2, lose
• Player picks door 2 --> shown 3 --> switch to 1, win
• Player picks door 3 --> shown 2 --> switch to 1, win
As you can see the probability of winning is 50% whether you switch or not. Furthermore it just goes against common sense that switching would be better.
Anonymous
Your mistake is to assume each of these events has a 25% possibility. Following is the correct probability of each event.
• Player picks door 1 (1/3) * shown 2 (1/2) = player loses (1/6)
• Player picks door 1 (1/3) * shown 3 (1/2) = player loses (1/6)
• Player picks door 2 (1/3) * shown 3 (1/1) = player wins (1/3)
• Player picks door 3 (1/3) * shown 2 (1/1) = player wins (1/3)
So losing events have a total probability of 2*(1/6) = 1/3 and winning events have a total probability of 2*(1/3)=2/3.
With five different toppings to choose from, how many different pizzas can you make, with any number of toppings?
Anonymous
There is 1 way with 0 toppings, 5 ways with 1 topping, 10 ways with 2 toppings, 10 ways with 3 toppings, 5 ways with 4 toppings, and 1 way with 5 toppings. So the answer is 1+5+10+10+5+1 = 32. Another way to solve is either topping can be used or not. So the total is 25 = 32.
I saw in the paper last week that the latest earthquake that ravaged Indonesia hit on December 26th. It also showed that of the eight deadliest earthquakes to hit over the last 100 years, three of them have been on December 26th. I was wondering what the odds are of having three massive quakes hit on the same day knowing these facts: Earthquakes of this magnitude (8.0 or larger) happen only once per year. The last big quake was exactly one year ago, 12/26/03 in Iran (back to back probabilities?) I look forward to hearing from you.
Steve A. from Fort Collins, CO
After discovering the claim that the Florida hurricanes only hit Bush voting counties was a hoax (see the October 17, 2004 column) I am going to be more skeptical about such alleged coincidences. According to the National Earthquake Information Center of the top 11 earthquakes since 1990 only the recent one of 2004 hit on a December 26. The Iranian earthquake you mention was only 6.7 in magnitude, which is far from making the top eight.
How many eggs do you start with if each day you sell 1/2 the eggs plus 1/2 an egg; after 3 days you have zero eggs? At the end of each day, the number of eggs is a whole number.
Anonymous
Let’s let d (for day) be the number of eggs at the beginning of the day and n (for night) be the number at the end. The problem tells us that d/2 - ? = n. So, let’s solve d in terms of n.
d/2 = n + ?
d= 2n + 1
So on the third day n=0, so d=1.
On the second day n=1, so d=3.
On the third day n=3, so d=7.
So there you have it, you started with 7 eggs.
Imagine a island that is inhabited by 10 people, and the politics is such that each day an islander is chosen at random to be chief for exactly one day; after the day has elapsed another islander is chosen at random (so the same islander who was just chief has a 1/10 chance of being chief again). The question to be solved: on average, how many days would have to elapse before each islander would have been chief at least once?
Anonymous
It will only take 1 day so that 1 person has served as chief. For the second day the probability of a new chief is 0.9. The expected number of days it will take to get a new chief, if the probability each day is 0.9 is 1/0.9 = 1.11. This is true for any probability: the expected number of trials until a success is 1/p. So after 2 people have served the probability of a new chief on the next day is 0.8. So the waiting period for a 3rd chief is 1/0.8 = 1.25 days. The answer is the sum of the waiting periods, which is 1/1 + 1/.9 + 1/.8 + ... + 1/.1 = 29.28968 days.
The radius of the circle is 1. The triangle is equilateral. Find the area of each colored region.
Anonymous
I don’t want to blow the answer for those who want to solve it for themselves. For the answer and solution visit my other web site mathproblems.info, problem 189.
Say you won a contest where at halftime of an NBA game you got to shoot a free throw and if you make it you win \$1 million. Further, you can keep shooting free throws, double or nothing, till you miss or choose to stop. If you’re a 75% free throw shooter, when would you stop? Can you ever? At some point the money starts to mean less and less. What would you do?
Pete from New York
At some point you should refuse a good bet because the stakes are too high. Personally I think a good measure of the enjoyment one gets from money is the log of the amount. The base of the log does not matter so let’s use 10. However we can’t take a log less than 10, so let’s say the enjoyment is 0 for any amount less than ten. So in your example let’s assume you have \$0 before winning the \$1,000,000 with your first throw. Now you have log(1,000,000) = 6 units of happiness. The expected value of your happiness taking another free throw is 0.75*log(2,000,000) + 0.25*0 = 4.975772. This is less than 6 so in this case you should take the million and walk. However it might be different if you already had some money. Let’s say you already have \$200,000. Then your happiness by walking is log(1,200,000) = 6.07918. Your happiness by risking the million and taking another shot is 0.75*log(2,200,000) + 0.25*log(200,000) = 6.082075, so you marginally take the second shot. If you were to win that one your choice would be between log(2,200,000) = 6.34242 and 0.75*log(4,200,000)+0.25*log(200,000) = 6.29269. In this case you should not take a third shot and instead walk with the \$2,000,000 win. The breakeven point for accepting the first double is an existing wealth of \$191,487. To accept two doubles you should have \$382,975 in other money.
I remember that if 22 people are in a room the odds are even that two will celebrate the same birthday [month and day, not year]. I have forgotten how to do the math to prove this. Could you please provide it.
Dean from Bainbridge Island, WA
I think I have answered this before but the 50/50 point is closer to 23. To make things simple let’s ignore leap years. The long answer is to order the 23 people somehow. The probability that person #2 has a different birthday from person #1 is 364/365. The probability person #3 has a different birthday from persons #1 and #2, assuming they are different from each other, is 363/365. Keep repeating until person 23. The probability is thus (364/365)*(363/365)*...*(343/365) = 49.2703%. So the probability of no match is 49.27% and of at least one match is 50.73%. Another solution is the number of permutations of 23 different birthdays divided by the total number of ways to pick 23 random numbers from 1 to 365, which is permut(365,23)/36523 = 42,200,819,302,092,400,000,000,000,000,000,000,000,000,000,000,000,000,000,000 / 85,651,679,353,150,300,000,000,000,000,000,000,000,000,000,000,000,000,000,000 = 49.27%.
The weekly salaries of teachers in one state are normally distributed with a mean of \$490 and a standard deviation of \$45. What is the probability that a randomly selected teacher earns more than \$525 a week? I can’t remember how to calculate a probability from just the mean and SD without the population.
Sue from Queen Creek
That would be \$35 above average, or 7/9 standard deviations. The probability of being more than 7/9 standard deviations above expectations would be 1-Z(7/9) = 1- 0.78165 = 0.21835.
Two people are playing rock paper scissors. It is presumed the game doesn’t involve strategy. If you are playing ’best of 3’, and player A wins the first round, what are the odds that player B will win the game?
Andrew from Pewaukee,WI
Player B would need to win the next two (not counting ties) so the probability is (1/2)*(1/2) = 1/4.
Hi, I thought I would ask you this since I cannot find it anywhere on the web. I hope you answer this; What are the odds of existing? Whether it be on Earth or somewhere else in the universe? It’s not a gambling question but an answer we should all know so we can appreciate what kind of odds we beat just being alive!
Andreas from Edmonton
The probability that intelligent life exists anywhere in the galaxy I believe is very high. The Drake Equation seeks to estimate the number of incidents of intelligent life in the galaxy, which depending on the numbers you put into comes up with a figure of about a million. However there is also no good evidence that these civilizations have ever visited us or made contact. So the famous Fermi Question is "Where is everybody?" I do think the lack of evidence of other intelligent life casts some doubt on the Drake Equation but I would still put the number of intelligent civilizations in our galaxy in the ballpark of 1000. That is just our galaxy, there are billions of galaxies out there. However the distance between galaxies is so vast there is really not much point in discussing travel or communication between them. So to answer your question I would say roughly 99.9%.
Suppose a hotel has 10,000,000 rooms and electronic 10,000,000 keys. Due to a computer mistake each key is programmed with a random code, having a 1 in 10,000,000 chance of being correct. The hotel is sold out. What is the probability at least one customer has a working key?
Danny from London, U.K.
The exact answer 1-(9,999,999/10,000,000)10,000,000 = 0.632121. This is also the same as (e-1)/e to seven decimal places.
There are 75 multiple choice questions in an exam. Each question contains 4 possible answers only 1 is correct. The exam pass mark is 50%. What are the chances of passing the exam by guessing each answer?
Wendy from London
1 in 635,241.
For in-running betting, if a tennis player has a chance "p" of winning a game, what chance does he have of winning a set?
Mike from Perth
As I understand the rules of tennis the winner of a set is the first to win six games, and by a margin of at least two games, except a 6-6 tie will result in a single tie-breaker game. The following table shows the probability of winning a set, given the probability of winning a game.
### Probabilities in Tennis
ProbabilityGame Win ProbabilitySet Win 0.05 0.000003 0.1 0.000189 0.15 0.001899 0.2 0.009117 0.25 0.028853 0.3 0.06958 0.35 0.138203 0.4 0.23687 0.45 0.361085 0.5 0.5 0.55 0.638915 0.6 0.76313 0.65 0.861797 0.7 0.93042 0.75 0.971147 0.8 0.990883 0.85 0.998101 0.9 0.999811 0.95 0.999997
The formula for any probability of winning a game p, and losing q, is 1*p6 + 6*p6*q + 21*p6*q2 + 56*p6*q3 + 126*p6*q4 + 252*p7*q5 + 504*p7*q6
You are in a boat with a rock, on a fresh water lake. You throw the rock into the lake. With respect to the land (shore), does the water level increase, decrease, or stay the same? My co-workers think that the water level will stay the same.
David
The water level relative to the shore will decrease. Inside the boat the rock is pressing down on the canoe and thus pushing up the water around it. The amount of water displaced is equal in weight to that of the rock. For example, a 10 pound rock will displace 10 pounds of water upward. When the rock is thrown overboard the weight will not matter but rather the volume of the rock. So the rock will push upward an amount of water equal in volume to the rock. The mass of a rock is greater than that of water so the rock displaces more water pushing down on it than in it. So the level of the lake will be higher with the rock in the canoe than at the bottom of the lake.
How does this work?
1. Grab a calculator. (you won’t be able to do this one in your head)
2. Key in the first three digits of your phone number (NOT THE AREA CODE)
3. Multiply by 80
5. Multiply by 250
8. Subtract 250
9. Divide number by 2
Chris M. from Las Vegas
Let’s call the first three digits in your phone number x, and the last four y. Now let’s see what I have at each step.
2. X
3. 80x
4. 80x+1
5. 250*(80x+1) = 20000x+250
6. 20000x+250+y
7. 20000x+250+2y
8. 20000x+250+2y-250 = 20000x+2y
9. (20000x+2y)/2 = 10000x+y
So that is of course going to equal your phone number. We need the 10000x to move the prefix four places to the left, and then we add on the last four digits.
There is a drawing for a \$27,000 car, with tickets being sold at six for \$500.00, or one for \$100.00. 68 tickets have been sold, and tomorrow is the deadline for purchase. I know that for a 50% probability of winning, I must spend \$5666.44, and for a 66.66% of winning, I must spend \$11,332.88 (Right?). How much should I spend (or how many tickets must I buy) to virtually ensure that I "win" the car? (90%? 95%?) Is this raffle worth playing, or must I spend the cost of the car?
Annette from Boise
You are right regarding the 1/2 and 2/3 probabilities. If you buy t tickets your probability of winning is t/(68+t). So for a 90% probability, solve for t as follows.
0.9 = t/(68+t)
0.9*(68+t) = t
61.2 = 0.1t
t = 612, or \$51,000
For 95%...
0.95= t/(68+t)
0.95(68+t) = t
64.6 = 0.05t
t = 1292, or \$107,666.67
Assuming the car is worth \$27,000 to you, you should quit buying tickets as soon as the next ticket sold does not increase your probability of winning enough to warrant the price.
For a ticket to be worth the price it should increase your probability of winning by p, where...
27000*p=(500/6)
p=0.003086
Let's say t is the number of tickets your purchased where you are indifferent to buying one more ticket.
[(t+1)/(t+68+1)] − [t/(t+68)] = 0.003086
[(t+1)/(t+69)] − [t/(t+68)] = 0.003086
[((t+1)*(t+68))/((t+69)*(t+68))] − [(t*(t+69))/((t+68)*(t+69))] = 0.003086
[((t2 +69t+68)/((t+69)*(t+68))] − [(t2+69t)/((t+68)*(t+69))] = 0.003086
68/((t+68)*(t+69)) = 0.003086
((t+68)*(t+69)) = 220.32
t2+137t+4692 = 22032
t2 +137t - 17340=0
t=(-137+/-(1372-4*1*-17340)2)/2
t = 79.9326
Let's test this by plugging in some values for tickets purchased, assuming the player can always buy tickets at \$500/6 = \$83.33 each.
At 79 tickets your cost is 79*(500/6) = \$6,583.33, your probability of winning is 79/(79+68) = 53.74%, your expected return is \$27,000*0.5374 = \$14,510.20, and your expected profit is \$14,510.20 - \$6,583.33 = \$7,926.87.
At 80 tickets your cost is 80*(500/6) = \$6,666.67, your probability of winning is 80/(80+68) = 54.04%, your expected return is \$27,000*0.5405 = \$14,594.59, and your expected profit is \$14,594.59 - \$6,666.67 = \$7,927.92
At 81 tickets your cost is 81*(500/6) = \$6,750.00, your probability of winning is 81/(81+68) = 54.36%, your expected return is \$27,000*0.5436 = \$14,677.85, and your expected profit is \$14,594.59 - \$6,750.00 = \$7,927.85.
So we can see that the maximum expected win peaks at 80 tickets.
I’m trying to compare the cost of replacing an old refrigerator now in order to save on electrical costs, vs. waiting until it dies to replace it. I can calculate how much cheaper it is to run the new fridge vs. the old one: \$37/yr., that’s easy. But how do I factor in the cost of the new fridge? Say the new fridge costs \$425. I can’t say that *all* of that \$425 is a new expense, because I’ll have to replace the old fridge *someday*, if not now, so I’ll have that new-fridge expense at some point anyway. Let’s say that a typical fridge lasts 14 years and my old fridge is 9 years old, so if I replaced it now I’d be replacing it in 5 years. I tried to make a two-column table, comparing the cost of keeping the current fridge for 9 years and then replacing it, vs. replacing it now, but I didn’t know how to make an apples-to-apples comparison because I didn’t know for how far into the future to consider the costs, and because the fridges are replaced in different years. How do I compare the economics of replacing now vs. replacing later? By the way, this isn’t for my own situation, because my current fridge is probably 30 years old. It’s for, uh, a friend.
Spanky McBluejay from Austin, TX
If you keep the current fridge then in five years you will have spent an extra \$37*5 = \$185 on electricity compared to a new one. If you replace it now you’ll be out \$425 but assuming linear depreciation after five years it will still be worth \$425*(9/14) = \$273.21. So you will have lost \$425*(5/14) = \$151.79 due to depreciation. So the cost of depreciation of the new fridge is less than the additional electricity expense of keeping the old one, so I favor buying a new one now.
If there are three people, then What is the probability that at least two persons have the birthday on the same date.
Manish from New Delhi
Ignoring leap day, the probability of all three different birthdays is (364/365)*(363/365) = 0.99179583. So the probability of at least one common birthday is 1 - 0.99179583 = 0.00820417.
Five persons are in a room. What is the probability that at least 2 of them were born in the same birth month?
Amy
To keep things simple let’s assume that each person has a 1/12 probability of being born in each month. The probability that all five people are born in different months is (11/12)*(10/12)*(9/12)*(8/12) = 0.381944. So the probability of a common month is 1 - 0.381944 = 0.618056.
We’ve been given a challenge at work -- just for fun, and none of us can work it out. A farmer has 5 trailers full of sheep. Four of the trailers contain sheep weighing 39kg and the 5th trailer contains sheep weighing 40kg. All of the sheep are identical. He goes to the market. He wants to find out which of the trailers contains the sheep weighing 40kg, and he can only use the large weighing scales once!!! How does he do it? Please help, it is driving us all mad at my work place -- it’s a vet’s!!
Becca
The answer is at the end of the column.
Anonymous
Take one sheep from trailer 1, two from trailer 2, three from trailer 3, four from trailer 4, and zero from trailer 5. If all the sheep weighed 39 kg then the total weight would be 39 * 10 = 390 kg. However 0 to 4 sheep are one kg heavier. If the total weight is 391, then there is one heavy sheep on the scale; thus it must have come from trailer 1. Likewise, if the total weight is 392, then there are two heavy sheep on the scale, which must have come from trailer 2. In the same manner a weight of 393 means the heavy sheep are in trailer 3, a weight of 394 means the heavy sheep are in trailer 4, and a weight of 390 means the heavy sheep are in trailer 5.
On an airplane with 180 seats, what are the odds of me sitting next to the good looking girl I see who will be on the same flight?
Ted H. from Salt Lake City
It depends on the number of seats in a cluster. Most domestic flights have three seats on either side of the aisle. That would make 60 3-seat clusters. After the first one of you is seated, there will be two seats in the same cluster out of the remaining 179, so the chances of being in the same cluster are 2/179 = 1.12%. Then you can’t have somebody else in the middle seat. The chances of the third person being in the middle seat are 1/3. So the answer is (2/179)*(2/3) = 0.74%, or 1 in 134.25.
Three logicians are playing a game. Each must secretly write down a positive integer. The logician with the lowest unique integer will win \$3. If all three have the same number, each will win \$1. The logicians are selfish, and each wishes to maximize his own winnings. Communication is not allowed. What strategy will each logician follow?
Matthew from Fort Wayne, IN
The answer will appear in the next column.
I read that Warren Buffet (the world’s third richest man) complained that he only paid a 17.7% federal tax rate, while his secretary paid 30%. This seems outrageous to me. Can you comment?
Joe from Nashville
Normally I would say this is out of my area. However, as a former government actuary for eight years, I know a thing or two about taxes. From what I’ve read, most of Warren Buffet’s income is defined as capital gains, which is taxed at only a 15% rate. Like it or not, the tax laws allow it. What puzzled me is why his secretary was paying as much as 30%. According to this video, he was counting “payroll and income taxes.” By “payroll taxes” he obviously meant Social Security and Medicare taxes. Let’s see if 30% is a reasonable total federal tax rate for his secretary.
In 2007 the highest tax bracket was taxed at 35%, but that only applies to income above \$349,700. The income up to that point is taxed much less. Let’s assume his secretary is single, with no dependent children, and her salary was \$100,000. First, let’s subtract the minimum deductions. In 2007 the standard deduction for single filers was \$5,350. The personal deduction was \$3,400. So, we’re left with \$100,000 - \$5,350 - \$3,400 = \$91,250 in income subject to income taxes. For single filers in 2007, the tax rate was 10% on the first \$7825 in income, then 15% up to \$31,850, then 25% up to \$77,100, and 28% up to \$160,850. So, her income tax would have been =0.1×\$7,825+0.15×(\$31,850-\$7825)+0.25×(\$77,100-\$31,850)+0.28×(\$91,250-\$77,100) = \$19,660.75. That is only 19.7% of her income. All my assumptions like her income, filing status, and not itemizing worked against her, or for a higher tax rate.
Now let’s do Social Security and Medicare. In 2007, the Social Security tax was 6.2%, up to incomes of \$97,500, when it completely shuts off. The 2007 Medicare tax rate was 1.45%, with no cap. So, her combined Social Security and Medicare tax would have been 6.2%*97,500 + 1.45%*100000 = \$7,495. Counting those taxes, her overall tax rate would have been (\$19,660.75 + \$7,495)/\$100,000 = 27.2%. Still we’re 2.8% short of 30%.
My best guess is that she is also considering the fact that ultimately she is the one paying the employer’s matching Social Security and Medicare tax. For those who don’t know, Social Security and Medicare taxes are really double that deducted from your checks. The employer pays the other half. However, some, including me, would argue that ultimately it is the employee who pays both. If the employer didn’t have to pay that tax, he would have more money to pay his employees. It is easy to feel that way when you’re self-employed, like I am, and have to pay both shares. If you double the Social Security/Medicate tax, the rate is now (\$19,660.75 + 2×\$7,495)/\$100,000 = 34.7%. I assume the 4.7% difference is because she makes less than \$100,000, is married, has dependents, itemizes deductions, or some combination.
The Social Security and Medicare taxes would not apply much to Warren Buffet. First, the Social Security cap of \$97,500 would be insignificant to him. Second, those taxes apply to wages, not capital gains, as he defines most of his income to be.
So, that is my best guess as to the math behind Mr. Buffet’s statement.
Update: Shortly after this column appeared I received the following response. In the interests of fairness, I present the following argument that Mr. Buffet is paying too much in taxes.
I read with interest your answer to the ’outraged’ person who thinks it is so unfair that Warren Buffet pays less percentage in taxes than his secretary. I was disappointed in your answer, which does not correct the misinformation that implies that Mr Buffet pays less tax than his secretary.
First, as you noted, investment income is indeed taxed at 15%. This is in effect double taxation as the earned income that Mr. Buffet invested was taxed at his marginal rate of 36%. Comparing apples to oranges (work income vs investment income).
Second, one should not look at the percentage. In gambling terms, one should look at the ’payout’ instead. I am very certain that Mr. Buffet paid millions of dollars in taxes in the same year that his secretary paid thousands of dollars. Shouldn’t your reader be more outraged that one citizen of the country is paying 1000’s of times more than other citizens for the same government services? Once could just as easily say "I heard that Warren Buffet paid 1,000,000 times more taxes than his secretary, that is outrageous!"
Just thought I’d point out that only looking at "percentage" and not "actual payout" is a fallacy. Similar to many of your gambling fallacies.
Best Regards,
Kevin A. (Dallas)
Thank you for your entertaining collection of math puzzles. My girlfriend and I came up with this variation on the pirate puzzle. What if all the pirates are of equal rank, and in each round the proposer of the division is chosen by lot? In this variation, assume that each pirate’s highest priority is to maximize his expected amount of coins received. I have what I think is the solution, but perhaps you’d like to try your hand at it first. Thanks again.
Jon S
You’re welcome. If there are only two pirates left, then the one chosen to make a suggestion has no hope, because the other pirate will vote no. The one drawn will get zero, and the other all 1000. So, before the draw, the expected value with two pirates left is 500 coins.
At the three pirate stage, the drawn pirate should suggest giving one of the other pirates 501, and 499 to himself. The one getting 501 will vote yes, because it is more than the expected value of 500 by voting no. Before the draw, with three pirates left, you have a 1/3 chance each of getting 0, 499, or 501 coins, for an average of 333.33.
At the four pirate stage the drawn pirate should choose to give 334 to any two of the other pirates, and 332 to himself. That will get him two ’yes’ votes from the pirates getting 334 coins, because they would rather have 334 than 333.33. Including your own vote, you will have 3 out of 4 votes. Before the draw, the expected value for each pirate is the average of 0, 334, 334, and 332, or 1000/4=250.
By the same logic, at the five pirate stage, the drawn pirate should choose to give 251 to any two pirates, and 498 to himself. Unlike the original problem, it isn’t necessary to work backwards. Just divide the number of coins by the number of pirates, not including yourself. Then give half of them (rounding down) that average, plus one more coin.
I need help with a puzzle called Eternity II. The prize for solving the puzzle is a staggering \$2,000,000, a considerable amount of money to me. Here's a link to an interview, including the game maker himself, Christopher Monckton (former adviser to Margaret Thatcher, among many things). The game is obviously not really about gambling at all, but despite this fact, maybe you could add a word or two on your web page about it.
The game maker brags about the puzzle to be insolvable, in that link given above. I'm starting to think that he's actually right, and that he himself is the only one who will eventually become rich from selling that (ridiculous but fascinating) game. How would you, being a mathematician and all, go about solving this type of puzzle?
Robert
I hope you’re happy; I’ve been obsessed by this puzzle for the last month or so. I was lucky (or perhaps unlucky) to find the 256-piece puzzle at the local Borders book store, but I had to buy the four clue puzzles on eBay, from a guy in Australia.
I wrote a program that can easily solve the four clue puzzles. It solved the 72-piece clue puzzle #4 in less than a second. The way I did it was with a simple brute-force recursive program. I mapped out a path on the board, starting with the border. At each position, the program looped through all the unused pieces, looking for one that fit. If it found one, it moved to the next square, if it didn’t, it moved back a square.
I have had two computers crank away at the 256-piece \$2 million puzzle for weeks, and neither have come anywhere close. I tend to agree with what the creator said in that video, that if you hooked up ten million of the world’s fastest computers, they still might not find the solution by the death of the universe. You would think I would have heeded his warning before starting, but in the face of a good puzzle, all consideration for practical use of my time goes out the window.
I have lots of ideas for shortcuts, but even if they sped up my program by a factor of a billion, it still probably wouldn’t help. I’m going to be extremely impressed if anybody solves this thing. What really nags at me is I feel there is some undiscovered branch of mathematics that could solve puzzles like this easily. Until then, I think glorified trial and error is the best we can do to solve it. Today’s computers are simply too slow, and the number of combinations too vast, for that to have much of a chance of success.
Suppose the distance between two cities is 1000 miles. In zero-wind, a plane can travel at 500 mph. Will it take longer to make the round trip with no wind, or a direct 100 mph tailwind in one direction, and equal head wind the other way?
Kevin from Portland, OR
In zero-wind it will take 2 hours each way, for a total of 4 hours. With the tailwind, the plane will travel at 600 mph, making the trip in 1000/600 = 1.667 hours. With the headwind, the plane will travel at 400 mph, taking 1000/400 = 2.5 hours. So, in the wind, the total time is 4.167 hours, or 10 minutes longer.
This just goes to show that it is dangerous to average averages. You can’t say the average rate of a trip is 500 mph, if it is 400 mph one way and 600 mph the other, because the 400 mph leg is over a longer period of time.
If this isn’t intuitive, consider a 500 mph wind. The plane would take 1 hour only with the wind, but it would stay in place the other way, taking forever.
I recently entered a raffle where there are 7,033 prizes and they say the odds of winning a prize are 1 in 13. I bought 5 tickets. What are my actual odds of winning something? Also, there are 40 big prizes. What are my odds of winning a big prize?
Anonymous from Mesa, AZ
For the sake of simplicity, let’s ignore the fact that the more tickets you buy the lower the value of each ticket becomes because you compete with yourself. That said, the probability of losing all five tickets is (12/13)5 = 67.02%. So the probability of winning at least one prize is 32.98%. There are 7033×13=91,429 total tickets in the drum before you buy any. 91,429-40=91,389 are not big prizes. The probability of not winning any big prizes with five tickets is (91,389/91429)5 = 99.78%. So the probability of winning at least one big prize is 0.22%, or 1 in 458.
I have a puzzle that I’ve been trying to solve for a few months, with absolutely no progress. Time permitting, I’m hoping you can indulge me, as it’s been keeping me up at night :-). Anyway, in the glossary of Beyond Counting -- Exhibit CAA, three sequences of numbers and letters are given as the glossary entry for "Magic Numbers." One of these numbers even graces the book’s cover, so I assume they’re of some importance. Do you have any thoughts?
Anonymous
It isn’t often I say this, but I have no idea. As you noted in another e-mail, they take the format of serial number on US currency, two letters, with a ten-digit number in between. Out of respect for copyright, I won’t indicate what the numbers are here.
I am curious to know what became of that Eternity II puzzle challenge. Was it solved? Are you still working on it?
Matt from Las Vegas
Thanks for asking. No, I haven’t touched that thing since I wrote about in the November 17, 2008 Ask the Wizard column. According to their web site, they will have "scrutiny dates" on December 31, 2009, and 2010 if necessary. In my opinion, it will never be solved.
Update: The Eternity II web size appears to no longer exist.
I read with fascination the Wizard's blog about Arnold Schwarzenegger’s veto letter. My question has to do with the governor’s ridiculous but predictable response. The governor stated that it was just a ’wild coincidence’. Notwithstanding the overwhelming circumstantial evidence (The bill’s sponsor and letter’s addressee was the person who had hurled insults at the governor a week earlier), do you have an estimate of what the odds are of an exactly seven-line letter spelling this phrase by chance? I think taking into account the letters used, it will be even more improbable than just assigning a 1 in 26 chance to each. It doesn't seem like U, Y, and especially K are common word-starting letters.
pocketaces
For the benefit of my readers who didn’t read that blog, look at the first letter of each line in this memo by California governor Arnold Schwarzenegger (PDF), starting with the line beginning with the letter F.
This was discussed in my companion site Wizard of Vegas. To find an answer, I found a frequency of each letter of the first word in the English Language at Wikipedia.
### Word Frequency by First Letter
Letter Frequency A 11.60% B 4.70% C 3.51% D 2.67% E 2.00% F 3.78% G 1.95% H 7.23% I 6.29% J 0.63% K 0.69% L 2.71% M 4.37% N 2.37% O 6.26% P 2.55% Q 0.17% R 1.65% S 7.76% T 16.67% U 1.49% V 0.62% W 6.66% X 0.01% Y 1.62% Z 0.05%
To estimate the probability that Arnold’s message was indeed just a coincidence would be Prob(F) × Prob(U) × ... × prob(U) = 0.0378 × 0.0149 × 0.0351 × 0.0069 × 0.0162 × 0.0626 × 0.0149 = 1 in 486,804,391,348. This doesn’t even factor in the fact that a line break conveniently was in the place of the space between the two words.
I’d like to thank Eliot J. and Jonathan F. for their input into this solution.
At the luggage carousel in the airport, the more bags I have to retrieve the longer I can expect to wait for all of them to come out. If I have one bag, I would have to wait until about half of the bags come out. If I take 2 bags, my wait is going to be longer and with 3, longer still. Assuming my bags are mixed up randomly among the others, what is a general formula for number of bags I’ll have to wait to come out to get all my bags, in terms of my number of bags and the total number of bags?
MrPogle
Let’s define some variables first, as follows:
n = number of your bags
b = total number of bags
As the number of total bags gets larger the answer will get closer to b×n/(n+1). For a large plane, that will give you a good estimate. However, if you want to be exact, the answer is
[b×combin(b,n)-(sum for i=n to b-1 of combin(i,n))]/combin(b,n)
For example, if there are 10 total bags, and four of them are yours, then the expected wait time =
[10×combin(10,4)-combin(4,4)-combin(5,4)-combin(6,4)-combin(7,4)-combin(8,4)-combin(9,4)]/combin(10,4) = 8.8 bags.
Solution:
The number of ways to pick n out of b bags is combin(b,n). So, the probability that all your bags come out within the first x bags is combin(x,n)/combin(b,n). The probability that your last bag is the xth bag to come out is (combin(x,n)-combin(x-1,n))/combin(b,n), for x>=n+1. For x=n it is 1/combin(b,n).
So, the ratio of the expected wait time to the total wait time is:
n×combin(n,n)/combin(b,n) +
(n+1)×(combin(n+1,n)-combin(n,n))/combin(b,n) +
(n+2)×(combin(n+2,n)-combin(n+1,n))/combin(b,n) +
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(b-1)×(combin(b-1,n)-combin(b-2,n))/combin(b,n) +
b×(combin(b,n)-combin(b-1,n))/combin(b,n)
Taking a telescoping sum, this can be simplified to:
[b×combin(b,n)-combin(b-1,n)-combin(b-2,n)-...-combin(n,n)]/combin(b,n)
A reader later wrote in saying that the answer can be simplified to n×(b+1)/(n+1). This can be shown by induction, a legitimate method, but always leaves me emotionally unsatisfied.
This question was raised and discussed in the forum of my companion site Wizard of Vegas.
I sell sculptures. On average, out of every seven sculpture sales, one will be a turtle, and the rest will be other types of sculptures. How many turtles do I need to have in stock if I want a 90% chance of not running out in the next 100 sculpture sales?
RbStimers
This is a good confidence interval kind of problem. In 100 sales the expected turtles sold will be 14.29. The standard deviation is sqrt(100×(1/7)×(6/7)) = 3.50.
Let t be the number of turtles made, and x the number sold.
pr(x<=t)=0.9
pr(x-14.29<=t-14.29)=0.9
pr((x-14.29)/3.5)<=(t-14.29)/3.5))=0.9
The left side of the inequality follows a standard normal distribution (mean of 0, standard deviation of 1). This next step takes an introductory statistics course, or some faith, to accept.
(t-14.29)/3.5 = normsinv(0.9) This is the Excel function.
(t-14.29)/3.5 = 1.282
t-14.29 = 4.4870
t = 18.77
Nobody is likely to buy 0.77 of a turtle statue, so I would round up to 19. According to the binomial distribution, the probability of selling 18 or less is 88.35%, and 19 or less is 92.74%. This question was raised and discussed in the forum of my companion site Wizard of Vegas.
Five sailors survive a shipwreck. The first thing they do is gather coconuts and put them into a big community pile. They meant to divide them up equally afterward, but after the hard work gathering the coconuts, they are too tired. So they go to sleep for the night, intending to divide up the pile in the morning.
However, the sailors don’t trust one another. At midnight one of them wakes up to take his fair share. He divides up the pile into five equal shares, with one coconut left over. He buries his share, combines the other four piles into a new community pile, and gives the remaining coconut to a monkey.
At 1:00 AM, 2:00 AM, 3:00 AM, and 4:00 AM each of the other four sailors does the exact same thing.
In the morning, nobody confesses what he did, and they proceed with the original plan to divide up the pile equally. Again, there is one coconut left over, which they give to the monkey.
What is the smallest possible number of coconuts in the original pile?
David Filmer from MA (Cantab)
"Scroll down 100 lines for the answer.
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There were 15,621 coconuts in the original pile. Scroll down another 100 lines for my solution.
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Let c be the number of coconuts in the original pile and f be the final share for each sailor after the last division.
After sailor 1 takes his share and gives the monkey his coconut there will be (4/5)×(c-1) = (4c-1)/5 left.
After sailor 2 takes his share and gives the monkey his coconut there will be (4/5)×(((4c-1)/5)-1) = (16c-36)/25 left.
After sailor 3 takes his share and gives the monkey his coconut there will be (4/5)×(((16c-36)/25)-1) = (64c-244)/125 left.
After sailor 4 takes his share and gives the monkey his coconut there will be (4/5)×(((64c-244)/125)-1) = (256c-1476)/625 left.
After sailor 5 takes his share and gives the monkey his coconut there will be (4/5)×(((256c-1476)/625)-1) = (1024c-8404)/3125 left.
In the morning each sailor’s share of the remaining pile will be f = (1/5)×(((1024c-8404)/3125)-1) = (1024c-11529)/15625 left.
So, the question is what is the smallest value of c such that f=(1024×c-11529)/15625 is an integer. Let’s express c in terms of f.
(1024×c-11529)/15625 = f
1024c - 11529 = 15625×f
1024c = 15625f+11529
c = (15625f+11529)/1024
c = 11+((15625×f+265)/1024)
c = 11+15×f+(265×(f+1))/1024
So, what is the smallest f such that 265×(f+1)/1024 is an integer? 265 and 1024 do not have any common factors, so f+1 by itself is going to have to be divisible by 1024. The smallest possible value for f+1 is 1024, so f=1023.
Thus, c = (15625×1023+11529)/1024 = 15,621.
Here is how many coconuts each person, and monkey, received:
### Coconut Problem
Sailor Coconuts 1 4147 2 3522 3 3022 4 2622 5 2302 Monkey 6 Total 15621
David Filmer, the one who challenged me the with question, already knew the answer. Actually, he asked me the formula for the general case of s sailors, but I had enough trouble with the specific case of 5 sailors. David notes the answer for the general case is c = ss+1 - s + 1.
I’ll leave that proof to the reader.
Here are some links to alternate solutions to the problem:
A man is presented with two envelopes full of money. One of the envelopes contains twice the amount as the other envelope. Once the man has chosen his envelope, opened and counted it, he is given the option of changing it for the other envelope. The question is, is there any gain to the man in changing the envelope?
It would appear that by switching the man would have a 50% chance of doubling his money should the initial envelope be the lesser amount and a 50% chance of halving it if the initial envelope is the higher amount. Thus, let x be the amount contained in the initial envelope and y be the value of changing it:
y = 0.5×(x/2) + 0.5×(2x) = 1.25x
Let’s say that the initial envelope contained \$100. So there should be a 50% chance that the other envelope contains 2 × \$100 = \$200 and a 50% chance that the other envelope contains (1/2) × \$100 = \$50. In such a case, the value of the envelope is:
0.5×(\$100/2) + 0.5×(2×\$100) = \$125
This implies that the man would, on average, increase his wealth by 25% simply by switching envelopes! How can this be?
DorothyGale
This appears to be a mathematical paradox, but is really just an abuse of the expected value formula. As you noted in the question, it seems like the other envelope should have 25% more than the one you chose. However, if you buy that, then you may as well pick the other envelope to begin with. Furthermore, you could use that argument to switch back and forth forever if you don’t get to open the envelopes before deciding to switch. Clearly there must be some flaw in the expected value argument. The question is, where is the flaw?
I have spent a lot of time reading about this problem and discussing it over the years. I’ve heard and read many explanations about why the y=.5x + .5*2x = 1.25x argument is wrong. Many have used many pages of advanced mathematics in the explanation, which I don’t think is necessary. It is a simple question that calls out for a simple answer. So, this is my crack at it.
You must be very careful with what you do with the stated fact that one envelope has twice the money as the other one. Let’s call the amount in the smaller envelope S, and the larger one L. So we have:
L=2×S
S=0.5×L
Notice how the 2 and 0.5 factors are applied to different envelopes. You can’t take both factors and apply them to the same amount. If the first envelope has \$100 then if it was the smaller envelope, the other one will have \$200. If the \$100 was the larger envelope, then the other one will have \$50. So the other envelope will have \$50 or \$200. However, you can’t jump from there to say there is a 50/50 chance of each. This is because that would be applying the 0.5 and 2 factors to the same amount, which you can’t do. Without knowing the prize distribution to begin with, you can’t assign possible amounts to the second envelope.
If the 0.5x/2x argument is wrong, then what would be the correct way to set up the expected value of the other envelope? The way I would do it is to say that the difference between the two envelopes is L-S = 2S-S = S. By switching you’ll either gain or lose S, whatever it is. If the two envelopes have \$50 and \$100, then switching will gain or lose \$50. If the two envelopes have \$100 and \$200, then switching will gain or lose \$100. Either way, the expected gain by switching is 0. I think I could say that if the first envelope has \$100, then there is a 50% chance the difference in the other envelope is \$50, and a 50% chance it is \$100. So the expected difference is \$75. Thus, the expected value of the other envelope is 0.5×(\$100+\$75) + 0.5×(\$100-\$75) = 0.5×(\$175+\$25) = \$100.
I hope that makes some sense. This problem always induces lots of comments. If you have one, please don’t write to me directly, but post it in my Wizard of Vegas forum. The link is below.
This question was raised and discussed in the forum of my companion site Wizard of Vegas.
Consider a truel (a three-way duel) with participants A, B, and C. They are fighting to the death over a woman. They are all gentlemen, and they all agree to the following rules.
1. The three participants form a triangle.
2. Each has one bullet only.
3. A goes first, then B, and C.
4. A’s probability of hitting an intended target is 10%.
5. B’s probability of hitting an intended target is 60%.
6. C’s probability of hitting an intended target is 90%.
7. There are no accidental shootings.
8. Shooting in the air (deliberately missing) and shooting yourself is allowed, and are always successful.
9. If two or three survivors remain after any round, then each is given a new bullet. They will then repeat taking turns shooting, in the same order, skipping anybody who already died.
10. All three participants are perfect logicians.
Who should A aim at initially? What is his probability of survival for each initial target?
Dween
This puzzle is discussed on the BBC show Quite Interesting. Scroll down 100 lines for the answer and solution.
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Here are my probabilities of A winning according to each initial target. As you can see, A’s probability of winning is maximized by deliberately firing in the air.
### Truel Odds
Strategy Prob. Win Air 13.887% A 0.000% B 12.560% C 13.094%
For the solution, let’s use the notation Pr(X) to denote the probability of group X, and only group X, remains after a round. Let’s use the terminololy Pr(X*) to denote the probability of group X eventually winning the round, after repeating until the game state changes by somebody getting hit. Let Pr(X**) be the probability that player X is the sole survivor. To find the final probabilities, let’s look at the two-player states first. It is obvious that each will shoot at the other.
A vs. B
• Pr(A) = 0.1
• Pr(B) = 0.9×0.6 = 0.54
• Pr(AB) = 0.9×0.4 = 0.36
If both survive then they will repeat until there is one survivor only. So the probabilities of being the final survivor are:
• Pr(A*) = Pr(A)/(1-Pr(AB)) = 0.1/0.64 = 0.15625
• Pr(B*) = Pr(B)/(1-Pr(AB)) = 0.54/0.64 = 0.84375
A vs. C
• Pr(A) = 0.1
• Pr(C) = 0.9×0.9 = 0.81
• Pr(AC) = 0.9×0.1 = 0.09
If both survive then they will repeat until there is one survivor only. So the probabilities of being the final survivor are:
• Pr(A*) = Pr(A)/(1-Pr(AC)) = 0.1/0.91 = 0.10989011
• Pr(C*) = Pr(B)/(1-Pr(AC)) = 0.81/0.91= 0.89010989
B vs. C
• Pr(B) = 0.6
• Pr(C) = 0.4×0.9 = 0.36
• Pr(BC) = 0.\$×0.1 = 0.04
If both survive then they will repeat until there is one survivor only. So the probabilities of being the final survivor are:
• Pr(B*) = Pr(A)/(1-Pr(BC)) = 0.6/.96 = 0.625
• Pr(C*) = Pr(B)/(1-Pr(BC)) = 0.36/.96= 0.375
Now we’re ready to analyze the three-player case. Let’s consider the situation where A aims at B.
Three Player — A Aims at B
If A hits B then C will definitely survive, and may or may not hit A. So two possible outcomes of hitting B are AC and C. If A misses B then B will aim at the greater threat C. If B hits C then A and B will survive. If B misses C then C will aim at the greater threat B. If C misses B then all three will survive. If C hits B then A and C will survive. So the possible outcomes are C, AB, AC, and ABC.
• Pr(A) = 0.
• Pr(B) = 0.
• Pr(C) = 0.1 × 0.9 = 0.09. This is achieved by A hitting B, and then C hitting A.
• Pr(AB) = 0.9 × 0.6 = 0.54. This is achieved by A missing B, and then B hitting C.
• Pr(AC) = 0.1 × 0.1 + 0.9 × 0.4 × 0.9 = 0.334. This can be achieved two ways. The first is A hitting B, and then C missing A. The second is A missing B, B missing C, and then C hitting B.
• Pr(BC) = 0.
• Pr(ABC) = 0.9 × 0.4 × 0.1 = 0.036. This is achieved by all three missing.
By the same logic as the two-player cases, we can divide each outcome by (1-Pr(ABC))=0.964 to find the probabilities of each state, assuming that the state of the game did change after the round.
• Pr(C*) = 0.09/0.964 = 0.093361.
• Pr(AB*) = 0.54/0.964 = 0.560166.
• Pr(AC*) = 0.334/0.964 = 0.346473.
From the two-player cases, we know if it comes down to A and B then A will win with probability 0.15625, and B 0.84375. If it comes down to A and C then A will win with probability 0.109890, and C 0.890110.
• Pr(A**) = (0.560165975 × 0.15625) + (0.346473029 × 0.10989011) = 0.125600. A can be the winner two ways: (1) getting to the AB state, and then winning, or (2) getting to the AC state and then winning.
• Pr(B**) = 0.560166 × 0.84375 = 0.472640. B will be the winner if it gets to the AB state, and then B wins.
• Pr(C**) = 0.093361 + (0.346473 × 0.890110) = 0.401760. C can win by A killing B, and then C killing A in the first round, or by it getting to state AC, and then C winning.
So, if A’s strategy is to aim at B at first, then his probability of being the sole survivor is 12.56%.
Three Player — A Aims at C
If A hits C then B will definitely survive, and may or may not hit A. So two possible outcomes of hitting C are AB and B. If A misses C then B will aim at the greater threat C. If B hits C then A and B will survive. If B misses C then C will aim at the greater threat B. If C misses B then all three will survive. If C hits B then A and C will survive. So the possible outcomes are B, AB, AC, and ABC.
• Pr(A) = 0.
• Pr(B) = 0.1 × 0.6 = 0.06.
• Pr(C) = 0.
• Pr(AB) = (0.1 × 0.4) + (0.9 × 0.6) = 0.04+0.54 = 0.58. This can be achieved two ways. The first is A hitting C, and then B missing A. The second is A missing B, and then B hitting C.
• Pr(AC) = 0.9 × 0.4 × 0.9 = 0.324. This is achieved by A missing C, B missing C, and C hitting B.
• Pr(BC) = 0.
• Pr(ABC) = 0.9 × 0.4 × 0.1 = 0.036. This is achieved by all three missing.
By the same logic as the two-player cases, we can divide each outcome by (1-Pr(ABC))=0.964 to find the probabilities of each state, assuming that the state of the game did change after the round.
• Pr(B*) = 0.06/0.964 = 0.062241.
• Pr(AB*) = 0.58/0.964 = 0.601660.
• Pr(AC*) = 0.324/0.964 = 0.336100.
By the same logic as the solution for the A aims at B case:
• Pr(A**) = (0.601660 × 0.15625) + (0.336100 × 0.10989011) = 0.130943.
• Pr(B**) = 0.062241 + 0.601660 × 0.84375 = 0.569891.
• Pr(C**) = 0.336100 × 0.890110 = 0.299166.
So, if A’s strategy is to aim at C at first, then his probability of being the sole survivor is 13.09%.
Three Player — A Misses Deliberately
After A deliberately misses then B will aim at the greater threat C. If B hits C then A and B will survive. If B misses C then C will aim at the greater threat B. If C misses B then all three will survive. If C hits B then A and C will survive. So the possible outcomes are AB, AC, and ABC.
• Pr(A) = 0.
• Pr(B) = 0.
• Pr(C) = 0.
• Pr(AB) = 0.6. This is achieved by B hitting C.
• Pr(AC) = 0.4 × 0.9 = 0.36. This is achieved by B missing C, and then C hitting B.
• Pr(BC) = 0.
• Pr(ABC) = 0.4 × 0.1 = 0.04. This is achieved by all three missing.
By the same logic as the two-player cases, we can divide each outcome by (1-Pr(ABC))=0.96 to find the probabilities of each state, assuming that the state of the game did change after the round.
• Pr(AB*) = 0.6/0.96 = 0.625.
• Pr(AC*) = 0.36/0.96 = 0.375.
By the same logic as the solution for the A aims at B case:
• Pr(A**) = (0.625 × 0.15625) + (0.375 × 0.109890) = 0.138865.
• Pr(B**) = 0.625 × 0.84375 = 0.527344.
• Pr(C**) = 0.375 × 0.890110 = 0.333791.
So, if A’s strategy is to aim at C at first, then his probability of being the sole survivor is 13.89%.
This question was raised and discussed in the forum of my companion site Wizard of Vegas.
Comparing two elements at a time, what is the fastest way to sort a list, minimizing the maximum number of comparisons?
Anon E. Mouse
There are several ways that are about equally as good. However, the one I find the easiest to understand is called the merge sort. Here is how it works:
1. Divide the list in two. Keep dividing each subset in two, until every subset is size 1 or 2.
2. Sort each subset of 2 by putting the smaller member first.
3. Merge pairs of subsets together. Keep repeating until there is just one sorted list.
The way to merge two lists is to compare the first member of each list, and put the smaller one in a new list. Then repeat, and put the smaller one after the smaller member from the previous comparison. Keep repeating until the two groups have been merged into one sorted group. If one of the original two lists is empty, then you can append the other list to the end of the merged list.
The following table shows the maximum number of comparisons necessary according to the number of elements in the list.
### Merge Sort
Elements Maximum Comparisons 1 0 2 1 4 5 8 17 16 49 32 129 64 321 128 769 256 1,793 512 4,097 1,024 9,217 2,048 20,481 4,096 45,057 8,192 98,305 16,384 212,993 32,768 458,753 65,536 983,041 131,072 2,097,153 262,144 4,456,449 524,288 9,437,185 1,048,576 19,922,945 2,097,152 41,943,041
This question was raised and discussed in the forum of my companion site Wizard of Vegas.
On October 29, 2010, the The Las Vegas Review Journal published a poll in the Reid-Angle Senate race. It said that on a survey of 625 likely voters, Angle won 49% and Reid won 45%. It also said the margin of error was 4%. Here are my questions:
1. What is Angle’s probability of winning?
2. What would be a 95% confidence interval for Angle’s share of the vote?
3. What does the margin of error mean?
Anon E. Mouse
My apologies for the dated reply on this. I wrote the following before the election.
First, I’m going to get rid of that other 6%, who are either undecided or will waste their votes on a third party candidate or "none of the above," which is an option in Nevada. Some may disagree with this assumption. To be honest, another reason for ignoring them is the math gets more complicated with more than two candidates. So, after rounding, that would leave us with 306 votes for Angle, and 281 votes for Reid, for a total of 587 in the sample.
I’m going to use the standard normal approximation to answer this question. If I were going to be a perfectionist, then I would use the T distribution, because the actual mean and variance are not known. However, in my opinion, a sample size of 587 is perfectly fine for the normal distribution.
Sample size = 306+281 = 587.
Angle sample mean is 306/587 = 0.521295.
The estimated standard deviation of the mean is (0.521295 × 0.478705 / (587-1))^0.5 = 0.0206361.
Angle’s share above 50% is (0.521295-0.5)/0.0206361 = 1.031917 standard deviations.
According to the normal distribution, the probability of Reid finishing 1.031917 standard deviations above expectations is 0.151055. This can be found in Excel with the function NORMSDIST(-1.031917). So Angle’s probability of winning is 1-0.151268 = 84.89%.
To create a 95% confidence interval, note that the 2.5% point on either side of the Gaussian curve is at 1.959964 standard deviations from the mean. This can be found in Excel with the function NORMSINV(0.975). As already noted, the estimated standard deviation of the sample mean is 0.0206361. So there is a 95% chance that either candidate will get within 0.0206361×1.959964 = 0.040446 standard deviations of the poll results. So Angle’s 95% confidence interval is 0.521295 +/- 0.040446 = 48.08% to 56.17%.
I’m told it would be mathematically incorrect to phrase that as "Angle’s share of all Angle/Reid votes has a 95% chance of falling between 48.08% and 56.17%." That was how I originally phrased my answer, but two statisticians recoiled in horror at my wording. To paraphrase their response, they said I had to use the passive voice, and say that "48.08% and 56.17% will surround Angle’s share with 95% probability." To be honest with you, it sounds the same to me. However, they stressed that the confidence interval is random and Angle’s share is immutable, and that my original wording implied the opposite. Anyway, I hope the frequentist statisticians out there will be satisfied with the second wording.
The "margin of error" is half the difference between the two ends of the 95% confidence interval. In this case (56.17% - 48.08%)/2 = 4.04%.
As a follow-up, here are the actual results:
Reid: 361,655
Angle: 320,996
Other: 21,979
So, not counting the "other" votes, Reid got 53.0% and Angle 47.0%. That is a comfortable 6% win for Reid. It begs the question of why the poll was so far off. Was it chance? Did voters change their minds? Or was it a bad poll to begin with? I leave those questions to the reader (I hate it when textbooks say that).
This question was raised and discussed in the forum of my companion site Wizard of Vegas.
I tried your actuarial calculator. Why is it that the probability of reaching my expected age at death is less than 50%?
pacomartin
You’re confusing the mean and the median. Let’s look at my situation as an example. I’m a 45-year-old male. My life expectancy is 78.11 years, yet I have a 50.04% chance of making it to age 80.
My age at death will be like throwing a dart at this graph. Notice how the left tail is a lot fatter than the right. That means my probability of death right now is quite low. However, as I get older, the probability of death in the next year will keep getting higher. For example, for a 45-year-old male the probability of surviving to 46 is quite high at 99.64%. However, at age 85 the probability of making it to 86 is 89.21% only. It is like nature slowly pushing a knife in your back. At first it probably won’t kill you, but with each passing year, the odds slowly increase that it will. However, once you get to the late seventies nature says enough with the games and really starts shoving it in.
So if a lot of 45-year-old men throw darts at this graph, 49.96% will hit between 45 and 79, and 50.04% will hit between 80 and 111. However, the lucky half who make it on the right side of the graph will probably not live much past 80. Once a male reaches 80 he can expect to live 7.78 more years only. Meanwhile, many in the unlucky half who don’t make it to 80 will die much younger than that. So it is the many young deaths that pull down average life expectancy.
For a similar situation, consider a die numbered 10, 20, 30, 31, 32, 33. The average is 26, yet there is a 2/3 chance of rolling more than that.
As an example of how the mean and median are different, suppose we add two more deaths to the sample. One death at 46 and one at 81. The probability of making it to 80 doesn’t change, but the average life expectancy at age 45 would go down.
This question was raised and discussed in the forum of my companion site Wizard of Vegas.
Imagine an infinitely elastic rubber band that is 1 km. long unstretched. It expands at a rate of 1 km. per second. Next, imagine an ant at one end of the rubber band. At the moment the rubber band starts expanding the ant crawls towards the other end at a speed, relative to his current position, of 1 cm. per second. Will the ant ever reach the other end? If so, when?
Anon E. Mouse
Yes, it will, after e100,000 -1 seconds. See my mathproblems.info site, problem 206, for two solutions.
This question was raised and discussed in the forum of my companion site Wizard of Vegas.
Do you think the better fuel efficiency is worth the additional cost of a hybrid? How many miles would you have to drive to break even?
Rob from Las Vegas
Good question. To answer it I considered a Toyota Highlander, a vehicle I am thinking of purchasing. The retail cost of the standard hybrid model is \$37,490. For the same four-wheel-drive vehicle non-hybrid the cost is \$29,995. So the hybrid engine adds \$7,495 to the cost.
The gas mileage of the hybrid is 28 mph, both city and highway. The mileage of the non-hybrid is 17 city and 22 highway. Let’s take the average at 19.5.
The general formula for the number of miles to break even is h×mh×mr/(g×(mh-mr)), where
h = Additional cost of the hybrid.
g = Cost for a gallon of gas.
mr = Mileage for non-hybrid (the "r" is for a regular car).
mh = Mileage for a hybrid.
The following table uses this formula to find the break even point for various prices of gas from \$2 to \$5 per gallon.
### Hybrid Break Even Point
Cost of Gas Number of Miles \$2.00 240,722 \$2.25 213,975 \$2.50 192,577 \$2.75 175,070 \$3.00 160,481 \$3.25 148,136 \$3.50 137,555 \$3.75 128,385 \$4.00 120,361 \$4.25 113,281 \$4.50 106,987 \$4.75 101,357 \$5.00 96,289
So, at the current price of \$3.00 per gallon here in Vegas, you would need to put more than 160,481 miles on the vehicle to come out ahead. This does not consider other expenses that may be associated with a hybrid, such as the expensive replacement cost of batteries, nor any perceived green points for consuming less fossil fuel.
This question was raised and discussed in the forum of my companion site Wizard of Vegas.
What ratio of genes would I have in common with a full brother or sister, other than identical twin?
HotBlonde
1/2.
If we used keno as a comparison, everybody would have 40 genes, each represented by a keno ball. However, each ball would have unique number. When two people, who are not related, mate it is like combining 80 balls between the two of them into a hopper, and randomly choosing 40 genes for the offspring of the mating.
So when you were conceived, you got half the balls in the hopper, and the other half were wasted. When your brother or sister was conceived he/she got half from the balls drawn when you were born, and half that were not drawn. So you are 50% genetically identical. Much for the same reason that if 40 numbers were drawn in keno, two consecutive draws would average 20 balls in common.
This question was raised and discussed in the forum of my companion site Wizard of Vegas.
A factory that produces tables and chairs is equipped with 10 saws, 6 lathes, and 18 sanding machines. It takes a chair 10 minutes on a saw, 5 minutes on a lathe, and 5 minutes of sanding to be completed. It takes a table 5 minutes on a saw, 5 minutes on a lathe, and 20 minutes of sanding to be completed. A chair sells for \$10 and a table sells for \$20. How many tables and chairs should the factory produce per hour to yield the highest revenue, and what is that revenue?
Anon E. Mouse
Let’s let c stand for the number of chairs made per hour, and t the number of tables. Revenue per hour will be 10×c + 20×t.
The 10 saws result in 600 minutes of sawing per hour. We were given that it takes a chair 10 minutes of saw time, and a table 5. So that limits hour hourly production to:
(1) 10c + 5t <= 600
The 6 lathes result in 360 minutes of lathing per hour. We were given that it takes a chair 5 minutes of saw time, and a table 5. So that limits hour hourly production to:
(2) 5c + 5t <= 360
The 18 sanding machines result in 1080 minutes of sanding per hour. We were given that it takes a chair 5 minutes of saw time, and a table 20. So that limits hour hourly production to:
(3) 5c + 20t <= 1080
The following graph shows the three constraints placed by the three sets of machinery. The factory may produce any combination of chairs and tables that is under all three lines. The question is where under the three lines results in the greatest revenue.
It stands to reason that the answer would be the intersection of two lines, make all chairs, or make all tables. So let’s find where the lines intersect. First, let’s find where equation (1) and (2) intersect. We can change the <= expression to just =, to use the machines to their maximum potential.
(1) 10c + 5t = 600
(2) 5c + 5t = 360
Subtract (2) from (1):
5c = 240
c = 48
Plugging 48 for c into equation (1):
10×48 + 5t = 600
5t = 120
t = 24
So, equations (1) and (2) meet at 48 chairs and 24 tables.
Next, let’s find where equations (2) and (3) meet:
(2) 5c + 5t = 360
(3) 5c + 20t = 1080
Subtracting (2) from (3):
15t = 720
t = 48
Putting that into (2) or (3) we can solve for c, which is 24.
So, equations (2) and (3) meet at 24 chairs and 48 tables.
We don’t need to bother finding where equations (1) and (3) meet, because we can see from the graph that where the saw and sanders lines meet is outside of the lathe constraint.
It is also possible that making chairs only is the right answer. The graph shows that the saws is the biggest limitation to making only chairs. From equation (1) if we put in 0 for the number of tables we get c=60.
Another possibility is making tables only. The graph shows that the sanders will be the biggest limitation. Putting in 0 chairs into equation (3) we find we can make no more than 54 tables.
The following graph shows the total revenue for each viable answer. Remember, the revenue is \$10 per chair and \$20 per table.
### Total Hourly Revenue
Chairs Tables Revenue 0 54 \$1,080 24 48 \$1,200 48 24 \$960 60 0 \$600
So, the greatest revenue possible is \$1,200, from making 24 chairs and 48 tables. Note that this would leave some saws unused part of the time. This question is also raised and discussed in the forum of my companion site MathProblems.info, problem #28.
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Interview Questions and Answers :: Cisco
Home > Experience Archives > Cisco > Interview Question Set 3
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Write a Program to insert a node in a linked list in a sorted manner.
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Write a Program to Search and delete a node from linked list based on data value.
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Write Program to reverse the linked list and display the reversed linked list nodes.
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Write a Program to swap the 2 nibbles of a byte size variable.
assuming 2 numbers are
a = 10
b = 20
# 1. Summing up two numbers and store in 'a'
a = a b
# 2. swap b by subtracting b value from above a value
b = a - b # b = 30 - 20 = 10
# 3. get a value swapped
a = a - b # a = 30 - 10 = 20
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Write a Program to count the number of SET bits in an integer.
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Allocating memory for a 2D array where number of rows and columns is dynamically decided.
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What are the contents of argc and argv[]?
argc contains count of argument
argv contains list of arguments
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What is the difference between array of pointers and pointer to an array. Write the declarations.
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Write a Program to implement strcat() or strstr() equivalents in your own program
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Briefly Introduce yourself.
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IOS versus Linux - important points.
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What is flat memory architecture?
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Write a sample program having two processes using semaphore to synchronize the access to a common resource. Use a C structure as the shared resource.
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Write a sample program having 2 processes communicating using message queues.
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fork(), exec(), signal() calls what do they do? Give examples in pseudo code
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Multiplying Decimals by Powers of 10 Video
Multiplying decimals by powers of 10 (10^(x))^( )
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Multiplying decimals by powers of 10 (10^(x))^( )
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https://www.daniweb.com/programming/software-development/threads/227917/having-trouble-with-the-fibonaccie-sequence | 1,529,928,234,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267867666.97/warc/CC-MAIN-20180625111632-20180625131632-00326.warc.gz | 779,581,220 | 11,888 | The following is my class project and I do not have a clue where to begin. Could someone at least point me in the right direction?
For this project we are going to work with computing the Fibonacci Sequence (also called Fibonacci Number). We will be using a menu to control this program’s execution. The program should clear the screen each time the menu is displayed. The menu should display the following options:
0. Exit Program
1. Fibonacci For
2. Fibonacci Do
3. Fibonacci While
The user should be able to enter a number between 0 and 3, each number will coorespond to one of the menu options listed above. The menu should loop continuously until the user enters 0 to exit the program. You should use a switch statement to control the calling of the appropriate function.
The program should have 3 functions one for the for loop, one for the do loop and one for the while loop.
Clear the screen, then ask the user each time how many iterations of the Fibonacci Sequence he/she would like to see before calling the function to display the output.
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Last Post by ccoleman52932
Which part do you want help with? for starters, determining the fib sequence will look something like this:
``````int[size] num;
num[0]=1;
num[1]=1;
int index=2;
while(notdone) {
num[index]=num[index-1]+num[index-2];
index++;
}``````
but you'll have to tweak it to fit what you want it to do. :)
Which part do you want help with? for starters, determining the fib sequence will look something like this:
``````int[size] num;
num[0]=1;
num[1]=1;
int index=2;
while(notdone) {
num[index]=num[index-1]+num[index-2];
index++;
}``````
but you'll have to tweak it to fit what you want it to do. :)
Thanks, the information that you provided was helpful.
This topic has been dead for over six months. Start a new discussion instead.
Have something to contribute to this discussion? Please be thoughtful, detailed and courteous, and be sure to adhere to our posting rules. | 478 | 2,014 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.0625 | 3 | CC-MAIN-2018-26 | latest | en | 0.887977 |
http://cn.metamath.org/mpeuni/df-pmap.html | 1,653,111,201,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662538646.33/warc/CC-MAIN-20220521045616-20220521075616-00649.warc.gz | 11,123,375 | 3,522 | Mathbox for Norm Megill < Previous Next > Nearby theorems Mirrors > Home > MPE Home > Th. List > Mathboxes > df-pmap Structured version Visualization version GIF version
Definition df-pmap 35108
Description: Define projective map for 𝑘 at 𝑎. Definition in Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 2-Oct-2011.)
Assertion
Ref Expression
df-pmap pmap = (𝑘 ∈ V ↦ (𝑎 ∈ (Base‘𝑘) ↦ {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)𝑎}))
Distinct variable group: 𝑘,𝑎,𝑝
Detailed syntax breakdown of Definition df-pmap
StepHypRef Expression
1 cpmap 35101 . 2 class pmap
2 vk . . 3 setvar 𝑘
3 cvv 3231 . . 3 class V
4 va . . . 4 setvar 𝑎
52cv 1522 . . . . 5 class 𝑘
6 cbs 15904 . . . . 5 class Base
75, 6cfv 5926 . . . 4 class (Base‘𝑘)
8 vp . . . . . . 7 setvar 𝑝
98cv 1522 . . . . . 6 class 𝑝
104cv 1522 . . . . . 6 class 𝑎
11 cple 15995 . . . . . . 7 class le
125, 11cfv 5926 . . . . . 6 class (le‘𝑘)
139, 10, 12wbr 4685 . . . . 5 wff 𝑝(le‘𝑘)𝑎
14 catm 34868 . . . . . 6 class Atoms
155, 14cfv 5926 . . . . 5 class (Atoms‘𝑘)
1613, 8, 15crab 2945 . . . 4 class {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)𝑎}
174, 7, 16cmpt 4762 . . 3 class (𝑎 ∈ (Base‘𝑘) ↦ {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)𝑎})
182, 3, 17cmpt 4762 . 2 class (𝑘 ∈ V ↦ (𝑎 ∈ (Base‘𝑘) ↦ {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)𝑎}))
191, 18wceq 1523 1 wff pmap = (𝑘 ∈ V ↦ (𝑎 ∈ (Base‘𝑘) ↦ {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)𝑎}))
Colors of variables: wff setvar class This definition is referenced by: pmapfval 35360
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https://mathexamination.com/lab/rational-trigonometry.php | 1,621,314,164,000,000,000 | text/html | crawl-data/CC-MAIN-2021-21/segments/1620243989820.78/warc/CC-MAIN-20210518033148-20210518063148-00499.warc.gz | 398,056,491 | 8,634 | Do My Rational Trigonometry Lab
As mentioned over, I utilized to compose an easy and also uncomplicated mathematics lab with only Rational Trigonometry Nevertheless, the easier you make your lab, the simpler it ends up being to obtain stuck at completion of it, then at the beginning. This can be extremely irritating, and all this can happen to you due to the fact that you are utilizing Rational Trigonometry and/or Modular Equations incorrectly.
With Modular Formulas, you are already utilizing the incorrect formula when you get stuck at the start, if not, after that you are most likely in a dead end, as well as there is no feasible escape. This will just become worse as the problem ends up being extra complicated, but after that there is the inquiry of just how to wage the trouble. There is no chance to properly tackle addressing this kind of mathematics issue without being able to right away see what is taking place.
It is clear that Rational Trigonometry as well as Modular Equations are hard to learn, as well as it does take practice to develop your very own sense of instinct. However when you wish to address a math problem, you need to make use of a tool, as well as the tools for learning are used when you are stuck, and also they are not made use of when you make the wrong action. This is where lab Aid Service can be found in.
For instance, what is wrong with the question is incorrect concepts, such as obtaining a partial worth when you do not have sufficient functioning components to finish the entire job. There is an excellent factor that this was wrong, as well as it refers reasoning, not instinct. Logic enables you to adhere to a step by step procedure that makes good sense, as well as when you make a wrong relocation, you are typically required to either try to go forward as well as remedy the error, or try to step as well as do an in reverse step.
One more instance is when the pupil does not understand an action of a procedure. These are both sensible failings, as well as there is no way around them. Also when you are embeded an area that does not enable you to make any sort of move, such as a triangle, it is still vital to recognize why you are stuck, so that you can make a much better step and go from the step you are stuck at to the following location.
With this in mind, the best method to solve a stuck circumstance is to just take the progression, as opposed to trying to go backward. The two processes are different in their strategy, yet they have some basic similarities. However, when they are attempted together, you can rapidly inform which one is much better at addressing the trouble, as well as you can also inform which one is much more powerful.
Allow's discuss the very first example, which associates with the Rational Trigonometry math lab. This is not as well difficult, so let's initial discuss just how to start. Take the following procedure of connecting a component to a panel to be used as a body. This would certainly need three measurements, and also would certainly be something you would need to attach as part of the panel.
Now, you would certainly have an added dimension, but that doesn't indicate that you can simply maintain that measurement as well as go from there. When you made your very first step, you can conveniently forget about the measurement, and afterwards you would have to go back and backtrack your steps.
However, instead of keeping in mind the extra measurement, you can utilize what is called a "mental shortcut" to aid you remember that added measurement. As you make your primary step, picture on your own taking the measurement as well as connecting it to the part you intend to attach to, and after that see exactly how that makes you really feel when you repeat the process.
Visualisation is a very effective technique, and also is something that you must not miss over. Imagine what it would certainly feel like to really affix the part as well as have the ability to go from there, without the measurement.
Currently, allow's look at the second instance. Let's take the exact same process as previously, and now the pupil has to keep in mind that they are going to move back one step. If you tell them that they need to return one step, however then you get rid of the concept of having to return one step, then they won't know how to proceed with the problem, they will not recognize where to search for that action, as well as the procedure will be a mess.
Rather, utilize a psychological shortcut like the psychological layout to emotionally show them that they are going to move back one action. and place them in a setting where they can move forward from there. without having to think of the missing out on an action.
Hire Someone To Do Your Rational Trigonometry Lab
" Rational Trigonometry - Required Help with a Math lab?" Unfortunately, numerous trainees have actually had a trouble understanding the principles of linear Rational Trigonometry. Fortunately, there is a new format for linear Rational Trigonometry that can be made use of to teach straight Rational Trigonometry to students who deal with this idea. Students can utilize the lab Help Service to help them learn new strategies in direct Rational Trigonometry without dealing with a hill of problems and also without needing to take an examination on their principles.
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All of the jobs for the lab Assist Solution are consisted of in the package. Pupils can login and complete their assigned work while having the pupil help offered behind-the-scenes to help them. The lab Aid Service can be a terrific aid for your pupils as they start to navigate the difficult university admissions and work hunting waters.
Students must be prepared to get utilized to their tasks as promptly as feasible in order to reach their major goal of entering into the college. They need to work hard enough to see outcomes that will enable them to walk on at the following degree of their research studies. Obtaining made use of to the procedure of completing their jobs is very essential.
Students are able to locate various ways to help them find out exactly how to utilize the lab Assist Service. Knowing just how to make use of the lab Assist Solution is essential to pupils' success in university and also work application.
Hire Someone To Take My Rational Trigonometry Lab
Rational Trigonometry is utilized in a lot of schools. Some instructors, nonetheless, do not use it really properly or utilize it incorrectly. This can have a negative influence on the student's discovering.
So, when assigning projects, use an excellent Rational Trigonometry assistance solution to aid you with each lab. These services offer a selection of helpful services, consisting of:
Tasks may need a great deal of reviewing and also looking on the computer system. This is when using a help service can be a great advantage. It permits you to get more work done, enhance your comprehension, and also stay clear of a lot of stress.
These kinds of homework services are a great means to begin working with the most effective kind of help for your demands. Rational Trigonometry is just one of one of the most challenging based on understand for students. Working with a service, you can make certain that your needs are met, you are educated correctly, as well as you comprehend the material effectively.
There are numerous manner ins which you can educate yourself to function well with the course and be successful. Use a correct Rational Trigonometry assistance service to direct you and also obtain the job done. Rational Trigonometry is among the hardest classes to discover however it can be quickly mastered with the appropriate assistance.
Having a research solution likewise aids to enhance the student's qualities. It permits you to include extra credit scores in addition to boost your GPA. Getting added credit is typically a massive advantage in many colleges.
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One advantage of having a research solution is that you can aid on your own. If you do not feel confident in your capacity to do so, after that an excellent tutor will certainly have the ability to help you. They will certainly be able to resolve the troubles you face and assist you understand them in order to get a far better quality.
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Keeping your qualities up can be difficult due to the fact that you typically need to examine a lot and take a lot of tests. You do not have time to service your qualities alone. Having a great tutor can be a terrific assistance because they can aid you and also your research out.
A help solution can make it less complicated for you to manage your Rational Trigonometry course. On top of that, you can find out more regarding on your own and help you prosper. Discover the most effective tutoring service and you will certainly have the ability to take your research study skills to the following degree. | 2,500 | 12,790 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.28125 | 3 | CC-MAIN-2021-21 | latest | en | 0.982382 |
http://book.caltech.edu/bookforum/showthread.php?s=0f6a35f1e481ed09688baa664c1f9385&t=4501 | 1,606,274,653,000,000,000 | text/html | crawl-data/CC-MAIN-2020-50/segments/1606141180636.17/warc/CC-MAIN-20201125012933-20201125042933-00434.warc.gz | 14,477,405 | 8,271 | LFD Book Forum Problem 2.15b
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#1
10-01-2014, 02:13 PM
cumings Junior Member Join Date: Oct 2014 Posts: 2
Problem 2.15b
Are we finding m(N) for our example in part (a) or for the overall hypothesis set containing all monotonically increasing functions?
#2
10-02-2014, 09:10 PM
magdon RPI Join Date: Aug 2009 Location: Troy, NY, USA. Posts: 597
Re: Problem 2.15b
For the entire set of monotonically increasing hypotheses.
(m(N) for a single hypothesis as in part (a) is 1 since a single hypothesis can only implement one dichotomy on any data set)
Quote:
Originally Posted by cumings Are we finding m(N) for our example in part (a) or for the overall hypothesis set containing all monotonically increasing functions?
__________________
Have faith in probability
#3
04-10-2018, 07:14 AM
k_sze Member Join Date: Dec 2016 Posts: 12
Re: Problem 2.15b
For a), am I correct in imagining a hypothesis where I have a 2D Cartesian plane, which is divided by a "stairs" line that goes from the top left to the bottom right? The region "above" the stairs would be +1, and the region below the stairs would be -1.
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Contact Us - LFD Book - Top | 606 | 2,498 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.59375 | 3 | CC-MAIN-2020-50 | latest | en | 0.808886 |
https://www.esaral.com/q/let-a-parabola-p-be-such-that-its-vertex-48086 | 1,718,673,747,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861746.4/warc/CC-MAIN-20240618011430-20240618041430-00055.warc.gz | 659,262,489 | 11,547 | # Let a parabola P be such that its vertex
Question:
Let a parabola P be such that its vertex and focus lie on the positive $x$-axis at a distance 2 and 4 units from the origin, respectively. If tangents are drawn from $\mathrm{O}(0,0)$ to the parabola $\mathrm{P}$ which meet $\mathrm{P}$ at $\mathrm{S}$ and $R$, then the area (in sq. units) of $\Delta S O R$ is equal to :
1. $16 \sqrt{2}$
2. 16
3. 32
4. $8 \sqrt{2}$
Correct Option: , 2
Solution:
Clearly RS is latus-rectum
$\because \mathrm{VF}=2=\mathrm{a}$
$\therefore \mathrm{RS}=4 \mathrm{a}=8$
Now $\mathrm{OF}=2 \mathrm{a}=4$
$\Rightarrow$ Area of triangle ORS $=16$ | 219 | 640 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.625 | 4 | CC-MAIN-2024-26 | latest | en | 0.725589 |
https://www.teacherspayteachers.com/Product/Big-Science-5-Density-10-Calculating-the-Density-of-Room-Temperature-Water-1230223 | 1,484,659,538,000,000,000 | text/html | crawl-data/CC-MAIN-2017-04/segments/1484560279915.8/warc/CC-MAIN-20170116095119-00287-ip-10-171-10-70.ec2.internal.warc.gz | 991,504,487 | 46,258 | Teachers Pay Teachers
# Big Science 5 Density 10 Calculating the Density of Room-Temperature Water
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### PRODUCT DESCRIPTION
This is an excerpt from Amazon's popular line of Bossy Brocci Math and Big Science workbooks! [it's pronounced like "Brawsee"]
Printing should be done in Landscape and DOUBLE-SIDED, with the flip being along the 'SHORT' side.
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Bossy Brocci worksheets.
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Students will:
1) Determine and Record the Average Density of Room Temperature water from 4 Volume samples:
10 mL
20 mL
30 mL
and
40 mL
2) Measure, Calculate and Record 5 physical properties in a Data Table:
Mass of Empty, Dry Graduated Cylinder (or other vessel)
Water Volume at Meniscus (depicted in Table)
Mass of Filled-to-mark Graduated Cylinder (or other vessel)
Mass of specified Volume of Water
DENSITY of specified volume of water
3) Fill-in a total of 20 cells in this first Data Table
4) Calculate & Record in a 2nd, mini Data Table:
Sum of four Density runs/results
Room-Temperature Water's Average Density
Room-Temperature Water's Average Density rounded to a WHOLE Number
5) Graph or Plot Water Mass (y) vs. Water Volume (x) from their 1st Data Table onto the provided and pre-scaled a Graph
6) Draw a Line of Best Fit (by eye).
7) Calculate and Record Unit Slope (simplified slope) of this Best Fit Line, rounded to a Whole Number
8) Answer 20 Fill-in-the-Blank and Multiple-choice questions based on their Calculations, Observations and Plots in the preceding Data Tables & Graph
9) Discover that Density is an Intensive Property: pouring more water increases volume and mass of water, but the density of water (at a particular temperature) is constant, invariant
10) Prove to themselves why we say the Density of water is approximately 1 g/mL
11) Be compelled to present their work in a neat & orderly format
12) Be trained to Calculate & Comprehend Density as a Constant, Intensive Property at a particular temperature methodically & systematically
Printing should be done in Landscape and double-sided, with the flip being along the 'short' side
Science Chemistry Physical Science Physical Property Particle Packing Density Mass Volume Calculating Density Identifying Elements Substances Density of irregular solids Density of Liquids Density of Water Density Anomaly Density Inversion Intensive Properties Extensive Properties Density as an Intensive Property Density as a Unique Substance-Specific Property Temperature's Effect on Density Calculating Density by Displacement Calculating Density of Regular Geometric Solids Rectangular Prisms Cubes Density of Elements Water's Density Anomaly Density Layers Density Columns Density Layer Columns Using Density to Identify Unknowns Identifying Unknowns by their with using Density
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Total:
0 ratings | 867 | 3,862 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.28125 | 3 | CC-MAIN-2017-04 | longest | en | 0.85093 |
https://www.mathlearnit.com/fraction-as-decimal/what-is-201-185-as-a-decimal | 1,723,191,455,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640762343.50/warc/CC-MAIN-20240809075530-20240809105530-00467.warc.gz | 694,812,257 | 7,250 | # What is 201/185 as a decimal?
## Solution and how to convert 201 / 185 into a decimal
201 / 185 = 1.086
Fraction conversions explained:
• 201 divided by 185
• Numerator: 201
• Denominator: 185
• Decimal: 1.086
• Percentage: 1.086%
201/185 or 1.086 can be represented in multiple ways (even as a percentage). The key is knowing when we should use each representation and how to easily transition between a fraction, decimal, or percentage. Both are used to handle numbers less than one or between whole numbers, known as integers. The difference between using a fraction or a decimal depends on the situation. Fractions can be used to represent parts of an object like 1/8 of a pizza while decimals represent a comparison of a whole number like \$0.25 USD. If we need to convert a fraction quickly, let's find out how and when we should.
201 / 185 as a percentage 201 / 185 as a fraction 201 / 185 as a decimal
1.086% - Convert percentages 201 / 185 201 / 185 = 1.086
## 201/185 is 201 divided by 185
The first step of teaching our students how to convert to and from decimals and fractions is understanding what the fraction is telling is. 201 is being divided into 185. Think of this as our directions and now we just need to be able to assemble the project! Fractions have two parts: Numerators and Denominators. This creates an equation. We must divide 201 into 185 to find out how many whole parts it will have plus representing the remainder in decimal form. This is our equation:
### Numerator: 201
• Numerators are the portion of total parts, showed at the top of the fraction. Any value greater than fifty will be more difficult to covert to a decimal. The bad news is that it's an odd number which makes it harder to covert in your head. Large two-digit conversions are tough. Especially without a calculator. Let's look at the fraction's denominator 185.
### Denominator: 185
• Denominators represent the total parts, located at the bottom of the fraction. Larger values over fifty like 185 makes conversion to decimals tougher. But the bad news is that odd numbers are tougher to simplify. Unfortunately and odd denominator is difficult to simplify unless it's divisible by 3, 5 or 7. Ultimately, don't be afraid of double-digit denominators. Now let's dive into how we convert into decimal format.
## How to convert 201/185 to 1.086
### Step 1: Set your long division bracket: denominator / numerator
$$\require{enclose} 185 \enclose{longdiv}{ 201 }$$
Use long division to solve step one. This method allows us to solve for pieces of the equation rather than trying to do it all at once.
### Step 2: Solve for how many whole groups you can divide 185 into 201
$$\require{enclose} 00.1 \\ 185 \enclose{longdiv}{ 201.0 }$$
How many whole groups of 185 can you pull from 2010? 185 Multiple this number by our furthest left number, 185, (remember, left-to-right long division) to get our first number to our conversion.
### Step 3: Subtract the remainder
$$\require{enclose} 00.1 \\ 185 \enclose{longdiv}{ 201.0 } \\ \underline{ 185 \phantom{00} } \\ 1825 \phantom{0}$$
If there is no remainder, you’re done! If there is a remainder, extend 185 again and pull down the zero
### Step 4: Repeat step 3 until you have no remainder
In some cases, you'll never reach a remainder of zero. Looking at you pi! And that's okay. Find a place to stop and round to the nearest value.
### Why should you convert between fractions, decimals, and percentages?
Converting fractions into decimals are used in everyday life, though we don't always notice. They each bring clarity to numbers and values of every day life. This is also true for percentages. It’s common for students to hate learning about decimals and fractions because it is tedious. But each represent values in everyday life! Here are just a few ways we use 201/185, 1.086 or 108% in our daily world:
### When you should convert 201/185 into a decimal
Sports Stats - Fractions can be used here, but when comparing percentages, the clearest representation of success is from decimal points. Ex: A player's batting average: .333
### When to convert 1.086 to 201/185 as a fraction
Cooking: When scrolling through pintress to find the perfect chocolate cookie recipe. The chef will not tell you to use .86 cups of chocolate chips. That brings confusion to the standard cooking measurement. It’s much clearer to say 42/50 cups of chocolate chips. And to take it even further, no one would use 42/50 cups. You’d see a more common fraction like ¾ or ?, usually in split by quarters or halves.
### Practice Decimal Conversion with your Classroom
• If 201/185 = 1.086 what would it be as a percentage?
• What is 1 + 201/185 in decimal form?
• What is 1 - 201/185 in decimal form?
• If we switched the numerator and denominator, what would be our new fraction?
• What is 1.086 + 1/2?
### Convert more fractions to decimals
From 201 Numerator From 185 Denominator What is 201/175 as a decimal? What is 191/185 as a decimal? What is 201/176 as a decimal? What is 192/185 as a decimal? What is 201/177 as a decimal? What is 193/185 as a decimal? What is 201/178 as a decimal? What is 194/185 as a decimal? What is 201/179 as a decimal? What is 195/185 as a decimal? What is 201/180 as a decimal? What is 196/185 as a decimal? What is 201/181 as a decimal? What is 197/185 as a decimal? What is 201/182 as a decimal? What is 198/185 as a decimal? What is 201/183 as a decimal? What is 199/185 as a decimal? What is 201/184 as a decimal? What is 200/185 as a decimal? What is 201/185 as a decimal? What is 201/185 as a decimal? What is 201/186 as a decimal? What is 202/185 as a decimal? What is 201/187 as a decimal? What is 203/185 as a decimal? What is 201/188 as a decimal? What is 204/185 as a decimal? What is 201/189 as a decimal? What is 205/185 as a decimal? What is 201/190 as a decimal? What is 206/185 as a decimal? What is 201/191 as a decimal? What is 207/185 as a decimal? What is 201/192 as a decimal? What is 208/185 as a decimal? What is 201/193 as a decimal? What is 209/185 as a decimal? What is 201/194 as a decimal? What is 210/185 as a decimal? What is 201/195 as a decimal? What is 211/185 as a decimal?
### Convert similar fractions to percentages
From 201 Numerator From 185 Denominator 202/185 as a percentage 201/186 as a percentage 203/185 as a percentage 201/187 as a percentage 204/185 as a percentage 201/188 as a percentage 205/185 as a percentage 201/189 as a percentage 206/185 as a percentage 201/190 as a percentage 207/185 as a percentage 201/191 as a percentage 208/185 as a percentage 201/192 as a percentage 209/185 as a percentage 201/193 as a percentage 210/185 as a percentage 201/194 as a percentage 211/185 as a percentage 201/195 as a percentage | 1,800 | 6,775 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.5625 | 5 | CC-MAIN-2024-33 | latest | en | 0.905876 |
https://litux.nl/Books/books/www.leothreads.com/e-book/oreillybookself/java%20enterprise/jnut/ch01_03.htm | 1,723,308,018,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640810581.60/warc/CC-MAIN-20240810155525-20240810185525-00451.warc.gz | 277,218,840 | 14,582 | ## 1.3. An Example Program
Example 1-1 shows a Java program to compute factorials.[4] The numbers at the beginning of each line are not part of the program; they are there for ease of reference when we dissect the program line-by-line.
[4]The factorial of an integer is the product of the number and all positive integers less than the number. So, for example, the factorial of 4, which is also written 4!, is 4 times 3 times 2 times 1, or 24. By definition, 0! is 1.
#### Example 1-1. Factorial.java: A Program to Compute Factorials
``` 1 /**
2 * This program computes the factorial of a number
3 */
4 public class Factorial { // Define a class
5 public static void main(String[] args) { // The program starts here
6 int input = Integer.parseInt(args[0]); // Get the user's input
7 double result = factorial(input); // Compute the factorial
8 System.out.println(result); // Print out the result
9 } // The main() method ends here
10
11 public static double factorial(int x) { // This method computes x!
12 if (x < 0) // Check for bad input
13 return 0.0; // if bad, return 0
14 double fact = 1.0; // Begin with an initial value
15 while(x > 1) { // Loop until x equals 1
16 fact = fact * x; // multiply by x each time
17 x = x - 1; // and then decrement x
18 } // Jump back to start of loop
19 return fact; // Return the result
20 } // factorial() ends here
21 } // The class ends here
```
### 1.3.1. Compiling and Running the Program
Before we look at how the program works, we must first discuss how to run it. In order to compile and run the program, you need a Java software development kit (SDK) of some sort. Sun Microsystems created the Java language and ships a free Java SDK for its Solaris operating system and for Microsoft Windows (95/98/NT) platforms. At the time of this writing, the current version of Sun's SDK is entitled Java 2 SDK, Standard Edition, Version 1.2.2 and is available for download from http://java.sun.com/products/jdk/1.2/ (Sun's Java SDK is still often called the JDK, even internally). Be sure to get the SDK and not the Java Runtime Environment. The JRE enables you to run existing Java programs, but not to write your own.
Sun supports its SDK only on Solaris and Windows platforms. Many other companies have licensed and ported the SDK to their platforms, however. Contact your operating-system vendor to find if a version of the Java SDK is available for your system. Linux users should visit http://www.blackdown.org/.
The Sun SDK is not the only Java programming environment you can use. Companies such as Borland, Inprise, Metrowerks, Oracle, Sybase, and Symantec offer commercial products that enable you to write Java programs. This book assumes that you are using Sun's SDK. If you are using a product from some other vendor, be sure to read that vendor's documentation to learn how to compile and run a simple program, like that shown in Example 1-1.
Once you have a Java programming environment installed, the first step towards running our program is to type it in. Using your favorite text editor, enter the program as it is shown in Example 1-1. Omit the line numbers, as they are just there for reference. Note that Java is a case-sensitive language, so you must type lowercase letters in lowercase and uppercase letters in uppercase. You'll notice that many of the lines of this program end with semicolons. It is a common mistake to forget these characters, but the program won't work without them, so be careful! If you are not a fast typist, you can omit everything from // to the end of a line. Those are comments ; they are there for your benefit and are ignored by Java.[5]
[5]I recommend that you type this example in by hand, to get a feel for the language. If you really don't want to, however, you can download this, and all examples in the book, from http://www.oreilly.com/catalog/javanut3/.
When writing Java programs, you should use a text editor that saves files in plain-text format, not a word processor that supports fonts and formatting and saves files in a proprietary format. My favorite text editor on Unix systems is emacs. If you use a Windows system, you might use Notepad or WordPad, if you don't have a more specialized programmer's editor. If you are using a commercial Java programming environment, it probably includes an appropriate text editor; read the documentation that came with the product. When you are done entering the program, save it in a file named Factorial.java. This is important; the program will not work if you save it by any other name.
After writing a program like this one, the next step is to compile it. With Sun's SDK, the Java compiler is known as javac. javac is a command-line tool, so you can only use it from a terminal window, such as an MS-DOS window on a Windows system or an xterm window on a Unix system. Compile the program by typing the following command line:[6]
[6]The "C:\>" characters represent the command-line prompt; don't type these characters yourself.
`C:\> javac Factorial.java`
If this command prints any error messages, you probably got something wrong when you typed in the program. If it does not print any error messages, however, the compilation has succeeded, and javac creates a file called Factorial.class. This is the compiled version of the program.
Once you have compiled a Java program, you must still run it. Unlike some other languages, Java programs are not compiled into native machine language, so they cannot be executed directly by the system. Instead, they are run by another program known as the Java interpreter. In Sun's SDK, the interpreter is a command-line program named, appropriately enough, java. To run the factorial program, type:
`C:\> java Factorial 4`
java is the command to run the Java interpreter, Factorial is the name of the Java program we want the interpreter to run, and 4 is the input data--the number we want the interpreter to compute the factorial of. The program prints a single line of output, telling us that the factorial of 4 is 24:
```C:\> java Factorial 4
24.0```
Congratulations! You've just written, compiled, and run your first Java program. Try running it again to compute the factorials of some other numbers.
### 1.3.2. Analyzing the Program
Now that you have run the factorial program, let's analyze it line by line, to see what makes a Java program tick.
The first three lines of the program are a comment. Java ignores them, but they tell a human programmer what the program does. A comment begins with the characters /* and ends with the characters */. Any amount of text, including multiple lines of text, may appear between these characters. Java also supports another type of comment, which you can see in lines 4 through 21. If the characters // appear in a Java program, Java ignores those characters and any other text that appears between those characters and the end of the line.
### 1.3.2.2. Defining a class
Line 4 is the beginning of the program. It says that we are defining a class named Factorial. This explains why the program had to be stored in a file named Factorial.java. That filename indicates that the file contains Java source code for a class named Factorial. The word public is a modifier ; it says that the class is publicly available and that anyone may use it. The open curly-brace character ({) marks the beginning of the body of the class, which extends all the way to line 21, where we find the matching close curly-brace character (}). The program contains a number of pairs of curly braces; the lines are indented to show the nesting within these braces.
A class is the fundamental unit of program structure in Java, so it is not surprising that the first line of our program declares a class. All Java programs are classes, although some programs use many classes instead of just one. Java is an object-oriented programming language, and classes are a fundamental part of the object-oriented paradigm. Each class defines a unique kind of object. Example 1-1 is not really an object-oriented program, however, so I'm not going to go into detail about classes and objects here. That is the topic of Chapter 3, "Object-Oriented Programming in Java". For now, all you need to understand is that a class defines a set of interacting members. Those members may be fields, methods, or other classes. The Factorial class contains two members, both of which are methods. They are described in upcoming sections.
### 1.3.2.3. Defining a method
Line 5 begins the definition of a method of our Factorial class. A method is a named chunk of Java code. A Java program can call, or invoke, a method to execute the code in it. If you have programmed in other languages, you have probably seen methods before, but they may have been called functions, procedures, or subroutines. The interesting thing about methods is that they have parameters and return values. When you call a method, you pass it some data you want it to operate on, and it returns a result to you. A method is like an algebraic function:
`y = f(x)`
Here, the mathematical function f performs some computation on the value represented by x and returns a value, which we represent by y.
To return to line 5, the public and static keywords are modifiers. public means the method is publicly accessible; anyone can use it. The meaning of the static modifier is not important here; it is explained in Chapter 3, "Object-Oriented Programming in Java". The void keyword specifies the return value of the method. In this case, it specifies that this method does not have a return value.
The word main is the name of the method. main is a special name. When you run the Java interpreter, it reads in the class you specify, then looks for a method named main().[7] When the interpreter finds this method, it starts running the program at that method. When the main() method finishes, the program is done, and the Java interpreter exits. In other words, the main() method is the main entry point into a Java program. It is not actually sufficient for a method to be named main(), however. The method must be declared public static void exactly as shown in line 5. In fact, the only part of line 5 you can change is the word args, which you can replace with any word you want. You'll be using this line in all of your Java programs, so go ahead and commit it to memory now![8]
[7]By convention, when this book refers to a method, it follows the name of the method by a pair of parentheses. As you'll see, parentheses are an important part of method syntax, and they serve here to keep method names distinct from the names of classes, fields, variables, and so on.
[8]All Java programs that are run directly by the Java interpreter must have a main() method. Programs of this sort are often called applications. It is possible to write programs that are not run directly by the interpreter, but are dynamically loaded into some other already running Java program. Examples are applets, which are programs run by a web browser, and servlets, which are programs run by a web server. Applets are discussed in Java Foundation Classes in a Nutshell (O'Reilly), while servlets are discussed in Java Enterprise in a Nutshell (O'Reilly). In this book, we consider only applications.
Following the name of the main() method is a list of method parameters, contained in parentheses. This main() method has only a single parameter. String[] specifies the type of the parameter, which is an array of strings (i.e., a numbered list of strings of text). args specifies the name of the parameter. In the algebraic equation f(x), x is simply a way of referring to an unknown value. args serves the same purpose for the main() method. As we'll see, the name args is used in the body of the method to refer to the unknown value that is passed to the method.
As I've just explained, the main() method is a special one that is called by the Java interpreter when it starts running a Java class (program). When you invoke the Java interpreter like this:
`C:\> java Factorial 4`
the string "4" is passed to the main() method as the value of the parameter named args. More precisely, an array of strings containing only one entry, "4", is passed to main(). If we invoke the program like this:
`C:\> java Factorial 4 3 2 1`
then an array of four strings, "4", "3", "2", and "1", are passed to the main() method as the value of the parameter named args. Our program looks only at the first string in the array, so the other strings are ignored.
Finally, the last thing on line 5 is an open curly brace. This marks the beginning of the body of the main() method, which continues until the matching close curly brace on line 9. Methods are composed of statements, which the Java interpreter executes in sequential order. In this case, lines 6, 7, and 8 are three statements that compose the body of the main() method. Each statement ends with a semicolon to separate it from the next. This is an important part of Java syntax; beginning programmers often forget the semicolons.
### 1.3.2.4. Declaring a variable and parsing input
The first statement of the main() method, line 6, declares a variable and assigns a value to it. In any programming language, a variable is simply a symbolic name for a value. Think back to algebra class again:
c2 = a2 + b2
The letters a, b, and c are names we use to refer to unknown values. They make this formula (the Pythagorean theorem) a general one that applies to arbitrary values of a, b, and c, not just a specific set like:
52 = 42 + 32
A variable in a Java program is exactly the same thing: it is a name we use to refer to a value. More precisely, a variable is a name that refers to a storage space for a value. We often say that a variable holds a value.
Line 6 begins with the words int input. This declares a variable named input and specifies that the variable has the type int; that is, it is an integer. Java can work with several different types of values, including integers, real or floating-point numbers, characters (e.g., letters, digits), and strings. Java is a strongly typed language, which means that all variables must have a type specified and can only refer to values of that type. Our input variable always refers to an integer; it cannot refer to a floating point number or a string. Method parameters are also typed. Recall that the args parameter had a type of String[].
Continuing with line 6, the variable declaration int input is followed by the = character. This is the assignment operator in Java; it sets the value of a variable. When reading Java code, don't read = as "equals," but instead read it as "is assigned the value." As we'll see in Chapter 2, "Java Syntax from the Ground Up", there is a different operator for "equals."
The value being assigned to our input variable is Integer.parseInt(args[0]). This is a method invocation. This first statement of the main() method invokes another method whose name is Integer.parseInt(). As you might guess, this method "parses" an integer; that is, it converts a string representation of an integer, such as "4", to the integer itself. The Integer.parseInt() method is not part of the Java language, but it is a core part of the Java API or Application Programming Interface. Every Java program can use the powerful set of classes and methods defined by this core API. The second half of this book is a quick-reference that documents that core API.
When you call a method, you pass values (called arguments) that are assigned to the corresponding parameters defined by the method, and the method returns a value. The argument passed to Integer.parseInt() is args[0]. Recall that args is the name of the parameter for main(); it specifies an array (or list) of strings. The elements of an array are numbered sequentially, and the first one is always numbered 0. We only care about the first string in the args array, so we use the expression args[0] to refer to that string. Thus, when we invoke the program as shown earlier, line 6 takes the first string specified after the name of the class, "4", and passes it to the method named Integer.parseInt(). This method converts the string to the corresponding integer and returns the integer as its return value. Finally, this returned integer is assigned to the variable named input.
### 1.3.2.5. Computing the result
The statement on line 7 is a lot like the statement on line 6. It declares a variable and assigns a value to it. The value assigned to the variable is computed by invoking a method. The variable is named result, and it has a type of double. double means a double-precision floating-point number. The variable is assigned a value that is computed by the factorial() method. The factorial() method, however, is not part of the standard Java API. Instead, it is defined as part of our program, by lines 11 through 19. The argument passed to factorial() is the value referred to by the input variable, which was computed on line 6. We'll consider the body of the factorial() method shortly, but you can surmise from its name that this method takes an input value, computes the factorial of that value, and returns the result.
### 1.3.2.6. Displaying output
Line 8 simply calls a method named System.out.println(). This commonly used method is part of the core Java API; it causes the Java interpreter to print out a value. In this case, the value that it prints is the value referred to by the variable named result. This is the result of our factorial computation. If the input variable holds the value 4, the result variable holds the value 24, and this line prints out that value.
The System.out.println() method does not have a return value, so there is no variable declaration or = assignment operator in this statement, since there is no value to assign to anything. Another way to say this is that, like the main() method of line 5, System.out.println() is declared void.
### 1.3.2.7. The end of a method
Line 9 contains only a single character, }. This marks the end of the method. When the Java interpreter gets here, it is done executing the main() method, so it stops running. The end of the main() method is also the end of the variable scope for the input and result variables declared within main() and for the args parameter of main(). These variable and parameter names have meaning only within the main() method and cannot be used elsewhere in the program, unless other parts of the program declare different variables or parameters that happen to have the same name.
### 1.3.2.8. Blank lines
Line 10 is a blank line. You can insert blank lines, spaces, and tabs anywhere in a program, and you should use them liberally to make the program readable. A blank line appears here to separate the main() method from the factorial() method that begins on line 11. You'll notice that the program also uses spaces and tabs to indent the various lines of code. This kind of indentation is optional; it emphasizes the structure of the program and greatly enhances the readability of the code.
### 1.3.2.9. Another method
Line 11 begins the definition of the factorial() method that was used by the main() method. Compare this line to line 5 to note its similarities and differences. The factorial() method has the same public and static modifiers. It takes a single integer parameter, which we call x. Unlike the main() method, which had no return value (void), factorial() returns a value of type double. The open curly brace marks the beginning of the method body, which continues past the nested braces on lines 15 and 18 to line 20, where the matching close curly brace is found. The body of the factorial() method, like the body of the main() method, is composed of statements, which are found on lines 12 through 19.
### 1.3.2.10. Checking for valid input
In the main() method, we saw variable declarations, assignments, and method invocations. The statement on line 12 is different. It is an if statement, which executes another statement conditionally. We saw earlier that the Java interpreter executes the three statements of the main() method one after another. It always executes them in exactly that way, in exactly that order. An if statement is a flow-control statement; it can affect the way the interpreter runs a program.
The if keyword is followed by a parenthesized expression and a statement. The Java interpreter first evaluates the expression. If it is true, the interpreter executes the statement. If the expression is false, however, the interpreter skips the statement and goes to the next one. The condition for the if statement on line 12 is x < 0. It checks whether the value passed to the factorial() method is less than zero. If it is, this expression is true, and the statement on line 13 is executed. Line 12 does not end with a semicolon because the statement on line 13 is part of the if statement. Semicolons are required only at the end of a statement.
Line 13 is a return statement. It says that the return value of the factorial() method is 0.0. return is also a flow-control statement. When the Java interpreter sees a return, it stops executing the current method and returns the specified value immediately. A return statement can stand alone, but in this case, the return statement is part of the if statement on line 12. The indentation of line 13 helps emphasize this fact. (Java ignores this indentation, but it is very helpful for humans who read Java code!) Line 13 is executed only if the expression on line 12 is true.
Before we move on, we should pull back a bit and talk about why lines 12 and 13 are necessary in the first place. It is an error to try to compute a factorial for a negative number, so these lines make sure that the input value x is valid. If it is not valid, they cause factorial() to return a consistent invalid result, 0.0.
### 1.3.2.11. An important variable
Line 14 is another variable declaration; it declares a variable named fact of type double and assigns it an initial value of 1.0. This variable holds the value of the factorial as we compute it in the statements that follow. In Java, variables can be declared anywhere; they are not restricted to the beginning of a method or block of code.
### 1.3.2.12. Looping and computing the factorial
Line 15 introduces another type of statement: the while loop. Like an if statement, a while statement consists of a parenthesized expression and a statement. When the Java interpreter sees a while statement, it evaluates the associated expression. If that expression is true, the interpreter executes the statement. The interpreter repeats this process, evaluating the expression and executing the statement if the expression is true, until the expression evaluates to false. The expression on line 15 is x > 1, so the while statement loops while the parameter x holds a value that is greater than 1. Another way to say this is that the loop continues untilx holds a value less than or equal to 1. We can assume from this expression that if the loop is ever going to terminate, the value of x must somehow be modified by the statement that the loop executes.
The major difference between the if statement on lines 12-13 and the while loop on lines 15-18 is that the statement associated with the while loop is a compound statement. A compound statement is zero or more statements grouped between curly braces. The while keyword on line 15 is followed by an expression in parentheses and then by an open curly brace. This means that the body of the loop consists of all statements between that opening brace and the closing brace on line 18. Earlier in the chapter, I said that all Java statements end with semicolons. This rule does not apply to compound statements, however, as you can see by the lack of a semicolon at the end of line 18. The statements inside the compound statement (lines 16 and 17) do end with semicolons, of course.
The body of the while loop consists of the statements on line 16 and 17. Line 16 multiplies the value of fact by the value of x and stores the result back into fact. Line 17 is similar. It subtracts 1 from the value of x and stores the result back into x. The * character on line 16 is important: it is the multiplication operator. And, as you can probably guess, the - on line 17 is the subtraction operator. An operator is a key part of Java syntax: it performs a computation on one or two operands to produce a new value. Operands and operators combine to form expressions, such as fact * x or x - 1. We've seen other operators in the program. Line 15, for example, uses the greater-than operator (>) in the expression x > 1, which compares the value of the variable x to 1. The value of this expression is a boolean truth value--either true or false, depending on the result of the comparison.
To understand this while loop, it is helpful to think like the Java interpreter. Suppose we are trying to compute the factorial of 4. Before the loop starts, fact is 1.0, and x is 4. After the body of the loop has been executed once--after the first iteration--fact is 4.0, and x is 3. After the second iteration, fact is 12.0, and x is 2. After the third iteration, fact is 24.0, and x is 1. When the interpreter tests the loop condition after the third iteration, it finds that x > 1 is no longer true, so it stops running the loop, and the program resumes at line 19.
### 1.3.2.13. Returning the result
Line 19 is another return statement, like the one we saw on line 13. This one does not return a constant value like 0.0, but instead returns the value of the fact variable. If the value of x passed into the factorial() function is 4, then, as we saw earlier, the value of fact is 24.0, so this is the value returned. Recall that the factorial() method was invoked on line 7 of the program. When this return statement is executed, control returns to line 7, where the return value is assigned to the variable named result.
### 1.3.3. Exceptions
If you've made it all the way through the line-by-line analysis of Example 1-1, you are well on your way to understanding the basics of the Java language.[9] It is a simple but nontrivial program that illustrates many of the features of Java. There is one more important feature of Java programming I want to introduce, but it is one that does not appear in the program listing itself. Recall that the program computes the factorial of the number you specify on the command line. What happens if you run the program without specifying a number?
[9] If you didn't understood all the details of this factorial program, don't worry. We'll cover the details of the Java language a lot more thoroughly in Chapter 2, "Java Syntax from the Ground Up" and Chapter 3, "Object-Oriented Programming in Java". However, if you feel like you didn't understand any of the line-by-line analysis, you may also find that the upcoming chapters are over your head. In that case, you should probably go elsewhere to learn the basics of the Java language and return to this book to solidify your understanding, and, of course, to use as a reference. One resource you may find useful in learning the language is Sun's online Java tutorial, available at http://java.sun.com/docs/books/tutorial/.
```C:\> java Factorial
java.lang.ArrayIndexOutOfBoundsException: 0
at Factorial.main(Factorial.java:6)
C:\> ```
And what happens if you specify a value that is not a number?
```C:\> java Factorial ten
java.lang.NumberFormatException: ten
at java.lang.Integer.parseInt(Integer.java)
at java.lang.Integer.parseInt(Integer.java)
at Factorial.main(Factorial.java:6)
C:\>```
In both cases, an error occurs or, in Java terminology, an exception is thrown. When an exception is thrown, the Java interpreter prints out a message that explains what type of exception it was and where it occurred (both exceptions above occurred on line 6 ). In the first case, the exception is thrown because there are no strings in the args list, meaning we asked for a nonexistent string with args[0]. In the second case, the exception is thrown because Integer.parseInt() cannot convert the string "ten" to a number. We'll see more about exceptions in Chapter 2, "Java Syntax from the Ground Up" and learn how to handle them gracefully as they occur. | 6,390 | 28,751 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.28125 | 3 | CC-MAIN-2024-33 | latest | en | 0.647911 |
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réponse | 1,546 | 4,847 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.6875 | 3 | CC-MAIN-2022-49 | latest | en | 0.598099 |
https://www.physicsforums.com/threads/nonnegative-solutions.223820/ | 1,607,104,447,000,000,000 | text/html | crawl-data/CC-MAIN-2020-50/segments/1606141740670.93/warc/CC-MAIN-20201204162500-20201204192500-00109.warc.gz | 807,206,799 | 14,151 | Nonnegative Solutions
Homework Statement
I need to find the number of nonnegative integer solutions to the equation a+2b+4c=10^30
The Attempt at a Solution
I was thinking of trying to find perhaps some sort of a multinomial generating function, but am not sure how that will help me. Any suggestions? Thanks.
Related Calculus and Beyond Homework Help News on Phys.org
Here are my thoughts:
You can find the total number of solutions by incrementally varying parameters. Since for any solution c will be the least likely to be an integer if you choose arbitrary a and b, varying c and trying to peg possible a's and b's would be the wisest way to proceed. For example:
If c=50, a+2b = 10^30-200. Because once again b is least likely to be an integer if we choose arbitrary a subject to a+2b=10^30-200, b would be the best to vary. Anywhere from b=(10^30-200)/2 to b=0 will have a corresponding a, so there are (10^30-200)/2+1 different solutions for the case c=50. Can you see how this result might generalize to all c?
Last edited:
So for all c, there would be (10^30-4c)/2+1 corresponding to every possible c value, but that would give an infinite number of solutions wouldn't it?
If 10^30-4c < 0, do you have any solutions for non-negative b and a? My approach is simply iteratively inspecting the sum and assuming a varying "chunk" of 10^30 is made up of 4c. The answer is definitely finite. | 362 | 1,403 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.765625 | 4 | CC-MAIN-2020-50 | latest | en | 0.922092 |
https://mathhelpboards.com/threads/stefans-question-about-a-line-integral.10827/ | 1,590,979,983,000,000,000 | text/html | crawl-data/CC-MAIN-2020-24/segments/1590347413901.34/warc/CC-MAIN-20200601005011-20200601035011-00072.warc.gz | 448,643,675 | 14,234 | # Stefan's question about a line integral.
#### Prove It
##### Well-known member
MHB Math Helper
Evaluate the line integral
\displaystyle \begin{align*} \int_{(0,0,0)}^{ \left( \frac{1}{3}, \frac{\pi}{2}, 1 \right) }{ \left[ 12\,z^2 + 4\,\mathrm{e}^{4x}\cos{(5y)} \right] \,\mathrm{d}x - \left[ 5\,\mathrm{e}^{4x}\sin{(5y)} \right] \,\mathrm{d}y + 24\,x\,z\,\mathrm{d}z } \end{align*}
The line starting at \displaystyle \begin{align*} \left( 0, 0, 0 \right) \end{align*} and ending at \displaystyle \begin{align*} \left( \frac{1}{3}, \frac{\pi}{2}, 1 \right) \end{align*} can be expressed in a parametric (vector) form as \displaystyle \begin{align*} \left( x, y, z \right) = \left( \frac{1}{3}t , \frac{\pi}{2}t, t \right) \end{align*} with \displaystyle \begin{align*} 0 \leq t \leq 1 \end{align*}. Thus
\displaystyle \begin{align*} x &= \frac{1}{3}t \implies \mathrm{d}x = \frac{1}{3}\mathrm{d}t \\ y &= \frac{\pi}{2}t \implies \mathrm{d}y = \frac{\pi}{2}\mathrm{d}t \\ z &= t \implies \mathrm{d}z = \mathrm{d}t \end{align*}
and so the line integral becomes
\displaystyle \begin{align*} &= \int_0^1{ \left\{ 12t^2 + 4\mathrm{e}^{ 4 \left( \frac{1}{3} t \right) } \cos{ \left[ 5 \left( \frac{\pi}{2}t \right) \right] } \right\} \frac{1}{3}\mathrm{d}t - \left\{ 5\mathrm{e}^{4 \left( \frac{1}{3}t \right) } \sin{ \left[ 5 \left( \frac{\pi}{2}t \right) \right] } \right\} \frac{\pi}{2}\mathrm{d}t + 24 \left( \frac{1}{3}t \right) t \, \mathrm{d}t } \\ &= \int_0^1{ 4\,t^2 + \frac{4}{3}\,\mathrm{e}^{ \frac{4}{3}t } \cos{ \left( \frac{5\pi}{2}t \right)} - \frac{5\pi}{2}\,\mathrm{e}^{\frac{4}{3}t}\sin{ \left( \frac{5\pi}{2}t \right) } + 8\,t^2 \, \mathrm{d}t } \\ &= \int_0^1{ 12\,t^2 + \frac{4}{3}\,\mathrm{e}^{\frac{4}{3}t}\cos{ \left( \frac{5\pi}{2}t \right) } - \frac{5\pi}{2}\,\mathrm{e}^{\frac{4}{3}t} \sin{ \left( \frac{5\pi}{2} t\right) } \, \mathrm{d}t } \\ \end{align*}
Now to integrate these, the product functions either require integration by parts, or use of the rules \displaystyle \begin{align*} \int{ \mathrm{e}^{b\,x}\sin{(a\,x)} \, \mathrm{d}x } = \frac{1}{a^2 + b^2} \, \mathrm{e}^{b\,x} \, \left[ b\sin{(a\,x)} - a\cos{(a\,x)} \right] + C \end{align*} and \displaystyle \begin{align*} \int{ \mathrm{e}^{b\,x}\cos{(a\,x)} \,\mathrm{d}x } = \frac{1}{a^2 + b^2}\,\mathrm{e}^{b\,x} \, \left[ a\sin{(a\,x)} + b\cos{(a\,x)} \right] + C \end{align*}. | 1,088 | 2,371 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.9375 | 4 | CC-MAIN-2020-24 | latest | en | 0.309089 |
http://freetofindtruth.blogspot.com/2016/02/33-41-223-322-obamas-planned-trip-to.html | 1,529,734,072,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267864943.28/warc/CC-MAIN-20180623054721-20180623074721-00159.warc.gz | 115,939,621 | 22,251 | ## Friday, February 19, 2016
### 33 41 223 322 | Obama's planned trip to Cuba for March 21 and March 22, 2016
This announcement comes on February 18, 2016, the 49th day of the year.
2/18/2016 = 2+18+20+16 = 56 (President)
2/18/2016 = 2+18+(2+0+1+6) = 29
2/18/2016 = 2+1+8+2+0+1+6 = 20
2/18/16 = 2+18+16 = 36
The date of the announcement, February 18, to the date of March 21, when he is to arrive in Cuba, is a span of 33-days.
March 22 will follow February 18 by 33-days, meaning it is a total span of 34-days.
3/21/2016 = 3+21+20+16 = 60 (Conspiracy)
3/21/2016 = 3+21+(2+0+1+6) = 33
3/21/2016 = 3+2+1+2+0+1+6 = 15
3/21/16 = 3+21+16 = 40 (United States)
The "countdown sequence" of March 21, or 3/21, reminds me of the connection between 'Obama' and 'Bomb'.
Obama = 15+2+1+13+1 = 32
Bomb = 2+15+13+2 = 32
3/22/2016 = 3+22+20+16 = 61 (Barry O's year of birth)
3/22/2016 = 3+22+(2+0+1+6) = 34 (Murder)
3/22/2016 = 3+2+2+2+0+1+6 = 16
3/22/16 = 3+22+16 = 41 (USA) (13th Prime) (Skull and Bones)
Of course the date March 22 also reminds me the Masonic Order, Skull and Bones.
Notice the name of Obama's deputy national security adviser.
Rhodes = 9+8+6+4+5+1 = 33/42
Ben = 2+5+5 = 12
Ben Rhodes = 45/54
https://news.vice.com/article/president-obama-says-he-will-make-a-historic-trip-to-cuba-this-spring?utm_source=vicenewsfb | 572 | 1,332 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.109375 | 3 | CC-MAIN-2018-26 | latest | en | 0.913909 |
https://www.infoapper.com/math-formulas/perimeter/perimeter-of-trapezoid/ | 1,590,559,215,000,000,000 | text/html | crawl-data/CC-MAIN-2020-24/segments/1590347392141.7/warc/CC-MAIN-20200527044512-20200527074512-00363.warc.gz | 786,327,968 | 7,168 | # Calculate perimeter of trapezoid
## ( Equivalent to: circumference of trapezoid )
In next fields, kindly type the dimensions of your shape in the text box under title [ Unknowns: ]. After you type your value, click "CALCULATE" button; the answer will be automatically calculated and displayed in the text box under title [ Answer: ]. Also, you will be able to animate your shape through buttons under the below figure.
## How to Calculate Perimeter Of Trapezoid
### Example: What is the perimeter of trapezoid with upper base = 24 units, lower base = 20 units, left side = 5 units and right side = 8 units?
As;
perimeter of trapezoid = upper base+lower base+left side+right side
By substituting the above given data in the previous function;
(i.e.) = 24+20+5+8
### Practice Question: Calculate perimeter of trapezoid for the following problems:
N.B.: After working out the answer of each of the next questions, click adjacent button of see the correct answer.
( i ) upper base = 21 units, lower base = 20 units, left side = 12 units and right side = 15 units
( ii ) upper base = 20 units, lower base = 25 units, left side = 13 units and right side = 10 units
( iii ) upper base = 22 units, lower base = 20 units, left side = 10 units and right side = 6 units
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Results: | 453 | 1,863 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.3125 | 3 | CC-MAIN-2020-24 | longest | en | 0.870578 |
https://blog.enterprisedna.co/mastering-exploratory-data-analysis-with-python-best-practices-and-tools/page/2/?et_blog | 1,723,750,908,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722641311225.98/warc/CC-MAIN-20240815173031-20240815203031-00340.warc.gz | 107,998,754 | 60,138 | # Comprehensive Guide to Effectively Performing Exploratory Data Analysis (EDA) Using Python
## Introduction to Exploratory Data Analysis
Exploratory Data Analysis (EDA) is a crucial step in the data science workflow. Its primary objective is to understand the dataset, discover patterns, identify anomalies, and check assumptions with the help of summary statistics and graphical representations.
### Prerequisites
Ensure you have the following Python libraries installed:
• pandas
• numpy
• matplotlib
• seaborn
You can install these libraries using pip:
``````pip install pandas numpy matplotlib seaborn
``````
### 1. Setting Up the Environment
Here’s a basic setup to start your EDA process:
``````# Import necessary libraries
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
# Configure visualization options
sns.set(style="whitegrid")
plt.rcParams['figure.figsize'] = (10, 6)
``````
### 2. Viewing the Dataset
First, inspect the basic structure and content of the dataset:
``````# Display the first few rows of the dataset
print("First 5 rows:")
# Display the last few rows of the dataset
print("Last 5 rows:")
print(df.tail())
# Display the structure
print("DataFrame Info:")
df.info()
``````
### 3. Summary Statistics
Generate summary statistics to quickly understand the central tendencies and distribution of the numerical variables:
``````# Summary of numerical columns
print("Summary statistics:")
print(df.describe(include=[np.number]))
# Summary of categorical columns
print("Summary statistics for categorical columns:")
print(df.describe(include=[np.object, pd.CategoricalDtype]))
``````
### 4. Checking for Missing Values
Identifying missing values is essential to decide on further actions like imputation or deletion:
``````# Check for missing values
print("Missing values in each column:")
print(df.isnull().sum())
``````
### 5. Univariate Analysis
Visualize the distribution of individual variables using histograms and box plots:
``````# Histogram for a numerical column
df['numerical_column'].hist(bins=30)
plt.title('Histogram of Numerical Column')
plt.xlabel('Value')
plt.ylabel('Frequency')
plt.show()
# Boxplot for a numerical column
sns.boxplot(x=df['numerical_column'])
plt.title('Boxplot of Numerical Column')
plt.show()
# Count plot for a categorical column
sns.countplot(x=df['categorical_column'])
plt.title('Count Plot of Categorical Column')
plt.show()
``````
### 6. Bivariate Analysis
Examine relationships between pairs of variables using scatter plots and correlation matrices:
``````# Scatter plot between two numerical columns
sns.scatterplot(data=df, x='numerical_column1', y='numerical_column2')
plt.title('Scatter Plot')
plt.show()
# Correlation matrix heatmap
corr_matrix = df.corr()
sns.heatmap(corr_matrix, annot=True, cmap='coolwarm', linewidths=0.5)
plt.title('Correlation Matrix Heatmap')
plt.show()
``````
### Conclusion
This section provided a foundation in EDA, introducing basic steps to start analyzing your dataset. After completing this guide, you should be able to set up your environment, inspect data, generate summary statistics, and perform basic visualizations.
# Setting Up Your Python Environment
To effectively perform Exploratory Data Analysis (EDA) using Python, you need a robust and well-configured setup of your Python environment. Below are the steps to set up your Python environment along with code snippets that will ensure you have all the necessary tools and libraries for EDA.
## 1. Install Python
Make sure Python is installed on your machine. You can download and install Python from the official Python website. Verify the installation by running:
``````python --version
``````
You should see the Python version printed in the terminal.
## 2. Set Up a Virtual Environment
Creating a virtual environment allows you to isolate your project dependencies. Use the following commands to set it up:
``````# Create a virtual environment named 'eda_env'
python -m venv eda_env
# Activate the virtual environment
# On Windows
eda_env\Scripts\activate
# On macOS/Linux
source eda_env/bin/activate
``````
## 3. Install Required Libraries
Once your virtual environment is activated, you need to install essential libraries for EDA. Create a `requirements.txt` file with the following content:
``````pandas
numpy
matplotlib
seaborn
jupyterlab
scipy
``````
Use the following command to install these libraries:
``````pip install -r requirements.txt
``````
## 4. Initialize a Jupyter Notebook
Jupyter Notebooks are beneficial for EDA as they allow you to combine code execution with visualization and narrative text. Start the Jupyter Notebook server:
``````jupyter lab
``````
## 5. Verify the Environment
In your Jupyter Notebook, create a new Python notebook and run the following code to verify your environment setup:
``````import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from scipy import stats
# Print library versions to confirm installation
print(f"Pandas version: {pd.__version__}")
print(f"NumPy version: {np.__version__}")
print(f"Matplotlib version: {plt.__version__}")
print(f"Seaborn version: {sns.__version__}")
print(f"SciPy version: {stats.__version__}")
# Test a basic plot
data = np.random.randn(100)
plt.figure(figsize=(8, 6))
sns.histplot(data, kde=True)
plt.title("Sample Histogram")
plt.show()
``````
This script will import the essential libraries and create a simple histogram plot to ensure that everything is working correctly.
## 6. Project Structure
Organize your project directory to keep your code and data organized. A suggested structure is:
``````your_project_name/
??? data/
? ??? raw/
? ??? processed/
??? notebooks/
? ??? exploratory_data_analysis.ipynb
??? scripts/
? ??? data_preprocessing.py
??? requirements.txt
``````
This setup will provide a clean and manageable workspace for your EDA project.
By following these steps, you will have a robust Python environment set up for performing effective Exploratory Data Analysis using best practices and powerful tools.
# 3. Data Collection and Cleaning Techniques
## Data Collection
In the real world, data collection might come from multiple sources like databases, CSV files, Excel files, APIs, etc. Below, you’ll find an example of how to collect data from a CSV file and an API.
### Collecting Data from a CSV File
``````import pandas as pd
# Load the CSV file into a DataFrame
file_path = 'path_to_your_file.csv'
``````
### Collecting Data from an API
``````import requests
import pandas as pd
# API endpoint
url = 'https://api.example.com/data'
# Send GET request
response = requests.get(url)
# Check if the request was successful
if response.status_code == 200:
data = response.json()
df_api = pd.DataFrame(data)
else:
print('Failed to retrieve data:', response.status_code)
``````
## Data Cleaning
After collecting the data, it’s crucial to clean it to ensure a smooth EDA process. The following steps will often be involved in data cleaning:
1. Handling missing values
2. Removing duplicates
3. Data type conversion
4. Removing/handling outliers
5. Standardizing data formats
### Handling Missing Values
#### Dropping Missing Values
``````# Drop rows with any missing values
df_cleaned = df_csv.dropna()
``````
#### Filling Missing Values
``````# Fill missing values with the mean of the column
df_cleaned = df_csv.fillna(df_csv.mean())
``````
### Removing Duplicates
``````# Remove duplicate rows
df_cleaned = df_cleaned.drop_duplicates()
``````
### Data Type Conversion
``````# Convert column 'date' to datetime
df_cleaned['date'] = pd.to_datetime(df_cleaned['date'])
# Convert 'price' column to float
df_cleaned['price'] = df_cleaned['price'].astype(float)
``````
### Removing/Handling Outliers
``````# Define a function to remove outliers based on Z-score
from scipy import stats
def remove_outliers(df, col):
z_scores = stats.zscore(df[col])
abs_z_scores = abs(z_scores)
filtered_entries = (abs_z_scores < 3).all(axis=1)
return df[filtered_entries]
df_cleaned = remove_outliers(df_cleaned, ['price'])
``````
### Standardizing Data Formats
``````# Convert column 'category' to lowercase
df_cleaned['category'] = df_cleaned['category'].str.lower()
# Strip any leading/trailing whitespace from text columns
df_cleaned['category'] = df_cleaned['category'].str.strip()
``````
Putting It All Together:
``````import pandas as pd
import requests
from scipy import stats
# Load the CSV file into a DataFrame
file_path = 'path_to_your_file.csv'
# Collecting Data from an API
url = 'https://api.example.com/data'
response = requests.get(url)
if response.status_code == 200:
data = response.json()
df_api = pd.DataFrame(data)
else:
print('Failed to retrieve data:', response.status_code)
# Assuming you are concatenating data from CSV and API
df = pd.concat([df_csv, df_api], ignore_index=True)
# Clean the data
df = df.dropna() # Handle missing values
df = df.drop_duplicates() # Remove duplicates
df['date'] = pd.to_datetime(df['date']) # Convert to correct data types
df['price'] = df['price'].astype(float)
df = remove_outliers(df, ['price']) # Remove outliers
df['category'] = df['category'].str.lower().str.strip() # Standardize formats
# Now df is clean and ready for EDA
``````
The above implementation shows practical techniques for data collection and cleaning that can be directly applied in a Python environment.
# Understanding and Handling Missing Data
### Introduction
Handling missing data is a crucial part of any data analysis process. It can impact the performance and validity of your model. This section provides practical ways to detect and handle missing data using Python.
### Detecting Missing Data
To identify missing data, you can use functions from pandas which is a powerful data manipulation library in Python.
``````import pandas as pd
# Summary of missing values
missing_values_summary = df.isnull().sum()
print(missing_values_summary)
# Percentage of missing values
missing_percentage = (df.isnull().mean() * 100).round(2)
print(missing_percentage)
# Visual summary
import seaborn as sns
import matplotlib.pyplot as plt
plt.figure(figsize=(10, 6))
sns.heatmap(df.isnull(), cbar=False, cmap='viridis')
plt.title('Heatmap of Missing Values')
plt.show()
``````
### Handling Missing Data
#### 1. Removing Missing Data
Sometimes, it makes sense to simply remove data with missing values.
``````# Remove rows with any missing values
df_dropna = df.dropna()
# Remove columns with any missing values
df_dropna_columns = df.dropna(axis=1)
``````
#### 2. Filling Missing Data
You can fill in missing data with different strategies such as mean, median, mode, or a specific value.
Filling with Specific Value:
``````# Fill missing values with 0
df_fillna = df.fillna(0)
``````
Filling with Mean/Median/Mode:
``````# Fill numerical columns with mean
df_filled_mean = df.fillna(df.mean())
# Fill numerical columns with median
df_filled_median = df.fillna(df.median())
# Fill categorical columns with mode
for column in df.select_dtypes(include=['object']).columns:
df[column].fillna(df[column].mode()[0], inplace=True)
``````
Forward Fill / Backward Fill:
``````# Forward fill
df_ffill = df.fillna(method='ffill')
# Backward fill
df_bfill = df.fillna(method='bfill')
``````
For more sophisticated imputation, you can use machine learning algorithms.
Example using Scikit-Learn’s IterativeImputer:
``````from sklearn.experimental import enable_iterative_imputer
from sklearn.impute import IterativeImputer
iterative_imputer = IterativeImputer()
df_imputed = iterative_imputer.fit_transform(df)
# Convert back to DataFrame
df_imputed = pd.DataFrame(df_imputed, columns=df.columns)
``````
### Conclusion
Missing data can be handled in various ways depending on the nature of your dataset and your analysis goals. The above methods provide practical implementations for detecting and handling missing data using Python, ensuring that you maintain the quality of your analysis.
# Data Visualization with Matplotlib and Seaborn
## Importing Required Libraries
``````import matplotlib.pyplot as plt
import seaborn as sns
import pandas as pd
import numpy as np
``````
## Initial Steps
Ensure you have your data ready, as this will be the starting point for the visualizations. Below is an example using a generic DataFrame `df`.
``````# Example DataFrame
df = pd.DataFrame({
'A': np.random.randn(100),
'B': np.random.randn(100),
'C': np.random.randn(100),
'D': np.random.randn(100),
'Category': np.random.choice(['Category1', 'Category2'], 100)
})
``````
## Matplotlib Visualizations
### Line Plot
``````plt.figure(figsize=(10, 6))
plt.plot(df['A'], label='Line A')
plt.plot(df['B'], label='Line B')
plt.xlabel('Index')
plt.ylabel('Values')
plt.title('Line Plot for A and B')
plt.legend()
plt.show()
``````
### Scatter Plot
``````plt.figure(figsize=(10, 6))
plt.scatter(df['A'], df['B'], c='blue', alpha=0.5)
plt.xlabel('A')
plt.ylabel('B')
plt.title('Scatter Plot between A and B')
plt.show()
``````
### Histogram
``````plt.figure(figsize=(10, 6))
plt.hist(df['A'], bins=30, alpha=0.5, label='A')
plt.hist(df['B'], bins=30, alpha=0.5, label='B')
plt.xlabel('Value')
plt.ylabel('Frequency')
plt.title('Histogram of A and B')
plt.legend()
plt.show()
``````
## Seaborn Visualizations
### Pair Plot
``````sns.pairplot(df)
plt.suptitle('Pair Plot of the DataFrame', y=1.02)
plt.show()
``````
### Box Plot
``````plt.figure(figsize=(10, 6))
sns.boxplot(x='Category', y='A', data=df)
plt.xlabel('Category')
plt.ylabel('A')
plt.title('Box Plot of A by Category')
plt.show()
``````
### Heatmap
``````plt.figure(figsize=(10, 6))
correlation_matrix = df.corr()
sns.heatmap(correlation_matrix, annot=True, cmap='coolwarm')
plt.title('Heatmap of Correlation Matrix')
plt.show()
``````
### Count Plot
``````plt.figure(figsize=(10, 6))
sns.countplot(x='Category', data=df)
plt.xlabel('Category')
plt.ylabel('Count')
plt.title('Count Plot of Categories')
plt.show()
``````
These examples cover a variety of common visualizations that can be used to explore and understand your data. Each plot provides unique insights and can help highlight different aspects of the dataset.
# Descriptive Statistics and Summary Measures in Python
## Import Necessary Libraries
``````import pandas as pd
import numpy as np
``````
``````# Assume we have a file called 'data.csv'
``````
## General Overview of the Data
``````# Display the first few rows of the data
# Display basic information about the dataset
print(df.info())
# Display summary statistics for numerical columns
print(df.describe())
# Display summary statistics for categorical columns
print(df.describe(include=[object]))
``````
## Descriptive Statistics Functions
``````# Mean, Median, Mode for a specific column
column_name = 'your_column'
mean_value = df[column_name].mean()
median_value = df[column_name].median()
mode_value = df[column_name].mode()[0]
print(f"Mean: {mean_value}, Median: {median_value}, Mode: {mode_value}")
# Variance and Standard Deviation
variance_value = df[column_name].var()
std_dev_value = df[column_name].std()
print(f"Variance: {variance_value}, Standard Deviation: {std_dev_value}")
# Skewness and Kurtosis
skewness_value = df[column_name].skew()
kurtosis_value = df[column_name].kurt()
print(f"Skewness: {skewness_value}, Kurtosis: {kurtosis_value}")
``````
## Summary Statistics for the Entire DataFrame
``````# Custom function to compute summary statistics
def summary_statistics(dataframe):
summary = pd.DataFrame()
summary['Mean'] = dataframe.mean()
summary['Median'] = dataframe.median()
summary['Mode'] = dataframe.mode().iloc[0]
summary['Variance'] = dataframe.var()
summary['Standard Deviation'] = dataframe.std()
summary['Skewness'] = dataframe.skew()
summary['Kurtosis'] = dataframe.kurt()
return summary
# Apply the function to the DataFrame
summary_stats = summary_statistics(df.select_dtypes(include=[np.number]))
print(summary_stats)
``````
``````# Minimum and Maximum values
min_values = df.min()
max_values = df.max()
print(f"Minimum values:\n{min_values}\n")
print(f"Maximum values:\n{max_values}\n")
# Quantiles
quantiles = df.quantile([0.25, 0.5, 0.75])
print(f"Quantiles:\n{quantiles}\n")
``````
## Handling Outliers
``````# Interquartile Range (IQR) to detect outliers for a specific column
Q1 = df[column_name].quantile(0.25)
Q3 = df[column_name].quantile(0.75)
IQR = Q3 - Q1
lower_bound = Q1 - 1.5 * IQR
upper_bound = Q3 + 1.5 * IQR
outliers = df[(df[column_name] < lower_bound) | (df[column_name] > upper_bound)]
print(f"Outliers in {column_name}:\n{outliers}")
``````
This comprehensive implementation covers how to compute and display descriptive statistics and summary measures using Python on your DataFrame. You can expand and apply these techniques to any dataset as needed.
### 7. Exploratory Data Analysis with Pandas
``````import pandas as pd
# 1. Preview the Data
print("Data Preview:")
# 2. Data Info
print("\nData Info:")
print(df.info())
# 3. Check for Missing Values
print("\nMissing Values:")
print(df.isnull().sum())
# 4. General Descriptive Statistics
print("\nDescriptive Statistics:")
print(df.describe())
# 5. Unique Values for Categorical Features
print("\nUnique Values for Categorical Features:")
categorical_features = df.select_dtypes(include=['object']).columns
for col in categorical_features:
print(f"{col}: {df[col].unique()}")
# 6. Correlation Matrix
print("\nCorrelation Matrix:")
print(df.corr())
# 7. Detecting Outliers
from scipy import stats
print("\nOutlier Detection:")
z_scores = stats.zscore(df.select_dtypes(include=[float, int]))
abs_z_scores = abs(z_scores)
outliers = (abs_z_scores > 3).all(axis=1)
print("Outliers detected (rows):")
print(df[outliers])
# 8. Value Counts for Key Features
print("\nValue Counts for Key Features:")
for col in ['important_feature_1', 'important_feature_2']: # Replace with your key features
print(f"Value counts for {col}:")
print(df[col].value_counts())
# 9. Grouped Statistics
print("\nGrouped Statistics:")
grouped = df.groupby('categorical_feature') # Replace with your categorical feature
print(grouped['numerical_feature'].agg(['mean', 'median', 'std']))
# 10. Further Exploration
# Example: Checking distribution of numerical features
import matplotlib.pyplot as plt
df.hist(bins=30, figsize=(15, 10))
plt.suptitle("Distribution of Numerical Features")
plt.show()
``````
### Summary of Steps
1. Preview the Data: Look at the first few rows to understand the structure.
2. Data Info: Get a concise summary of the DataFrame.
3. Check for Missing Values: Identify columns with missing values.
4. General Descriptive Statistics: Summary statistics for numerical columns.
5. Unique Values for Categorical Features: Examine unique values in categorical columns.
6. Correlation Matrix: Check correlations between numerical columns.
7. Detecting Outliers: Use Z-scores to identify outlier rows.
8. Value Counts for Key Features: Count occurrences of values in specified columns.
9. Grouped Statistics: Aggregate statistics based on a categorical feature.
10. Further Exploration: Plot distributions of numerical features for deeper insights.
Apply each step systematically to unearth valuable patterns and relationships within the dataset.
# Univariate and Bivariate Analysis
## 1. Univariate Analysis
Univariate analysis involves examining each variable individually. The best practice is to understand the distribution, central tendency, and spread of the data.
### Practical Implementation:
``````import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
# Assuming `df` is your DataFrame
# Summary statistics
summary_stats = df.describe()
print(summary_stats)
# Distribution of a single variable (numerical)
sns.histplot(df['numerical_column'])
plt.title('Distribution of Numerical Column')
plt.xlabel('Numerical Column')
plt.ylabel('Frequency')
plt.show()
# Distribution of a single variable (categorical)
sns.countplot(x='categorical_column', data=df)
plt.title('Count of Categorical Column')
plt.xlabel('Categorical Column')
plt.ylabel('Count')
plt.show()
``````
## 2. Bivariate Analysis
Bivariate analysis involves the simultaneous analysis of two variables, typically to discover relationships. This can be done using scatter plots, bar plots, or correlation matrices.
### Practical Implementation:
``````# Scatter plot for two numerical variables
sns.scatterplot(x='numerical_column_1', y='numerical_column_2', data=df)
plt.title('Scatter Plot of Numerical Column 1 vs Numerical Column 2')
plt.xlabel('Numerical Column 1')
plt.ylabel('Numerical Column 2')
plt.show()
# Box plot for a numerical vs categorical variable
sns.boxplot(x='categorical_column', y='numerical_column', data=df)
plt.title('Box Plot of Numerical Column by Categorical Column')
plt.xlabel('Categorical Column')
plt.ylabel('Numerical Column')
plt.show()
# Correlation matrix for numerical variables
correlation_matrix = df.corr()
print(correlation_matrix)
# Heatmap of the correlation matrix
sns.heatmap(correlation_matrix, annot=True, cmap='coolwarm', linewidths=0.5)
plt.title('Heatmap of Correlation Matrix')
plt.show()
``````
Now you can apply these practical implementations to perform univariate and bivariate analysis on your dataset effectively. This will help in better understanding the data and revealing any possible relationships or patterns.
# Correlation and Causation Analysis
In this section, we will focus on practical implementation steps to analyze correlation and causation using Python. We will use `Pandas` for data manipulation, `Seaborn` and `Matplotlib` for visualization, and `Statsmodels` for statistical analysis.
## Correlation Analysis
### Step 1: Import Libraries
``````import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
import statsmodels.api as sm
import numpy as np
``````
For our example, assume you have a dataset named `data.csv`.
``````df = pd.read_csv('data.csv')
``````
### Step 3: Calculate Correlation Matrix
Use the `.corr()` method to calculate the correlation coefficients.
``````correlation_matrix = df.corr()
``````
### Step 4: Visualize Correlation Matrix
Use a heatmap to visualize the correlation matrix.
``````plt.figure(figsize=(10,8))
sns.heatmap(correlation_matrix, annot=True, cmap='coolwarm', fmt=".2f")
plt.title('Correlation Matrix')
plt.show()
``````
## Causation Analysis
Correlation does not imply causation. To explore causation, we use statistical methods such as Linear Regression.
### Step 1: Define Variables
Assuming `x` is the independent variable and `y` is the dependent variable.
``````X = df['x']
y = df['y']
``````
Add a constant to the independent variable set for statistical purposes.
``````X = sm.add_constant(X)
``````
### Step 3: Fit Linear Regression Model
``````model = sm.OLS(y, X).fit()
``````
### Step 4: Summarize the Model
Get a summary of the regression model to analyze the significance and coefficients.
``````print(model.summary())
``````
### Step 5: Analyze Residuals
Check the distribution of residuals to validate the assumptions of linear regression.
``````residuals = model.resid
sns.histplot(residuals, kde=True)
plt.title('Residuals Distribution')
plt.show()
``````
### Step 6: Visualize Regression Line
``````plt.figure(figsize=(10,6))
sns.regplot(x='x', y='y', data=df, line_kws={"color":"r","alpha":0.7,"lw":2})
plt.title('Regression Line')
plt.show()
``````
### Step 7: Conduct Granger Causality Test (if applicable)
For time series data, you might want to conduct a Granger Causality Test.
``````from statsmodels.tsa.stattools import grangercausalitytests
# Assume 'data' contains time series data with columns 'x' and 'y'
grangercausalitytests(data[['x', 'y']], maxlag=5)
``````
This test checks if past values of one variable contain information that helps predict another variable.
# Conclusion
This section provided a thorough, practical implementation of correlation and causation analysis using Python. Ensure you interpret the results correctly and understand the implications of the statistical outputs.
# Reporting and Communicating EDA Results
After conducting Exploratory Data Analysis (EDA), effectively reporting and communicating your results is crucial. It ensures that findings are easily understood by stakeholders. Below is a practical implementation highlighting key methods to succinctly report and communicate your EDA results using Python.
## 1. Generate Summary Reports
### 1.1. Summary Statistics
You can use Pandas to create a summary table of your dataset’s key statistics.
``````import pandas as pd
# Assuming df is your DataFrame
summary_stats = df.describe().transpose()
summary_stats.to_csv('summary_statistics.csv') # Save to a CSV file
print(summary_stats)
``````
## 2. Visual Reports
Visualizations are an effective way to communicate your findings. You can use Matplotlib and Seaborn for generating various plots.
### 2.1. Histograms
``````import matplotlib.pyplot as plt
df.hist(figsize=(10, 8))
plt.tight_layout()
plt.savefig('histograms.png') # Save the figure
plt.show()
``````
### 2.2. Correlation Heatmap
``````import seaborn as sns
plt.figure(figsize=(12, 10))
correlation_matrix = df.corr()
sns.heatmap(correlation_matrix, annot=True, cmap='coolwarm', linewidths=0.5)
plt.title('Correlation Matrix Heatmap')
plt.savefig('correlation_heatmap.png') # Save the figure
plt.show()
``````
### 2.3. Boxplots
``````plt.figure(figsize=(10, 8))
sns.boxplot(data=df)
plt.title('Boxplot of Variables')
plt.savefig('boxplot.png') # Save the figure
plt.show()
``````
## 3. Detailed EDA Report using Pandas Profiling
For a comprehensive and interactive EDA report, you can use `pandas_profiling`.
``````from pandas_profiling import ProfileReport
# Generate the report
profile = ProfileReport(df, title='EDA Report', explorative=True)
profile.to_file('eda_report.html') # Save as an HTML file
``````
## 4. Creating a Jupyter Notebook Report
Using Jupyter Notebooks to combine narrative text, code, and visualizations is a powerful way to report your findings.
``````# In a Jupyter Notebook cell
# Imports and initial setup
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
from pandas_profiling import ProfileReport
# Load and explore the data
# Display summary statistics
print(df.describe().transpose())
# Generate and display histograms
df.hist(figsize=(10, 8))
plt.tight_layout()
plt.show()
# Generate and display correlation heatmap
plt.figure(figsize=(12, 10))
sns.heatmap(df.corr(), annot=True, cmap='coolwarm', linewidths=0.5)
plt.title('Correlation Matrix Heatmap')
plt.show()
# Generate an interactive EDA report
profile = ProfileReport(df, title='EDA Report', explorative=True)
profile
# Save the report
profile.to_file('eda_report.html')
``````
## 5. Automating Report Generation in Python Script
### `eda_report.py`
``````# eda_report.py
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
from pandas_profiling import ProfileReport
def generate_eda_report(data_path):
# Summary Statistics
summary_stats = df.describe().transpose()
summary_stats.to_csv('summary_statistics.csv')
# Histograms
df.hist(figsize=(10, 8))
plt.tight_layout()
plt.savefig('histograms.png')
# Correlation Heatmap
plt.figure(figsize=(12, 10))
sns.heatmap(df.corr(), annot=True, cmap='coolwarm', linewidths=0.5)
plt.title('Correlation Matrix Heatmap')
plt.savefig('correlation_heatmap.png')
# Boxplot
plt.figure(figsize=(10, 8))
sns.boxplot(data=df)
plt.title('Boxplot of Variables')
plt.savefig('boxplot.png')
# Pandas Profiling Report
profile = ProfileReport(df, title='EDA Report', explorative=True)
profile.to_file('eda_report.html')
if __name__ == "__main__":
generate_eda_report('your_data.csv')
``````
Execute the script with:
``````python eda_report.py
``````
By the end of this process, you will have multiple artifacts (`summary_statistics.csv`, `histograms.png`, `correlation_heatmap.png`, `boxplot.png`, `eda_report.html`) to communicate your EDA results effectively.
# Final Thoughts
Exploratory Data Analysis (EDA) is a crucial step in the data science workflow that allows analysts and data scientists to gain valuable insights from their datasets. Throughout this comprehensive guide, we’ve covered the essential aspects of performing EDA using Python, from setting up the environment to advanced analysis techniques and effective reporting.
By mastering the tools and techniques discussed in this blog post, including data loading, cleaning, visualization, and statistical analysis, you’ll be well-equipped to tackle complex datasets and uncover meaningful patterns. Remember that EDA is an iterative process, and the insights gained often lead to new questions and further exploration.
As you apply these best practices in your projects, keep in mind that the goal of EDA is not just to generate statistics and plots, but to develop a deep understanding of your data. This understanding will inform your subsequent modeling decisions and help you communicate your findings effectively to stakeholders.
Whether you’re a beginner or an experienced data professional, continual practice and experimentation with different datasets will help you refine your EDA skills. As the field of data science evolves, stay curious and open to learning new techniques and tools that can enhance your exploratory analysis capabilities.
By leveraging the power of Python and its rich ecosystem of data analysis libraries, you’re now ready to dive deep into your data, ask insightful questions, and extract valuable knowledge that can drive informed decision-making in your organization.
## Analyzing Sales Data for Retail Business
A comprehensive business intelligence project in Python using Google Colab to analyze and visualize retail sales data.
## Mastering Efficient Management of Colab Notebooks
A comprehensive guide to maximizing productivity and efficiency in managing Google Colab Notebooks.
## Mastering Collaboration in Google Colab
A comprehensive guide to effectively collaborate using Google Colab.
## Data Visualization Basics in Google Colab
A comprehensive guide to mastering data visualization using Google Colab.
## Basic Data Manipulation in Google Colab
Learn the fundamentals of data manipulation using Python in Google Colab.
## Writing and Running Code Cells in Google Colab
This guide introduces beginners to the basics of using Google Colab for coding and computation.
## Google Colab Interface Deep Dive
A comprehensive guide to learning and navigating the Google Colab environment.
## Mastering the Google Colab Environment Setup
A thorough guide to effectively setting up and utilizing the Google Colab environment.
## A Comprehensive Guide to Google Colab
Learn the basics and advanced techniques of Google Colab for Python programming.
## Google Colab Best Practices for Optimal Usage
An exhaustive guide to effectively use Google Colab for data analysis, machine learning, and more. | 7,287 | 31,187 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2024-33 | latest | en | 0.673233 |
https://freethoughtblogs.com/pharyngula/2019/01/24/algorithms-are-only-magical-oracles-if-you-dont-understand-them/ | 1,718,464,088,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861594.22/warc/CC-MAIN-20240615124455-20240615154455-00760.warc.gz | 230,359,729 | 20,388 | # Algorithms are only magical oracles if you don’t understand them
I saw a lot of news flashes about twins comparing DNA testing services and finding that they weren’t perfectly identical, and that the services didn’t produce identical results. I didn’t bother to look any deeper, because yes, identical twins do have a small number of genetic differences, and those testing services don’t sequence your genome, they rely on chips to identify some short sequences from a subset of your genome, and there is naturally some sampling error in the process. So this shouldn’t be a surprise.
Fortunately, Larry Moran explains the sources of error.
The main problem by far is due to the way the tests are done which is by hybridizing the customers’ DNA to DNA on a microchip and reading the chip to see if there’s a match. (Ancestry.com uses the latest Illumina microchip that assays 700,000 SNPs.) I think the rate of false positives is quite low but the rate of false negatives is about 2% according to 23andMe. The absence of a match where there should be one can be due to bad luck and differences in the threshold level of binding that constitutes a “hit.” It’s these “no-reads” that makes up most of the false negatives. Because of these limitations of the assay the twins’ DNA results could differ by 2-4% of the SNPs being tested.
So no surprise that they reported some variation. What I found odd is that anyone found this odd at all.
The different testing services also reported different patterns of ancestry. Why would anyone find that to be unexpected?
While he can’t say for certain what accounts for the difference, Gerstein suspects it has to do with the algorithms each company uses to crunch the DNA data.
“The story has to be the calculation. The way these calculations are run are different.”
Heh. I believe I’ve mentioned this very point here: that saying something is an “algorithm” doesn’t mean it’s bias-free. The inputs and the weights on the data and the processing used are all choices by the person who designed the algorithms, and different companies will have different pools of data they are drawing on to make their decisions. Some people don’t get that, though.
This should be used as a nice example of how datasets and algorithms can color the interpretation of data. Maybe we’ll see fewer asshats buying into digital reductionism, as if everything that comes out of a computer is inarguable truth.
1. call me mark says
I take it that Saavedra has never heard of “Garbage in – Garbage out”
2. says
BTW, Saavedra has apparently complained extensively that social media algorithms are biased against Conservatives.
3. ardipithecus says
He’s doing us a favour by demonstrating his complete and utter failure to understand how data crunching algorithms work.
4. whheydt says
My wife wandered past and commented that the title for this post was an application of Clarke’s Third Law. (And for those unfamiliar with it, it’s “Any sufficiently advanced technology is indistinguishable from magic.”)
5. mond says
Funny how the algorithm I use to make a cup of tea seems to produce a cuppa just the way I like.
It’s almost as if I put my own preferences and biases into the algorithm.
6. Artor says
“Socialist Rep. Alexandria Ocasio-Cortez (D-NY) claims that algorithms, which are driven by math, are racist.”
Algorithms- driven by math, but written by people.
7. robert79 says
One of my main frustrations about reporting is that many media (both online and offline) seem to think the word “algorithm” is a synonym for “something that cannot be understood”, “stuff with maths” or “magic”. The long division you learned in elementary school is an algorithm, as is the algorithm for a cup of tea that mond described above.
I forbid my maths students to use the word algorithm unless they can actually describe the algorithm.
8. chris61 says
This should be used as a nice example of how datasets and algorithms can color the interpretation of data.
Clever.
9. dexitroboper says
There is a mathematical term for bias, it’s called weighting.
10. chrislawson says
Artor@6–
Algorithms are not necessarily driven by math. Algorithms are simply sets of discrete, ordered* instructions. Algorithms are crucially important in mathematics and vice versa, but do not map 1:1. My definition of an algorithm is something along these lines, adapted from this hackernoon article:
Algorithm: a sequence of steps that describes an idea for solving a problem, an abstract recipe for a solution independent of implementation.
vs.
Code: a set of instructions for a computer. A concrete implementation of the algorithm on a specific platform in a specific programming language.
An example of non-mathematical algorithms is Western sheet music, which tells you the sequence of notes to be played and includes control commands (e.g. time signature, key, dynamic instructions like “diminuendo”) and recursive elements like repeat brackets.
*the order does not have to be linear; almost all algorithms in practical use are non-linear and multiply recursive.
11. bryanfeir says
My favourite story about this was something I heard back in University (so over 25 years ago now). I may have mentioned it before. And no, I don’t remember enough details to be able to pin this down, so feel free to take this with however many grains of salt required.
So, a school’s grad program had been under fire for racism and sexism in their selection process. One common method of outside checking for this sort of thing involves sending c.v.’s that are pretty much identical except for the names, and seeing if the ‘obviously’ female or non-WASP names get lower rates of follow-ups. This school had failed a test like that pretty badly, and decided to try and automate the first pass of the selection and remove the human element from it, as part of renovating their image.
So, what they did was they took a bunch of their old c.v.’s and selection criteria, fed them into a neural network along with whether or not the student was accepted, and trained the neural network in which sets of criteria were acceptable for further review by a human. (Some of you have already figured out where this is going.)
When the same sort of test was performed on the new system, it failed again. The administrators claimed that wasn’t possible, because the machine couldn’t be racist or sexist. But, of course, when they trained the neural network, they had included the whole c.v…. which included the name of the applicant. So the neural network had learned that given two otherwise identical c.v.’s, the one with the female or non-WASP-sounding name was to be given lower priority.
Needless to say, the network had to be completely retrained with anonymized data.
12. mikehuben says
An excellent expert on algorithmic biases is Cathy O’Neill, author of “Weapons of Math Destruction.”
13. methuseus says
@mikehuben #12:
I just started reading this a week ago thanks to PZ mentioning someone hating having to read her book for a data science class.
As I was thinking, this just again goes to this whole thing: people are actually surprised that we can write programs that reflect our internal biases??
14. John Morales says
methuseus, yes, because algorithms are but information machines, and machines don’t have consciousness, so they can’t be biased.
(The information upon which they are constituted, though…)
15. I second mikehuben’s recommendation for “Weapons of Math Destruction.” A math seminar class I took last year at Central Washington University used it as a launching point to discuss how technology that aims to end bias can actually make it more deeply entrenched by hiding it behind a black box. Who gets paroled, who gets employed, who gets an apartment, who gets in to school… it’s really insidious. | 1,662 | 7,829 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2024-26 | latest | en | 0.947884 |
https://www.scribd.com/doc/111545987/Solv-Samp | 1,521,403,990,000,000,000 | text/html | crawl-data/CC-MAIN-2018-13/segments/1521257645943.23/warc/CC-MAIN-20180318184945-20180318204945-00153.warc.gz | 902,846,911 | 27,392 | # 115108768.xls.
ms_office Quick Tour of Microsoft Excel Solver
Month Seasonality Units Sold Sales Revenue Cost of Sales Gross Margin Salesforce Advertising Corp Overhead Total Costs Prod. Profit Profit Margin Product Price Product Cost Q1
0.9 3,592 \$143,662 89,789 53,873 8,000 10,000 21,549 39,549 \$14,324 10% \$40.00 \$25.00
Q2
1.1 4,390 \$175,587 109,742 65,845 8,000 10,000 26,338 44,338 \$21,507 12%
Q3
0.8 3,192 \$127,700 79,812 47,887 9,000 10,000 19,155 38,155 \$9,732 8%
Q4
1.2 4,789 \$191,549 119,718 71,831 9,000 10,000 28,732 47,732 \$24,099 13%
Total
15,962 \$638,498 399,061 239,437 34,000 40,000 95,775 169,775 \$69,662 11%
Color Coding Target cell Changing cells Constraints
The following examples show you how to work with the model above to solve for one value or several values to maximize or minimize another value, enter and change constraints, and save a problem model. Row 3 5 Contains Fixed values Explanation Seasonality factor: sales are higher in quarters 2 and 4, and lower in quarters 1 and 3.
=35*B3*(B11+3000)^0.5 Forecast for units sold each quarter: row 3 contains the seasonality factor; row 11 contains the cost of advertising. =B5*\$B\$18 =B5*\$B\$19 =B6-B7 Fixed values Fixed values =0.15*B6 =SUM(B10:B12) =B8-B13 =B15/B6 Fixed values Fixed values Sales revenue: forecast for units sold (row 5) times price (cell B18). Cost of sales: forecast for units sold (row 5) times product cost (cell B19). Gross margin: sales revenues (row 6) minus cost of sales (row 7). Sales personnel expenses. Advertising budget (about 6.3% of sales). Corporate overhead expenses: sales revenues (row 6) times 15%. Total costs: sales personnel expenses (row 10) plus advertising (row 11) plus overhead (row 12). Product profit: gross margin (row 8) minus total costs (row 13). Profit margin: profit (row 15) divided by sales revenue (row 6). Product price. Product cost.
6 7 8 10 11 12 13 15 16 18 19
This is a typical marketing model that shows sales rising from a base figure (perhaps due to the sales personnel) along with increases in advertising, but with diminishing returns. For example, the first \$5,000 of advertising in Q1 yields about 1,092 incremental units sold, but the next \$5,000 yields only about 775 units more. You can use Solver to find out whether the advertising budget is too low, and whether advertising should be allocated differently over time to take advantage of the changing seasonality factor.
Solving for a Value to Maximize Another Value
One way you can use Solver is to determine the maximum value of a cell by changing another cell. The two cells must be related through the formulas on the worksheet. If they are not, changing the value in one cell will not change the value in the other cell.
Page 1
115108768. so your problem is to determine the number of each product to build from the inventory on hand in order to maximize profits. speaker cones.9.155 Speaker 100 0 0 1 0 1 \$2. Parts are limited.ms_office Example 1: Product mix problem with diminishing profit margin. Number to build value must be greater than or equal to 0. This change also makes the problem linear.xls. using a common parts inventory of power supplies. Your company manufactures TVs. Number of parts used must be less than or equal to the number of parts in inventory. Used 450 250 800 450 600 200 100 500 200 400 By Product Total 100 1 1 2 1 2 Profits: \$4.0 to indicate that profit per unit remains constant with volume.732 \$10. and then click Solve again. the optimal solution will change. stereos and speakers. If you change H15 to 1. The formulas for profit per product in cells D17:F17 include the factor ^H15 to show that profit per unit diminishes with volume. which makes the problem nonlinear. Color Coding Target cell Changing cells TV set Part Name Chassis Picture Tube Speaker Cone Power Supply Electronics Number to Build-> Inventory No.095 Stereo 100 1 0 2 1 1 \$3. etc.208 Diminishing Returns Exponent: 0. Parts are in limited supply and you must determine the most profitable mix of products to build. each with a different profit margin per unit. H15 contains 0. Problem Specifications Target Cell Changing cells Constraints D18 D9:F9 C11:C15<=B11:B15 D9:F9>=0 Goal is to maximize profit. Units of each product to build. Page 4 .9 Constraints This model provides data for several products using common parts. But your profit per unit built decreases with volume because extra price incentives are needed to load the distribution channel.
xls.ms_office Target cell Changing cells Constraints Page 5 .115108768.
A problem of this type has an optimum solution at which amounts to ship are integers.xls. while not exceeding the supply available from each plant and meeting the demand from each metropolitan area. The problem is to determine the amounts to ship from each plant to each warehouse at minimum shipping cost in order to meet the regional demand. Total shipped must be less than or equal to supply at plant. Number to ship must be greater than or equal to 0. but it obviously costs more to ship goods over long distances than over short distances. Minimize the costs of shipping goods from production plants to warehouses near metropolitan demand centers. Problem Specifications Target cell Changing cells Constraints B20 C8:G10 B8:B10<=B16:B18 C12:G12>=C14:G14 C8:G10>=0 Goal is to minimize total shipping cost. Totals shipped to warehouses must be greater than or equal to demand at warehouses.ms_office Example 2: Transportation Problem. if all of the supply and demand constraints are integers. Carolina Tennessee Arizona Shipping: 310 260 280 \$83 10 6 3 \$19 8 5 4 \$17 6 4 5 \$15 5 3 5 \$13 4 6 9 \$19 The problem presented in this model involves the shipment of goods from three plants to five regional warehouses. Amount to ship from each plant to each warehouse. Number to ship from plant x to warehouse y (at intersection): San Fran Denver Chicago Dallas New York 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ----------3 3 3 3 3 Color Coding Target cell Changing cells Constraints Plants: S. Carolina Tennessee Arizona Totals: Total 5 5 5 Demands by Whse --> 180 80 200 160 220 Plants: Supply Shipping costs from plant x to warehouse y (at intersection): S. Goods can be shipped from any plant to any warehouse.115108768. You can solve this problem faster by selecting the Assume linear model check box in the Solver Options dialog box before clicking Solve. Page 6 . while not exceeding the plant supplies.
ms_office Target cell Changing cells Constraints Page 7 .xls.115108768.
Employees working each day must be greater than or equal to the demand. Number of employees must be greater than or equal to 0. Sch. 1 means employee on that schedule works that day. Number of employees must be an integer. For employees working five consecutive days with two days off. Problem Specifications Target cell Changing cells Constraints D20 D7:D13 D7:D13>=0 D7:D13=Integer F15:L15>=F17:L17 Possible schedules Rows 7-13 Goal is to minimize payroll cost. Wed. Each employee works five consecutive days. In this example.280 The goal for this model is to schedule employees so that you have sufficient staff at the lowest cost. Thursday Thursday. Saturday Saturday. you use an integer constraint so that your solutions do not result in fractional numbers of employees on each schedule.. followed by two days off. Sunday Schedule Totals: Total Demand: Employees Sun Mon Tue Wed Thu Fri Sat Color Coding 4 4 4 6 6 4 4 32 0 1 1 1 1 1 0 24 22 0 0 1 1 1 1 1 24 17 1 0 0 1 1 1 1 24 13 1 1 0 0 1 1 1 22 14 1 1 1 0 0 1 1 20 15 1 1 1 1 0 0 1 22 18 1 1 1 1 1 1 0 28 24 Pay/Employee/Day: Payroll/Week: \$40 \$1. Wed. Selecting the Assume linear model check box in the Solver Options dialog box before you click Solve will greatly speed up the solution process. so by minimizing the number of employees working each day. Tuesday Tuesday. In this example. Employees on each schedule. all employees are paid at the same rate. you also minimize costs. Monday Monday. find the schedule that meets demand from attendance levels while minimizing payroll costs.Staff Scheduling Example 3: Personnel scheduling for an Amusement Park. A B C D E F G Days off Sunday. Page 8 . Friday Friday.
Staff Scheduling Target cell Changing cells Constraints Page 9 .
and cash needed for company operations for each month. and then it subtracts from this amount the total investment.000 100.400 (15.000 110. click Solve. are reinvested in new three-month CDs.000 -290000 (10. one of your tasks is to manage cash and short-term investments in a way that maximizes interest income.000 10. restore the original values and then click Solver on the Tools menu.ms_office Example 4: Working Capital Management. To add this constraint.700 End \$125. Ending cash must be greater than or equal to \$100. 2.000 1. and 6 months). weighted by the maturities (1. This solution satisfies all of the constraints. If this quantity is zero or less.000 (20.400 100. You have a total of nine decisions to make: the amounts to invest in one-month CDs in months 1 through 6. while keeping funds available to meet expenditures. outflows for new CDs.000 Interest Earned: \$7. type 0 in the Constraint box.000. weighted by 4.300 Color Coding 100. Suppose.000 Term 1 3 6 Month 2 \$205. and the amount to invest in six-month CDs in month 1. Yield 1. according to the present plan.000) \$125.400 120.000 1.0% 4. however. you can keep the cash instead of reinvesting. 3.000 100.000 10. This model calculates ending cash based on initial cash (from the previous month). Click Add.000 1.0% Month 1 \$400.000 \$187. The \$56.000 Month 3 \$216. The optimal solution determined by Solver earns a total interest income of \$16.000) \$237.531 by investing as much as possible in six-month and three-month CDs. Solver shifts funds from six-month CDs to three-month CDs. If you need the funds. the average maturity will not exceed four months.000 1. Problem Specifications Target cell Changing cells Constraints H8 B14:G14 B15. however.000 10. and then click OK.400 100.896 turning over in month 4 is more than sufficient for the equipment payment in month 5. E15.000 100. inflows from maturing certificates of deposit (CDs). To satisfy the four-month maturity constraint.400 60.000 100.000 75.000 \$109. Determine how to invest excess cash in 1-month. 4.000 1-mo CDs: 3-mo CDs: 6-mo CDs: Month: Init Cash: Matur CDs: Interest: 1-mo CDs: 3-mo CDs: 6-mo CDs: Cash Uses: End Cash: Total Month 6 \$109.000 100.000 \$205. Type b20 in the Cell Reference box.000 100. B15.0% 9.000 50.115108768.000) \$216. and then turns to one-month CDs.000 80.400 Month 5 \$158. Add a constraint that the average maturity of the investments held in month 1 should not be more than four months. B16 B14:G14>=0 B15:B16>=0 E15>=0 B18:H18>=100000 Goal is to maximize interest earned. The shifted funds now mature in month 4 and. that you want to guarantee that you have enough cash in month 5 for an equipment payment. Investment in each type of CD must be greater than or equal to 0.xls.400 100. 3.000 1. and B16). the amounts to invest in three-month CDs in months 1 and 4. You've traded about \$460 in interest income to gain this flexibility.000 Purchase CDs in months: 1. Dollars invested in each type of CD.700 If you're a financial officer or a manager. Page 10 .000 \$158. To solve the problem. 5 and 6 1 and 4 1 Month 4 \$237. 3-month and 6-month CDs so as to maximize interest income while meeting company cash requirements (plus safety margin). The formula in cell B20 computes a total of the amounts invested in month 1 (B14.000 2. You must trade off the higher interest rates available from longer-term investments against the flexibility provided by keeping funds in short-term investments.
115108768.xls.ms_office Page 11 .
115108768.xls.ms_office Color Coding Target cell Changing cells Constraints Page 12 .
you can earn a rate of return that represents the average of the returns from the individual stocks.0% 20. 0.002 0.070768 5 TRUE TRUE TRUE TRUE TRUE TRUE TRUE One of the basic principles of investment management is diversification. select cells D21:D29 on the worksheet.0% Return 16.160 3. Use Solver to try different allocations of funds to stocks and T-bills to either maximize the portfolio rate of return for a specified level of risk or minimize the risk for a given rate of return.00 Market variance Maximum weight Weight 20.360 0. while reducing your risk that any one stock will perform poorly. To load these problem specifications into Solver. Problem Specifications Target cell Changing cells Constraints E18 E10:E14 E10:E14>=0 E16=1 G18<=0.016 0.04 0.0% 20. As you can see.80 2. Initially equal amounts (20 percent of the portfolio) are invested in each security.4 percent and the variance is 7. you can also use the Markowitz method if you have covariance terms available.160 0. Weights must be greater than or equal to 0. Solver finds portfolio Page 13 . assumed to have a risk-free rate of return and a variance of zero. your portfolio includes investments in Treasury bills (T-bills). or that maximizes the rate of return for a given level of risk. Weight of each stock. the portfolio return is 16. Weights must equal 1.115108768. Find the weightings of stocks in an efficient portfolio that maximizes the portfolio rate of return for a given level of risk.071 Beta for each stock Variance for each stock B10:B13 C10:C13 Goal is to maximize portfolio return.1 percent.030 Variance 7.000 0. click Options. click Load Model. for example.ms_office Example 5: Efficient stock portfolio. By holding a portfolio of several stocks. In addition. With the initial allocation of 20 percent across the board.1% Color Coding Target cell Changing cells Constraints Total Portfolio Totals: Maximize Return: A21:A29 0.0% 20. This worksheet contains figures for beta (market-related risk) and residual variance for four stocks.0% 100. Using this model.12 0.0% 20.200 0. click Solver on the Tools menu.00 6.0% 100.0% *Var.005 0. This worksheet uses the Sharpe single-index model.4% *Beta 0.40 0.000 1.071.20 0. Click Solve.440 0.0% 15. and then click OK until the Solver Parameters dialog box is displayed. you can use Solver to find the allocation of funds to stocks that minimizes the portfolio risk for a given rate of return.80 1.008 0. Cells D21:D29 contain the problem specifications to minimize risk for a required rate of return of 16.xls. Risk-free rate Market rate Beta 0.1644 5 TRUE TRUE TRUE TRUE TRUE TRUE TRUE Stock A Stock B Stock C Stock D T-bills Minimize Risk: D21:D29 0.00 1.20 0. Variance must be less than or equal to 0.4 percent.0% ResVar 0.
select cells A21:A29 on the worksheet. These two allocations both represent efficient portfolios.ms_office allocations in both cases that surpass the rule of 20 percent across the board. click Solver on the Tools menu. and then click OK. Page 14 .xls. Cells A21:A29 contain the original problem model. click Options. You can earn a higher rate of return (17. To reload this problem. click Load Model. Solver displays a message asking if you want to reset the current Solver option settings with the settings for the model you are loading. Click OK to proceed.115108768.1 percent) for the same risk. or you can reduce your risk without giving up any return.
ms_office Example 6: Value of a resistor in an electrical circuit.9375 3. Resistor. resistor. This problem and solution are appropriate for a narrow range of values. With the switch in the left position.07203653 -0.115108768. The formula relates the charge q[t] at time t to the inductance L.973947 0. the battery charges the capacitor. Use Solver to pick an appropriate value for the resistor R (given values for the inductor L and the capacitor C) that will dissipate the charge to one percent of its initial value within one-twentieth of a second after the time the switch is thrown. Switch-> q0 = q[t] = t= L= C= R= 9 0.52445064 300 ohms q[t] = 0. resistance R.05 8 0 volts volts seconds henrys farads Color Coding Target cell Changing cells Constraints Battery Capacitor (C) Inductor (L) Resistor (R) 1/(L*C) (R/(2*L))^2 SQRT(B15-B16) COS(T*B17) -R*T/(2*L) Q0*EXP(B19) 1250 351. Problem Specifications Target cell Changing cell Constraints G15 G12 D15:D20 Goal is to set to value of 0. and inductor.25 This model depicts an electrical circuit containing a battery. capacitor. you can formulate and solve a differential equation to determine how the charge on the capacitor varies over time. Using Kirchhoff's second law. the function represented by the charge on the capacitor over time is actually a damped sine wave.xls. the capacitor discharges through the inductor and the resistor.09 0. Algebraic solution to Kirchhoff's law. When the switch is thrown to the right. Page 15 . Find the value of a resistor in an electrical circuit that will dissipate the charge to 1 percent of its original value within one-twentieth of a second after the switch is closed. switch.09. and capacitance C of the circuit elements. both of which dissipate electrical energy.5625 29. | 4,438 | 17,084 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.390625 | 3 | CC-MAIN-2018-13 | latest | en | 0.739369 |
http://www.hexdictionary.com/hex/2D5 | 1,481,227,256,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698542655.88/warc/CC-MAIN-20161202170902-00316-ip-10-31-129-80.ec2.internal.warc.gz | 507,215,806 | 2,618 | As Decimal: 725
As Binary: 1011010101
This value can also be written as 0x2D5. This is correct hexadecimal syntax
The value 2 occurs 1 times. The value D occurs 1 times. The value 5 occurs 1 times.
The decimal representation of the hexadecimal value 2D5 is 725. The binary representation of the hexadecimal value 2D5 is 1011010101
The Number value of the hexadecimal value 2D5 is 725, or 2D5 in hexadecimal
The Cosine value of the hexadecimal value 2D5 is -0.75972711649105, or 0 in hexadecimal
The Hyperbolic Cosine value of the hexadecimal value 2D5 is INF, or 8000000000000000 in hexadecimal
The Binary value of the hexadecimal value 2D5 is 1011010101, | 193 | 659 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2016-50 | longest | en | 0.703631 |
http://www.transum.org/Maths/Exercise/Graph/Graph_Points.asp?Level=1 | 1,544,772,284,000,000,000 | text/html | crawl-data/CC-MAIN-2018-51/segments/1544376825495.60/warc/CC-MAIN-20181214070839-20181214092339-00016.warc.gz | 508,525,598 | 11,155 | # Equation of a Line Through Points
## Match the equations of the straight line graphs to the clues about gradients and points.
##### Level 1Level 2Level 3Level 4MixedExam-StyleDescriptionHelpMore...
This is level 1: find the equation of the line given its gradient and a point it passes through. Type in the letter of the equation that is the answer for each question. You can earn a virtual trophy if you do this activity online.
Gradient 1,Passing through (3,3) Gradient 1,Passing through (2,4) Gradient -1,Passing through (1,2) Gradient $$\frac12$$,Passing through (4,4) Gradient -2,Passing through (2,-3) Gradient $$\frac23$$,Passing through (3,7) Bonus question: From the letters chosen as answers to the first six questions make a six letter mathematical word.
Check
This is Equation of a Line Through Points level 1. You can also try:
Graph Match Level 2 Level 3 Level 4 Mixed Questions
## Instructions
Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.
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## Transum.org
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#### ChrisMaths
Christmas activities make those December Maths lessons interesting, exciting and relevant. If students have access to computers there are some online activities to keep them engaged such as Christmas Ornaments and Christmas Light Up.
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## Go Maths
Learning and understanding Mathematics, at every level, requires learner engagement. Mathematics is not a spectator sport. Sometimes traditional teaching fails to actively involve students. One way to address the problem is through the use of interactive activities and this web site provides many of those. The Go Maths page is an alphabetical list of free activities designed for students in Secondary/High school.
## Maths Map
Are you looking for something specific? An exercise to supplement the topic you are studying at school at the moment perhaps. Navigate using our Maths Map to find exercises, puzzles and Maths lesson starters grouped by topic.
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## Description of Levels
Close
Level 1 - Find the equation of the line given its gradient and a point it passes through
Level 2 - Find the equation of the line given two points it passes through
Level 3 - Find the equation of the line given the equation of a parallel line and a point it passes through
Level 4 - Find the equation of the line given the equation of a perpendicular line and a point it passes through
Mixed Questions - A multi-level online exercise about the equation and features of a straight line graph.
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## Example
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Close | 1,386 | 6,652 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.015625 | 3 | CC-MAIN-2018-51 | latest | en | 0.907063 |
https://derangedphysiology.com/main/cicm-fellowship-exam/past-papers/2005-paper-1-saqs/question-13 | 1,669,534,985,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710218.49/warc/CC-MAIN-20221127073607-20221127103607-00096.warc.gz | 240,406,570 | 4,871 | # Question 13
For each of the following terms, provide a definition, outline their derivation and outline their role:
• Sensitivity,
• Specificity,
• Positive Predictive Value,
• Negative Predictive Value.
[Click here to toggle visibility of the answers]
## College Answer
Test Disease Present Disease Absent Positive A B A+B Negative C D C+D A+C B+D A+B + C+D
Using the presence or absence of a disease, and the result a specific test as an example: Sensitivity = proportion of patients with disease detected by positive test = A/(A+C). Very high values essential if wish to catch all with disease, and allow a negative result to virtually rule out the diagnosis.
Specificity = proportion of patients without disease detected by negative test = D/(B+D). Very high values of specificity essential if wish to catch all without the disease, and allow a positive result to rule in the diagnosis.
Positive predictive value = proportion of patients with positive test who have disease = A/(A+B). PPV allows estimate of certainty around positive result.
Negative predictive value = proportion of patients with negative test who do not have disease = D/(C+D). NPV allows estimate of certainty about a negative result.
## Discussion
Later papers focus merely on the candidate's ability to apply the formulae.
One can make a strong argument for a return to questions which test one's understanding of the actual concept, rather than demanding the regurgitation of rote-learned equations.
To rote-learn the abovemention equations, here is a helpful list.
Sensitivity = true positives / (true positives + false negatives)
This is the proportion of patients in whom disease which was correctly identified by the test.
Specificity = true negatives / (true negatives + false positives)
This is the proportion of patients in whom the disease was correctly excluded
Positive predictive value = (true positives / total positives)
This is the proportion of patients with positive test results who are correctly diagnosed.
Negative predictive value = (true negatives / total negatives)
This is the proportion of patients with negative test results who are correctly diagnosed.
## References
Altman, Douglas G., and J. Martin Bland. "Statistics Notes: Diagnostic tests 2: predictive values." Bmj 309.6947 (1994): 102. | 490 | 2,325 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.953125 | 4 | CC-MAIN-2022-49 | latest | en | 0.836039 |
http://math.stackexchange.com/questions/37829/computing-the-integral-of-log-sin-x | 1,469,448,214,000,000,000 | text/html | crawl-data/CC-MAIN-2016-30/segments/1469257824226.79/warc/CC-MAIN-20160723071024-00168-ip-10-185-27-174.ec2.internal.warc.gz | 159,819,947 | 25,221 | # Computing the integral of $\log(\sin x)$
How to compute the following integral? $$\int\log(\sin x)\,dx$$
Motivation: Since $\log(\sin x)'=\cot x$, the antiderivative $\int\log(\sin x)\,dx$ has the nice property $F''(x)=\cot x$. Can we find $F$ explicitly? Failing that, can we find the definite integral over one of intervals where $\log (\sin x)$ is defined?
-
I'm pretty sure this is an integral that can't be expressed in terms of elementary functions (that is, the functions of 1st-year calculus). See, for example, reference.wolfram.com/legacy/v5/TheMathematicaBook/… about halfway down the page. – Gerry Myerson May 8 '11 at 13:00
Yes, the dilogarithm seems to be required here... – J. M. May 8 '11 at 13:03
@Kolya: Do you actually want to compute $\int_a^b {\log (\sin (x))\,{\rm d}x}$ for certain $a$ and $b$? – Shai Covo May 8 '11 at 13:32
For $a=0$ and $b=\pi/2$ or $b=\pi$, for example... – Did May 8 '11 at 13:59
Although this integral may cannot be expressed in elementary function, but it may can be expressed in series form. For example, ∫sin(sin x)dx and ∫cos(cos x)dx can both be evaluated in series form. – ᴊ ᴀ s ᴏ ɴ Jul 12 '12 at 8:21
You can calculate $$\int_0^\pi\log(\sin x)\,dx = -\pi\log2$$ and integrating up to $\pi/2$ would give half of this.
Note that integrating $\log(\sin x)$ from 0 to $\pi/2$ is the same as integrating $\log(\cos x)$ so that \begin{align} \int_0^{\pi/2}\log(\sin x)\,dx &= \frac12\int_0^{\pi/2}\log(\sin x\cos x)\,dx\\ &= \frac12\int_0^{\pi/2}\log(\sin 2x)\,dx - \frac{\pi}{4}\log 2. \end{align} After a change of variables, this can be rearranged to get the result.
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Actually, as the OP hasn't come back to say if it was the definite or indefinite integral that he was after, I'm not sure if this fully answers the question. – George Lowther May 8 '11 at 17:41
Also, I'm not sure what the appropriate amount of detail is for a homework question. The value of the integral is no secret anyway, as Wolfram alpha knows it. – George Lowther May 8 '11 at 17:45
Yes, and in Abramowitz and Stegun, too. – J. M. May 8 '11 at 17:46
(should have said he/she above. The ability to edit comments runs out too quickly.) – George Lowther May 8 '11 at 17:50
I was wondering just last night whether $$\int_{0}^{\pi/2}\ln^{k}(\sin{x})\;{dx}$$ where $k\in\mathbb{N}$, can be calculated! – Lyrebird May 8 '11 at 18:52
I think it worth mentioning the history of (essentially) this function, tracing back to work of Lobachevsky in the beginnings of non-Euclidean geometry. See the pdf here for Milnor's survey, the function is discussed from page 9 onward.
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Series expansion can be used for this integral too.
We use the following identity; $$\log(\sin x)=-\log 2-\sum_{k\geq 1}\frac{\cos(2kx)}{k} \phantom{a} (0<x<\pi)$$ This identity gives $$\int_{a}^{b} \log(\sin x)dx=-(b-a)\log 2-\sum_{k\ge 1}\frac{\sin(2kb)-\sin(2ka)}{2k^2}$$ ($a, b<\pi$)
For example, $$\int_{0}^{\pi/4}\log(\sin x)dx=-\frac{\pi}{4}\log 2-\sum_{k\ge 1}\frac{\sin(\pi k/2)}{2k^2}=-\frac{\pi}{4}\log 2-\frac{1}{2}K$$ $$\int_{0}^{\pi/2} \log(\sin x)dx=-\frac{\pi}{2}\log 2$$ $$\int_{0}^{\pi}\log(\sin x)dx=-\pi \log 2$$ ($K$; Catalan's constant ... $\displaystyle K=\sum_{k\ge 1}\frac{(-1)^{k-1}}{(2k-1)^2}$)
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I discovered the identity you used above as $\sin^2(x)=\dfrac{1-\cos(2x)}{2}=\dfrac{(1-e^{2ix})(1-e^{-2ix})}{4}$ while answering this question. I was lead here via a series of links. Nice answer (+1). – robjohn Mar 11 '14 at 14:43
@hunminpark, How did you derive that identitiy in the beginning of this answer? – Amad27 Dec 15 '14 at 7:57
An excellent discussion of this topic can be found in the book The Gamma Function by James Bonnar. Consider just two of the provably equivalent definitions of the Beta function: $$\begin{eqnarray} B(x,y)&=& 2\int_0^{\pi/2}\sin(t)^{2x-1}\cos(t)^{2y-1}\,dt\\ &=& \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}. \end{eqnarray}$$
Directly from this definition we have
$$B(n+\frac{1}{2},\frac{1}{2}): \int_0^{\pi/2}\sin^{2n}(x)\,dx=\frac{\sqrt{\pi} \cdot\Gamma(n+1/2)}{2(n!)}$$ $$B(n+1,\frac{1}{2}): \int_0^{\pi/2}\sin^{2n+1}(x)\,dx=\frac{\sqrt{\pi} \cdot n!}{2 \Gamma(n+3/2)}$$ Hence the quotient of these two integrals is $$\begin{eqnarray} \frac{ \int_0^{\pi/2}\sin^{2n}(x)\,dx}{\int_0^{\pi/2}\sin^{2n+1}(x)\,dx}&=& \frac{\Gamma(n+1/2)}{n!}\frac{\Gamma(n+3/2)}{n!}\\ &=& \frac{2n+1}{2n}\frac{2n-1}{2n}\frac{2n-1}{2n-2}\cdots\frac{3}{4}\frac{3}{2}\frac{1}{2}\frac{\pi}{2} \end{eqnarray}$$ where the quantitiy $\pi/2$ results from the fact that $$\frac{\int_0^{\pi/2}\sin^{2\cdot 0}(x)\,dx}{\int_0^{\pi/2}\sin^{2\cdot 0+1}x\,dx}=\frac{\pi/2}{1}=\frac{\pi}{2}.$$ So we have that $$\int_0^{\pi/2}\sin^{2n}(x)\,dx=\frac{2n-1}{2n}\frac{2n-3}{2n-2}\cdots\frac{1}{2}\frac{\pi}{2}=\frac{(2n)!}{4^n (n!)^2}\frac{\pi}{2}.$$ Hence an analytic continuation of $\int_0^{\pi/2}\sin^{2n}(x)\,dx$ is $$\int_0^{\pi/2}\sin^{2z}(x)\,dx=\frac{\pi}{2}\frac{\Gamma(2z+1)}{4^z \Gamma^2(z+1)}=\frac{\pi}{2}\Gamma(2z+1)4^{-z}\Gamma^{-2}(z+1).$$ Now differentiate both sides with respect to $z$ which yields
$$\begin{eqnarray} 2\int_0^{\pi/2}\sin^{2z}(x)\log(\sin(x))\,dx =\frac{\pi}{2} \{2\Gamma'(2z+1)4^{-z}\Gamma^{-2}(z+1)\\ +2\Gamma(2z+1)4^{-z}\Gamma^{-3}(z+1)\Gamma'(z+1)\\ -\log(4)\Gamma(2z+1)4^{-z}\Gamma^{-2}(z+1)\}. \end{eqnarray}$$
Finally set $z=0$ and note that $\Gamma'(1)=-\gamma$ to complete the integration: $$\begin{eqnarray} 2\int_0^{\pi/2}\log(\sin(x))\,dx&=&\frac{\pi}{2}(-2\gamma+2\gamma-\log(4))\\ &=& -\frac{\pi}{2}\log(4)=-\pi\log(2). \end{eqnarray}$$ We conclude that $$\int_0^{\pi/2}\log(\sin(x))\,dx=-\frac{\pi}{2}\log(2).$$
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Nice solution..... – juantheron May 31 '14 at 4:01
(I am assuming that the OP is interested in the definite integral).
The following argument is not completely rigorous $\displaystyle \int_0^{\pi/2} \log(\sin(x)) dx = - \dfrac{\pi}2 \log 2$ but I think it can be made rigorous.
From integration by parts/ other techniques, we have that $$\int_0^{\pi/2} \sin^{2k}(x) dx = \frac{2k-1}{2k}\frac{2k-3}{2k-2} \cdots \frac{1}{2} \frac{\pi}{2} = \dfrac{(2k)!}{4^k (k!)^2} \dfrac{\pi}2 = \dfrac{\Gamma(2k+1)}{4^k \Gamma^2(k+1)} \dfrac{\pi}2$$
Hence, a possible analytic extension to $\displaystyle \int_0^{\pi/2} \sin^{2z}(x) dx$ is $\dfrac{\Gamma(2z+1)}{4^z \Gamma^2(z+1)} \dfrac{\pi}2$.
Now differentiate both sides with respect to $z$, and set $z=0$, to get $$2 \int_0^{\pi/2} \log(\sin(x)) = -\dfrac{\pi}2 \log(4)$$ Hence, we get that $$\int_0^{\pi/2} \log(\sin(x)) dx = -\dfrac{\pi}2 \log(2)$$ This also provides you a way to evaluate $\displaystyle \int_0^{\pi/2} \sin^{n}(x) \log(\sin(x)) dx$.
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The differentiation under the integral sign is fine, I think, so it seems to me that the only gap is to justify the expression for $\int_0^{\pi/2} \sin^{2\alpha}(x)\mathrm dx$ for noninteger $\alpha$... – J. M. Jul 12 '12 at 8:11
@J.M. Actually thinking about it since the domain is only from $0$ to $\pi/2$, $\sin^{2 \alpha}(x)$ is well defined even for non-integer $\alpha$. So I think this does it. Hence, the analytic extension is the analytic extension. – user17762 Jul 12 '12 at 8:13
There was a duplicate posted a while ago. Since I think my answer might be of some interest, here it goes:
By substituting $\sin{x}=t$, we can write it as: \begin{align*} \int_{0}^{\pi/2} \, \log\sin{x}\, dx &= \int_{0}^{1} \, \frac{\log{t}}{\sqrt{1-t^2}}\, dt \tag{1} \end{align*}
Now, consider:
\begin{align*} I(a) &= \int_{0}^{1} \, \frac{t^a}{(1-t^2)^{1/2}}\, dt \\ &= \mathrm{B}\left(\frac{a+1}{2},\; \frac{1}{2}\right) \\ \frac{\partial }{\partial a}I(a) &= \frac{1}{4}\left(\psi\left(\frac{a+1}{2}\right)-\psi\left(\frac{a+2}{2}\right)\right)\mathrm{B}\left(\frac{a+1}{2},\; \frac{1}{2}\right) \\ \implies I'(0) &= \frac{1}{4}\left(\psi\left(\frac{1}{2}\right)-\psi\left(1\right)\right)\mathrm{B}\left(\frac{1}{2},\; \frac{1}{2}\right) \tag{2} \end{align*} Putting the values of digamma and beta functions. \begin{align*} \psi\left(\frac{1}{2}\right) &= -2\log{2}-\gamma \\ \psi\left(1\right) &= -\gamma \\ \mathrm{B}\left(\frac{1}{2}, \frac{1}{2}\right) &= \pi \end{align*}
Hence, from $(1)$ and $(2)$, \begin{align*} \boxed{\displaystyle \int_{0}^{\pi/2} \, \log\sin{x}\, dx = -\frac{\pi}{2}\log{2}} \end{align*}
Using a CAS, we can derive for higher powers of $\ln\sin{x}$, e.g. \begin{align*} \int_{0}^{\pi/2} \, \left(\log\sin{x}\right)^2\, dx &= \frac{1}{24} \, \pi^{3} + \frac{1}{2} \, \pi \log\left(2\right)^{2} \\ \int_{0}^{\pi/2} \, \left(\log\sin{x}\right)^3\, dx &= -\frac{1}{8} \, \pi^{3} \log\left(2\right) - \frac{1}{2} \, \pi \log\left(2\right)^{3} - \frac{3}{4} \, \pi \zeta(3)\\ \int_{0}^{\pi/2} \, \left(\log\sin{x}\right)^4\, dx &= \frac{19}{480} \, \pi^{5} + \frac{1}{4} \, \pi^{3} \log\left(2\right)^{2} + \frac{1}{2} \, \pi \log\left(2\right)^{4} + 3 \, \pi \log\left(2\right) \zeta(3) \end{align*}
We can also observe another interesting thing, for small values of $n$
\begin{align*} \displaystyle \int_{0}^{\pi/2} \, \left(\log\sin{x}\right)^n\, dx \approx \displaystyle (-1)^n\, n! \end{align*}
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For the indefinite integral, you have this closed form:
$$\frac{i{x}^{2}}{2}+x\ln \left( \cos \left( x \right) \right) -x\ln \left( 1+{{\rm e}^{2\,ix}} \right) +\frac{i}{2} Li_2 ( -{ {\rm e}^{2\,ix}} ),$$
where $Li_2$ is a polylogarithm.
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Simply stating a closed form without a derivation seems mostly useless. – Carl Mummert Aug 18 '14 at 23:40
@Downvoter: What's the down vote for? – Mhenni Benghorbal Aug 18 '14 at 23:41
@CarlMummert: It tells people there exists a closed form and whoever is interested in proving it can put some effort to find it. Giving detailed answers all the time is not useful. – Mhenni Benghorbal Aug 18 '14 at 23:43 | 3,839 | 9,666 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.90625 | 4 | CC-MAIN-2016-30 | latest | en | 0.88901 |
http://jde27.uk/la/08_operations.html | 1,701,580,359,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100484.76/warc/CC-MAIN-20231203030948-20231203060948-00878.warc.gz | 25,352,619 | 4,859 | # 08. Other operations
## 08. Other operations
In this video, we'll define some further operations you can do to produce new matrices. The first is matrix addition If we have two $m$ -by-$n$ matrices $A$ and $B$ with entries $A_{ij}$ and $B_{ij}$ , we can form a new matrix $A+B$ with $(A+B)_{ij}=A_{ij}+B_{ij}.$ In other words, you take the $ij$ th entries of both matrices and add them.
$\begin{pmatrix}1&0\\ 1&1\end{pmatrix}+\begin{pmatrix}1&1\\ 0&-1\end{pmatrix}=\begin{pmatrix}2&1\\ 1&0\end{pmatrix}.$
This is most useful when $A$ and $B$ are both column vectors, i.e. $m$ -by-$1$ matrices. Let's see what it means in for vectors in $\mathbf{R}^{2}$ . The formula is $\begin{pmatrix}x\\ y\end{pmatrix}+\begin{pmatrix}a\\ b\end{pmatrix}=\begin{pmatrix}x+a\\ y+b\end{pmatrix}.$
Geometrically, we add two vectors $v=\begin{pmatrix}x\\ y\end{pmatrix}$ and $w=\begin{pmatrix}a\\ b\end{pmatrix}$ by translating $w$ to the tip of $v$ and drawing the arrow from the tail of $v$ to the tip of $w$ . One can see from the picture that the $x$ - (respectively $y$ -) coordinate of this arrow is the sum of the $x$ - (respectively $y$ -) coordinates of $v$ and $w$ .
### Rescaling
Given a number $\lambda$ and a matrix $A$ , you can form the matrix $\lambda A$ whose entries are $\lambda$ times the entries of $A$ .
$2\begin{pmatrix}1&2\\ 3&4\end{pmatrix}=\begin{pmatrix}2&4\\ 6&8\end{pmatrix}$ .
### Matrix exponentiation
The exponential of a number $x$ is defined by the Taylor series of $\exp$ : $\exp(x)=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots=\sum_{n=0}^{\infty}\frac{% x^{n}}{n!}$ We can use the same definition to define the exponential of a matrix: $\exp(M)=\sum_{n=0}^{\infty}\frac{1}{n!}M^{n}.$ Here, $M^{0}$ is understood to mean the identity matrix $I$ (the analogue for matrices of the number $1$ ).
Consider $M=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}$ . Since $M^{2}=0$ , all the higher powers of $M$ vanish (the name for this is nilpotence: some power of $M$ is zero), so the matrix exponential becomes $\exp(M)=I+M=\begin{pmatrix}1&1\\ 0&1\end{pmatrix}.$ So we get the matrix for a shear as the exponential of a nilpotent matrix.
In fact, $\exp\begin{pmatrix}0&t\\ 0&0\end{pmatrix}=\begin{pmatrix}1&t\\ 0&1\end{pmatrix},$ so we get a whole family of matrices which shear further and further to the right as $t$ varies.
Take $M=\begin{pmatrix}0&-t\\ t&0\end{pmatrix}$ . We have $M=t\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}$ $M^{2}=\begin{pmatrix}-t^{2}&0\\ 0&-t^{2}\end{pmatrix}=-t^{2}I$ $M^{3}=-t^{3}\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}$ $M^{4}=t^{4}I$ etc.
and in the end we get $\exp(M)=I+t\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}$ $-\frac{t^{2}}{2!}I-\frac{t^{3}}{3!}\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}$ $+\frac{t^{4}}{4!}I+\frac{t^{5}}{5!}\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}$ etc. The coefficient of $I$ is the Taylor series for $\cos t$ ; the coefficent of $\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}$ is the Taylor series for $\sin t$ , so overall we get $\exp(M)=\begin{pmatrix}\cos t&-\sin t\\ \sin t&\cos t\end{pmatrix}.$ So we get a general rotation matrix in 2-d by exponentiating this very simple matrix. | 1,175 | 3,141 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 61, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.84375 | 5 | CC-MAIN-2023-50 | latest | en | 0.684544 |
http://www.gradesaver.com/textbooks/science/physics/conceptual-physics-12th-edition/chapter-15-think-and-explain-page-299-300/59 | 1,481,410,651,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698543577.51/warc/CC-MAIN-20161202170903-00146-ip-10-31-129-80.ec2.internal.warc.gz | 483,012,577 | 35,856 | ## Conceptual Physics (12th Edition)
Published by Addison-Wesley
# Chapter 15 - Think and Explain: 59
#### Answer
The iron, because it has the smaller specific heat capacity, will have a greater rise in temperature.
#### Work Step by Step
The equation $Q = cm \Delta T$ relates the heat transferred to the temperature change. For the same mass and heat, the temperature change is greater if the specific heat capacity, c, is smaller. This is discussed on pages 289-290.
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | 143 | 636 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.625 | 3 | CC-MAIN-2016-50 | longest | en | 0.868511 |
https://nolongerset.com/gift-counts-recursion-solution/ | 1,726,036,329,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651344.44/warc/CC-MAIN-20240911052223-20240911082223-00859.warc.gz | 396,450,526 | 11,039 | # Calculating Gift Counts for "The Twelve Days of Christmas": Recursion Edition
In the second solution to my recent reader challenge, I use recursion to calculate the cumulative number of gifts given for each day of the "12 Days of Christmas."
In Wednesday's article, I posted a reader challenge:
How many total gifts does one's true love deliver by the nth day of Christmas?
SPOILER ALERT: If you would like to try figuring it out on your own, follow the link above to read the original article before scrolling down to see one of my solutions.
The goal is to write a function that takes as input the first column (Day of Christmas) and returns as output the last column (Cumulative gifts):
## Using Recursion to Calculate the Total
In general, recursion is harder for programmers to wrap their head around than non-recursive solutions (myself included).
In some situations, though, it maps more closely to how our brains naturally solve a problem. The classic example is with any sort of hierarchical tree structure, such as Windows subfolders.
It turns out that it also works well with pear tree structures, as we have here.
## Explaining the Algorithm
The first part of the function is a loop that counts the number of gifts given on a single day.
The next section of the function adds in the number of gifts given on the previous day. It does this by calling itself but passing the previous day of Christmas.
The final section acts as a brake to terminate the recursion. Once `DayOfXmas = 1` (i.e., the first day of Christmas), the function returns a literal `1` rather than calling itself again.
``````'--== Recursive Version ==--
Function GiftsByXmasDay(DayOfXmas As Long) As Long
'Calculate the number of gifts given for the current day
Dim GiftsThisDay As Long, NumberOfItems As Long
For NumberOfItems = 1 To DayOfXmas
Next NumberOfItems
If DayOfXmas > 1 Then
'Use recursion to call the function for each previous day
Else
'With recursion, you always need a code path where
' the function does not call itself, otherwise you
' end up with infinite recursion and
' "Out of stack space" errors | 477 | 2,125 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.765625 | 4 | CC-MAIN-2024-38 | latest | en | 0.872853 |
https://it.mathworks.com/matlabcentral/cody/problems/45401-zigzag-02/solutions/2188933 | 1,606,730,012,000,000,000 | text/html | crawl-data/CC-MAIN-2020-50/segments/1606141211510.56/warc/CC-MAIN-20201130065516-20201130095516-00187.warc.gz | 350,295,574 | 17,294 | Cody
# Problem 45401. ZigZag - 02
Solution 2188933
Submitted on 2 Apr 2020 by bainhome
• Size: 78
• This is the leading solution.
This solution is locked. To view this solution, you need to provide a solution of the same size or smaller.
### Test Suite
Test Status Code Input and Output
1 Pass
a=[1,2,3,4,5; 6,7,8,9,1; 1,1,1,1,1]; y=[1,2,3,4,5,8,1,1,1,1,1]; assert(isequal(z_mat_02(a),y))
2 Pass
a=reshape(1:24,3,[]); y=[1:3:22,3:3:24]; assert(isequal(z_mat_02(a),y))
3 Pass
a=reshape(1:24,4,[]); y=[1:4:21,4:4:24]; assert(isequal(z_mat_02(a),y))
4 Pass
a=reshape(30:65,12,[]) y=[ 30 42 54 41 53 65]; assert(isequal(z_mat_02(a),y))
a = 30 42 54 31 43 55 32 44 56 33 45 57 34 46 58 35 47 59 36 48 60 37 49 61 38 50 62 39 51 63 40 52 64 41 53 65
5 Pass
a=magic(5); y=[ 17 24 1 8 15 14 13 12 11 18 25 2 9]; assert(isequal(z_mat_02(a),y))
6 Pass
a=reshape(magic(9),3,[]); y=[47 77 26 58 7 28 69 18 39 80 20 50 1 31 61 12 42 72 23 53 74 34 55 4 45 66 15 41 67 16 37 78 27 48 8 29 59 10 40 70 21 51 81 32 62 2 43 64 13 54 75 24 56 5 35]; assert(isequal(z_mat_02(a),y))
7 Pass
a=eye(5); aa=[a;a;a]; y=[ 1 0 0 0 0 1 0 0 0 0 1]; assert(isequal(z_mat_02(aa),y))
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Start Hunting! | 625 | 1,305 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.484375 | 3 | CC-MAIN-2020-50 | latest | en | 0.48906 |
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1. Nov 6, 2013
### 012anonymousx
1. The problem statement, all variables and given/known data
A monument is made from stone blocks of density 3800kg/m^3. The monument is 15.7m high, 64.8m wide at the base, and 3.6m thick from front th back. How much work was required to build the monument? (Hint: find ycm).
2. Relevant equations
ycm = (1/M) * ∫ydm, M = mass total
3. The attempt at a solution
Take a cross section and get a rectangle.
The length will be: (64.8/15.7) * y
The width will be: (3.6/15.7) * y
Thus, dm = density * length * width * dy
M total = density * volume = 6958742.8
ycm = (1/M) * ∫y * density * length * width * dy (from 0 to 15.7)
Okay, so this solution is wrong. It gives the the ycm as 7.85m. But ycm is actually a third of the height (as it is for triangles).
My question is: what is fundamentally wrong with my approach?
I have the solution though. I don't need that.
2. Nov 6, 2013
### Staff: Mentor
Your formulas for the length and width have them both zero for y = 0. Unless the monument is supposed to be standing on its head, that could be problematical
3. Nov 6, 2013
### 012anonymousx
That shouldn't matter.
Take the center of mass from 7.85m from the top.
Interestingly. 7.85m is the middle of the triangle.
4. Nov 6, 2013
### 012anonymousx
Got it got it got it! 3.6 is constant. Brb
[EDIT] Works. You made me think of it. I was looking at integrating from the top and wrote out eqn to reverse and realized... wait a minute, i'm scaling the width but its constant...
TYVM. | 482 | 1,581 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.71875 | 4 | CC-MAIN-2017-43 | longest | en | 0.938221 |
https://discuss.codechef.com/t/help-me-in-solving-noseq-problem/113996 | 1,702,100,524,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100800.25/warc/CC-MAIN-20231209040008-20231209070008-00468.warc.gz | 250,892,334 | 4,584 | Help me in solving NOSEQ problem
My issue
Judge error on multiple already accepted and new solutions;
My code
``````// author: imtheonly1
// time: 2023-11-14 18:06:31
// URL: https://www.codechef.com/problems/NOSEQ
// Time Limit: 1000 ms
//
#include "bits/stdc++.h"
#define ll long long
using namespace std;
void solve() {
ll n, k, s;
cin >> n >> k >> s;
vector<int> a(n);
for (int i = 0; i < n; i++) {
if (s % k == 0) {
a[i] = 0;
s /= k;
} else if ((s - 1) % k == 0) {
a[i] = 1;
s = (s - 1) / k;
} else if ((s + 1) % k == 0) {
a[i] = -1;
s = (s + 1) / k;
} else {
break;
}
}
if (!s) {
for (int i = 0; i < n; i++) {
cout << a[i] << ' ';
}
} else {
cout << -2;
}
cout << '\n';
}
int main() {
ios_base::sync_with_stdio(false);
cin.tie(nullptr);
int tc = 1;
cin >> tc;
while (tc--) {
solve();
}
return 0;
}
``````
Problem Link: NOSEQ Problem - CodeChef | 325 | 857 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.65625 | 3 | CC-MAIN-2023-50 | latest | en | 0.319895 |
http://www.jiskha.com/display.cgi?id=1119711395 | 1,498,382,589,000,000,000 | text/html | crawl-data/CC-MAIN-2017-26/segments/1498128320476.39/warc/CC-MAIN-20170625083108-20170625103108-00424.warc.gz | 577,394,962 | 4,069 | # Maths!
posted by .
There has just been a planetary alignment of five planets in the system Blogg, all the astrologers have gone loopy, and they want to know when the next occurance of these five planets will happen again.
Krypton the nearest planet to the sun takes 6 days to orbit the sun.
Zenon the next planet takes 23 days to orbit the sun.
Argon the middle planet take 37 days.
Neon the largest planet takes 68 days.
And Radon the last planet take 137 days to orbit the sun.
25000 + 1200
its wrong anyways :/
The answer is 23,783,748 days. There will certainly be an alignment after
6 X 23 X 37 X 68 X 137 = 47,567,496 days. However, the first alignment
will occur after the smallest number of days that can be divided by 6,
23, 37, 68, and 137 to give an integer. To find this numer, we look at the
factors of all of these numbers: 23, 37, and 137 are prime, 6 = 2 X 3,
and 68 = 2 X 2 X 17. Thus we need 2 X 2 X 3 X 17 X 23 X 37 X 137 =
23,783,748 days.
The answer is 23,783,748 days. There will certainly be an alignment after
6 X 23 X 37 X 68 X 137 = 47,567,496 days. However, the first alignment
will occur after the smallest number of days that can be divided by 6,
23, 37, 68, and 137 to give an integer. To find this numer, we look at the
factors of all of these numbers: 23, 37, and 137 are prime, 6 = 2 X 3,
and 68 = 2 X 2 X 17. Thus we need 2 X 2 X 3 X 17 X 23 X 37 X 137 =
23,783,748 days. | 453 | 1,414 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.0625 | 4 | CC-MAIN-2017-26 | latest | en | 0.939976 |
https://www.instructables.com/id/Origami-A-wing/ | 1,576,334,149,000,000,000 | text/html | crawl-data/CC-MAIN-2019-51/segments/1575541157498.50/warc/CC-MAIN-20191214122253-20191214150253-00391.warc.gz | 725,909,371 | 21,749 | # Origami A-wing
4,600
1
4
Following my X-wing Instructions comes this A-wing, another ship from the movie StarWars -- though I suppose everybody knows that. This is more difficult than the X-wing, so I apologize if you're a newbie to the origami universe and really want to make this piece. Even the better origamists should have some trouble with this unless you're really into sink folds (Which is like deer liking lion meat). Well, anyway, here are the instructions -- and don't forget to show me the results! Thank you.
### Teacher Notes
Teachers! Did you use this instructable in your classroom?
Add a Teacher Note to share how you incorporated it into your lesson.
## Step 1: Start Like This
Get out your square paper and place it with the colored side up. Fold it in half for the crease, and unfold it. Then fold the edges almost to the center crease, but an ample distance away. KEEP IT SYMMETRICAL! . . . I mean, keep it symmetrical. This is slightly difficult, but the more symmetrical it is, the better looking your ship will become.
Flip your slab over and fold the top corners diagonally, creating an X at the top of the slab. Then mountain fold (Fold behind) the top part, putting its crease in the middle of the X. Squish this into a Water Bomb Base, and you're good to go.
## Step 2: First Sink
Sink the top corner of the base in, and then fold the lower edge up to the bottom of water bomb base. If you are awesome or just a complete no-lifer, than the bottom should look to have squares. Fold the squares diagonally in half. If they're not squares, fold them until they reach the bottom of the base.
With that, inside reverse fold them inside the structure.
## Step 3: Like the Titanic
Now take the bottom flap and fold it over. This will make headway for the upper half, where you will make the slanted edge meet the nearest vertical edge. Now the paper behind it must follow, except this time you must sink it into the body. This is a difficult process, but I found that I could do this by lifting the top flaps out and shoving the paper in.
Now fold the flap you folded over in the second sink in front of the spike that had been in front of the paper you sunk. Repeat this on the other side.
## Step 4: Elementary
Take the bottom part and fold over the top of it to get a bar. Mountain fold the entire thing in half and fold the top parts as diagonally as you can. Repeat this on the other side. This creates a bump that resembles a cockpit. Now inside reverse the small corner in the bottom.
## Step 5: Little Things
Unfold the ship and fold the bar in to the center.
## Step 6: Origami A-Wing
Voila, your fandom is complete. Now go to that comic con dressed as a wookie and do something creepy. . . again. That will be about it for me, so thanks for reading -- and remember, don't talk to strangers unless you know them.
81 12K
1 26 2.5K
83 7.5K | 690 | 2,891 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.6875 | 3 | CC-MAIN-2019-51 | latest | en | 0.944546 |
http://solverworld.com/sum-of-a-geometric-series/ | 1,702,321,205,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679516047.98/warc/CC-MAIN-20231211174901-20231211204901-00366.warc.gz | 41,850,583 | 10,207 | # Sum of a Geometric Series
What is the sum of the following geometric series?
$\sum_{i=0}^{k}a^i=1+a+a^2+…+a^k$
We will frequently need a simple formula for this finite series. It is called a geometric series because each term is related by a multiple to the previous one. If each term was related by a fixed difference, it would be called an arithmetic series; but that’s for another day.
Let us define the sum as $$S$$. Then writing the equation for $$S$$ and $$aS$$ with some clever alignment:
\begin{array}{rrrccc}
S=&1&+a&+a^2&+…&+a^k \\
aS=&&a&+a^2&+…&+a^k&+a^{k+1}
\end{array}
Subtract the equations
$S(1-a)=1-a^{k+1}$
or
\begin{align}
S=
\begin{cases}
\frac{1-a^{k+1}}{1-a} &a \neq 1 \\
k+1 &\text{otherwise}
\end{cases}
\end{align}
where we have to be careful to divide by $$1-a$$ only if $$a\neq 1$$, and the answer for $$a=1$$ is determined by inspection.
Q.E.D. | 305 | 876 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.375 | 4 | CC-MAIN-2023-50 | latest | en | 0.863288 |
http://forum.math.toronto.edu/index.php?action=profile;area=showposts;sa=messages;u=2201 | 1,611,318,871,000,000,000 | text/html | crawl-data/CC-MAIN-2021-04/segments/1610703529331.99/warc/CC-MAIN-20210122113332-20210122143332-00102.warc.gz | 39,335,855 | 6,758 | ### Show Posts
This section allows you to view all posts made by this member. Note that you can only see posts made in areas you currently have access to.
### Messages - Joy Zhou
Pages: [1]
1
##### Quiz-5 / LEC5101 Quiz5
« on: November 01, 2019, 02:12:58 PM »
$y''+4y'+4y=t^{-2}e^{-2t}, t>0$
The homogeneous differential equation is as follows:
$$y^{\prime\prime}+4 y^{\prime}+4 y=0$$
The characteristic equation for homogeneous differential equation is
\begin{aligned} r^{2}+4 r+4 &=0\\ (r+2)^{2} &=0 \\ r &=-2,-2 \end{aligned}
since the roots of characteristic equation are real and repeating.
Therefore, the solution of differential equation is as follows:
$${y_{c}(t)=c e^{-2 t}+c_{2} te^{-2t}}$$
($c$ and $c_{2}$ are constants)
From above solution, the two fundamental solutions of the differential equation are
$y_{1}(t)=e^{-2 t}$ and $y_{2}(t)=t e^{-2 t}$
Then, use the variation of parameters method to determine the particular solution $Y(t) .$
Find the Wronskian as follows:
\begin{aligned} W\left(y_{1}, y_{2}\right)(t) &=\left|\begin{array}{cc}{y_{1}(t)} & {y_{2}(t)} \\ {y_{1}^{\prime}(t)} & {y_{2}^{\prime}(t)}\end{array}\right| \\ &=\left|\begin{array}{cc}{e^{-2 t}} & {te^{-2t}} \\ {-2 e^{-2 t}} & {-2 t e^{-2 t}+e^{-2 t}}\end{array}\right| \\ &=e^{-2 t}\left(-2 t e^{-2 t}+e^{-2 t}\right)-\left(-2 e^{-2 t}\right)\left(t e^{-2 t}\right) \\ &=-2 t e^{-4 t}+e^{-4 t}+t e^{-4 t} \end{aligned}
$W\left(y_{1}, y_{2}\right)(t)=e^{-4t}$
The particular solution is given as follows:
$$Y(t)=u_{1}(t) y_{1}(t)+u_{2}(t) y_{2}(t)$$
Here, $u_{1}(t)$ and $u_{2}(t)$ are the parameters defined as follows:
$u_{1}(t)=-\int \frac{y_{2}\left(t^{-2} e^{-2 t}\right)}{W\left(y_{1}, y_{2}\right)(t)} d t$ and $u_{2}(t)=\int \frac{y_{1}\left(t^{-2} e^{-2 t}\right)}{W\left(y_{1}, y_{2}\right)(t)} d t$
\begin{aligned} u_{1}(t) &=-\int \frac{y_{2}\left(t^{-2} e^{-2 t}\right)}{W\left(y_{1}, y_{2}\right)(t)} d t \\ &=-\int \frac{t e^{-2 t}t^{-2}e^{-2t}}{e^{-4 t}} d t \\ &=-\int \frac{t^{-1} e^{-4 t}}{e^{-4 t}} d t \\ &=-\int t^{-1} d t \\&=-\ln t \end{aligned}
\begin{aligned} u_{2}(t) &=\int \frac{y_{1}\left(t^{-2} e^{-2 t}\right)}{W\left(y_{1}, y_{2}\right)(t)} d t \\ &=\int \frac{e^{-2 t} t^{-2} e^{-2 t}}{e^{-4 t}} d t \\ &=\int \frac{e^{-4 t}t^{-2}}{e^{-4 t}} d t \\ &=\int t^{-2} d t \\ &=-t^{-1} \end{aligned}
Put all the values in equation $Y(t)=u_{1}(t) y_{1}(t)+u_{2}(t) y_{2}(t)$ for particular solution.
\begin{aligned} Y(t) &=u_{1}(t) y_{1}(t)+u_{2}(t) y_{2}(t) \\ &=(-\ln t)\left(e^{-2 t}\right)+\left(t e^{-2 t}\right)\left(-t^{-1}\right) \\ &=-e^{-2 t} \ln t-e^{-2 t} \end{aligned}
So, the general solution of the equation is:
$$\begin{array}{l}{y=y_{c}(t)+Y(t)} \\ {=c e^{-2 t}+c_{2} t e^{-2 t}-e^{-2 t} \ln t-e^{-2 t}} \\ {=(c-1) e^{-2 t}+c_{2} t e^{-2 t}-e^{-2 t} \ln t} \\ {=c_{1} e^{-2 t}+c_{2} t e^{-2 t}-e^{-2 t} \ln t}\end{array}$$
$$(c-1=c_1)$$
Therefore, the required general solution is
$$y(t)=c_{1} e^{-2 t}+c_{2} t e^{-2 t}-e^{-2 t} \ln t$$
2
##### Quiz-4 / TUT0601 Quiz4
« on: October 18, 2019, 09:44:50 PM »
Find the general solution of given equation:
$$y^{\prime \prime}+y^{\prime}-6 y=12 e^{3 t}+12 e^{-2 t}$$
Step1. To find the general solution, get the homogeneous equation
$$y^{\prime \prime}+y^{\prime}-6 y=0$$
The characteristic equation is
$$r^{2}+r-6=0$$
$$(r+3)(r-2)=0$$
$$r=2, -3$$
Thus, the solution of homogeneous differential equation $y^{\prime \prime}+y^{\prime}-6 y=0$ is
$$y_{c}(t)=c_{1} e^{-3 t}+c_{2} e^{2 t}$$
Step2. To find the particular solution, use undetermined coefficients method.
Suppose $y_{1}(t)=A e^{3_{t}}$ is a function satisfying the equation
$$y_{1}^{\prime \prime}+y_{1}^{\prime}-6 y_{1}=12 e^{3t}$$
Then,
$$\begin{array}{l}{y_{1}(t)=A e^{3 t}} \\ {y_{1}^{\prime}(t)=3 A e^{3 t}} \\ {y_{1}^{\prime \prime}(t)=9 A e^{3 t}}\end{array}$$
$$9 A e^{3 t}+3 A e^{3 t}-6 A e^{3 t}=12 e^{3 t}$$
$$6 A e^{3 t}=12 e^{3 t}$$
$$A=2$$
So, the solution for $y_{1}(t)$ is
$$y_{1}(t)=2 e^{3 t}$$
Now suppose $y_{2}(t)=B e^{-2 t}$ to satisfies the equation
$$y_{1}^{\prime \prime}+y_{1}^{\prime}-6 y_{1}=12 e^{-2 t}$$
Then,
$$y_{2}(t)=B e^{-2 t}$$
$$y_{2}^{\prime}=-2 B e^{-2 t}$$
$$y_{2}^{\prime \prime}=4 B e^{-2 t}$$
$$4 B e^{-2 t}-2 B e^{-2 t}-6 B e^{-2 t}=12 e^{-2 t}$$
$$-4 B e^{-2 t}=12 e^{-2 t}$$
$$B=-3$$
So, the solution for $y_{2}(t)$ is
$$y_{2}(t)=-3 e^{-2 t}$$
Therefore, the general solution for the non-homogeneous differential equation is
$$y=y_{c}(t)+y_{1}(t)+y_{2}(t)$$
$$y=c_{1} e^{-3 t}+c_{2} e^{2 t}+2 e^{3 t}-3 e^{-2 t}$$
3
##### Quiz-3 / TUT0601 Quiz3
« on: October 11, 2019, 09:47:00 PM »
Find the Wronskian of two solutions of the given differential equation without solving the
equation.
$$\cos (t) y^{\prime \prime}+\sin (t) y^{\prime}-t y=0$$
First, we divide both sides of the equation by $\cos (t):$
$$y^{\prime \prime}+\tan (t) y^{\prime}-\frac{t}{\cos (t)} y=0$$
Now the given second-order differential equation has the form:
$$L[y]=y^{\prime \prime}+p(t) y^{\prime}+q(t) y=0$$
Noting if we let $p(t)=\tan (t)$ and $q(t)=-\frac{t}{\cos (t)},$ then $p(t)$ is continuous
everywhere except at $\frac{\pi}{2}+k \pi,$ where $k=0,1,2, \ldots$ and $q(t)$ is also continuous
everywhere except at $t=0$.
Therefore, by Abel's Theorem: the Wronskian $W\left[y_{1}, y_{2}\right](t)$ is given by
\begin{aligned} W\left[y_{1}, y_{2}\right](t) &=cexp\left(-\int p(t) d t\right) \\ &=cexp\left(-\int \tan (t) d t\right) \\ &=c e^{\ln |\cos (t)|} \\ &=c\cos (t) \end{aligned}
4
##### Quiz-2 / TUT0601 Quiz2
« on: October 05, 2019, 06:31:35 PM »
Find an integrating factor and solve the given equation.
$$\left(3 x+\frac{6}{y}\right)+\left(\frac{x^{2}}{y}+3 \frac{y}{x}\right) \frac{d y}{d x}=0$$
We want to find an integrating factor $\mu$ as a function of $xy$ such that
$(\mu M)_{y}=(\mu N)_{x}$, Let $z=xy$. Thus, $\mu(x y)=\mu(z(x, y))$ Then
$\mu_{x}(x y)=\frac{d \mu}{d z} \frac{\partial z}{\partial x}=y \frac{d \mu}{d z}$
$\mu_{y}(x y)=\frac{d \mu}{d z} \frac{\partial z}{\partial y}=x \frac{d \mu}{d z}$
Therefore,
$$(\mu M)_{y}=(\mu N)_{x}$$
$$\mu M_{y}+x M \frac{d \mu}{d z}=\mu N_{x}+y N \frac{d \mu}{d z}$$
$$\mu\left(M_{y}-N_{x}\right)=\frac{d \mu}{d z}(y N-x M)$$
$$\frac{\mathrm{d} \mu}{\mathrm{d} z}=\mu\left(\frac{N_{x}-M_{y}}{x M-y N}\right)$$
Therefore,
$\mu(z)=\exp \left(\int R(z) \mathrm{d} z\right)$
\quad where $R(z)=R(x y)=\frac{N_{x}-M_{y}}{x M-y N}$
Returning to our original differential equation, let
$M(x, y)=3 x+\frac{6}{y}$ \quad and \quad $N(x, y)=\frac{x^{2}}{y}+3 \frac{y}{x}=0$
Then
$\frac{\partial}{\partial y} M(x, y)=\frac{-6}{y^{2}}$ \quad and \quad $\frac{\partial}{\partial x} N(x, y)=\frac{2 x}{y}-\frac{3 y}{x^{2}}$
We can see that this equation is not exact, however, note that
$$\frac{N_{x}-M_{y}}{x M-y N}=\frac{\frac{2 x}{y}-\frac{3 y}{x^{2}}+\frac{6}{y^{2}}}{x\left(3 x+\frac{6}{y}\right)-y\left(\frac{x^{2}}{y}+3 \frac{y}{x}\right)}=\frac{\frac{2 x}{y}-\frac{3 y}{x^{2}}+\frac{6}{y^{2}}}{2 x^{2}+\frac{6 x}{y}-\frac{3 y^{2}}{x}}=\frac{\frac{2 x}{y}-\frac{3 y}{x^{2}}+\frac{6}{y^{2}}}{x y\left(\frac{2 x}{y}-\frac{3 y}{x^{2}}+\frac{6}{y^{2}}\right)}=\frac{1}{x y}$$
Let $xy=z$
Thus, we have an integrating factor
$$\mu(x y)=\exp \left(\int \frac{1}{z} \mathrm{d} z\right)=e^{\log |z|}=z=x y$$
Multiplying the original differential equation through by our integrating factor, we have
$$\left(3 x^{2} y+6 x\right)+\left(x^{3}+3 y^{2}\right) \frac{d y}{d x}=0$$
We can see that this differential equation is exact because
$$\frac{\partial}{\partial y}\left(3 x^{2} y+6 x\right)=3 x^{2}=\frac{\partial}{\partial x}\left(x^{3}+3 y^{2}\right)$$
Thus, there exists a function $\psi(x, y)$ such that
\psi_{x}(x, y)=3 x^{2} y+6x
\tag{1}
\psi_{y}(x, y)=x^{3}+3 y^{2}
\tag{2}
Integrating (1) with respect to $x$, we get
$$\psi(x, y)=x^{3} y+3 x^{2}+h(y)$$
for some function $h$ of $y$. Next, differentiating with respect to $y,$ and equating
with ( 2)$,$ we get
$$\psi_{y}(x, y)=x^{3}+h^{\prime}(y)$$
Therefore,
$$h^{\prime}(y)=3 y^{2}$$
$$h(y)=y^{3}$$
and we have
$$\psi(x, y)=x^{3} y+3 x^{2}+y^{3}$$
Thus, the solutions of the differential equation are given implicitly by
$$x^{3} y+3 x^{2}+y^{3}=C$$
5
##### Quiz-1 / TUT0601 Quiz1
« on: September 27, 2019, 11:35:07 PM »
Solve the given differential equation.
The given differential equation is
$\frac{d y}{d x}=\frac{x-e^{-x}}{y+e^{y}}$
i.e. $\left(y+e^{y}\right) \frac{d y}{d x}=\left(x-e^{-x}\right)$
or $\left(e^{-x}-x\right)+\left(y+e^{y}\right) \frac{d y}{d x}=0$
This equation is of the form $M(x)+N(y) \frac{d y}{d x}=0$ and hence is separable
On rewriting this equation we have
$\left(y+e^{y}\right) d y=\left(x-e^{-x}\right) d x$
On integrating $\int\left(y+e^{y}\right) d y=\int\left(x-e^{-x}\right) d x$
i.e. $\frac{y^{2}}{2}+e^{y}=\frac{x^{2}}{2}+e^{-x}+c^{\prime}$
where $c'$ is an arbitrary constant of integration
i.e. $y^{2}+2 e^{y}=x^{2}+2 e^{-x}+2 c^{\prime}$
Put $2 c^{\prime}=c$, another arbitrary
i.e. $y^{2}-x^{2}+2\left(e^{y}-e^{-x}\right)=c$ is the required solution
where $y+e^{y} \neq 0$
Pages: [1] | 3,894 | 9,048 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.1875 | 4 | CC-MAIN-2021-04 | latest | en | 0.624112 |
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# ENGINEERING ELECTROMAGNETICS SOLUTION MANUAL
Engineering Electromagnetics 7th Edition William H. Hayt
Internet Archive BookReader Engineering Electromagnetics 7th Edition William H. Hayt Solution Manual
Solution Manual of Engineering Electromagnetics.. - scoop
Nov 28, 2018Solution Manual of Engineering Electromagnetics 8th Edition by William H. Hayt, John A. Buck Chapter Buy Chapter Buy Free Sample Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Coming Soon Chapter 7 Chapter 8 Chapter 9 Chapter 10 Coming Soon. Solution Manual of Engineering Electromagnetics 8th Edition by | Solution ManualsAuthor: Arsalan Malik
Solutions Manual for Engineering Electromagnetics 8th
Download Full Solutions Manual for Engineering Electromagnetics 8th Edition by William H. Hayt. ISBN-13 9780073380667 ISBN-10 0073380660. Solutions Manual (Answers Key) Mean? By Solution Manual (SM) we mean Comprehensive solutions to end of each chapter’s problems which also called as Instructor Solution Manual (ISM).
engineering electromagnetics 8th edition solution manual
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(PDF) Solutions Manual Engineering Electromagnetics 8th
Solutions Manual Engineering Electromagnetics 8th Edition Hayt. Mohammed Ksheer. Full file at https://fratstock CHAPTER 2 2.1. Three point charges are positioned in the x-y plane as follows: 5nC at y = 5 cm, -10 nC at y = 5 cm, 15 nC at x = 5 cm. Find the required x-y coordinates of a 20-nC fourth charge that will produce a zero electric
Solutions Manual for Engineering Electromagnetics and - US
Solutions Manual for Engineering Electromagnetics and Waves, 2nd Edition Download Instructor's Solutions Manual - PDF (application/zip) (25) Download Addendum - PDF (application/zip) (1)Availability: LiveISBN-13: 9780132662796Format: On-line SupplementOnline purchase price: \$0
Engineering Electromagnetics - 7th Edition - William H | 620 | 2,525 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.8125 | 3 | CC-MAIN-2019-22 | latest | en | 0.753914 |
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May 19, 2013
# Posts by Ben
Total # Posts: 598
french
please unscramble this word.. the word is eeienhtocasr
french
thanks
french
can you unscamble eeienhtocasr
math
two thirds of 9 is 6 and one third of 18 is 6 if you draw a gragh of 9 divide it into three sections.each section is a third,two of those sections equal two thirds which is 6.now a graph of 18 divided into 3 sections each section is 6.each section represents one third. 6 is th...
chemistry
Write the major products with the correct regiochemistry and stereochemistry for 1- ethylcyclohexene and the following reagents: a) NBS, H2O, DMSO if someone could atleast direct me to a site or anything that would be great.
Physics
In the figure below, two blocks are connected over a pulley. The mass of block A is 12 kg, and the coefficient of kinetic friction between A and the incline is 0.22. Angle θ is 30°. Block A slides down the incline at constant speed. What is the mass of block B?
calculus
Consider the graph of y2 = x(4 − x)2 (see figure). Find the volumes of the solids that are generated when the loop of this graph is revolved about each of the following
chemistry
Mechanism for the reaction between 3-methyl-2-pentene and HCl. is this right? %img836.imageshack.us/i/questionk%.jpg/% * delete the percentages for the link to work, its the only way i could post a link
chemistry
Write down the correct mechanism for the reaction between 3-methyl-2-pentene and HCL?
Econ
Consider the following consumption function. C = 200 + .75(DPI), Where C is consumption, autonomous spending is 200, the MPC is .75, and DPI is disposable personal income. Using the Graph below, graph the Consumption Function. C 100 0 = Income On the Graph, label the point whe...
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https://brainly.com/question/96912 | 1,485,214,084,000,000,000 | text/html | crawl-data/CC-MAIN-2017-04/segments/1484560283301.73/warc/CC-MAIN-20170116095123-00449-ip-10-171-10-70.ec2.internal.warc.gz | 798,719,394 | 9,164 | 2014-08-19T21:56:43-04:00
### This Is a Certified Answer
Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
5Q - 7 = 2/3
You could begin by adding 7 to each side :
5Q = 7 and 2/3
The rest of it will be a lot easier if you change the right side
to a single fraction. Do you remember how to do that ?
7 is the same as 21/3 . So 7 and 2/3 is the same as 23/3 .
5Q = 23/3
Divide each side by 5 :
Q = 23 / (3 x 5) = 23/15 = 1 and 8/15 . | 214 | 672 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.875 | 4 | CC-MAIN-2017-04 | latest | en | 0.904687 |
https://www.bartleby.com/solution-answer/chapter-7-problem-1caq1-cardiopulmonary-anatomy-and-physiology-7th-edition/9781337794909/case-1-in-the-emergency-department-even-though-the-patients-paco2-was-very-high-539-mm-hg-the/8d229d1c-6664-11e9-8385-02ee952b546e | 1,571,856,829,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570987835748.66/warc/CC-MAIN-20191023173708-20191023201208-00283.warc.gz | 807,919,908 | 38,678 | Chapter 7, Problem 1CAQ1
### Cardiopulmonary Anatomy & Physiolo...
7th Edition
Des Jardins + 1 other
ISBN: 9781337794909
Chapter
Section
### Cardiopulmonary Anatomy & Physiolo...
7th Edition
Des Jardins + 1 other
ISBN: 9781337794909
Textbook Problem
# Case 1In the emergency department, even though the patient's Pa co 2 was very high (539 mm Hg), the CO Hb level of 47 percent (enhanced ______; impaired______) the hemoglobin's ability to carry oxygen.
Summary Introduction
To review:
The given blank space in the statement, “In the emergency department, even though the patient’s PaCO2 was very high (539 mm Hg), the COHb level of 47 percent (enhanced……; impaired…….) the hemoglobin’s ability to carry oxygen”.
Introduction:
The measurement of the partial pressure of carbon dioxide (PaCO2) and carboxyhemoglobin (COHb) in blood is important to test the function of the lungs and the risk of any future lung diseases. The level of high carbon dioxide and carbon monoxide will cause a decrease inflow of oxygen into the blood through the alveoli, which leads to less binding of oxygen with the hemoglobin molecule. This condition could cause conditions like anemia.
Explanation
Carbon dioxide and carbon monoxide are the gases that are not required by our body. The accumulation of these gases in the blood could be fatal. A high amount of carbon dioxide in blood causes hypercapnia and a high level of carbon monoxide in blood could cause carbon monoxide poisoning. In both the conditions, the inflow of oxygen decreases, which hinders the ability of hemoglobin to carry oxygen. The normal range of partial pressure of carbon dioxide in the blood is 40-45 mm Hg. The concentration of more than 40% of carbon monoxide in blood causes death
### Still sussing out bartleby?
Check out a sample textbook solution.
See a sample solution
#### The Solution to Your Study Problems
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Get Started | 476 | 2,021 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.03125 | 3 | CC-MAIN-2019-43 | latest | en | 0.837021 |
https://www.jiskha.com/display.cgi?id=1241652194 | 1,503,143,103,000,000,000 | text/html | crawl-data/CC-MAIN-2017-34/segments/1502886105341.69/warc/CC-MAIN-20170819105009-20170819125009-00430.warc.gz | 896,982,211 | 3,865 | # English
posted by .
1. I want something for writing;any pen or pencil will be OK.
2. I want something for writing with;any pen or pencil will be OK.
3.I want something for writing.
4. I want something for writing with.
(Are they correct? If I remove the latter part, are they correct? Check the four sentences, please.)
• English -
It was Winston Churchill who said "Never end a sentence a preposition with!" #2, I want something to write with = is more common.
#4 again, I want something to write with = is more common.
Sra
## Similar Questions
1. ### Writing
HI, I AM WRITING A RESEARCH PAPER ABOUT SOMETHING IN WASHINGTON DC. BUT THE PAPER HAS A TWIST TO IT. IT HAS TO BE A TOPIC THAT IS CONTOVERSIAL/CAN BE ARGUED ON BOTH SIDES. I WAS THINKING ABOUT DOING SOMETHING ON THE TOMB OF THE UNKNOWN …
2. ### English
1. I want something for writing;any pen or pencil will be OK. 2. I want something for writing with;any pen or pencil will be OK. 3.I want something for writing. 4. I want something for writing with. (Are they correct?
3. ### English
I have a question about quoting. If I have a sentence like: Something something something something "something". Do I quote it this way?
4. ### math
Upon examining the contents of 38 backpacks, it was found that 23 contained a black pen, 27 contained a blue pen, and 21 contained a pencil, 15 contained both a black pen and a blue pen, 12 contained both a black pen and a pencil, …
5. ### Critiques cont'd
So, if I spend about 5-6 minutes planning what else would I write except for a couple of bullet points about my two main examples?
6. ### Math
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7. ### Math
A pen costs twice as much as a pencil. If Susan spends \$ 1.50 for a pen and a pencil, how much does a pencil cost ?
8. ### Math
A pen costs twice as much as a pencil. If Susan spends \$ 1.50 for a pen and a pencil, how much does a pencil cost ?
9. ### linear equation and one variable
A pen costs 7 times as that of a pencil. If raju buys 3 pencils and 2pen what is the cost of pen and pencil
10. ### English
1. She wrote with a pencil. 2. She wrote in pencil. 3. She is writing a letter in pen. 4. She is writing a letter with a pen. 5. She is writing a letter in a pen. -------------------------------- Are they all grammatical except #5?
More Similar Questions | 657 | 2,529 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.609375 | 3 | CC-MAIN-2017-34 | latest | en | 0.913858 |
https://www.aqua-calc.com/one-to-one/density/gram-per-us-gallon/stone-per-cubic-yard/1 | 1,571,114,135,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570986655864.19/warc/CC-MAIN-20191015032537-20191015060037-00518.warc.gz | 836,202,715 | 8,642 | # 1 gram per US gallon [g/US gal] in stones per cubic yard
## g/US gal to st/yd³ unit converter of density
1 gram per US gallon [g/gal] = 0.03 stone per cubic yard [st/yd³]
### grams per US gallon to stones per cubic yard density conversion cards
• 1
through
25
grams per US gallon
• 1 g/gal to st/yd³ = 0.03 st/yd³
• 2 g/gal to st/yd³ = 0.06 st/yd³
• 3 g/gal to st/yd³ = 0.1 st/yd³
• 4 g/gal to st/yd³ = 0.13 st/yd³
• 5 g/gal to st/yd³ = 0.16 st/yd³
• 6 g/gal to st/yd³ = 0.19 st/yd³
• 7 g/gal to st/yd³ = 0.22 st/yd³
• 8 g/gal to st/yd³ = 0.25 st/yd³
• 9 g/gal to st/yd³ = 0.29 st/yd³
• 10 g/gal to st/yd³ = 0.32 st/yd³
• 11 g/gal to st/yd³ = 0.35 st/yd³
• 12 g/gal to st/yd³ = 0.38 st/yd³
• 13 g/gal to st/yd³ = 0.41 st/yd³
• 14 g/gal to st/yd³ = 0.45 st/yd³
• 15 g/gal to st/yd³ = 0.48 st/yd³
• 16 g/gal to st/yd³ = 0.51 st/yd³
• 17 g/gal to st/yd³ = 0.54 st/yd³
• 18 g/gal to st/yd³ = 0.57 st/yd³
• 19 g/gal to st/yd³ = 0.6 st/yd³
• 20 g/gal to st/yd³ = 0.64 st/yd³
• 21 g/gal to st/yd³ = 0.67 st/yd³
• 22 g/gal to st/yd³ = 0.7 st/yd³
• 23 g/gal to st/yd³ = 0.73 st/yd³
• 24 g/gal to st/yd³ = 0.76 st/yd³
• 25 g/gal to st/yd³ = 0.8 st/yd³
• 26
through
50
grams per US gallon
• 26 g/gal to st/yd³ = 0.83 st/yd³
• 27 g/gal to st/yd³ = 0.86 st/yd³
• 28 g/gal to st/yd³ = 0.89 st/yd³
• 29 g/gal to st/yd³ = 0.92 st/yd³
• 30 g/gal to st/yd³ = 0.95 st/yd³
• 31 g/gal to st/yd³ = 0.99 st/yd³
• 32 g/gal to st/yd³ = 1.02 st/yd³
• 33 g/gal to st/yd³ = 1.05 st/yd³
• 34 g/gal to st/yd³ = 1.08 st/yd³
• 35 g/gal to st/yd³ = 1.11 st/yd³
• 36 g/gal to st/yd³ = 1.14 st/yd³
• 37 g/gal to st/yd³ = 1.18 st/yd³
• 38 g/gal to st/yd³ = 1.21 st/yd³
• 39 g/gal to st/yd³ = 1.24 st/yd³
• 40 g/gal to st/yd³ = 1.27 st/yd³
• 41 g/gal to st/yd³ = 1.3 st/yd³
• 42 g/gal to st/yd³ = 1.34 st/yd³
• 43 g/gal to st/yd³ = 1.37 st/yd³
• 44 g/gal to st/yd³ = 1.4 st/yd³
• 45 g/gal to st/yd³ = 1.43 st/yd³
• 46 g/gal to st/yd³ = 1.46 st/yd³
• 47 g/gal to st/yd³ = 1.49 st/yd³
• 48 g/gal to st/yd³ = 1.53 st/yd³
• 49 g/gal to st/yd³ = 1.56 st/yd³
• 50 g/gal to st/yd³ = 1.59 st/yd³
• 51
through
75
grams per US gallon
• 51 g/gal to st/yd³ = 1.62 st/yd³
• 52 g/gal to st/yd³ = 1.65 st/yd³
• 53 g/gal to st/yd³ = 1.69 st/yd³
• 54 g/gal to st/yd³ = 1.72 st/yd³
• 55 g/gal to st/yd³ = 1.75 st/yd³
• 56 g/gal to st/yd³ = 1.78 st/yd³
• 57 g/gal to st/yd³ = 1.81 st/yd³
• 58 g/gal to st/yd³ = 1.84 st/yd³
• 59 g/gal to st/yd³ = 1.88 st/yd³
• 60 g/gal to st/yd³ = 1.91 st/yd³
• 61 g/gal to st/yd³ = 1.94 st/yd³
• 62 g/gal to st/yd³ = 1.97 st/yd³
• 63 g/gal to st/yd³ = 2 st/yd³
• 64 g/gal to st/yd³ = 2.04 st/yd³
• 65 g/gal to st/yd³ = 2.07 st/yd³
• 66 g/gal to st/yd³ = 2.1 st/yd³
• 67 g/gal to st/yd³ = 2.13 st/yd³
• 68 g/gal to st/yd³ = 2.16 st/yd³
• 69 g/gal to st/yd³ = 2.19 st/yd³
• 70 g/gal to st/yd³ = 2.23 st/yd³
• 71 g/gal to st/yd³ = 2.26 st/yd³
• 72 g/gal to st/yd³ = 2.29 st/yd³
• 73 g/gal to st/yd³ = 2.32 st/yd³
• 74 g/gal to st/yd³ = 2.35 st/yd³
• 75 g/gal to st/yd³ = 2.39 st/yd³
• 76
through
100
grams per US gallon
• 76 g/gal to st/yd³ = 2.42 st/yd³
• 77 g/gal to st/yd³ = 2.45 st/yd³
• 78 g/gal to st/yd³ = 2.48 st/yd³
• 79 g/gal to st/yd³ = 2.51 st/yd³
• 80 g/gal to st/yd³ = 2.54 st/yd³
• 81 g/gal to st/yd³ = 2.58 st/yd³
• 82 g/gal to st/yd³ = 2.61 st/yd³
• 83 g/gal to st/yd³ = 2.64 st/yd³
• 84 g/gal to st/yd³ = 2.67 st/yd³
• 85 g/gal to st/yd³ = 2.7 st/yd³
• 86 g/gal to st/yd³ = 2.74 st/yd³
• 87 g/gal to st/yd³ = 2.77 st/yd³
• 88 g/gal to st/yd³ = 2.8 st/yd³
• 89 g/gal to st/yd³ = 2.83 st/yd³
• 90 g/gal to st/yd³ = 2.86 st/yd³
• 91 g/gal to st/yd³ = 2.89 st/yd³
• 92 g/gal to st/yd³ = 2.93 st/yd³
• 93 g/gal to st/yd³ = 2.96 st/yd³
• 94 g/gal to st/yd³ = 2.99 st/yd³
• 95 g/gal to st/yd³ = 3.02 st/yd³
• 96 g/gal to st/yd³ = 3.05 st/yd³
• 97 g/gal to st/yd³ = 3.09 st/yd³
• 98 g/gal to st/yd³ = 3.12 st/yd³
• 99 g/gal to st/yd³ = 3.15 st/yd³
• 100 g/gal to st/yd³ = 3.18 st/yd³
• g/gal stands for g/US gal
#### Foods, Nutrients and Calories
AWARD WINNING SALSA, UPC: 702218347693 weigh(s) 236.7 gram per (metric cup) or 7.9 ounce per (US cup), and contain(s) 18 calories per 100 grams or ≈3.527 ounces [ weight to volume | volume to weight | price | density ]
BITE SIZE CRACKERS, UPC: 705194124098 contain(s) 400 calories per 100 grams or ≈3.527 ounces [ price ]
#### Gravels, Substances and Oils
CaribSea, Marine, Arag-Alive, Bimini Pink weighs 1 441.7 kg/m³ (90.00239 lb/ft³) with specific gravity of 1.4417 relative to pure water. Calculate how much of this gravel is required to attain a specific depth in a cylindricalquarter cylindrical or in a rectangular shaped aquarium or pond [ weight to volume | volume to weight | price ]
Arsenic trioxide, claudetite [As2O3] weighs 3 740 kg/m³ (233.48057 lb/ft³) [ weight to volume | volume to weight | price | mole to volume and weight | density ]
Volume to weightweight to volume and cost conversions for Refrigerant R-417C, liquid (R417C) with temperature in the range of -51.12°C (-60.016°F) to 68.34°C (155.012°F)
#### Weights and Measurements
A microinch per second squared (µin/s²) is a non-SI (non-System International) measurement unit of acceleration
The fuel consumption or fuel economy measurement is used to estimate gas mileage and associated fuel cost for a specific vehicle.
g/Ų to oz/pm² conversion table, g/Ų to oz/pm² unit converter or convert between all units of surface density measurement.
#### Calculators
Online Food Calculator. Food Weight to Volume Conversions | 2,652 | 5,452 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.921875 | 3 | CC-MAIN-2019-43 | latest | en | 0.25318 |
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### Evil spirits
Andrey Selivanov's S#3 from the "Show me your problem" section reminds me of the following S#2 which keeps haunting me from its first day of appearance, yowling at me: "unfinished business, am I not?"
To translate this into a concrete question: can the idea be shown without the Zeroposition?
A "normal" twin should do the trick, not to speak of a simple try (or set) & solution.
Could anyone make the evil spirits go away?
Yochanan AFEK
Uri AVNER *
Die Schwalbe 1982
3rd Prize
(= 10+10 )
s#2
Zeroposition
a) +WPa4
b) +BPb5
a)
1.d3!
1…Kxb6 2.Qc5+ Kxc5# (A)
1…Rxb6 2.Qb5+ Rxb5# (B)
1…axb6 2.Bxa6 b5# (C)
b)
1.d4!
1…Kxb6 2.Qxb5+ Kxb5# (B)
1…Rxb6 2.Ba6 R~# (C)
1…axb6 2.Qc5+ bxc5# (A)
* Order of names above the diagram has been swapped by Administrator after the request by author of this post.
Hi Uri,
Because of the complexity of the scheme it's very hard to imagine that it's possible to find position without zero position. The best position that I found without zero positions is:
(= 9+9 )
a) diagram
b) -bPb5
a) 1.Ng7!
b) 1.Ndc3!
Of course this change a genre I am sure this is not what you expected but anyway I decided to reply. This may help somebody to come up with an ortodox version without zero position twin.
Miodrag , in diagram position bPf7 is not necessary. In b) works bPb5->f7.
(4) Posted by Uri Avner [Wednesday, Nov 1, 2006 00:39]
Hi Misha,
Thanks for trying. This, evidently, is not exactly what I had in mind, but it serves to show how close the orthodox position is to the possibility of a "regular" twin. Not much is needed, and yet...
Sorry for the late response (I have just joined as a member): Another evil spirit should wonder: was it the original order of composers above the diagram? apparently not and not without a good reason.
To shake off at least certain spirits, I have no problem to switch names.
Matter settled | 624 | 2,037 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.53125 | 3 | CC-MAIN-2022-21 | latest | en | 0.893787 |
https://www.calculatoratoz.com/en/residual-shear-stress-in-shaft(r-lies-between-material-constant-and-r2)-calculator/Calc-37291 | 1,675,228,815,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764499911.86/warc/CC-MAIN-20230201045500-20230201075500-00866.warc.gz | 653,550,288 | 48,026 | ## Residual shear stress in shaft(r lies between material constant and r2) Solution
STEP 0: Pre-Calculation Summary
Formula Used
ΞΆep_res = π0*(1-(4*r*(1-((1/4)*(Ο/r2)^3)-(((3*r1)/(4*Ο))*(r1/r2)^3)))/(3*r2*(1-(r1/r2)^4)))
This formula uses 6 Variables
Variables Used
Residual shear stress in Elasto plastic yielding - (Measured in Pascal) - Residual shear stress in Elasto plastic yielding can be defined as the algebraic sum of applied stress and recovery stress.
Yield stress in shear - (Measured in Pascal) - Yield stress in shear is the yield stress of the shaft in shear condition.
Radius Yielded - (Measured in Meter) - Radius Yielded is the yielded portion of shaft under load.
Radius of plastic front - (Measured in Meter) - Radius of plastic front is the difference between the outer radius of shaft and depth yields plastically.
Outer radius of shaft - (Measured in Meter) - Outer radius of shaft is the external radius of shaft.
Inner radius of shaft - (Measured in Meter) - Inner radius of shaft is the internal radius of shaft.
STEP 1: Convert Input(s) to Base Unit
Yield stress in shear: 145 Megapascal --> 145000000 Pascal (Check conversion here)
Radius Yielded: 60 Millimeter --> 0.06 Meter (Check conversion here)
Radius of plastic front: 80 Millimeter --> 0.08 Meter (Check conversion here)
Outer radius of shaft: 100 Millimeter --> 0.1 Meter (Check conversion here)
Inner radius of shaft: 40 Millimeter --> 0.04 Meter (Check conversion here)
STEP 2: Evaluate Formula
Substituting Input Values in Formula
ΞΆep_res = π0*(1-(4*r*(1-((1/4)*(Ο/r2)^3)-(((3*r1)/(4*Ο))*(r1/r2)^3)))/(3*r2*(1-(r1/r2)^4))) --> 145000000*(1-(4*0.06*(1-((1/4)*(0.08/0.1)^3)-(((3*0.04)/(4*0.08))*(0.04/0.1)^3)))/(3*0.1*(1-(0.04/0.1)^4)))
Evaluating ... ...
ΞΆep_res = 44047619.0476191
STEP 3: Convert Result to Output's Unit
44047619.0476191 Pascal -->44.0476190476191 Megapascal (Check conversion here)
44.0476190476191 Megapascal <-- Residual shear stress in Elasto plastic yielding
(Calculation completed in 00.031 seconds)
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## < 7 Residual Stresses For Idealized Stress Strain Law Calculators
Residual shear stress in shaft(r lies between r1 and material constant)
β
## Residual shear stress in shaft(r lies between r1 and material constant)
Formula
"ΞΆ"_{"ep_res"} = (("π"_{"0"}*"r"/"Ο")- (((4*"π"_{"0"}*"r")/(3*"r"_{"2"}*(1-("r"_{"1"}/"r"_{"2"})^4)))*(1-(1/4)*("Ο"/"r"_{"2"})^3-((3*"r"_{"1"})/(4*"Ο"))*("r"_{"1"}/"r"_{"2"})^3)))
Example
"7.797619MPa"=(("145MPa"*"60mm"/"80mm")- (((4*"145MPa"*"60mm")/(3*"100mm"*(1-("40mm"/"100mm")^4)))*(1-(1/4)*("80mm"/"100mm")^3-((3*"40mm")/(4*"80mm"))*("40mm"/"100mm")^3)))
Calculator
LaTeX
Residual angle of twist for Elasto plastic case
β
## Residual angle of twist for Elasto plastic case
Formula
"ΞΈ"_{"res"} = ("π"_{"0"}/("G"*"Ο"))*(1-((4*"Ο")/(3*"r"_{"2"}))* ((1-(1/4)*("Ο"/"r"_{"2"})^3-((3*"r"_{"1"})/(4*"Ο"))*("r"_{"1"}/"r"_{"2"})^3)/(1-("r"_{"1"}/"r"_{"2"})^4)))
Example
"0.001547rad"=("145MPa"/("84e3MPa"*"80mm"))*(1-((4*"80mm")/(3*"100mm"))* ((1-(1/4)*("80mm"/"100mm")^3-((3*"40mm")/(4*"80mm"))*("40mm"/"100mm")^3)/(1-("40mm"/"100mm")^4)))
Calculator
LaTeX
Residual shear stress in shaft(r lies between material constant and r2)
β
## Residual shear stress in shaft(r lies between material constant and r2)
Formula
"ΞΆ"_{"ep_res"} = "π"_{"0"}*(1-(4*"r"*(1-((1/4)*("Ο"/"r"_{"2"})^3)-(((3*"r"_{"1"})/(4*"Ο"))*("r"_{"1"}/"r"_{"2"})^3)))/(3*"r"_{"2"}*(1-("r"_{"1"}/"r"_{"2"})^4)))
Example
"44.04762MPa"="145MPa"*(1-(4*"60mm"*(1-((1/4)*("80mm"/"100mm")^3)-(((3*"40mm")/(4*"80mm"))*("40mm"/"100mm")^3)))/(3*"100mm"*(1-("40mm"/"100mm")^4)))
Calculator
LaTeX
Residual angle of twist in fully plastic case
β
## Residual angle of twist in fully plastic case
Formula
"ΞΈ"_{"res"} = ("π"_{"0"}/("G"*"r"_{"1"}))*(1-((4*"r"_{"1"})/(3*"r"_{"2"}))*((1-("r"_{"1"}/"r"_{"2"})^3)/(1-("r"_{"1"}/"r"_{"2"})^4)))
Example
"0.021046rad"=("145MPa"/("84e3MPa"*"40mm"))*(1-((4*"40mm")/(3*"100mm"))*((1-("40mm"/"100mm")^3)/(1-("40mm"/"100mm")^4)))
Calculator
LaTeX
Recovery Elasto plastic Torque
β
## Recovery Elasto plastic Torque
Formula
"T"_{"rec"} = -(pi*"π"_{"0"}*(("Ο"^3/2)*(1-("r"_{"1"}/"Ο")^4)+(2*"r"_{"2"}^3/3)*(1-("Ο"/"r"_{"2"})^3)))
Example
"-257526821.790267N*mm"=-(pi*"145MPa"*(("80mm"^3/2)*(1-("40mm"/"80mm")^4)+(2*"100mm"^3/3)*(1-("80mm"/"100mm")^3)))
Calculator
LaTeX
Residual shear stress in shaft for fully plastic case
β
## Residual shear stress in shaft for fully plastic case
Formula
"ΞΆ"_{"f_res"} = "π"_{"0"}*(1-((4*"r"*(1-("r"_{"1"}/"r"_{"2"})^3))/(3*"r"_{"2"}*(1-("r"_{"1"}/"r"_{"2"})^4))))
Example
"33.57143MPa"="145MPa"*(1-((4*"60mm"*(1-("40mm"/"100mm")^3))/(3*"100mm"*(1-("40mm"/"100mm")^4))))
Calculator
LaTeX
Recovery Torque in Fully plastic case
β
## Recovery Torque in Fully plastic case
Formula
"T"_{"f_rec"} = -((2/3)*pi*"r"_{"2"}^3*"π"_{"0"}*(1-("r"_{"1"}/"r"_{"2"})^3))
Example
"-284251303.296805N*mm"=-((2/3)*pi*"100mm"^3*"145MPa"*(1-("40mm"/"100mm")^3))
Calculator
LaTeX
Fully plastic recovery torque = -((2/3)*pi*Outer radius of shaft^3*Yield stress in shear*(1-(Inner radius of shaft/Outer radius of shaft)^3))
## Residual shear stress in shaft(r lies between material constant and r2) Formula
ΞΆep_res = π0*(1-(4*r*(1-((1/4)*(Ο/r2)^3)-(((3*r1)/(4*Ο))*(r1/r2)^3)))/(3*r2*(1-(r1/r2)^4)))
## How residual stresses are generated in shafts?
When a shaft is twisted, it starts yielding once the shear stress crosses its yield limit. Torque applied may be elasto-plastic or fully plastic. This process is called LOADING. When the shaft so twisted is applied with a torque of same magnitude in the opposite direction, then the recovery of stress takes place. This process is called UNLOADING. The process of UNLOADING is always assumed to be elastic following a linear stress-strain relation. But for a plastically twisted shaft, the recovery doesnβt take place fully. Therefore some amount of stresses are left over or locked. Such stresses are called the residual stresses.
## How to Calculate Residual shear stress in shaft(r lies between material constant and r2)?
Residual shear stress in shaft(r lies between material constant and r2) calculator uses Residual shear stress in Elasto plastic yielding = Yield stress in shear*(1-(4*Radius Yielded*(1-((1/4)*(Radius of plastic front/Outer radius of shaft)^3)-(((3*Inner radius of shaft)/(4*Radius of plastic front))*(Inner radius of shaft/Outer radius of shaft)^3)))/(3*Outer radius of shaft*(1-(Inner radius of shaft/Outer radius of shaft)^4))) to calculate the Residual shear stress in Elasto plastic yielding, The Residual shear stress in shaft(r lies between material constant and r2) formula is defined as the algebraic sum of applied stress and recovery stress. Residual shear stress in Elasto plastic yielding is denoted by ΞΆep_res symbol.
How to calculate Residual shear stress in shaft(r lies between material constant and r2) using this online calculator? To use this online calculator for Residual shear stress in shaft(r lies between material constant and r2), enter Yield stress in shear (π0), Radius Yielded (r), Radius of plastic front (Ο), Outer radius of shaft (r2) & Inner radius of shaft (r1) and hit the calculate button. Here is how the Residual shear stress in shaft(r lies between material constant and r2) calculation can be explained with given input values -> 44.04762 = 145000000*(1-(4*0.06*(1-((1/4)*(0.08/0.1)^3)-(((3*0.04)/(4*0.08))*(0.04/0.1)^3)))/(3*0.1*(1-(0.04/0.1)^4))).
### FAQ
What is Residual shear stress in shaft(r lies between material constant and r2)? | 2,755 | 7,930 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.90625 | 4 | CC-MAIN-2023-06 | latest | en | 0.758779 |
https://www.gradesaver.com/textbooks/math/algebra/algebra-a-combined-approach-4th-edition/chapter-12-section-12-5-logarithmic-functions-exercise-set-page-875/54 | 1,534,493,423,000,000,000 | text/html | crawl-data/CC-MAIN-2018-34/segments/1534221211933.43/warc/CC-MAIN-20180817065045-20180817085045-00463.warc.gz | 880,165,059 | 14,459 | ## Algebra: A Combined Approach (4th Edition)
$x=-3$
We know that if $b\gt0$ and $b\ne1$, then $y=log_{b}x$ is equivalent to $x=b^{y}$ (where $x\gt0$ and $y$ is a real number). Therefore, $log_{5}\frac{1}{125}=x$ is equivalent to $5^{x}=\frac{1}{125}$. Therefore, $x=-3$, because $5^{-3}=\frac{1}{5^{3}}=\frac{1}{125}$. | 129 | 320 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.125 | 4 | CC-MAIN-2018-34 | longest | en | 0.643042 |
https://forum.ngsolve.org/t/dimensions-dont-match-between-numpy-array-and-ngsolve-lf/2385 | 1,695,949,432,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233510462.75/warc/CC-MAIN-20230928230810-20230929020810-00036.warc.gz | 289,037,022 | 4,367 | Dimensions don't match between numpy array and ngsolve LF
Hello,
I have a problem of shapes with array on NGSolve / Numpy.
The problem is the following, I have a “pwc” Compressed on a certain specific region (reducing the size of all element linked to pwc). Then I calculate an Integral element_wise, but the dimensions of this integral is on all the mesh and not only where I defined it…
``````pwc = Compress(L2(mesh,order=0,definedon=mesh.Materials(VARIABLE)))
gfRho = GridFunction(pwc)
Ates = Integrate(gfRho,mesh, element_wise=True,definedon=mesh.Materials(VARIABLE))
Ates = np.array(Ates)
dJdrho = LinearForm(pwc)
dJdrho.vec.FV().NumPy()[:] = Ates # ERROR, dimensions don't match Ates >> dJdrho
``````
The previous code is a simplified one, I really need to convert the Integrate to a numpy array, I also really need to compress over the definition domain pwc. Also gfRho has a certain value I don’t show in the previous simplified code, but is also defined on pwc…
The question is : How can I “compress” the integrale on a specific domain, so that Ates and dJdrho can have the same size.
Thank you a lot !
Yes elementwise integral gives value for each element. You can create a mask for the elements of your material in numpy:
``````# Get array with 1 == material, 0 other material
mat = mesh.Materials(VARIABLE)
els = sum((mesh.ngmesh.Elements2D().NumPy()["index"] == index+1 for index, val in enumerate(mat.Mask()) if val), start=np.zeros(mesh.ne))
# Set only on selected elements
dJdrho.vec.FV().NumPy()[:] = Ates.NumPy()[els.astype(bool)]
``````
Best
Christopher | 429 | 1,585 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.859375 | 3 | CC-MAIN-2023-40 | latest | en | 0.734719 |
http://ncvm3.books.nba.co.za/chapter/unit-5-solve-quadratic-inequalities/ | 1,723,207,049,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640763425.51/warc/CC-MAIN-20240809110814-20240809140814-00847.warc.gz | 21,948,426 | 27,459 | Functions and algebra: Solve algebraic equations and inequalities
# Unit 5: Solve quadratic inequalities
Dylan Busa
### Unit outcomes
By the end of this unit you will be able to:
• Find the critical values.
• Solve quadratic inequalities using the table and graphical methods.
• Represent the solution using set builder notation, interval notation and on a number line.
## What you should know
Before you start this unit, make sure you can:
Here is a short self-assessment to make sure you have the basic skills you need to proceed with this unit.
Solve for $\scriptsize x$ and show your answer as indicated:
1. $\scriptsize 2-3x\le 3-5x$ (show your answer on a number line)
2. $\scriptsize \displaystyle \frac{{7x+3}}{4} \lt 4(2x-7)$ (show your answer using interval and set builder notation)
Solutions
1. .
\scriptsize \begin{align*}2-3x&\le 3-5x\\ \therefore -3x & \le 1-5x&& \text{Subtract }2\text{ from both sides of the inequality}\\ \therefore 2x & \le 1 &&\text{Add }5x\text{ to both sides of the inequality}\\ \therefore x & \le \displaystyle \frac{1}{2}\end{align*}
2. .
\scriptsize \begin{align*}\displaystyle \frac{{7x+3}}{4} &\lt 4(2x-7)\\ \therefore 7x+3 & \lt 16(2x-7)\\ \therefore 7x+3 & \lt 32x-112\\ \therefore -25x & \lt -115 &&\text{Remember, we need to change the direction of the inequality sign}\\ &&& \text{when multipling by a negative number}\\ \therefore x & \gt \displaystyle \frac{{115}}{{25}}\\ \therefore x & \gt \displaystyle \frac{{23}}{5}\end{align*}Interval notation: $\scriptsize x\in (\displaystyle \frac{{23}}{5},\infty )$
Set builder notation: $\scriptsize \{x|x\in \mathbb{R},x \gt \displaystyle \frac{{23}}{5}\}$
## Introduction
We learnt how to solve linear inequalities in level 2 subject outcome 2.3 unit 3. We saw that solving for the unknown in an inequality is very similar to solving for an unknown in an equation.
In this unit, we are going to learn how to solve quadratic inequalities. Again, the basic techniques for solving inequalities and equations are very similar. The biggest difference is still the fact that we have to flip the inequality sign around if we multiply both sides of an inequality by a negative number.
## Solve simple quadratic inequalities
If $\scriptsize a{{x}^{2}}+bx+c=0$ is the standard form of a quadratic equation, then it stands to reason that the following are the standard forms of a quadratic inequality:
• $\scriptsize a{{x}^{2}}+bx+c\ge 0$
• $\scriptsize a{{x}^{2}}+bx+c \gt 0$
• $\scriptsize a{{x}^{2}}+bx+c\le 0$
• $\scriptsize a{{x}^{2}}+bx+c \lt 0$
We know that quadratic equations give us two solutions or roots. But how does this translate into the solutions to quadratic inequalities? Are there always two ranges?
A useful way to think about the solutions to a quadratic inequality is to think about where the graph of the quadratic equation lies in relation to the x-axis. Think about the quadratic function $\scriptsize f(x)={{x}^{2}}+3x-4$ (see figure 1).
We know that the solution to $\scriptsize {{x}^{2}}+3x-4=0$ gives us the x-intercepts (see Figure 1). This is where the function $\scriptsize f(x)=0$. The x-values of the x-intercepts are the roots. They are the specific values that make the function value zero.
Now, if we solve the quadratic inequality of $\scriptsize {{x}^{2}}+3x-4\le 0$, what x-values do you think will satisfy this inequality? What x-values will make $\scriptsize f(x)\le 0$?
You should be able to see that any value for $\scriptsize x$ that lies between the x-intercepts will result in $\scriptsize f(x)\le 0$ (that part of the graph of the function below the x-axis) (see Figure 2).
We can present the solution to the quadratic inequality $\scriptsize {{x}^{2}}+3x-4\le 0$ on a number line (see Figure 3).
We can also write the solution as $\scriptsize x\in [-4,1]$ (interval notation) or $\scriptsize \{x|x\in \mathbb{R},-4\le x\le 1\}$ (set builder notation). Note that because we are also interested in where the function is equal to zero, we include the roots in our solution.
Now, what do you think the solution to the quadratic inequality $\scriptsize {{x}^{2}}+3x-4 \gt 0$ will be? Note that we are interested in where $\scriptsize f(x) \gt 0$ (in other words where the graph of the function is above the x-axis) and that we need to exclude the values where $\scriptsize f(x)=0$ (in other words the roots).
Figure 4 shows the solution to $\scriptsize {{x}^{2}}+3x-4 \gt 0$ graphically. We can see that we need x-values to the left of $\scriptsize -4$ and to the right of $\scriptsize 1$.
We can represent this on a number line (see Figure 5), using interval notation $\scriptsize x\in (-\infty ,-4)\bigcup (1,\infty )$ or set builder notation $\scriptsize \{x|x\in \mathbb{R},x \lt -4,x \gt 1\}$.
When representing the solution on a number line, we use a closed dot to represent values that are included (i.e. $\scriptsize \le \text{ and }\ge \text{ }$ or $\scriptsize [\text{ and }\!\!]\!\!\text{ }$). We use an open dot to represent values that are not included (i.e. or $\scriptsize (\text{ and )}$).
### Take note!
The $\scriptsize \bigcup$ symbol in $\scriptsize x\in (-\infty ,-4)\bigcup (1,\infty )$ is called Union and it lets us combine different intervals of values together into a single solution set.
Sketching the graph of the quadratic function each time we want to solve a quadratic inequality would be effective but not very efficient. Instead, we use a simplified representation of a table of values to figure out where the quadratic function is positive or negative and then build our solution from here. Take a look at Example 5.1 to see how this works.
### Example 5.1
Solve for $\scriptsize x$ in $\scriptsize {{x}^{2}}+3x-4\ge 0$.
Solution
Notice that we are dealing with the same quadratic as above, so feel free to have a look at the figures above as you work through this solution.
Step 1: Get the inequality into standard form
The first step is always to get the quadratic inequality into standard form. Remember to flip the direction of the inequality sign if you ever multiply through by a negative number. Our inequality is already in standard form.
$\scriptsize {{x}^{2}}+3x-4\ge 0$
Step 2: Determine the critical values
The critical values are the roots of the corresponding quadratic equation. Remember these are the x-values where the function is zero as it goes from being positive to negative or negative to positive.
\scriptsize \begin{align*}{{x}^{2}}+3x-4 & \ge 0\\\therefore (x+4)(x-1) & \ge 0\end{align*}
Our critical values are, therefore, $\scriptsize x=-4$ and $\scriptsize x=1$.
### Note
Almost always, the quadratic inequalities that you will be expected to solve will factorise easily. It would be a very nasty question that expects you to find the critical values by using the quadratic formula. If you find that your quadratic expression does not factorise, first go back to check your work before using the quadratic formula.
Step 3: Complete a table of signs
Create a table like this:
$\scriptsize x \lt -4$ $\scriptsize x=-4$ $\scriptsize -4 \lt x \lt 1$ $\scriptsize x=1$ $\scriptsize x \gt 1$ $\scriptsize (x+4)(x-1)$
We know that each factor will be zero at its respective critical value and therefore, the whole quadratic expression will be zero at these values. So, fill this into your table.
$\scriptsize x \lt -4$ $\scriptsize x=-4$ $\scriptsize -4 \lt x \lt 1$ $\scriptsize x=1$ $\scriptsize x \gt 1$ $\scriptsize (x+4)(x-1)$ $\scriptsize 0$ $\scriptsize 0$
Next, you need to determine whether the quadratic expression is positive or negative on either side of each critical value. Start by picking any value less than $\scriptsize -4$, say $\scriptsize -5$.
$\scriptsize (x+4)=-5+4=-1 \lt 0$ and $\scriptsize (x-1)=-5-1=-6 \lt 0$. Therefore, the overall product will be positive. Fill this into your table.
$\scriptsize x \lt -4$ $\scriptsize x=-4$ $\scriptsize -4 \lt x \lt 1$ $\scriptsize x=1$ $\scriptsize x \gt 1$ $\scriptsize (x+4)(x-1)$ $\scriptsize +$ $\scriptsize 0$ $\scriptsize 0$
Complete the rest of the table by choosing the appropriate values and testing to see whether the product of the factors is positive or negative. The final table will look like this:
$\scriptsize x \lt -4$ $\scriptsize x=-4$ $\scriptsize -4 \lt x \lt 1$ $\scriptsize x=1$ $\scriptsize x \gt 1$ $\scriptsize (x+4)(x-1)$ $\scriptsize +$ $\scriptsize 0$ $\scriptsize -$ $\scriptsize 0$ $\scriptsize +$
Step 4: Read off the solution.
Now we can read off the solution to our inequality $\scriptsize {{x}^{2}}+3x-4\ge 0$. We need all those ranges of values that are greater than or equal to zero. These are:
$\scriptsize x\in (-\infty ,-4]\bigcup [1,\infty )$
$\scriptsize \{x|x\in \mathbb{R},x\le -4,x\ge 1\}$
### Example 5.2
Solve for $\scriptsize x$ in $\scriptsize -{{x}^{2}}+6x\ge -7$.
Solution
Step 1: Get the inequality into standard form
\scriptsize \begin{align*}-{{x}^{2}}+6x & \ge -7\\ \therefore -{{x}^{2}}+6x+7 & \ge 0\quad \text{Leave the inequality like this or multiply through by }-1\\ \therefore {{x}^{2}}-6x-7 & \le 0\quad \text{If you do multiply through by }-1\text{ remember to flip the inequality}\end{align*}
Step 2: Determine the critical values
\scriptsize \begin{align*}{{x}^{2}}-6x-7 & \le 0\\ \therefore (x-7)(x+1) & \ge 0\end{align*}
Our critical values are, therefore, $\scriptsize x=-1$ and $\scriptsize x=7$.
Step 3: Complete a table of signs
$\scriptsize x \lt -1$ $\scriptsize x=-1$ $\scriptsize -1 \lt x \lt 7$ $\scriptsize x=7$ $\scriptsize x \gt 7$ $\scriptsize (x-7)(x+1)$ $\scriptsize +$ $\scriptsize 0$ $\scriptsize -$ $\scriptsize 0$ $\scriptsize +$
Step 4: Read off the solution
$\scriptsize {{x}^{2}}-6x-7 \le 0$. Therefore:
$\scriptsize x\in [-1,7]$
$\scriptsize \{x|x\in \mathbb{R},-1\le x\le 7\}$
### Exercise 5.1
Solve for $\scriptsize x$ in the following inequalities, representing your answer as indicated:
1. $\scriptsize {{x}^{2}}+2x-15 \lt 0$ (represent your answer using set builder notation)
2. $\scriptsize 10{{x}^{2}}\le 31x-15$ (represent your answer using interval notation)
3. $\scriptsize 9x(x+1) \gt 10$ (represent your answer using interval notation)
4. $\scriptsize 5(2{{x}^{2}}-7) \lt 11x$ (represent your answer using set builder notation)
5. $\scriptsize \displaystyle \frac{{40x-21}}{{{{x}^{2}}}}\ge 16$ (represent your answer on a number line)
The full solutions are at the end of the unit.
## Solve simple quadratic inequalities graphically
Let’s do a few more examples to show how to solve for $\scriptsize x$ graphically.
### Example 5.3
Determine graphically for which values of $\scriptsize x$ $\scriptsize f(x)\ge 0$ if $\scriptsize f(x)=-4{{x}^{2}}+4x+3$.
Solution
The first thing to notice is that we need to solve for $\scriptsize x$ graphically, which means we have to draw a sketch of the function and interpret this sketch to come to the solution.
Step 1: Sketch the function
If you need help sketching quadratic functions like this refer to level 3 subject outcome 2.1 unit 1.
Step 2: Read off the solution
We were asked to find where $\scriptsize f(x)\ge 0$, in other words, where the function $\scriptsize f$ is above the x-axis. From the graph, we can see that $\scriptsize f(x)\ge 0$ between $\scriptsize -\displaystyle \frac{1}{2}$ and $\scriptsize \displaystyle \frac{3}{2}$. Therefore, $\scriptsize x\in [-\displaystyle \frac{1}{2},\displaystyle \frac{3}{2}]$ or $\scriptsize \{x|x\in \mathbb{R}\text{,}-\displaystyle \frac{1}{2}\le x\le \displaystyle \frac{3}{2}\}$
There is something important we need to recognise about solving quadratic inequalities from Example 5.3. In this case, $\scriptsize f(x)=-4{{x}^{2}}+4x+3$ and we were asked where $\scriptsize f(x)\ge 0$ or above the x-axis. We can see that the graph has a maximum turning point because $\scriptsize a \lt 0$ and so the solution is as shown in Figure 6.
However, if we were solving this inequality algebraically, we would probably proceed as follows:
\scriptsize \begin{align*}-4{{x}^{2}}+4x+3 & \ge 0\quad \text{Multiply through by }-1\text{ and reverse the inequality}\\\therefore 4{{x}^{2}}-4x+3 & \le 0\end{align*}
Now you might be wondering how changing the function from $\scriptsize f(x)=-4{{x}^{2}}+4x+3$ to another function, for example $\scriptsize g(x)=4{{x}^{2}}-4x-3$, will not result in a completely different solution. The answer lies in the importance of reversing the inequality sign whenever we multiply both sides of an inequality by a negative number. If we look at a sketch of $\scriptsize g(x)=4{{x}^{2}}-4x-3$ we can see why (see Figure7).
Because we reversed the inequality sign, we are now interested in where $\scriptsize 4{{x}^{2}}-4x-3\le 0$, where $\scriptsize g(x)\le 0$ or where $\scriptsize g(x)$ is below the x-axis.
We can see that we get exactly the same solution: $\scriptsize x\in [-\displaystyle \frac{1}{2},\displaystyle \frac{3}{2}]$ or $\scriptsize \{x|x\in \mathbb{R}\text{,}-\displaystyle \frac{1}{2}\le x\le \displaystyle \frac{3}{2}\}$.
## Solve inequalities with unknowns in the denominator
The process of solving quadratic inequalities when there are fractions involved and where the denominators contain variables is slightly different.
If we were asked to solve for $\scriptsize x$ in $\scriptsize \displaystyle \frac{4}{{x-3}}=\displaystyle \frac{5}{{x+4}}$, we would multiply both sides by the LCD of $\scriptsize (x-3)(x+4)$ to get:
\scriptsize \begin{align*}\displaystyle \frac{4}{{x-3}} & =\displaystyle \frac{5}{{x+4}},x\ne 3,x\ne -4\\\therefore 4(x+4) & =5(x-3)\\\therefore 4x+16 & =5x-15\\\therefore x & =31\end{align*}
We can do this because we don’t need to worry about whether $\scriptsize (x-3) \lt 0$ or whether $\scriptsize (x+4) \lt 0$. But if we have an inequality, we know we have to flip the inequality around if we multiply through by a negative number. Because we don’t know if $\scriptsize (x-3) \lt 0$ or if $\scriptsize (x+4) \lt 0$ we don’t know whether to flip the inequality around or not. So, we need a different method.
### Example 5.4
Solve for $\scriptsize x$ in $\scriptsize \displaystyle \frac{4}{{x-3}}\le \displaystyle \frac{5}{{x+4}}$.
Solution
Step 1: Get one side of the inequality equal to zero
To make sure that we don’t have to multiply through by the LCD, we rearrange the inequality to get one side equal to zero.
\scriptsize \begin{align*}\displaystyle \frac{4}{{x-3}} & \le \displaystyle \frac{5}{{x+4}}\quad \text{Subtract }\displaystyle \frac{5}{{x+4}}\text{ from both sides}\\\therefore \displaystyle \frac{4}{{x-3}}-\displaystyle \frac{5}{{x+4}} & \le 0\end{align*}
Step 2: Add the fractions
Next, we need to subtract the fractions on the other side of the inequality.
\scriptsize \begin{align*}\displaystyle \frac{4}{{x-3}}-\displaystyle \frac{5}{{x+4}} & \le 0\\\therefore \displaystyle \frac{{4(x+4)-5(x-3)}}{{(x-3)(x+4)}} & \le 0\\\therefore \displaystyle \frac{{4x+16-5x+15}}{{(x-3)(x+4)}} & \le 0\\\therefore \displaystyle \frac{{-x+31}}{{(x-3)(x+4)}} & \le 0\quad \text{We can leave the LHS as it is, or multiply through by }-1\\\therefore \displaystyle \frac{{-(x-31)}}{{(x-3)(x+4)}} & \le 0\\\therefore \displaystyle \frac{{(x-31)}}{{(x-3)(x+4)}} & \ge 0\quad \text{Remember to flip the inequality sign}\end{align*}
Step 3: Determine the critical values
The critical values now are not just the roots (those values that make the expression on the LHS equal to zero) but also the restrictions (those values that make the expression on the LHS undefined).
Our critical values are $\scriptsize x=31$, $\scriptsize x=3$ and $\scriptsize x=-4$.
Step 4: Complete a table of signs
Notice how we need to add additional columns to accommodate the extra critical value. As before, choose values for $\scriptsize x$ to complete the table.
$\scriptsize x \lt -4$ $\scriptsize x=-4$ $\scriptsize -4 \lt x \lt 3$ $\scriptsize x=3$ $\scriptsize 3 \lt x \lt 31$ $\scriptsize x=31$ $\scriptsize x \gt 31$ $\scriptsize \displaystyle \frac{{(x-31)}}{{(x-3)(x+4)}}$ $\scriptsize -$ Undefined $\scriptsize +$ Undefined $\scriptsize -$ $\scriptsize 0$ $\scriptsize +$
Step 5: Read off the solution
We are solving for $\scriptsize \displaystyle \frac{{(x-31)}}{{(x-3)(x+4)}} \ge 0$. Which ranges result in a positive expression? These are:
$\scriptsize x\in (-4,3)\bigcup [31,\infty )$
$\scriptsize \{x|x\in \mathbb{R},-4 \lt x \lt 3,x\ge 31\}$
Note: Even though we are solving for a greater than or equal to inequality, we cannot include $\scriptsize -4$ or $\scriptsize 3$ in our solutions because these values make the expression undefined.
### Exercise 5.2
Solve for $\scriptsize x$ in the following inequalities and represent your answers using interval and set builder notation:
1. $\scriptsize \displaystyle \frac{4}{{x-3}}\ge 1$
2. $\scriptsize \displaystyle \frac{{15-x}}{x}\ge 2x$
3. $\scriptsize \displaystyle \frac{{-4}}{{(x-3)(x-1)}} \lt 3$
4. $\scriptsize \displaystyle \frac{{{{x}^{2}}+3}}{{3x-2}}\le 0$
The full solutions are at the end of the unit.
## Summary
In this unit you have learnt the following:
• How to find the critical values.
• How to solve quadratic inequalities using the table method.
• How to solve quadratic inequalities using the graphical method.
• How to represent the solution to a quadratic inequality on a number line.
• How to solve inequalities with fractions where there are unknowns in the denominator.
# Unit 5: Assessment
#### Suggested time to complete: 60 minutes
1. Solve for $\scriptsize x$ in the following inequalities and represent your answers as indicated:
1. $\scriptsize 2+x-{{x}^{2}}\le 0$ (represent your answer on a number line)
2. $\scriptsize (x-1)(x-2) \lt 6$ (represent your answer using set builder notation)
3. $\scriptsize -10x \lt -{{x}^{2}}$ (represent your answer using set builder notation)
4. $\scriptsize \displaystyle \frac{{{{x}^{2}}+4}}{{x-7}}\le 0,x\ne 7$ (represent your answer using interval notation)
5. $\scriptsize {{x}^{2}}+4x \lt -4$ (represent your answer using set builder notation)
6. $\scriptsize \displaystyle \frac{{x+2}}{x} \gt 1,x\ne 0$ (represent your answer using interval notation)
2. Solve for $\scriptsize x$ in $\scriptsize -6\le {{x}^{2}}-5x\le 6$ and represent your answer on a number line.
Hint: Split the expression into two separate inequalities first.
The full solutions are at the end of the unit.
# Unit 5: Solutions
### Exercise 5.1
1. .
\scriptsize \begin{align*}{{x}^{2}}+2x-15 & \lt 0\\\therefore (x-3)(x+5) & \lt 0\end{align*}
Critical values: $\scriptsize x=-5$ and $\scriptsize x=3$
$\scriptsize x \lt -5$ $\scriptsize x=-5$ $\scriptsize -5 \lt x \lt 3$ $\scriptsize x=3$ $\scriptsize x \gt 3$ $\scriptsize (x-3)(x+5)$ $\scriptsize +$ $\scriptsize 0$ $\scriptsize -$ $\scriptsize 0$ $\scriptsize +$
$\scriptsize \{x|x\in \mathbb{R},-5 \lt x \lt 3\}$
2. .
\scriptsize \begin{align*}10{{x}^{2}} & \le 31x-15\\\therefore 10{{x}^{2}}-31x+15 & \le 0\\\therefore (2x-5)(5x-3) & \le 0\end{align*}
Critical values: $\scriptsize x=\displaystyle \frac{5}{2}$ and $\scriptsize x=\displaystyle \frac{3}{5}$
$\scriptsize x \lt \displaystyle \frac{3}{5}$ $\scriptsize x=\displaystyle \frac{3}{5}$ $\scriptsize \displaystyle \frac{3}{5} \lt x \lt \displaystyle \frac{5}{2}$ $\scriptsize x=\displaystyle \frac{5}{2}$ $\scriptsize x \gt \displaystyle \frac{5}{2}$ $\scriptsize (2x-5)(5x-3)$ $\scriptsize +$ $\scriptsize 0$ $\scriptsize -$ $\scriptsize 0$ $\scriptsize +$
$\scriptsize x\in [\displaystyle \frac{3}{5},\displaystyle \frac{5}{2}]$
3. .
\scriptsize \begin{align*}9x(x+1) & \gt 10\\\therefore 9{{x}^{2}}+9x-10 & \gt 0\\\therefore (3x-2)(3x+5) & \gt 0\end{align*}
Critical values: $\scriptsize x=\displaystyle \frac{2}{3}$ and $\scriptsize x=-\displaystyle \frac{5}{3}$
$\scriptsize x \lt -\displaystyle \frac{5}{3}$ $\scriptsize x=-\displaystyle \frac{5}{3}$ $\scriptsize -\displaystyle \frac{5}{3} \lt x \lt \displaystyle \frac{2}{3}$ $\scriptsize x=\displaystyle \frac{2}{3}$ $\scriptsize x \gt \displaystyle \frac{2}{3}$ $\scriptsize (3x-2)(3x+5)$ $\scriptsize +$ $\scriptsize 0$ $\scriptsize -$ $\scriptsize 0$ $\scriptsize +$
$\scriptsize x\in (-\infty ,-\displaystyle \frac{5}{3})\bigcup (\displaystyle \frac{2}{3},\infty )$
4. .
\scriptsize \begin{align*}5(2{{x}^{2}}-7) & \lt 11x\\\therefore 10{{x}^{2}}-11x-35 & \lt 0\\\therefore (5x+7)(2x-5) & \lt 0\end{align*}
Critical values: $\scriptsize x=-\displaystyle \frac{7}{5}$ and $\scriptsize x=\displaystyle \frac{5}{2}$
$\scriptsize x \lt -\displaystyle \frac{7}{5}$ $\scriptsize x=-\displaystyle \frac{7}{5}$ $\scriptsize -\displaystyle \frac{7}{5} \lt x \lt \displaystyle \frac{5}{2}$ $\scriptsize x=\displaystyle \frac{5}{2}$ $\scriptsize x \gt \displaystyle \frac{5}{2}$ $\scriptsize (5x+7)(2x-5)$ $\scriptsize +$ $\scriptsize 0$ $\scriptsize -$ $\scriptsize 0$ $\scriptsize +$
$\scriptsize \{x|x\in \mathbb{R}\text{,}-\displaystyle \frac{7}{5} \lt x \lt \displaystyle \frac{5}{2}\}$
5. .
\scriptsize \begin{align*}\displaystyle \frac{{40x-21}}{{{{x}^{2}}}} & \ge 16,{{x}^{2}}\ne 0\therefore x\ne 0\\\text{You can multiply both sides by }{{x}^{2}}\text{ because we know that }{{x}^{2}} \gt 0\\\therefore 40x-21 & \ge 16{{x}^{2}}\\\therefore 16{{x}^{2}}-40x+21 & \le 0\quad \text{Be careful of the direction of the inequality}\\\therefore (4x-3)(4x-7) & \le 0\end{align*}
Critical values: $\scriptsize x=\displaystyle \frac{3}{4}$ and $\scriptsize x=\displaystyle \frac{7}{4}$
$\scriptsize x \lt \displaystyle \frac{3}{4}$ $\scriptsize x=\displaystyle \frac{3}{4}$ $\scriptsize \displaystyle \frac{3}{4} \lt x \lt \displaystyle \frac{7}{4}$ $\scriptsize x=\displaystyle \frac{7}{4}$ $\scriptsize x \gt \displaystyle \frac{7}{4}$ $\scriptsize (4x-3)(4x-7)$ $\scriptsize +$ $\scriptsize 0$ $\scriptsize -$ $\scriptsize 0$ $\scriptsize +$
Back to Exercise 5.1
### Exercise 5.2
1. .
\scriptsize \begin{align*}\displaystyle \frac{4}{{x-3}} & \ge 1,x\ne 3\\\therefore \displaystyle \frac{4}{{x-3}}-1 & \ge 0\\\therefore \displaystyle \frac{{4-(x-3)}}{{(x-3)}} & \ge 0\\\therefore \displaystyle \frac{{-x+7}}{{(x-3)}} & \ge 0\quad \text{Multiply through by }-1\text{ and reverse the inequality sign}\\\therefore \displaystyle \frac{{x-7}}{{(x-3)}} & \le 0\end{align*}
Critical values: $\scriptsize x=7$ and $\scriptsize x=3$
$\scriptsize x \lt 3$ $\scriptsize x=3$ $\scriptsize 3 \lt x \lt 7$ $\scriptsize x=7$ $\scriptsize x \gt 7$ $\scriptsize \displaystyle \frac{{(x-7)}}{{(x-3)}}$ $\scriptsize +$ Undefined $\scriptsize -$ $\scriptsize 0$ $\scriptsize +$
$\scriptsize x\in (3,7]$
$\scriptsize \{x|x\in \mathbb{R},3 \lt x\le 7\}$ Remember that we cannot include $\scriptsize 3$ in our solution.
2. .
\scriptsize \begin{align*}\displaystyle \frac{{15-x}}{x} & \ge 2x,x\ne 0\\\therefore \displaystyle \frac{{15-x}}{x}-2x & \ge 0\\\therefore \displaystyle \frac{{15-x-2{{x}^{2}}}}{x} & \ge 0\\\therefore \displaystyle \frac{{(3-x)(5+2x)}}{x} & \ge 0\end{align*}
Critical values: $\scriptsize x=3$, $\scriptsize x=-\displaystyle \frac{5}{2}$ and $\scriptsize x=0$
$\scriptsize x \lt -\displaystyle \frac{5}{2}$ $\scriptsize x=-\displaystyle \frac{5}{2}$ $\scriptsize -\displaystyle \frac{5}{2} \lt x \lt 0$ $\scriptsize x=0$ $\scriptsize 0 \lt x \lt 3$ $\scriptsize x=3$ $\scriptsize x \gt 3$ $\scriptsize \displaystyle \frac{{(3-x)(5+2x)}}{x}$ $\scriptsize +$ $\scriptsize 0$ $\scriptsize -$ Undef $\scriptsize +$ $\scriptsize 0$ $\scriptsize -$
$\scriptsize x\in (-\infty ,-\displaystyle \frac{5}{2}]\bigcup (0,3]$
$\scriptsize \{x|x\in \mathbb{R},x\le -\displaystyle \frac{5}{2},0 \lt x\le 3\}$
3. .
\scriptsize \begin{align*}\displaystyle \frac{{-3}}{{(x-3)(x-1)}} & \lt 3,x\ne 3,x\ne 1\\\therefore \displaystyle \frac{{-3}}{{(x-3)(x-1)}}-3 & \lt 0\\\therefore \displaystyle \frac{{-3-3(x-3)(x-1)}}{{(x-3)(x-1)}} & \lt 0\\\therefore \displaystyle \frac{{-3-3({{x}^{2}}-4x+3)}}{{(x-3)(x-1)}} & \lt 0\\\therefore \displaystyle \frac{{-3-3{{x}^{2}}+12x-9}}{{(x-3)(x-1)}} & \lt 0\\\therefore \displaystyle \frac{{-3{{x}^{2}}+12x-12}}{{(x-3)(x-1)}} & \lt 0\quad \text{Multiply through by }-1\text{ and reverse the inequality}\\\therefore \displaystyle \frac{{3{{x}^{2}}-12x+12}}{{(x-3)(x-1)}} & \gt 0\quad \text{You can take out a common factor of }3\text{ or factorise as is}\\\therefore \displaystyle \frac{{(3x-6)(x-2)}}{{(x-3)(x-1)}} & \gt 0\\\therefore \displaystyle \frac{{3(x-2)(x-2)}}{{(x-3)(x-1)}} & \gt 0\end{align*}
Critical values: $\scriptsize x=1$, $\scriptsize x=2$ and $\scriptsize x=3$
$\scriptsize x \lt 1$ $\scriptsize x=1$ $\scriptsize 1 \lt x \lt 2$ $\scriptsize x=2$ $\scriptsize 2 \lt x \lt 3$ $\scriptsize x=3$ $\scriptsize x \gt 3$ $\scriptsize \displaystyle \frac{{3(x-2)(x-2)}}{{(x-3)(x-1)}}$ $\scriptsize +$ Undef $\scriptsize -$ $\scriptsize 0$ $\scriptsize -$ Undef $\scriptsize +$
$\scriptsize x\in (-\infty ,1)\bigcup \text{(3,}\infty )$
$\scriptsize \{x|x\in \mathbb{R},x \lt 1,x \gt 3\}$
.
You could also recognise that the numerator $\scriptsize 3{{(x-2)}^{2}}\ge 0$. Therefore for the fraction to be positive, the denominator must be positive. Hence, you could consider a condensed table of critical values.
$\scriptsize x \lt 1$ $\scriptsize x=1$ $\scriptsize 1 \lt x \lt 3$ $\scriptsize x=3$ $\scriptsize x \gt 3$ $\scriptsize \displaystyle \frac{{3(x-2)(x-2)}}{{(x-3)(x-1)}}$ $\scriptsize +$ Undef $\scriptsize -$ Undef $\scriptsize +$
4. .
$\scriptsize \displaystyle \frac{{{{x}^{2}}+3}}{{3x-2}}\le 0,x\ne \displaystyle \frac{2}{3}$
$\scriptsize {{x}^{2}}\ge 0\therefore {{x}^{2}}+3\ge 3$. Therefore, the numerator is always positive.
Critical values: $\scriptsize x=\displaystyle \frac{2}{3}$
$\scriptsize x \lt \displaystyle \frac{2}{3}$ $\scriptsize x=\displaystyle \frac{2}{3}$ $\scriptsize x \gt \displaystyle \frac{2}{3}$ $\scriptsize \displaystyle \frac{{{{x}^{2}}+3}}{{3x-2}}$ $\scriptsize -$ Undef $\scriptsize +$
$\scriptsize x\in (-\infty ,\displaystyle \frac{2}{3})$
$\scriptsize \{x|x\in \mathbb{R},x \lt \displaystyle \frac{2}{3}\}$
Back to Exercise 5.2
### Unit 5: Assessment
1. .
1. .
\scriptsize \begin{align*}2+x-{{x}^{2}} & \le 0\\\therefore {{x}^{2}}-x-2 & \ge 0\\\therefore (x+1)(x-2) & \ge 0\end{align*}
Critical values: $\scriptsize x=-1$ and $\scriptsize x=2$
$\scriptsize x \lt -1$ $\scriptsize x=-1$ $\scriptsize -1 \lt x \lt 2$ $\scriptsize x=2$ $\scriptsize x \gt 2$ $\scriptsize (x+1)(x-2)$ $\scriptsize +$ $\scriptsize 0$ $\scriptsize -$ $\scriptsize 0$ $\scriptsize +$
2. .
\scriptsize \begin{align*}(x-1)(x-2) & \lt 6\\\therefore {{x}^{2}}-3x+2-6 & \lt 0\\\therefore {{x}^{2}}-3x-4 & \lt 0\\\therefore (x+1)(x-4) & \lt 0\end{align*}
Critical values: $\scriptsize x=-1$ and $\scriptsize x=4$
$\scriptsize x \lt -1$ $\scriptsize x=-1$ $\scriptsize -1 \lt x \lt 4$ $\scriptsize x=4$ $\scriptsize x \gt 4$ $\scriptsize (x+1)(x-4)$ $\scriptsize +$ $\scriptsize 0$ $\scriptsize -$ $\scriptsize 0$ $\scriptsize +$
$\scriptsize \{x|x\in \mathbb{R},-1 \lt x \lt 4\}$
3. .
\scriptsize \begin{align*}-10x & \lt -{{x}^{2}}\\\therefore {{x}^{2}}-10x & \lt 0\\\therefore x(x-10) & \lt 0\end{align*}
Critical values: $\scriptsize x=0$ and $\scriptsize x=10$
$\scriptsize x \lt 0$ $\scriptsize x=0$ $\scriptsize 0 \lt x \lt 10$ $\scriptsize x=10$ $\scriptsize x \gt 10$ $\scriptsize x(x-10)$ $\scriptsize +$ $\scriptsize 0$ $\scriptsize -$ $\scriptsize 0$ $\scriptsize +$
$\scriptsize \{x|x\in \mathbb{R},0 \lt x \lt 10\}$
4. .
$\scriptsize \displaystyle \frac{{{{x}^{2}}+4}}{{x-7}}\le 0,x\ne 7$
$\scriptsize {{x}^{2}}\ge 0\therefore {{x}^{2}}+4\ge 4$. The numerator is always positive.
Critical value: $\scriptsize x=7$
$\scriptsize x \lt 7$ $\scriptsize x=7$ $\scriptsize x \gt 7$ $\scriptsize \displaystyle \frac{{{{x}^{2}}+4}}{{x-7}}$ $\scriptsize -$ Undefined $\scriptsize +$
$\scriptsize x\in (-\infty ,7)$
5. .
\scriptsize \begin{align*}{{x}^{2}}+4x & \lt -4\\\therefore {{x}^{2}}+4x+4 & \lt 0\\\therefore (x+2)(x+2) & \lt 0\end{align*}
Critical value: $\scriptsize x=-2$
$\scriptsize x \lt -2$ $\scriptsize x=-2$ $\scriptsize x \gt -2$ $\scriptsize (x+2)(x+2)$ $\scriptsize +$ $\scriptsize 0$ $\scriptsize +$
No solution
.
Alternative approach:
\scriptsize \begin{align*}{{x}^{2}}+4x & \lt -4\\\therefore {{x}^{2}}+4x+4 & \lt 0\\\therefore (x+2)(x+2) & \lt 0\\\therefore {{(x+2)}^{2}} & \lt 0\end{align*}
But $\scriptsize {{(x+2)}^{2}}$ is always positive. Therefore, there is no solution.
6. .
\scriptsize \begin{align*}\displaystyle \frac{{x+2}}{x} & \gt 1,x\ne 0\\\therefore \displaystyle \frac{{x+2}}{x}-1 & \gt 0\\\therefore \displaystyle \frac{{x+2-x}}{x} & \gt 0\\\therefore \displaystyle \frac{2}{x} & \gt 0\end{align*}
Critical value: $\scriptsize x=0$
$\scriptsize x \lt 0$ $\scriptsize x=0$ $\scriptsize x \gt 0$ $\scriptsize \displaystyle \frac{2}{x}$ $\scriptsize -$ Undefined $\scriptsize +$
$\scriptsize x\in (0,\infty )$
2. .
$\scriptsize -6\le {{x}^{2}}-5x\le 6$
$\scriptsize -6\le {{x}^{2}}-5x$ (1)
$\scriptsize {{x}^{2}}-5x\le 6$ (2)
.
From (1):
\scriptsize \begin{align*}-6 & \le {{x}^{2}}-5x\\\therefore {{x}^{2}}-5x+6 & \ge 0\\\therefore (x-2)(x-3) & \ge 0\end{align*}
Critical values: $\scriptsize x=2$ and $\scriptsize x=3$
$\scriptsize x \lt 2$ $\scriptsize x=2$ $\scriptsize 2 \lt x \lt 3$ $\scriptsize x=3$ $\scriptsize x \gt 3$ $\scriptsize (x-2)(x-3)$ $\scriptsize +$ $\scriptsize 0$ $\scriptsize -$ $\scriptsize 0$ $\scriptsize +$
From (2):
\scriptsize \begin{align*}{{x}^{2}}-5x & \le 6\\\therefore {{x}^{2}}-5x-6 & \le 0\\\therefore (x+1)(x-6) & \le 0\end{align*}
Critical values: $\scriptsize x=-1$ and $\scriptsize x=6$
$\scriptsize x \lt -1$ $\scriptsize x=-1$ $\scriptsize -1 \lt x \lt 6$ $\scriptsize x=6$ $\scriptsize x \gt 6$ $\scriptsize (x-1)(x-6)$ $\scriptsize +$ $\scriptsize 0$ $\scriptsize -$ $\scriptsize 0$ $\scriptsize +$
If we plot the results of the two sign tables above on the same number line, we get the following. We can see that the regions overlap (make $\scriptsize -6\le {{x}^{2}}-5x$ AND $\scriptsize {{x}^{2}}-5x\le 6$ at the same time) when $\scriptsize -1\le x\le 2$ or when $\scriptsize 3\le x\le 6$.
Therefore, the final solution is $\scriptsize x\in [-1,2]\bigcup [3,6]$.
Back to Unit 5: Assessment | 10,168 | 30,074 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 4.8125 | 5 | CC-MAIN-2024-33 | latest | en | 0.774747 |
https://www.theanalysisfactor.com/tag/multiple-imputation/page/2/ | 1,680,030,566,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296948868.90/warc/CC-MAIN-20230328170730-20230328200730-00409.warc.gz | 1,131,118,404 | 14,523 | # Multiple Imputation
### Is Multiple Imputation Possible in the Context of Survival Analysis?
May 27th, 2011 by
Sure. One of the big advantages of multiple imputation is that you can use it for any analysis.
It’s one of the reasons big data libraries use it–no matter how researchers are using the data, the missing data is handled the same, and handled well.
I say this with two caveats. (more…)
### Quiz Yourself about Missing Data
May 3rd, 2010 by
Do you find quizzes irresistible? I do.
Here’s a little quiz about working with missing data:
True or False?
1. Imputation is really just making up data to artificially inflate results. It’s better to just drop cases with missing data than to impute.
2. I can just impute the mean for any missing data. It won’t affect results, and improves power.
3. Multiple Imputation is fine for the predictor variables in a statistical model, but not for the response variable.
4. Multiple Imputation is always the best way to deal with missing data.
5. When imputing, it’s important that the imputations be plausible data points.
6. Missing data isn’t really a problem if I’m just doing simple statistics, like chi-squares and t-tests.
7. The worst thing that missing data does is lower sample size and reduce power.
### Answers to the Missing Data Quiz
May 3rd, 2010 by
In my last post, I gave a little quiz about missing data. This post has the answers.
If you want to try it yourself before you see the answers, go here. (It’s a short quiz, but if you’re like me, you find testing yourself irresistible).
True or False?
1. Imputation is really just making up data to artificially inflate results. It’s better to just drop cases with missing data than to impute. (more…)
### Multiple Imputation: 5 Recent Findings that Change How to Use It
March 24th, 2010 by
Missing Data, and multiple imputation specifically, is one area of statistics that is changing rapidly. Research is still ongoing, and each year new findings on best practices and new techniques in software appear.
The downside for researchers is that some of the recommendations missing data statisticians were making even five years ago have changed.
Remember that there are three goals of multiple imputation, or any missing data technique: Unbiased parameter estimates in the final analysis (more…)
### Multiple Imputation of Categorical Variables
June 1st, 2009 by
Most Multiple Imputation methods assume multivariate normality, so a common question is how to impute missing values from categorical variables.
Paul Allison, one of my favorite authors of statistical information for researchers, did a study that showed that the most common method actually gives worse results that listwise deletion. (Did I mention I’ve used it myself?) (more…)
### Missing Data: Criteria for Choosing an Effective Approach
May 20th, 2009 by
In choosing an approach to missing data, there are a number of things to consider. But you need to keep in mind what you’re aiming for before you can even consider which approach to take.
There are three criteria we’re aiming for with any missing data technique:
1. Unbiased parameter estimates: Whether you’re estimating means, regressions, or odds ratios, you want your parameter estimates to be accurate representations of the actual population parameters. In statistical terms, that means the estimates should be unbiased. If all the (more…) | 748 | 3,426 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.5625 | 3 | CC-MAIN-2023-14 | latest | en | 0.89272 |
https://www.physicsforums.com/threads/iterated-limit-double-limit.88825/ | 1,675,807,813,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764500641.25/warc/CC-MAIN-20230207201702-20230207231702-00526.warc.gz | 932,222,628 | 15,926 | # Iterated limit? double limit?
hanson
Hello!
I am doing something about complex variables, and learning to derive the Cauchy-Riemann Equation.
The concept of "iterated limit" to me is that, supossing 2 independt variables x&y, it is taking the limit either x first then y or y first then x. Geometrically, it is representing two paths to approach the right destination, right?
But I can't really visualise how "double limit" really work.
What is a double limit? I am told that it is "x and y go togather in any manner"
What does it mean?
I can think of some cases that x and y will approach the destination togather.
Say, the path is y=mx as x->0. In this case, x and y will approach to 0 togather. So, is this a double limit?
If this is, there is also a problem.
There is a statement that "the limit, representing the derivative of a complex function, must exist as a double limit for delta z= delta x+i(delta y) approaching zero."
I don't see why the term "double limit" is used here.
To me, the complex derivative exists when all double limits and all iterated limits gives the same value so that the limit can be said of being independent of any paths. Am I correct?
Gold Member
hanson said:
Hello!
I am doing something about complex variables, and learning to derive the Cauchy-Riemann Equation.
The concept of "iterated limit" to me is that, supossing 2 independt variables x&y, it is taking the limit either x first then y or y first then x. Geometrically, it is representing two paths to approach the right destination, right?
But I can't really visualise how "double limit" really work.
What is a double limit? I am told that it is "x and y go togather in any manner"
What does it mean?
I can think of some cases that x and y will approach the destination togather.
Say, the path is y=mx as x->0. In this case, x and y will approach to 0 togather. So, is this a double limit?
If this is, there is also a problem.
There is a statement that "the limit, representing the derivative of a complex function, must exist as a double limit for delta z= delta x+i(delta y) approaching zero."
I don't see why the term "double limit" is used here.
To me, the complex derivative exists when all double limits and all iterated limits gives the same value so that the limit can be said of being independent of any paths. Am I correct?
You are correct. The Cauchy-Riemann equations simply impose the restriction that the limit is the same whether $\Delta z$ appraoches zero along the real and imaginary axis. It can be proven that this is enough to guarantee that the limit is the same along all paths in the complex plane.
In terms of a double limit in general, that could mean something like:
$$\lim_{\substack{x\rightarrow 0\\y\rightarrow 0}} x^y=1$$
$$\lim_{\substack{y\rightarrow 0\\x\rightarrow 0}} x^y=0$$
This really is not the same thing as delta z approaching zero, having real and imaginary parts. z is only one number, and that is how it is treated in the analytic functions of complex analysis. You could write a real number in terms of rational and irrational parts, like: $x = a + b\sqrt{2}$ Then you could take the limit as x approaches zero as only one number, because in real analysis we do not treat functions that perform different operations on the rational and irrational parts. Just so, in complex analysis we deal with functions that treat z as one indivisible entity.
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hanson
don't we need to specific a path for double limit?
hanson
Don't we need to specify a path for double limit?
like y=mx, x->0, y->0?
And how can the two examples of double limit be calculated?
Homework Helper
Well, the limit only exists if it is "path independent"
let us fix that we're taking a limit as x,y both tend to 0 of some function f(x,y), let us further suppose that we want to show that limit is zero
then we need to show that for all e>0 there is a d>0 such that for all |(x,y)|<0 that |f(x,y)<0 where | | means distance eg the euclidean distance from the origin.
it is natrually easier to show that a limit does not exist by showing two different paths that have different limits.
however this is a situation when we are talking about (x,y) tends to zero.
this strictly different from a double limit
lim x tends to 0 of lim y tends to 0 of f(x,y)
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https://case-maze.com/data-capture-form-hbr-case-solution-and-analysis-essay-writing-help-service-7 | 1,537,662,052,000,000,000 | text/html | crawl-data/CC-MAIN-2018-39/segments/1537267158766.65/warc/CC-MAIN-20180923000827-20180923021227-00388.warc.gz | 491,243,652 | 5,471 | # Data Capture Form , HBR Case Solution And Analysis - Essay Writing Help Service
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every time the click reference pendulum swings through its centre placement, it releases one particular tooth of your escape wheel (g). The force from the clock's mainspring or a driving body weight hanging from the pulley, transmitted from the clock's equipment train, triggers the wheel to show, and a tooth presses against one of several pallets (h), giving the pendulum a brief thrust.
Kater crafted a reversible pendulum (shown at ideal) consisting of the brass bar with two opposing pivots crafted from limited triangular "knife" blades (a) near either finish. it may be swung more tips here from both pivot, Using the knife blades supported on agate plates. Rather than make a single pivot adjustable, he attached the pivots a meter aside and in its place adjusted the intervals using a moveable bodyweight on the pendulum rod (b,c).
Von Sterneck and Mendenhall gravimeters: In 1887 Austro-Hungarian scientist Robert von Sterneck created a little gravimeter pendulum mounted in a temperature-managed vacuum tank to remove the Data Capture Form , consequences of temperature and air stress. It made use of a "half-second pendulum," aquiring a period close to a person 2nd, about 25 cm very long. The pendulum was nonreversible, And so the instrument was employed for relative gravity measurements, but their tiny measurement created them smaller and transportable. | 831 | 4,234 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.5625 | 3 | CC-MAIN-2018-39 | latest | en | 0.909276 |
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Newton’s laws of motion project-based learning provide a platform for students to apply their knowledge on energy, force and motion. Students may work in groups using a jigsaw strategy- details included in the assignment as a Microsoft Word link in PowerPoint presentation, and also in the file. The activity is designed for students to cooperate with peers as they brainstorm and find answers to the issues presented in the assignments.
Before beginning the assignment, students will review basic information on force and motion using 10 questions, answers are provided.
Embedded in the PowerPoint is a video on the 3 laws of motion and before each assignment is the background knowledge on each law of motion.
Students will therefore be able to explain why a body is jerked forward when a moving vehicle comes to a sudden stop, identify how tension, stress, shear, bending and torsion influence bridges construction and also apply the concept of action and reaction to the NASA space mission.
Instructor should allow creativity and other community issues that students raise that relate to the laws of motion.
Standards covered
SPS7. Students will determine relationships among force, mass, and motion.
a. Apply Newton’s three laws to everyday situations by explaining the following:
• Inertia
• Relationship between force, mass and acceleration
• Equal and opposite forces
S8P5: Students will recognize characteristics of gravity, electricity, and magnetism as major kinds of forces acting in nature.
a. Recognize that every object exerts gravitational force on every other object and that the force exerted depends on how much mass the objects have and how far apart they are.
S8P3. Students will investigate relationship between force, mass, and the motion of objects.
a. Determine the relationship between velocity and acceleration.
b. Demonstrate the effect of balanced and unbalanced forces on an object in terms of gravity, inertia, and friction.
Based on a work at http://www.teacherspayteachers.com/Store/Tides.
Permissions beyond the scope of this license may be available at www.tidesinc.org/contact.
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N/A | 505 | 2,378 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 3.71875 | 4 | CC-MAIN-2020-34 | latest | en | 0.920647 |
http://dienekes.blogspot.com/2008/10/estimating-tmrca-for-pair-of-y-str.html?showComment=1223910240000 | 1,529,708,668,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267864822.44/warc/CC-MAIN-20180622220911-20180623000911-00187.warc.gz | 87,512,646 | 23,442 | ## October 13, 2008
### Estimating TMRCA for a pair of Y-STR haplotypes using Average Squared Distance (ASD) cont'd
This is a continuation of On the use of average squared distance (ASD) to estimate the time to most recent common ancestor (TMRCA) of a pair of Y-STR haplotypes inspired by James Heald's interesting comments. I suggest that you read that post first if you want to make sense of this one.
Summary of first post
In the first post I studied via simulation, the distribution P(gest | g) where g is the real TMRCA, and gest = ASD(hta, htb)/2μest is its estimate via average squared distance of two haplotypes hta and htb.
It was shown that the expected value E[gest | g] = g, and also that gest varies by quite a lot around g.
E[g | gest]
What can we say about E(g | gest), i.e. the expected value of the TMRCA if we have an estimate of its age? In other words, the expected real age, given its apparent age.
To study this, I sample g from a prior distribution P(g) and determine for each sample (via simulation) the corresponding gest. Thus, for each gest I have a set {g1, g2, ..., gn} (real ages), whose apparent age is gest.
Then:
E[g | gest] = mean{g1, g2, ..., gn}
Figuring out the prior distribution P(g) is of course the real trick. Bruce Walsh suggests (Genetics 158: 897–912) using p(t) = λexp(-λt) with λ=1/Ne, where t is the TMRCA and Νe is the effective population size. He cites an estimate for this Ne=5000 for humans, and discovers that the posterior distribution of t is not very sensitive to the prior.
Various kinds of belief or evidence can be incorporated in the prior. For example, we can set P(g)=0 if g>2,500 since two Y-chromosomes cannot have an MRCA older than "Y-chromosome Adam" -- if we are convinced that 2,500 generations is an upper limit on the age of Y-chromosome Adam.
But, if we are comparing Y-chromosomes of patrilineal descendants of a known founder (e.g., Genghis Khan), then we can set P(g) = 0 if g>40. Conversely, if we are comparing an R1a and R1b haplotype then we can set P(g) = 0 for g less than e.g., 160 generations, since the MRCA of R1a and R1b must be older than the time in which R1a1 has been detected in ancient DNA.
The point is that different real ages can lead to the same observed apparent age. To go from the apparent age to the real one, we can use prior information about it, i.e., the P(g) distribution.
It is important to note that P(g) can be seen as our prior belief about the distribution of g, that is: given two Y-chromosomes, what is our guess about their TMRCA before we see their haplotypes?
But, it can also be seen as the actual distribution of g in the collection of haplotypes from which we have sampled a pair. This will be different e.g., in a rapidly expanding population vs. a static one, or in a population expanding early or late in its history, etc.
Simulation
In the following I use a prior P(g) that is Uniform(1, 2500). I take a million samples from P(g). The number of markers is 50, and the mutation rate is .0025. In the present, I ignore uncertainty about the mutation rate that was the topic of the previous post.
Note that my primary concern isn't to motive this prior as realistic, since P(g) is dependent on prior knowledge (ancient DNA/population history) as mentioned in the previous section. Rather, my goal is to show (i) that E[g | gest] is not generally equal to gest, and (ii) that the choice of prior affects this quantity.
In the following I plot E[g | gest]/gest as a function of gest. For better visualization, and since a particular gest value may not be observed in the simulation, I group gest's into 100-generation long bins, i.e., I show the expected value of g, given that gest is between 1 and 100 generations, 101 and 200, and so on.
It is fairly obvious that E[g | gest] is not equal to gest. For small g it is greater than gest, while for large g it is smaller than gest.
It is easy to see why: consider for example ASD=0 and hence gest=0. It is clearly the case that ASD=0 is compatible with many real ages, all of them are greater or equal to 1. Hence gest=0 is clearly an underestimate of the real age. On the other hand an apparent age of 2,500 generations corresponds to real ages less or equal to 2,500 generations, and hence the expected real age is greater than the apparent age.
Now, consider a different prior: Uniform(1, 1000).
Or Uniform(200, 2500):
It is clear that E[g | gest] is not generally equal to gest and moreover depends on P(g).
Conclusion
To put the conclusions succinctly:
• If the age g of the common ancestor is known, then ASD is expected to be 2μg.
• If ASD is known, then the expected age of the common ancestor is not in general expected to be ASD/2μ.
• The expected age (given an ASD value) varies with ASD, and the way in which it varies depends on the prior estimate of the TMRCA.
In short, expected ASD grows linearly with age; expected age does not grow linearly with ASD.
This adds an extra level of uncertainty about the TMRCA, namely population history. It does not seem to be the case that an unbiased estimate of TMRCA can be estimated from ASD.
Of course, the way forward is, once again, to increase nm and pin down the mutation rate more accurately. This will increase the effect of the "evidence" in a Bayesian analysis, making it less susceptible to the background knowledge or belief represented by the prior.
#### 1 comment:
McG said...
I applaud your modelling efforts and have said so before. I also respect Jim Heald. My comment is how real is this simulation?? 1. P(g): I use the number 30 for years to gen conversion. At this point each 1 year "error" is about 3%+ error in estimating time to a specific occurrance. Since events are usually dated by year, the spread in years can be very large. 2. You use a fixed number for mutation rate; I estimate as much as a 30:1 range over the "slow" mutators and approaching 100:l over all dys loci. 3. I have studied the Ian Cam of the FtDNA Clan Gregor extensively. They all have a common ancestor who was born about 1300 AD. (note there have been 24 chieftains since him or an average number of years per chieftain of c. 30). The range of the number of mutations of this data set ranges from: 0 for the clan chieftain, a direct descendant to 7 for some some cousins whose ancestors travelled to Australia. (this is over 37 dys loci). How would one ever estimate the number of g's back to a common ancestor??
I have come to the conclusion that, "weak" as it is, TMRCA is the best approach and the larger the data set the better the estimate. Somehow?? on an individual basis, this doesn't make much sense due to multiple step mutations and apparent increase in rate with change of country?? (climate?). The only way it seems to work is if you use the group of descendants together?
So I can demonstrate with just one data set that "modelling" the mutation process still requires some inputs we don't understand or have descriptions of??
Maybe you can argue that my examples are all acceptable results within the framework of your model?? I just don't see it. | 1,759 | 7,112 | {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0} | 2.96875 | 3 | CC-MAIN-2018-26 | latest | en | 0.913137 |
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