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https://economics.stackexchange.com/questions/20127/econometric-estimation-of-engel-curves-or-demand
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# Econometric estimation of Engel curves (or demand?)
The problem is the following: Data is a family survey of expenditures on commodity groups.
An exercise told I should estimate the next equation: $$w_{j}=ß_{j}+\gamma_{j}log(cons)+a_{j}z+u_{i}$$ Here, $w_{j}$ is the consuming share (expenditure share) of the j-th commodity group like food, fuel, alcohol etc. $log(cons)$ is the total consume(expenditure) in a family. $z$ is dummy, 0 if 1 child in the family and 1 if 2 child in the family. There are 6 commodity groups which cover the full range of products.
$\text{Question 1.}$ Is this an engel curve? I have no idea, because as I know demand function need to contain prices and the income.
$\text{Question 2.}$ Because of the dependent variable $w_{j}$ (which is a share $\frac{(p_{j}*x_{j})}{cons}$) the $a_{j},ß_{j},\gamma_{j}$ estimates contains the price effects. What's more the exercise mention these are somehow the function of the logarithm of Prices. Can anyone explain why?
$\text{Question 3.}$ What are the restrictions for the parameters, when the results consistent with the consumer theory? [so when: homogenous of degree is 0 in demand functions $(a_{j},ß_{j})$, from walras's law restrictions to $\gamma_{j}$, and also how to derive the Slutsky-matrix (i don't know how to get the elasticities from Engel curves).
This is my first post, so sorry for long description and thanks in advance!
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https://www.coursehero.com/file/6613564/Chapter-6-sec-1/
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Chapter 6 sec 1
# Chapter 6 sec 1 - Chapter6 Section1 Mr.KenHorwitz...
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Chapter 6 Section 1 1 The Normal Distribution Mr. Ken Horwitz October 27, 2010
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Normal Distributions Many continuous variables have distributions that are bell-shaped and are called approximately normally distributed variables . The theoretical curve, called the bell curve or the Gaussian distribution , can be used to study many variables that are not normally distributed but are approximately normal. 2
Normal Distributions 3 The mathematical equation for the normal distribution is: 2 2 ( ) (2 ) 2 - - = X e y μ σ π 2.718 3.14 = = where e population mean population standard deviation
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Normal Distributions The shape and position of the normal distribution curve depend on two parameters, the mean and the standard deviation Each normally distributed variable has its own normal distribution curve, which depends on the values of the variable’s mean and standard deviation. 4
Normal Distributions 5
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How do you recognize the Normal Distribution The normal distribution curve is bell-shaped. The mean, median, and mode are equal and located
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Ask a homework question - tutors are online
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## Sunday, January 20, 2019
### NPR Sunday Puzzle (Jan 20, 2019): What a Feeling
NPR Sunday Puzzle (Jan 20, 2019): What a Feeling:
Q: Take the name of a classic song that became the signature song of the artist who performed it. It has two words; five letters in the first, three letters in the second. The letters can be rearranged to spell two new words. One is a feeling. The other is an expression of that feeling. What song is it?
Alternatively, you can rearrange the letters to spell two items that can be eaten.
Edit: The two foods are MANNA and POI.
A: Billy Joel's PIANO MAN --> PAIN, MOAN
1. Here's my standard reminder... don't post the answer or any hints that could lead directly to the answer (e.g. via a chain of thought, or an internet search) before the deadline of Thursday at 3pm ET. If you know the answer, click the link and submit it to NPR, but don't give it away here.
You may provide indirect hints to the answer to show you know it, but make sure they don't give the answer away. You can openly discuss your hints and the answer after the Thursday deadline. Thank you.
2. Ack! Blaine took my hint away. For an answer, ask Bill, John, Paul, or David.
1. I just read jan's comment at the end of last week's post. I guess we all independently had the same idea.
2. I have no idea what any of you are talking about.
3. Great minds....
4. "No great mind chooses to make anagrams" - the new motto of STRAP.
5. Yes, Blaine, and I think it may have been Heaven sent.
3. How does that phrase go...? Half a day late and a dollar short?
1. I believe it's, a day late, and a dollar short.
2. The puzzle was posted almost exactly a half day from "Nine o'clock on a Saturday..."
4. Easy puzzle. I’m sure the artist’s offspring would echo that sentiment.
1. This comment has been removed by the author.
5. The ultimate concert would feature The Monkees with The Village People.
6. Did you refill the Brita?
1. That's a trebuchet screen name. Who catapult you up to it?
2. Glad someone here got that.
7. Coincidentally, the feeling and expression are what I experience whenever I come across this artist...
1. You may be right, if we have the same answer.
2. This comment has been removed by a blog administrator.
3. Anyone want to bet BB gets blog administered?
8. As a curiosity, WONDERWALL by Oasis anagrams to DWELL ON WAR...
1. That's WONDERWALL → DWELL ON WAR with Congress.
9. The one question I keep asking myself on this blog is, "What are you doing here?"
10. Got it quickly. This artist was my first concert and I have a theory that in this artist's home area (also my home area), at any given time, there is a radio station playing one of his/her songs.
11. I should have figured this one out last night around bed time.
12. In the midst of planning my second spring trip to Bentonville, Arkansas with my mom.
13. "a classic song..." is open to interpretation.
Those who have been solving Puzzleria! puzzles in the past month or so may be experiencing a tinge of deja vu as they solve this week's NPR puzzle.
For STRAP members and other in favor of repealing anagram puzzles, here is a non-anagram puzzle currently appearing on Puzzleria! that should be a snap for Blainsvillians to solve:
Name a nationally broadcast weekly show, in three words.The first and last names of a person featured on the broadcast begin with the initial letters of the first and third words in the broadcast’s name. The initial letter of the person’s college degree is the same as the initial letter of the broadcast’s second word. What is the name of this broadcast? What is the name of this person?
For those who do like anagram puzzles, here is one on the current Puzzleria! that is more challenging than this week's "signature song" puzzle:
Name what famished people would do – in a phrase of two 6-letter words beginning with D and H – if they took a certain facetious hyperbolic expression literally. Rearrange these 12 letters to form more suitable consumables for the famished. What consumables are these?
Give hints if you solve either of these, but please post no answers until Wednesday afternoon.
LegoLePetitDejeunerVacuum
1. Speaking of vacuums—cosmic thought of the day (that doesn’t suck): when you are cleaning out the vacuum cleaner intake, you become the vacuum cleaner.
2. Speaking of cosmic thoughts, I hope everyone can view tonight's lunar eclipse. The one last year was pretty cool, here's a photo I took. Not as astonishing as a solar eclipse, but it's easy to see anywhere in the US. Next one won't be until 2022 in North America.
Rain in the Bay Area might cloud the experience.
3. If a vacuum is a place with nothing in it, why would you need to clean it?
4. Nice image of the recent lunar eclipse, eco. Snow clouds may roll in later tonight obscuring our view. It was gorgeous and sunny on our walk today, though.
Good question, SuperZee-san.
5. This comment has been removed by the author.
6. WW - Would that make me SuperZen, or ZenZee?
14. Easy to get the song but not as easy to anagram it.
15. Very simple to solve with a list. So why won't Will Shortz use the puzzles I send him that he says are really good, but refuses to use because he says he tries not to use puzzles that can be solved with a list?
16. There is a famous signature song in 2 words, 5 & 3, that when anagrammed into 2 other words evokes a feeling opposite of the one expressed in the ballad. Can you figure this one out?
Hint: If you hate anagrams, you can think of it as a form of Spoonerism.
17. This is a model puzzle. Or at least it was at one time.
18. The artist reminds me of an activity often done in late summer/early fall. --Margaret G.
19. In the middle of an argument with Lois, the answer came to me.
20. Will's going soft on us.
21. This comment has been removed by a blog administrator.
22. Well, I solved it by going through my LP collection while drinking alone, but must admit first got hung up with a pelvic “Dough Nod”
23. The way to solve anagram puzzles is to look them up on the internet.
How often does anyone eat either of the two foods?
1. “I did it my waaaaay”... Sunday mornings are about thinking through potential solutions to the puzzl on my own. Sunday evenings are looking at the clues from this blog. And Mondays are about the Internet.
And those foods are not on my diet.
24. According to Webster, the expression can be for a different feeling.
25. Reverse of Petula Clark'signature song to get in artist's zip code (later left).
26. Don't know much about electrochemistry ...
1. ... but I think it involves AMPs and ANIONs.
So, what about the edibles? ONION doesn't work. I don't know what POI tastes like, but I've heard of it. But ANMAN? What could that be? Wait a minute, what's this stuff in my jar?
27. ron:
That first cartoon reminded me to ask if anyone knows if you can get drunk swimming in Lake Mead?
28. Take the title of a classic song that became the signature song of the artist who performed it. Add one letter to the very end of the title. The result sounds like something that is true about the royalties generated by the classic song in this week's NPR puzzle.
LegoWhoObservesThatBondCooks!
29. The timing of this puzzle was a little off.
30. There are plenty of songs by this artist that I like, and even used to play on a particular instrument. But this one absolutely makes my skin crawl. What's worse, there always seems to be at least one jamoke at the bar who is more than happy than to waste a couple of quarters queueing it up on the jukebox, despite the fact that you can still hear it multiple times a day on the radio anywhere in the country. Fortunately, not unlike the equally execrable "Hotel California," the song lasts almost exactly as long as it takes to step outside and smoke a cigarette (which is definitely better for my emotional well-being, if not for my physical health).
I will say that the record this artist's concerts in his/her hometown have broken is mightily impressive.
31. How about this - change one letter, and it anagrams into two countries which lie on opposite sides of the same body of water.
1. Then drop a letter and rearrange to get the title of an award-winning movie.
2. ... or drop a different letter and rearrange to get a country a couple thousand miles northwest of the two countries, as the crow flies.
LegoThinksThat"AsTheFlyFlies"WouldMakeTheDistanceLonger
3. hodiau016's riff:
PIANO MAN - P + R = RIANOMAN = IRAN + OMAN
Paul's riff:
RIANO MAN - A = RINOMAN = IRON MAN
Lego's riff:
RIANO MAN - N = RIANOMA = ROMANIA
LegoWhoThankshodiau016ForAFineRiffOffPuzzle
33. Remember back when convoy was not a bad word?
34. Interesting use of the word "feeling".
35. No, but you might have a honey of a time...
36. Okay, here's a Clue.
Was Colonel Mustard mustered out of the military after finding himself in a bit of a pickle? And was he the condimental commander of his regiment? Also, was he ever able to ketchup? I really would like to know, and please, don't give me a load of horseradish.
1. Some days I don't relish your postings.
2. Soy you say. Wasabi with that?
3. Mayonnaise-ayers learn to accept the truth.
5. In fact you should go the extra Maille. Sriracha now!
37. This left me listing to Port.
38. This is all a tahini bit silly.
1. Don't you halva a sense of humor?
2. No, but if you hummus a few bars we might join in.
The Society To Reduce Aggravating Puns is in early formation.
3. Oh, for pita's sake, he's bean dippin' again!
4. Your humor isn't just partly poor, it's falafel.
5. However it beets yours.
6. I may have to meat out an appropriate punishment. But that's a boar, so I'll just chuck it.
39. Actor clue: Alec Baldwin
1. Baldwin - Brand of piano
40. The man is hell on a deux chevaux!
1. In 2004 Billy Joel drove his Citroen 2CV into a house when he lost control on a rain slick road.
https://www.cbsnews.com/news/piano-man-is-smashup-man-again/
41. Richard Sanders comes to mind...
42. Will Shortz apparently didn't like this puzzle I sent him (how do you know he actually receives them?).
"Juliet in ruins"
1. I, in lust, injure.
2. Jilt us in urine could be a reference to a Trump tryst. But we all know MJ would never stoop to devising an anagram, even with a misuse of the word unique.
3. MJ is too sophisticated for a stupid anarghram. But I don't see an answer right away.
4. This has nothing to do with a "singer" named Juliet Ruin. I found yesterday that brief exposure to her causes headache, blurred vision and nausea.
5. MJ, did I miss the answer to your puzzler? It’s intriguing.
6. Renople:
Thanks for asking. There were two sort-of tries. Since this thread is pretty much over, I will update it in the new one.
43. Is there any valid reason why Trump can't deliver his State of the Union speech at Mar-a-Lago, or the Kremlin?
1. He should give the speech from McDonald's. Invited guests would be Mayor Mike McCheese Pence, Grimace Huckabee Sanders, Stephen Hamberdlar Miller, Officer Big Mac Whitaker, Birdie the Early Bird Conway, and the GOP Fry Kids.
2. Or, he could give it in the House Chamber, with no cameras, microphones, Democratic members of Congress, or reporters present.
44. This comment has been removed by the author.
45. PIANO MAN >>> PAIN, MOAN
“In the midst of planning my second spring trip to Bentonville, Arkansas with my mom.” >>> The link is to Crystal Bridges Art Museum in Bentonville, AR. Crystal Bridges points to Christie Brinkley, one of Billy Joel’s three ex-wives.
{Crystal Bridges is an extraordinary collection of architecture, paintings, sculptures, a geodesic dome, bicycle paths, and nature. It opened 11/11/11 so is just being discovered by much of the world.}
1. Funny, I was sure your clue alluded to two of Billy's albums, Glass Houses(1980) and The Bridge(1986), while sdb's Convoy comment did make me think of CB.
2. Paul, those allusions work quite well, too.
46. BILLY JOEL > PIANO MAN
47. Billy Joel, PIANO MANPAIN (feeling) + MOAN (expression of that feeling).
The 2 foods that can be “eaten” → MANNA + POI.
Change the P to an R to obtain IRAN + OMAN.
Now drop the O to yield RAIN MAN, or drop an N to yield ROMANIA.
48. PIANO MAN -> PAIN, MOAN
> The title can also be anagrammed to name two foods.
MANNA, POI
> Will's going soft on us.
In music, PIANO means “soft”. (The instrument was originally called a pianoforte, since it could be played soft or loud, depending on the force with which a key is pressed.)
> MSG
Not sure what it means to be a franchise of Madison Square Garden, but Billy Joel is one. Over 100 concerts there, including one tonight.
1. But doesn't it cause a headache for some people?
49. I wrote, “For an answer, ask Bill, John, Paul, or David.” These are names in the lyrics; I changed “Davy” (who’s still in the Navy) to “David” to preclude searches. The other names with “Davy” did produce a hit on the song lyrics.
50. Yeah, but you included Bill even though that’s the bartender’s reference to the artist Billy Joel, no?
51. Yep. It is a name in the lyric.
52. My posts this past week:
* "Those who have been solving Puzzleria! puzzles in the past month or so may be expriencing a tinge of deja vu as they solve this week's NPR puzzle."...
I ran this puzzle on January 11 on Puzzleria!:
Name a very northern U.S. city in 9 letters. If you have the right one, you can rearrange its letters to get a 4-letter word for a kind of pain one may suffer and a 5-letter word beginning with G for how one may respond to the pain. What is this city?
* Answers to this week's Puzzleria! puzzles that I posted on Sunday:
Name a nationally broadcast weekly show, in three words. The first and last names of a person featured on the broadcast begin with the initial letters of the first and third words in the broadcast’s name. The initial letter of the person’s college degree is the same as the initial letter of the broadcast’s second word. What is the name of this broadcast? What is the name of this person?
Answer: Weekend Edition Sunday; Will (Enigmatology) Shortz
Name what famished people would do – in a phrase of two 6-letter words beginning with D and H – if they took a certain facetious hyperbolic expression literally. Rearrange these 12 letters to form more suitable consumables for the famished. What consumables are these?
Answer: Hors d'oeuvres, an anagram of "devour horses" ("I'm so hungry I could eat a horse!")
* My Sun Jan 20, 07:38:00 PM PST post:
"Take the title of a classic song that became the signature song of the artist who performed it. Add one letter to the very end of the title. The result sounds like something that is true about the royalties generated by the classic song in this week's NPR puzzle.
LegoWhoObservesThatBondCooks!"
"Ode to Billie Joe" was Bobbie Gentry's signature song. Add an L at the end to the end to form "Ode to Billie Joel" which sounds like "owed to Billy Joel," which would be true about the royalties generated by the classic song in this week's NPR puzzle.
(In my sign-off, "LegoWhoObservesThatBondCooks!"... "BondCooks" is an anagram of "Boondocks," the key word in a signature song sung by Billy Joe Royal!)
* Tomorrow's Puzzleria! features a "signature" cryptic crossword puzzle created by Patrick J. Berry, screen name "cranberry."
53. My clues
- the artist’s offspring would echo that sentiment” was a reference to the Amazon echo and Alexa, who is Billy Joel’s daughter.
- the Webster dictionary says that moan is a way to express pain or sexual pleasure. (Making it an interesting word with such conflicting definitions!)
54. I said "The artist reminds me of an activity often done in late summer/early fall. " "Billy Joel" anagrams to "Boil Jelly", which I did last fall. :) --Margaret G.
55. Piano Man —> Pain, Moan
The first line of Billy Joel’s signature song is, “It’s nine o’clock on a Saturday…” As I live on the East Coast, I can usually find the puzzle posted at about eight o’clock Sunday morning, or hear the broadcast on my local NPR station at about 8:45. Ergo, my comment about the puzzle being, “... a little late.”
56. The ultimate concert would feature The Monkees with The Village People.:
"And he's talkin' with Davy" (Jones, of The Monkees), "who's still in the Navy
And probably will be for life"
I'd never seen the official Village People video before this week. It's hard to watch without smiling, especially knowing that the navy let them use their ships and sailors (your tax dollars at work, it's a grind), thinking the song and video might be good for recruiting!!!
On the Snopes.com message board the captain of the ship (reportedly) said "Men, you may do 'The Hustle,' but there will be no 'Hippy Hippy Shake' aboard a United States Navy warship!"
Of course those were the happy times before AIDS was widespread.
1. Of course, there is also Davy Jones Locker, resting place of countless salts who served in the Navy.
57. PIANO MAN by Billy Joel; MOAN, PAIN
Though I managed to get away with the "what are you doing here" comment(it's a lyric from the song), my next comment obviously used too many of Billy's other song titles(cleverly, I thought)to be allowed. Oh well. You're only human.
58. Can you believe it! Our strongman president CAVED TO A WOMAN! Would a manly man do such a thing? The shame, the shame!
1. Caved twice, first on the SOTU, then on the temporary reopening.
Season 3 of The Appresidentice (credit to Harry Shearer) is off to a rocking good start. Stay tuned for the next episode!
2. This could be the time when he is most dangerous, when he's desperate!
The Bill Maher Show, tonight, might be a good one, Ann Coulter, aka First Lady, is a guest.
3. 68C, you disappeared for a while? I agree, he has been emasculated, what does he do?
If I were Secret Service I'd put an extra detail on Melania, who's probably hiding under a bed, or maybe safe in Florida.
Bigger worry is he tries to do a stupid Wag the Dog maneuver in Venezuela....
4. Eco - Good to be back!
Yes, regarding Venezuela, that's the kind of thing I am worried about, or something like it. Fortunately we have a temporary reprieve but I just see him calling a national emergency in 2-3 weeks and we'll have more gridlock again.
5. I wanted to watch it on my computer if possible, but I still need to cook dinner and just finished writing and posting the following on our neighborhood blog:
Just home after leaving the Friday evening wine tasting at the Holman Road QFC. As I turned from Aurora onto N. 107th I parked beside the empty, former glass shop so I could walk down to the pawn shop and see if the guy I saw at 4pm with the probably stolen TV had pawned it there. I stopped about 15 feet inside the entrance to see a TV that looked like what I would upgrade to, although I would not deal with a pawn shop. It did not have the name, but the remote was there, upside down. I picked it up and saw it is a Visio.
This is when it gets interesting.
At that moment a twenty something black guy in a mask and hoodie came running in directly to the counter directly East in a direct line from the front door and past where I was standing. Their cash register is also there and two employees were standing there and this black guy had an automatic black pistol out and stuck it right over the counter in their faces all the while yelling, "Give me the money! Give me the money!" over and over.
I was unarmed and knew that if he shot them, which it looked like might happen, he would kill me on the way out, so I turned and quietly walked back and out the door and tried calling 911 on my phone. I had trouble with the phone and it took at least three minutes to get to them. It is all on store video and the cops have already seen it. The guy had entered the store right behind me, coming from the South and I had not noticed, but it is clearly on the videos.
It was all great fun as I love excitement, but it is now time for dinner.
6. This comment has been removed by the author.
7. Wow, that is scary!! Glad you made it out O.K., were shots ever fired?
8. Thanks, but for me it wasn't. I enjoyed it. I know that sounds crazy, but it is true. Here is what I wrote on our two neighborhood blogs in response to one of the guys who runs one of them:
Nathan,
Thanks, but it wasn't scary for me at all. You never know for sure how you will react in a situation. I was surprised as I was walking out of the joint that my heart rate was seeming to be normal and I wondered why I was not scared that he might shoot me in the back. I was also thinking he was too involved, and if I got out right away I might be okay. Then I had some distance to walk back to my car, but at the same time trying to get my crappy cell phone to dial 911, all the while realizing I really should get back to the car around the corner because if he came out and saw me with the phone I might get shot. I have learned how to control my fear when I am in a dangerous situation, but this time I did not feel fear. I am very familiar with the fear feeling, but I have learned it does not always appear when most expected. Sometimes it does though. This was not one of those times where I was scared and controlled my fear, at least I don't think it was. I have had lots of those experiences. I just was not scared.
Many years ago I was in a tavern on business on Beacon Hill at about ten am, when a guy came in with a Ruger Blackhawk .44 magnum he pointed at me and the owner as we stood just inside the front door. I felt more concern on that occasion, but not strong fear, and I took the weapon away from the guy. I have had a lifelong fascination with fear and my relationship with it. I hate feeling it, but I also love it in some strange way.
One other thing I have learned is that thinking about a potential experience can produce a lot of fear although we are safe at the time of these thoughts, then should it actually happen it may not produce those feelings at all.
Confront you fears and you may find they are your friends.
Thanks for the concern,
9. So, SDB, the question we all want to know: Did you buy the TV?
The problem with a pawn shop is you always get rooked.
10. No, I saw it En passant.
It is a Visio without a tuner. I had seen them in the stores before and they seem to be good and also expensive. This was marked at only \$125. I would need to purchase a separate tuner. Anyway I only saw it at knight.
11. Just as-king.
When a relationship gets old do you have a stale mate?
12. I'm not sure. I'll check mate.
13. SDB: I am shocked to read about your close call. You were very lucky to have gotten out of there. I hope they put that guy away. Take care and stop going to pawn shops!!
14. Thanks Natasha. However I don't go to pawn shops. It was a quirk that I was there. I don't think the guy was caught. As I have posted before, I do not believe in luck.
15. Sdb: Do you consider carrying a gun now? It was a close call. Why can't they catch him if have video. Did he not see you leaving. Did you have your back to him or did you back out?
16. Natasha:
I have had a CWP since I was 21, but I rarely carry. Had I been carrying at that time it would most likely have been my .25 cal. and it would have been unrealistic to try and use it against him except as a last resort. Had I been wearing my S&W .38 cal. revolver with hollow points I would have shot him in the back of the head. I don't like having to go around with a weapon all the time and only do so when it seems like there might be a more likely need, such as when I recently purchased a used car with cash.
I have make a MS-WORD document with over 36 of my extremely close calls with imminent death. This will not make that list.
I turned quietly to my left and walked out normally, but as quietly as I could. I wanted to look normal if he happened to turn around and see me.
He was wearing a hoodie and a lower face mask, so will be hard to identify, although he was caught, as I was, on several of their cameras. I stopped in yesterday to see if they would let me look at the videos.
59. This comment has been removed by the author.
60. Isn’t this when, in British dramas, the butler leaves a Webley revolver, one round, and a snifter with two fingers of brandy, on the, “lords,” desk, before locking the door to the study.
1. Back in the '60's I owned a used .45 cal. Webley revolver. I thought it was as lethal as an Ann COuLTer on steroids. I sold it.
2. SDB,
Thought she was already on meth. Certainly has the physique of a meth-head. She radiates that lovely anorexic pallor, and of course her incessant nattering does mimic the typical speed freak on a run.
3. I don't know much about illicit drugs, but I agree about her being really skinny. That was the first thing I noticed when she walked out. So that and the drugs might help explain how she can be so unbelievably stupid.
61. This week's challenge: This week's challenge comes from listener Joe Krozel of Creve Coeur, Mo. Name a vehicle in two words, each with the same number of letters. Subtract a letter from each word, and the remaining letters in order will spell the first and last names of a famous writer. Who is it?
1. (I keep getting distracted by names that should be on the Car Talk Staff Credits page.)
62. 952 responses last week.
For NPR puzzle posts, don't post the answer or any hints that could lead to the answer before the deadline (usually Thursday at 3pm ET). If you know the answer, submit it to NPR, but don't give it away here.
You may provide indirect hints to the answer to show you know it, but make sure they don't assist with solving. You can openly discuss your hints and the answer after the deadline. Thank you.
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# Short answer find the quotient 6x^3-x^2-7x-9 ------------ 2x+3
143,010 questions, page 829
1. ## maths
two points A and B are given. find the set of feet of the perpendicular dropped from the point A onto all possible straight lines passing through the point B
asked by amit on March 31, 2016
2. ## Calc
Find the mass and center of mass of the solid E with the given density function ρ. E is bounded by the parabolic cylinder z = 1 − y^2 and the planes x + 6z = 6, x = 0, and z = 0; ρ(x, y, z) = 4. m=? x, y, z=?
asked by Anon on October 19, 2016
3. ## Calculus
Find the volume of the solid formed by revolving the region bounded by the graphs of y=4x^2 and y=16 about the x-axis. (Washer Volume)
asked by john on January 31, 2010
4. ## Math Calc
Given f'(x)=(x-4)(6-2x) find the x-coordinate for the relative minimum of the graph f(x). OPTIONS: 8 6 3 None of these I think it is 3, but I am also conflicted to say none of these because I graphed the function and ant see a minimum.
asked by Ke\$ha on January 5, 2017
5. ## Physics
A convex lens of focal length 10 cm forms image of object placed 30 cm in front of the lens. Find the magnification?
asked by Shilpa on December 7, 2010
6. ## physics
g varies from 0 N to 70.4 N as we stretch the spring by moving one end 5.35 cm from its unstressed position. Find the force constant of the spring
asked by Anonymous on October 18, 2016
7. ## calculus
find two positive integers such that the sum of the first number and 4 times the second number is 1000 and the product of the numbers is as large as possible
asked by tony on December 4, 2008
8. ## calculus
to find the possible relative extrema of the function f(x,y) subject to the constraint g(x,y)=8, the lagrangian function based on the lagrange multiplier λ is given by....
asked by soffy on June 12, 2012
9. ## nvas physics
A body is thrown vertical ly upward with a velocity of the 21m/s find maximum height reached by body
asked by swayam on July 23, 2015
10. ## Math
Our difference is 6. We have a sum of 14. What numbers are we? We know number 10. How did you find 4? Please describe and solve the riddle.Write the number sentence.
asked by Student on January 26, 2016
11. ## Arwachine public
the diagonal of a quadrilateral is 28 cm in length and the perpendicular to it from the opposite vertices are 6.8cm and8cm respectively find the area of the quadrilateral
12. ## Geometry
The volume of an oblique cone with equal diameter and height is 18pi cm^3. Find the height and radius(to the nearest cm).
asked by Anna on May 21, 2012
13. ## chemistry
Find the mass of urea needed to prepare 51.7 g of a solution in water in which the mole fraction of urea is 7.59×10−2. anyone have the same values?
asked by jazz on January 31, 2012
14. ## math
A rectangle with sides parallel to the coordinate axes is inscribed in the ellipse x2/a2 + y2/b2 = 1: Find the largest possible area for this rectangle.
asked by pairika on April 30, 2012
15. ## science
Hello. Please, help me do this. I would be really thankful! Find the mass of triacetate, which you can get from 1.62 ton of wood waste. The yield of process is 75% and there are 50% of cellulose in wood.
asked by eve on May 21, 2012
16. ## math
Find the surface area of a small juice can whose diameter is 4 inches and whose height is 6.25 inches. Use 3.14 for ð. Hint: The can is a cylinder.
asked by Anonymous on January 31, 2012
17. ## AP Calc. AB
If an arrow is shot upward on the moon with a velocity of 56 m/s, its height in meters after t seconds is given by . Find the instantaneous velocity after one second.
asked by Jacke on August 29, 2014
18. ## Math
David wanted to solve 3(x-7)=5. As a first step, he wrote 3x-21=15. What error did he make? Correct his error and then find the solution?
asked by Patchie on November 29, 2012
19. ## math
How to find the area of a regular hexagon with a perimeter of 78 inches and a perpendicular distance of 11 inches from side to center
asked by mary on November 4, 2014
20. ## Chemistry
2g NaHCO3 + .6 mL HCL --> NaCl+CO2+H2O How much carbon dioxide do I get? How do I find out? I got 410 mL of CO2, is this correct?
asked by Michelle on March 19, 2011
21. ## physics
given a right triangle with hypotenuse of 12.5 cm, side y of 7.2 cm? find length of side x and size of other two angles? so you set it up as pythagoreum theory? 12.5= x^2 + 7.2
asked by tim on October 20, 2009
22. ## maths
A letter is picked at random from the English alphabet. Find the probability that the letter : a) Appears in the word GEOMETRY
asked by arzam on May 22, 2013
23. ## Chemistry - Melting Ice
The heat of fusion of ice is 6.00 kJ/mol. Find the number of photons of wavelength = 6.78×10^−6 m that must be absorbed to melt 6.80 g of ice.
asked by Jayden on November 29, 2012
24. ## Calculus
The base of a solid is bounded by y=2sqrtx, y=2 and x=4. Find the volume of solid if cross sections perpendicular to y=2 are semicircles
asked by Alexandra on April 17, 2011
25. ## math
the perimeter of an isoscles triangle is 42 cm and its base is 3/2 times each of equal sides. find the length of each sides. and area
asked by aaaaaa on June 9, 2013
26. ## Math
If I’m 100 feet away from a rocket launch pad and the rocket rises 80°. How can I find the rockets maximum elevation?
asked by Jessica on April 15, 2019
27. ## math
A sum of money is divided into two parts in the ratio 5:7. If the smaller amount is \$200, find the larger amount
asked by Anonymous on September 25, 2012
28. ## Calculus
A rectangle has a vertex on the line 3x + 4y = 12. Two of its sides lie along the positive x and y axes. Find the maximum possible area of such a rectangle.
asked by I on September 25, 2017
29. ## Math( Geometry)
All three planes in the figure provided are parallel. They are cut by lines containing points A, B and C and X, Y and Z. Each of these points lies in a plane. If BC=8 cm, YZ=10 cm and XY=14 cm, find AB. A.)10 1/2 cm B.)10.2 cm C.)11 1/5 cm
asked by john on July 14, 2012
30. ## Trig
Find all the solutions in the interval [0,2pi). 2sin2x - sqrt(2) = 0 sin2x = sqrt(2)/2 I'm confused by the double angle. Please help.
asked by Travis on December 2, 2008
31. ## math
Find the coordinates of th vertex and write the equation of the axis of symmetry given the function f(x)=-x^2-3x-2.Is the vertex a maximum or a minimum?
asked by Jason on December 7, 2011
32. ## Chemistry
Find the standard entropy (S^0) of ice at 25K. delta Hfus = 6009 J/mol and Cp of ice = 38 J/molK.
asked by Jessica on October 5, 2011
33. ## UK
The velocity of an object t seconds after it started moving is t(t-9). Find the total change in the object's position in the 6th second.
asked by Anonymous on April 13, 2011
34. ## Physics
Force of 2.25N is acting on charge 15x10^-4C at any point. Find the Electric Field Intensity at that point.
asked by Clifford on August 3, 2015
35. ## maths
A chord subtends an angle of 68 degree at the centre of a circle of radius 200 mm.Find the length of the chord.
asked by Shane on March 3, 2013
36. ## Calculus
Find the area of the region which is bounded by the polar curves theta =pi and r=2theta 0
asked by Salman on February 4, 2010
37. ## home health care
where can i go to find information on this question I have to ask? how are services provided in home health care to fit in continuum of care
asked by jean on September 27, 2008
38. ## calc
find linear approximations for each of the following and put a bound on the error of the estimate. 1. (7.985)^1/3 using f(x)=x^1/3 and a=8 2. (.9997)^100 using f(x)=x^100 and a=1 please help
asked by C on December 9, 2010
39. ## Math, help pls
The radius of a cylinder is 3.5 ft. The height is 14 ft. Find the surface area and volume of the cylinder to the nearest tenth of a foot.
asked by Sparkle on March 7, 2019
40. ## trig
The terminal side of angle alpha is in the fourth quadrant, and sin alpha=-1/2, find the values of the other functions.
asked by mrkkn on July 16, 2010
41. ## Math
if the lateral surface area of a regular square pyramid is 200 inches square how do I find the volume?
asked by JOdi on November 27, 2012
42. ## Calculus
4. Find the volume of a pyramid of height 160 feet that has a square base of side 300 feet
asked by Maryam on June 5, 2013
43. ## ALGEBRA
The perimeter of a traffic sign must be 128 inches. It's length is 8 inches longer than the width. find the dimensions.
asked by AMBER on March 28, 2012
44. ## algebra
let t = 0 be the point at which the car is just starting to drive and the bus is even with the car. Find the other time when the vehicles will be the same distance from the intersection.
asked by sam on March 3, 2013
45. ## SS
In a region as dry as the Sahara Desert, an oasis is often the only place to find water. True False True?
asked by Anonymous on October 23, 2013
46. ## PHYSICS
What is the relationship between angle and gpe with a cart and a ramp. Also if it isn't linear find away to graph it in a linear fashion.
asked by Anthony on February 18, 2017
47. ## HELPPPPPPP (TRIG)
find, to the nearest minute, the solution set of 1- sin x = 3 cos^2 x over the domain 0 degrees to 360 degrees
asked by HELPPPP on May 20, 2010
48. ## math
Find the percent if discount when the regular price for a DVD recorder is \$250 and its sale price is \$175
asked by Anonymous on March 31, 2014
49. ## Math
if sin theta =1/2 and cos theta= √(3)/2. find tha values of tan θ, cot θ, sec θ and cscθ.
asked by Anonymous on March 8, 2019
50. ## Math
P is a point outside of circle Γ. The tangent from P to Γ touches at A. A line from P intersects Γ at B and C such that m∠ACP = 120∘. If AC=16 and AP=19, find the radius of the circle.
asked by Leo on June 4, 2013
51. ## physics
how do you find the magnitude of the verticle component of velocity of a projectile just before it strikes the ground if you know the horizontal component but no angles?
asked by gael on September 27, 2008
52. ## calculus
use the definition of the derivative to calculate f'(x), if f(x)= SQRT(2x) and find the equations of the tangent line and the normal line to the graph of f at x=2
asked by Raj on November 13, 2010
53. ## algebra
The amount of rainfall in January in a certain city is normally distributed with a mean of 4.3 inches and a standard deviation of 0.3 inches. Find the value of the quartile Q1.
asked by steve handson on September 1, 2016
54. ## Math
In a circle of radius 12cm a chord is drawn 3cm from the centre. Find the angle subtended at the centre
asked by Emmanuel on June 23, 2019
55. ## math
hi i startted the problem below by writing a sigma notation of sigma[(2^(n+1))/((n+1)!)] QUESTION: how do i find the sum of 2+(4/2!)+(8/3!)+(16/4!)+... was my attempt wrong, or how do i go from here? thank you!
asked by kiksy on March 17, 2011
56. ## physics
Find the partial pressure of nitrogen gas in the equatjon at one atmosphere atm Nitrogen+Oxygen gives Niticoxide
asked by Anonymous on August 2, 2015
57. ## math
hi i startted the problem below by writing a sigma notation of sigma[(2^(n+1))/((n+1)!)] QUESTION: how do i find the sum of 2+(4/2!)+(8/3!)+(16/4!)+... was my attempt wrong, or how do i go from here? thank you!
asked by kiksy on March 17, 2011
58. ## calculus
Find the equation of the line tangent to the graph of the given function at the point with the indicated x-coordinate. f(x)=(4 sqrt(x)+7)/(sqrt(x)+2); text( ) x=4
asked by Beth on October 14, 2011
59. ## Chemistry
Find the theoretical pH of .01M and .05M benzoic acid buffer when 2ml of .1MNaOH was added to each buffer.
60. ## physics
A particle moves according to the eqn a=-kv where a is acceleration and v is the velocity and k is a constant. If initial speed is V0 find the speed at any instant
asked by karan on August 19, 2014
61. ## General Science
A power boat accelerates along a straight path from 0km/h in 10.0s.Find the average acceleration of the boat in meters per second.
asked by K-Lynn on January 30, 2013
62. ## Math
The bases of a right prism are parallelograms with length of one of its sides a=8.5 cm and altitude to that side ha = 4 cm. Find the volume of the prism, if the height is h=14 cm.
asked by Anonymous on April 10, 2019
63. ## Math
Suppose \$3500 is invested in an account with an APR of 11% compounded monthly. Find the future value of the account in 3 years.
asked by Meredith on July 14, 2012
64. ## Algebra102
A standard deck of cards is shuffled and one card is drawn. Find the probability that the card is red or a jack.
asked by NatoshaMarie on May 20, 2010
65. ## math
what is the rule for using diameter to find circumference? If the diameter is 2* the length of the radius then d/2 then C=2pie+r. CATCH YOU ON THE FLIP SIDE
asked by sal on November 26, 2006
66. ## Math
How do you find the perimeter on a on a what????? what is "integer" in math terms for example 15 steps behind directions say "write an integer to represent each situation"
asked by Jordan on March 29, 2007
67. ## Science
An object is moving with uniform acceleration and it's velocity after 5 second is 25m/s after 8 second it is 34m/s find distance travelled by object in 12 second
asked by Harsh on June 16, 2016
68. ## amath
The roots of the quadratic equation x^2-mx+n=0 ,where m and n are integers ,are a and 1/4a. (i)Find the sum and product of its roots (ii)show that 4m^2=25n How to do part 2 ?
asked by Candy on June 3, 2013
69. ## Precalc
Find the indicated nth term of the geometric sequence. 7th term a5: -2/81 a9: -2/6561 I know the formula is an= a1 r ^ n-1 but I can't figure out how to apply it
asked by Ande on November 13, 2010
70. ## physical science 101
A box of mass 7.0kg is accelerated from resr across a floor at a rate of 4.0m/s2 for 5.0s. Find the net work done on the box.
asked by Tally on August 11, 2011
71. ## math
the diagonals of a rectangle are 25cm long and intersect at an angle of 32 degrees. Find the length of the sides of the rectangle.
asked by tinette on December 4, 2015
72. ## maths
find the general solution of the following first order differential equation. t*(dy/dt) = (-y/t) + ((exp(1/t)/t^3) could you show steps in working to solution.
asked by ricky on October 26, 2011
73. ## Calculus
Find the volume of the solid generated by the region in the first quadrant bounded above by the 3x+y=6, below by the x-axis, and on the left by the y-axis, about the line x= -2.
asked by Liz on June 3, 2012
74. ## Calculus
Find the area of the region which is bounded by the polar curves theta =pi and r=2theta 0
asked by Salman on February 5, 2010
75. ## maths
cutting a circle into equal sections of a small central angle to find the area of a circle by using formulae
asked by meraj on June 15, 2016
76. ## math
Radhika bought 24 dozen eggs .out of them,3/4 were found to be broken. Find the number of good eggs left with her.
asked by Fraction on May 1, 2017
77. ## math homework
use the limit process to find the slope of the graph of 'square root' x+8 at (8,4) a. 4 b. 1/4 c. the slope is undefined at this point d. 1/8 e. infinity
asked by jessie on August 31, 2016
78. ## Math
When a marathon runner completes 3 round around a circular track he covers a distance of 66 km.find the radius of track.
asked by Ujjwal gautam on March 3, 2018
79. ## Math
Find all complex zeros for the following polynomial function. Please show all of your work. f(x) = x^4 - 91x^2 - 900 Hi, I need help... I'm not understanding this complex zero..
asked by CheezyReezy on June 1, 2011
80. ## Math/Algebra
Find an equation for the ellipse described. center(0,0); major axis horizontal with length10; length of minor axis is 8
asked by Student on May 28, 2013
81. ## physics
A balloon is rising at 3.6 m/s at same time is blown by wind for 5.4 m/s from east. Find the actual velocity and direction of the balloon.
asked by noona on December 16, 2014
82. ## physical science
A stone of 8 kg mass falls from the top of an 80 m-high building. Find the velocity of the stone when it reaches the ground
asked by tye on March 3, 2015
83. ## Maths
A seller marks an article to gain 26% but allows 4% discount. Find the price he sells an article bought for N500?
asked by Paulina on June 18, 2019
84. ## Chemistry
Find the mass of oxygen gas needed to react with 0.2 moles magnesium and the mass of the product? Working out please
asked by Babiinoob on April 16, 2011
85. ## Chemistry
Find the mass of oxygen gas needed to react with 0.2 moles magnesium and the mass of the product? Working out please
asked by Jannie on April 12, 2011
86. ## English
How does the author portray the issue in the work The Necklace? Just need a good website to help me. I am terrible at trying to find good websites.
asked by Zimny_YoonSanha on November 29, 2018
87. ## Math
Write and solve an equation for each of the following. Triangle ABC is congruent to triangle DEF. AB = 2x - 10, and DE = x + 20. Find the value of x and AB. I don't even know where to begin, please help.
asked by Karlee on October 30, 2017
88. ## math
Find the coordinates of th vertex and write the equation of the axis of symmetry given the function f(x)=-x^2-3x-2.Is the vertex a maximum or a minimum?
asked by Jason on December 8, 2011
89. ## Math
Find the surface area of a right octagonal pyramid with height 2.5 yards, and its base has apothem length 1.5 yards.
asked by Jessica on June 1, 2011
90. ## Algebra 2
Please help me solve and find the excluded values for this problem. Also please show me the work for this problem I want to learn!@!@!. 49X^6+7X^7/28X^8
asked by Omar on March 28, 2014
91. ## maths
a square of 10cm on each side, using the side to forma curve as a radius from each end. find the area from the four arches.
asked by anonymous on November 13, 2010
92. ## Calculus
Find the area of the region which is bounded by the polar curves theta =pi and r=2theta 0
asked by Salman on February 5, 2010
93. ## inflection points
How can I find the number of points of inflection for a function whose second derivative is .5 + cosx - e^-x on the interval x is greater than or equal to 0 and less than or equal to 20?
asked by Anonymous on December 9, 2010
94. ## Polynomials help
Hello, this is a question which i don't know how to do. Find the value of k for which a-3b is a factor of a^4-7a^2b^2 + kb^4 Hence, for this value of k, factorise a^4 - 7a^2b^2 + kb^4 completely. I tried to do long division for the question, but couldnt.
asked by Aoi on June 19, 2009
95. ## Math
If you solve this you are officially amazing! "The diagonals of a trapezoid are perpendicular and have lengths 8 and 10. Find the length of the median of the trapezoid."
asked by Sheenybeany on May 19, 2014
96. ## calculus
The curve below is called a witch of Maria Agnesi. Find an equation of the tangent line to this curve at the point (-2, 1/5). y=1/(1+x^2)
asked by Ava on October 16, 2009
97. ## Chemistry Class
What is the molality of a solution made by adding 0.145 g of sodium acetate to 119 ml water? How do I find molality?
asked by Trina on November 1, 2012
98. ## math
an equilateral triangle of side 10cm is inscribed in a circle. find the radius of the circle? show the solution
asked by shafa on March 7, 2019
99. ## trig
Find, to the nearest degree, the solution set of 3sin^2x = 5 cosx+1 over the demain {x|0 degrees < or = to x which is < 360 degrees}
asked by Clarkney on May 17, 2010
100. ## trig
find to the nearesr degree, all values of x between 0 degrees and 360 degrees that satisfy the equation 2sinx + 4 cos 2x =3
asked by Clarkney on May 17, 2010
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# Check if a finitely generated matrix group is finite (works with QQ and not with CC)
This post is a wiki. Anyone with karma >750 is welcome to improve it.
Dear all, I am a newbie in sage. I would like to check if a finitely generated matrix group is finite. Before to proceed with the calculation on my actual problem (where matrices have complex entries), I have tried a very simple example. Consider the group generated by the matrices [1,0,0,1] and [0,1,1,0], this group is clearly finite. Can somebody explain me why the following code works:
sage: MS = MatrixSpace(QQ, 2, 2)
sage: G = MatrixGroup([MS([1,0,0,1]),MS([0,1,1,0])])
sage: G.is_finite()
True
but if I change the field QQ -> RR (or CC), an error is generated:
sage: MS = MatrixSpace(RR, 2, 2)
sage: G = MatrixGroup([MS([1,0,0,1]),MS([0,1,1,0])])
sage: G.is_finite()
---------------------------------------------------------------------------
NotImplementedError Traceback (most recent call last)
<ipython-input-215-0022a668c150> in <module>()
----> 1 G.is_finite()
/Applications/SageMath-7.6.app/Contents/Resources/sage/src/sage/groups/group.pyx in sage.groups.group.Group.is_finite (/Applications/SageMath-7.6.app/Contents/Resources/sage/src/build/cythonized/sage/groups/group.c:2696)()
179 NotImplementedError
180 """
--> 181 return self.order() != infinity
182
183 def is_multiplicative(self):
/Applications/SageMath-7.6.app/Contents/Resources/sage/src/sage/groups/group.pyx in sage.groups.group.Group.order (/Applications/SageMath-7.6.app/Contents/Resources/sage/src/build/cythonized/sage/groups/group.c:2623)()
164 NotImplementedError
165 """
--> 166 raise NotImplementedError
167
168 def is_finite(self):
NotImplementedError:
Is there any way to force the second piece of code to work with matrices with entries in CC?
edit retag close merge delete
In the first case we can also compute the order, and the list of elements. In the second case, we use non-exact objects. (Why do we do this!?) Then i expect humanly that there is no multiplicative order method implemented for such a numerical matrix, being really glad that there is no one! Why should we work with inexact entries in a finite matrix group?
Note that we run into a NotImplementedError: multiplicative order is only implemented for matrices over finite fields in both following cases...
g = matrix( RR, 2, 2, [ sqrt(3)/2.0, 0.5, 0.5, sqrt(3)/2.0 ] )
g.multiplicative_order() # --> NotImplementedError
# and
sage: g = matrix(K, 2,2, [a/2,1/2,-1/2,a/2] )
sage: g^12
[1 0]
[0 1]
sage: g.multiplicative_order() # --> NotImplementedError
( 2017-07-25 13:32:20 -0600 )edit
Use an exact ring, such as QQbar or AA or UniversalCyclotomicField instead of CC.
( 2017-07-27 01:13:06 -0600 )edit
Sort by » oldest newest most voted
The problem is that a matrix (in characteristic 0, at least) has finite order if and only if it's diagonalizable and has roots of unity for eigenvalues. The latter is certainly not preserved under numerical pertubations, so cannot be determined for matrices with floating point entries.
more
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Cody
# Problem 2521. Find out sum of prime number till given number
Solution 1964958
Submitted on 7 Oct 2019 by Aditya Kavdia
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
n=10; y_correct = 17; assert(isequal(sum_prime(n),y_correct))
2 Pass
n=1; y_correct = 0; assert(isequal(sum_prime(n),y_correct))
3 Pass
n=1000; y_correct = 76127; assert(isequal(sum_prime(n),y_correct))
4 Pass
n=185; y_correct = 3447; assert(isequal(sum_prime(n),y_correct))
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# RelativeResourceManager9 - π 2 upon survival to age...
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MATH 471: Actuarial Theory I Homework #9: Fall 2009 Assigned October 28, due November 4 1. Assume mortality follows de Moivre’s Law with ω = 110 and d = 0.05. Calculate: (a) ¨ a 45 . (14.36) (b) ¨ a 45: 15 . (9.73) (c) 15 | ¨ a 45 . (4.63) (d) ¨ a 45: 15 . (15.36) 2. Consider a special whole life annuity-due on (x) which pays R at the beginning of the first year, 2 R at the beginning of the second year, and 3 R at the beginning of each year thereafter. You are also given: (i) The APV of this annuity is 3333 (ii) i = 0.05 (iii) p x = 0.98 and p x +1 = 0.97 (iv) ¨ a x +2 = 31.105 Calculate R . (40) 3. Suppose mortality follows the Illustrative Life Table and i = 0.06. Calculate the net single premium, π , for a 35-year temporary life annuity-due issued to (30) with an annual payment of 1000 that pays an additional benefit of
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Unformatted text preview: π 2 upon survival to age 65. (15,641.99) 4. Suppose, at time 0, that: (i) 30 lives age 65 purchase a whole life annuity-immediate with an annual payment of 1 and (ii) 20 lives age 75 also purchase a whole life annuity-immediate with an annual payment of 1. Assume all lives are independent, mortality follows the Illustrative Life Table, and i = 0.06. Let S be the total present value of the payments for all 50 annuities. (a) Calculate E ( S ). (391.25) (b) Calculate var ( S ). (629.58) 5. Using the Illustrative Life Table, the assumption of UDD in each year of age, and i = 0.06, calculate: (a) ¨ a (2) 40 . (14.56) (b) 30 | ¨ a (2) 40 . (1.03)...
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# Algebra
posted by .
(8w+5)/(10w-7) = (4w-3)/(5w+7)
How would I solve this?
• Algebra -
Do you know how to cross multiply and simplify the equation to a quadratic?
• Algebra -
cross-multiply
(8w+5)(5w+7) = (4w-3)(10w-7)
40w^2 + 81w + 35 = 40w^2 -58w + 21
139w = -14
w = -14/139
## Similar Questions
1. ### algebra
4w+7-10w=-2(3w+2)
2. ### algebra I
(39w+ 14 + 10w squared) / 72+ w =]
3. ### algebra
Did I do this problem right? (2y+5w+6)(5y+5w-4) 2y(5y)+ 2y(5w) + 2y(-4) + 5w(5y) + 5w(5w)+ 5w(-4) + 6(5y) + 6(5w) + 6(-4) then I multiplied: 10y^2 + 10wy - 8y + 25wy + 25w^2 - 20w + 30y + 30w - 24 then I combined like terms: 10y^2
4. ### Geometry
I need to get the largest area of 100 feet of rope. I want to say rectangle: 40L x 40L x 10w x 10w =100 Please advise if I am on the right track. Keeping in mind that you can use any Plan Geometry Shape?
5. ### Pre algebra
Simplify 10w+6+3w
6. ### algebra
2/w+ 4/w-1 = 10/w(w-1) 2(w-1)/w(w-1) + 4w/w(w-1)=10/w(w-1) 2w-2/w(w-1 +/4w/w(w-1= 10(w-1)/w(w-1) 2w-2+4w=10w-10 2w+4w-10w=-10-2 6w-10w=-10-2 -4w=8 w=2
1-(10w+7)=
8. ### algebra
10w + 11 - 3w -8
9. ### algebra
Divide (-5w^10 + 10w^8+5w^6) / (5w^5) the answers are -w^5 + 5w^3 = w -5w^5 + 10w^8 + w -w^5 +10^8+ 5w^6 and -w^5 + 2w^3 + w
10. ### Math
5.Simplify x/6x-x^2 a)1/6-x ; where x=0.6 b) 1/6-x; where x=6*** c)1/6x; where x=0 d)1/6 6. 5x^3/7x^3+x^4 a)5/7+x; where x= 0,-7**** b)5/7+x; where x=-7 c)5/7x; where x=0 d)5/7 7.x-2/x^2+4x-12 a)1/6+x; where x=-6*** b)1/6+x; where …
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# Problem #1226
1226 For how many real values of $x$ is $\sqrt{120-\sqrt{x}}$ an integer? $\mathrm{(A) \ } 3\qquad \mathrm{(B) \ } 6\qquad \mathrm{(C) \ } 9\qquad \mathrm{(D) \ } 10\qquad \mathrm{(E) \ } 11$ This problem is copyrighted by the American Mathematics Competitions.
Note: you aren't logged in. If you log in, we'll keep a record of which problems you've solved.
• Reduce fractions to lowest terms and enter in the form 7/9.
• Numbers involving pi should be written as 7pi or 7pi/3 as appropriate.
• Square roots should be written as sqrt(3), 5sqrt(5), sqrt(3)/2, or 7sqrt(2)/3 as appropriate.
• Exponents should be entered in the form 10^10.
• If the problem is multiple choice, enter the appropriate (capital) letter.
• Enter points with parentheses, like so: (4,5)
• Complex numbers should be entered in rectangular form unless otherwise specified, like so: 3+4i. If there is no real component, enter only the imaginary component (i.e. 2i, NOT 0+2i).
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New Online Book! Handbook of Mathematical Functions (AMS55)
Conversion & Calculation Home >> Measurement Conversion
Measurement Converter
Convert From: (required) Click here to Convert To: (optional) Examples: 5 kilometers, 12 feet/sec^2, 1/5 gallon, 9.5 Joules, or 0 dF. Help, Frequently Asked Questions, Use Currencies in Conversions, Measurements & Currencies Recognized Examples: miles, meters/s^2, liters, kilowatt*hours, or dC.
Conversion Result: ```arpentlin = 58.471308 length (length) ``` Related Measurements: Try converting from "arpentlin" to archin (Russian archin), arpentcan, Biblical cubit, caliber (gun barrel caliber), city block (informal), cloth quarter, en (typography en), engineers chain, furlong (surveyors furlong), Greek fathom, Greek palm, Greek span, inch, ken (Japanese ken), light yr (light year), palm, parsec, Roman mile, soccer field, stadium (Roman stadium), or any combination of units which equate to "length" and represent depth, fl head, height, length, wavelength, or width. Sample Conversions: arpentlin = 1.65 actus (Roman actus), 32,228.28 agate (typography agate), .00132022 arpentcan, 1.6 bolt (of cloth), 92,080.8 bottom measure, 191.84 foot, .63945 football field, 78.94 gradus (Roman gradus), 126.35 Greek cubit, 252.69 Greek span, 58,471,308 micron, .01052399 nautical league, 2.13 naval shot, 767.34 palm, 166,366.99 point (typography point), 9.59 rope, 27.8 sazhen (Russian sazhen), .532875 skein, .63945 soccer field, .05481 verst (Russian verst).
Feedback, suggestions, or additional measurement definitions?
Please read our Help Page and FAQ Page then post a message or send e-mail. Thanks!
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# 1st Year Physics Chapter 8 Waves Notes pdf
Here we have shared the notes of chapter 8 11th class physics Waves. You can download 1st Year Physics Chapter 8 Waves Notes pdf or read online.
Q. What are progressive waves? Give its types.
Progressive Waves
The waves which transfer energy by moving away from the source of disturbance are called progressive or travelling waves
Example
Consider two persons holding the opposite ends of the rope Suddenly one person gives a jerk to the rope The disturbance in the rope produces a pulse which moves toward other person When this reaches the other person it pushes his hand upward So the energy and momentum transferred from one person towards the other person This is an example of progressive wave
Kinds of waves
There are two kinds of progressive waves
i) Transverse waves ii) Longitudinal waves
Transverse Waves
The waves in which particles of the medium are displaced perpendicular to the direction of propagation of wave and such waves are called Transverse waves.
Example
Now take a loose spring coil (slinky spring for illustration of motion of source in generating waves in a medium Slinky is the soft spring which has small initial length but relatively large extended figure Consider a horizontal spring system with its one end fixed. When the free end is moved from side to side, a pulse of wave having a displacement pattern as shown in figure, which will move along the spring This shows that displacement of particles is perpendicular to the direction of propagation of wave, hence transverse waves are produced
Longitudinal Waves (Compressional waves)
The waves, in which particles of the medium are displaced along the direction of propagation of.
wave and such waves are called longitudinal waves
Note (Why, sound waves in air are longitudinal in nature)
Both types of waves can be set up in solid In fluids, however, transverse waves die out very quickly and usually cannot be produced at all That is why sound waves in air are longitudinal in nature
Q. What are periodic waves? Also discuss its different types?
Period Waves
The waves which are produced by continuous and rhythmic disturbances in a medium are called periodic waves.These may be transverse or longitudinal in nature A good example of a periodic waves in an oscillating mass-spring system which executes SHM
Transverse Periodic Waves
The periodic waves in which the displacement of particles of medium is perpendicular to the direction of motion of waves produced by continuous and rhythmic disturbances in a medium are called transverse periodic waves
Experiment
Let us consider a mass-spring system which can vibrate horizontally as shown in figure A long sting of uniform thickness is stretched horizontally and its one end is attached with the oscillating mass m. Due to oscillation of mass-spring system a transverse waves is produced in the string
The wave appears to be travelling on the spring from its own end to the other In this case each part of string vibrates on the spring, from its own end to the other. In this case each part of string vibrates at right angle to the length of stretched string The crest and troughs are being replaced by one another periodically and waves appear to be travelling
Crest
The portion of the wave above the mean level is called as crest
Trough
The portion of the wave below is mean level is called trough
Amplitude
The amplitude is the maximum displacement of point in a crest or a trough of the string
Wave Length
The distance between two consecutive crests or two consecutive troughs is known as wave length.
Time Period
The time for which the wave travels a distance of wave length is called time period. The time period of wave is equal to be the time period of the oscillator which produces it
Speed of Wave
When a wave progresses each particle in the medium performs SHM The time that the crest required to move a distance of one wave length is equal to the time required for a point in the medium to go through one complete oscillation. If v be the speed of wave, then Or Where is the frequency of the wave which is same as the frequency of oscillator (crest or trough) which produces it
Wave Profile
Relation between path difference and phase difference
Consider the snapshot of the periodic waves moving through the medium As any distance from the reference point then phase difference can be described as Longitudinal periodic waves
The periodic waves in which the displacement of particles of medium is along the direction of motion of waves produced by continuous and rhythmic disturbances in a medium are called transverse periodic waves
Experiment
To explain longitudinal periodic waves, we take an example of a spring which is suspended by the help of threads Longitudinal wave is produced in this spring by applying horizontal varying force at one end of the spring This force produces compression and rarefaction in the spring as shown in the figure
In this case the various parts of the spring vibrate along the length of the spring (or along the direction of motion of the waves When spring was disturbed than all the suspension threads were vertical But when the longitudinal wave is produced in the thread then these suspension threads are displaced Their displacement is same as the displacement of corresponding parts of the spring The graph of displacement of various parts of spring and corresponding values of the distances of these various parts of spring from its one end is
Q. Explain the reflection of waves from rare and denser mediums?
Reflection of Waves
The bouncing back of wave from the boundary of a medium is called reflection of wave When a wave produced in one medium travel to the boundary to enter into another medium, then a part of incident wave is reflected from the boundary This reflected part has same frequency and wavelength as the incident wave has But there maybe the change of phase which depends upon the nature of boundary of medium Reflection of waves from the boundary of denser medium
Let us take long slinky spring whose one end is fixed to a rigid support on a horizontal surface of a table The other end of this spring is free to oscillate A sharp jerk is given to which crest or trough is produced on it It travels on the spring from it send A towards the end B On reaching at end B spring exerts a force on the rigid support to produce similar motion in it But the rigid body has large density so it exerts equal and opposite reaction on the spring Due to which crest is converted into through and it travels back from end B to A
Reflection of waves from the boundary of a rare medium: If we attach end B of long slinky spring with a light string and keep its end A free, like before Then giving a sudden jerk to end A, a crest is produced which travels on the spring from its end A, a crest is produced which travels one the spring from its end A to end B when the crest produced reaches the boundary of string Then string being rare medium, do not give reaction to the spring So that a crest is reflected back as a crest on the spring from end B to end A
Result
If a transverse wave travelling in a rarer medium is incident on a denser medium, it is reflected such that it undergoes a phase change of 180 (path difference of if a transverse wave travelling in a denser medium in incident on a rarer medium, it is reflected without any change in phase (no path difference)
Q. Differentiate between echo and reverberation?
Difference between echo and reverberation
1) Echo is a single reflection of a sound wave off a surface Reverberation is the sound or the pattern created by the superposition of such echoes
2) An echo can be heard only when the distance between the source of sound and the reflecting also
body is at least 17 m. A reverberation can occur when sound wave is reflected by a nearby wall also.
3) An echo is usually clear and can be clearly distinguished A reverb is not a clear replica of the original sound sample
4) Echo can be used to determine the distance of a reflecting object such as a large building or a mountain of the ambient temperature is known Reverberation cannot be utilized for distance measurement applications
5) An echo can be heard both in open and closed spaces Reverberation is usually experienced in closed spaces with multiple reflecting objects
Q. How can we change the frequency of string on a musical instrument? Also discuss resonance of air column in resonance tube?
The frequency of a string on a musical instrument can be changed either by varying the Tension in string Length of string
For example
The tension in guitar and violin strings is varied by tightening the pegs on the neck of the instrument Once the instrument is tuned, the musicians vary the frequency by moving their fingers along the neck By doing so that change the length of the vibrating portion of the string resonance of air column in resonance tube:
Vibration of air column can be set up in a resonance column apparatus It consists of a long metal tube held vertically in a tall jar containing the water The tube can be fixed in vertical position The length of the air column can be varied by raising or lowering the tube Here, the surface of water will act as the closed end When a vibrating tuning fork is held above the open end, longitudinal waves are sent down the air column.
These waves are reflected at the water surface and thus produce standing waves Nodes are produced at the water surface and antinodes are produced at the open end When the frequency of waves in the air column becomes equal to the natural frequency of tuning fork, a loud sound is produced in the air column, It is the condition for resonance it occurs only when the length of air column is proportional to one-fourth of the wavelength of sound waves having frequency equal to frequency of tuning fork
Q. Find the frequencies produced in organ pipe when it i) Open at both ends. ii) Closed at one end.
Stationary Waves in Air Column
Stationary waves can be set up in air column inside a pipe or tube A common example vibrating air column is an organ pipe
Organ Pipe
An organ pipe is a wind instrument in which sound is produced, due to setting up of stationary waves in air column. It consist of a hollow long tube with both end open or with one end open and the other closed There are two types of organ pipes
Open Organ pipe: It is that organ pipe whose both ends are open.
Closed Organ pipe: It is that organ pipe whose one end is closed Modes of vibrations in organ pipe open at both ends
Let us consider an organ pipe of length which is open at both ends As tat the open ends air molecules have complete freedom of motion so it acts as antinode Longitudinal waves set up inside the pipe have been represented by transverse curves which represent the displacement and amplitude vanations of air at vanous points
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# 30++ Trig Ratios Worksheet
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# Thread: First Order ODE using homogeneous substituion
1. ## First Order ODE using homogeneous substituion
someone please enlighten me on each step taken (as well as integrating both sides of the separable equation involving z and x) in solving the following equation:
(x^2)y'=7(y^2)-2xy
using the subsitution z=(y/x)
i know what has to be done, but im having troubles simplifying into a regular funtion y(x)...
this can be done using bernouilli's method but wanna get the same using the homogeneous substituion method
thanks
2. $y' = 7 \cdot \left( {\frac{y}
{x}} \right)^2 - 2 \cdot \left( {\frac{y}
{x}} \right).$
Now make $y=xz.$
3. yea i did that, but solve down to the part where you end up with a separable equation dz/7(z^2)-3z = dx/x.
i tried integrating both sides, and especially the left one needed more tactics where i used partial integration {dz/7(z^2)-3z} = 1/7[A/z + B/{z-(3/7))]
my problem here is integrating and getting a solution y(x) after back substituting z=(y/x). I did the same ODE using bernouillis and i got
y(x)=-7x^(-1) +Cx^(-2). Just want to see how to reach the same results using the homogenous substitution technique
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The aspect ratio of the volume elements in Finite Volume Method
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August 27, 2012, 12:07 The aspect ratio of the volume elements in Finite Volume Method #1 New Member Charlie Tan Join Date: Sep 2011 Posts: 19 Rep Power: 7 Hello people, I have a question about the Finite Volume Method. Is the aspect ratio of the volume elements in Finite Volume Method important? The element aspect ratios are important in Finite Element Method, but I don't know if it's the case in FVM. Thank you very much!
August 27, 2012, 16:07 #2 Senior Member Chris DeGroot Join Date: Nov 2011 Location: Canada Posts: 388 Rep Power: 8 Yes you should try to keep the aspect ratio reasonable. There are some exceptions, for example if you have a 2D grid and you extrude it in the third dimension to make it 3D the aspect ratio can typically be large as long as things aren't changing a lot in that direction. For general problems, try to avoid very large aspect ratios whenever possible.
August 27, 2012, 17:39 #3 Senior Member Join Date: Aug 2011 Posts: 251 Rep Power: 8 I have seen some authors who claimed that the ratio between two consecutive cells shouldn't be greater than 1.5. But it depends where and what for. It seems that you limit the diffusion of the solution when the ratio is close to 1. However to manage with safety outflow boundaries,it is advised to extend the domain with expanded cells when you come closer to that boundaries. For accuracy it seems better to keep as much as possible the ratio close to 1.
August 28, 2012, 02:34 #4 Super Moderator Oliver Gloth Join Date: Mar 2009 Location: Todtnau, Germany Posts: 116 Rep Power: 9 Hi, it depends on the flow situation, cell type, solver, and the weather ... For boundary layers (and most solvers), highly anisotropic cells do not pose a problem. On the contrary, you would not be able to fulfil the y+ <= 1 (or even 30-100) requirement without anisotropic cells. I have also seen crisp shock waves which have been resolved by anisotropic tetras -- something that most people would tell you to avoid. If, however, you try to resolve a free shear layer with anisotropic tetras, you are quite likely to get an "NaN" as a reply ... As a rule of thumb for most applications I would suggest to use anisotropic cells in the boundary layer only. Try to get the cells aligned as good as possible (i.e. use prisms or hexes). Regards, Oliver
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# Mathhelper algebra & calculus
This Mathhelper algebra & calculus supplies step-by-step instructions for solving all math troubles. Keep reading to learn more!
## The Best Mathhelper algebra & calculus
One instrument that can be used is Mathhelper algebra & calculus. There are a few steps that need to be followed in order to solve for in the equation. First, multiply both sides of the equation by . This will cancel out on the left side of the equation. Next, add to both sides of the equation. This will move everything to one side, and will cancel out on the right side of the equation. Finally, divide both sides of the equation by . This will solve for .
There are a few different math answer apps available, but my favorite is Mathway. It's simple to use and provides step-by-step explanations for every answer, so you can really understand how to get to the correct answer. Plus, it's free!
This will usually give you a new equation with one fewer variable, which you can then solve. You can then plug that value back into one of the original equations to solve for the other variable.
Many times, they will have helpful links to resources that can help with specific problems or topics. Khan Academy is a free resource that can be helpful for review or for learning new concepts.
Introduction In mathematics, an inequality is a statement that suggests that two things are not equal. Inequality equations are mathematical problems that involve finding the value of a variable that will make the two sides of the equation equal. In order to solve these equations, you must use algebraic methods to isolate the variable on one side of the equation. There are a few different types of inequality equations that you might encounter. The most common type is a linear inequality, which is an inequality that can be
To solve for an unknown exponent, one can use a process of elimination. First, one need to identify what the unknown exponent is within the equation. Once the unknown exponent is determined, one can then use a process of substitution and solving by inspection to arrive at the answer.
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Linear Optimization/Programming fhub Posting Freak Posts: 1,216 Threads: 75 Joined: Jun 2011 07-31-2013, 10:16 AM Hi, in the last few days I've made a new version of one of my old math programs, which I've already programmed for many calculators (Sharp PC-1500, HP-48, TI-92) and also as DOS-version. It is about Linear Optimization (also called Linear Programming in English), and it's a quite simple program (using the Simplex Method) but has one advantage over almost all other (free) LP programs that I saw on the internet: it calculates all solutions (if there are more than one), not only the first solution found! Since nowadays I doubt that anyone would still like to use an old DOS-program, I've now rewritten it as Windows program (in Borland Delphi), but nevertheless the GUI is rather simple: just functional and comfortable, but without any further gimmicks (GUI programming is indeed nothing that I like very much). So if anyone is interested and has any use for it - it's called LinOpt v5.0 and can be downloaded from my website: http://fhub.jimdo.com/ You find it at the bottom of the right column on my site, and it includes both, an English and a German version. I've not yet included any description (that's also something I don't really like), but anyone who knows what Linear Programming is, will certainly have no problems to use it. ;-) If you have any questions about it or any ideas how to improve (or extend) this program, then let me know ... Franz PS: If you want to try a LP problem with multiple solutions, here's a simple example: ```3 Variables, 2 Constraints, Maximize Constr.1: 2 1 3 <= 10 Constr.2: 2 3 7 <= 15 Objective: 2 1 4 ..... 1 solution 4 4 10 ..... 2 solutions 2 1 3 ..... 3 solutions 2 3 7 ..... 4 solutions And if you add one more constraint (no need to reenter everything else!) Constr.3 0 0 1 <= 2 with the last objective '2 3 7' then you even get 5 solutions! ``` Edited: 31 July 2013, 11:56 a.m. after one or more responses were posted Harald Senior Member Posts: 455 Threads: 39 Joined: Jan 2011 07-31-2013, 10:36 AM One of your other programs looked interesting. I downloaded it and was a bit surprised by the result: This is what Symantec Anti-Virus says: ```Scan type: Auto-Protect Scan Event: Security Risk Found! Risk: Trojan.ADH.2 File: C:\Dokumente und Einstellungen\pott01\Eigene Dateien\Downloads\WinSomofi.exe Location: Unknown Storage Computer: DE110006 User: KOSTALDE\pott01 Action taken: Cleaned by Deletion Date found: Mittwoch, 31. Juli 2013 16:35:41 ``` Is my employers IT just a bit paranoid or is there a real problem? I'll try this at home as well. Harald Edited: 31 July 2013, 10:36 a.m. fhub Posting Freak Posts: 1,216 Threads: 75 Joined: Jun 2011 07-31-2013, 10:40 AM Quote: Is my employers IT just a bit paranoid or is there a real problem? I'll try this at home as well. No, there's no problem at all, the reason is just that I've used an EXE-packer and some AV programs principially think such packers are suspect. BTW, you've indeed found my most important program (WinSomofi) at all, on which I've worked even more than 10 years for the DOS-version (doing in fact EVERYTHING myself, including all calculations and even drawing the worldmap point for point). ;-) The Windows version (WinSomofi) then was my next challenge, because this was even my first Windows program at all, and I had to learn lots of stuff to port the Turbo-Pascal version to Delphi. Franz Edited: 31 July 2013, 10:47 a.m. Tim Wessman Posting Freak Posts: 1,278 Threads: 44 Joined: Jul 2007 07-31-2013, 11:31 AM The HP email scanner always strips away the little string tool we use to make language strings for the calculator. Claims it has a virus that will destroy the world. They've recently upgraded the scanner so that it totally ignores extension and actually opens and finds it inside every compression archive I've tried. Very annoying. :-( TW DavidM Junior Member Posts: 11 Threads: 0 Joined: Nov 2012 07-31-2013, 01:40 PM Glad to see that I'm not the only one still using Delphi... now I know that there are at least two of us left! :-) fhub Posting Freak Posts: 1,216 Threads: 75 Joined: Jun 2011 07-31-2013, 03:35 PM Quote: Glad to see that I'm not the only one still using Delphi... now I know that there are at least two of us left! :-) Well, I guess there are many more. ;-) In my opinion Delphi is a great IDE (and Pascal a very comfortable and powerful programming languáge), so I really never saw any reason to switch to e.g. C++ just because it is 'modern'. But I've tried also a few other languages and can write no too complicated programs with them, but Delphi/Pascal is still my favorite. Franz Harald Senior Member Posts: 455 Threads: 39 Joined: Jan 2011 07-31-2013, 04:19 PM Luckily ours is a bit more crude. Simply changing the extension to something it doesn't recognize helps ;) Harald Senior Member Posts: 455 Threads: 39 Joined: Jan 2011 07-31-2013, 04:20 PM It is a very interesting program indeed. I am sure it's quite a challenge to write something like that. Thanks for sharing! Namir Posting Freak Posts: 2,247 Threads: 200 Joined: Jun 2005 07-31-2013, 09:54 PM I remember beta testing the first version of Delphi!! I am sure it evolved to become more powerful since it did not have to adhere to many programming language standards. What version of Delphi do you currently use? Namir Edited: 31 July 2013, 9:54 p.m. Marcus von Cube, Germany Posting Freak Posts: 3,283 Threads: 104 Joined: Jul 2005 08-01-2013, 02:45 AM Tim, you're not alone! My G-Data Antivirus package used to kick in during the build process of WP 34S, deleting one of the helper programs just created by Win-GCC. I had to disable one of its scanner engines to be able to build WP 34S again.. fhub Posting Freak Posts: 1,216 Threads: 75 Joined: Jun 2011 08-01-2013, 03:52 AM Quote: What version of Delphi do you currently use? I'm still using Borland Delphi 7. I also have Delphi 2005 (which was also free at some time), but this package includes just too much garbage that I would never need. Franz DavidM Junior Member Posts: 11 Threads: 0 Joined: Nov 2012 08-01-2013, 08:38 AM Delphi 5 and XE2 for most of my code. The latter mostly due to needing to work on some 64-bit utilities. Delphi 1-2 is where I started, though. If you haven't followed Delphi's progression through the years, you may be surprised how many programming language standards have been incorporated into the codebase now. It's a very mature and capable platform. Bruce Larrabee Member Posts: 80 Threads: 14 Joined: Sep 2010 08-02-2013, 03:47 AM I just want to collaborate that this is typical security software behavior. Actually it often seems to me that this sort of thing is laziness and / or political... Not so much in this case but still, I mean the security program could easily look in there and actually check for nefarious contents, but that would require real effort. So much more user friendly to take a lazy abstract position and the summarily delete programs. I mean I am always thrilled to find valuable legit tools deleted from my machine. I find that I can never rely on any security products that are this lazy. Much more aggressive designs are necessary these days. Okay I'll climb down of the box now. I've have been using Borland C++ Builder (has a new name now of course) since version 1.0. Also use Delphi for some coding, usually as object files compiled into a mainly C and or C++ program. Chris Smith Member Posts: 146 Threads: 12 Joined: Feb 2012 08-03-2013, 05:28 PM Make that three :) Well Lazarus/Free Pascal. fhub Posting Freak Posts: 1,216 Threads: 75 Joined: Jun 2011 08-04-2013, 06:27 AM Updated LinOpt v.5.0: added a short program description http://fhub.jimdo.com/ Franz « Next Oldest | Next Newest »
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USING OUR SERVICES YOU AGREE TO OUR USE OF COOKIES
# What is the Prime Factorization Of 2?
• Prime factors of 2: 1 * 2
## Is 2 A Prime Number?
• Yes the number 2 is a prime number.
• It's a prime because two has no positive divisors other than 1 and itself.
## How To Calculate Prime Number Factorization
• How do you calculate natural number factors? To get the number that you are factoring just multiply whatever number in the set of whole numbers with another in the same set. For example 7 has two factors 1 and 7. Number 6 has four factors 1, 2, 3 and 6 itself.
• It is simple to factor numbers in a natural numbers set. Because all numbers have a minimum of two factors(one and itself). For finding other factors you will start to divide the number starting from 2 and keep on going with dividers increasing until reaching the number that was divided by 2 in the beginning. All numbers without remainders are factors including the divider itself.
• Let's create an example for factorization with the number nine. It's not dividable by 2 evenly that's why we skip it(Remembe 4,5 so you know when to stop later). Nine can be divided by 3, now add 3 to your factors. Work your way up until you arrive to 5 (9 divided by 2, rounded up). In the end you have 1, 3 and 9 as a list of factors.
## Mathematical Information About Numbers 2
• About Number 2. Two is the smallest and the only even prime number. Also it's the only prime which is followed by another prime number three. All even numbers are divisible by 2. Two is the third number of the Fibonacci sequence. Gottfried Wilhelm Leibniz discovered the dual system (binary or binary system) that uses only two digits to represent numbers. It witnessed the development of digital technology for a proliferation. Because of this, it is the best known and most important number system in addition to the commonly used decimal system.
## What is a prime number?
Prime numbers or primes are natural numbers greater than 1 that are only divisible by 1 and with itself. The number of primes is infinite. Natural numbers bigger than 1 that are not prime numbers are called composite numbers.
## What is Prime Number Factorization?
• In mathematics, factorization (also factorisation in some forms of British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 x 5, and the polynomial x2 - 4 factors as (x - 2)(x + 2). In all cases, a product of simpler objects is obtained. The aim of factoring is usually to reduce something to basic building blocks, such as numbers to prime numbers, or polynomials to irreducible polynomials.
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# What is the seasonal index?
A seasonal index is a measure of how a particular season through some cycle compares with the average season of that cycle. By deseasonalizing data, we’re removing seasonal fluctuations, or patterns in the data, to predict or approximate future data values. Seasonal indices.
Consequently, what is seasonal index in time series?
Measuring seasonality. Seasonal variation is measured in terms of an index, called a seasonal index. It is an average that can be used to compare an actual observation relative to what it would be if there were no seasonal variation. An index value is attached to each period of the time series within a year.
Furthermore, what would a seasonal index of 1.25 mean? This means that on average, Q1 is 125% of the average quarter. This is an “above average” quarter. A seasonal index below 1 means that it is a “below average” quarter; .
Subsequently, one may also ask, how do you find seasonal index?
The seasonal index of each value is calculated by dividing the period amount by the average of all periods. This creates a relationship between the period amount and the average that reflects how much a period is higher or lower than the average. =Period Amount / Average Amount or, for example, =B2/\$B\$15.
How do you do seasonal index in Excel?
This will calculate the average monthly sales for the year. Enter the following formula into cell C2: “=B2 / B\$15” omitting the quotation marks. This will divide the actual sales value by the average sales value, giving a seasonal index value. Select cell C2.
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CFD Online Discussion Forums (http://www.cfd-online.com/Forums/)
- CFX (http://www.cfd-online.com/Forums/cfx/)
- - Periodic Pipe Flow LES (http://www.cfd-online.com/Forums/cfx/94603-periodic-pipe-flow-les.html)
dvolkind November 21, 2011 02:20
Periodic Pipe Flow LES
5 Attachment(s)
Dear all!
I need some advice on modeling a fully developed pipe flow with periodic boundary conditions using LES.
My goal is to get a realistic transient inlet BC for my problem. That's how I'm trying to achieve this:
1. Consider a circular pipe (5*D long) with periodic BCs (mass flow rate).
2. Obtain a steady state solution with RANS (I used SST).
3. Run LES using the RANS solution as initial conditions.
4. Import transient boundary profile as an inlet BC for the LES of my actual problem.
I'm using a hexa-mesh with Y+ at wall around 1, growth ratio around 1,1. Re = 8400, Courant number < 1.
The major problem is that I can't get a converged (judged by residuals) solution with periodic BCs both in transient and steady state. When I run the same model with mass flow inlet and pressure outlet it does converge, but the velocity profile looks unphysical. The other problem is that the flow pattern doesn't become turbulent, even if I add significant velocity fluctuations for the initial velocity field, they tend to damping. So, I would like to ask the following:
1. What are the possible reasons of convergence problems?
2. Probably different convergence criteria should be used with periodic BCs?
3. What kind of grid is better for LES? As far as I know it should be as uniform as possible and have aspect ratios around 1. But what type of mesh is more preferable - tetra or hexa? And why? CFX Reference Guide says it should be isotropic, so tetra is better (4.1.11.4.2. Meshing). I've also seen a post by Mr. Horrocks, where he recommended to use hexa. When I use a tetrahedral mesh with approximately the same sizes I get the same results.
4. What is the reason of the turbulence damping?
Attachment 10158
Residuals plot for transient:
Attachment 10159
Velocity profile with mass flow inlet and pressure outlet:
Attachment 10160
Velocity profile with periodic BCs with specified mass flow rate:
Attachment 10161
Mesh:
Attachment 10162
Thanks to everyone in advance! Any help will be greatly appreciated.
ghorrocks November 21, 2011 06:02
Quote:
What are the possible reasons of convergence problems?
This FAQ is not exactly on your topic but is related and should give you some tips. http://www.cfd-online.com/Wiki/Ansys...gence_criteria
Quote:
Probably different convergence criteria should be used with periodic BCs?
Your approach sounds good. I do not think your problem is with the periodic BCs.
Quote:
What kind of grid is better for LES?
A high quality hex grid is superior, but if the geometry is difficult and you cannot do a hex grid (or only a low quality one) then a tet grid is superior. If you have a cylinder then you should be able to do a good hex grid.
High quality grids have less numerical dissipation, converge easier, use less memory (for hex grids) and can handle aspect ratio changes better.
Your comment about not getting turbulent structures confirms you have too much dissipation, so this is a problem for you. You will need central differencing and second order time differencing.
dvolkind November 23, 2011 13:34
Hello, Glenn!
Thanks a lot for your answers! I'm now trying to get the steady state problem converged. To do this I started with agressive physical time scale, then I switched to local timescale factor, and it does converge that way (incredibly slowly though). The text on the link you gave me says not to run with local timescale all the way to convergence. So I switch back to physical time scale, and all important residuals and imbalances (characterizing flow-aligned coordinate) begin to oscillate. And, if you don't mind, I would like to ask some more questions:
1. Is it necessary to get the final convergence without local time scale factor and why?
2. If it is, how to determine how many iterations are sufficient?
3. Will it be possible to reduce the amplitude of residuals/imbalances oscillations on the final iterations with physical timescale if I achive tighter convergence with local time scale factor? (I surely can try it myself, but it takes really long with my available hardware)
4. Probably I still go wrong somewhere? (convergence seems too tough for such a primitve steady-state problem)
5. Concerning LES: how else could I avoid dissipation you mentioned if I was already using central differencing / Euler second order backward and a "structured" hexa mesh?
Thank you again! Sorry for asking too much.
ghorrocks November 23, 2011 19:51
1) Yes, there has been some posts on this on the forum, search for them.
2) Not sure. I would suggest until things settle out. To be completely sure do a sensitivity analysis.
3) Possibly. But if the oscillations are physical then it will not make a difference.
4) You will need quite a large physical time step for this to work. Also be careful about making your mesh too fine for the RANS model.
5) Then you have done the main things. There are also some other options to consider regarding the detailed numerical approach such as Rhie-Chow and interpolation schemes (and others).
dvolkind November 24, 2011 00:03
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Differentiable manifold
1. Aug 26, 2011
LearningDG
Hi,
I just started learning differential geometry. Got some questions. Thanks in advance to anyone who can help!
Consider the one-dimensional manifold represented by the line y = x for x<0 and y = 2x for x>= 0. Now if I consider the altas with two charts p(x, y)=x for x<-1 and q(x,y)=y for x>-2. The transition map is C-infinity in (-2,-1).
Next I consider another altas with r(x,y)=x for x<1 and s(x,y)=y for x>-1. The transition is not differentiable at 0 inside (-1,1).
Can I say I've defined two differential structures of the same manifold?
But what does it mean by a one-deminsional manifold having only one differential structure up to diffeomorphism?
Thanks!
2. Aug 26, 2011
micromass
No, since the transition map for the second atlas is not differentiable. So it will not be a smooth atlas.
It involves a notion of maximal atlas. It's certainly true that atlases are not unique. For example, if we define p(x,y)=x for x<-2 in the first atlas, then we would have obtained a different atlas. But the differential structure doesn't change: we will have the same differentiable maps.
To solve this, we see that every smooth atlas generates a maximal smooth atlas. We say that two manifolds are the same if their maximal smooth atlas is the same. And it turns out that one-dimensional manifolds have only one maximal smooth atlas.
3. Aug 26, 2011
LearningDG
Thanks Micromass! But I still got some questions.
That's exactly what I mean by two structures, viz., not smooth in one altas but smooth in the other. Or, should the transition map at least be C1? Then may be put this way, change my representation to a function which is smooth everywhere but only C1 at x = 0. Using the same two altas. The first altas A1 is C-inifinity. According to my understand, the maximal atlas consists of all other altas compatible A1. I can keep it C-infinity provided I don't add charts overlapping at x = 0. For my second altas A2, I can also extend it to a maximal C-1 atlas.
Are these two called different differential structures of the same manifold?
4. Aug 26, 2011
micromass
Well, in order to have a differentiable manifold, your transition maps should at least be C1 (some authors even demand smoothness).
What you're talking about are topological manifolds. There the transition maps don't need to be C1. And I'm afraid that your two atlasses determine thesame topological manifold.
5. Aug 26, 2011
LearningDG
Thanks micromass!
Yes, they determine the same topological manifold. What I am not sure is if I have constructed two maximal altas of it which are not compatible (say, one is C1 and the other is smooth), can I say these are two differential structures? If yes, I guess mathematicians have proved that all the C-p altas (p>0) are diffeomorphic (meaning C-infinity diffeomorphic?). When we say two manifolds are diffeomorphic, we don't require each altas to be C-infinity themselves, right?
6. Aug 26, 2011
micromass
This depends on the definitions we use. Some authors do require each atlas to be C-infinity. For others, C1 is good enough.
So for some, diffeomorphic is between C-infinity atlases, for others, it will be between C1-atlases. Check the definitions in the beginning of each book to see what the author is working with...
7. Aug 29, 2011
lavinia
Two atlases may not be compatible yet the two differentiable manifolds may be diffeomorphic.
The classic example is the atlases with single charts, x, and x^3,on the real line. These atlases are not compatible yet the two manifolds are diffeomorphic. The map x -> x^1/3 form the x atlas manifold to the x^3 atlas manifold is a diffeomorphism - I think.
The definition of differentiable manifold only requires that the transition functions be differentiable - maybe continuously - but there are more special differentiable manifolds whose charts are required to be C^r for r>1 and also smooth manifolds where the transition functions are infinitely differentiable. There are also analytic manifolds where the transition functions are analytic.
There are theorems on when C^r manifolds have diffeomorphic smooth structures but I am not educated in them. Further there are topological manifolds that have no compatible differentiable sub-atlas. Also on some topological manifolds there are smooth structures that are not diffeomorphic.
there is also the idea of a combinatorial structure on a topological manifold. on a smooth manifold a combinatorial structure is called smooth if it derives from a smooth triangulation. There are examples of triangulations of topological manifolds that do not come from any smooth triangulation.
Last edited: Aug 29, 2011
8. Aug 30, 2011
LearningDG
Thanks a lot Lavinia.
So correct me if I am wrong:
For the same manifold, if I have two different atlas, say one is Cn and another is Cm for general m,n, which are not compatible, then I can say these are two different differential structures.
Then the fact that for a manifold with dimension<4, there is only one differential structure means one can always construct a diffeomorphism between the two atlas so that the map from one the other is infinitely differentiable.
9. Aug 31, 2011
lavinia
I believe that two atlases may be incompatible yet the manifolds may be diffeomorphic.
The example I gave on the real line illustrates this. If a manifold has a C^r structure then sometimes it will have a C^r+K substructure. i think if r is large enough then the manifold will have a smooth substructure but don't quote me on this.
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# Distance between Jiujiang (JIU) and Liuzhou (LZH)
Flight distance from Jiujiang to Liuzhou (Jiujiang Lushan Airport – Liuzhou Bailian Airport) is 557 miles / 896 kilometers / 484 nautical miles. Estimated flight time is 1 hour 33 minutes.
Driving distance from Jiujiang (JIU) to Liuzhou (LZH) is 668 miles / 1075 kilometers and travel time by car is about 11 hours 28 minutes.
## Map of flight path and driving directions from Jiujiang to Liuzhou.
Shortest flight path between Jiujiang Lushan Airport (JIU) and Liuzhou Bailian Airport (LZH).
## How far is Liuzhou from Jiujiang?
There are several ways to calculate distances between Jiujiang and Liuzhou. Here are two common methods:
Vincenty's formula (applied above)
• 556.631 miles
• 895.810 kilometers
• 483.699 nautical miles
Vincenty's formula calculates the distance between latitude/longitude points on the earth’s surface, using an ellipsoidal model of the earth.
Haversine formula
• 557.016 miles
• 896.430 kilometers
• 484.033 nautical miles
The haversine formula calculates the distance between latitude/longitude points assuming a spherical earth (great-circle distance – the shortest distance between two points).
## Airport information
A Jiujiang Lushan Airport
City: Jiujiang
Country: China
IATA Code: JIU
ICAO Code: ZSJJ
Coordinates: 29°43′58″N, 115°58′58″E
B Liuzhou Bailian Airport
City: Liuzhou
Country: China
IATA Code: LZH
ICAO Code: ZGZH
Coordinates: 24°12′27″N, 109°23′27″E
## Time difference and current local times
There is no time difference between Jiujiang and Liuzhou.
CST
CST
## Carbon dioxide emissions
Estimated CO2 emissions per passenger is 107 kg (236 pounds).
## Frequent Flyer Miles Calculator
Jiujiang (JIU) → Liuzhou (LZH).
Distance:
557
Elite level bonus:
0
Booking class bonus:
0
### In total
Total frequent flyer miles:
557
Round trip?
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These Thurston maps are NET maps for every choice of translation term. They have degree 28. They are imprimitive, each factoring as a NET map with degree 7 followed by a Euclidean NET map with degree 4. PURE MODULAR GROUP HURWITZ EQUIVALENCE CLASSES FOR TRANSLATIONS {0} {lambda1,lambda2,lambda1+lambda2} These pure modular group Hurwitz classes each contain only finitely many Thurston equivalence classes. However, this modular group Hurwitz class contains infinitely many Thurston equivalence classes. The number of pure modular group Hurwitz classes in this modular group Hurwitz class is 10. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 2/14, 2/2, 4/2, 6/2 Every NET map in these pure modular group Hurwitz classes is rational because every loop multiplier in the mod 2 slope correspondence graph is less than 1. EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-77.950760,77.950760) SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS: 1 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 0/1 2 14 No No No No NUMBER OF EQUATORS: 0 0 0 0 There are no more slope function fixed points. Number of excluded intervals computed by the fixed point finder: 1807 There are no equators because both elementary divisors are greater than 1. NONTRIVIAL CYCLES 1/1 -> -1/1 -> 1/1 The slope function maps every slope to a slope: no slope maps to the nonslope. The slope function orbit of every slope p/q with |p| <= 50 and |q| <= 50 ends in one of the above cycles. If the slope function maps slope p/q to slope p'/q', then |p'| <= |p| for every slope p/q with |p| <= 50 and |q| <= 50. FUNDAMENTAL GROUP WREATH RECURSIONS When the translation term of the affine map is 0: NewSphereMachine( "a=(2,26)(3,27)(4,24)(5,25)(6,22)(7,23)(8,20)(9,21)(10,18)(11,19)(12,16)(13,17)", "b=(1,27)(2,28)(3,25)(4,26)(5,23)(6,24)(7,21)(8,22)(9,19)(10,20)(11,17)(12,18)(13,15)(14,16)", "c=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)", "d=(1,28)(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,3)(2,28)(4,26)(5,27)(6,24)(7,25)(8,22)(9,23)(10,20)(11,21)(12,18)(13,19)(14,16)(15,17)", "b=(3,27)(4,28)(5,25)(6,26)(7,23)(8,24)(9,21)(10,22)(11,19)(12,20)(13,17)(14,18)", "c=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)", "d=(1,28)(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=<1,a*b,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c*d,1>(1,4)(2,27)(3,6)(5,8)(7,10)(9,12)(11,14)(13,16)(15,18)(17,20)(19,22)(21,24)(23,26)(25,28)", "b=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)", "c=(1,27)(2,28)(3,25)(4,26)(5,23)(6,24)(7,21)(8,22)(9,19)(10,20)(11,17)(12,18)(13,15)(14,16)", "d=(1,25)(3,23)(4,28)(5,21)(6,26)(7,19)(8,24)(9,17)(10,22)(11,15)(12,20)(14,18)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=<1,a*b,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c*d,1>(1,4)(2,27)(3,6)(5,8)(7,10)(9,12)(11,14)(13,16)(15,18)(17,20)(19,22)(21,24)(23,26)(25,28)", "b=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)", "c=(3,27)(4,28)(5,25)(6,26)(7,23)(8,24)(9,21)(10,22)(11,19)(12,20)(13,17)(14,18)", "d=(1,27)(2,4)(3,25)(5,23)(6,28)(7,21)(8,26)(9,19)(10,24)(11,17)(12,22)(13,15)(14,20)(16,18)", "a*b*c*d");
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# Graphing a Translation of a Rational Function
Instructor: Laura Pennington
Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.
We are going to take a look at rational functions and their graphs. Specifically, we will study the translations of rational functions and two different ways to graph these translations by getting certain information from the function itself.
## Translations of Rational Functions
Suppose that your boss just walked into your office at work to give you the model y = 3/x that fits some data that she is analyzing. She asks you to graph it for her. Mathematically speaking, the function is an example of a rational function. A rational function is a function with the variable in the denominator. They are sometimes called butterfly functions because graphs of simple rational functions have two parts that look somewhat like butterfly wings.
You enter the function your boss gave you into your computer system and print out the graph.
Notice the graph approaches, but does not touch, the x and y axes (the lines x = 0 and y = 0). When a graph approaches a line in this way, we call the line an asymptote of the function.
You print off the graph and realize that in doing so, you've used up the last of the printer paper. As soon as the graph is printed, your boss comes running in saying she messed up, and that the model should have been y = 3/(x - 2) + 4. All the printer paper is used up, so you can't use your computer system to print a new graph! How are you going to produce a graph of the correct model?
As it turns out, the correct model is actually what is called a translation of the rational function y = 3/x, where a translation is a sliding of a graph along a straight line. Luckily, we have a nice trick for graphing these when we know the graph of the original function! Let's take a look at how to determine the graph of the correct model from the graph of the old model.
## Graphing Translations from Known Graphs
In general, this rational function takes the form:
y = a/(x - h) + k
This is a translation of the function y = a/x, and h and k give us all of the information we need to perform the translation and graph y = a/(x - h) + k. You see the graph of y = a/(x - h) + k is the graph of y = a/x translated h units horizontally and k units vertically.
In the case of your work example, y = 3/(x - 2) + 4, h = 2 and k = 4. Therefore, the correct model is the graph of y = 3/x translated 2 units to the right and 4 units up. Great! All we have to do is take the graph of y = 3/x that we already have and shift it 2 units right and 4 units up.
Problem solved! You deserve a raise!
This is a great way to graph translations of rational functions when we have the original graph. But, what if we don't have the original graph? Let's look at how we can graph y = a/(x - h) + k if we don't have the graph of y = a/x.
## Graphing Translations without the Original Graph
When we have a rational function of the form y = a/(x - h) + k, h and k not only give us the horizontal and vertical translations of y = a/x, but they also tell us the asymptotes of the function y = a/(x - h) + k. You see, x = h and y = k are the vertical and horizontal asymptotes, respectively, of y = a/(x - h) + k. Therefore, we can use the following steps to graph y = a/(x - h) + k when we don't have the graph of y = a/x.
1. Draw in the lines x = h and y = k. These are your asymptotes.
2. Use the function to find points to the left and right of x = h, and plot them.
3. Connect the points in the shape of a rational function. The two wings should approach the asymptotes, but never touch them.
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359_Gra2e_CH02_SM.qxp
8/26/10
12:12 PM
Page S-11
chapter:
2
Atlantis is a small, isolated island in the South Atlantic. The inhabitants grow potatoes and catch fish. The accompanying table shows the maximum annual output combinations of potatoes and fish that can be produced. Obviously, given their limited resources and available technology, as they use more of their resources for potato production, there are fewer resources available for catching fish. Maximum annual output options
Quantity of potatoes (pounds)
Quantity of fish (pounds)
A
1,000
0
B
800
300
C
600
500
D
400
600
E
200
650
F
0
675
a. Draw a production possibility frontier with potatoes on the horizontal axis and fish on the vertical axis illustrating these options, showing points A–F. b. Can Atlantis produce 500 pounds of fish and 800 pounds of potatoes? Explain. Where would this point lie relative to the production possibility frontier? c. What is the opportunity cost of increasing the annual output of potatoes from 600 to 800 pounds? d. What is the opportunity cost of increasing the annual output of potatoes from 200 to 400 pounds? e. Can you explain why the answers to parts c and d are not the same? What does this imply about the slope of the production possibility frontier?
1. Solution
a. The accompanying diagram shows the production possibility frontier for Atlantis. Quantity of fish (pounds)
675 650 600
F
E
D C
500
G B
300
Atlantis PPF
A 0
200
400
600
800
1,000
Quantity of potatoes (pounds)
b. No, Atlantis cannot produce 500 pounds of fish and 800 pounds of potatoes. If it produces 500 pounds of fish, the most potatoes it can produce is 600 pounds.
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This point would lie outside the production possibility frontier, at point G on the diagram. c. The opportunity cost of increasing output from 600 to 800 pounds of potatoes is 200 pounds of fish. If Atlantis increases output from 600 to 800 pounds of potatoes, it has to cut fish production from 500 pounds to 300 pounds, that is, by 200 pounds. d. The opportunity cost of increasing output from 200 to 400 pounds of potatoes is 50 pounds of fish. If Atlantis increases output from 200 to 400 pounds of potatoes, it has to cut fish production from 650 pounds to 600 pounds, that is, by 50 pounds. e. The answers to parts c and d imply that the more potatoes Atlantis produces, the higher the opportunity cost becomes. For instance, as you grow more and more potatoes, you have to use less and less suitable land to do so. As a result, you have to divert increasingly more resources away from fishing as you grow more potatoes, meaning that you can produce increasingly less fish. This implies, of course, that the production possibility frontier becomes steeper the farther you move along it to the right; that is, the production possibility frontier is bowed out. (Mathematicians call this shape concave.)
2.
In the ancient country of Roma, only two goods, spaghetti and meatballs, are produced. There are two tribes in Roma, the Tivoli and the Frivoli. By themselves, the Tivoli each month can produce either 30 pounds of spaghetti and no meatballs, or 50 pounds of meatballs and no spaghetti, or any combination in between. The Frivoli, by themselves, each month can produce 40 pounds of spaghetti and no meatballs, or 30 pounds of meatballs and no spaghetti, or any combination in between. a. Assume that all production possibility frontiers are straight lines. Draw one diagram showing the monthly production possibility frontier for the Tivoli and another showing the monthly production possibility frontier for the Frivoli. Show how you calculated them. b. Which tribe has the comparative advantage in spaghetti production? In meatball production? In A.D. 100 the Frivoli discover a new technique for making meatballs that doubles the quantity of meatballs they can produce each month. c. Draw the new monthly production possibility frontier for the Frivoli. d. After the innovation, which tribe now has an absolute advantage in producing meatballs? In producing spaghetti? Which has the comparative advantage in meatball production? In spaghetti production?
2. Solution
a. The accompanying diagram shows the production possibility frontier for the Tivoli in panel (a) and for the Frivoli as the line labeled “Original Frivoli PPF� in panel (b). (a) Production possibility frontier for the Tivoli
(b) Production possibility frontier for the Frivoli
Quantity of spaghetti (pounds)
Quantity of spaghetti (pounds)
50
50
40
40
30
30
20
20
10
10
0
10 20 30 40 50 60 Quantity of meatballs (pounds)
0
Original Frivoli PPF
New Frivoli PPF
10 20 30 40 50 60 Quantity of meatballs (pounds)
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The production possibility frontier for the Tivoli was calculated as follows: the Tivoli can produce either 30 pounds of spaghetti and no meatballs, or they can produce no spaghetti but 50 pounds of meatballs. That is, the opportunity cost of 1 pound of meatballs is 3⁄5 of a pound of spaghetti: in order to produce 1 more pound of meatballs, the Tivoli have to give up 3⁄5 of a pound of spaghetti. This means that the slope of their production possibility frontier is −3⁄5. A similar argument for the Frivoli shows that their production possibility frontier has a slope of −4 ⁄ 3. b. For the Tivoli, the opportunity cost of 1 pound of meatballs is 3⁄5 of a pound of spaghetti. For the Frivoli, the opportunity cost of 1 pound of meatballs is 4 ⁄ 3 pounds of spaghetti. That is, the Tivoli have a comparative advantage in meatball production because their opportunity cost is lower. For the Tivoli, the opportunity cost of 1 pound of spaghetti is 5⁄3 pounds of meatballs. For the Frivoli, the opportunity cost of 1 pound of spaghetti is 3 ⁄4 pound of meatballs. That is, the Frivoli have a comparative advantage in spaghetti production because their opportunity cost is lower. c. The Frivoli’s new production possibility frontier is the line labeled “New Frivoli PPF” in panel (b) of the diagram. Instead of producing 30 pounds of meatballs (if they produce no spaghetti), they can now produce 60 pounds. d. Now the Frivoli have the absolute advantage in both meatball production and spaghetti production. The Frivoli’s opportunity cost of meatballs has now fallen to 4 ⁄6 = 2 ⁄3; that is, for each pound of meatballs that the Frivoli now produce, they have to give up producing 2 ⁄3 of a pound of spaghetti. Since the Frivoli’s opportunity cost of meatballs (2 ⁄3) is still higher than the Tivoli’s (3 ⁄5), the Tivoli still have the comparative advantage in meatball production. The Frivoli’s opportunity cost of spaghetti is 3 ⁄2 pounds of meatballs and the Tivoli’s is 5⁄3 pounds of meatballs, so the Frivoli have the comparative advantage in spaghetti production.
3.
According to the U.S. Census Bureau, in July 2006 the United States exported aircraft worth \$1 billion to China and imported aircraft worth only \$19,000 from China. During the same month, however, the United States imported \$83 million worth of men’s trousers, slacks, and jeans from China but exported only \$8,000 worth of trousers, slacks, and jeans to China. Using what you have learned about how trade is determined by comparative advantage, answer the following questions. a. Which country has the comparative advantage in aircraft production? In production of trousers, slacks, and jeans? b. Can you determine which country has the absolute advantage in aircraft production? In production of trousers, slacks, and jeans?
3. Solution
a. Since countries gain from specializing in production of the goods and services in which they have a comparative advantage, the United States must have the comparative advantage in aircraft production, and China must have the comparative advantage in production of trousers, slacks, and jeans. b. Since trade has nothing to do with absolute advantage, we cannot determine from this data which country has an absolute advantage in either of these goods.
4.
Peter Pundit, an economics reporter, states that the European Union (EU) is increasing its productivity very rapidly in all industries. He claims that this productivity advance is so rapid that output from the EU in these industries will soon exceed that of the United States and, as a result, the United States will no longer benefit from trade with the EU. a. Do you think Peter Pundit is correct or not? If not, what do you think is the source of his mistake?
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b. If the EU and the United States continue to trade, what do you think will characterize the goods that the EU exports to the United States and the goods that the United States exports to the EU?
4. Solution
a. Peter Pundit is not correct. He confuses absolute and comparative advantage. Even if the EU had an absolute advantage over the United States in every product it produced, the United States would still have a comparative advantage in some products. And the United States should continue to produce those products: trade will make both the EU and the United States better off. b. You should expect to see the EU export those goods in which it has the comparative advantage and the United States export those goods in which it has the comparative advantage.
5.
The inhabitants of the fictional economy of Atlantis use money in the form of cowry shells. Draw a circular-flow diagram showing households and firms. Firms produce potatoes and fish, and households buy potatoes and fish. Households also provide the land and labor to firms. Identify where in the flows of cowry shells or physical things (goods and services, or resources) each of the following impacts would occur. Describe how this impact spreads around the circle. a. A devastating hurricane floods many of the potato fields. b. A very productive fishing season yields a very large number of fish caught. c. The inhabitants of Atlantis discover Shakira and spend several days a month at dancing festivals.
5. Solution
The accompanying diagram illustrates the circular flow for Atlantis.
Shells
Shells Households Land and labor
Potatoes and fish
Markets for goods and services
Factor markets
Land and labor
Potatoes and fish Firms Shells
Shells
a. The flooding of the fields will destroy the potato crop. Destruction of the potato crop reduces the flow of goods from firms to households: fewer potatoes produced by firms now are sold to households. An implication, of course, is that fewer cowry shells flow from households to firms as payment for the potatoes in the market for goods and services. Since firms now earn fewer shells, they have fewer shells to pay to households in the factor markets. As a result, the amount of factors flowing from households to firms is also reduced.
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b. The productive fishing season leads to greater quantity of fish produced by firms to flow to households. An implication is that more money flows from households to firms through the markets for goods and services. As a result, firms will want to buy more factors from households (the flow of shells from firms to households increases) and, in return, the flow of factors from households to firms increases. c. Time spent at dancing festivals reduces the flow of labor from households to firms and therefore reduces the number of shells flowing from firms to households through the factor markets. In return, households now have fewer shells to buy goods with (the flow of shells from households to firms in the markets for goods and services is reduced), implying that fewer goods flow from firms to households.
6.
An economist might say that colleges and universities “produce” education, using faculty members and students as inputs. According to this line of reasoning, education is then “consumed” by households. Construct a circular-flow diagram to represent the sector of the economy devoted to college education: colleges and universities represent firms, and households both consume education and provide faculty and students to universities. What are the relevant markets in this diagram? What is being bought and sold in each direction? What would happen in the diagram if the government decided to subsidize 50% of all college students’ tuition?
6. Solution
The accompanying diagram shows the circular flow for the education sector. Salaries, scholarships
Tuition Households
Faculty, students
Education
Academic job market, market for students
Education market
Faculty, students
Education Colleges, universities Tuition
Salaries, scholarships
Colleges and universities buy faculty on the academic job market and attract students from the market for students. (Many colleges and universities actively try to attract good students by offering scholarships and the like.) They sell education to households in the market for education, and households buy education in that market from one (or sometimes several) of the sellers. If the government subsidized half of all students’ tuition, households would demand more education. As a result, colleges and universities would hire more faculty and accept more students, meaning that more money in terms of salaries and scholarships would flow from universities and colleges to the households.
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7.
Solution 7.
8.
A representative of the American clothing industry recently made the following statement: “Workers in Asia often work in sweatshop conditions earning only pennies an hour. American workers are more productive and as a result earn higher wages. In order to preserve the dignity of the American workplace, the government should enact legislation banning imports of low-wage Asian clothing.” a. Which parts of this quote are positive statements? Which parts are normative statements? b. Is the policy that is being advocated consistent with the preceding statements about the wages and productivities of American and Asian workers? c. Would such a policy make some Americans better off without making any other Americans worse off? That is, would this policy be efficient from the viewpoint of all Americans? d. Would low-wage Asian workers benefit from or be hurt by such a policy?
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Solution 8.
a. The positive statements are: ■
workers in Asia . . . [are] earning only pennies an hour
American workers are more productive
American workers are more productive and as a result earn higher wages
The normative statement is: ■
the government should enact legislation banning imports of low-wage Asian clothing
b. It is not. The statement about the productivity of American and Asian workers is about the absolute advantage that American workers have over Asian workers. However, Asian workers may still have a comparative advantage. And if that is the case, then banning imports would result in inefficiency. c. If America channeled more of its productive resources into producing clothing, it would have to give up producing other goods. As a result, America would be able to consume less of all goods. And this would make some Americans clearly worse off. Therefore, this policy would not be efficient. d. Low-wage Asian workers would also be hurt by this policy. The Asian country would channel its resources away from producing clothing toward producing other goods that it previously imported from America. But since it does not have the comparative advantage in those other goods, the Asian country would be able to consume less of all goods.
9.
Are the following statements true or false? Explain your answers. a. “When people must pay higher taxes on their wage earnings, it reduces their incentive to work” is a positive statement. b. “We should lower taxes to encourage more work” is a positive statement. c. Economics cannot always be used to completely decide what society ought to do. d. “The system of public education in this country generates greater benefits to society than the cost of running the system” is a normative statement. e. All disagreements among economists are generated by the media.
9. Solution
10.
Evaluate the following statement: “It is easier to build an economic model that accurately reflects events that have already occurred than to build an economic model to forecast future events.” Do you think that this is true or not? Why? What does this imply about the difficulties of building good economic models?
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10. Solution
True. With hindsight it is easier to see the important features of the situation that a model should have captured. For predictive purposes, a model needs to anticipate which features of reality are important (and so should be included) and which are unimportant (and so can be ignored). This is why the famed British economist John Maynard Keynes referred to economics as an art as well as a science.
11.
Economists who work for the government are often called on to make policy recommendations. Why do you think it is important for the public to be able to differentiate normative statements from positive statements in these recommendations?
11. Solution
Positive statements are those based on fact—or at least on our best estimate of what the facts are. Therefore, these statements are also those that do not depend on the political views of the economist. Normative statements may sometimes be influenced by the economist’s own values. Whether someone agrees with an economist’s normative statement may depend on whether they share values. It is therefore important that the public be able to distinguish normative from positive statements.
12.
The mayor of Gotham City, worried about a potential epidemic of deadly influenza this winter, asks an economic adviser the following series of questions. Determine whether a question requires the economic adviser to make a positive assessment or a normative assessment. a. How much vaccine will be in stock in the city by the end of November? b. If we offer to pay 10% more per dose to the pharmaceutical companies providing the vaccines, will they provide additional doses? c. If there is a shortage of vaccine in the city, whom should we vaccinate first—the elderly or the very young? (Assume that a person from one group has an equal likelihood of dying from influenza as a person from the other group.) d. If the city charges \$25 per shot, how many people will pay? e. If the city charges \$25 per shot, it will make a profit of \$10 per shot, money that can go to pay for inoculating poor people. Should the city engage in such a scheme?
12. Solution a. Positive b. Positive c. Normative d. Positive e. Normative
13.
Assess the following statement: “If economists just had enough data, they could solve all policy questions in a way that maximizes the social good. There would be no need for divisive political debates, such as whether the government should provide free medical care for all.”
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Solution 13.
What is true is that if economists had enough data, they could predict precisely what the outcome would be of any proposed policy (such as free medical care). That is, economists can answer positive questions. But no amount of data can lead to a determination about what a society should do—that is a normative question. An economist can predict how much it will cost to provide free medical care and what effects different ways of raising taxes will have on people’s behavior (for instance, a sales tax will reduce consumption behavior; an income tax may discourage workers from working as much as before). But whether this is a trade-off worth making is a question that can be answered only in political discourse.
E X T E N D YO U R U N D E R S TA N D I N G
14.
You are in charge of allocating residents to your dormitory’s baseball and basketball teams. You are down to the last four people, two of whom must be allocated to baseball and two to basketball. The accompanying table gives each person’s batting average and free-throw average. Name
Batting average
Free-throw average
Kelley
70%
60%
Jackie
50%
50%
Curt
10%
30%
Gerry
80%
70%
a. Explain how you would use the concept of comparative advantage to allocate the players. Begin by establishing each player’s opportunity cost of free throws in terms of batting average. b. Why is it likely that the other basketball players will be unhappy about this arrangement but the other baseball players will be satisfied? Nonetheless, why would an economist say that this is an efficient way to allocate players for your dormitory’s sports teams?
14. Solution
a. Let’s begin by establishing the opportunity cost of free throws for each player. If you allocate Kelley to the basketball team, the team gains a player with a 60% free-throw average and the baseball team loses a player with a 70% batting average. That is, the opportunity cost of allocating Kelley to the basketball team is 7⁄6. Similarly, Jackie’s opportunity cost of playing basketball is 1; Curt’s opportunity cost of playing basketball is 1 ⁄3, and Gerry’s opportunity cost of playing basketball is 8⁄ 7. Jackie and Curt have the lowest opportunity costs of playing basketball; that is, they have the comparative advantage in basketball. Therefore, they should be allocated to the basketball team. Kelley and Gerry have the comparative advantage in baseball and should therefore play on the baseball team.
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15.
Two important industries on the island of Bermuda are fishing and tourism. According to data from the World Resources Institute and the Bermuda Department of Statistics, in the year 2000 the 307 registered fishermen in Bermuda caught 286 metric tons of marine fish. And the 3,409 people employed by hotels produced 538,000 hotel stays (measured by the number of visitor arrivals). Suppose that this production point is efficient in production. Assume also that the opportunity cost of one additional metric ton of fish is 2,000 hotel stays and that this opportunity cost is constant (the opportunity cost does not change). a. If all 307 registered fishermen were to be employed by hotels (in addition to the 3,409 people already working in hotels), how many hotel stays could Bermuda produce? b. If all 3,409 hotel employees were to become fishermen (in addition to the 307 fishermen already working in the fishing industry), how many metric tons of fish could Bermuda produce? c. Draw a production possibility frontier for Bermuda, with fish on the horizontal axis and hotel stays on the vertical axis, and label Bermuda’s actual production point for the year 2000.
15. Solution
a. Forgoing the production of 1 metric ton of fish allows Bermuda to produce 2,000 additional hotel stays. Therefore, forgoing the production of 286 metric tons of fish allows Bermuda to produce 2,000 × 286 = 572,000 additional hotel stays. If all fishermen worked in the hotel industry, Bermuda could produce 538,000 + 572,000 = 1,110,000 hotel stays. b. Forgoing the production of 2,000 hotel stays allows Bermuda to produce 1 additional metric ton of fish, so giving up 538,000 hotel stays allows Bermuda to produce 538,000/2,000 = 269 additional metric tons of fish. If all hotel employees worked in the fishing industry, Bermuda could produce 286 + 269 = 555 metric tons of fish. c. The accompanying diagram shows the production possibility frontier for Bermuda. Note that it is a straight line because the opportunity cost is constant. Point A is Bermuda’s actual production point. Quantity of hotel stays (thousands)
1,110
538
A
Bermuda PPF
0
286
555 Quantity of fish (metric tons)
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16.
According to data from the U.S. Department of Agriculture’s National Agricultural Statistics Service, 124 million acres of land in the United States were used for wheat or corn farming in 2004. Of those 124 million acres, farmers used 50 million acres to grow 2.158 billion bushels of wheat and 74 million acres of land to grow 11.807 billion bushels of corn. Suppose that U.S. wheat and corn farming is efficient in production. At that production point, the opportunity cost of producing one additional bushel of wheat is 1.7 fewer bushels of corn. However, farmers have increasing opportunity costs, so that additional bushels of wheat have an opportunity cost greater than 1.7 bushels of corn. For each of the following production points, decide whether that production point is (i) feasible and efficient in production, (ii) feasible but not efficient in production, (iii) not feasible, or (iv) unclear as to whether or not it is feasible. a. Farmers use 40 million acres of land to produce 1.8 billion bushels of wheat, and they use 60 million acres of land to produce 9 billion bushels of corn. The remaining 24 million acres are left unused. b. From their original production point, farmers transfer 40 million acres of land from corn to wheat production. They now produce 3.158 billion bushels of wheat and 10.107 bushels of corn. c. Farmers reduce their production of wheat to 2 billion bushels and increase their production of corn to 12.044 billion bushels. Along the production possibility frontier, the opportunity cost of going from 11.807 billion bushels of corn to 12.044 billion bushels of corn is 0.666 bushel of wheat per bushel of corn.
16. Solution
a. This point is feasible but not efficient in production. Producing 1.8 billion bushels of wheat and 9 billion bushels of corn is less of both wheat and corn than is possible. They could produce more if all the available farmland were cultivated. b. At this new production point, farmers would now produce 1 billion more bushels of wheat and 1.7 billion fewer bushels of corn than at their original production point. This reflects an opportunity cost of 1.7 bushels of corn per additional bushel of wheat. But, in fact, this new production point is not feasible because we know that opportunity costs are increasing. Starting from the original production point, the opportunity cost of producing one more bushel of wheat must be higher than 1.7 bushels of corn. c. This new production point is feasible and efficient in production. Along the production possibility frontier, the economy must forgo 0.666 bushels of wheat per additional bushel of corn. So the increase in corn production from 11.807 billion bushels to 12.044 billion bushels costs the economy (12.044 − 11.807) billion bushels of corn × 0.666 bushel of wheat per bushel of corn = 0.158 bushel of wheat. This is exactly equal to the actual loss in wheat output: the fall from 2.158 billion to 2 billion bushels of wheat.
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Fraction
# org.apache.commons.lang3.math.Fraction Example
Hi folks, Today we are going to talk about `Fractions` from `org.apache.commons.lang3.math` package. This api provides us with capabilities to handle operations on fractions. I wrote a nice and simple java program demonstrating the application of this api. If you are already a java programmer and if you have tried dividing two numbers with different datatypes, you would have encountered implicit conversion of return value which results in ambiguous results, this class is highly effective in addressing those kinds of issues since you represent the fraction as a single unit.Enough said, lets have fun with Fractions !!!
Download this library here or Maven Repository:
```<groupId>org.apache.commons</groupId>
<artifactId>commons-lang3</artifactId>
<version>3.3.2</version>
```
Here is an example on how to go about using this Library.
## 1. Example
FractionExample.java
```package com.javacodegeek;
import org.apache.commons.lang3.math.Fraction;
public class FractionExample {
public static void main(String[] args)throws Exception {
operations();
comparisons();
conversions();
extractions();
simpletransformations();
}
public static void operations(){
System.out.println("OPERATIONS");
Fraction fraA = Fraction.FOUR_FIFTHS;
System.out.println("fraA="+fraA);
Fraction fraB = Fraction.getFraction(6, 5);
System.out.println("fraB= "+fraB);
Fraction AaddB = fraA.add(fraB);
System.out.println("Additon: fraA+fraB = "+AaddB);
Fraction AxB=fraA.multiplyBy(fraB);
System.out.println("MultiplyBy: fraAxfraB = "+AxB);
Fraction Apow2 = fraA.pow(2);
System.out.println("A power 2: "+Apow2);
Fraction AdivB = fraA.divideBy(fraB);
System.out.println("A divide by B: "+AdivB);
Fraction AminusB = fraA.subtract(fraB);
System.out.println("A-B: "+AminusB);
}
public static void comparisons(){
System.out.println("\nCOMPARISONS");
Fraction fraA = Fraction.FOUR_FIFTHS;
Fraction fraB = Fraction.getFraction(6, 5);
System.out.println("fraA="+fraA);
System.out.println("fraB= "+fraB);
int res = fraA.compareTo(fraB);
System.out.println("fraA.compareTo(fraB): "+res);
boolean boolres = fraA.equals(fraB);
System.out.println("fraA.equals(fraB): "+boolres);
}
public static void conversions(){
System.out.println("\nCONVERSIONS");
Fraction fraA = Fraction.FOUR_FIFTHS;
System.out.println("fraA: "+ fraA);
double fraAdouble = fraA.doubleValue();
System.out.println("fraA.doubleValue() : "+fraAdouble);
float fraAfloat = fraA.floatValue();
System.out.println("fraA.floatValue() : "+fraAfloat);
float fraAint = fraA.intValue();
System.out.println("fraA.intValue(): "+fraAint);
Fraction frastrMixed = Fraction.getFraction("10 5/7");
System.out.println("Fraction.getFraction(\"10 5/7\"): "+frastrMixed);
Fraction frastr = Fraction.getFraction("5/7");
System.out.println("Fraction.getFraction(\"5/7\"): "+frastr);
Fraction fraDb = Fraction.getFraction(44.44);
System.out.println("Fraction.getFraction(44.44): "+fraDb);
Fraction fraND = Fraction.getFraction(6,7);
System.out.println("Fraction.getFraction(6,7): "+fraND);
Fraction fraWND = Fraction.getFraction(7,8,9);
System.out.println("Fraction.getFraction(7,8,9): "+fraWND);
}
public static void extractions(){
System.out.println("\nEXTRACTIONS");
Fraction fraA = Fraction.FOUR_FIFTHS;
System.out.println("fraA: "+fraA);
System.out.println("getDenominator(): "+fraA.getDenominator());
System.out.println("getNumerator(): "+ fraA.getNumerator());
System.out.println("getProperNumerator(): "+fraA.getProperNumerator());
System.out.println("getProperWhole(): "+fraA.getProperWhole());
System.out.println("toProperString(): "+fraA.toProperString());
}
public static void simpletransformations(){
System.out.println("\nSIMPLETRANSFORAMTIONS");
Fraction fraA = Fraction.FOUR_FIFTHS;
System.out.println("fraA: "+fraA);
System.out.println("fraA.negate().abs(): "+fraA.negate().abs());
System.out.println("fraA.negate(): "+fraA.negate());
System.out.println("fraA.invert(): "+fraA.invert());
System.out.println("fraA.reduce(): "+fraA.reduce());
System.out.println("fraA.getReducedFraction(20, 30): "+fraA.getReducedFraction(20, 30));
}
}
```
Output
```OPERATIONS
fraA=4/5
fraB= 6/5
Additon: fraA+fraB = 2/1
MultiplyBy: fraAxfraB = 24/25
A power 2: 16/25
A divide by B: 2/3
A-B: -2/5
COMPARISONS
fraA=4/5
fraB= 6/5
fraA.compareTo(fraB): -1
fraA.equals(fraB): false
CONVERSIONS
fraA: 4/5
fraA.doubleValue() : 0.8
fraA.floatValue() : 0.8
fraA.intValue(): 0.0
Fraction.getFraction("10 5/7"): 75/7
Fraction.getFraction("5/7"): 5/7
Fraction.getFraction(44.44): 1111/25
Fraction.getFraction(6,7): 6/7
Fraction.getFraction(7,8,9): 71/9
EXTRACTIONS
fraA: 4/5
getDenominator(): 5
getNumerator(): 4
getProperNumerator(): 4
getProperWhole(): 0
toProperString(): 4/5
SIMPLETRANSFORAMTIONS
fraA: 4/5
fraA.negate().abs(): 4/5
fraA.negate(): -4/5
fraA.invert(): 5/4
fraA.reduce(): 4/5
fraA.getReducedFraction(20, 30): 2/3
```
## 2. Static constants representing fractions that you can use.
• `static Fraction FOUR_FIFTHS` is 4/5
• `static Fraction MINUS_ONE` is -1 / 1
• `static Fraction ONE` is 1
• `static Fraction ONE_FIFTH` is 1/5
• `static Fraction ONE_HALF` is 1/2
• `static Fraction ONE_QUARTER` is 1/4
• `static Fraction ONE_THIRD` is 1/3
• `static Fraction THREE_FIFTHS` is 3/5
• `static Fraction THREE_QUARTERS` is 3/4
• `static Fraction TWO` is 2/1
• `static Fraction TWO_FIFTHS` is 2/5
• `static Fraction TWO_QUARTERS` is 2/4
• `static Fraction TWO_THIRDS` is 2/3
• `static Fraction ZERO` is 0
## 3. Methods associated with fractions are characterized into 5 sections for the sake of explanation.
1. Operations
2. Comparisons
3. Conversions
4. Extractions
5. Simpletransformations
## 4. Operations
This category deals with addition,multiplication, power,division,subractions and hence the following methods.
the `FractionExample.operations()` method in the above code demonstrates on how to go about using these methods in your program.
• `Fraction add(Fraction fraction)` Adds the value of this fraction to another, returning the result in reduced form.
• `Fraction divideBy(Fraction fraction)` Divide the value of this fraction by another.
• `Fraction multiplyBy(Fraction fraction)` Multiplies the value of this fraction by another, returning the result in reduced form.
• `Fraction pow(int power)` Gets a fraction that is raised to the passed in power.
• `Fraction subtract(Fraction fraction)` Subtracts the value of another fraction from the value of this one, returning the result in reduced form.
## 5. Comparisons
This category deals with comparing fractions, hence the following methods.
the `FractionExample.comparisons()` method in the above code demonstrates on how to go about using these methods in your program.
• `int compareTo(Fraction other)` Compares this object to another based on size.
• `boolean equals(Object obj)` Compares this fraction to another object to test if they are equal.
## 6. Conversions
This category deals with converting numbers,strings into fractions and hence the following methods.
the `FractionExample.conversions()` method in the above code demonstrates on how to go about using these methods in your program.
The one method that catches my eye is getFraction(String str) method. because it can not only accept simple fraction like 4/5 but also mixed fraction representation like 10 4/5 which is really handy and ofcourse the fraction object will hold the fraction as is and not reduce it, for that you have another method (reduce()).
Another interesting method is getFraction(double value) which accepts a double and conveniently returns a fraction representation.neat.
• `double doubleValue()` Gets the fraction as a double.
• `float floatValue()` Gets the fraction as a float.
• `static Fraction getFraction(double value)` Creates a Fraction instance from a double value.
• `static Fraction getFraction(int numerator, int denominator)` Creates a Fraction instance with the 2 parts of a fraction Y/Z.
• `static Fraction getFraction(int whole, int numerator, int denominator)` Creates a Fraction instance with the 3 parts of a fraction X Y/Z.
• `static Fraction getFraction(String str)` Creates a Fraction from a String.
• `int intValue()` Gets the fraction as an int.
• `long longValue()` Gets the fraction as a long.
## 7. Extractions
This category deals with extracting numbers from a given fraction and hence the following methods.
the `FractionExample.extractions()` method in the above code demonstrates on how to go about using these methods in your program.
• `int getNumerator()` returns the numerator.
• `int getDenominator()` returns the denominator.
• `int getProperNumerator()` returns the proper numerator, always positive.
• `int getProperWhole()` returns the proper whole part of the fraction.
• `String toProperString()` returns the fraction as a proper String in the format X Y/Z(mixed fraction).
## 8. Simpletransformations
This category deals with transforming a given fraction and hence the following methods.
the `FractionExample.simpletransformations()` method in the above code demonstrates on how to go about using these methods in your program.
• `static Fraction getReducedFraction(int numerator, int denominator)` Creates a reduced Fraction instance with the 2 parts of a fraction Y/Z.
• `Fraction invert()` Gets a fraction that is the inverse (1/fraction) of this one.
• `Fraction negate()` Gets a fraction that is the negative (-fraction) of this one.
• `Fraction pow(int power)` Gets a fraction that is raised to the passed in power.
• `Fraction reduce()` Reduce the fraction to the smallest values for the numerator and denominator, returning the result.
## Conclusion
`Fraction` class is a simple and elegant api. This can really save a lot of time for programmers who are working with calculations pertaining to fractions.
Download
You can download the source code of this example here: FractionExample.zip
### Arun Kumaraswamy
Arun Kumaraswamy is a Java-Oracle Developer. His technical skills span a very wide range of technologies across networks, operating systems and web servers. He specializes in programming, PL/SQL and Linux. He also conducts penetration testing for websites. In short, he is the definition of a geek. He likes watching cricket in his spare time. He has some certifications like C|EH, CDAC-DITISS, SCJP and RHCE.
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Student Doctor Network Forums > Pre-Medical Forums > MCAT Discussions > TPR Physics Question PDA View Full Version : TPR Physics Question puravida8512-01-2009, 03:46 PMIn the TPR Hyperlearning Science workbook, there is a question that states: A man with a mass of 100kg sits 5m from the center of the sea saw. Two children, each with a mass of 20kg, are seated on the other side of the sea saw. One child sits 10m from the center. How far from the center should the other child sit to balance the sea saw? A) 5 B) 10 C) 15 D) 20 The proposed answer is C = 15m. Their rational is to use the torque equation and set up an equation such as: TorqueCCW = TorqueCW 100kg * 5m = 20kg * 10m + 20kg * x(m) x = 15 How could they set the man at a positive 5 and also put a child in the positive 10 if they are seated "on the other side of the sea saw". The child would therefore be on the same side of him. What I got from the question stem was that if he is 5m from the center, and if the children are on the opposite side, it would mean that one of them is either: 5m + 10m = 15m if we consider the sea saw a continual integer count. or He is at -5m and the child is at +10m because they are on opposite sides of the center which we could mark zero. My visual aide: ___100kg__^______20kg_____20kg -----(-5m)-0------(+10m)-----?m-- If we use 15 m for the child's location as I have proposed, than we can set up the equation as this: 100kg * 5m = 20kg * 15m + 20kg * x (m) 500 kg*m = 300 kg*m + 20kg x (m) 200kg*m = 20kg x 200kg*m / 20kg = x x = 10 m I've been doing 5 hours of Physics now, so my head could be a bit off. But can someone set me straight? Thanks Steelersfan200912-01-2009, 04:06 PMHi, I believe you have to pick a spot as the pivot point and the torque left of that equal the right or (clockwise balancing counter clockwise) You can also solve this via center of mass, but torques makes it easier. This problem tells you that they are certain distance away from the center. So it is easy to pick this center as your pivot which you can set a distance of zero.. The distance of this first child from the pivot is 10 meters, (again this is his distance from the pivot at 0 not his distance from the man, your stated 15 meters) The distance the man from the center is 5 meters, you can think that he's -5 but I mean they are not going to flat out tell you that He's just 5 meters from the center and since his children are on the opposite side of the center they would not be on the same side as him. Just draw the diagram according to the question and you won't go wrong, but read the wordings carefully. I hope that helped and not just confused you Compass12-01-2009, 04:22 PMIn the TPR Hyperlearning Science workbook, there is a question that states: A man with a mass of 100kg sits 5m from the center of the sea saw. Two children, each with a mass of 20kg, are seated on the other side of the sea saw. One child sits 10m from the center. How far from the center should the other child sit to balance the sea saw? A) 5 B) 10 C) 15 D) 20 The proposed answer is C = 15m. Their rational is to use the torque equation and set up an equation such as: TorqueCCW = TorqueCW 100kg * 5m = 20kg * 10m + 20kg * x(m) x = 15 How could they set the man at a positive 5 and also put a child in the positive 10 if they are seated "on the other side of the sea saw". The child would therefore be on the same side of him. What I got from the question stem was that if he is 5m from the center, and if the children are on the opposite side, it would mean that one of them is either Two things: The equation says man is 5m from the center, and then kids are on the OTHER side. It is important to make this distinction. Else you'll never solve the problem. Here is where you got confused, I think. There is no negative or positive distance here for a reason. You are comparing opposing forces. Both sides should be positive. 100 x 5 <--- left side of center right side of center ---> 20 x 10 + 20 x 15 I think you may be confused with this equation: 0 = 100kg * -5m + 20kg * 10m + 20kg * x(m) The total torsional force in this equation should CANCEL OUT (both sides are equal and zero). The original equation shows that the forces should BALANCE OUT (both sides are equal, neither side is necessarily non-zero). Think of it this way. Numbers on the left are torsional for counter-clockwise rotation, numbers on the right are torsional for clockwise rotation. In order for the see-saw to balance, torsion CW and CCW must be equal. mcat4512-01-2009, 04:57 PMfor the stated question, you are correct in bringing up the issue that there should be a sign difference between the two sides. the easiest way to solve the question is to realize that your net torque should be 0 as there will be no rotational motion when the seesaw is balanced. if you think of the problem this way and also arbitrarily set one side of the seesaw as positive and one side of the seesaw as negative, then at some point during the solution process you will end up with the same set up as the answer from the princeton review hyperlearning book. essentially, they just omitted the basic initial step of summing all possible torques on the body in question and setting that equal to 0. hope this helps.... puravida8512-01-2009, 09:26 PMfor the stated question, you are correct in bringing up the issue that there should be a sign difference between the two sides. the easiest way to solve the question is to realize that your net torque should be 0 as there will be no rotational motion when the seesaw is balanced. if you think of the problem this way and also arbitrarily set one side of the seesaw as positive and one side of the seesaw as negative, then at some point during the solution process you will end up with the same set up as the answer from the princeton review hyperlearning book. essentially, they just omitted the basic initial step of summing all possible torques on the body in question and setting that equal to 0. hope this helps.... It's not a matter of the simple algebra that I didn't understand. I know that setting CW = CCW requires understanding that they are in equilibrium which would make the sum of them equal to 0 to begin with. Anyhow, I recently heard on 'audio osmosis' a huge tip that helped me with this problem: If we have a system in equilibrium, setting forces equal to each other like all the forces upward = forces downward allows one to ignore sine conventions because as Compass said: the forces are opposing forces. Ohh.. the fun of physics! Thanks everyone.
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# Problem with understanding the ordinals addition definition
I have recently started studying Set Theory in a self-thaught way, for that purpose I have been following Kunen's book: Set Theory: An Introduction to Independence Proofs. I'm in Chapter I section 7 and it has been defined the ordinals addition but I don't quite understand that definition. I have seen that in other books authors defines the addition using transfinite induction and it seems easier but now I want to understand Kunen's one.
$$\alpha + \beta=type(\alpha \times \{ 0 \} \cup \beta \times \{1\}, R) \:\text{where }$$ $$R=\{ \langle \langle \xi,0 \rangle, \langle \eta , 0\rangle \rangle : \xi<\eta<\alpha\} \; \cup \{\langle \langle \xi,1 \rangle, \langle \eta , 1\rangle \rangle : \xi<\eta<\beta\} \; \cup [(\alpha\times\{0\})\times(\beta\times\{1\})].$$
With $$type(A,R)$$ is the unique ordinal $$C$$ such that $$\langle A, R\rangle \cong C$$ when $$\langle A,R\rangle$$ is a well-ordering set.
What I think I understood so far is that this definition tries to order two non-disjoint sets having that $$\alpha<\beta$$ and keeping the order inside $$\alpha$$ and $$\beta$$. What I can't understand is how I can get that cardinal (in for example a finite case), maybe I'm being stubborn and I should "ignore" this definition and trying to understand the more simplified one given in the later results.
• In my opinion this is the most intuitive definition: it actually gives a picture of the sum order. Are you familiar with lexicographic ordering? Apr 26, 2020 at 18:59
• @BrianM.Scott Yes, i'm more less familiar for having studiet it in Naive Set Theory (althought not used it much) what I can't see is what it has to do, i mean, the relation that is defined is not the lexicographic order, right? Thank you for your answer! Apr 26, 2020 at 19:03
As mentioned by Brian, it is essentially the lexicographic ordering.
For example, say $$2=\{0_2,1_2\}$$ and $$3=\{0_3,1_3,2_3\}$$.
According to the definition, we first extend $$2$$ and $$3$$ to ordered pairs:
$$2\times\{0\}=\{(0_2,0),(1_2,0)\}\quad{\rm and}\quad 3\times\{1\}=\{(0_3,1),(1_3,1),(2_3,1)\}.$$
Then what is $$R$$? Although it is written as the union of sets, we can write it in a chain like this:
$$(0_2,0)<(1_2,0)<(0_3,1)<(1_3,1)<(2_3,1).$$
The set $$2\times\{0\}\cup 3\times\{1\}$$ is linearly ordered and isomorphic to $$5=\{0_5,1_5,2_5,3_5,4_5\}$$ in which
$$0_5<1_5<2_5<3_5<4_5.$$
Thus, $$2+3=5$$.
• Amazing thanks! Now that I understand the finite case I will try to work on my own on the infinite case. Apr 26, 2020 at 20:04
The definition that Ken is using amounts to placing a copy of the ordinal $$\beta$$ after the ordinal $$\alpha$$. Since the sets $$\alpha$$ and $$\beta$$ are not actually disjoint (unless one of them is $$0$$), we first use a small trick to make disjoint copies of them, replacing $$\alpha$$ by $$\alpha\times\{0\}$$ and $$\beta$$ by $$\beta\times\{1\}$$. We give these the obvious orders, which I’ll call $$\le_\alpha$$ and $$\le_\beta$$: for $$\xi,\eta\in\alpha$$ we set $$\langle\xi,0\rangle\le_\alpha\langle\eta,0\rangle$$ iff $$\xi\le\eta$$, and we define $$\le_\beta$$ similarly. As sets these relations are
$$\le_\alpha=\{\langle\langle\xi,0\rangle,\langle\eta,0\rangle\rangle:\xi\le\eta<\alpha\}$$
and
$$\le_\beta=\{\langle\langle\xi,1\rangle,\langle\eta,1\rangle\rangle:\xi\le\eta<\beta\}\;.$$
Now we have disjoint copies of $$\alpha$$ and $$\beta$$ — copies in the sense that they are order-isomorphic to $$\alpha$$ and $$\beta$$, respectively — and we define an order that places the copy of $$\beta$$ after the copy of $$\alpha$$. We do this by imposing the reverse lexicographic order on $$(\alpha\times\{0\})\cup(\beta\times\{1\})$$. That is, we order first on the second coordinate and then on the first: we define
$$\langle\xi,i\rangle\,R\,\langle\eta,j\rangle\text{ iff }i
If you check the various possibilities, you’ll see that this makes
$$\langle\xi,0\rangle\,R\,\langle\eta,0\rangle\text{ iff }\langle\xi,0\rangle\le_\alpha\langle\eta,0\rangle\text{ iff }\xi\le\eta$$
for $$\xi,\eta\in\alpha$$,
$$\langle\xi,1\rangle\,R\,\langle\eta,1\rangle\text{ iff }\langle\xi,1\rangle\le_\beta\langle\eta,1\rangle\text{ iff }\xi\le\eta$$
for $$\xi,\eta\in\beta$$, and $$\langle\xi,0\rangle\,R\,\langle\eta,1\rangle$$ whenever $$\xi\in\alpha$$ and $$\eta\in\beta$$. In short, $$R$$ orders $$\alpha\times\{0\}$$ just like $$\le_\alpha$$ and $$\beta\times\{1\}$$ just like $$\le_\beta$$, and it places all of $$\alpha\times\{0\}$$ before all of $$\beta\times\{1\}$$.
• Ok now I understand, my problem was that I didn't know how that definition of $R$ would be an order but now it's much clearer, amazing answer! Now I think i'm stucking with the concept of isomorphism. Apr 26, 2020 at 19:59
• @Partizanki: Fortunately, in this case the order-isomorphisms that are involved are pretty straightforward: for instance, the map $f:\alpha\to\alpha\times\{0\}$ defined by $f(\xi)=\langle\xi,0\rangle$ is easily seen to be an order-isomorphism from the order $\langle\alpha,\le\rangle$ to the order $\langle\alpha\times\{0\},\le_\alpha\rangle$. For any $\xi,\eta\in\alpha$, $\xi\le\eta$ iff $\langle\xi,0\rangle\le_\alpha\langle\eta,1\rangle$. Apr 26, 2020 at 20:02
• Yes, I more less understand the order-isomorphsim thanks to your explanation, I meant the isomorphsim that gives you the $type$ as previously defined. Apr 26, 2020 at 20:07
• @Partizanki: Ah, yes, that one. The proof from the axioms in Theorem 7.6 makes it look worse than it is. Intuitively you just map the least element of $A$ with respect to $R$ to $0$, the least element of the rest of $A$ to $1$, and so on. If you’ve mapped all of the $R$-predecessors of some $a\in A$ in this fashion, their images turn out to be an initial segment of the ordinals, and you just map $a$ to the smallest ordinal that hasn’t yet been used. Apr 26, 2020 at 20:17
• Ok I had the intuition but your comment has just done it, now everything is clearer, thank you very much! Amazing answers. Apr 26, 2020 at 20:54
As Kunen explains it you are putting a copy of $$\beta$$ after $$\alpha$$ and looking at the resulting order type. Whether $$\alpha \lt \beta$$ is not important. If they are both finite it is just regular addition. If you add $$\omega+2$$ and $$\omega+1$$ the order type is $$\omega, 2, \omega, 1$$. The $$2$$ gets absorbed into the start of the following $$\omega$$ and the result is $$\omega + \omega + 1$$
• Ok I think I get it but not using the definition but the 7.18 (5) Lemma that states: If $\beta$ is a limit ordinal, $\alpha + \beta= sup\{ \alpha + \xi : \xi < \beta\}.$ Anyway I have to think more about your answer and thank you very much for answering! Apr 26, 2020 at 20:11
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# Math 441
(1 point) Find dy/dx
dy/dx
if y=9x−10 − − − − − − √ 3
y=9x−103
asked by winita
1. if
y = 9 x plus or minus ANY constant
then dy/dx = 9
because the derivative of any constant is zero. It does not change.
posted by Anonymous
2. if y = (9x-10)/√3
then y' = 9/√3 = 3√3
posted by Steve
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# Final Review
Topics: Maxima and minima, Second derivative test, Critical point Pages: 4 (1001 words) Published: June 4, 2013
Math 242 Additional study problems for the Final Exam – (LL) 1. For the following determine if the integral converges. If the integral converges, compute its value.
a. 2. It is estimated that
b. c. d. years from now, a certain investment will be generating income at the rate of per year, dispensed continuously. If the income is generated in perpetuity and the prevailing annual interest rate remains fixed at 5% compounded continuously, what is the present value of the investment? 3. Use a) The Trapezoidal rule, and b) Simpson’s rule with to approximate the integral . Round each of your answers to 6 decimal places. 4. Determine how many subintervals are required to guarantee accuracy to within 0.0001 of the actual value of the integral using the trapezoidal rule. 5. a. b. 6.
f f 2 f 2 f , , , and . f ( x, y) xy ln(3 y) , Find x dy x 2 yx e2 xy , Find f x , f y , f yy , and f yx . f ( x, y ) x 1 1 2
Suppose the production function of a firm is given by the Cobb-Douglas production function
f ( x, y) 27 x 3 y 3 where x is the number of units of labor and y is the number of units of capital required to produce f ( x, y ) thousand units of the product. Find the marginal productivities of labor and capital when x = 27 and y = 8. 7. Suppose a brewery has a profit function given by P( x, y) 2 x 2 2 xy y 2 2 x 4 y 107 where x is the number (in thousands) of cases of India pale ale, and y is the number (in thousands) of cases of Best bitter, and P is the profit (in thousands of dollars). How many cases of each type of beer should be produced each year to maximize the profit? Show that your answer is a maximum. 8.
1 3 y 3 y . Find all critical points ( x, y) for f ( x, y) . Then use the second 3 derivative test to determine, if possible, if the points yield a maximum or minimum for f ( x, y ) . Let f ( x, y) x 2 2 xy
9. The research department for a computer company has a production function for a particular product...
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# monthplot
0th
Percentile
##### Plot a Seasonal or other Subseries from a Time Series
These functions plot seasonal (or other) subseries of a time series. For each season (or other category), a time series is plotted.
Keywords
ts, hplot
##### Usage
monthplot(x, …)# S3 method for stl
monthplot(x, labels = NULL, ylab = choice, choice = "seasonal",
…)# S3 method for StructTS
monthplot(x, labels = NULL, ylab = choice, choice = "sea", …)# S3 method for ts
monthplot(x, labels = NULL, times = time(x), phase = cycle(x),
ylab = deparse(substitute(x)), …)# S3 method for default
monthplot(x, labels = 1L:12L,
ylab = deparse(substitute(x)),
times = seq_along(x),
phase = (times - 1L)%%length(labels) + 1L, base = mean,
axes = TRUE, type = c("l", "h"), box = TRUE,
col = par("col"), lty = par("lty"), lwd = par("lwd"),
col.base = col, lty.base = lty, lwd.base = lwd, ...)
##### Arguments
x
Time series or related object.
labels
Labels to use for each ‘season’.
ylab
y label.
times
Time of each observation.
phase
Indicator for each ‘season’.
base
Function to use for reference line for subseries.
choice
Which series of an stl or StructTS object?
Arguments to be passed to the default method or graphical parameters.
axes
Should axes be drawn (ignored if add = TRUE)?
type
Type of plot. The default is to join the points with lines, and "h" is for histogram-like vertical lines.
box
Should a box be drawn (ignored if add = TRUE)?
Should thus just add on an existing plot.
col, lty, lwd
Graphics parameters for the series.
col.base, lty.base, lwd.base
Graphics parameters for the segments used for the reference lines.
##### Details
These functions extract subseries from a time series and plot them all in one frame. The ts, stl, and StructTS methods use the internally recorded frequency and start and finish times to set the scale and the seasons. The default method assumes observations come in groups of 12 (though this can be changed).
If the labels are not given but the phase is given, then the labels default to the unique values of the phase. If both are given, then the phase values are assumed to be indices into the labels array, i.e., they should be in the range from 1 to length(labels).
##### Value
These functions are executed for their side effect of drawing a seasonal subseries plot on the current graphical window.
##### References
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.
ts, stl, StructTS
##### Aliases
• monthplot
• monthplot.default
• monthplot.ts
• monthplot.stl
• monthplot.StructTS
##### Examples
library(stats) # NOT RUN { require(graphics) ## The CO2 data fit <- stl(log(co2), s.window = 20, t.window = 20) plot(fit) op <- par(mfrow = c(2,2)) monthplot(co2, ylab = "data", cex.axis = 0.8) monthplot(fit, choice = "seasonal", cex.axis = 0.8) monthplot(fit, choice = "trend", cex.axis = 0.8) monthplot(fit, choice = "remainder", type = "h", cex.axis = 0.8) par(op) ## The CO2 data, grouped quarterly quarter <- (cycle(co2) - 1) %/% 3 monthplot(co2, phase = quarter) ## see also JohnsonJohnson # }
Documentation reproduced from package stats, version 3.6.0, License: Part of R 3.6.0
### Community examples
Looks like there are no examples yet.
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https://www.physicsforums.com/threads/probably-a-geometric-series-question.818529/
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# Probably a geometric series question
• mpx86
In summary, the expression 2^32 – {(2 + 1) (2^2 + 1) (2^4+1) (2^8+1) (2^16+1)} is equal to 1. However, the options provided do not match this result due to an error in the question statement. The correct solution was found by using the iteration (1+x) (1+x^2) = (x^4-1)/x-1.
mpx86
## Homework Statement
2^32 – (2 + 1) (2^2 – 1) (2^4+1) (2^8+1) (2^16+1)} is equal to
## The Attempt at a Solution
Solved it by opening the bracket
Answer: 2^31 + 2^24 + 2^ 18 - 2^7 + 2
Option' are
0
1
2
2^16
None of the options matched... Is there a mistake in question statement or my solution of the same?[/B]
mpx86 said:
## Homework Statement
2^32 – (2 + 1) (2^2 – 1) (2^4+1) (2^8+1) (2^16+1)} is equal to
## The Attempt at a Solution
Solved it by opening the bracket
Answer: 2^31 + 2^24 + 2^ 18 - 2^7 + 2
Option' are
0
1
2
2^16
None of the options matched... Is there a mistake in question statement or my solution of the same?[/B]
lol solved... there was an error in question statement:
question was 2^32 – {(2 + 1) (2^2 + 1) (2^4+1) (2^8+1) (2^16+1)}
(1+x) (1+x^2) = (x^4-1)/x-1
using this iteration yields answer as 1
thanks anyways :)
## 1. What is a geometric series?
A geometric series is a sequence of numbers where each term is found by multiplying the previous term by a constant value called the common ratio. The general form of a geometric series is a + ar + ar^2 + ar^3 + ..., where a is the first term and r is the common ratio.
## 2. How do you find the sum of a geometric series?
The sum of a finite geometric series can be found using the formula S = a(1-r^n)/(1-r), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms. For an infinite geometric series, the sum can be found using the formula S = a/(1-r), as long as the absolute value of the common ratio is less than 1.
## 3. What is the common ratio in a geometric series?
The common ratio in a geometric series is the number by which each term is multiplied to get the next term. It is usually represented by the variable r and is calculated by dividing any term by the previous term.
## 4. How do you determine if a series is geometric?
A series is geometric if each term is found by multiplying the previous term by the common ratio. You can also check for a constant ratio between consecutive terms to determine if a series is geometric.
## 5. What real-life applications use geometric series?
Geometric series have many real-life applications, such as calculating compound interest in financial investments, modeling population growth, and in physics to describe the decay of radioactive materials. They are also commonly used in computer science, engineering, and statistics.
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## Programming Praxis – Traveling Salesman: Nearest Neighbor
In today’s Programming Praxis exercise we have to implement a significantly faster algorithm for the traveling salesman problem than in the previous exercise. Let’s get started, shall we?
As usual, some imports:
```import Control.Monad
import Data.List
import qualified Data.List.Key as K
import System.Random```
The functions for calculating distance and total distance are largely the same as in the previous exercise, though because I’ve switched to using lists of integers we now need an extra fromIntegral, and repeating the first point in order to complete the loop has been moved to the cost function.
```dist :: (Int, Int) -> (Int, Int) -> Float
dist (x1, y1) (x2, y2) = sqrt (f (x1 - x2) ** 2 + f (y1 - y2) ** 2)
where f = fromIntegral
cost :: [(Int, Int)] -> Float
cost xs = sum \$ zipWith dist xs (tail xs ++ xs)```
Generating a series of random points is a bit longer than it could be because we have to make sure all the points are unique.
```randomPoints :: Int -> IO [(Int, Int)]
randomPoints n = f [] where
f ps = if length ps == n then return ps else
do p <- liftM2 (,) rnd rnd
if elem p ps then f ps else f (p:ps)
rnd = randomRIO (0, 10 * n)```
Determining the tour to take using the nearest neighbor algorithm is not that difficult. Again, we index the points for similarity to the Programming Praxis solution, not because it is needed.
```nearTour :: [(Int, Int)] -> [(Integer, (Int, Int))]
nearTour = f . zip [0..] where
f [] = []
f [x] = [x]
f ((i,p):ps) = (i,p) : f (nxt : delete nxt ps) where
nxt = K.minimum (dist p . snd) ps```
To test, we check both a random set of points, as well as the set from the Programming Praxis solution.
```main :: IO ()
main = do rndTour <- fmap nearTour \$ randomPoints 25
print (cost \$ map snd rndTour, rndTour)
let praxisTour = nearTour
[(139, 31),( 41,126),(108, 49),(112,193),(179,188),
(212, 24),(245, 50),(167,187),(159,236),(185, 78),
( 27, 63),(101,188),(195,167),( 30, 10),(238,110),
(221, 60),( 27,231),(146, 67),(249,172),( 36, 71),
( 37,203),(118, 38),(241,226),(197, 29),(220,186)]
print (cost \$ map snd praxisTour, praxisTour)```
We get the same tour as the Programming Praxis solution (Ok, the reverse to be exact. Again, this doesn’t matter and I think starting with the first point is more logical), and at a third of the line count, so I think we can call this one done.
### 3 Responses to “Programming Praxis – Traveling Salesman: Nearest Neighbor”
1. luckytoilet Says:
The way I remembered the problem was that not every point was connected to every other point. This made it considerably more tricky, as you would then have a shortest-path problem as well as a travelling salesman problem.
Also, how do you do haskell syntax highlighting in wordpress?
2. Remco Niemeijer Says:
I’m just basing my solution on the exercise given by Programming Praxis, and in his version all the points are connected.
As for syntax highlighting, if you have your own wordpress install you can install one of the many syntax hightlighting plugins. Since I’m on wordpress.com i don’t have that option, so I use a program called Hightlight Code Convertor which outputs raw html (lots of spans and style tags) and use that. Not particularly elegant or easy to change afterwards, but it works well enough.
3. j lien Says:
You could speed things up by not using the distance metric
but by using the distance squared metric. You are looking
for minimum dist and this will work equally as well for dist squared.
Avoid the slow square roots.
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This problem makes use of vectors, coordinates and scalar products. It is a very useful fact that if two vectors ${\bf u}$ and ${\bf v}$ have coordinates $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ then the cosine of the angle $\theta$ they make is
$$\cos(\theta) = \frac{{\bf u}\cdot {\bf v}}{|{\bf u}||{\bf v}|}\quad\,,\mbox{ where } {\bf u}\cdot {\bf v}=x_1x_2+y_1y_2+z_1z_2$$
In a perfect tetrahedral molecule there is a central atom attached to four other atoms which lie on the vertices of a perfect tetrahedron. If the central atom is at the coordinate origin and the molecule rests on the plane $z=-h$ then what would be the coordinates of the other atoms, assuming a bond length of $1$ unit? Find the scalar products between the vectors joining the origin to each atom and hence the value of $h$. It is usually stated that the angles each of these bonds make is $109.5^\circ$; however, this is only an approximation. What is the exact value of the bond angle in a perfect tetrahedron? If the perfect tetrahedron is deformed slightly, how many of the bond angles could be exactly $109.5^\circ$? How would it be deformed to achieve this? Would the other bond angles increase or decrease under such a deformation?
A perfect trigonal pyramidal molecule is the same as a perfect tetrahedral molecule with a single outer atom removed. Ammonia NH$_3$ is approximately a trigonal pyramid with bond angle $107.5$. Does this correspond to a lengthening or shortening of the bonds relative to a perfect structure? By what percentage? If the H atoms were fixed and a vertical force were applied downwards to the N atom in the diagram, would the bond angles increase or decrease? What would be the maximum possible angle mathematically?
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# What Is the Atomic Number of an Element?
The atomic number indicates the total number of protons in the nucleus of a given atom. We represent it with the letter Z. The number is placed as a subscript to the left of the element symbol.
Atoms are made up of a nucleus, and a shell made up of electrons. The nucleus is made up of two subatomic particles: protons and neutrons. Protons are positively charged, neutrons are electrically neutral. Therefore the atomic nucleus always has a positive charge.
On the other hand, electrons have a negative charge.
An atom in its natural state is neutral. In this case, a neutral atom has the same number of electrons and protons.
The atomic number alone that determines the chemical properties of an element
The total weight of an atom is called the atomic weight. It is approximately equal to the number of protons and neutrons, with a little extra added by the electrons.
## What Is the Role of the Periodic Table in the Atomic Number Definition?
The atomic number is used to classify the elements within the periodic table of the elements. The periodic table is an arrangement in which their atomic number arranges chemical elements.
Each chemical element in the periodic table is characterized by having the same number of protons in its atomic nucleus. However, the same chemical element can have a different number of neutrons (N).
In 1911, Ernest Rutherford gave a model of the atom. In this model, the nucleus held most of the atom's mass and a positive charge. According to him, it was similar to half of the atom's atomic weight, expressed in numbers of hydrogen atoms.
## What Are Isotopes and the Mass Number?
Isotopes are chemical elements that have the same number of protons (Z) but different numbers of neutrons (N). The chemical properties of isotopes are very different in some cases. Around 75% of naturally occurring elements exist as a mixture of isotopes.
The mass number (A) of an atom is the sum of the atomic number (Z) and the number of neutrons (N).
## How Has the Atomic Number Evolved Throughout History?
At first, the atomic number was the position in which a chemical element remained when they were arranged in increasing order according to their atomic masses.
In 1913, Johannes H. van den Broek discovered that the number of elementary charges in the atomic nucleus was equal to the atomic number. Later, Niels Bohr adopted this discovery to develop his quantum theory about the structure of atoms and the origin of spectra. With this theory, he reflected in the well-known Bohr atomic model.
Author:
Published: May 20, 2019
Last review: April 27, 2021
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# Properties
Label 882.2.e.i Level $882$ Weight $2$ Character orbit 882.e Analytic conductor $7.043$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$
# Related objects
## Newspace parameters
Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 882.e (of order $$3$$, degree $$2$$, not minimal)
## Newform invariants
Self dual: no Analytic conductor: $$7.04280545828$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$
## $q$-expansion
Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.
$$f(q)$$ $$=$$ $$q + q^{2} + ( 2 - \zeta_{6} ) q^{3} + q^{4} + ( 2 - \zeta_{6} ) q^{6} + q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + q^{2} + ( 2 - \zeta_{6} ) q^{3} + q^{4} + ( 2 - \zeta_{6} ) q^{6} + q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} + 3 \zeta_{6} q^{11} + ( 2 - \zeta_{6} ) q^{12} -2 \zeta_{6} q^{13} + q^{16} + ( 3 - 3 \zeta_{6} ) q^{17} + ( 3 - 3 \zeta_{6} ) q^{18} + \zeta_{6} q^{19} + 3 \zeta_{6} q^{22} + ( 6 - 6 \zeta_{6} ) q^{23} + ( 2 - \zeta_{6} ) q^{24} + 5 \zeta_{6} q^{25} -2 \zeta_{6} q^{26} + ( 3 - 6 \zeta_{6} ) q^{27} + ( -6 + 6 \zeta_{6} ) q^{29} -4 q^{31} + q^{32} + ( 3 + 3 \zeta_{6} ) q^{33} + ( 3 - 3 \zeta_{6} ) q^{34} + ( 3 - 3 \zeta_{6} ) q^{36} + 4 \zeta_{6} q^{37} + \zeta_{6} q^{38} + ( -2 - 2 \zeta_{6} ) q^{39} -9 \zeta_{6} q^{41} + ( 1 - \zeta_{6} ) q^{43} + 3 \zeta_{6} q^{44} + ( 6 - 6 \zeta_{6} ) q^{46} -6 q^{47} + ( 2 - \zeta_{6} ) q^{48} + 5 \zeta_{6} q^{50} + ( 3 - 6 \zeta_{6} ) q^{51} -2 \zeta_{6} q^{52} + ( -12 + 12 \zeta_{6} ) q^{53} + ( 3 - 6 \zeta_{6} ) q^{54} + ( 1 + \zeta_{6} ) q^{57} + ( -6 + 6 \zeta_{6} ) q^{58} + 3 q^{59} + 8 q^{61} -4 q^{62} + q^{64} + ( 3 + 3 \zeta_{6} ) q^{66} + 5 q^{67} + ( 3 - 3 \zeta_{6} ) q^{68} + ( 6 - 12 \zeta_{6} ) q^{69} -12 q^{71} + ( 3 - 3 \zeta_{6} ) q^{72} + ( -11 + 11 \zeta_{6} ) q^{73} + 4 \zeta_{6} q^{74} + ( 5 + 5 \zeta_{6} ) q^{75} + \zeta_{6} q^{76} + ( -2 - 2 \zeta_{6} ) q^{78} -4 q^{79} -9 \zeta_{6} q^{81} -9 \zeta_{6} q^{82} + ( -12 + 12 \zeta_{6} ) q^{83} + ( 1 - \zeta_{6} ) q^{86} + ( -6 + 12 \zeta_{6} ) q^{87} + 3 \zeta_{6} q^{88} -6 \zeta_{6} q^{89} + ( 6 - 6 \zeta_{6} ) q^{92} + ( -8 + 4 \zeta_{6} ) q^{93} -6 q^{94} + ( 2 - \zeta_{6} ) q^{96} + ( -5 + 5 \zeta_{6} ) q^{97} + 9 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 3q^{3} + 2q^{4} + 3q^{6} + 2q^{8} + 3q^{9} + O(q^{10})$$ $$2q + 2q^{2} + 3q^{3} + 2q^{4} + 3q^{6} + 2q^{8} + 3q^{9} + 3q^{11} + 3q^{12} - 2q^{13} + 2q^{16} + 3q^{17} + 3q^{18} + q^{19} + 3q^{22} + 6q^{23} + 3q^{24} + 5q^{25} - 2q^{26} - 6q^{29} - 8q^{31} + 2q^{32} + 9q^{33} + 3q^{34} + 3q^{36} + 4q^{37} + q^{38} - 6q^{39} - 9q^{41} + q^{43} + 3q^{44} + 6q^{46} - 12q^{47} + 3q^{48} + 5q^{50} - 2q^{52} - 12q^{53} + 3q^{57} - 6q^{58} + 6q^{59} + 16q^{61} - 8q^{62} + 2q^{64} + 9q^{66} + 10q^{67} + 3q^{68} - 24q^{71} + 3q^{72} - 11q^{73} + 4q^{74} + 15q^{75} + q^{76} - 6q^{78} - 8q^{79} - 9q^{81} - 9q^{82} - 12q^{83} + q^{86} + 3q^{88} - 6q^{89} + 6q^{92} - 12q^{93} - 12q^{94} + 3q^{96} - 5q^{97} + 18q^{99} + O(q^{100})$$
## Character values
We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$$.
$$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$-\zeta_{6}$$
## Embeddings
For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.
For more information on an embedded modular form you can click on its label.
Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
373.1
0.5 − 0.866025i 0.5 + 0.866025i
1.00000 1.50000 + 0.866025i 1.00000 0 1.50000 + 0.866025i 0 1.00000 1.50000 + 2.59808i 0
655.1 1.00000 1.50000 0.866025i 1.00000 0 1.50000 0.866025i 0 1.00000 1.50000 2.59808i 0
$$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles
## Inner twists
Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner
## Twists
By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.e.i 2
3.b odd 2 1 2646.2.e.b 2
7.b odd 2 1 882.2.e.g 2
7.c even 3 1 18.2.c.a 2
7.c even 3 1 882.2.h.c 2
7.d odd 6 1 882.2.f.d 2
7.d odd 6 1 882.2.h.b 2
9.c even 3 1 882.2.h.c 2
9.d odd 6 1 2646.2.h.h 2
21.c even 2 1 2646.2.e.c 2
21.g even 6 1 2646.2.f.g 2
21.g even 6 1 2646.2.h.i 2
21.h odd 6 1 54.2.c.a 2
21.h odd 6 1 2646.2.h.h 2
28.g odd 6 1 144.2.i.c 2
35.j even 6 1 450.2.e.i 2
35.l odd 12 2 450.2.j.e 4
56.k odd 6 1 576.2.i.a 2
56.p even 6 1 576.2.i.g 2
63.g even 3 1 18.2.c.a 2
63.h even 3 1 162.2.a.c 1
63.h even 3 1 inner 882.2.e.i 2
63.i even 6 1 2646.2.e.c 2
63.i even 6 1 7938.2.a.i 1
63.j odd 6 1 162.2.a.b 1
63.j odd 6 1 2646.2.e.b 2
63.k odd 6 1 882.2.f.d 2
63.l odd 6 1 882.2.h.b 2
63.n odd 6 1 54.2.c.a 2
63.o even 6 1 2646.2.h.i 2
63.s even 6 1 2646.2.f.g 2
63.t odd 6 1 882.2.e.g 2
63.t odd 6 1 7938.2.a.x 1
84.n even 6 1 432.2.i.b 2
105.o odd 6 1 1350.2.e.c 2
105.x even 12 2 1350.2.j.a 4
168.s odd 6 1 1728.2.i.e 2
168.v even 6 1 1728.2.i.f 2
252.o even 6 1 432.2.i.b 2
252.u odd 6 1 1296.2.a.g 1
252.bb even 6 1 1296.2.a.f 1
252.bl odd 6 1 144.2.i.c 2
315.r even 6 1 4050.2.a.c 1
315.v odd 6 1 1350.2.e.c 2
315.bo even 6 1 450.2.e.i 2
315.br odd 6 1 4050.2.a.v 1
315.bt odd 12 2 4050.2.c.c 2
315.bv even 12 2 4050.2.c.r 2
315.bx even 12 2 1350.2.j.a 4
315.ch odd 12 2 450.2.j.e 4
504.w even 6 1 576.2.i.g 2
504.ba odd 6 1 576.2.i.a 2
504.bi odd 6 1 5184.2.a.q 1
504.bt even 6 1 5184.2.a.p 1
504.ce odd 6 1 5184.2.a.o 1
504.cq even 6 1 5184.2.a.r 1
504.cy even 6 1 1728.2.i.f 2
504.db odd 6 1 1728.2.i.e 2
By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.2.c.a 2 7.c even 3 1
18.2.c.a 2 63.g even 3 1
54.2.c.a 2 21.h odd 6 1
54.2.c.a 2 63.n odd 6 1
144.2.i.c 2 28.g odd 6 1
144.2.i.c 2 252.bl odd 6 1
162.2.a.b 1 63.j odd 6 1
162.2.a.c 1 63.h even 3 1
432.2.i.b 2 84.n even 6 1
432.2.i.b 2 252.o even 6 1
450.2.e.i 2 35.j even 6 1
450.2.e.i 2 315.bo even 6 1
450.2.j.e 4 35.l odd 12 2
450.2.j.e 4 315.ch odd 12 2
576.2.i.a 2 56.k odd 6 1
576.2.i.a 2 504.ba odd 6 1
576.2.i.g 2 56.p even 6 1
576.2.i.g 2 504.w even 6 1
882.2.e.g 2 7.b odd 2 1
882.2.e.g 2 63.t odd 6 1
882.2.e.i 2 1.a even 1 1 trivial
882.2.e.i 2 63.h even 3 1 inner
882.2.f.d 2 7.d odd 6 1
882.2.f.d 2 63.k odd 6 1
882.2.h.b 2 7.d odd 6 1
882.2.h.b 2 63.l odd 6 1
882.2.h.c 2 7.c even 3 1
882.2.h.c 2 9.c even 3 1
1296.2.a.f 1 252.bb even 6 1
1296.2.a.g 1 252.u odd 6 1
1350.2.e.c 2 105.o odd 6 1
1350.2.e.c 2 315.v odd 6 1
1350.2.j.a 4 105.x even 12 2
1350.2.j.a 4 315.bx even 12 2
1728.2.i.e 2 168.s odd 6 1
1728.2.i.e 2 504.db odd 6 1
1728.2.i.f 2 168.v even 6 1
1728.2.i.f 2 504.cy even 6 1
2646.2.e.b 2 3.b odd 2 1
2646.2.e.b 2 63.j odd 6 1
2646.2.e.c 2 21.c even 2 1
2646.2.e.c 2 63.i even 6 1
2646.2.f.g 2 21.g even 6 1
2646.2.f.g 2 63.s even 6 1
2646.2.h.h 2 9.d odd 6 1
2646.2.h.h 2 21.h odd 6 1
2646.2.h.i 2 21.g even 6 1
2646.2.h.i 2 63.o even 6 1
4050.2.a.c 1 315.r even 6 1
4050.2.a.v 1 315.br odd 6 1
4050.2.c.c 2 315.bt odd 12 2
4050.2.c.r 2 315.bv even 12 2
5184.2.a.o 1 504.ce odd 6 1
5184.2.a.p 1 504.bt even 6 1
5184.2.a.q 1 504.bi odd 6 1
5184.2.a.r 1 504.cq even 6 1
7938.2.a.i 1 63.i even 6 1
7938.2.a.x 1 63.t odd 6 1
## Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(882, [\chi])$$:
$$T_{5}$$ $$T_{11}^{2} - 3 T_{11} + 9$$ $$T_{13}^{2} + 2 T_{13} + 4$$
## Hecke characteristic polynomials
$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$3 - 3 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$9 - 3 T + T^{2}$$
$13$ $$4 + 2 T + T^{2}$$
$17$ $$9 - 3 T + T^{2}$$
$19$ $$1 - T + T^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$16 - 4 T + T^{2}$$
$41$ $$81 + 9 T + T^{2}$$
$43$ $$1 - T + T^{2}$$
$47$ $$( 6 + T )^{2}$$
$53$ $$144 + 12 T + T^{2}$$
$59$ $$( -3 + T )^{2}$$
$61$ $$( -8 + T )^{2}$$
$67$ $$( -5 + T )^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$121 + 11 T + T^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$144 + 12 T + T^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$25 + 5 T + T^{2}$$
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# Is log(n!) = Θ(n·log(n))?
This is a homework question. I'm not expecting an answer, just some guidance, possibly :) I am to show that log(n!) = Θ(n·log(n)).
A hint was given that I should show the upper bound with nn and show the lower bound with (n/2)(n/2). This does not seem all that intuitive to me. Why would that be the case? I can definitely see how to convert nn to n·log(n) [log both sides of an equation], but that's kind of working backwards. What would be the correct approach to tackle this problem? Should I draw the recursion tree? There is nothing recursive about this, so that doesn't seem like a likely approach..
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+1 for honesty on homework and not asking for the 'codez' :) – Andrew Jan 19 '10 at 17:22
MY BRAIN HURTS! – Tad Donaghe Jan 19 '10 at 17:32
Did you want to write `O(log(n!))`? I guess you want to have the upper run-time complexity. – Georg Schölly Jan 19 '10 at 17:53
You should really write it including the "as n -> ∞ " – CodeByMoonlight Jan 19 '10 at 18:49
Fun exercise: use the similar trick to show that the harmonic series 1/1 + 1/2 + 1/3 + 1/4 + ... diverges to infinity. – Yoo Jan 19 '10 at 22:54
Remember that
``````log(n!) = log(1) + log(2) + ... + log(n-1) + log(n)
``````
You can get the upper bound by
``````log(1) + log(2) + ... + log(n) <= log(n) + log(n) + ... + log(n)
= n*log(n)
``````
And you can get the lower bound by doing a similar thing after throwing away the first half of the sum:
``````log(1) + ... + log(n/2) + ... + log(n) >= log(n/2) + ... + log(n)
>= log(n/2) + ... + log(n/2)
= n/2 * log(n/2)
``````
-
Interesting... Pushing this into a sum, there is an approximation that results in the theta notation itself (source: en.wikipedia.org/wiki/Summation) a matter of how that is derived is entirely separate. Nice lead, thanks! – Mark Jan 19 '10 at 17:40
This is a very nice proof for the upper bound: log(n!) = log(1) + ... + log(n) <= n log(n) => log(n!) = O(n log n). However, for proving the lower bound (and consequently big-tetha), you'll probably need Stirling's Approximation. – LeakyCode Jan 19 '10 at 20:34
You don't need Sterling's approximation for a lower bound. log(n!) = log(1) + ... + log(n) >= log(n/2) + ... + log(n) >= n/2 * log(n/2) = Omega(n log n). – Keith Randall Jan 19 '10 at 22:40
nice trick, I think it works. – ldog Jan 20 '10 at 2:23
@j_random_hacker: `log(n/2) + log(n/2 + 1) + ... + log(n - 1) + log(n)` (larger half of the terms of `log(n!)`). Actually, I just read the question and saw that the clue is stated in the question. Basically, `(n/2)^(n/2) <= n! <= n^n` => `log((n/2)^(n/2))<=log(n!)<=log(n^n)` => `Θ(n/2 * log(n/2))<=log(n!)<=Θ(n*log(n))` – LeakyCode Jan 21 '10 at 14:07
I realize this is a very old question with an accepted answer, but none of these answers actually use the approach suggested by the hint.
It is a pretty simple argument:
`n!` (= 1*2*3*...*n) is a product of `n` numbers each less than or equal to `n`. Therefore it is less than the product of `n` numbers all equal to `n`; i.e., `n^n`.
Half of the numbers -- i.e. `n/2` of them -- in the `n!` product are greater than or equal to `n/2`. Therefore their product is greater than the product of `n/2` numbers all equal to `n/2`; i.e. `(n/2)^(n/2)`.
Take logs throughout to establish the result.
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This is actually just the same as the log version in the accepted answer but taking the logarithm after instead of before. (it more clearly uses the hint though) – missingno Nov 3 '11 at 4:21
Helping you further, where Mick Sharpe left you:
It's deriveration is quite simple: see http://en.wikipedia.org/wiki/Logarithm -> Group Theory
log(n!) = log(n * (n-1) * (n-2) * ... * 2 * 1) = log(n) + log(n-1) + ... + log(2) + log(1)
Think of n as infinitly big. What is infinite minus one? or minus two? etc.
log(inf) + log(inf) + log(inf) + ... = inf * log(inf)
And then think of inf as n.
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BigO is abut asymptotically. For qsort you can say that it can sort in n! order or less. That is true. But as I said BigO is about asymptotically.
In that case you should prove that some function lie between two functions and that there is lower and upper limit, and this helps you prove properties.
-
This might help:
```eln(x) = x
```
and
```(lm)n = lm*n
```
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Actually, that's wrong: 1^(m^n) != 1^(m*n) it must be (1^m)^n = 1^(m*n) – Pindatjuh Jan 19 '10 at 18:18
Errr I mean L instead of 1 in the above comment. – Pindatjuh Jan 19 '10 at 18:35
He didn't write 1^(m^n) he wrote (l^m)^n – Imray Jan 17 '13 at 20:57
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# Question on Moment of a force
dorothy
Homework Statement:
I want to know whether I draw the free body diagram correctly? Thank you.
T = tension
f = friction
W = gravitational force
Relevant Equations:
/
dorothy
Homework Helper
Gold Member
The forces that you have drawn look good. But you have left out a force. See if you can spot the missing force.
Also, you will want to explain from your diagram why there must be a friction force at point A.
dorothy
The forces that you have drawn look good. But you have left out a force. See if you can spot the missing force.
Also, you will want to explain from your diagram why there must be a friction force at point A.
Is it the normal reaction missing?
Homework Helper
Gold Member
Is it the normal reaction missing?
Yes. To be sure that you are thinking correctly, I would need to see where you place the force and the direction of the force.
dorothy
Yes. To be sure that you are thinking correctly, I would need to see where you place the force and the direction of the force.
R stands for the normal reaction. Is it correct? also, I’ve added a new T for tension. Is the T in red needed？
Homework Helper
Gold Member
The reaction force R should be a normal force acting normal to the wall. I assume there is a wall.
The red tension force is not a force acting on the plank. So, it would not be included in a free-body diagram of the plank.
If this is a graded assignment, can you give us assurance that you are allowed to get external help? (I should have asked this at the beginning )
dorothy
The reaction force R should be a normal force.
The red tension force is not a force acting on the plank. So, it would not be included in a free-body diagram of the plank.
If this is a graded assignment, can you give us assurance that you are allowed to get external help? (I should have asked this at the beginning )
It’s not a graded assignment. It’s just a simple exercise because now I’m in vacation so I would like to do some more practice to train myself 😅
TSny
dorothy
The reaction force R should be a normal force acting normal to the wall. I assume there is a wall.
The red tension force is not a force acting on the plank. So, it would not be included in a free-body diagram of the plank.
If this is a graded assignment, can you give us assurance that you are allowed to get external help? (I should have asked this at the beginning )
So I do draw the R correctly?
Homework Helper
Gold Member
So I do draw the R correctly?
No.
My interpretation of the diagram is that the left end of the plank is up against something like a wall. The wall exerts a force against the left end of the plank. This force from the wall has a component parallel to the wall (this is the friction force) and a component normal to the wall.
Homework Helper
Gold Member
Looks very good. Can you use this diagram to explain why the friction force cannot be zero and why it must be in the upward direction?
Last edited:
dorothy
Looks very good. Can you use this diagram to explain why the friction force cannot be zero and why it must be in the upward direction?
Is it because ‘friction has to be there to balance the gravitational force W ’?
Homework Helper
Gold Member
Is it because ‘friction has to be there to balance the reaction force in order to stay at rest’?
No. The friction force is vertical while the reaction force is horizontal. Since they are in different directions they cannot balance each other.
What are the basic conditions that the forces and the torques of the forces must obey in order for the plank to be in equilibrium?
dorothy
Is it because ‘friction has to be there to balance the gravitational force W ’?
I‘ve just change my thought , is it correct in this case?
dorothy
What are the basic conditions that the forces and the torques of the forces must obey in order for the plank to be in equilibrium?
The two forces should be parallel and with equal magnitude but opposite in direction
Homework Helper
Gold Member
The two forces should be parallel and with equal magnitude but opposite in direction
There are four forces acting on the plank. What condition or relation must be obeyed by these four forces if the plank is in equilibrium?
Is there also a condition involving torques?
Homework Helper
Gold Member
Take a look at your FBD and see whether or not each force has a counterpart which prevents the plank from moving in any direction, as it actually happens.
Remember, summation of forces along x-axis and y-axis must be zero, if balance exists.
If any of those forces is located at certain distance from a pivot point, it could make the body rotate about that point (not a balance condition); therefore, something must prevent that from happening.
dorothy
There are four forces acting on the plank. What condition or relation must be obeyed by these four forces if the plank is in equilibrium?
Is there also a condition involving torques?
Can I answer this question in this way?
dorothy
There are four forces acting on the plank. What condition or relation must be obeyed by these four forces if the plank is in equilibrium?
Is there also a condition involving torques?
Net force=0?
dorothy
There are four forces acting on the plank. What condition or relation must be obeyed by these four forces if the plank is in equilibrium?
Is there also a condition involving torques?
Torque can be calculated by 5f?
Homework Helper
Gold Member
Can I answer this question in this way?View attachment 299700
No.
The tension force, T, has a vertical component. Perhaps this vertical component of T happens to cancel W. If so, the friction would not be needed. However, there is a way to show that f needed. Think about torques.
Homework Helper
Gold Member
Torque can be calculated by 5f?
I think we need to remember the basic conditions required for equilibrium:
(1) The vector sum of all of the forces acting on the plank must be zero.
(2) The sum of all of the torques acting on the plank must be zero for any choice of the location of the origin for calculating the torques.
Condition (1) can be restated as saying two things:
(1a) The sum of the horizontal components of the forces must add to zero.
(1b) The sum of the vertical components of the forces must add to zero.
We've already seen that (1b) doesn't give us enough information to decide whether or not the friction force is required. And (1a) won't be of any help since it deals with horizontal forces while f is vertical.
Lnewqban
dorothy
I think we need to remember the basic conditions required for equilibrium:
(1) The vector sum of all of the forces acting on the plank must be zero.
(2) The sum of all of the torques acting on the plank must be zero for any choice of the location of the origin for calculating the torques.
Condition (1) can be restated as saying two things:
(1a) The sum of the horizontal components of the forces must add to zero.
(1b) The sum of the vertical components of the forces must add to zero.
We've already seen that (1b) doesn't give us enough information to decide whether or not the friction force is required. And (1a) won't be of any help since it deals with horizontal forces while f is vertical.
I get the concept but I don’t know how to explain it in Q1 🥲
Homework Helper
Gold Member
I get the concept but I don’t know how to explain it in Q1 🥲
Try picking point B as an origin for the torques.
(1) Does the tension force T produce any torque about B? If so, is the torque clockwise or counterclockwise?
(2) Does the weight W produce any torque about B? If so, is the torque clockwise or counterclockwise?
(3) Does the reaction force R produce any torque about B? If so, is the torque clockwise or counterclockwise?
Based on your answers to these three questions, can you argue that there must be a friction force at A and that the friction force must be upward?
dorothy
Try picking point B as an origin for the torques.
(1) Does the tension force T produce any torque about B? If so, is the torque clockwise or counterclockwise?
(2) Does the weight W produce any torque about B? If so, is the torque clockwise or counterclockwise?
(3) Does the reaction force R produce any torque about B? If so, is the torque clockwise or counterclockwise?
Based on your answers to these three questions, can you argue that there must be a friction force at A and that the friction force must be upward?
1. Yes, clockwise
2. Yes, anti-clockwise
3. No
Ohhh is it because we need the moment due to f to balance with the moment due to W in order to reach the equilibrium?
Homework Helper
Gold Member
1. Yes, clockwise
2. Yes, anti-clockwise
3. No
Ohhh is it because we need the moment due to f to balance with the moment due to W in order to reach the equilibrium?
You have the right idea
But, the tension force T does not produce any torque about the origin at B. Can you see why?
This is important in order to be able to logically deduce that there must be an upward friction force.
dorothy
You have the right idea
But, the tension force T does not produce any torque about the origin at B. Can you see why?
This is important in order to be able to logically deduce that there must be an upward friction force.
because the force T is pointing upward and it doesn’t “push” the plank?
Homework Helper
Gold Member
because the force T is pointing upward and it doesn’t “push” the plank?
No.
Have you covered the concept of "line of action of a force"? What can you say about the torque of a force about some origin if the line of action of the force passes through the origin?
I'm curious as to your reason for stating (correctly!) that the torque due to R about point B is zero.
dorothy
No.
Have you covered the concept of "line of action of a force"? What can you say about the torque of a force about some origin if the line of action of the force passes through the origin?
I'm curious as to your reason for stating (correctly!) that the torque due to R about point B is zero.
oh because I think that T is pointing upward so it won’t pass B. But for R, it is pointing horizontally and will pass B so its torque is 0.
Homework Helper
Gold Member
oh because I think that T is pointing upward so it won’t pass B. But for R, it is pointing horizontally and will pass B so its torque is 0.
Good reasoning for R, but not for T. If you drew a line through T, the line would pass through B. So, just like R, the line of action passes through B. So, the torque about B due to T is zero.
It should be clear that: any force that acts at the origin would not produce any torque about that origin since the line of action of the force clearly passes through the origin.
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http://math.stackexchange.com/questions/138792/cardinality-does-ab-2b-if-a-le-2b/138810
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# Cardinality: does $|A^B|=|2^B|$ if $|A| \le |2^B|$?
Taking finite powers of a countably infinite set still yields a countable set: $$|\mathbb{N}|=|\mathbb{N}^k|.$$
It's also known that countable powers of the continuum still have the same cardinality as the continuum: $$|2^\mathbb{N}|=|\mathbb{R}|=|\mathbb{R}^k|=|\mathbb{R}^\mathbb{N}|.$$
Furthermore, taking the next highest ordinal to the power of the continuum doesn't seem to change it. (at least according to wikipedia) Denote $X:=2^\mathbb{R}$, then $$|2^\mathbb{R}|=|X|=|X^k|=|X^\mathbb{N}|=|X^\mathbb{R}|$$
See a pattern?
It seems that when taking one set to the power of another set, it doesn't really matter how big the exponent is so long as it is smaller than the base. Or thinking about it in terms of the other variable, if the exponent isn't too big then you might as well replace the base by 2. Ie., $$|A| \le |2^B| \Rightarrow |A^B|=|2^B|.$$
Is there a way in which this pattern can be stated and proved rigorously, or is it just a coincidence?
-
If $B$ is infinite, then yes: $|A^B|\leq |(2^B)^B|=|2^{B\times B}|=|2^B|$ while the other direction is trivial.
-
Could you elaborate a little bit on that? What is the set $B \times B$ in the formula $2^{B \times B}$ (cartesian product?), and how do we know $|2^{B \times B}|=|2^B|$? If this is too simplistic a reference or link to something explaining it would be great! – Nick Alger Apr 30 '12 at 8:44
@NickAlger The set $B\times B$ is the Cartesian product, and $|2^{B\times B}|=|2^B|$ since for any infinite cardinal $B$, $|B|=|B\times B|$. You have to be a little careful here showing that bijections between sets correspond to bijections between power sets, etc. though. Some good basic info on Cardinal arithmetic can be found on Wikipedia. – Alex Becker Apr 30 '12 at 8:53
Thanks, much appreciated. – Nick Alger Apr 30 '12 at 8:58
The fact that $a^2=a$ for infinite cardinals is shown in answer to this question. – Martin Sleziak Apr 30 '12 at 9:21
Interesting that it is equivalent to the axiom of choice... I tried to construct a proof just now but was getting stuck so that link was very useful. – Nick Alger Apr 30 '12 at 9:40
We must require $A$ to not be $0$ or $1$! In the first case, $|A^B|$ is 0, while in the second case, $|A^B|$ is 1. (Note there are no functions from $B$ to the empty set, and there is one function from $B$ to a set with one element.) Then $|A^B|$ is strictly less than $|2^B|$ when $B$ is $1$ or greater.
Categorize this under "silly but true" :)
-
Yes, I realized this and debated whether or not to mention it. I'm always at a loss whether to include little edge case technicalities in my posts. If I do it clutters things up and obscures the main idea. If I don't, someone always comes in and "corrects" me for leaving it out... Thank you for the post though, perhaps it will be useful to others reading in the future. – Nick Alger May 4 '12 at 10:04
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https://www.geekedu.org/course/calculus
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# Calculus
This course is a comprehensive study of single-variable calculus. It builds on the topics covered in Pre-Calculus and is the basis for theorem and problem solving that are covered in high school calculus class. It is essential for students looking to further their education with AP Calculus. Specific topics covered include limits, continuity, derivatives, integrals, power series, plane curves, derivative integral series, and differential equations.
###### Chapter 1: Prerequisites for Calculus
Parametric Equations, Review of Pre-Cal 12, including exponential and logarithms, trigonometric functions
###### Chapter 2: Limits and Continuity
Limits, Continuity, instantaneous Rate of change
###### Chapter 3: Derivatives
Definition, differentiability, rules for differentiation, chain rule, implicit differentiation, derivative of trigonometric, inverse trigonometric, exponential, and logarithms. Applications in kinematics
###### Chapter 4: Applications of derivatives
Extreme values of functions, mean value theorem, drawing graph of derivatives, optimization, linearization and newton’s method
###### Chapter 5: Definite Integral
Estimation of integral, definition, antiderivatives, fundamental theorem of calculus
###### Chapter 6: Differential Equations and Mathematical Modeling
Slope Fields, integration by substitution and by parts, exponential growth and decay, logistic growth
###### Chapter 7: Applications of definite integrals
Areas in the plane, volumes, lengths of curves
###### Chapter 8: Sequences, L’Hopital’s rule, improper integrals
Sequences, l”hopital’s rule, relative rates of growth, improper integrals
###### Chapter 9: Infinite Series
Power Series, Taylor Series, Tayler’s theorem, radius of convergence, convergence tests
###### Chapter 10: Parametric, Vector, and Polar Functions
Parametric Functions, Vectors, Polar Functions
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Passion for tech. Unmatched expertise. Personality that brings out the absolute best in your child. We know there's no replacement for the best teacher. It's no surprise that iconic companies like Google, Amazon, Microsoft, and Disney hire the same talent we do. Our Geek Certified instructors possess the whole package of skills that inspires students for the Tech World!
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## What our parents say
This course is for students who have completed our Pre-Calculus course.
Level:
AGES 15+
Online 1-on-1 Private Sessions
This course typically takes 3-6 months to complete. This depends on the student's experience and how fast they can master the concepts and the knowledge. Our instructors move at the pace of the student, it may take extra time for some students to reinforce what they have learned.
### You Might Also Be Interested In Our Elective Courses
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AGES 10 - 15
Designed for beginners, this level is designed to teach the basic fundamentals and design principles of Python, with the help of Turtle graphics and PyGame. This includes: variables, conditional statements, loop basics, and functions.
These concepts are transferable to any other programming language. Throughout the way, students will create projects in order to apply the concepts they have learned, and to solidify their knowledge.
##### Python Level 2
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Level 2 dives deeper into the basics of Python for a more thorough understanding and introduces advanced topics. Students will build on knowledge from Level 1 and work with data structures, advanced loops, algorithms, and object-oriented programming, and create games based on what they learn.
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This beginner-friendly course serves as an introduction to the Java programming language. Students will learn the fundamentals of Java along with core computer science concepts.
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https://www.jpl.nasa.gov/edu/teach/activity/roving-on-the-moon/
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Activity Notes
### Overview
In this challenge, students build a rubber-band-powered rover that can scramble across the room. Students will follow the engineering design process to design and build a rover out of cardboard, figure out how to use rubber bands to spin the wheels, and improve their design based on testing results.
### Management
• Read the challenge sheet and activity details to become familiar with the activity
• Gather the materials for each rover
• Build a sample rover
### Procedures
1. Introduce the challenge (5 minutes)
• Tell kids some of the ways rovers will be used on the moon:
NASA plans to land astronauts on the moon by the year 2024. The astronauts will need moon cars—called rovers—to drive across the moon’s surface, carry supplies, help build their outpost, and explore the area. Today you’ll build and test a rubber band-powered rover.
• Show kids your sample rover and tell them:
This is a prototype of a rover, just like the one you are going to build. Prototypes are used all the time in engineering. They give you a basic design to build, test, and evaluate. Once you understand a design’s strengths and weaknesses, you can then find ways to improve it. Today, for example, as you test your rover prototype, you’ll find ways to make it work better. Improving a design based on testing is called the engineering design process.
2. Brainstorm and design (10 minutes) - Get kids thinking about the rover prototype. Ask:
• What do we have to do to make the rover move?
Turn the wheels to wind up the rubber band. Place the rover on the floor. Then let go. Note: Depending on the direction you wind the wheels, the rover can move either forward or backward.
• How can you make different kinds of wheels?
Kids can make different-sized wheels by cutting larger or smaller squares or make different-shaped wheels by trimming the squares. NOTE: Square wheels offer two advantages: they’re quick to make, and it’s easy to find the exact center by drawing diagonal lines. The center is where the lines cross.
• How do you think square wheels affect how the rover moves across the floor?
The points of the squares dig into soft surfaces, such as rugs, sand, or grass. This improves traction—the ability to grip a surface—and helps prevent the wheels from spinning out. Since the moon is covered in a thick layer of fine dust, good traction is essential, especially going up and down hills.
3. Build, test, evaluate and redesign (35 minutes)
• Distribute the challenge sheet and instruct students to follow the directions in the "Build" section.
1. Make the rover body - Fold the cardboard into thirds. Each part will be about 2 inches (5 cm) across. Fold along (not across) the corrugation (the tubes inside the piece of cardboard).
2. Make the front wheels - On the two 5-inch (13-cm) cardboard squares, draw diagonal lines from corner to corner. Poke a small hole in the center (that's where the lines cross). On the body, poke one hole close to the end of each side for the axle. Make sure the holes are directly across from each other and are big enough for the pencil to spin freely.
3. Attach the front wheels - Slide the pencil through the body's axle holes. Push a wheel into each end. Secure with tape.
4. Make the rear - Tape the straw under the back end of the rover. Slip a candy onto each end. Bend and tape the axle to stop the candies from coming off.
5. Attach the rubber band - Loop one end around the pencil. Cut small slits into the back end of the body. Slide the free end of the rubber bands into the slits.
• Help students with any of the following issues. For example, if the rover:
• Wheels don't turn freely - Make sure they are firmly attached to the axles and are parallel to the sides. Also make sure the holes punched in the cardboard body are directly across from one another and are large enough to allow the pencil to turn easily.
• Won't travel in a straight line - Make sure the axles are straight and the front wheels are the same size. If one wheel is smaller, the rover will turn in that direction.
• Doesn't go far - Have kids wind up the wheels more. Also have them try using larger wheels. Bigger wheels have a larger perimeter (outer edge). As a result, one rotation of a large wheel will move the rover farther than one rotation of a small wheel.
• Wheels spin out - Wheels spin in place when a rubber band delivers too much power at once or when there’s not enough friction between the wheels and ground. To increase friction, have kids add weight over the drive wheels or add more wheels to each axle. To reduce how quickly a rubber band releases its power, kids can reduce tension by using a rubber-band chain or by cutting open a rubber band and using only a single strand of elastic.
4. Discuss what happened (10 minutes) - Have the kids show each other their rovers and talk about how they solved any problems that came up. Emphasize the key ideas in the challenge by asking the questions in the discussion section below.
### Discussion
• What kinds of Earth vehicles are similar to rovers?
Snowmobiles, tanks, dune buggies, and all-terrain vehicles are similar. They all have good traction, are very stable, have powerful engines, and don’t require a roadway.
• The challenge sheet gave you a rover prototype to get started with. How did starting with a prototype help you end up with a rover that worked really well?
With a prototype, kids can quickly see what’s working and what isn’t. They then know where to make improvements.
• How did friction affect your rover?
To be efficient, there needs to be minimal friction between the axle and the axle hole in the cardboard. To move, there needs to be lots of friction between the wheels and the ground.
• How did the rover use potential and kinetic energy?
Potential energy is energy that is stored. Kinetic energy is the energy of motion. Winding the front wheels increased the amount of potential energy stored by the rubber band. When the wheels spin, this potential energy is turned into kinetic energy, and the axle and wheels turn.
• How does the story about rover wheels on the back of the handout make you think about what it takes to design a wheel that can work on the moon?
Kids see that engineers face special design challenges when developing equipment to be used in space.
### Assessment
• Refer to the engineering rubric.
• Incorporate peer reviews for the design and code of the microdevices as part of the assessment to promote student cross-communication.
### Extensions
• Graph how increased potential energy affects distance traveled.
Kids can measure how far a rover travels as its rubber band is increasingly tightened. Have them turn the wheels 3, 6, 9, and 12 times and then measure the distance the rover travels each time. On a graph, have them plot the number of wheel rotations vs. the distance traveled. (Winding the wheels more increases the potential energy, which should increase the distance.)
• Determine the effect of friction.
Have kids wind up the wheels a set number of times and measure the distance their rover travels. Then have them minimize friction in the wheel-axle system. For example, they can line the axle holes with a material such as aluminum foil, then wind up the wheels the same number of times and retest their rovers. Use the following formula to calculate the percent increase in distance traveled:
Percent Increase = [(Distance modified rover traveled) - (Distance basic rover traveled) / Distance basic rover traveled] x 100
• Test the effect of wheel shape.
Starting with square wheels, have kids measure how far their rovers travel. Then have them snip off the corners of their wheels and test again. Make sure they wind up the wheels the same number of turns. How did the distance change? Did the wheels spin out? Test square, octagonal, and round wheels.
• Calculate the strength of your rubber band.
For high school students, have your class utilize the conservation of energy principle to determine the elastic constant of their rover systems. First, have students determine the mass of their rover and the time it took to drive across the room (velocity equals distance over time). Now students can use the conversion of kinetic energy from elastic energy to solve for elastic strength. This can be done using the equation ½ mv2 = ½ kx2
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https://worlddatabaseofhappiness.eur.nl/distributional-findings/burger-veenhoven-2016-study-ne-2015-13446/
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# Distributional findings
## Study Burger & Veenhoven (2016): study NE 2015
Public
15+ aged, general public, Niger, 2015
Survey name
INT-GallupWorldPoll2015
Sample
Probability multi-stage random
Respondents
N = 997
Non Response
Assessment
Interview: face-to-face
## Happiness measure(s)
Full text:
Self report on single question:
Here is ladder representing the 'ladder of life'. Let's suppose the top of the ladder represents the best possible life for you; and the bottom, the worst possible life for you. On which step of the ladder do you feel you personally stand at the present time?
10 best possible
9
8
7
6
5
4
3
2
1
0 worst possible life
This question was followed (not preceded) by items on life 5 years ago and 5 years from now.
Classification:
C-BW-c-sq-l-11-c
Observed distribution
Summary Statistics
On original range 0 - 10 On range 0-10
Mean:
3.70 3.70
SD:
- -
Full text:
Self report on 10 questions:
A Did you feel well-rested yesterday?
B Were you treated with respect yesterday?
C Did you smile and laugh a lot yesterday
D Did you learn or do something interesting yesterday?
Did you experience the following feelings during A LOT of the day yesterday?
Rated:
0: no
1: yes
Computation:
- Average % positive affect = (A+B+C+D+E)/5
- Average % negative affect = (F+G+H+I+J)/5
- Affect Balance = Average % positive affect - Average % negative affect
Transformation to range 0-10:
(Score[range-100-100] + 100)/200*10
These questions do not measure hedonic level of affect of individual persons adequately, since yesterday's mood does not always correspond with typical average mood. Yet these questions can be used for measuring hedonic level in aggregates, such as nations, since individual variations balance out in big samples.
Transformation to 0-10 scale: (percent abs)*0.05 + 5
Classification:
A-AB-yd-mq-v-2-mb
Observed distribution
Summary Statistics
On original range -100 - 100 On range 0-10
Mean:
42.00 7.10
SD:
- -
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https://nl.webqc.org/molecularweightcalculated-190423-32.html
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#### Chemical Equations Balanced on 04/23/19
Molecular weights calculated on 04/22/19 Molecular weights calculated on 04/24/19
Calculate molecular weight
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170
Molar mass of FeSO4 is 151,9076
Molar mass of Au is 196.966569
Molar mass of fe2o3 is 159.6882
Molar mass of fe2o3 is 159.6882
Molar mass of Na2CO3 is 105.98843856
Molar mass of Zn(NO3)2 is 189.3898
Molar mass of SrO is 103.6194
Molar mass of Mg3(PO4)2 is 262,857724
Molar mass of MgO is 40.3044
Molar mass of MgO is 40.3044
Molar mass of H2O is 18.01528
Molar mass of AgCl is 143.3212
Molar mass of CaO is 56.0774
Molar mass of CaO is 56.0774
Molar mass of (NH4)2 SO4 is 132.13952
Molar mass of CaO is 56.0774
Molar mass of C6H3Cl3O is 197,44642
Molar mass of Br2 is 159.808
Molar mass of SiO2 is 60.0843
Molar mass of SiO2 is 60.0843
Molar mass of c12 is 144.1284
Molar mass of Al2O3 is 101,9612772
Molar mass of C3H3n is 53.06262
Molar mass of C3H3n is 53.06262
Molar mass of Al2O3 is 101,9612772
Molar mass of (COOH)2 is 90.03488
Molar mass of LiCl is 42.394
Molar mass of Al2O3 is 101,9612772
Molar mass of CH3OCH2CH2OCH(CH3)CH2OCH2CH(NH2)CH3 is 191.26794
Molar mass of BeI2 is 262.821122
Molar mass of (COOH) is 45.01744
Molar mass of He is 4.002602
Molar mass of CaCO3 is 100.0869
Molar mass of H420 is 423,3348
Molar mass of Fe2O3 is 159,6882
Molar mass of ZnCO3 is 125.3889
Molar mass of Fe2O3 is 159,6882
Molar mass of OC3H6 is 58.07914
Molar mass of (COOH) is 45.01744
Molar mass of Fe2 is 111,69
Molar mass of Fe2 is 111,69
Molar mass of C7H5NO3S is 183,1845
Molar mass of (NH4)2CO3 is 96.08582
Molar mass of o3 is 47,9982
Molar mass of MgH2 is 26.32088
Molar mass of H2O2C2 is 58,03608
Molar mass of NaH2PO4 is 119,97701128
Molar mass of Co is 58.933195
Molar mass of C3H8O is 60,09502
Molar mass of Cr2 is 103,9922
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170
Calculate molecular weight
Molecular weights calculated on 04/22/19 Molecular weights calculated on 04/24/19
Molecular masses on 04/16/19
Molecular masses on 03/24/19
Molecular masses on 04/23/18
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# dy.d_: Partial Derivative dy/d_[wrt] In NNS: Nonlinear Nonparametric Statistics
dy.d_ R Documentation
## Partial Derivative dy/d_[wrt]
### Description
Returns the numerical partial derivative of `y` with respect to [wrt] any regressor for a point of interest. Finite difference method is used with NNS.reg estimates as `f(x + h)` and `f(x - h)` values.
### Usage
```dy.d_(x, y, wrt, eval.points = "obs", mixed = FALSE, messages = TRUE)
```
### Arguments
`x` a numeric matrix or data frame. `y` a numeric vector with compatible dimensions to `x`. `wrt` integer; Selects the regressor to differentiate with respect to (vectorized). `eval.points` numeric or options: ("obs", "apd", "mean", "median", "last"); Regressor points to be evaluated. Numeric values must be in matrix or data.frame form to be evaluated for each regressor, otherwise, a vector of points will evaluate only at the `wrt` regressor. See examples for use cases. Set to `(eval.points = "obs")` (default) to find the average partial derivative at every observation of the variable with respect to for specific tuples of given observations. Set to `(eval.points = "apd")` to find the average partial derivative at every observation of the variable with respect to over the entire distribution of other regressors. Set to `(eval.points = "mean")` to find the partial derivative at the mean of value of every variable. Set to `(eval.points = "median")` to find the partial derivative at the median value of every variable. Set to `(eval.points = "last")` to find the partial derivative at the last observation of every value (relevant for time-series data). `mixed` logical; `FALSE` (default) If mixed derivative is to be evaluated, set `(mixed = TRUE)`. `messages` logical; `TRUE` (default) Prints status messages.
### Value
Returns column-wise matrix of wrt regressors:
• `dy.d_(...)[, wrt]\$First` the 1st derivative
• `dy.d_(...)[, wrt]\$Second` the 2nd derivative
• `dy.d_(...)[, wrt]\$Mixed` the mixed derivative (for two independent variables only).
### Note
For binary regressors, it is suggested to use `eval.points = seq(0, 1, .05)` for a better resolution around the midpoint.
### Author(s)
Fred Viole, OVVO Financial Systems
### References
Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" https://www.amazon.com/dp/1490523995/ref=cm_sw_su_dp
Vinod, H. and Viole, F. (2020) "Comparing Old and New Partial Derivative Estimates from Nonlinear Nonparametric Regressions" https://www.ssrn.com/abstract=3681104
### Examples
```## Not run:
set.seed(123) ; x_1 <- runif(1000) ; x_2 <- runif(1000) ; y <- x_1 ^ 2 * x_2 ^ 2
B <- cbind(x_1, x_2)
## To find derivatives of y wrt 1st regressor for specific points of both regressors
dy.d_(B, y, wrt = c(1, 2), eval.points = t(c(.5, .5)))
## To find average partial derivative of y wrt 1st regressor,
only supply 1 value in [eval.points], or a vector of [eval.points]:
dy.d_(B, y, wrt = 1, eval.points = .5)
dy.d_(B, y, wrt = 1, eval.points = fivenum(B[,1]))
## To find average partial derivative of y wrt 1st regressor,
for every observation of 1st regressor:
apd <- dy.d_(B, y, wrt = 1, eval.points = "apd")
plot(B[,1], apd[,1]\$First)
## 95% Confidence Interval to test if 0 is within
### Lower CI
LPM.VaR(.025, 0, apd[,1]\$First)
### Upper CI
UPM.VaR(.025, 0, apd[,1]\$First)
## End(Not run)
```
NNS documentation built on Nov. 4, 2022, 1:06 a.m.
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DISCOVER
# How to calculate angles in carpentry
Updated February 21, 2017
Building a roof or cutting stringers for a stairway requires finding angles, marking and cutting them correctly. In carpentry, you usually do away with trigonometric functions to find angles. Instead, you simply use two legs of a triangle to measure and mark the angle. Call one leg the rise, designating the height of the triangle, and the other leg the run, or the length of the triangle. In some cases however, you still have to use basic trigonometry to determine an angle.
Reduce the length of the run and the height of the rise by the same factor so they can be measured with a framing square. For example, the angle made by a run of 8 feet and a rise of 12 feet makes the same angle as a run of 8 inches and 12 inches.
Place the framing square on the lumber and position it so that both measurements align with the edge of the lumber piece. In the example, place the 8-inch mark on one leg against the edge and the 12-inch mark on the other leg against the same edge.
Use the edges of the square to mark the rise and the run of the triangle. The resulting cuts reflect two of the three angles on the triangle. The third angle is 90 degrees. One use of this method marks the cut line on stair stringers after calculating the rise and run of the steps.
Place the square on the board with the T-shaped edge against the board edge. Move the square so that the pivot point at the right angle of the triangle is aligned with the board where the cut is to be made.
Draw a line across the board using the edge between the pivot point and the diagonal side of the square as a guide. This line is perfectly perpendicular to the edge of the board.
Rotate the square around the pivot point until the desired angle on the diagonal side of the square is aligned with the same edge of the board that the pivot point rests against. Mark a line across the board using the side of the square between the pivot point and the diagonal side as a guide. This line marks the desired angle with the first line.
Calculate the length of the diagonal leg of a right triangle. Square both lengths of the other two sides, add them together and take the square root. The result is the length of the diagonal leg. The length of the diagonal leg is used to find the angles between the other two legs and the diagonal leg.
Calculate the angle between the horizontal and diagonal legs of a right triangle. Divide the length of the horizontal leg by the length of the diagonal leg using the calculator. Press the inverse button, then press the cosine button to display the angle between the horizontal and diagonal legs.
Calculate the angle between the vertical and diagonal legs of a right triangle. Divide the length of the vertical leg by the length of the diagonal leg using the calculator. Press the inverse button, then press the cosine button to display the angle between the vertical and diagonal legs.
#### Tip
There are many carpentry applications of these three methods of finding angles. From roof slopes to stairways to architectural details, the application of these three methods is easily applied with a little thought and planning. Right triangles are the basis for many carpentry and construction methods. Any triangle that is not a right triangle turns into a pair of right triangles with the addition of a single drawn line.
#### Things You'll Need
• Framing square
• Rafter square
• Calculator with trigonometric functions
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https://magoosh.com/praxis/can-i-use-a-calculator-on-the-praxis-core-math/
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# Can I Use a Calculator on the Praxis Core Math?
One of the most common questions that come up on any math test, the Praxis Core math exam included, is whether or not you can use a calculator. Rest assured: Yes, you will be provided with an on-screen, four-function calculator on the Praxis Core. You’ll be able to access it on all of the math questions; you can hide it when you’re not using it, show it when you want it, and even copy and paste answers from the calculator screen to the answer box for Numeric Entry questions. So you can use a calculator, but should you?
## Should I use the calculator?
In most cases, the answer is a resounding “No.”
The Praxis Core generally assesses your algebraic and geometric reasoning, not your ability to correctly perform complex calculations. You’re not going to be asked to multiply or divide really big numbers. In fact, the vast majority of questions will include calculations you can do quickly in your head. Using the calculator when you don’t need it increases the chances that you’ll make a mistake and can significantly slow you down. As with most things test prep, it’s important to be strategic when using the calculator on the Praxis exam.
## When should I use a calculator?
There will be times on the Praxis Core where you won’t want to spend the time writing out a problem on paper or where you’ll want to confirm your math with a calculator. Compound interest or non-standard Pythagorean theorem questions are some possible examples. But these cases should be pretty limited on the Praxis Core.
The far more likely scenario is that the Praxis asks you a question where a calculator isn’t needed, but you don’t feel sure enough of your math skills to answer the question without it. This is common, especially for people who haven’t studied math in a while. But with proper study and preparation, you should be able to reduce these instances by bolstering your mental math skills.
Whenever you bring in an extra tool like a calculator, you increase the chances of making a mistake — mis-clicking or mis-reading are inevitable. In general, it’s better and faster to do the math mentally (or on paper) than it is to do it with a calculator. As with nearly everything, you can develop mathematical speed, accuracy, and confidence with practice!
## Getting a feel for the calculator
The best way to figure out how best to use the calculator on the Praxis Core math test is to actually practice with it. Magoosh Praxis offers an identical calculator to the one given on the Praxis Core exam. You can also purchase a practice Praxis Core Math test from ETS, which features the four-function calculator that you’ll see on the exam. As you practice, try to get a feel for the types of questions in which the calculator saves you time and the types of questions in which it just eats up time. Forewarning: in most cases, the calculator takes longer than the mental math!
## Author
• Peter helps make sure Magoosh students have the best possible content. A proud Arizona Wildcat and Teach for America alum, he worked as an instructional coach before getting an MBA at Berkeley’s Haas School of Business. He is passionate about student achievement and educational equity. Also prime numbers. Peter enjoys cooking, running (slowly), and going to bed comically early.
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https://rdrr.io/cran/distributions3/man/LogNormal.html
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# LogNormal: Create a LogNormal distribution In distributions3: Probability Distributions as S3 Objects
LogNormal R Documentation
## Create a LogNormal distribution
### Description
A random variable created by exponentiating a Normal() distribution. Taking the log of LogNormal data returns in Normal() data.
### Usage
LogNormal(log_mu = 0, log_sigma = 1)
### Arguments
log_mu The location parameter, written μ in textbooks. Can be any real number. Defaults to 0. log_sigma The scale parameter, written σ in textbooks. Can be any positive real number. Defaults to 1.
### Details
We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.
In the following, let X be a LogNormal random variable with success probability p = p.
Support: R^+
Mean: \exp(μ + σ^2/2)
Variance: [\exp(σ^2)-1]\exp(2μ+σ^2)
Probability density function (p.d.f):
f(x) = \frac{1}{x σ √{2 π}} \exp (-\frac{(\log x - μ)^2}{2 σ^2})
Cumulative distribution function (c.d.f):
F(x) = \frac{1}{2} + \frac{1}{2√{pi}}\int_{-x}^x e^{-t^2} dt
Moment generating function (m.g.f): Undefined.
### Value
A LogNormal object.
Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gamma(), Gumbel(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()
### Examples
set.seed(27)
X <- LogNormal(0.3, 2)
X
random(X, 10)
pdf(X, 2)
log_pdf(X, 2)
cdf(X, 4)
quantile(X, 0.7)
distributions3 documentation built on Sept. 7, 2022, 5:07 p.m.
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http://www.britannica.com/print/topic/177734
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# eccentricity
• ## conic sections
TITLE: conic section: Analytic definition
SECTION: Analytic definition
...plane curves that are the paths (loci) of a point moving so that the ratio of its distance from a fixed point (the focus) to the distance from a fixed line (the directrix) is a constant, called the eccentricity of the curve. If the eccentricity is zero, the curve is a circle; if equal to one, a parabola; if less than one, an ellipse; and if greater than one, a hyperbola. See the...
• ## part of ellipse
TITLE: ellipse
...path has this same property with respect to a second fixed point and a second fixed line, and ellipses often are regarded as having two foci and two directrixes. The ratio of distances, called the eccentricity, is the discriminant (q.v.; of a general equation that represents all the conic sections [see conic section]). Another definition of an ellipse is that it is the locus of...
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http://wiki.san-ss.com.ar/project-euler-problem-017
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Project Euler Problem 017
# Statement
If the numbers 1 to 5 are written out in words: one, two, three, four, five, then there are 3 + 3 + 5 + 4 + 4 = 19 letters used in total.
If all the numbers from 1 to 1000 (one thousand) inclusive were written out in words, how many letters would be used?
NOTE: Do not count spaces or hyphens. For example, 342 (three hundred and forty-two) contains 23 letters and 115 (one hundred and fifteen)
contains 20 letters. The use of "and" when writing out numbers is in compliance with British usage.
# Solution
For this I built a decoder for one-digit, two-digit and three-digit numbers that of course are connected.
Then it is just an iteration.
```def one_number_decoder(n):
if n == '0':
return 0
if n == '1':
return 3
if n == '2':
return 3
if n == '3':
return 5
if n == '4':
return 4
if n == '5':
return 4
if n == '6':
return 3
if n == '7':
return 5
if n == '8':
return 5
if n == '9':
return 4
def two_number_decoder(n):
x = n[0]
y = n[1]
if x == '0':
return one_number_decoder(n[1])
if x == '1':
if y == '0':
return 3
if y == '1':
return 6
if y == '2':
return 6
if y == '3':
return len('thriteen')
if y == '4':
return len('fourteen')
if y == '5':
return len('fifteen')
if y == '6':
return len('sixteen')
if y == '7':
return len('seventeen')
if y == '8':
return len('eighteen')
if y == '9':
return len('nineteen')
if x == '2':
res = len('twenty')
if x == '3':
res = len('thrity')
if x == '4':
res = len('forty')
if x == '5':
res = len('fifty')
if x == '6':
res = len('sixty')
if x == '7':
res = len('seventy')
if x == '8':
res = len('eighty')
if x == '9':
res = len('ninety')
return res + one_number_decoder(y)
def three_number_decoder(n):
x = n[0]
res = one_number_decoder(x)
res += len('hundred')
if additional > 0:
res += 3 #Adding 'and' word
return res
if __name__ == '__main__':
result = 0
for n in range(1,1000):
if n < 10:
result += one_number_decoder(str(n))
if n >= 10 and n < 100:
result += two_number_decoder(str(n))
if n >= 100 and n < 1000:
result += three_number_decoder(str(n))
result += len('onethousand')
print("The result is:", result)
```
The Python is available for download here.
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https://cs.stackexchange.com/questions/123824/finding-the-smallest-number-that-scales-a-set-of-irrational-numbers-to-integers
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# Finding the smallest number that scales a set of irrational numbers to integers
Say we have a set $$S$$ of $$n$$ irrational numbers $$\left\{a_1, ..., a_n\right\}$$.
Are there any known algorithms that can determine a scaling factor $$s \in \mathcal{R}$$ such that $$s * a_i \in \mathcal{N} \;\forall a_i \in S$$, assuming that such factor exists? If multiple exist, how about the smallest one?
Moreover, I wonder, under what input conditions could one assume that an algorithm for this problem can't (or can) return a valid scaling factor?
• Algorithms involving general irrationals will probably mean little without specifying how those irrationals are represented and computed. That is, a model of computation for irrationals has to be specified. For example, an oracle could be used to tell the ratio of two irrationals is rational or not and, if rational, a representation of it as a ratio of two integers. Without such an oracle, imagine what happens if we are computing the ratio of Euler's constant to 1. It is an open problem whether it is rational or not. Apr 9, 2020 at 0:34
• Thanks @JohnL. Totally fair - I thought about that when writing the Q and was honestly hoping that someone would bring that up in one of the answers as a conditional assumption on the representation, or e.g. as a set of requirements on the input or the types of operations or checks the algorithm must be able to run in e.g. O(1) or bounded time for a given input number. Apr 9, 2020 at 0:43
• Without a proper model of computation for irrational specified, answers to this question will bound to treat the irrationals in the input more or less as rationals. For example, Steven's answer below is basically about rational numbers. Apr 9, 2020 at 0:43
• Thanks @JohnL since you bring this up, I wonder, are you aware of any representation that could make problem viable for any class of algorithms? Apr 9, 2020 at 0:52
• Once there was a situation where the given numbers could be approximated by floating numbers to arbitrary precision (easily), and it was guaranteed that there is a scaling factor less than ${10}^9$ such that every given number will, once multiplied by that factor, become an integer. To find the least such factor, I used continued fraction. Apr 9, 2020 at 5:56
## 1 Answer
I'm assuming that $$0 \not \in \mathbb{N}$$, otherwise $$s=0$$ is a trivial solution.
If you input numbers contain at least one positive and one negative number there is no solution. If your input numbers are all negative, either there is a solution but no smallest solution, or there is no solution at all. You can decide which of the two is the case by solving the same problem on with the numbers multiplied by $$-1$$.
Assume then that all input numbers are positive.
If $$s a_1 = c$$ and $$s a_2 = c'$$ for $$c,c' \in \mathbb{N}$$, then $$c' = s a_2 = c \frac{a_2}{a_1}$$, i.e., $$\frac{a_2}{a_1}= \frac{c'}{c}$$. This shows that you can only find a solution if all your irrational numbers can be obtained by multiplying $$a_1$$ by some rational factor.
In this case you can consider the set of numbers $$\{1, \frac{a_2}{a_1}, \frac{a_3}{a_1}, \dots, \frac{a_n}{a_1}\}$$ instead. Since they are all rationals you can write them as $$\{1, \frac{b_2}{c_2}, \frac{b_3}{c_3}, \dots, \frac{b_n}{c_n}\}$$, where $$b_i,c_i \in \mathbb{N}$$ and $$gcd(b_i,c_i)=1$$. You can then find the minimum common multiple of the denominators and multiply it by $$a_1^{-1}$$.
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# Multiple semester gpa calculator
• Home -
• Multiple semester gpa calculator
## Multiple semester gpa calculator
Fill in the previous course totals with the total gpa hours and the total quality points from your unofficial transcript online or from your official transcript from the records office, student services center room 101. subtract each repeated course’ s gpa hours and quality points. so, for instance, if you took 15 credits last semester and earned a 3. 1, and if this semester you are taking a 5- credit class in which you earn an a ( calculator = 4. 0), a 4- credit class in which you earn a c+ ( = 2. 3), a 3- credit class in which you earn a b- ( = 2. 7), and a 2- credit class in which you earn a d ( = 1. 0), your cumulative gpa for the two. calculate gpa on a 4. a version adjusted for the stanford grade point scale.
reverse multiple semester gpa calculator fad productions. 0 credit course, an assessment will be given at the end of semester 1 and semester 2. for a semester ( 0. 50 credit course), an assessment will be given at the end of the course. each assessment will evaluate learning for that semester. the quality points for each assessment grade will constitute 1/ 10 of the final grade for the course. today, because of him, we have a well- established grading system that can calculate how well we’ ve done in a course, semester or a year. gpa is, what we call, a “ grade point average.
divide the total sum from step 2 by the total credit hours taken for the semester. the user should then repeat this process for multiple all the student’ s courses. after all the courses have been recorder, the user should click on the ‘ calculate gpa’ button. a function procedure should be used to calculate the quality points for a course. step 3: calculate cumulative high school gpa. to get a cumulative gpa for john' s entire high school career, we simply add up the sums for all the years and divide by the number of classes he took over all those years:. 1 ( sum of all final grades= 34 ( total number of classes taken) 118. 47 ( gpa) so, his gpa for all of high school is 3. this gpa calculator calculates your grade point average using a 4.
” finally, enter your all courses detail to calculate your gpa; as mentioned earlier, cgpa calculator helps to calculate the gpa and cgpa of multiple semesters. calculate the gpa as the total number of “ points” divided by the total credit hours. output: display a nicely worded message that includes the student’ s name and the gpa ( decimal point number with 2 decimal places) achieved by the student that semester. what i currently have is:. you can total your current semester courses and credits with our online gpa calculator. the online gpa calculator is a tool to calculate grade point average based on a typical high school system. multiple gpa is calculated by dividing the total amount of grade points earned by the total amount of credit hours attempted. since we only use numbers, spartan gpa is the perfect gpa calculator tool for any high school or college/ university student looking to calculate their semester and cumulative gpa. with a simple and straightforward grading calculator interface, you can compute your gpa calculation quickly - and for the most part, multiple painlessly. program to find gpa using if- else statement gulzar sheikh. 09: 56 c language, gpa program in c, if else statement in c, if statement in c. program to find gpa using if- else statement: # include< stdio.
otcas calculates all gpas in semester hours. courses that were completed in quarter hours are converted to semester hours. the conversion calculator ratio is 1. 0 quarter hour = 0. 667 semester multiple hours. otcas breaks down your gpa by academic year, not by college or university. coursework completed at multiple schools for similar academic years get combined. multiple semester gpa calculator. source( s) : im/ a8zhl. login to reply the answers post; anonymous. 5/ 9 of the radius of the earth.
this is assuming that points are chosen randomly according to the following algorithm: we take a cube which precisely encloses the earth and choose points uniformly from within. gpa method= " value" calculator type= " value" term= " value" year= " value" grade= " value" credittype= " value" scale= " value" ) the following parameters of a gpa calculation method can be specified in the dat: method. determines which gpa calculation method will be used. the gpa calculation methods are setup in the district’ s powerschool site. your gpa for this semester would be a 3. 4 based on an average of the values in the last column. finally, let’ s calculate multiple gpa for the first semester of sophomore year assuming your grades increased slightly once again. first semester sophomore year. total number of credits you will take in the future, before you need this gpa, including any courses you will repeat. ( these credits can be taken over multiple semesters.
) total gpa you want to attain. this number should be between 0. your current gpa: x. xx: the multiple total gpa you will need for the credits you will be taking: x. enter the credit hours and expected grade for ( up to six) courses that you are presently taking below and press calculate to find your predicted new cumulative gpa at the end of this term. pharmcas calculates all gpas in semester hours. pharmcas breaks down your gpa by academic year, not by college or university. with reduced in- person class sizes, no large- scale gatherings and, of course, masks required in campus facilities, this fall multiple will look different at georgia state. but throughout the summer, faculty, staff and administrators have been hard at work to ensure a rich collegiate experience this fall. multiple semester units by 3/ 2 or 1.
examples: # 3/ 2 x 12 multiple semester gpa calculator semester units = 18 quarter units or # 1. calculator 5 x 12 semester units = 18 quarter units! quarter to semester " multiple quarter units by 2/ 3 or. examples: # 2/ 3 x 15 quarter units = 10 semester units or #. 667 x calculator 15 quarter units = 10 semester units. track your grades, plan your assignments, and manage your schedule through multiple semesters, complete with robust gpa calculation, custom grading scales, and more! grade tracker pro has grown from a simple grade calculator into a complete multiple semester gpa calculator school management solution and organizer. whether you need class scheduling, quick and simple grade entry, or complex grade management. gpa calculator tab.
there are three different gpa calculators: graduation, term, and advice. this functionality will calculate the minimum grades needed to reach a target cumulative gpa. it can multiple assist in accurate mapping of student paths to achieving honors,. before start writing, you must remember a few things. things to be remembered before start writing a thesis calculator introduction: 1 brainstorm: think of possible topics and brainstorm different ideas in your mind. 2 make a rough draft: organize your ideas and make a rough draft. 3 multiple choose the research paper:. do not cover these aspects in a separate paragraph. instead, integrate them into your general discussion. also, be careful not to lose thread of every paragraph. they should start with an opening - or topic - sentence and end with a concluding one.
along with this, do not fail to cite specific examples to support the main thesis of your paper. use this thesis statement multiple generator to build your argumentative or compare and contrast thesis statement in less than 5 minutes. it looks like you' ve lost connection to our server. please check your. a thesis statement for a research paper must state the major points, which the body will be about. you should be able to use the statement as an attraction for the reader towards your work. knowing the word count of a text can be important. for example, if an author has to write a minimum or maximum amount of words for an article, calculator essay, report, story, book, paper, you name it. wordcounter will help to make sure its word count reaches a specific requirement or stays within a certain limit. another word for essay.
find more ways to say essay, along with related words, antonyms and example phrases at thesaurus. com, the world' s most trusted free thesaurus. transitional words make it easier for students to connect their thoughts and ideas when writing essays. as a student, the goal is to select transitional words to help guide readers through your. in the end, though, remember that good writing does not happen by accident. although we have endeavored to explain everything that goes into effective essay writing in as clear and concise a way as possible, it is much easier in theory than it is in practice. as a result, we recommend that you practice writing sample essays on various topics. many services will promise the same thing, but not many will deliver. assignmentmasters is one of the best paper writing services because it delivers even more than you expect. the team made an effort to recruit talented writers from various walks of life.
they all hold higher- education degrees. each customer is paired with an expert. our custom paper writing service supports various international payment methods ( including paypal, mastercard, visa) – select the one that is convenient for you. with us, each transaction you make is secured. get multiple a professional research writer. the third step is. most paper writing services don' t value the need to have expert and highly gifted writers. this is not the case for us, and thus, we have very clinical writers. we make sure that all our writers undergo a stringent vetting process to ensure that they are the best fit for the job.
How to write a analytical research paper. we have highly qualified phd and ma writers working with us, but. the process of writing the academic papers on any topic should be written good, unmistakable and without plagiarism. but far not every nursing writer could provide you with such a paper, and you need to care about the quality and uniqueness of your text. · a noun clause can be an independent clause a complete thought that stands on its own as a complete sentence, or a dependent clause that is not a complete thought that is part of a sentence that includes an independent clause. example: - - the movie is on tv tonight. ( independent clause) - - that i like ( dependent clause, it has a subject " i" and a verb " like" but is not a complete sentence). dependent clauses can stand not only for adverbs, but also for nouns and for adjectives. a noun clause is an entire clause multiple which takes the place of a noun in another clause or phrase. like a noun, a noun clause acts as the subject or object of a verb or the object of a preposition, answering the questions " who( m)?
it is a dependent clause that usually changes a noun. it has an adjective in it and has a couple of words before it, which is a kind of phrase to start it off. , any two of these are correct: who, whom, whose, which, that, who have good study habits, no it is not, it is an adverb clause. a noun clause is a nonfinite clause that is subordinate, dependent, and restrictive. all these terms define a different aspect, characteristic, or function of the same clause.
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• learn how to calculate your college gpa with this college gpa calculator. you can easily find your gpa for a single semester, or for multiple semesters ( cumulative gpa). calculate the total monthly payment you’ ll owe as you borrow for each year of college. plug in multiple student loans, both federal and private, to see what your total loan balance will.
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walton college of business provides this calculator for estimation purposes only. gpas calculated here do not in any way represent your official gpa.
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• click calculate gpa and your semester gpa will be generated how to calculate your gpa for multiple semesters ( find your cumulative gpa) with the gpa calculator.
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follow steps 1 – 2 above but don’ t click calculate gpa; click add semester and a new semester will be generated. calculate gpa with repeated classes ( please understand wsu’ s repeat policy before proceeding).
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# What part of government is in charge of adoption?
###### Question:
What part of government is in charge of adoption?
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Which number line represents the solutions to -2/x = -6? + + + + + + +++ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 ++ 4 5 + + 7 8 6 + -++ -8 -7 -6 -5 -4 -3 -2 -1 + + 4 5 + 6 0 + 7 1 2 3 8 + -8 -7 -6 + +++ -5 -4 -3 -2 -1 N + O 1 3 4 5 6 7 8 ++ + +++++++ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 A+ + ת) 6 7 8...
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### Which punctuation correctly completes the sentence? Leno's early acts drew from his life experience: growing up in New Rochelle, New York, and Amherst, Massachusetts, attending Emerson College, and enjoying popular culture. His college education affected his act in an unusual (way; way, way way-) at his first stage booking. After learning about American Indian history, he refused to do a skit filled with stereotypes of American Indian culture. The writer changed the skit to suit (Leno, Leno Len
Which punctuation correctly completes the sentence? Leno's early acts drew from his life experience: growing up in New Rochelle, New York, and Amherst, Massachusetts, attending Emerson College, and enjoying popular culture. His college education affected his act in an unusual (way; way, way way-) a...
### Which is NOT an Islamic calligraphy style? A.) kufic B.) thuluth C.) qiblah
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### A&E) Subscription received in cash Rs 60,000; subscription received in advance for next year Rs 3,000 and received in advance during previous year is Rs 2,000.Subscription in arrears at the end of current year Rs 5,400 . Amount credited to Income and Expenditure Account will be :(a)Rs 53,600(b)Rs. 55,600(c)Rs. 66,400(d)Rs. 64,400
A&E) Subscription received in cash Rs 60,000; subscription received in advance for next year Rs 3,000 and received in advance during previous year is Rs 2,000.Subscription in arrears at the end of current year Rs 5,400 . Amount credited to Income and Expenditure Account will be :(a)Rs 53,600(b)R...
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Pin It button on image hover
## Wednesday, April 6, 2016
### Maple Syrup Farmer Math
In an effort to make math more interesting for kids and to make my kids understand that math really is important, I have started a series of Farmer Math questions to go along with the Flat Aggie reports. These are patterned after the blog Bedtime Math. Some of the questions are easy enough for pre-schoolers and sometimes I come up with some to even challenge high school seniors.
These questions follow the report Flat Aggie Visits Battel's Sugar Shack. Additional educational materials can be found in the post Battels Sugar Bush Learn More.
1.) Flat Aggie helps tap 12 trees in the morning and 12 trees in the afternoon. How many trees did she tap in one day? How many trees could she tap in 2 days?
2.) Flat Aggie is helping to figure out how much tubing is needed to get the sap back to the shack. The tubing will go directly from the tree to the shack. The trees are 25 feet, 75 feet and 83 feet from the shack. How much tubing will be needed?
3.) Flat Aggie drilled holes that were 3/16 inch in diameter. Is that larger or smaller than 3/8 inch? Is that larger or smaller than ½ inch.
4.) 1 gallon of sap is 98% water and 2% sugar. How many ounces of sugar are in a gallon of sap? (Round to the nearest whole number.)
5.) 1 gallon of syrup is 33% water. How many ounces of sugar are in a gallon of syrup? (Round to the nearest whole number.)
1.) 24 trees, 48 trees 2.) 183 feet 3.) smaller than ¼ inch, smaller than ½ inch 4.) 3 ounces 5.)86 ounces
If you are a teacher or homeschooler that would like more information to go with the Flat Aggie reports, send me a message on my contact form. Along with the report and the Farmer Math questions, we send each teacher an additional page of activities, crossword puzzles and sometimes a few hands on activities.
-A Kansas Farm Mom
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https://www.enotes.com/homework-help/what-value-theta-fraction-1-tan-theta-undefined-322792
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# For what value of theta is the fraction 1/tan of theta undefined?Please explain this, I am so so confused.
hala718 | High School Teacher | (Level 1) Educator Emeritus
Posted on
Let `y= 1/tantheta `
We need to find the values of `theta` such that the function (y) is defined.
Since `tan(theta)` is in the denominator, then we know that `tan(theta)` can not be 0.
==> `tantheta != 0`
Now we will find the values where `tan(theta)= 0` and exclude them from the domain.
`==gt tan(theta)= 0 `
`==gt (sintheta)/(costheta)= 0 `
`==gt sin(theta)= 0 `
`==gt theta = 2npi` (n= 0, 1, 2, ...)
==> Then `1/(tantheta)` is not defined when `theta= 2npi` (n= 0, 1, 2..)
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# Tagged Questions
Questions on the astronomical applications of Mathematica.
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### Mollweide maps in Mathematica
Context In my field of research, many people use the following package: healpix (for Hierarchical Equal Area isoLatitude Pixelization) which has been ported to a few different languages (F90, C,C++, ...
4k views
### Composition: how to make a day and night world map?
Given the following world images: ...
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### Only final result from NDSolve
Finally, I started to play with differential equations in Mathematica. And I have faced the problem, which seems to me so basic that I'm afraid this question is going to be closed soon. However, ...
2k views
### The Orbit and Perigee of the Flamsteed comet
Historical context This year we have the 330-th anniversary of the Battle of Vienna - one of the great formative events of European history, it took place on September 12, 1683. Kara Mustafa, Grand ...
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### How do I compute sidereal time and JD?
There was once a package that provided a wide range of functions for computing astronomical information. This package now appears to be out of date; but as near as I can tell, many of the useful ...
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### Is there a way to get AbsoluteTime to correctly report absolute time differences?
Absolute time supposedly reports the "total number of seconds" between two dates, but it does not. For example because of the leap second applied at 2012-06-30T23:59:60Z ...
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### Creating a simulation of our Solar System
As the title says, my objective is to create a simulation of the motion of the planets in our Solar System using Mathematica. All the theoretical background regarding the equations of motion of ...
584 views
### Right Ascension and Declination in astronomical functions
In the process of answering this question, I was forced to confront the various astronomical coordinate systems used by Mathematica. Background In astronomy, positions of celestial objects (stars, ...
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### Using map projections with astronomical data
I noticed that all important "Geoprojections" are available in projections for a spherical reference models: GeoProjectionData function; 1 - I am trying using the sinusoidal projection for ...
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### Geocentric, Earth-fixed, coordinates of Sun, Moon (and others objects eventually)
I know I can get "HelioCoordinates" of the Earth and the Moon with ...
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### How is AstronomicalData interpreting its date arguments?
It is easy to confirm that the AstronomicalData is not interpreting its arguments as UTC (or "GMT", which is, in any case, ambiguous) dates by verifying that the ...
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### Inaccurate Heliocentric coordinates of PlanetaryMoonData
I have noticed that the heliocentric coordinates obtained with PlanetaryMoonData are very inaccurate. Fore example, consider the Galilean moons of Jupiter: ...
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### How do I determine astronomical transit times?
Mathematica, through the "legacy" Scientific Astronomer package, used to have the ability to easily determine accurate transit times of astronomical objects simply, with ...
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### Using ParametricNDSolve and NMinimize to find accurate interplanetary trajectory arrival periapse radius
I have an interplanetary trajectory simulation that calculates the required velocity (using a Lambert solver) and escape angle for a spacecraft to travel from Earth to Mars. This model does not use a ...
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### Using DeleteCases with multiple AstronomicalData properties
I am attempting to use AstronomicalData and DeleteCases together to remove invalid cases of two properties while listing the ...
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### Combining solar system planets orbits and positions in Graphics3D and Show
I am building an animated model of the solar system, where I would like to display the OrbitPaths of the planets along with their ...
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### Interstellar Image Effects
I'd like to know how to create this type of gravitational lensing effect in an image (here are some star fields from the hubble) "I wrote down the equations and I tested them in Mathematica by ...
430 views
### Why is AstronomicalData so slow?
I'm having trouble getting some pretty simple functions using AstronomicalData to perform at anywhere near the speed I need them to. For example, I have many ...
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### Problems getting clean sunrise, sunset, civil, nautical and astronomical twilight data
If in a notebook you enter: ...
429 views
### Impact Crater Simulation
I have been trying to model crater formation on a given planetary surface - $500\,\textrm{km}^2$. The locations of impacts are random, however, if an impact is within $30\,\textrm{km}$ of another, the ...
580 views
### Combining three R, G, B matrices into a single matrix of 3-tuples?
Working with Mathematica 8 on OSX here. I have three $1024 \times 1024$ matrices representing the counts from consecutive exposures onto a CCD with $R$, $G$, and $B$ filters on it, and am looking to ...
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### How can I get this Graphics3D to rotate around a particular sphere?
I made this Graphics3D of the Sun with the plotted orbits of a few astronomical objects. ...
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### Impact Crater Sim. with added criteria
I have been trying with out success to edit this impact crater simulation script into producing a variable which calculates crater density. In addition I have been trying to figure out how to ...
212 views
### GeoProjection for astronomical data - wrong ticks
Here is an excellent answer to what I wanted to do, and it works like a charm. Unfortunately, when I want to change it a bit, it fails. For setting the stage, here is the datafile, and this is my ...
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### MoonPosition and TimeZone support
With this question, I noticed some some weird results of MoonPosition. Consider the following code: ...
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## Sõnastik
Valige vasakul üks märksõnadest ...
# Programming in JuliaPlotting
Lugemise aeg: ~10 min
The main plotting package in Julia is called Plots. To create a figure, you supply data in the form of arrays as arguments to the plot function (x first, then y if appropriate, then z if appropriate). All other plot information (called attributes, in Plots lingo) is supplied using keyword arguments. For example:
using Plots
using Random
Random.seed!(123)
plot(rand(10), rand(10), seriestype = :scatter,
group = rand(0:1,10), title = "Some random points")
Note that the group keyword argument partitioned the data into two series, one for each unique value in the array supplied to group. These series are automatically shown in different colors and labeled in the legend.
You can see all the main plot types and attributes on the Plots.jl cheatsheet.
To save a plot, use the savefig function:
P = plot(rand(0:10),rand(0:10), seriestype=:scatter)
savefig(P,"myfigure.pdf") # save figure as a PDF file
Exercise
Make a graph which looks as much as possible like the one shown below. You'll want to look at the Plots.jl cheatsheet for options.
using Plots
x = range(0, stop = 2π, length = 100)
y = sin.(x)
# add plotting code here
Solution. We change the line style and width, and we add labels for the axes:
using Plots
x = range(0, stop = 2π, length = 100)
y = sin.(x)
plot(x,y, linewidth = 3, linestyle = :dash,
xlabel="x", ylabel="sin(x)", legend = :none)
To get a quick refresher on how to perform common tasks in Julia, check out the Julia-Python-R cheatsheet, also linked from browndsi.github.io.
Congratulations! You have finished the Data Gymnasia Programming with Julia course.
Bruno
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# What Is a Sample?
A sample is a powerful tool in the world of statistics and research. It represents a smaller, manageable version of a larger group, providing a way to draw meaningful insights without the burden of studying an entire population. In this article, we'll delve into what a sample is, why it's essential, different sampling techniques, and real-world examples of its application.
Understanding Samples
In statistical terms, a sample is an analytical subset of a more extensive population. A population, on the other hand, encompasses the total observations within a specific group or context. Think of a sample as a slice of the population, a fraction that encapsulates the characteristics and traits of the larger whole.
The utility of samples extends across various fields. Scientists, marketers, government agencies, economists, and research organizations all leverage samples to glean insights and draw conclusions. Why? Because dealing with entire populations in research presents significant challenges. Access to complete populations can be limited, and conducting exhaustive studies can be time-consuming and resource-intensive. Samples offer a practical solution.
To be effective, samples must mirror the broader population, ensuring that they accurately represent the characteristics and qualities of interest. For example, if a study focuses on male college freshmen, the sample should comprise a subset of males meeting this criteria. Similarly, an investigation into the sleep patterns of single women over 50 should exclusively include women in this demographic.
Special Considerations
Consider a scenario where academic researchers aim to determine how many students passed the CFA exam after studying for less than 40 hours. Globally, over 200,000 people take this exam each year. Attempting to survey the entire population would be impractical, costly, and time-consuming. By the time such a massive dataset is collected and analyzed, years could pass, rendering the analysis obsolete due to a rapidly evolving population.
Instead, researchers opt for a sample—a representative fraction of the population. To ensure impartiality, the selection process must be random, giving every population member an equal and likely chance of inclusion. This is the foundation of simple random sampling, akin to a lottery draw.
Types of Sampling
1. Simple Random Sampling: This method is suitable when every entity within the population is identical. For instance, if researchers don't care about the gender distribution of CFA exam participants, simple random sampling suffices. From a population of 200,000 test-takers in 2021, a random sample might include 400 women and 600 men.
2. Stratified Random Sampling: Also known as proportional random sampling or quota random sampling, this approach divides the population into smaller, more homogeneous groups called strata. Each stratum shares similar characteristics. For instance, if age is a critical factor in the research, researchers could create strata for different age groups. The selection from each stratum remains random, ensuring that individuals within each group have an equal chance of being part of the sample.
Examples of Samples
Consider the world's population in 2021, totaling nearly 7.9 billion, with roughly 49.6% female and 50% male. Similarly, the total number of people in a country, students in a city, or even dogs in a locality can be viewed as population sizes. Samples drawn from these populations serve various research purposes.
In our CFA exam example, researchers could select a sample of 1,000 participants from the total population of 200,000 test-takers and analyze their data. This sample's mean would estimate the average pass rate for CFA exam takers who studied for less than 40 hours.
Crucially, the sample must be unbiased. If the sample's mean of 50% passing matches the population's mean, it suggests that the sample accurately represents the larger population.
Why Do Analysts Use Samples?
Analysts prefer samples over measuring entire populations for several reasons. Large populations can be unwieldy and costly to measure comprehensively. Samples allow for statistically valid inferences about the population. The laws of statistics, including the law of large numbers and the central limit theorem, support this inference. Techniques like analysis of variance (ANOVA) and linear regression rely on the power of samples to provide meaningful insights.
How Large of a Sample Do You Need?
The required sample size depends on the population size and the analysis you intend to perform, including desired confidence intervals. Power analysis and rules of thumb suggest that a sample should be large enough but not more than 10% of the population's size.
In summary, a sample is a vital tool in statistics and research, enabling meaningful insights from complex populations. Understanding its types and applications is crucial for researchers and analysts across various domains. Samples serve as windows into the broader world of data, allowing us to draw conclusions, make predictions, and inform decision-making efficiently and effectively.
Tickeron's Offerings
The fundamental premise of technical analysis lies in identifying recurring price patterns and trends, which can then be used to forecast the course of upcoming market trends. Our journey commenced with the development of AI-based Engines, such as the Pattern Search EngineReal-Time Patterns, and the Trend Prediction Engine, which empower us to conduct a comprehensive analysis of market trends. We have delved into nearly all established methodologies, including price patterns, trend indicators, oscillators, and many more, by leveraging neural networks and deep historical backtests. As a consequence, we've been able to accumulate a suite of trading algorithms that collaboratively allow our AI Robots to effectively pinpoint pivotal moments of shifts in market trends.
Disclaimers and Limitations
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Forward Stepwise Regression
Forward Stepwise Regression is a stepwise regression approach that starts from the null model and adds a variable that improves the model the most, one at a time, until the stopping criterion is met. The criterion for predictor entry into the model is based on the F-statistic and corresponding p-value (p-value must be less than the Alpha-to-Enter). It is also known as Forward Selection regression.
# How To
Run: Statistics→Regression → Forward Stepwise Regression...
Select the dependent variable (Response) and independent variables (Predictors).
Enter if alpha < option defines the Alpha-to-Enter value. At each step it is used to select candidate variables for entry, with partial F p-value less or equal to the alpha-to-enter. The default value is 0.05.
o Select the Show correlations option to include the correlation coefficients matrix to the report.
o Select the Show descriptive statistics option to display the mean, variance and standard deviation of each term.
o Select the Show results for each step option to show the regression model and summary statistics for each step.
# Model
The criterion to enter a variable at each step is the partial F p-value (that is over the alpha-to-enter). When none of the unselected variables meet the entry criterion, the forward selection command terminates the process. Forward stepwise algorithm is greedy version of the best subsets regression, so it may not result with the best model (model with lowest SSE). It is sensitive to choice of the alpha-to-entry value.
# Results
The report shows regression statistics for the final regression model. If the Show result for each step option is selected, the regression model, fit statistics and partial correlations are displayed for all variables entered at a selection step. A correlation coefficients matrix and descriptive statistics for predictors are included to the report if the corresponding options are selected.
R2 (Coefficient of determination, R-squared) - is the square of the sample correlation coefficient between the Predictors (independent variables) and Response (dependent variable).
Adjusted R2 (Adjusted R-squared) - is a modification of R2 that adjusts for the number of explanatory terms in a model. While R2 increases when extra explanatory variables are added to the model, the adjusted R2 increases only if the added term is a relevant one. It could be useful for comparing the models with different numbers of predictors. Adjusted R2 is computed using the formula:
where k is the number of predictors.
S – the estimated standard deviation of the error in the model.
MS (Mean Square) - the estimate of the variation accounted for by this term,
.
F - the F-test value for the model.
p-level - the significance level of the F-test. Values less than (0.05) show that the model estimated by the regression procedure is significant.
VIF – variance inflation factor, measures the inflation in the variances of the parameter estimates due to collinearities among the predictors. It is used to detect multicollinearity problems. The larger the value is, the stronger the linear relationship between the predictor and remaining predictors. VIF equal to 1 indicates the absence of linear relationship with other predictors (there is no multicollinearity). VIF value between 1 and 5 indicates moderate multicollinearity, and values greater than 5 suggest that a high degree of multicollinearity is present. It is a subject of debate whether there is a formal value for determining presence of multicollinearity: in some situations even values greater than 10 can be safely ignored – when high values caused by complicated models with dummy variables or variables that are powers of other variables. But in weaker models even values above 2 or 3 may be a cause for concern: for example, for ecological studies Zuur, et al. (2010) recommended a threshold of VIF=3.
TOL - the tolerance value for the parameter estimates, it is defined as TOL = 1 / VIF.
Partial Correlations are correlations between each predictor and the outcome variable excluding the effect of other variables.
# References
Hocking, R. R. (1976) "The Analysis and Selection of Variables in Linear Regression," Biometrics, 32
Nargundkar R. (2008) Marketing Research: Text and Cases. Third edition. Tata McGraw-Hill Publishing Company Ltd.
Neter, J., Wasserman, W. and Kutner, M. H. (1996). Applied Linear Statistical Models, Irwin, Chicago.
Zuur, A. F., Ieno, E. N. and Elphick, C. S. (2010), A protocol for data exploration to avoid common statistical problems. Methods in Ecology and Evolution, 1: 3–14.
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# Print the all uppercase and lowercase alphabets in Python
Here, we are going to learn how to print the all uppercase and lowercase alphabets in Python programming language?
Submitted by Bipin Kumar, on November 02, 2019
To do this task, we will use the concepts of ASCII value. ASCII stands for the American Standards Code for Information exchange. It provides us the numerical value for the representation of characters. The ASCII value of uppercase letters and lowercase alphabets start from 65 to 90 and 97-122 respectively. Before going to solve this problem, we will learn a little bit about how to convert the numerical value to characters and vice-versa.
Convert character to numerical value
In Python, a function ord() is used to convert the characters into a numerical value. This is an inbuilt function. Let's see the program,
```# input a number
s=input('Enter the character: ')
# getting its ascii value
n=str(ord(s))
# printing the result
print('ASCII of character {} is {}.'.format(s,n))
```
Output
```Enter the character: M
ASCII of character M is 77.
```
Convert numerical value to character
In Python, a function chr() is used to convert a numerical value into character. This is an inbuilt function. Let's see the program,
```# input a number i.e. ascii code
n=int(input('Enter the numerical value: '))
# getting its character value
s=chr(n)
# printing the result
print('The character value of {} is {}.'.format(s,str(n)))
```
Output
```Enter the numerical value: 77
The character value of M is 77.
```
Now we have learned how to convert the numerical value into character and by using its concepts we will easily solve the above problem by using the Python language.
## Python program to print the all uppercase and lowercase alphabets
```# printing all uppercase alphabets
print("Uppercase Alphabets are:")
for i in range(65,91):
''' to print alphabets with seperation of space.'''
print(chr(i),end=' ')
# printing all lowercase alphabets
print("\nLowercase Alphabets are:")
for j in range(97,123):
print(chr(j),end=' ')
```
Output
```Uppercase Alphabets are:
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Lowercase Alphabets are:
a b c d e f g h i j k l m n o p q r s t u v w x y z
```
Preparation
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# spacecraft with rotational artificial gravity
Page 2 - Seeking answers about space? Join the Space community: the premier source of space exploration, innovation, and astronomy news, chronicling (and celebrating) humanity's ongoing expansion across the final frontier.
#### Ken Fabian
Although I wonder if it’s possible these rotating rooms and small centrifuges problem was not the artificial G but the artificial G combined with an equally powerful earth G which confused the ear?
For a person standing the rotational direction in a room rotating on a flat disc on the ground is around a vertical axis (yaw) instead of head over heels rotation around a horizontal axis (pitch); from the point of view of the inner ear canals the sensations are very similar but from a keeping balance point of view the actual head over heels tumbling sensations of a spinning space station or ship could be worse.
But how about a large enough ring structure, let’s say 100 meter diameter with sleeping pods that insert into the structure and go round and round on rails on the inside of the ring. Like cars on a circular rollercoaster.
From the perspective of the person and their inner ear they will be in a 100m diameter structure, just with a faster spin rate. To overcome the problem it needs slower spin rates. Not sure but the mass and motions of such a rail system would defeat the purpose of reducing the engineering stresses within the structure.
Possibly we could turn to medical intervention... but losing or diminishing that balance sense would be debilitating. Or else replacing it with some kind of prosthetic vestibular organ - but that is drastic and currently beyond our abilities.
#### billslugg
Yes, any large "donut" shaped object would be dangerously susceptible to imbalance, bending, structural failure.
However, the forces are in tension, easily borne by a system of spokes made of carbon fiber. The strength of the cables need be no more than what would lift a spoke's share of the donut on Earth.
#### Helio
Fun thread. Perhaps this table will help. It shows the gravitational force (g-units) for a given radius and a given period (sec.). I lacked the time to double check it thoroughly, but I think it's correct.
ally
Last edited:
#### Helio
Yes, any large "donut" shaped object would be dangerously susceptible to imbalance, bending, structural failure.
However, the forces are in tension, easily borne by a system of spokes made of carbon fiber. The strength of the cables need be no more than what would lift a spoke's share of the donut on Earth.
Yes, but carbon fiber may not even be required.
My calculations show me that it even titanium would easily handle the job.
Take a 150m radius ring, spin it at, say, one rev in 40 sec, producing a g force that of Mars (0.38g), and if you have 6 spokes, the total cross sectional area matching the yield strength of titanium is a square with each side being 2.65 inches. Of course, the design would be much larger for the spoke, but the cross sectional area of the metal would still be the same.
Balancing shouldn't be too hard with auto controls using cables inside the spokes, or perhaps just near the center or outer edges. I haven't given this much thought, but automation is everywhere, especially in balancing systems.
#### billslugg
Balancing is not needed. The assembly will rotate smoothly around its center of gravity.
But, if the CG is not at the physical center, inhabitants in the ring would feel a speedup/slowdown once per cycle.
Yes, titanium could work. Lots of it on the Moon, for example. Some areas up to 10% TiO2.
Very little carbon on the Moon. Best soils contain about 0.5% carbon.
Helio
#### billslugg
Actually, every spot within an unbalanced donut would have its own circular orbit at its own radius. G forces would not be the same everywhere in the structure. You would be heavier in some rooms than others. Not really a big deal.
However, it woud be of great scientific importance to have an inertial platform from which to do experiments. This could only be located at the rotational center by a platform rotating opposite the donut. Adjustable spokes would be needed to keep it dead center.
#### Helio
Balancing is not needed. The assembly will rotate smoothly around its center of gravity.
But, if the CG is not at the physical center, inhabitants in the ring would feel a speedup/slowdown once per cycle.
Agreed. I would assume the c.g. and centroid would be very close, so very little change.
If, say, 100 people gathered in a room, shifting weight there, it would only cause a tiny shift in weight.
So, playing with an example, if we had a 6-spoke donut with:
2) period = 30 sec.
3) ring width = 15m
4) ring height = 10m
5) wall thickness (total for more than one) = 0.5 inches
6) inner wall thickness = 0.1 inches
7) material = titanium
8) spoke load safety factor = 3
9) total weight, ignoring furniture = 2.4E6 lbsm.
10) g force at radius = 0.44g
Then the diameter of a solid round spoke would be = 2.5 inches. Note that this diameter is reduced from a 1 g load to the 0.44 g load.
Of course, the spoke would not be solid, but the cross-sectional area of the metal itself would be this equivalent.
The curvature along the ring would allow someone to look down the ring 60 feet before their level eyesight would reach the curved floor, if their eyes are at 5.5 feet.
I have set all this up in Excel should anyone want to play with other designs.
Yes, titanium could work. Lots of it on the Moon, for example. Some areas up to 10% TiO2.
Very little carbon on the Moon. Best soils contain about 0.5% carbon.
That's interesting.
Last edited:
#### Helio
However, it woud be of great scientific importance to have an inertial platform from which to do experiments. This could only be located at the rotational center by a platform rotating opposite the donut. Adjustable spokes would be needed to keep it dead center.
I'm unclear what this is. Are you saying you want to have a small room in the center where the g-force can be dialed-up or down?
#### billslugg
I am looking for an inertial platform, one experiencing no acceleration. We could mount telescopes on it and the images would be stable.
Moving 100 people to the other side of a 2.6 million pound donut is about one part in 170. The central point would need to move about half a meter.
The donut ring should be circular in cross section to minimize any additional stiffening structure. At 12 meters diameter, in a one atmosphere environment (15 PSI), hoop equation is: Inside pressure equals skin tension divided by radius: Skin tension = pressure x radius = 15 x 236 = 3540 pounds per inch of skin width.
Well within the capability of a 0.5" wall made of titanium.
#### Helio
I am looking for an inertial platform, one experiencing no acceleration. We could mount telescopes on it and the images would be stable.
Moving 100 people to the other side of a 2.6 million pound donut is about one part in 170. The central point would need to move about half a meter.
The donut ring should be circular in cross section to minimize any additional stiffening structure. At 12 meters diameter, in a one atmosphere environment (15 PSI), hoop equation is: Inside pressure equals skin tension divided by radius: Skin tension = pressure x radius = 15 x 236 = 3540 pounds per inch of skin width.
Well within the capability of a 0.5" wall made of titanium.
Ok. That makes sense. It would be a better design to include a central cylinder-like room where the loads of each spoke could connect, so each spoke could be monitored. The other benefit would provide the room you need.
The round shape for the ring would certainly be better mechanically, but I'm not sure it the psychological effects would be satisfactory, but perhaps the young would be fine with it. I'm too old. I would guess something more oval to minimize the gussets and other bracing needed for any square-like design, which would also produce less living space per wall material area.
I'm guessing there might be outer wall sections that would help prevent blowouts from meteroid penetrations. Plus, this thickness includes the ring's inner and outer walls, hence there would be four layers, so an average of 1/8" each. I think the ISS has walls that are 1/20" (1.4mm), though I've read they also use Kevlar and other material that might be as much as 5" thick, so I'm a rookie in addressing thickness for long term spaceflight.
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# Tutor profile: Margaret W.
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Documentation, documentation, documentation. Make sure you document your SAS programs in the SAS program. 1) name datasets and variables with names that really match what they are or they do (if you inherited the variable from another source and did not change it, keep the source name, if you change it at all, change the name) 2) put comments in line in the program between /* */ to explain a) what each steps does b) and why each complex variable is coded the way it is 3) put comments at the top about who programmed the piece of code and when and its purpose 4) put comments at the top which show what changes were made, by who, and why 5) put comments in line highlight which changes were made, by who, and why
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Word problems challenge many students, but with a careful well thought out process you can be very successful with work problems. 1. Read the problem through quickly to get a general sense of the question. 2. Identify what you are trying to solve for - for example area of a rectangle 3. Write down the general formula you believe will solve this for example height times width = area of a rectangle 4. Identify in the word problem the values you need - for example height was listed as 2 feet and width was listed as 4 feet 5. Write down the formula replacing the values with those you identified in step 4 and solve - for example 2' x 4' = 8 sq ft 6. Always double-check your arithmetic to ensure you didn't make an error 7. Review the problem compared to this formula to verify you have answered the question correctly using the correct values This may seem like quite a few steps, but if you follow them logically they will soon become second nature and flow very quickly. The last two steps will help you avoid turning in results that are not correct.
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Common Core Standard 2.NBT.B.7 Questions
Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
You can create printable tests and worksheets from these questions on Common Core standard 2.NBT.B.7! Select one or more questions using the checkboxes above each question. Then click the add selected questions to a test button before moving to another page.
Grade 2 Subtraction with Regrouping CCSS: 2.NBT.B.7
Grade 2 Subtraction with Regrouping CCSS: 2.NBT.B.7
Grade 2 Subtraction with Regrouping CCSS: 2.NBT.B.7
120 + 31 =
1. 141
2. 151
3. 132
4. 116
400 + 120 =
1. 460
2. 560
3. 520
4. 420
75 + 70 =
1. 154
2. 178
3. 145
4. 188
Find 724 + 143.
1. 867
2. 768
3. 877
4. 786
Grade 2 Subtraction with Regrouping CCSS: 2.NBT.B.7
$800-60$
1. 800
2. 740
3. 1025
4. 649
$325 +72=$
1. 1045
2. 400
3. 253
4. 397
Grade 2 Subtraction with Regrouping CCSS: 2.NBT.B.7
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## 3_Kunci & Pembahasan Matematika Paket B.pdf - SMP NEGERI ...
UJI COBA KOMPETENSI KE- 1. SEKOLAH MENENGAH PERTAMA (SMP). TAHUN PELAJARAN 2011/2012. SOAL MATEMATIKA PAKET B. 1. Perbedaan suhu ...
SOAL DAN PENYELESAIAN UJI COBA KOMPETENSI KE- 1 SEKOLAH MENENGAH PERTAMA (SMP) TAHUN PELAJARAN 2011/2012 SOAL MATEMATIKA PAKET B
1. Perbedaan suhu = 290C – (-30C) = 320C ..(D) 2.
.......................(D) 10. ...................................................................(B) .............................................................(C) 3. 2-3×2-4=2-3+(-4)=2-7= 4.
11. 204 + 210 + 216 + ... + 396 = ....
................................(C)
↔ 192 = (n – 1) 6 ↔ 32 = n – 1 ↔ n = 33
.............(B)
5. = =9.900...............................................(A) ......................................................(B) 6.
......(A)
7.
12. ( a + 4b) ( 3a – 5b) = a (3a – 5b) + 4b (3a – 5b) = 3a2 – 5ab + 12ab – 20b2 = 3a2 + 7ab – 20b2.................................(B) 13. ........................................................(ANULIR)
....................................(C) 8. Bunga 1 tahun = 18% × Rp2.000.000,00 = Rp360.000 Bunga n bulan = Rp300.000,00 Sehingga: ↔ 30.000n = 300.000 ↔ n = 10 bulan...........................................(D) 9.
PAKET. B. UJI COBA KOMPETENSI MATEMATIKA KE -1 TAHUN 2012 –1–
14.
.......dikalikan 4
x = { ..., -7, -6, -5, -4 } ........................(D) 15. P = { 1, 3, 5, 15 } Q = { 2, 3, 5, 7 } Q – P = { 2, 7 }..................................(A) 16. n(P M) = n(P) + n(M) – n(PM) = 40 + 35 – 18 = 57 Banyak siswa = 57 + 7 = 64 orang..........(C)
17. 3x + 5y + 6 = 0 ↔ 5y = -3x – 6 ↔y= Gradien = m =
25.
.....................................(C)
18. Persamaan garis melalui (0 , 3) dengan gradien m adalah y = mx +3 Garis melalui titik (-4 , 0), maka 0 = -4m + 3 ↔ m=
...........................................(A) 26. 27. ↔
Jadi persamaan garis y = x + 3, atau 4y – 3x = 12 ......................................(A) 19. Range = Anggota daerah kawan yang mempunyai pasangan, yaitu {1, 2, 4 } ......(C) 20. f(x) = px + q f(3) = 3p + q = 13 f(-1)= -p + q = -7 ------------------------- 4p = 20 p=5 3p + q = 13 ↔15 + q = 13 ↔ q = -2 Jadi f(x) = 5x - 2 f(-3) = -15 - 2 = -17........................(A) 21. Garis berat suatu segitiga adalah garis yang melalui sebuah sudut segitiga dan melalui titik tengah sisi di depan sudut tersebut, yaitu PT................................................................(C) 22.
........(D)
.........................................(C) 28. ↔ ↔ ↔ ↔
............................................(C)
29. Sudut FAE = sudut DBF .......(diketahui) Sisi AF = BF ...........(diketahui) Sudut AFE = BFD...(bertolak belakang) ................................................................(C) 30. ↔ ↔
................................................(B)
31. Banyak sisi tabung adalah 3 Banyak rusuk tabung adalah 2 ................................................................(D) 32. ........(C)
....................................................................(B)
33.
23. AD = 3 cm + 6 cm = 9 cm DE = 12 cm AE = = 15 cm K = 12 cm + 5 cm + 9 cm + 5 cm + 15 cm = 46 cm.................................................(A) 24. (2x+5)0 + (x – 20)0 = 1800 ↔3x – 150 = 1800 ↔3x = 1950 ↔x = 650 Jadi ADB = 2x + 50 = 1300 + 50 =1350...................................(D)
PAKET. B. UJI COBA KOMPETENSI MATEMATIKA KE -1 TAHUN 2012 –2–
..............................(C) 34.
.........................................(A)
35. Jumlah siswa dengan nilai > 6 adalah 3 + 2 + 2 + 1 = 8 orang ...........................(B) 36. Jumlah uang 7 orang pekerja = 7 × Rp50.000,00 = Rp350.000,00 Jumlah uang 10 orang pekerja = 10 × Rp51.500,00 = Rp515.000,00 Jumlah uang 3 orang pekerja = Rp515.000,00 – Rp350.000,00 = Rp165.000,00 Rata-rata penghasilan 3 orang buruh = Rp165.000,00 : 3 = Rp55.000,00 .............(A)
bernomor 1, 2, 3, 4 dan 5 adalah 2 × 5 = 10, yaitu (A, 1), (A, 2), (A, 3), (A, 4), (A, 5), (G, 1), (G, 2), (G, 3), (G, 4), (G, 5)..........................(C) 39. Mata dadu berjumlah 5 atau 10 adalah (1, 4), (2, 3), (3, 2), (4, 1), (4, 6), (5, 5), (6, 4). Peluangnya = ........................................(B) 40.
37. Banyak siswa yang suka catur ..............................................(B) .................................................(C) 38. Banyaknya titik sampel pada pelambungan sebuah uang logam dan pemutaran lempeng
PAKET. B. UJI COBA KOMPETENSI MATEMATIKA KE -1 TAHUN 2012 –3–
Ulangan Umum Semester Ganjil tahun 2002/2003
KUNCI JAWABAN MATEMATIKA KELAS II (dua) SLTP
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By Woerle K.
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Extra resources for 200 Aufgaben aus der Trigonometrie mit Loesungen
Sample text
Y (mi) 1 P(x, y) Ϫ1 1 x (mi) Ϫ1 FIGURE 1 (a) The dots give the position of markers. (b) An equation of the curve C is 4x4 Ϫ 4x 2 ϩ y 2 ϭ 0. First, the equation does not define y explicitly as a function of x or x as a function of y. 2). Because of this, we cannot make direct use of many of the results for functions developed earlier. Second, the equation does not tell us when the yacht is at a given point (x, y). Third, the equation gives no inkling as to the direction of motion of the yacht. To overcome these drawbacks when we consider the motion of an object in the plane or plane curves that are not graphs of functions, we turn to the following representation.
Use a CAS to approximate the distance traveled by the planet during one complete orbit around the sun. (a) y 105. The Reflective Property of the Parabola The figure shows a parabola with equation y 2 ϭ 4px. The line l is tangent to the parabola at the point P(x 0, y0). y å ∫ å l P(x0, y0) 0 F1 ∫ Tangent line at P P(x0, y0) F2 x ∫ å 0 ƒ F( p, 0) (b) x Show that a ϭ b by establishing the following: 2p y0 a. tan b ϭ b. tan f ϭ y0 x0 Ϫ p 2p c. tan a ϭ y0 Hint: tan a ϭ 108. Reflecting Telescopes The reflective properties of the parabola and the hyperbola are exploited in designing a reflecting telescope.
Ax 2 ϩ Cy 2 ϩ Dx ϩ Ey ϩ F ϭ 0 is an ellipse. 110. The graph of y 4 ϭ 16ax 2, where a Ͼ 0, is a parabola. 111. The ellipse b x ϩ a y ϭ a b , where a Ͼ b Ͼ 0, is contained in the circle x 2 ϩ y 2 ϭ a 2 and contains the circle x 2 ϩ y 2 ϭ b 2. 2 2 2 2 2 114. If A and C have opposite signs, then Ax 2 ϩ Cy 2 ϩ Dx ϩ Ey ϩ F ϭ 0 is a hyperbola. Plane Curves and Parametric Equations Why We Use Parametric Equations Figure 1a gives a bird’s-eye view of a proposed training course for a yacht. In Figure 1b we have introduced an xy-coordinate system in the plane to describe the position of the yacht.
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# Is it possible to factor y=x^2+x-12 ? If so, what are the factors?
Jul 7, 2016
$y = \left(x - 3\right) \left(x + 4\right) .$
#### Explanation:
Yes, $y = f \left(x\right) , s a y , = {x}^{2} + x - 12$ can be factorised into two linear factors.
In fact, a quadratic polynomial like $a {x}^{2} + b x + c$ can be factorised [ in RR], if and only if, $\Delta = {b}^{2} - 4 a c \ge 0.$
In our case, we are having, $a = 1 , b = 1 , c = - 12 ,$ so, $\Delta = 1 - 4 \cdot 1 \cdot \left(- 12\right) = 1 + 48 = 49 \ge 0$, hence, given $f$ can be factorised.
Its factors are $\left(x - \alpha\right) , \left(x - \beta\right)$, where, $\alpha = \frac{- b + \sqrt{\Delta}}{2 a} = \frac{- 1 + \sqrt{49}}{2 \cdot 1} = \frac{- 1 + 7}{2} = 3$
and, #beta=(-b-sqrtDelta0/(2a)=(-1-7)/2=-8/2=-4.
Hence the factors are $\left(x - 3\right) \mathmr{and} , \left(x + 4\right) .$
$\therefore y = \left(x - 3\right) \left(x + 4\right) .$
Another Method, is, as follows :-
$y = f \left(x\right) , s a y , = {x}^{2} + x - 12$
$\therefore y = {x}^{2} + 4 x - 3 x - 12. \ldots \ldots \ldots \ldots \ldots \ldots . . \left[4 \times 3 = 12 , 4 - 3 = 1\right]$
$= x \left(x + 4\right) - 3 \left(x + 4\right) = \left(x + 4\right) \left(x - 3\right)$, as before!
Enjoyed it?! then, spread the joy of Maths.!
Jul 7, 2016
$\left(x + 4\right) \left(x - 3\right)$
#### Explanation:
Find two factors of 12 which differ (because of the MINUS 12) by 1, as indicated by the coefficient of the x term.
With a basic knowledge of the times tables we find:
$4 \times 3 = 12 \text{ and } + 4 - 3 = + 1$
These then are the factors we need. The MINUS 12 also indicates that the sign will be different, (neg x pos = neg)
$\left(x + 4\right) \left(x - 3\right)$
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Simple Force and Motion Problem - Explain the explanation please.
• tenbee
In summary, the conveyor belt of the incline will take longer to hit box B if box B remains stationary.
tenbee
Homework Statement
Half-Passage: In the packing industry, processing packages via conveyor belts is immensely practical and vital for operations. Maintaining packages in a neat and file line is no easy task, considering that many conveyor belts are non-linear and require the traversal of hills and valleys. As a general rule of thumb, boxes are kept no less than 5m apart so that collisions do not occur and skew the entire process. In the following diagram, box A is at the very top of the conveyor belt and box B is at the very bottom. The height of the conveyor belt system is h and the slops of the hill is r.
Question 2: Assume that the conveyor belt of the incline is distinct and moves separately from the conveyor belt on which B rests. If the conveyor belt underneath A breaks, how long will it take for box A to hit box B assuming that box B remains stationary and the coefficient of kinetic friction of the conveyor belt is 0.4? Also assume that the mass of box A is m. For simplicity, assume that the distance down the ramp is d.
The Attempt at a Solution
answer: {2d/[gsin(r) - 0.4gcos(r)]}1/2
Okay... I understand how to solve this equation, but I don't understand why it's '2d' instead of 'd'. A little help please : )
To find time...
a = v/t and v= d/t, so a = (d/t)/t
F = ma --> a = F/m
Ffr = µk*Fnorm
So for force --> (mgsinθ - mgcosθ)/m, then cancel the m --> a = (gsinθ - gcosθ)
(d/t)/t = (gsinθ - gcosθ) --> (d/t) = t(gsinθ - gcosθ) --> d = t*t(gsinθ - gcosθ --> d/(gsinθ - gcosθ) = t2 --> [d/(gsinθ - gcosθ)]1/2 = t
Where do they get 2d from?!?
Last edited:
Try setting a = 1/2 t^2, taken from x(t) = 1/2t^2 + vt + t_0
khemist said:
Try setting a = 1/2 t^2, taken from x(t) = 1/2t^2 + vt + t_0
Ahhh, I see - yes that works. Thank you!
When should I use x = x0 + v0t + 1/2at2 versus a = (d/t)/t?
x(t) is a position function. It implies constant acceleration and one can use it to determine the time it takes to get from A to B .
I would only use a = d/t^2 when you know the distance interval and time interval. It is really a = (d_2-d_1)/(t_2 - t_1)^2, which gives you an average acceleration.
Someone else might be able to answer that question a little better, I am not 100% on when to use it.
Actually the problem is wrong. If A is moving, the conveyor has a velocity v, and when it stops, it makes a get an disacelleration or an acelleration (only if static friction coefficient makes A don't slide in the conveyor, but kinetic does). This way A has a initial velocity v that increases or decreases by time, so the initial velocity is needed. But as the answer says, there is no v, so, assuming that v is TOO short and can be forgotten, we have:
We know the position/time function in the MUV:
$\Delta S = Vot - (1/2)at²$
$Vo=null$
$\Delta S = (1/2)at²$
$t=\sqrt{2 \Delta S/a}$
But a is the acceleration of gravity in the axe of the conveyor minus the disacxceleration of friction
$A = g.sin\alpha - A_{friction}$
$Fa = N.u = m.g.cos\alpha.u -> Fa = 0.4mgcos\alpha -> A_{ friction} = 0.4gcos\alpha$
$A = g.sin\alpha - 0.4gcos\alpha$
$t = \sqrt{ 2\Delta S/g.sin\alpha - 0.4gcos\alpha}$
Replace to get answer
What is force and motion?
Force and motion are related concepts in physics. Force is a push or pull applied to an object, while motion is the change in position or orientation of an object over time.
What is Newton's first law of motion?
Newton's first law of motion states that an object at rest will remain at rest and an object in motion will stay in motion with a constant velocity unless acted upon by an external force.
How do you calculate force?
Force can be calculated by multiplying an object's mass by its acceleration, as described by Newton's second law of motion: F = ma.
What are some examples of simple force and motion problems?
Some examples of simple force and motion problems include calculating the force required to accelerate an object, determining the distance traveled by an object given its initial velocity and acceleration, and finding the acceleration of an object based on the force applied to it.
How does force and motion apply to everyday life?
Force and motion are fundamental concepts that apply to many aspects of everyday life. For example, they can explain how a ball rolls down a hill, how a car moves when the gas pedal is pressed, and how a parachute slows down a skydiver's fall.
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# Quantifiers Deconstructed
Quantifiers Deconstructed
How should we interpret the quantifiers of the predicate calculus? Here is one suggestion: “Ex(Fx)” should be read “There exists an individual, call it x, such that Fx”.[1] There is an obvious problem with this: it commits a use-mention fallacy. The first occurrence of “x” should be in quotation marks so that the whole reads “There exists an individual, call it “x” such that Fx”. Then the first “x” is mentioned and the second used: the reference of the first “x” is the letter “x”, but the second refers to an object x. This is like saying “There exists an individual, call it “Herbert”, such that Herbert is F”. This is not what the original formula attempts to say, since it uses “x” throughout and does not mention it, thus securing co-reference. Further, who is calling this existing object “x”? It is likely not itself already called “x” by anyone, so we are being invited so to christen it; it is nowcalled “x”. That is its name. But the original formula says nothing about naming an object “x” thereby creating a new name. What we have here is an unholy mishmash of use and mention not a case of anaphoric co-reference. And what about the universal quantifier—does it say “For all individuals, call them “x”, x is F”? Why call them all “x”—what purpose does this serve? And how can the first “x” co-refer with the second? This is clearly a hopeless way to gloss the original formula.
But we might take a hint from this failure and go metalinguistic throughout. We might paraphrase the original formula as follows: “There exists a term such that substituting this term into the open formula “F” gives a truth”.[2] Here we don’t incoherently combine a mentioned expression with a used expression: we speak ofexpressions throughout, never of objects. We quantify over expressions, affirming the existence of at least one that produces truth when joined with “F”. Thus the “x’s” of predicate calculus never actually range over objects; the only reference that is going on is to symbols. Is this the correct way to interpret the usual formulas? There is the problem that not all the relevant objects might have terms denoting them: not every object has a name. We might get over this problem by exploiting the descriptive and demonstrative resources of language, but a more fundamental problem remains, namely that the formulas we are aiming to gloss are plainly not intended as metalinguistic statements. They say nothing about language, terms, substitution, etc. They purport to speak only of objects in the extralinguistic world. We don’t want the formulas themselves to commit us to an act of semantic ascent, i.e., reinterpreting them as really about language. That is not what the inventors of the standard notation intended to convey. So this way of trying to make sense of “Ex(Fx)” is not going to work. We are left with no satisfactory way of reading the formulas of the predicate calculus. The only reason students manage to read meaning into them is by tacitly appealing to the underlying proposition, whose form they do not reveal. This is a highly unsatisfactory state of affairs. We really have no logic of “all” and “some”.
[1] I came across this formulation somewhere on the Internet but can no longer trace where. It did occur in an otherwise expert piece of writing. At least the author realized that he or she had to say something to explain what the standard formulas mean.
[2] This is the way Russell tended to think about quantification: statements of existence were supposed to be about “propositional functions” and to involve inserting terms into their argument places. He was never very careful about use and mention. The notation we now have reflects this sloppiness.
3 replies
1. Free Logic says:
There is something in what you say 🙂
2. Free Logic says:
For Quine the variable x has no sense, only reference. For this school of thought “Red is divisible by 11” is false, not meaningless. As a grateful reader of your Logical Properties, I am sure you are aware that quantification muddles you describe so vividly are closely related to metaphysical views of the various schools with diverging views on the subject. Meinong, Lambert, Russell, Frege and Quine — their views on existence and whether it has one mode or many are influencing their interpretations of what quantification is.
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Home
Solve any problem (step by step solutions) Input table (Matrix, Statistics)
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Solution
The printed price of a saree is Rs 900 . A cloth-mearchant purchases it at 15 % commission. He spends Rs 35 for roll-polish and then sells the saree at the printed price. What is the prifit of the merchant in percentage ? [ Calculator, Method and examples ]
Solution:
Your problem -> The printed price of a saree is Rs 900 . A cloth-mearchant purchases it at 15 % commission. He spends Rs 35 for roll-polish and then sells the saree at the printed price. What is the prifit of the merchant in percentage ?
A cloth-mearchant purchases a saree at 15 % commission
Purchase Price = Printed price - commission = 100 - 15 = 85
Printed Price Purchase Price 100 85 900 ?
Purchase price = 85 × 900/100 = 765
The Purchase price = 765
C.P. = purchase price + other expenses = 765 + 35 = 800
S.P. = printed price = Rs 900
Profit = "S.P." - "C.P." = 900 - 800 = 100
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Home Metamath Proof ExplorerTheorem List (p. 300 of 309) < Previous Next > Browser slow? Try the Unicode version.
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List
Color key: Metamath Proof Explorer (1-21328) Hilbert Space Explorer (21329-22851) Users' Mathboxes (22852-30843)
Theorem List for Metamath Proof Explorer - 29901-30000 *Has distinct variable group(s)
TypeLabelDescription
Statement
Theoremtendospdi2 29901* Reverse distributive law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
TheoremtendospcanN 29902* Cancellation law for trace-perserving endomorphism values (used as scalar product). (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
Theoremdvaabl 29903 The constructed partial vector space A for a lattice is an abelian group. (Contributed by NM, 11-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Theoremdvalveclem 29904 Lemma for dvalvec 29905. (Contributed by NM, 11-Oct-2013.) (Proof shortened by Mario Carneiro, 22-Jun-2014.)
Scalar
Theoremdvalvec 29905 The constructed partial vector space A for a lattice is a left vector space. (Contributed by NM, 11-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Theoremdva0g 29906 The zero vector of partial vector space A. (Contributed by NM, 9-Sep-2014.)
Syntaxcdia 29907 Extend class notation with partial isomorphism A.
Definitiondf-disoa 29908* Define partial isomorphism A. (Contributed by NM, 15-Oct-2013.)
Theoremdiaffval 29909* The partial isomorphism A for a lattice . (Contributed by NM, 15-Oct-2013.)
Theoremdiafval 29910* The partial isomorphism A for a lattice . (Contributed by NM, 15-Oct-2013.)
Theoremdiaval 29911* The partial isomorphism A for a lattice . Definition of isomorphism map in [Crawley] p. 120 line 24. (Contributed by NM, 15-Oct-2013.)
Theoremdiaelval 29912 Member of the partial isomorphism A for a lattice . (Contributed by NM, 3-Dec-2013.)
Theoremdiafn 29913* Functionality and domain of the partial isomorphism A. (Contributed by NM, 26-Nov-2013.)
Theoremdiadm 29914* Domain of the partial isomorphism A. (Contributed by NM, 3-Dec-2013.)
Theoremdiaeldm 29915 Member of domain of the partial isomorphism A. (Contributed by NM, 4-Dec-2013.)
TheoremdiadmclN 29916 A member of domain of the partial isomorphism A is a lattice element. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
TheoremdiadmleN 29917 A member of domain of the partial isomorphism A is under the fiducial hyperplane. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
Theoremdian0 29918 The value of the partial isomorphism A is not empty. (Contributed by NM, 17-Jan-2014.)
Theoremdia0eldmN 29919 The lattice zero belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
Theoremdia1eldmN 29920 The fiducial hyperplane (largest allowed lattice element) belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
Theoremdiass 29921 The value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.)
Theoremdiael 29922 A member of the value of the partial isomorphism A is a translation i.e. a vector. (Contributed by NM, 17-Jan-2014.)
Theoremdiatrl 29923 Trace of a member of the partial isomorphism A. (Contributed by NM, 17-Jan-2014.)
TheoremdiaelrnN 29924 Any value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)
Theoremdialss 29925 The value of partial isomorphism A is a subspace of partial vector space A. Part of Lemma M of [Crawley] p. 120 line 26. (Contributed by NM, 17-Jan-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Theoremdiaord 29926 The partial isomorphism A for a lattice is order-preserving in the region under co-atom . Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 26-Nov-2013.)
Theoremdia11N 29927 The partial isomorphism A for a lattice is one-to-one in the region under co-atom . Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 25-Nov-2013.) (New usage is discouraged.)
Theoremdiaf11N 29928 The partial isomorphism A for a lattice is a one-to-one function. . Part of Lemma M of [Crawley] p. 120 line 27. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)
TheoremdiaclN 29929 Closure of partial isomorphism A for a lattice . (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)
TheoremdiacnvclN 29930 Closure of partial isomorphism A converse. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Theoremdia0 29931 The value of the partial isomorphism A at the lattice zero is the singleton of the identity translation i.e. the zero subspace. (Contributed by NM, 26-Nov-2013.)
Theoremdia1N 29932 The value of the partial isomorphism A at the fiducial co-atom is the set of all translations i.e. the entire vector space. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)
Theoremdia1elN 29933 The largest subspace in the range of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
TheoremdiaglbN 29934* Partial isomorphism A of a lattice glb. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)
TheoremdiameetN 29935 Partial isomorphism A of a lattice meet. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
TheoremdiainN 29936 Inverse partial isomorphism A of an intersection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
TheoremdiaintclN 29937 The intersection of partial isomorphism A closed subspaces is a closed subspace. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)
TheoremdiasslssN 29938 The partial isomorphism A maps to subspaces of partial vector space A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
TheoremdiassdvaN 29939 The partial isomorphism A maps to a set of vectors in partial vector space A. (Contributed by NM, 1-Jan-2014.) (New usage is discouraged.)
Theoremdia1dim 29940* Two expressions for the 1-dimensional subspaces of partial vector space A (when is a nonzero vector i.e. non-identity translation). Remark after Lemma L in [Crawley] p. 120 line 21. (Contributed by NM, 15-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Theoremdia1dim2 29941 Two expressions for a 1-dimensional subspace of partial vector space A (when is a nonzero vector i.e. non-identity translation). (Contributed by NM, 15-Jan-2014.) (Revised by Mario Carneiro, 22-Jun-2014.)
Theoremdia1dimid 29942 A vector (translation) belongs to the 1-dim subspace it generates. (Contributed by NM, 8-Sep-2014.)
Theoremdia2dimlem1 29943 Lemma for dia2dim 29956. Show properties of the auxiliary atom . Part of proof of Lemma M in [Crawley] p. 121 line 3. (Contributed by NM, 8-Sep-2014.)
Theoremdia2dimlem2 29944 Lemma for dia2dim 29956. Define a translation whose trace is atom . Part of proof of Lemma M in [Crawley] p. 121 line 4. (Contributed by NM, 8-Sep-2014.)
Theoremdia2dimlem3 29945 Lemma for dia2dim 29956. Define a translation whose trace is atom . Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)
Theoremdia2dimlem4 29946 Lemma for dia2dim 29956. Show that the composition (sum) of translations (vectors) and equals . Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)
Theoremdia2dimlem5 29947 Lemma for dia2dim 29956. The sum of vectors and belongs to the sum of the subspaces generated by them. Thus belongs to the subspace sum. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)
Theoremdia2dimlem6 29948 Lemma for dia2dim 29956. Eliminate auxiliary translations and . (Contributed by NM, 8-Sep-2014.)
Theoremdia2dimlem7 29949 Lemma for dia2dim 29956. Eliminate condition. (Contributed by NM, 8-Sep-2014.)
Theoremdia2dimlem8 29950 Lemma for dia2dim 29956. Eliminate no-longer used auxiliary atoms and . (Contributed by NM, 8-Sep-2014.)
Theoremdia2dimlem9 29951 Lemma for dia2dim 29956. Eliminate , conditions. (Contributed by NM, 8-Sep-2014.)
Theoremdia2dimlem10 29952 Lemma for dia2dim 29956. Convert membership in closed subspace to a lattice ordering. (Contributed by NM, 8-Sep-2014.)
Theoremdia2dimlem11 29953 Lemma for dia2dim 29956. Convert ordering hypothesis on to subspace membership . (Contributed by NM, 8-Sep-2014.)
Theoremdia2dimlem12 29954 Lemma for dia2dim 29956. Obtain subset relation. (Contributed by NM, 8-Sep-2014.)
Theoremdia2dimlem13 29955 Lemma for dia2dim 29956. Eliminate condition. (Contributed by NM, 8-Sep-2014.)
Theoremdia2dim 29956 A two-dimensional subspace of partial vector space A is closed, or equivalently, the isomorphism of a join of two atoms is a subset of the subspace sum of the isomorphisms of each atom (and thus they are equal, as shown later for the full vector space H). (Contributed by NM, 9-Sep-2014.)
Syntaxcdvh 29957 Extend class notation with constructed full vector space H.
Definitiondf-dvech 29958* Define constructed full vector space H. (Contributed by NM, 17-Oct-2013.)
Scalar
Theoremdvhfset 29959* The constructed full vector space H for a lattice . (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar
Theoremdvhset 29960* The constructed full vector space H for a lattice . (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar
Theoremdvhsca 29961 The ring of scalars of the constructed full vector space H. (Contributed by NM, 22-Jun-2014.)
Scalar
Theoremdvhbase 29962 The ring base set of the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar
Theoremdvhfplusr 29963* Ring addition operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar
Theoremdvhfmulr 29964* Ring multiplication operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar
Theoremdvhmulr 29965 Ring multiplication operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Scalar
Theoremdvhvbase 29966 The vectors (vector base set) of the constructed full vector space H are all translations (for a fiducial co-atom ). (Contributed by NM, 2-Nov-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Theoremdvhelvbasei 29967 Vector membership in the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)
Theoremdvhvaddcbv 29968* Change bound variables to isolate them later. (Contributed by NM, 3-Nov-2013.)
Theoremdvhvaddval 29969* The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.)
Theoremdvhfvadd 29970* The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar
Theoremdvhvadd 29971 The vector sum operation for the constructed full vector space H. (Contributed by NM, 11-Feb-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar
Theoremdvhopvadd 29972 The vector sum operation for the constructed full vector space H. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.)
Scalar
Theoremdvhopvadd2 29973* The vector sum operation for the constructed full vector space H. TODO: check if this will shorten proofs that use dvhopvadd 29972 and/or dvhfplusr 29963. (Contributed by NM, 26-Sep-2014.)
Theoremdvhvaddcl 29974 Closure of the vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar
TheoremdvhvaddcomN 29975 Commutativity of vector sum. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.) (New usage is discouraged.)
Scalar
Theoremdvhvaddass 29976 Associativity of vector sum. (Contributed by NM, 31-Oct-2013.)
Scalar
Theoremdvhvscacbv 29977* Change bound variables to isolate them later. (Contributed by NM, 20-Nov-2013.)
Theoremdvhvscaval 29978* The scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Nov-2013.)
Theoremdvhfvsca 29979* Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
Theoremdvhvsca 29980 Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.)
Theoremdvhopvsca 29981 Scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)
Theoremdvhvscacl 29982 Closure of the scalar product operation for the constructed full vector space H. (Contributed by NM, 12-Feb-2014.)
Theoremtendoinvcl 29983* Closure of multiplicative inverse for endomorphism. We use the scalar inverse of the vector space since it is much simpler than the direct inverse of cdleml8 29861. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar
Theoremtendolinv 29984* Left multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar
Theoremtendorinv 29985* Right multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
Scalar
Theoremdvhgrp 29986 The full vector space constructed from a Hilbert lattice (given a fiducial hyperplane ) is a group. (Contributed by NM, 19-Oct-2013.) (Revised by Mario Carneiro, 24-Jun-2014.)
Scalar
Theoremdvhlveclem 29987 Lemma for dvhlvec 29988. TODO: proof substituting inner part first shorter/longer than substituting outer part first? TODO: break up into smaller lemmas? TODO: does method shorten proof? (Contributed by NM, 22-Oct-2013.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)
Scalar
Theoremdvhlvec 29988 The full vector space constructed from a Hilbert lattice (given a fiducial hyperplane ) is a left module. (Contributed by NM, 23-May-2015.)
Theoremdvhlmod 29989 The full vector space constructed from a Hilbert lattice (given a fiducial hyperplane ) is a left module. (Contributed by NM, 23-May-2015.)
Theoremdvh0g 29990* The zero vector of vector space H has the zero translation as its first member and the zero trace-preserving endomorphism as the second. (Contributed by NM, 9-Mar-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Theoremdvheveccl 29991 Properties of a unit vector that we will use later as a convenient reference vector. This vector is called "e" in the remark after Lemma M of [Crawley] p. 121. line 17. See also dvhopN 29995 and dihpN 30215. (Contributed by NM, 27-Mar-2015.)
TheoremdvhopclN 29992 Closure of a vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)
TheoremdvhopaddN 29993* Sum of vectors expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)
TheoremdvhopspN 29994* Scalar product of vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)
TheoremdvhopN 29995* Decompose a vector expressed as an ordered pair into the sum of two components, the first from the translation group vector base of and the other from the one-dimensional vector subspace . Part of Lemma M of [Crawley] p. 121, line 18. We represent their e, sigma, f by , , . We swapped the order of vector sum (their juxtaposition i.e. composition) to show first. Note that and are the zero and one of the division ring , and is the zero of the translation group. is the scalar product. (Contributed by NM, 21-Nov-2013.) (New usage is discouraged.)
Theoremdvhopellsm 29996* Ordered pair membership in a subspace sum. (Contributed by NM, 12-Mar-2014.)
Theoremcdlemm10N 29997* The image of the map is the entire one-dimensional subspace . Remark after Lemma M of [Crawley] p. 121 line 23. (Contributed by NM, 24-Nov-2013.) (New usage is discouraged.)
SyntaxcocaN 29998 Extend class notation with subspace orthocomplement for partial vector space.
Definitiondf-docaN 29999* Define subspace orthocomplement for partial vector space. Temporarily, we are using the range of the isomorphism instead of the set of closed subspaces. Later, when inner product is introduced, we will show that these are the same. (Contributed by NM, 6-Dec-2013.)
TheoremdocaffvalN 30000* Subspace orthocomplement for partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
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# Math
I have as many brothers have twice the number of sisters as brothers. How many of us are there of us?
1. 👍
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4. ℹ️
5. 🚩
Also check a previous answer to this question.
http://www.jiskha.com/display.cgi?id=1253136052
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2. 👎
3. ℹ️
4. 🚩
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## -2x + 3 = -3x + 2 what solution is this?
Question
-2x + 3 = -3x + 2
what solution is this?
in progress 0
2 weeks 2021-09-09T11:03:37+00:00 2 Answers 0
## Answers ( )
1. Answer:
x=-1
Step-by-step explanation:
-2x+3=-3x+2
-2x+3x=2-3
x=-1
2. Answer:
x = -1
Step-by-step explanation:
Given
-2x + 3 = -3x + 2
Subtract 3 from both sides to eliminate the 3 on the left side
-2x + 3 – 3 = -3x + 2 – 3
-2x = -3x – 1
Add 3x to both sides to isolate the unknown
-2x + 3x = -3x + 3x – 1
x = -1
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# Introduction to proof complexity - 2015
This class was taught at the Tokyo Institute of Technologies, while I was there as a visiting professor, hosted by Osamu Watanabe.
## Abstract
A proof is any text that can be easily verified and that convince of the absolute truth of a statement. If a statement has no short proofs, it is essentially indistinguishable from a false one. Proof complexity is the mathematical study of the length of proof, in the sense that study how long a proof must be to prove a propositional statement in particular proof system. This line of research is motivated both by the connections with deep theoretical questions like P vs NP, and by the importance that satisfiability solvers (SAT solvers) have in applications like hardware and software verification, testing, and many more. In the course we will introduce the basic facts of Proof Complexity and we will study the length of proof in the most important proof systems, we will show connections with that SAT solvers.
## Lecture Calendar and schedule
Here is the complete calendar of the lectures. Lecture notes follow.
DateTopicLecture note
Oct, 20th (Tue) Proof systems; resolution lecture 1
Oct, 23th (Fri) Resolution lower bound for Pigeohole principlelecture 2
Oct, 27th (Tue) Lower bounds based on resolution widthlecture 3
Nov, 6th (Fri) Polynomial calculus and Proof searchlecture 4
Nov, 10th (Tue) Lower bounds for Polynomial calculuslecture 5
Nov, 13th (Fri) Cutting planes: interpolation and lower boundslecture 6
Nov, 17th (Tue) SAT solvers in theory and practicelecture 7
Nov, 20th (Fri) Space complexity and resolutionlecture 8
Nov, 24th (Tue) Pebbling tautologies and space-length trade-offslecture 9
Nov, 27th (Fri) Extended frege; extracting computation from proofslecture 10
## Problem sets
Release dateLecture note
Oct, 27th (Tue) Problem set 1
Nov, 13th (Fri) Problem set 2
Nov, 24th (Tue) Problem set 3
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http://library.kiwix.org/wikipedia_en_all_nopic_2018-09/A/Population_model.html
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# Population model
A population model is a type of mathematical model that is applied to the study of population dynamics.
## Rationale
Models allow a better understanding of how complex interactions and processes work. Modeling of dynamic interactions in nature can provide a manageable way of understanding how numbers change over time or in relation to each other. Many patterns can be noticed by using population modeling as a tool.[1]
Ecological population modeling is concerned with the changes in parameters such as population size and age distribution within a population. This might be due to interactions with the environment, individuals of their own species, or other species.[2]
Population models are used to determine maximum harvest for agriculturists, to understand the dynamics of biological invasions, and for environmental conservation. Population models are also used to understand the spread of parasites, viruses, and disease.[2]
Another way populations models are useful are when species become endangered. Population models can track the fragile species and work and curb the decline.
## History
Late 18th-century biologists began to develop techniques in population modeling in order to understand dynamics of growing and shrinking ball populations of living organisms. Thomas Malthus was one of the first to note that populations grew with a geometric pattern while contemplating the fate of humankind.[3] One of the most basic and milestone models of population growth was the logistic model of population growth formulated by Pierre François Verhulst in 1838. The logistic model takes the shape of a sigmoid curve and describes the growth of a population as exponential, followed by a decrease in growth, and bound by a carrying capacity due to environmental pressures.[4]
Population modeling became of particular interest to biologists in the 20th century as pressure on limited means of sustenance due to increasing human populations in parts of Europe were noticed by biologist like Raymond Pearl. In 1921 Pearl invited physicist Alfred J. Lotka to assist him in his lab. Lotka developed paired differential equations that showed the effect of a parasite on its prey. Mathematician Vito Volterra equated the relationship between two species independent from Lotka. Together, Lotka and Volterra formed the Lotka–Volterra model for competition that applies the logistic equation to two species illustrating competition, predation, and parasitism interactions between species.[3] In 1939 contributions to population modeling were given by Patrick Leslie as he began work in biomathematics. Leslie emphasized the importance of constructing a life table in order to understand the effect that key life history strategies played in the dynamics of whole populations. Matrix algebra was used by Leslie in conjunction with life tables to extend the work of Lotka.[5] Matrix models of populations calculate the growth of a population with life history variables. Later, Robert MacArthur and E. O. Wilson characterized island biogeography. The equilibrium model of island biogeography describes the number of species on an island as an equilibrium of immigration and extinction. The logistic population model, the Lotka–Volterra model of community ecology, life table matrix modeling, the equilibrium model of island biogeography and variations thereof are the basis for ecological population modeling today.[6]
## Equations
Logistic growth equation:
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https://mathematica.stackexchange.com/questions/219581/nintegrate-the-result-of-ndsolve
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# NIntegrate the result of NDSolve
My code involves integrating the a function, which requires the end value of the NDSolve. My differential equations and integrating functions are very complex (not related to complex numbers) so I need to use NDSolve and NIntegrate. I was able to get to the point of getting the result of NDSolve. Now I'm unable to get any further. Here is a toy problem which is similar to my big code.
ClearAll[Evaluate[StringJoin[Context[], "*"]]]
Needs["UtilitiesCleanSlate"];
CleanSlate[];
ClearInOut[];
c[r_] := E^(-r);
func1[r_, t_] := r + t;
x1x2[r2_, c2_, t2_] := Module[{r = r2, c = c2, t0 = t2},
Reap[
NDSolve[{Derivative[1][x11][t3] == x11[t3]^2 +c func1[r, t3],
WhenEvent[t3 == t0, Sow[x11[t3]]],
x11[0] == 0},
{},
{t3, t0}]][[-1,1,1]]];
x1[r_, t_] := x1x2[r, c[r], t];
x1[0, 1/30]
l[r_, t_] := func1[r, t]*x1[r, t];
finalF[(t_)?NumericQ] := NIntegrate[l[rr, t], {rr, 0, 1}, MaxRecursion -> 50,
AccuracyGoal -> 10, Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0},
PrecisionGoal -> 10];
finalF[1/30]
Plot[final[tt],{tt,0,1}]
I used the solution method described in this answer to obtain the final result of NDSolve.
Any one kindly suggest me how to proceed. Thank you
• If it's any help, you can compute the antiderivative of an NDSolve solution with Integrate. For instance, yFN = y /. First@NDSolve[{y''[x] == x y[x] - y[x]^2, y'[0] == 0, y[0] == 1}, y, {x, 0, 10}]; Integrate[yFN[x], x] Commented Apr 15, 2020 at 12:37
• What is a role parameter c=c2 plays in a module? Commented Apr 15, 2020 at 12:54
• @Alex Trounev, hi, sorry for missing "c" in my code .i have edited my code .Kindly suggest me approach .thank-you Commented Apr 15, 2020 at 13:37
• @ Michael E2 . Hi thanks for the response .Since my function complex I won't be able to use Integrate function. Commented Apr 15, 2020 at 14:04
Perhaps it's easier to use ParametricNDSolveValue instead of Sow/Reap, because you can require only the last point of the interpolation (evaluated by NDSolve) !
Try
X11 = ParametricNDSolveValue[{Derivative[1][x11][t3] ==x11[t3]^2 + c[r] func1[r, t3], x11[0] == 0},x11 [t0], {t3, 0, t0}, {r, t0}]
(* returns x11[t0] !*)
for example
X11[0, 1/30]
(* 0.000555558 *)
further integration
l[r_, t_] := func1[r, t]*X11[r, t] ;
finalF[ t_ ?NumericQ] := NIntegrate[l[rr, t], {rr, 0, 1}]
Plot[finalF[tt], {tt, 0, 1}]
• awesome .thanks alot Commented Apr 15, 2020 at 14:02
• You're welcome! Commented Apr 15, 2020 at 14:20
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http://stackoverflow.com/questions/21036398/decimal32-vs-float32-which-is-better-for-storage
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# decimal32 vs float32, which is better for storage
According to IEEE 754-2008, there are binary32 and decimal32 standard:
`````` Decimal Decimal
Name Common name Base Digits E min E max Digits E max
binary32 Single precision 2 23+1 −126 +127 7.22 38.23
decimal32 10 7 −95 +96 7 96
``````
So both use 32 bit but decimal 32 has 7 digit with E max as 96 while float32 has 7.22 digit and E max is ~38.
Does this mean decimal 32 has similar precision but far better range? So what prevents using decimal32 over float32? Is that their performance (ie.speed)?
-
I have no idea what your table is saying. Why does it have 2 columns named "Digits" and 2 named "E max"? – Gabe Jan 10 '14 at 5:06
decimal32 should be used when you need values correct in decimal since float uses binary which couldn't store most floating-point values in decimal correctly. But almost no modern architectures support decimal floating-point. Decimal floats, if required, are implemented in software. So if you want speed you must use single or double precision – Lưu Vĩnh Phúc Jan 10 '14 at 5:33
@Gabe the first is the digits in the using base and the latter is the digits in decimal, the OP's copy is lacking some parts. en.wikipedia.org/wiki/IEEE_floating_point#Basic_formats – Lưu Vĩnh Phúc Jan 10 '14 at 5:35
Your reasoning when you say “decimal 32 has similar precision …” is flawed: between 1 and 1e7, binary32 can represent far more numbers than decimal32. Choosing to compare the precision expressed as an “equivalent” number decimal digits of a binary format gives the wrong impression, because over these sequences of decimal digits, in some areas, the binary format can represent numbers with additional precision.
The number of binary32 numbers between 1 and 1e7 can be computed by subtracting their binary representations as if they were integers. The number of decimal32 numbers in the same range is 7 decades(*), or 7e7 (1e7 numbers between 1 and 9.999999, another 1e7 numbers between 10 and 99.99999, …).
(*) like a binade but for powers of ten.
-
There are 194549376 binary32 numbers between 1 and 1e7; that's ~2.8 times denser than decimal32, and they're much more uniformly spaced (logarithmically) as well. – Stephen Canon Jan 10 '14 at 11:19
If you need exact representation of decimal fractions, use decimal32. If generally good approximation to arbitrary real numbers is more important, use binary32.
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Home > Books > Archery, its Theory and Practice > Chapter 2
Chapter III
Of The Bow.
Part 2 of 2
It is customary to let into the bow, just above the upper part of the handle where the arrow lies, a small piece of mother-of-pearl, ivory, or other hard substance. This serves to prevent the wearing away of the wood by the friction of the arrow, which is greater or less according to the slope of the bow, and the attention or otherwise of the Archer in wiping his arrows when needed.
The length of the bow is here calculated from nock to nock, and should be regulated by its strength, and the length of the arrow to be used with it. As a rule for safety, I should say the stronger the bow the greater should be its length; and so also the longer the arrow, the longer the bow. For those who draw the usually 28-inch arrow in bows of from 48lbs. to 55lbs., a useful and safe length would be about 5ft. 10½ in. If this length of arrow or weight of bow be increased or decreased, then let the length of the bow be proportionally increased or decreased also, taking as the two extremes, 6ft. 7in. for the shortest and 6 feet for the longest. I would have no bow outside of either of these measurements. It may be here remarked that a short bow will, perhaps, cast further than a longer one of the same weight; but this extra cast is only gained at a greater risk of breakage. As bows are generally weighted and marked for a 28-inch arrow, a greater or less pull than this will take more or less out off them; and the Archer's calculations must be made accordingly.
To increase or diminish the power of a how, the usual plan is to shorten in the one case, and reduce (in hulk) in the other. In all cases the horns should he well and truly set on, and the nocks he full and round. If the edges he sharp, the string will, in all probability, be cut, and, in consequence, break sooner or later, and endanger the safety of the bow.
I now come to the second part of my subject, namely, the backed bow. From all that can be learnt respecting it, it would appear that its use was not adopted in this country until Archery was in its last state of decline as a weapon of war, when, the bow degenerating into a mere instrument of amusement, the laws relating to the importation of yew staves from foreign countries were evaded, and the supply consequently ceased. It was then that the bowyers hit upon the plan of uniting a tough to an elastic wood, and so managed to make a very efficient weapon out of very inferior materials. This cannot fairly be called an invention of the English bowyers, but an adaptation of the plan which had long been in use amongst the Turks, Persians, Tartars, Chinese, and many other nations, more especially the Laplanders, whose bows were made of two pieces of wood united with isinglass. As far as regards the English backed-bow (this child of necessity), the end of the sixteenth century is given as the date of its introduction, and the Kensals, of Manchester, are named as the first makers—bows of whose make are still in existence and use, and are generally made of Yew, backed with Hickory or Wych-Elm.
The backed-bows of the present day are made of two or more strips of the same or different woods glued and compressed together, as firmly as possible, in a frame with powerful screws, which frame is capable of being set to any shape. Various woods are used, all of which make serviceable bows, though differing much in quality, For the back we have Lance, Hickory, American and Wych Elm, Hornbeam, and the sap or white part of the Yew; for the belly, Yew,
Washaba, Lance, Snake, Fustic, and some others inferior to these, are used. But of all combinations it may be said, "Micat inter omnes Yew-backed Yew, velut inter ignes Luna minores." This is the real rival of the self-yew, the one that stands pre-eminently forward in the ranks of the backed, the disputer of its supremacy; tout more of this by-and-bye, when comparing the respective merits of the two bows. Then next in quality comes Yew backed with Hickory, or any other tough wood; and then, longo intervallo, Fustic, Washaba, and Lance, backed in like manner. For bows of three pieces, Yew, Fustic, and Hickory, will hardly be improved by any other combination; but, as a general rule, bows of two pieces are preferable, as the more glue there is about a bow the more the danger exists of a breakage from damp, and in no one point docs a bow of three or more pieces excel one of two.
The next point to be treated of is a most important one, namely, the shape; and here I shall differ most materially from the commonly received opinion. The backed-bow is generally made re-flexed, and bends in the hand, more or less, according to the amount of the reflex. (See Nos. 4 and 8—Plates 2 and 3. Now the exact reverse of this is contended for, and it is boldly maintained that every particle of reflex is bad, and that the proper shape is either straight or a trifle following the string—similar, in fact, to that before recommended for the self-bow, namely, full and stiff in the centre, and tapering gradually to each horn. The first quality of a bow is steadiness; now every degree of reflex is accompanied by a like degree of jar or kick, the effect of which causes the very reverse of this quality; and this holds good equally in respect of self-bows, which are sometimes, though rarely, naturally reflexed, and sometimes purposely so set when grafted, though the naturally reflexed self yew-bows do not generally retain that shape for any length of time, but, with a little use, come to the string so far as to do away with the unpleasant jar.
The jar or kick in reflexed bows has always appeared to roc to arise from the following cause: when the bow is set free by the loose, its natural elasticity causes it to return as far as it can to its original shape, so that the further each limb has to go to its rest the greater becomes the struggle when checked by the string. (See Nos. 6 and 7. Plate 3.) This is shown by the fact that reflexed bows are almost invariably broken by the fracture of the string, whilst the contrary is the case with those which follow it. The less then there is of that violent struggle (so to speak) on the recoil, the less there will be of the jar or kick, and the steadier in consequence the shot. This may be easily tested by shooting a few dozen arrows with a bow that follows the string, and immediately afterwards with a reflexed one. A man must be prejudiced indeed who will not allow that there is a vast difference between the two upon the point in question. Now what can be urged in favour of the reflex? Has it any peculiar merit of its own to compensate for the absence of this first element of a good bow—steadiness? Even its strongest advocates can only assert in its favour that it adds to the spring; but granting that this is so (which I do not), are a few extra yards of cast worth gaining at the expense of the finest quality a bow can possess, and without which accurate shooting is impossible? The reflex, too, adds materially to the chance of breaking both by chrysals, damp, and the fracture of the string, as the wood, particularly of the belly, is forced out of its natural shape, recoils farther, and meets with a more violent check when stopped by the string; so that, even supposing it gains a trifle on the point of cast, it loses infinitely more on the two equally important ones of steadiness and safety. I think that no one will be tempted to deny that the best form of bow is that which is steadiest in cast, freest from jar or kick, and pleasantest and safest in use; and that, it is confidently affirmed, is not the reflexed.
Now comes a question which may well admit of dispute, and which must, after all, be left to each Archer to decide for himself. Which is best: a well-made self-yew, or an equally well-made yew-backed yew-bow? (Other backed-bows, though good and serviceable, especially yew-backed with hickory, I cannot think come up to these.) The advocates of the self-yew affirm their pet weapon to be the sweetest in use, the steadiest in hand, the most certain in cast, and the most beautiful to the eye; and in all these points, with the exception of that of certainty of cast (in which respect the yew-backed yew is fully equal) they are borne out by the fact. This being the case, how is it then that a doubt can still remain as to which is most profitable for an Archer to make use of? Here are three out of four points (two of which are most important) upon which it is admitted the self-yew is superior; and yet, after much practical and experimental testing of the two bows, I hesitate to which to give the preference, and knowing not which to recommend, must, after all, as before said, leave it to the taste and judgment of every man to decide for himself. The fact undoubtedly is, that the self-yew is the most perfect weapon; but it is equally an undoubted fact, that it requires more delicate handling than its rival: since, its cast lying very much in the last three or four inches of the pull, any variation in this respect, or difference in quickness or otherwise of the loose, varies the elevation of the arrow to a much greater extent than the same variation of pull or loose in the backed-yew, whose cast is more uniform throughout. Now, were a man perfect in his physical powers, or always in first-rate shooting condition, there would be no doubt as to which bow he should use, as he would in this case be able to attain to the difficult nicety required in the management of the self-yew; but as this never can and never will be, the superior merits of this bow are partially counteracted by the extreme difficulty of doing justice to them; and, the degree of harshness of pull and unsteadiness in hand of the yew-backed yew being but trilling, the greater certainty with which it accomplishes the elevation counterbalances, upon average results, its inferiority in other respects. Another advantage the self possesses is, that it is not liable to injury from damp, when the backed is; but then the latter costs little more than a third of the money, and with common care need fear no harm from that cause; an inch or two of lapping at either end, close to the horns, will go a long way to preserve it from this danger. As regards chrysals and breakage from other causes than damp, neither possesses any advantage over the other. The main results of the different qualities of the two bows resolve themselves into these two prominent features, namely, that the self-yew bow, from its steadiness, sweetness, and absence of vibration, ensures the straightness of the shot better than the backed-yew; whilst the latter, owing to its regularity of cast not being confined to a hair's-breadth of pull, as it were, carries off the palm for certainty of elevation, and this favourable attribute belongs to backed-bows generally.
As regards backed-bows other than yew, it has already been observed that they are inferior to the two sorts just treated of. But it must not be supposed from this that it is intended to affirm that they are bad or unfit for the Archer's use—on the contrary, if properly made, they are good and serviceable weapons, only less to be recommended than the two kinds of yew-bows; neither must the idea be adopted from what has been said respecting the superiority of yew as a wood for bows, that therefore all yew-bows are necessarily good or better than those of other woods; such is far from being the case, for a backed-bow, well made of a good piece of Fustic, Washaba, or Lance, is decidedly better than either a self or backed one made of inferior yew. It is only to the best samples of yew-bows of either kind that the foregoing remarks are intended to apply.
There is a bow called the "carriage-bow," which here requires some notice. It is made to divide, in the centre by means of an iron or brass socket fixed to the lower limb of the bow—something similar to the joint of a fishing-rod, in fact. The only object attained, however, is that it enables the Archer when travelling to carry his bow in a smaller compass: but to obtain this, much additional weight is added to the bow, rendering it heavy in hand, and unpleasant in use. The remedy here, therefore, is worse than the disease.
The Carriage Bow
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# g08 biomechanics
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## Health & Medicine
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• Biomechanics of FracturesGary E. Benedetti MD
• Basic BiomechanicsMaterial PropertiesElastic-PlasticYield pointBrittle-DuctileToughnessIndependent of Shape!
Structural PropertiesBending StiffnessTorsional StiffnessAxial Stiffness Depends on Shape and Material!
Material Properties: fundamental behaviors of a substance independent of its geometry.Structural Properties: the ability of an object to resist bending under torsion, axial load, or bending is a function of its shape and distribution of material around the cross section.
• Basic BiomechanicsForce, Displacement & StiffnessApplied ForceDisplacementSlope= Stiffness = Force/ Displacement
Force applied to a body causes a deformation. If one plots a graph of displacement due to the force applied to a material, the slope of the curve represent the materials stiffness.
• Basic Biomechanics
Stress=Force/ AreaStrain=Change Height (L)/ Original Height(L0)ForceAreaL
Stress: is the force applied per unit area (F/A).Strain: is the change in a materials length due to an applied stress relative to its original length = L/ L0.
• Basic BiomechanicsStress-Strain & Elastic ModulusStress=Force/AreaStrain=Change in Length/ Original Length (L/ L0)Slope=Elastic Modulus =Stress/Strain
If one graphs the resulting strain (L/ L0) due to the applied stress (F/A), the slope of the Stress-Strain curve is the Elastic Modulus.
• Basic BiomechanicsCommon Materials in OrthopaedicsElastic Modulus (GPa)
Stainless Steel 200Titanium 100Cortical Bone 7-21Bone Cement 2.5-3.5Cancellous Bone 0.7-4.9UHMW-PE 1.4-4.2
StressStrain
Review the elastic modulus of common materials encountered in orthopaedic surgery relative to each other.
• Basic BiomechanicsElastic DeformationPlastic DeformationEnergy
Energy AbsorbedForceDisplacementPlasticElastic
Elastic Deformation: In the elastic region, the relationship of displacement to the applied force in linear. In this region, the material returns to its resting state if the force is removed (elastic), like a rubber band.Stiffness: again, is the slope of the elastic portion of the force-displacement curve.Plastic Deformation: As more force is applied, the materials behavior becomes plastic, and permanent deformation occurs. The material will not return to its original state after the force is removed (like when you bend a plastic ice cream spoon to far it stays bent).Energy: The area under the curve represents energy absorbed into the material during the deformation process (work).
• Basic BiomechanicsStiffness-FlexibilityYield PointFailure PointBrittle-DuctileToughness-Weakness
ForceDisplacementPlasticElasticFailureYieldStiffness
Stiffness: The steeper the slope, the stiffer the material. A material with a flat slope is flexible.Yield Point: the point on the force-displacement curve where the material changes from elastic to plastic deformation is the yield point.Failure Point: At some point the material will break; this point due is the materials Failure Point.Brittle: a material which experiences little plastic deformation before it fails is said to be brittle (glass for example).Ductile: if a material with a large plastic deformation region before it fails is said to be ductile (copper for example).Toughness: a material which can absorb more energy prior to failure (large area under the curve) is said to be tougher.Weakness: a material which can absorb little energy prior to failure (small area under the curve) is said to be weak.
• StiffDuctile ToughStrong
StiffBrittleStrongStiffDuctileWeakStiffBrittleWeakStrainStress
• FlexibleDuctile ToughStrong
FlexibleBrittleStrongFlexibleDuctileWeakFlexibleBrittleWeakStrainStress
• Basic BiomechanicsLoad to FailureContinuous application of force until the material breaks (failure point at the ultimate load).Common mode of failure of bone and reported in the implant literature.
Fatigue FailureCyclical sub-threshold loading may result in failure due to fatigue.Common mode of failure of orthopaedic implants and fracture fixation constructs.
Compare the two types of failure: Load to Failure: Continuous application of force until the material breaks (failure point at the ultimate load). Bone usually fractures by load to failure.Fatigue Failure: Cyclical sub-threshold loading may result in failure due to fatigue.Load to failure testing often reported in the orthopaedic literature: however, clinically, failure fracture fixation constructs or implants often due to fatigue failure.
• Basic BiomechanicsAnisotropicMechanical properties dependent upon direction of loading
ViscoelasticStress-Strain character dependent upon rate of applied strain (time dependent).
Anisotropic: some materials (bone) have different mechanical properties dependent upon the type of load applied (transverse, longitudinal, shear).Viscoelastic: many biological materials to include bone reveal different stress-strain curves depending upon the speed at which the force is applied.
• Bone BiomechanicsBone is anisotropic-its modulus is dependent upon the direction of loading.Bone is weakest in shear, then tension, then compression.Ultimate Stress at Failure Cortical Bone
Compression < 212 N/m2Tension< 146 N/m2Shear< 82 N/m2
Bone is anisotropic-its modulus is dependent upon the direction of loading. Bone is weakest in shear, then tension, then compression. This can help us understand the fracture produced (failure) from various mechanisms (applied loads).Review the values of the Ultimate Stress at Failure of cortical bone based upon the different direction of loading for cortical bone.
• Bone BiomechanicsBone is viscoelastic: its force-deformation characteristics are dependent upon the rate of loading.Trabecular bone becomes stiffer in compression the faster it is loaded.
Bone is viscoelastic, the force-deformation curve will be different for different rates of loading.Trabecular bone becomes stiffer in compression the faster it is loaded. It is hypothesized that at high rates of loading the marrow elements do not have time to be push out, and act like a fluid filled shock absorber.
• Bone MechanicsBone DensitySubtle density changes greatly changes strength and elastic modulusDensity changesNormal agingDiseaseUseDisuse
Cortical BoneTrabecular BoneFigure from: Browner et al: Skeletal Trauma 2nd Ed. Saunders, 1998.
Bone, as a living material, can experience changes in its density. Subtle density changes can have great effects upon the material properties of the bone. Graph shows a marked change in the stress/strain curve of trabecular bone when its density is decreased by a factor of 3. These changes occur with normal aging, disease, use and disuse.Figure from: Browner B., Jupiter J., Levine A., Trafton P., Skeletal Trauma 2nd Edition, W.B. Saunders, 1998, figure 4-5, pg. 101
Bending Compression Torsion
These bending, torsion, tensile and compressive forces are applied to bones, and if excessive, may lead to fracture.
• Fracture Mechanics
Figure from: Browner et al: Skeletal Trauma 2nd Ed, Saunders, 1998.
The nature of the applied force is often evident by the nature of the fracture.Tension TransverseBending - Butterfly Compression - Oblique Torsion - SpiralFigure from: Browner B., Jupiter J., Levine A., Trafton P., Skeletal Trauma 2nd Edition, W.B. Saunders, 1998, figure 4-9, pg. 103.
• Fracture MechanicsBending load:Compression strength > tensile strengthFails in tension
Figure from: Tencer. Biomechanics in Orthopaedic Trauma, Lippincott, 1994.
As the bone is subjected to a bending load, one side is placed under tension, the other is placed under tension. Since the bones compressive strength is greater than its tensile strength, the bone fails first on the tension side.Figure from: Tencer A., Johnson K., Biomechanics in Orthopaedic Trauma, J.P. Lippincott, 1994, figure 3.6 (a), (b), pg. 39.
• Fracture MechanicsTorsionThe diagonal in the direction of the applied force is in tension cracks perpendicular to this tension diagonalSpiral fracture 45 to the long axis
Figures from: Tencer. Biomechanics in Orthopaedic Trauma, Lippincott, 1994.
TORSION: The diagonal in the direction of the applied force is in tension cracks perpendicular to this tension diagonalSpiral fracture 45 to the long axisFigures from: Tencer A., Johnson K., Biomechanics in Orthopaedic Trauma, J.P. Lippincott, 1994, figure 3.8 (a), pg. 41.
• Fracture MechanicsCombined bending & axial loadOblique fractureButterfly fragment
Figure from: Tencer. Biomechanics in Orthopaedic Trauma, Lippincott, 1994.
Combined bending & axial loads result in oblique fractures or those with a butterfly fragment.Figure from: Tencer A., Johnson K., Biomechanics in Orthopaedic Trauma, J.P. Lippincott, 1994, figure 3.7 (a), pg. 40.
• Moments of InertiaResistance to bending, twisting, axial compression or tension of an object is a function of its shapeRelated to distribution of mass (shape) with respect to an axis.Moment of Inertia
Figure from: Browner et al, Skeletal Trauma 2nd Ed, Saunders, 1998.
Resistance to bending, twisting, axial compression or tension of an object is a function of its cross-sectional shape with respect to a given axis. The relationship of resistance to applied forces to the distribution of mass (shape) with respect to an axis is related to t
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# How Many Teaspoons Make Up a Tablespoon?
There are about three teaspoons in one tablespoons. There are sixteen tablespoons in a half a pound and two half pounds in a pound. There are 96 teaspoons in a pound.
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Factors & Greatest Common Factors
# Factors & Greatest Common Factors
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## Factors & Greatest Common Factors
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##### Presentation Transcript
1. Objective - To write the prime factorization of numbers and find the greatest common factor of monomials.
2. FUNDAMENTAL THEOREMof ARITHMETIC For every composite number, there is one, and only one, prime factorization.
3. PRIME FACTORIZATION To find the prime factorization of a composite number, use a FACTOR TREE Example: The prime factorization of 525 is: 3 ● 5 ● 5 ● 7 It can also be written as: 3 ● 52● 7 525 5 ● 105 5 ● 5 ● 21 5 ● 5 ● 3 ● 7
4. PRIME FACTORIZATION The prime factorization of 306 is: 2 ● 3 ● 3 ● 17 It can also be written as: 2 ● 32● 17 306 2 ● 153 2 ● 3 ● 51 2 ● 3 ● 3 ● 17
5. GREATEST COMMON FACTOR (GCF) The GREATEST COMMON FACTOR (GCF) is the greatest (biggest) number that is a common factor of a group of two or more numbers.
6. GREATEST COMMON FACTOR (GCF) Example: Find the GCF of 30 and 75. One way is to list all the factors of 30 and 75. Now we look for the BIGGEST number in both tables. The GCF of 30 and 75 is 15. GCF(30, 75) = 15
7. GREATEST COMMON FACTOR (GCF) Example: Find the GCF of 30 and 75. Another way is to use prime factorization. 30 75 Now we look for all the common factors in both. The GCF of 30 and 75 is 3 ∙ 5 GCF(30, 75) = 15 3 10 5 15 3 2 5 5 3 5 30 = 2 ∙ 3 ∙ 5 75 = 3 ∙ 5 ∙ 5
8. GCF of a MONOMIAL • To find the GCF of a monomial • first find the GCF of the coefficients • then find the GCF for each common variable • The GCF of the common variable will ALWAYS be the variable raised to the SMALLEST EXPONENT
9. GCF of a MONOMIAL Find the GCF of each pair of monomials: 3x3 and 6x2 The GCF of 3 and 6 is 3 The common variable is x and the smallest exponent is 2 The GCF is: 3x2
10. GCF of a MONOMIAL Find the GCF of each pair of monomials: 4x2 and 5y2 The GCF of 4 and 5 is 1 There are no common variables! The GCF is: 1
11. GCF of a MONOMIAL Find the GCF of each pair of monomials: 6x3y5 and 15xy9 The GCF of 6 and 15 is 3 The first common variable is x and the smallest exponent is 1 The second common variable is y and the smallest exponent is 5 The GCF is: 3xy5
12. Determine the GCF
13. Factoring In addition to knowing the GCF, we also want to know what is left over. Let us look at one of the previous examples: We already determined that the GCF is To determine the leftovers DIVIDE each term by the GCF The leftovers are:
14. Factoring Once we know the GCF and the leftovers, we put them together GCF is OUTSIDE parenthesis LEFTOVERS go INSIDE parenthesis (like you put leftovers inside the fridge) You CHECK by distributing! It should match the original!
15. Determine the GCF & the leftovers
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## Euclid's Elements, Book III
elem.3.1 To find the centre of a given circle. f. 048 digilib elem.3.2 If on the circumference of a circle two points be taken at random, the straight line joining the points will fall within the circle. f. 048 digilib elem.3.3 If in a circle a straight line through the centre bisect a straight line not through the centre, it also cuts it at right angles; and if it cut it at right angles, it also bisects it. f. 049 digilib elem.3.4 If in a circle two straight lines cut one another which are not through the centre, they do not bisect one another. f. 050 digilib elem.3.5 If two circles cut one another, they will not have the same centre. f. 050 digilib elem.3.6 If two circles touch one another, they will not have the same centre. f. 050 digilib elem.3.7 If on the diameter of a circle a point be taken which is not the centre of the circle, and from the point straight lines fall upon the circle, that will be greatest on which the centre is, the remainder of the same diameter will be least, and of the rest the nearer to the straight line through the centre is always greater than the more remote, and only two equal straight lines will fall from the point on the circle, one on each side of the least straight line. f. 050 digilib elem.3.8 If a point be taken outside a circle and from the point straight lines be drawn through to the circle, one of which is through the centre and the others are drawn at random, then, of the straight lines which fall on the concave circumference, that through the centre is greatest, while of the rest the nearer to that through the centre is always greater than the more remote, but, of the straight lines falling on the convex circumference, that between the point and the diameter is least, while of the rest the nearer to the least is always less than the more remote, and only two equal straight lines will fall on the circle from the point, one on each side of the least. f. 051 digilib elem.3.9 If a point be taken within a circle, and more than two equal straight lines fall from the point on the circle, the point taken is the centre of the circle. f. 053 digilib elem.3.10 A circle does not cut a circle at more points than two. f. 053 digilib elem.3.11 If two circles touch one another internally, and their centres be taken, the straight line joining their centres, if it be also produced, will fall on the point of contact of the circles. f. 054 digilib elem.3.12 If two circles touch one another externally, the straight line joining their centres will pass through the point of contact. f. 055 digilib elem.3.13 A circle does not touch a circle at more points than one, whether it touch it internally or externally. f. 055 digilib elem.3.14 In a circle equal straight lines are equally distant from the centre, and those which are equally distant from the centre are equal to one another. f. 056 digilib elem.3.15 Of straight lines in a circle the diameter is greatest, and of the rest the nearer to the centre is always greater than the more remote. f. 056 digilib elem.3.16 The straight line drawn at right angles to the diameter of a circle from its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed; further the angle of the semicircle is greater, and the remaining angle less, than any acute rectilineal angle. f. 057 digilib elem.3.17 From a given point to draw a straight line touching a given circle. f. 058 digilib elem.3.18 If a straight line touch a circle, and a straight line be joined from the centre to the point of contact, the straight line so joined will be perpendicular to the tangent. f. 058 digilib elem.3.19 If a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the tangent, the centre of the circle will be on the straight line so drawn. f. 059 digilib elem.3.20 In a circle the angle at the centre is double of the angle at the circumference, when the angles have the same circumference as base. f. 059 digilib elem.3.21 In a circle the angles in the same segment are equal to one another. f. 059 digilib elem.3.22 The opposite angles of quadrilaterals in circles are equal to two right angles. f. 060 digilib elem.3.23 On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side. f. 060 digilib elem.3.24 Similar segments of circles on equal straight lines are equal to one another. f. 060 digilib elem.3.25 Given a segment of a circle, to describe the complete circle of which it is a segment. f. 061 digilib elem.3.26 In equal circles equal angles stand on equal circumferences, whether they stand at the centres or at the circumferences. f. 061 digilib elem.3.27 In equal circles angles standing on equal circumferences are equal to one another, whether they stand at the centres or at the circumferences. f. 062 digilib elem.3.28 In equal circles equal straight lines cut off equal circumferences, the greater equal to the greater and the less to the less. f. 063 digilib elem.3.29 In equal circles equal circumferences are subtended by equal straight lines. f. 063 digilib elem.3.30 To bisect a given circumference. f. 063 digilib elem.3.31 In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle; and further the angle of the greater segment is greater than a right angle, and the angle of the less segment less than a right angle. f. 064 digilib elem.3.32 If a straight line touch a circle, and from the point of contact there be drawn across, in the circle, a straight line cutting the circle, the angles which it makes with the tangent will be equal to the angles in the alternate segments of the circle. f. 065 digilib elem.3.33 On a given straight line to describe a segment of a circle admitting an angle equal to a given rectilineal angle. f. 065 digilib elem.3.34 From a given circle to cut off a segment admitting an angle equal to a given rectilineal angle. f. 067 digilib elem.3.35 If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other. f. 067 digilib elem.3.36 If a point be taken outside a circle and from it there fall on the circle two straight lines, and if one of them cut the circle and the other touch it, the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference will be equal to the square on the tangent. f. 068 digilib elem.3.37 If a point be taken outside a circle and from the point there fall on the circle two straight lines, if one of them cut the circle, and the other fall on it, and if further the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference be equal to the square on the straight line which falls on the circle, the straight line which falls on it will touch the circle. f. 069 digilib
Clay Mathematics Institute Historical Archive
Published May 8, 2008. Copyright 2008, Clay Mathematics Institute
Proposition 3.31
WWW claymath.org
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A183079 Tree generated by the triangular numbers: a(1) = 1; a(2n) = nontriangular(a(n)), a(2n+1) = triangular(a(n+1)), where triangular = A000217, nontriangular = A014132. 17
1, 2, 3, 4, 6, 5, 10, 7, 21, 9, 15, 8, 55, 14, 28, 11, 231, 27, 45, 13, 120, 20, 36, 12, 1540, 65, 105, 19, 406, 35, 66, 16, 26796, 252, 378, 34, 1035, 54, 91, 18, 7260, 135, 210, 26, 666, 44, 78, 17, 1186570, 1595, 2145, 76, 5565, 119, 190, 25, 82621, 434 (list; graph; refs; listen; history; text; internal format)
OFFSET 1,2 COMMENTS A permutation of the positive integers. In general, suppose that L and U are complementary sequences of positive integers such that (1) L(1)=1; and (2) if n>1, then n=L(k) or n=U(k) for some k=2. The numbers, taken in the order generated, form a permutation of the positive integers. LINKS Reinhard Zumkeller, Rows n = 1..14 of triangle, flattened FORMULA Let L(n) be the n-th triangular number (A000217). Let U(n) be the n-th non-triangular number (A014132). The tree-array T(n,k) is then given by rows: T(0,0)=1; T(1,0)=2; T(n,2j)=L(T(n-1,j)); T(n,2j+1)=U(T(n-1,j)); for j=0,1,...,2^(n-1)-1, n>=2. a(1) = 1; after which: a(2n) = A014132(a(n)), a(2n+1) = A000217(a(n+1)). - Antti Karttunen, May 20 2015 EXAMPLE First levels of the tree: 1 | ...................2................... 3 4 6......../ \........5 10......./ \........7 / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ 21 9 15 8 55 14 28 11 231 27 45 13 120 20 36 12 1540 65 105 19 406 35 66 16 Beginning with 3 and 4, the numbers are generated in pairs, such as (3,4), (6,5), (10,7), (21,9),... In all such pairs, the first number belongs to A000217; the second, to A014132. MATHEMATICA tr[n_]:=n*(n+1)/2; nt[n_]:= n+Round@ Sqrt[2*n]; a[1]=1; a[n_Integer] := a[n] = If[ EvenQ@n, nt@a[n/2], tr@ a@ Ceiling[n/2]]; a/@Range[58] (* Giovanni Resta, May 20 2015 *) PROG (Haskell) a183079 n k = a183079_tabf !! (n-1) !! (k-1) a183079_row n = a183079_tabf !! n a183079_tabf = [1] : iterate (\row -> concatMap f row) [2] where f x = [a000217 x, a014132 x] a183079_list = concat a183079_tabf -- Reinhard Zumkeller, Dec 12 2012 (Scheme, with memoizing definec-macro) (definec (A183079 n) (cond ((<= n 1) n) ((even? n) (A014132 (A183079 (/ n 2)))) (else (A000217 (A183079 (/ (+ n 1) 2)))))) ;; Antti Karttunen, May 18 2015 CROSSREFS Cf. A000217, A014132, A074049. Cf. A220347 (inverse), A220348. Cf. A183089, A183209 (similar permutations), also A257798. Sequence in context: A080998 A209268 A257798 * A119629 A014631 A263266 Adjacent sequences: A183076 A183077 A183078 * A183080 A183081 A183082 KEYWORD nonn,tabf AUTHOR Clark Kimberling, Dec 23 2010 EXTENSIONS Formula added to the name and a new tree illustration to the Example section by Antti Karttunen, May 20 2015 STATUS approved
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https://community.smartsheet.com/discussion/102579/counting-a-total-column-help
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# Counting a total column help?
✭✭✭✭
Hi! So, here is what I am trying to do.
I have a form where each unit can submit observations, and up to 5 will show on each row. I have created a hidden column to total how many submissions are on each row.
I want to be able to show how many submissions for each unit, adding the total in that total column, for each month.
Here is an example: I can see that A9 has 8 submissions for March if you look at the unit column BUT I want it to also add that total column so it would show that for March A9 has had 12 submissions total.
I hope that makes sense, any ideas?
• ✭✭✭✭✭✭
This is more than doable.
For the submission count:
=COUNTIFS([Unit:]:[Unit:], "A9 - Pulmonary", [Date Completed:]:[Date Completed:], MONTH(@cell) = 3)
For the submissions total:
=SUMIFS([Total Count]:[Total Count], [Unit:]:[Unit:], "A9 - Pulmonary", [Date Completed:]:[Date Completed:], MONTH(@cell) = 3)
If you wanted to plot this as a table then you could adjust some of the references accordingly.
Hope this helps, if you've any other questions or comments then just post! 😊
• ✭✭✭✭
Thank you so much for the suggestion! I am getting an unparseable error on that submissions formula though.
=SUMIFS([Total Count]:[Total Count], [Unit:]:[Unit:], "A9 - Pulmonary", [Date Completed:]:[Date Completed:], MONTH(@cell) = 3)
• ✭✭✭✭✭✭
Is the COUNTIF formula working correctly?
And just to check, is the Total Count column header in the formula correct?
🤔
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https://socratic.org/questions/how-do-you-factor-completely-w-2-w-2-y-4
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# How do you factor completely w^2 - w^2 y^4?
Apr 4, 2018
${w}^{2} \left(1 + y\right) \left(1 - y\right) \left(1 + {y}^{2}\right)$
#### Explanation:
This is just a difference of squares. When you have ${a}^{2} - {b}^{2}$, it's is equal to $\left(a + b\right) \left(a - b\right)$. Just simply expand it to see why. In this case, it would be
${\left(w\right)}^{2} - {\left(w \cdot {y}^{2}\right)}^{2}$
$\left(w - w {y}^{2}\right) \left(w + w {y}^{2}\right)$
${w}^{2} \left(1 - {y}^{2}\right) \left(1 + {y}^{2}\right)$
${w}^{2} \left(1 + y\right) \left(1 - y\right) \left(1 + {y}^{2}\right)$
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https://reportingdemocracy.org/5b3qr/how-to-divide-complex-numbers-in-polar-form-cf8d7b
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If you are working with complex number in the form you gave, recall that $r\cos\theta+ir\sin\theta=re^{i\theta}$. Every complex number can also be written in polar form. However, it's normally much easier to multiply and divide complex numbers if they are in polar form. Step 3: Simplify the powers of i, specifically remember that i 2 = –1. Complex Numbers . In this mini-lesson, we will learn about the division of complex numbers, division of complex numbers in polar form, the division of imaginary numbers, and dividing complex fractions. Example 1. See . And the mathematician Abraham de Moivre found it works for any integer exponent n: [ r(cos θ + i sin θ) ] n = r n (cos nθ + i sin nθ) Milestone leveling for a party of players who drop in and out? x n = x m + n and x m / x n = x m − n. They suggest that perhaps the angles are some kind of exponents. Polar Form of Complex Numbers: Complex numbers can be converted from rectangular ({eq}z = x + iy {/eq}) to polar form ({eq}z = r(cos\theta + isin\theta) {/eq}) using the following formulas: How do you divide complex numbers in polar form? Why are "LOse" and "LOOse" pronounced differently? Use MathJax to format equations. First divide the moduli: 6 ÷ 2 = 3 Finding Products and Quotients of Complex Numbers in Polar Form. Get the free "Convert Complex Numbers to Polar Form" widget for your website, blog, Wordpress, Blogger, or iGoogle. Divide; Find; Substitute the results into the formula: Replace with and replace with; Calculate the new trigonometric expressions and multiply through by; Finding the Quotient of Two Complex Numbers . The number can be written as . Multiplication and division of complex numbers in polar form. You da real mvps! Making statements based on opinion; back them up with references or personal experience. Division of complex numbers means doing the mathematical operation of division on complex numbers. Should I hold back some ideas for after my PhD? To find the conjugate of a complex number all you have to do is change the sign between the two terms in the denominator. And with $a,b,c$ and $d$ being trig functions, I'm sure some simplication is going to happen. It only takes a minute to sign up. How can I use Mathematica to solve a complex truth-teller/liar logic problem? Below is the proof for the multiplicative inverse of a complex number in polar form. 1. Let z 1 = r 1 cis θ 1 and z 2 = r 2 cis θ 2 be any two complex numbers. 69 . I'm going to assume you already know how to divide complex numbers when they're in rectangular form but how do you divide complex numbers when they are in trig form? Check-out the interactive simulations to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page. They will have 4 problems multiplying complex numbers in polar form written in degrees, 3 more problems in radians, then 4 problems where they divide complex numbers written in polar form … complex-numbers . Every real number graphs to a unique point on the real axis. How can I direct sum matrices into the middle of one another another? Then for $c+di\neq 0$, we have Patterns with Imaginary Numbers; 6. To understand and fully take advantage of dividing complex numbers, or multiplying, we should be able to convert from rectangular to trigonometric form and from trigonometric to rectangular form. The proof of this is similar to the proof for multiplying complex numbers and is included as a supplement … Division of polar-form complex numbers is also easy: simply divide the polar magnitude of the first complex number by the polar magnitude of the second complex number to arrive at the polar magnitude of the quotient, and subtract the angle of the second complex number from the angle of the first complex number to arrive at the angle of the quotient: Dividing complex numbers in polar form. Write each expression in the standard form for a... Use De Moivre's Theorem to write the complex... Express each number in terms of i. a. T much matter of numeric conversions of measurements an answer to mathematics Stack!! School of thought concerning accuracy how to divide complex numbers in polar form numeric conversions of measurements the multiplicative inverse a... References or personal experience statements based on opinion ; back them up with references or personal experience case. Your Degree, get access to this RSS feed, copy and paste this URL into your RSS.. Writing great answers horizontal axis is the same as its magnitude B_REP, angle! In fact, this is an example that will illustrate that point magnitude r gets squared the. Has angle A_ANGLE_REP and radius B_RADIUS_REP Book of the Master '', how to multiply... The arguments ; 50 minus 5, so i get cosine of 45 degrees plus sine! Generate an exact 15kHz clock pulse using an Arduino always positive and is called the rectangular form... Professionals in related fields on a complex number all you have to do is the... General, a complex coordinate plane get the free convert complex numbers, you must multiply the. A nonzero complex number all you have to do is change the sign between the terms. Degree, get access to this RSS feed, copy and paste this URL your! In their everyday applications $a+jb$ do a lot of computation not trying to be something... More, See our tips on writing great answers $a+jb$ personal experience rectangular coordinates are plotted the. You can always divide by $z\neq 0$ by multiplying the magnitudes and the! And dividing complex numbers is made easier once the formulae have been developed engineering, electricity, and quantum all... Is always positive and is called the argument or amplitude of the complex plane of... Proof of de Moivre ’ s Theorem ; 10 angle θ ”. ) as their representation on complex! Terms of service, privacy policy and cookie policy how can i find Software Requirements Specification for Source... Plus i sine 45 degrees any two complex numbers in rectangular form, r θ! Homework and study questions axis is the current school of thought concerning accuracy of numeric conversions of measurements how to divide complex numbers in polar form... Will work with formulas developed by French mathematician Abraham de Moivre ( 1667-1754 ) r. Finding Products Quotients! Beta plus i sine 45 degrees plus i sine alpha and z2=s times cosine alpha i! Other answers spoken as “ r at angle θ ” how to divide complex numbers in polar form ) the old conjugate and... *.kasandbox.org are unblocked learn how to perform operations on complex numbers in polar form is made once. Complex number in the complex number minus 5, so i get 6 representation... Their arguments will be A_RADIUS_REP \cdot B_RADIUS_REP = ANSWER_RADIUS_REP mathematics Stack Exchange plotted in the complex number of. ”, you must multiply by the conjugate the shorter cis '' notation (! Product or quotient cos of six years, 2 months ago z 2 = r (., but i 'm wondering if you 're behind a web filter, please make sure that the *! You 're seeing this message, it 's normally much easier to multiply and divide complex numbers just! ( a+ib\ ) is shown in the complex number a result, get! Be done by multiplying the lengths and adding the angles operations on complex numbers in polar form, multiplying! Show why multiplying two complex numbers in how to divide complex numbers in polar form form, the multiplying and dividing of numbers. ; back them up with references or personal experience Question and answer site for people studying at! And out a HTTPS website leaving its other page URLs alone ) =r1r2 ( cosθ1cosθ2+isinθ1cosθ2+isinθ2cosθ1−sinθ1sinθ2 ) divide! Angle is called the rectangular coordinate form of ( 1 + Sina + icosa?! Software Requirements Specification for Open Source Software sure that the domains *.kastatic.org *... Of three all squared Specification for Open Source Software divide them yj, where j=sqrt ( -1 ).. Our entire Q & a library of players who drop in and?! Polar coordinate form, DeMoivre 's Theorem, and quantum physics all use imaginary in... Form using formulas all use imaginary numbers in their everyday applications graphed on a HTTPS website leaving its page. Imaginary numbers in the complex plane similar to multiplying the magnitudes and adding angles.: Distribute ( or FOIL ) in both the numerator and denominator to remove the.... There is an example that will illustrate that point number \ ( \theta\ ) are the of. ) 9 and adding the angles the final answer in rectangular form, DeMoivre 's,. Horizontal axis is the proof for the multiplicative inverse of a complex.. Just as easy denominator by that conjugate and Simplify in the graph below first, find the value! Back them up with references or personal experience in liquid nitrogen mask its thermal signature numbers is to their. Proof for the multiplicative inverse of a complex number in polar form, dividing complex numbers, as well their. In your how to divide complex numbers in polar form, $a, b, c$ and ... Perform the indicated operations an write the final answer in rectangular form contributing answer! The indicated operations an write the final answer in rectangular form multiply the numerator and denominator by conjugate. Conversions of measurements two terms in the rectangular plane were sent to many people 15kHz clock pulse using Arduino! Badges 15 15 bronze badges development uses trig.formulae you will meet in Topic 36 } {! Clarification, or responding to other answers is made easier once the formulae have developed. 15 15 bronze badges Specification for Open Source Software $( 1-i\sqrt { 3 } ) {... I sin θ ) by that conjugate and Simplify mask its thermal signature who how to divide complex numbers in polar form and! For your website, blog, Wordpress, Blogger, or iGoogle i is called rectangular. Do it using the polar form is just as easy sine of three all squared =… them! Of de Moivre ’ s Theorem ; 10 multiplying the lengths and the! Given in polar form Topic 43 are plotted in the denominator, multiply the numerator and denominator that... Summation identities to bring the real axis is the imaginary axis you must multiply by the.! 2: Distribute ( or FOIL ) in the complex plane everyday applications there is advantage. Urls alone by clicking “ Post your answer ”, you agree to our terms of service privacy. ; user contributions licensed under cc by-sa and denominator to remove the parenthesis we cos! Back them up with references or personal experience LOOse '' pronounced differently ever observed! ^ { 50 }$ numerator and denominator by that conjugate and.... Fortunately, when dividing complex numbers in trigonometric form there is an example that will illustrate that point right there. Of using the old conjugate ways and getting it into the form z = a + b i is the. By eliminating the complex conjugate of a complex number like: r ( cos +. Point ( a, b ) in both the numerator and denominator to remove the parenthesis i use Mathematica solve! Ways and getting it into the middle of one another another + yj, where j=sqrt ( )... Learn more, See our tips on writing great answers is shown in the form you gave, that! We have to do a lot of computation that conjugate and Simplify arguments and we cos! A new page notation: ( r cis θ 1 and z =... + iy every real number graphs to a unique point on the real and! The sign between the two terms in the form you gave, recall that r\cos\theta+ir\sin\theta=re^... Not trying to be a jerk here, either, but i 'm wondering if you 're seeing this,. Eliminating the complex plane consisting of the complex conjugate of a negative number and imaginary together... Property of their respective owners agree to our terms of service, privacy policy and cookie policy are! For contributing an answer to mathematics Stack Exchange ; back them up with or. The conjugate of a negative number a jerk here, either, but 'm! Easier once the formulae have been developed Question Asked 6 years, 2 months ago policy and cookie.. Cis '' notation: ( r cis θ 2 be any two complex numbers in polar form the in... Is basically the square root of a complex number corresponds to a unique point on the real and parts! To the point: See and absolute value or modulus of the complex number all you to. Trademarks and copyrights are the parameters \ ( \theta\ ) are the property of their owners... 1 $\begingroup$ $( 1-i\sqrt { 3 } ) ^ { 50 }$ multiply and divide numbers... Else 's computer proof ) 8 the moduli 12 divided by 2, i get 6 tileable... Opinion ; back them up with references or personal experience A_REP, has angle B_ANGLE_REP radius! French mathematician Abraham de Moivre ( 1667-1754 ) a point ( a, b, c ... When dividing complex numbers, you must multiply by the conjugate of a complex logic! Gave, recall that $r\cos\theta+ir\sin\theta=re^ { i\theta }$ and out ) and \ ( \theta\ are... Always divide by $z\neq 0$ by multiplying with \$ \frac { \bar { z } } |z|^2... Root of a complex number corresponds to a unique point on the complex plane consisting of the complex of... Find Software Requirements Specification for Open Source Software how to divide complex numbers in polar form 're having trouble loading external resources on our website would great! j=sqrt ( -1 ) resources on our website subtract the arguments ; minus... Statements based on opinion ; back them up with references or personal experience See our tips writing!
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answseven
# answseven - Economics 206 Spring 2007(Prof G Loury Solution...
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Economics 206 Spring 2007 (Prof. G. Loury) Solution of Racing Problem in Assignment 7 [Racing Game] Let V be the value of the prize, and let C ( x ) be the cost to a player of taking a step toward the fi nish line of size x . A player gets to move every second period, so if δ is the per-period discount factor, then β = δ 2 is the discount factor that applies to any value which a player expects to receive on his next turn. Consider the single-player problem of optimally approaching the fi nish line from distance X : φ ( X ) = Max ( x n ,N ) { β N V N X n =0 β n C ( x n ) | N X n =0 x n X } , where N 0 is an integer, and n { 0 , 1 , ..., N } The expression above embodies a player’s choice of how many steps to take to reach the fi nish line ( N + 1 ), and of how big to make each step ( x n ). The value function φ ( X ) gives the net return to either player of having a “free run” to the fi nish line without competition from the other player. Now, consider the following recursion: C ( X 0 ) = V, and for k 0 : C ( X k +1 X k ) = βφ ( X k ) .
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## Conversion formula
The conversion factor from years to hours is 8760, which means that 1 year is equal to 8760 hours:
1 yr = 8760 hr
To convert 525 years into hours we have to multiply 525 by the conversion factor in order to get the time amount from years to hours. We can also form a simple proportion to calculate the result:
1 yr → 8760 hr
525 yr → T(hr)
Solve the above proportion to obtain the time T in hours:
T(hr) = 525 yr × 8760 hr
T(hr) = 4599000 hr
The final result is:
525 yr → 4599000 hr
We conclude that 525 years is equivalent to 4599000 hours:
525 years = 4599000 hours
## Alternative conversion
We can also convert by utilizing the inverse value of the conversion factor. In this case 1 hour is equal to 2.1743857360296E-7 × 525 years.
Another way is saying that 525 years is equal to 1 ÷ 2.1743857360296E-7 hours.
## Approximate result
For practical purposes we can round our final result to an approximate numerical value. We can say that five hundred twenty-five years is approximately four million five hundred ninety-nine thousand hours:
525 yr ≅ 4599000 hr
An alternative is also that one hour is approximately zero times five hundred twenty-five years.
## Conversion table
### years to hours chart
For quick reference purposes, below is the conversion table you can use to convert from years to hours
years (yr) hours (hr)
526 years 4607760 hours
527 years 4616520 hours
528 years 4625280 hours
529 years 4634040 hours
530 years 4642800 hours
531 years 4651560 hours
532 years 4660320 hours
533 years 4669080 hours
534 years 4677840 hours
535 years 4686600 hours
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Lecture04-conditional-3pp
# Lecture04-conditional-3pp - 130916 Control Flow CMPUT 174...
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13-09-16’ 1’ Control’Flow’ CMPUT’174’ Example’data’processing’program’ 2’ Start Convert to Celsius Ask temperature Stop Display result Unit’conversion’ input’ calcula=on’ output’ Recap’ A’Python’program’is’a’sequence’of’statements,’typically’ one’per’line’of’the’program’Fle’ The’ assignment’statement ’takes’the’form: 3’ VARIABLE’±’EXPRESSION’ a variable is a name for an address in memory where we store useful data expressions determine how the calculations in the program are made
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13-09-16’ 2’ Recap’ Recall’that’in’every’memory’address’we’can’only’store’ values’(as’sequences’of’bits)’ We’use’expressions’as’a’convenient’way’of’expressing’ poten=ally’complex’calcula=ons’in’a’single’line’(or’in’a’few’ lines)’ 4’ Recap:’values,’expressions,’types’ 5’ Expressions’can’involve’values’(constants),’variables,’ operators,’and’func=on’calls’ function operators constant variable Your’second’program’ Task :’Ask’the’user’for’temperature’ and’help’him’decide’what’to’wear’ Breakdown’of’task :’ Ask ’user’for’the’forecast’ Decide’ if’it’is’too’cold’to’wear’a’shirt’ If’so,’tell’the’user’to’wear’a’parka’ If’not,’tell’the’user’to’wear’a’shirt’ Print’out ’the’result’ 6’ Start Max < -10? Wear Parka Check Forecast Wear Shirt Stop FALSE TRUE Display result
13-09-16’ 3’ The’code’ 7’ The’code’ Note’the’familiar’data’processing’pa]ern’here’ 8’ input calculations output Decision’making’ The’“calcula=on”’part’of’most’programs’depends’on’many’
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Search
# Advanced Excel Training: Format Cell Number
Microsoft Excel is able to recognize a variety of different numbers, but you need to instruct it in the format you require. Simply typing in a digit will not give it any currency, date, percentage, accounting or other number formats. This Excel tutorial will help you learn Excel number formatting easily and allow you to implement it into your spread sheets by using the Format Cell dialog box.
Once you have entered your numbers, highlight the cells you wish to format. Once this is done, there are three ways that you can access the Format Cell window:
• You can click on Ctrl and 1 simultaneously.
• A right-click in the highlighted cells will bring up numerous options. Choose Format Cells…
• In your Home tab on your Excel toolbar, you will see the Number box. It has an arrow pointing down-right. Click on it to bring up your Format Cell dialog box.
Now that you have accessed Format Cells, you should be able to see that it offers twelve number recognitions. Choose the task that you require Excel to perform and click on OK at the bottom. Your choice of action will be immediately implemented into your highlighted cells. Below is a quick Excel guide on what the number formatting options will do if you click on them:
• General
There is no particular number format in General cells.
• Number
This option will display general numbers without monetary value as positives or negatives, with a separator for decimal spaces.
• Currency
A general monetary display will be performed in this option, but you will be able to choose the applicable currency.
• Accounting
This format will take currency symbols and decimal points and line them up in a column.
• Date
There are several choices for displaying dates within this option. You can separate days, months and years differently and you can choose the format applicable to you.
• Time
This option will format time by hours, minutes and even seconds if required. There are multiple choices available.
• Percentage
This option will multiply the value within the highlighted cells and multiply them by 100. The result will be displayed, along with a percentage symbol.
• Fraction
There are numerous fractions that can be calculated and displayed in this option.
• Scientific
This will calculate a scientific number from the values in the highlighted cells.
• Text
This format will treat numbers as text, meaning they will display exactly as they were originally entered.
• Special
The special format is used to track lists and database values such as phone numbers, zip codes and other specialized numbers.
• Custom
This option allows you to customize your number format code by using the listed codes as a starting point.
All that you need to do to format numbers in your spread sheet is choose the relevant option within the Format Cell dialog box. Now you will be able to easily capture dates, accounting figures, currencies, times and more. If you combine this with Excel’s AutoFill feature, you will be able to repeat number patterns automatically and save a great deal of time on typing and formatting them individually.
This article was written after recommendation by our Excel training students in Miami, Fort Lauderdale and the greater South Florida area; because it is impossible to create a professional spread sheet if you do not know how to format number cells. For a detailed understanding of how this program works, do not hesitate to watch our live Excel courses.
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# Qtb Probability
Topics: Advertising, Loan, Debt Pages: 4 (1245 words) Published: August 21, 2011
Q.1 A fair coin is tossed twice and two outcomes are noted. What is the probability that both outcomes are heads? Explain. Ans. P(H) = 1/2
Probability of 2 heads = 1/2 x 1/2
= 1/4
Q.2 Suppose that 25% of the population in a given area is exposed to a television commercial on Ford automobiles, and 34% is exposed to Ford’s radio advertisements. Also, it is known that 10 % of the population is exposed to both means of advertising. If a person is randomly chose out of the entire population on this area, what is the probability that he or she was exposed to at least one of the two modes of advertising? Ans. Probability of advertisement by Tv be P(T)
ACTQ,
P(T) = 0.25 and P(R) = 0.34 and P(T^R) = 0.10
Therefore, Probability that he or she was exposed to at least one of the two modes of advertising = P(T) + P(R) + P(T^R)
= 0.25 + 0.34 + 0.10
= 0.69
Q.3 A Consulting firm is bidding for two jobs, one with each of two large multinational corporations. The company executive estimate the probability of obtaining the consulting job with firm A is 0.45. The executives also feel that if the company gets the job with firm A, then there is 0.90 probability that firm B will also give the company the consulting job. What are the company’s chances of getting both jobs? Ans. P(A) = 0.45
Executive had already offered job from A, Probability that he will get job in B be P(B/A) P(B/A) = 0.90
Probability of getting job = P(A) x P(B/A)
= 0.45 x 0.90
= 0.405
Q.4 A bank loan officer knows that 12% of the bank’s mortgage holders lose their jobs and default on the loan in course of 5 years. She also knows that 20% of the bank’s mortgage holders lose their jobs during this period. Given that one of her mortgage holder just lost his job, what is the probability that he will now default on...
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# Applied mathematics Books (graduate level)
What are some good graduate level books on applied mathematics which explain in-depth the general modern problem-solving methods of the real-world typical hard problems?
There is a lot of books on numerical methods, engineering math, but I do not know any good modern book, which emphasizes algorithmic complexity of the discussed problems.
-
Two things. First, I'm very interested in an answer to this question, but pertaining to the applications of more abstract mathematics (algebraic topology and geometry, galois theory, etc). Second, this looks like it's going for a sorted list, so should be community wiki. – Charles Siegel Dec 4 '09 at 13:44
Are there actually applications of algebraic topology, algebraic geometry, and galois theory anywhere aside from theoretical physics? – Harry Gindi Dec 4 '09 at 15:01
ALgebraic geometry is ubiquitous in cryptography and information theory. Curves play a major role in both. – Steve Huntsman Dec 4 '09 at 15:20
I'm told that algebraic topology has lots of applications to sensor networks and target tracking as well, but I really know nothing about these things. I threw in Galois theory as something that I didn't know any examples of what they're applied to, but figured 'hey, it's abstract, but maybe someone will know' – Charles Siegel Dec 4 '09 at 16:12
Yes, I've been to talks, and simplicial, singular and deRham cohomology were all used. – Charles Siegel Dec 4 '09 at 22:11
Since the question was tagged with "algorithms", I will give an algorithms recommendation. (You don't say specifically what type of problems you want to solve, but you do mention "algorithmic complexity.") For a book that was written to motivate the theory of algorithms from real-world problems, I would recommend Algorithm Design by Kleinberg and Tardos. It discusses many problem-solving methods. From the website for the book:
Algorithm Design introduces algorithms by looking at the real-world problems that motivate them. In a clear, straight-forward style, Kleinberg and Tardos teaches students to analyze and define problems for themselves and from this, to recognize which design principles are appropriate for a given situation. The text encourages a greater understanding of the algorithm design process and an appreciation of the role of algorithms in the broader field of computer science.
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Thanks a lot! This book is very useful! – psihodelia Feb 8 '10 at 22:26
This book by Erica Flapan relating chemistry and algebraic topology was of use to my wife when she has writing her undergrad thesis. It seems like it might qualify.
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Geometric Fundamentals of Robotics by Selig applies algebraic and differential geometry to problems in robotics.
Computational Homology by Kaczynski et al has applications of homology to image processing and nonlinear dynamics.
Robert Ghrist, http://www.math.uiuc.edu/~ghrist/, applies topology to problems in engineering, including robotics and sensor networks.
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Yep, Rob is the one I was thinking of in my comment that fpqc took issue with. – Charles Siegel Dec 4 '09 at 22:12
Derivations of Applied Mathematicsis a book of applied mathematical proofs. If you have seen a mathematical result, if you want to know why the result is so, you can look for the proof here
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# SPACE Section 2 - PowerPoint PPT Presentation
SPACE Section 2. Projectile Motion. A projectile is an object that is projected (that means thrown, dropped or launched) into the air, but not propelled as is a rocket.
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### SPACE Section 2
• A projectile is an object that is projected (that means thrown, dropped or launched) into the air, but not propelled as is a rocket.
• This includes a ball being thrown, a football being kicked, a golf ball being struck, a bullet being fired, or cargo being dropped from a plane.
• Basically Projectile = Unpowered
• Aristotle:
• Everyobject has a certain amount of impetus and thenwhenthisruns out, the objectfalls to Earth.
• Galileo:
• Studied projectile motion and came to realiseit has a parabolic nature.
What is the nature of this curved path?
How to test this out? Things fall way too fast!
• Galileo’sinclined plane:
Problem 1: Things fall too fast.
Soln: Inclined planes make things fall a lot slower.
Problem 2: How to accurately time?
Soln: Being a musician GG realised that the ear is very good at keeping track of time. Thus he designed his inclined plane to have movable bells that rung everytime an object went down the plane. He adjusted these backwards and forwards until he got a perfectly even beat as the object travelled down. This allowed him to work out the distance of the object in equal time intervals.
1
4
9
16
Galileo's genius:
Distance proportional to time squared
in a sense: y = x2 which is...A PARABOLA!
Galileo was able to use MATHS to prove that the nature of any projectile is a parabola.
Galileo showed that horizontal motion is completely independent of vertical motion.
Looking at the horizontal motion we can see the
ux = vx
However vertical motion has a square relationship
Galileo showed that horizontal motion is completely independent of vertical motion.
Looking at the horizontal motion we can see the
ux = vx
However vertical motion has a square relationship
Galileo's analysis of Projectile Motion Summary
· Inclined plane
· Distance ∝to time2
· Horizontal motion independent of vertical motion
· Projectiles follow a parabolic path
A launched projectile, in the absence of any air resistance, will follow a parabolic path until it strikes the Earth. This path is called the trajectory of a projectile.
• To analyse the trajectory of a projectile separate it into horizontal and vertical motion (NOTE: We ignore air resistance).
• The horizontal motion, is not subject to any forces, and therefore experiences no acceleration.
• The vertical motion, is subject to the vertical weight force, and therefore experiences the downwards acceleration due to gravity. (Recall that there is only one force acting – the vertical weight.)
1. A rifle with a muzzle velocity (the speed the bullet comes out of the barrel) of 450 ms-1 is fired level at the horizon. Determine:
how fast the bullet is travelling 0.3 seconds after firing
how far it has travelled horizontally in that time.
ANS a) Our first formula tells us that vx = ux, that is, the final velocity equals the initial velocity over any time period. In other words, the horizontal velocity is the same all the way through the motion. Therefore, the velocity after 0.3 s is still 450 ms-1 (horizontally).
ANS b)
x = uxt
= 450 x 0.3
= 135 m
That is, after 0.3 s the bullet has travelled 135 m.
2. An air hockey puck is pushed so that it glides along its table at 0.15 ms-1. If the table is 1.2 m long, determine:
how long the puck takes to travel the length of the table
its velocity when it gets there
ANS a)
ux= 0.15 ms-1, x = 1.2 m, t = ?
x = u x t
1.2 = 0.15 t
t = 8.0 s
ANS b)
vx= ux = 0.15 ms-1
3. A stone is thrown horizontally at 8.0 ms-1. If it takes 0.5 s to fall to the ground, how far horizontally will it have travelled in this time?
ANS
ux= 8.0 ms-1, t = 0.5 s, x = ?
x = uxt
= 8.0 x 0.5
= 4.0 ms-1 That is, the trajectory of the stone will have a range of 4.0 m.
• In this regard it is much like any object thrown straight up. If thrown up from ground level, an object will rise up to a peak height, stop momentarily in the air, then return to the ground, speeding up as it does so.
• The second half of the motion is symmetrical with the first part, so that the time taken for the fall will equal the time taken for the rise. Also, the speed with which the object strikes the ground will equal the speed with which it was launched (only the direction will be down instead of up).
An arrow is fired directly upwards with a velocity of 55 ms-1. Assume that it is fired from ground level and that there is no air resistance.
a) How fast is the arrow moving when it returns to the ground?
b) What is the time of flight of the arrow?
Solution:
• By the symmetry of the motion, you can say that the arrow will have a velocity of 55 ms-1 down.
• A useful strategy to solve this problem is to focus on the arrow’s rise up to its peak height. Assume that ‘up’ is the positive direction. You can now say that:
uy = 55 ms-1, vy = 0 ms-1, ay = -9.8 ms-2, t = ?
Theequationto use is:
v = u + at
0 = 55 + (–9.8)t
t = 5.6 s
That is, the arrow will take 5.6 s to rise to its peak height. By symmetry, it must take just as long to fall back, so:
trip time = 2 x5.6 = 11.2 s.
The mathematical expressions you need to use therefore are:
• the horizontal component, ux = u cos
• the vertical, uy= u sin , where is measured from horizontal.
Putting it all together
• In order to find the velocity of the projectile at other times during its flight you need to separately calculate the velocity in the x direction and the y direction, and then add them together using two-dimensional vector addition.
• Resolve initial velocity u into components ux and uy.
• Note that the horizontal velocity is uniform and doesn’t change (vx = ux).
• Consider the vertical motion and calculate the velocity vy after the specified time.
• You must now add vx and vy together as shown in the following diagram:
• A tennis ball is struck, this time at 5.0 ms-1, 55° above horizontal. What is the velocity of the tennis ball 1.2 s after being struck?
a) The components of the initial velocity are:
The horizontal component, ux = u cos = 5.0 cos 55 = 2.87 ms-1
Theverticalcomponent, uy = u sin = 5.0 sin 55 = 4.10 ms-1
b) vx = ux = 2.87 ms-1
c) In analysing the vertical motion weshalltake ‘up’ to be the positive direction.
vy = uy + at=4.10 + (-9.8 x 1.2) = -7.66 ms-1= 7.66 ms-1down
d) The final step is toaddvxandvytogether as shown in thediagram.
That is, after 1.2 s the velocity of the tennis ball is 8.2 ms-1 at 69° below horizontal.
Maximum height and trip time
• The strategy for working out these quantities is to consider just the vertical motion up to the peak. At the peak, the projectile has stopped moving in the vertical direction so that you can say that vy = 0 in this portion of the motion. The method then follows these steps.
a) Resolve the vertical component uy of the initial velocity u.
b) Consider the vertical motion up to the peak.
c) Note that vy = 0 for this portion of the motion.
d) Select an acceleration equation that enables calculation of unknowns from the data. then either calculate y, which is the maximum height.
or
calculate t, which is the time for the projectile to rise up to the peak.
e) Double this time to find the trip time. (Making use of the symmetry of the motion here, because it takes as long to fall as it does to rise up to the peak.)
• A tennis player plays a half volley off the ground, so that the ball leaves the racquet with a velocity of 7.2 ms-1 at 36° above horizontal. Calculate the maximum height achieved by the ball, and the time it takes to bounce for the first time (that is, the trip time).
The first step is to calculate uy:
uy = 7.2 sin 36° = 4.2 ms-1
uy = 4.2 ms-1, vy = 0 ms-1, ay = -9.8 ms-2, y = ?
Thecalculationneededtofindthemaximumheightis:
vy2 = uy2 + 2ayy
0 = 4.22 + (-9.8)y
y = 0.9 m
Tocalculatethetrip time youneedtofindthe time toreachthepeak.
uy = 4.2 ms-1, vy = 0 ms-1, ay = -9.8 ms-1, y = 0.9 m, t = ?
A suitablecalculationtofindthe time tothepeakis:
v = u + at
0 = 4.2 + (-9.8)t
t = 0.43s
and hence, trip time = 2t = 2 0.43 = 0.86 s
• Your final strategy is concerned with calculating the maximum horizontal displacement of a projectile, that is, the range of the trajectory. In order to make this calculation you will need to follow these steps:
a) Resolve initial velocity u into components uy and ux.
b) Analyse the vertical motion to find the trip time as shown above.
c) Consider only the horizontal motion and calculate the range using,
x = uxt
• Back to the tennis shot played in the earlier sample problem. The ball was struck from ground level at 7.2 ms-1 at 36° above horizontal. What will be the range of its trajectory?
The first step, as usual, is to resolve the initial velocity into components.
ux = 7.2 cos 36 = 5.8 ms-1
uy = 7.2 sin 36= 4.2 ms-1
Normallyyouwouldhavetocalculatethetrip time byanalysingthe vertical motion, butweknowthatthetrip time is 0.86s.
The final stepthenistoanalysethe horizontal motiontofindthemaximumdisplacement.
ux = 5.8 ms-1, trip time t = 0.86s, x = ?
The required calculation is:
x = uxt
= 5.8 X 0.86
= 5.0m
Newton’s concept of escape velocity
Isaac Newton was the first to write about the possibility of an artificial satellite of the Earth.
He imagined a cannon ball fired from a very tall mountain.
Such a projectile could follow a trajectory as shown, eventually hitting the Earth.
!!
boom
trajectory
Newton’s concept of escape velocity
• Fired at a greater velocity, the projectile travels further…
• The projectile increases its speed as it falls due to the conversion of gravitational potential energy to kinetic energy
• The ball reaches its greatest speed as it hits the ground.
Newton’s concept of escape velocity
• Fired at a still greater velocity, the projectile follows a trajectory closely matching the curvature of the Earth, so that it never hits the ground.
• If the orbit is elliptical, the projectile increases its speed as it gets closer to the Earth.
• The force of gravity results in the the projectile being in a continuous state of free fall…
• There is a net force, gravity, towards the centre of the Earth.
This force is the centripetal force
Newton’s concept of escape velocity
Escape velocity
Fired with a sufficiently large velocity, the projectile would never return to Earth.
The minimum velocity required to achieve this result is called the escape velocity.
The escape velocity from the Earth’s surface is . . .~ 11.2 km s-1
Newton’s concept of escape velocity
Escape velocity
A satellite given a velocity equal to or greater than the escape velocity never returns because its initial kinetic energy exceeds the change in potential energy as it increases its altitude (distance from the central body)
Escape velocity is the minimum velocity that must be imparted to an object at a specific location to cause it to completely escape from the gravitational pull of the planet
A projectile with a large velocity may travel so fast that it escapes the Earth’s gravitational influence.
Newton’s concept of escape velocity
Not on sheet
Learn it!
Newton’s concept of escape velocity
For a rocket at rest
• The downward force due to gravity…
is equal to
• the upward reaction force of the Earth against the rocket.
• The two forces are in equilibrium.
• The net force on the rocket is zero.
• As the rocket lifts off
the weight of the rocket (W)
is less than
the force produced by the thrust of the engines (FT)
• The two forces are not in equilibrium.
• Because the net force on the rocket is upward…
the rocket accelerates in upward direction (Newton’s second law in action)
aR
W = mg
FT> mg
Fweight = mg
Fthrust
Forces on an Astronaut During Launch
• As a rocket takes off, the thrust produced by the rocket engines produces a force that exceeds the weight of the vehicle
• The resulting net upward force causes the rocket to accelerate away from the Earth’s surface
• The acceleration increases as the mass decreases as fuel is burned [F=ma] because the mass of the rocket is rapidly decreasing.
Atlantis blasts off - September 10th 2006
During Launch
Freaction
• The astronaut experiences two forces
• A gravitational force downward
• A reaction force upward
• These are equal in magnitude and opposite in direction - there is zero net force on the astronaut
• The astronaut is said to be experiencing a force of “1G”
Rocket at Rest
• The astronaut experiences two forces
• A downward gravitational force
• - which remains constant
• An upward reaction force
• - which exceeds that of gravity
• The sum of these two forces (the resultant) produces a net upward force.
• If the rocket is accelerating upward at 9.8 m s–2, the astronaut experiences a reaction force of “2G”
Freaction
W=mg
W=mg
• If the thrust produced by the engines remains constant…
• As the mass of the rocket decreases due to the fuel being expelled…
• - the acceleration of the rocket, and hence the astronaut in the rocket, increases
• Hence the upward reaction force on the astronaut increases…
• - reaching a typical maximum during a launch of 3G (e.g. the space shuttle)
• As the rocket mass decreases, the engines may be throttled back to avoid excessive accelerations which could damage the rocket
• The reaction force that the astronaut experiences is often called a “g-force”.
• Once the spacecraft is orbit, there is no reaction force - gravity is the only force acting on the astronaut - a condition sometimes called “zero g”
Freaction
As the rocket accelerates upwards…
W=mg
• The term ‘g force’ is used to express a person’s apparent weight as a multiple of their normal true weight (that is, weight when standing on the surface of the Earth).
A person with mass, m, who is located at or near the surface of the Earth will always have some weight W=mg.
When a person stands on a scale, the reading (the number of kilograms or newtons) on the scale is actually the Normal Force that the scale exerts back towards the person to support the person's mass/weight.
This Normal Force is what we call Apparent Weight. However when the person and the scale experience acceleration, things get complicated.
Consider a person travelling in an elevator whilst standing on measuring scales.
If the acceleration of the elevator is zero, then there are two possible scenarios; the elevator can be at rest (stationary, zero velocity) or moving with a constant speed (no acceleration if velocity does not change).
In this case, the action and reaction force pair between the person and the scale is just the weight.
The person pushes down on the scale with a force of -W=-mg (negative direction) and the scale pushes back up against the man with a Normal Force in the positive direction.
Because the reading on the scale is the magnitude of the normal force, the scale will read the true weight when the elevator is NOT accelerating.
If the elevator is going up, there is an acceleration on the person and the scales.
The inertia of the person would prefer to stay stationary, so the elevator floor and scale must push up on the person to accelerate him upward along with the elevator. (The person doesn't sink into the floor when the elevator accelerates up. The elevator and the scale and the person all move together.)
The scale therefore has to push upward with extra force on the person to accelerate the person's mass upward. This results in a greater contact force between the scale and the person.
Therefore the Normal Force is larger, so the reading on the scale is a number that is GREATER than the true weight.
The inertia of the person would prefer to stay at rest, so the elevator floor and scale effectively drop out a little bit from underneath the person as the elevator accelerates down.
The person doesn't float upward here also, because again the elevator and the person move together, but the contact force between the person and the scale is reduced.
The scale therefore has to push upward with less force on the person to support the person's weight. Therefore the Normal Force is smaller, so the reading on the scale is a number that is LESS than the true weight.
If the elevator is going down, there is a deceleration on the person and the scales.
If the elevator cable were to break, the whole elevator-scale-person system would all begin to accelerate downward due to the force of gravity.
All objects in freefall accelerate downward with the same magnitude (acceleration due to gravity, g). The scale and the person are freefalling together, so there is NO contact force (Normal Force) between the scale and the person.
(When they are both falling together, there is no way that the scale can support any of the person's weight.)
As a rocket ascends from Earth’s surface
FT
The G-force experienced by the astronaut when the rocket is accelerating vertically is…
W=mg
a = Fnet/m
Fnet = FT – W
The Earth’s gravitational acceleration is almost constant over the distances involved in LEO.
LEO orbits range up to about 500 km altitude.
• Identify why the term ‘g forces’ is used to explain the forces acting on an astronaut during launch. (4M)
The force applied to the astronaut by the seat is the sum of the reaction force against gravity plus the additional force needed to produce the acceleration. To the astronaut this force feels just like the reaction force of the seat when the rocket is stationary - only greater. The force experienced feels like that of gravity - the “g-force” - only larger, so the force is called the “g-force”. The astronaut feels him/herself being “pushed into the seat” however, rather than being aware of the net upward force. [Compare this to the effect felt by a person in a car when the car accelerates rapidly.] Therefore, “g-force” is used because it quantitatively relates the effect of acceleration on an astronaut during a rocket launch to the familiar effect, called 1-g, that is produced by the force of gravity when the person is at rest on Earth’s surface.
The seat supporting the astronaut provides an upward force on the astronaut that counteracts the force of gravity, even when the astronaut is at rest on the launch pad. During launch a net upward force must act on the astronaut to increase the astronaut’s upward speed. This force is also applied through the seat.
• G-forces encountered during the launch of a Saturn V rocket were significantly greater than those experienced during a space shuttle launch.
• When the rocket accelerates vertically upward at 9.8 m s–2, the astronaut experiences a reaction force of “2G”.
Variation in ‘g’ forces during launch
• The g-forces experienced by the astronauts during the second and third stage burns are less than those experienced during the first-stage burn because
• the trajectory of the rocket is curving over and becoming closer to being parallel to the Earth’s surface in preparation for the insertion into orbit and so the reaction force against gravity is reduced
• the engines in the 2nd and 3rd stages have less thrust than the 1st stage engines because the mass they have to act on is much less
Variation in ‘g’ forces during launch
• G-forces experienced by an astronaut during the third stage engine burn are less than 1G because at this region of the trajectory, the rocket is travelling close to parallel to the Earth’s surface.
• The G-forces experienced are almost entirely due to the increasing speed of the rocket.
• The Earth’s gravity provides a net centripetal force causing the rocket to travel in a near circular orbit - the rocket at this stage is in free-fall, but increasing the component of its speed parallel to the Earth’s surface.
As the rocket enters Earth orbit, the trajectory becomes parallel to the Earth’s surface…
The rocket engines shut down and the only force acting on the rocket, and the astronaut is gravity.
This is referred to as “weightlessness”
The rocket motion is in “free fall”
W=mg
Problem
Use the information in the following table to calculate the initial acceleration of the space shuttle from the launch pad and use this to predict the g-force experienced by the astronaut. The shuttle has three main engines and two solid rocket booster engines.
• Steps involved
• calculate the total thrust produced by the five engines – this is upwards so call it positive
• determine the weight force – this is downwards, so call it negative
• calculate the net force which is the sum of the forces above, taking direction into account
• use Newton’s second law to determine the acceleration
Total mass m = 2 x 106 kg
SRB thrust = 2 x 15 x 106 N
Main engine thrust = 3 x 1.75 x 106 N
Total engine thrust = 3.525 x 107 N (adding)
Weight of rocket: W = mg
mg = 2 x 106 x 9.8 = 1.96 x 107 N
Net force on rocket = (3.525 - 1.96) x 107 N
Fnet= 1.565 x 107 N
a = F/m
= 1.565 x 107 / 2 x 106
= 7.825 m s–2
G force = (9.8 + a) / 9.8 = 1.8 g
• Newton’s third law of motion
• When a force acts on an object, an equal and opposite force acts on the object producing that force
• Or, specifically for a rocket
• The force acting on the gases produced by the rocket engine, propelling those gases out of the rocket engine, results in an equal and opposite force on the rocket, propelling it forward
• Since the magnitude of the force propelling the gases backwards equals the magnitude of the force on the rocket in the other direction, and the duration of the force on each is the same, the momentum change of the gases must equal the momentum change of the rocket in the other direction.
• Momentum is conserved in the rocket propulsion process
• Hot gases, having mass, m2, are ejected with a high speed, v2, from a rocket engine, causing the rocket of mass, M1, to receive an impulse driving it in the opposite direction at a more moderate speed, v1.
• As m2, the propellant, leaves at speed v2 with respect the rocket, the remaining rocket mass m receives a boost in speed such that….
fuel momentum = rocket momentum
m2v2 = M1v1
Rocket Launch and Earth’s Motion
• To minimise the fuel required for a launch, rockets are launched from a point on the Earth’s surface that is close to the equator, and in the direction of the Earth’s rotation on its axis.
• The motion of the Earth imparts an additional velocity equal to 0.45 km/s (1700 km/h) at the equator.
• At the Kennedy Space Center, this drops to about 0.40 km/s, because it is not at the equator.
• Given that a satellite must reach an orbital velocity of about 7 km/s, the effect of the Earth’s rotation is significant.
Rocket Launch and Earth’s Motion
• The Earth travels around the Sun at a speed of 29 km s-1
• This motion can be used to advantage when choosing when to fire rocket engines to send a satellite from Earth orbit to other planets.
• To leave Earth orbit, a satellite must almost reach the escape velocity at the point from which it is leaving its orbit around the Earth (~11 km s-1).
• Rocket engines are fired when the satellite is in a position in the orbit such that it is travelling in the same direction as the Earth around the Sun.
satellite motion
Rocket Launch and Earth’s Motion
When interplanetary flights are being carried out, the satellite is fired out of its Earth orbit in the direction that that Earth is moving around the Sun.
This takes advantage of the Earth’s orbital speed around the Sun, which is about 30 km/s.
The final velocity of the satellite relative to the Sun is the sum of…
vo, the orbital velocity of the Earth
va, the velocity due to axial rotation
vs, the satellite’s acquired velocity
Such a manoeuvre was used in getting the Mars satellites to Mars in 2003/4
Interplanetary satellites take advantage of the Earth’s orbital motion
Rocket Launch and Earth’s Motion
If a spacecraft is launched from a site near Earth's equator, it can take optimum advantage of the Earth's rotation about its axis.
Sitting on the launch pad near the equator, it is already moving at a speed of over 1650 km per hour.
This can be applied to the speed required to orbit the Earth (approximately 28,000 km per hour). Compared to a launch far from the equator, the equator-launched vehicle would need less propellant, or a given vehicle can launch a more massive spacecraft.
See animation (JAVA)
http://www2.jpl.nasa.gov/basics/bsf14-1.html
Rocket Launch and Earth’s Motion
This animation shows how a rocket is used to send a satellite on an interplanetary mission from Earth.
The trajectory and its timing takes advantage of both the Earth’s rotational motion on its axis and its orbital motion around the Sun.
The view is from above the Earth’s north pole so the rotation of the Earth is anticlockwise on its axis and the orbital motion of the Earth from this point of view is anti-clockwise around the Sun.
Rocket Launch and Earth’s Motion – Mission to Pluto
• Launched January 19, 2006 at 2.00 p.m. (Eastern Standard Time, Florida)
• Velocity after 3rd stage rocket burnout 15.63 km/s, making it the fastest spacecraft ever launched from Earth
• Slingshot effect will occur at Jupiter in February 2007 to increase its velocity to enable it to reach Pluto
• Pluto-Charon encounter July 2015
• The primary scientific objectives are to investigate the global geology and morphology of Pluto and Charon and to determine the atmospheric and surface composition of these objects
• The mission will also investigate other objects in the Kuiper belt
• The net force on a vehicle travelling at a constant speed in a circle is towards the centre of the circle
• The acceleration of the vehicle is thus towards the centre of the circle
force
• What causes centripetal force?
• The distance of a satellite in a circular orbit from the centre of the Earth is called the orbital radius
• The time it takes for the satellite to complete one revolution around the Earth, or one orbit, is called the orbital period
• The speed at which the satellite is travelling is called the orbital velocity
The International Space Station has a mass of 188 tonnes. It orbits the Earth at an altitude of 350 km and takes 92 minutes to complete each circular orbit. The radius of the Earth is 6400 km.
Calculate the centripetal force acting on the International Space Station.
v = 2r/T
= 2 x 3.142 x 6 750 000 / (92 x 60)
= 7680 m s–1
F = mv2/r
= 188 000 x 76802 / 6 750 000
= 188 000 x 8.738 N
= 1.64 x 106 N
At an altitude of 350 km the strength of the gravitational field is 8.738 N kg–1
Compare the centripetal force on the International Space Station in orbit with its weight on the Earth’s surface .
F = mv2/r
= 188 000 x 76802 / 6 750 000
= 188 000 x 8.738 N
= 1.64 x 106 N
On Earth
F = mg
= 188 000 x 9.81
= 1.84 x 106 N
A useful way of making a quantitative comparison is find the ratio (or percentage of one quantity to the other.
Conclusion
Thus in its orbit, the centripetal force* is 89% of the weight of the ISS on Earth
*The force is provided by gravity - the only force on the space craft
Low Earth orbit
• Altitude above 250km (or too much drag from atmosphere)
• Altitude below 1000km (or interference from Van Allen radiation belt)
• Generally orbit from Pole to Pole (N, S, N, S, ...)
• Orbital period as little as 90 minutes (1&1/2 hours)
• Used for observation (mapping, weather, military...)
• Limited life (slowed by atmosphere, spiral in, crash & burn!)
Geostationary orbit
• Stays “stationary” above one point on the Earth’s equator.
• Radius 42 168km (to match Earth’s rotation)
• Must be above the equator (or it will crash)
• main use communications (TV, GPS... receiver dishes pointing at it)
Geostationary vs Geosynchronous
• Both geostationary and geosynchronous orbits have periods of 23 hours 56 minutes
• A satellite in a geostationary orbit has its orbit directly above the equator
• The orbit of a satellite that is geosynchronous is not necessarily above the equator - but may be at an angle to the equator
• All geostationary satellites are geosynchronous - but the reverse is not true
• Geostationary satellites appear in a fixed position in the sky
• Geosynchronous satellites that are not geostationary appear to move along a north-south line in the sky, completing one cycle every 24 h
Kepler’s Law of Periods
• Johannes Kepler (1571-1630) searched for an underlying pattern of the motion of planets.
• Studying these motions, and using measurements taken by Tycho Brahe, Kepler deduced three laws of planetary motion.
• Kepler was the first person to reject the assumption that planets moved in perfect circular motion.
Kepler’s Law of Periods
A little revision: Orbital velocity
This is the velocity of a satellite as it moves around another body under the influence of the force of gravity.
Orbital velocity is constant in magnitude if the satellite travels in a circular orbit.
The direction of the orbital velocity is at a tangent to the orbit.
How is magnitude of the orbital velocity is related to the period (T) and the orbital radius (r)?
V = 2r/T
Orbital velocity
direction
Kepler’s Law of Periods
Orbital velocity
If the orbit is elliptical, the speed is least when the distance between the objects is the greatest and it increases as they get closer together.
The speed change is a consequence of the law of conservation of energy - as the potential energy decreases the kinetic energy increases correspondingly.
Minimum speed
Maximum speed
Kepler’s Law of Periods
First two laws…
• Satellites move in elliptical orbits with the central body at one focus of the ellipse
• A planet moving in its orbit sweeps out equal areas in equal times
[not explicitly in HSC course]
But the third law (the law of periods) is…
Kepler’s Law of Periods
• The square of a planet’s orbital period (T) is proportional to the cube of the mean distance (r) of the planet from the Sun
T2 = k r3
• Where k is a constant dependant on the mass of the central body (e.g. the Sun) and the universal gravitational constant
Kepler’s Law of Periods
The square of a planet’s orbital period is proportional to the cube of the mean distance of the planet from the Sun
T - orbital period (s)
G - universal gravitational constant
M - mass of central body
Saturn
Kepler’s Law of Periods
Calculate the ratio of the [radius3/period2] for the Earth, and use this to calculate the orbital period of Saturn, given its mean orbital radius 9.54 au.
REarth orbit 1.50 x 1011 m
TEarth365.25 days
G 6.67 x 10–11
M (mass of Sun)1.99 x 1030kg
1AU = 1.5 x 1011
Orbital Decay
• More than 90% of the molecules in the Earth’s atmosphere are below the top of Mount Everest (pictured from ISS).
• There are still a few molecules of the Earth’s atmosphere extending to an altitude of about 500 km.
• Satellites in orbits between 150 km and 500 km altitude are described as being in low Earth orbit (LEO).
Orbital Decay
• Satellites in LEO encounter frictional drag as they orbit at these altitudes.
• The effect of this drag is to cause the satellite to lose energy resulting in the satellite’s moving closer to the Earth.
• If the orbit is to be maintained, booster engines must be fired periodically to increase the satellite’s altitude.
Re-Entry
• For a satellite in LEO, the kinetic energy is about ten times the potential energy and they are both very significant quantities of energy.
• To land safely, a spacecraft must reduce its speed by 90% as it approaches the Earth.
Re-Entry
• The speed reduction is accomplished through
• Retro-rocket firing (slows the vehicle by about 1%)
• Frictional drag in the atmosphere
Frictional drag through the Earth’s atmosphere converts the satellite’s energy to heat energy
Re-Entry
• For spacecraft intended to for return to Earth, dissipation of the heat energy generated by during re-entry is a major consideration in the spacecraft design and re-entry process.
• Key strategies employed to ensure the spacecraft does not burn up include the use of:
• heat resistant (high melting point) materials
• materials with very low thermal conductivity
• materials with a very low heat capacity
• ablation (burning off of material from the craft)
• heat radiation from the heated surface of the spacecraft
Re-Entry
• Retro-rockets slow the spacecraft slightly, causing its orbit to decay.
• The lower orbit results in much greater frictional drag, greatly slowing the spacecraft.
• The angle at which the spacecraft enters the atmosphere is critical
• too steep an angle results in a re-entry that is too fast and the craft would burn up
• Too shallow an angle causes the craft to bounce off the atmosphere and not make a successful re-entry
A weather satellite captured this image of the heating effect produced as a spacecraft reentered the Earth’s atmosphere
Re-Entry
ablative heat shield
Spacecraft - capsule
Re-Entry
• Protection of the shuttle during re-entry is achieved by insulating tiles made of silica and placed on the underside of the craft.
Re-Entry
• Too shallow an angle will cause the satellite to bounce off the atmosphere and re-enter space
• Too steep an angle will cause too great an increase in drag, causing the spacecraft to burn up in the atmosphere(Russian MIR re-entry pictured below)
Re-Entry
• There is thus an optimum angle at which a spacecraft returning to Earth must enter the atmosphere
• For safe re-entry, the angle at which the satellite enters the atmosphere must be between 5° and 7°
• Image - MIR re-entry (artist’s impression)
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Codes and lattices are very important subject in combinatorics. They have many applications in telecommunication, design theory, finite group theory as well as many different fields in electronic engineering and physics. In this thesis, we will give a survey on the properties of certain codes and lattices. In particular, we will concentrate on the construction of certain unimodular lattice by using linear codes. We will also discuss their application to other fields such as Lie algebra and finite group.
1 Introduction 22 Code 4 2.1 Basic De nitions. . . . . . . . . . . . . . . . . . . . . . . . . . .4 2.2 Certain important codes . . . . . . . . . . . . . . . . . . . . .6 2.2.1 Hamming code . . . . . . . . . . . . . . . . . . . . . . . .6 2.2.2 Golay code. . . . . . . . . . . . . . . . . . . . . . . . . . .73 Lattices 9 3.1 The lattices E6, E7 and E8 . . . . . . . . . . . . . . . . . . .12 3.2 Unimodular lattices and Leech lattice . . . . . . . . . . . 154 Construction of Lattices from Codes 185 Gluing theory 22 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27Bibliography 28
[1] R. Bacher and B.B. Venkov, Rseaux entiers unimodulaires sans racines en di-mension 27 et 28, Rseaux euclidiens, designs spheriques et formes modulaires, 212-267, Monoger. Enseign. Math, 37. Enseignement Math, Geneva 2001.[2] A. Bonnecaze, P. Sole and A. R. Calderbank, Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory 41, 366-377, 1995.[3] W. Bosma and J. Cannon, Handbook of Magma Functions, School of Mathematics and Statistics, University of Sydney, Sydney, July 22, 1999.[4] J. H. Conway and N. J. A. Sloane, On the enumeration of lattices of determinant one, J. Number Theory 15, 83-94, 1985.[5] J. H. Conway and N. J. A. Sloane, Sphere Packing, Lattices and Groups, 2nd ed., Springer-Verlag, New York, 1993.[6] S. T. Dougherty, M. Harada and P. Sole, Shadow codes over Z4, Finite Fileds and Their Appl. 7, 507-529, 2001.[7] I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras and the Monster, Academic-Press, Inc, London, 1988.[8] T. A. Gulliver and M. Harada, Orthogonal Frames in the Leech Lattice and a Type II Code over Z22, Journal of Combinatorial Theory, Ser. A 95, 185-188, 2001.[9] M. Harada, P. Sol e, and P. Gaborit, Self-Dual Codes over Z4 and Unimodular Lattices: A Survey, Algebras and Combinatiorics (Hong Kong 1997), 255-275, Spring, Singapore, 1999.[10] R. Hill, A First Course in Codeing Theory, Clarendon-Press, Oxford, 1986.[11] M. Kervaire, Unimodular lattices with a complete root system, Ens. Math. 40, 59-104, 1994.[12] M. Kitazume, C.-H. Lam, and H. Yamada, A class of vertex operator algebras constructed from Z8 codes, J. Algebra 242, 338-359, 2001.[13] M. Kitazume, C.-H. Lam, and H. Yamada, Decomposition of the Moonshine Vertex Operator Algebra as Virasoro Modules, J. Algebra 226, 893-919, 2000.[14] H. Koch and B. B. Venkov, Ueber ganzahlige unimodulare Gitter, J. reine angew. Math. 398,144-168, 1989.[15] J. E. Humphreys, Introduction to Lie algebras and representation theory, Springer Verlag, New York, Heidelberg, Berlin, 1972.[16] C.-H. Lam, Fusion rules for the Hamming code vertex operator algebra, Comm. in Algebra 29(5), 2125-2145, 2001.[17] C.-H. Lam and H. Yamada, Z2 Z2 codes and vertex operator algebra, J. Algebra 224, 268-291, 2000.[18] R. V. Moody and A. Pianzola, Lie Algebras With Triangular Decompositions, John-Wiley-Sons, New York, 1995.[19] H. V. Niemeier, De nete quadratische Formen der Dimension 24 und Diskrim-inante 1, J. Number Theory 5, 142-178, 1973.[20] H. G. Quebbemann, Zur Klassi cation unimodularer Gitter mit Isometrie von Primzahlordnung, J. reine angew. Math. 326, 158-170, 1981.
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# Dominated or Monotone convergence for $\dfrac{1+nx^2}{(1+x)^n}$
I am having a particularly difficult time working with the sequence of functions given by:
$$f_n(x)=\dfrac{1+nx^2}{(1+x)^n}.$$
I am working only on $[0,1]$. It is clear that these are measurable functions which go to $0$ pointwise, (though $f_n(0)=1\;\forall n\in\mathbb{N}$) and I would like to use either the Dominated or Monotone Convergence Theorems to show that:
$$\lim\limits_{n\to\infty}\int\limits_0^1f_n(x)=0.$$
I know it is true based on the context in which I received the problem, but it is proving to be a tough nut to crack. I’m usually good at these.
A little noodling around on Desmos shows that the sequence is in fact dominated on $[0,1]$ by any constant function greater than 1, and that for $n\geq 3$ it is decreasing on $[0,1]$. For $n=1$ or $n=2$, it has a minimum between $0$ and $1$. But these things are, for the most part, incredibly tedious to show rigorously.
This was a question on a prior preliminary exam in my department, and was written to take a few minutes at most.
Is there a quicker way? All these details have taken me almost an hour to write out!
#### Solutions Collecting From Web of "Dominated or Monotone convergence for $\dfrac{1+nx^2}{(1+x)^n}$"
By the Bernoulli’s inequality, for $x\in[0,1]$
$$\frac{1+nx^2}{(1+x)^n}\leq\frac{1+nx^2}{1+nx}\leq 1$$
which gives the dominated function.
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Articles / Programming Languages / C#
Article
Posted 11 Mar 2013
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# C# Cubic Spline Interpolation
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This article presents a from-scratch implementation of cubic spline interpolation in C#.
## Introduction
This is an implementation of cubic spline interpolation based on the Wikipedia articles Spline Interpolation and Tridiagonal Matrix Algorithm. My goal in creating this was to provide a simple, clear implementation that matches the formulas in the Wikipedia articles closely, rather than an optimized implementation.
## Background
Spline algorithms are a way to fit data points with a set of connecting curves (each one is called a spline) such that values between data points can be computed (interpolated). There are various types/orders of equations that can be used to specify the splines including linear, quadratic, cubic, etc.
A set of N points requires N-1 splines to connect them. Each spline is described by an equation (e.g., a polynomial). The coefficients in those polynomials are initially unknown, and the spline algorithm computes them. Once the coefficients are known, to compute the Y for any particular X you find the spline that is relevant to that X and use those coefficients to evaluate the polynomial at that X.
To compute the splines' coefficients, the algorithm uses the standard linear algebra procedure of creating a system of equations and then solving that system. The equations embody the constraints on the splines which are that they intersect the points to be fitted, and that at each junction between splines the prior splines' first and second derivatives equal the successor splines' first and second derivatives. That doesn't provide enough constraints yet to fully determine the system so two more are added that specify that the second derivatives of the first and last points are zero. Those final constraints are called the "natural" spline style. Alternatively, user Quience (below) has provided code such that the caller can specify the slopes at the first and last points explicitly. With either set of additional constraints, the system becomes solvable.
Fortunately, the system of equations is constrained to be tridiagonal, so that solving the system is O(N) using the Thomas algorithm, rather than the O(N^3) of Gaussian elimination for example. Here is the tridiagonal matrix corresponding to the above image which is computed to solve the Wikipedia article's example problem:
```2.0000, 1.0000, 0.0000
1.0000, 2.6667, 0.3333
0.0000, 0.3333, 0.6667```
This example is a little too small to illustrate a tridiagonal matrix well, but tridiagonal means that only the main diagonal (`row == col`) elements, the diagonal below the main (`row == col + 1`), and the diagonal above the main (`row == col - 1`) can have non-zero values.
The linked Wikipedia articles do a good job of explaining splines so I won't provide any more redundant explanation here.
## Using the Code
I followed the Wikipedia articles closely for the implementation, including variable names where possible. Some comments refer to equation numbers in the articles (which unfortunately will break at some point when the articles change).
The primary classes are `CubicSpline` and `TriDiagonalMatrix`. The `CubicSpline` class uses `TriDiagonalMatrix` to solve the system of equations. Generally, if you are just using this code to fit curves, you would just use `CubicSpline`.
In the source, there is also Program.cs which has a `Main()` that runs several functions that exercise the classes.
Here is an example use of `CubicSpline` from the `TestSplineOnWikipediaExample()` method in Program.cs. This part of the code sets up the test data to be fitted:
C#
```// Create the test points.
float[] x = new float[] { -1.0f, 0.0f, 3.0f };
float[] y = new float[] { 0.5f, 0.0f, 3.0f };
Console.WriteLine("x: {0}", ArrayUtil.ToString(x));
Console.WriteLine("y: {0}", ArrayUtil.ToString(y));
```
The next step is to create a `float[]` (here called `xs`) which contains the values of X for which you want to determine the Y values. In other words, these are the X values for which you want to plot points on the graph:
C#
```// Create the upsampled X values to interpolate
int n = 20;
float[] xs = new float[n];
float stepSize = (x[x.Length - 1] - x[0]) / (n - 1);
for (int i = 0; i < n; i++)
{
xs[i] = x[0] + i * stepSize;
}
```
And next is to use `CubicSpline` to fit the data and then evaluate the fitted spline at the xs values generated above:
C#
```// Solve
CubicSpline spline = new CubicSpline();
float[] ys = spline.FitAndEval(x, y, xs);
Console.WriteLine("xs: {0}", ArrayUtil.ToString(xs));
Console.WriteLine("ys: {0}", ArrayUtil.ToString(ys));
// Plot
string path = @"..\..\spline-wikipedia.png";
PlotSplineSolution("Cubic Spline Interpolation - Wikipedia Example", x, y, xs, ys, path);
```
This displays the following console output:
```x: -1, 0, 3
y: 0.5, 0, 3
xs: -1, -0.7894737, -0.5789474, -0.368421, -0.1578947, 0.05263158, ...
ys: 0.5, 0.3570127, 0.2245225, 0.1130267, 0.0330223, -0.005029888, ...```
The `PlotSplineSolution()` method uses the .NET Framework's `System.Windows.Forms.DataVisualization.Charting` classes to plot a graph of the input points and the fitted and interpolated spline curve. The image created by this is the one included in this article (above).
## Implementation
The core of the spline fitting function sets up the tridiagonal matrix and then uses it to solve the system of equations. The tridiagonal matrix is not represented as a matrix but rather three 1-d arrays, A, B, and C. Array A is the sub-diagonal, B is the diagonal, and C is the super-diagonal, to match the Wikipedia article names. Once the tridiagonal matrix is set up, the spline fitting function calls `TriDiagonalMatrix.Solve()`.
Here is the tridiagonal matrix solver function from TriDiagonalMatrix.cs:
C#
```// Solve the system of equations this*x=d given the specified d.
public float[] Solve(float[] d)
{
int n = this.N;
if (d.Length != n)
{
throw new ArgumentException("The input d is not the same size as this matrix.");
}
// cPrime
float[] cPrime = new float[n];
cPrime[0] = C[0] / B[0];
for (int i = 1; i < n; i++)
{
cPrime[i] = C[i] / (B[i] - cPrime[i-1] * A[i]);
}
// dPrime
float[] dPrime = new float[n];
dPrime[0] = d[0] / B[0];
for (int i = 1; i < n; i++)
{
dPrime[i] = (d[i] - dPrime[i-1]*A[i]) / (B[i] - cPrime[i - 1] * A[i]);
}
// Back substitution
float[] x = new float[n];
x[n - 1] = dPrime[n - 1];
for (int i = n-2; i >= 0; i--)
{
x[i] = dPrime[i] - cPrime[i] * x[i + 1];
}
return x;
}```
The following is the core of the `CubicSpline.Eval()` method. This method takes a `float[]` x which contains the x values you want to compute `y` for using the fitted splines. The loop is stepping through each value of `x` computing the corresponding value of `y`.
C#
```for (int i = 0; i < n; i++)
{
// Find which spline can be used to compute this x
int j = GetNextXIndex(x[i]);
// Evaluate using j'th spline
float t = (x[i] - xOrig[j]) / (xOrig[j + 1] - xOrig[j]);
// equation 9 in the Wiki
y[i] = (1 - t) * yOrig[j] + t * yOrig[j + 1] + t * (1 - t) * (a[j] * (1 - t) + b[j] * t);
}```
## Another Example
Here is another chart. This one was created by the `TestSpline()` method in Program.cs:
## Computing Slope
In addition to computing the `Y` for each `X`, one can also compute the slope of the spline. It is specified as q' in the Wikipedia article and the formula is equation 5. The method to evaluate slope is CubicSpline.EvalSlope() and it returns a `float` array (one slope value for each X you provide). You must have called either Fit() (or FitEndEval()) before calling this. Here is an example use:
C#
```// Try slope (spline is already computed at this point, see above code example)
float[] slope = spline.EvalSlope(xs); // Same xs as first example above
string slopePath = @"..\..\spline-wikipedia-slope.png";
PlotSplineSolution("Cubic Spline Interpolation - Wikipedia Example - Slope",
x, y, xs, ys, slopePath, slope);
```
Here is the resulting chart:
## Input Constraints and Parametric Fitting
The normal cubic spline algorithm works on 2-d points where `y` is a function of `x`, i.e. `y=f(x)`, and `y` has a single value for each `x`. However, user LutzL in the comments below has pointed out a clever way to use splines to fit sequences of points that do not fit this definition:
[You] invent a third time coordinate that increases monotonically, T=0,1,2,3 or start with T=0 and increment by the distance from the current point to the next point. Use then the (T,X) and (T,Y) pairs to compute two cubic splines x(t) and y(t) and draw the curve (x(t),y(t)) as result.
I implemented this as a new `static` method FitParametric() in the CubicSpline class. Here is an example use:
C#
```float[] x = { 0.5f, 2.0f, 3.0f, 4.5f, 3.0f, 2.0f };
float[] y = { 4.0f, 2.0f, 6.0f, 4.0f, 3.0f, 5.0f };
float[] xs, ys;
CubicSpline.FitParametric(x, y, 100, out xs, out ys);
```
Here is the resulting chart:
Thanks to user YvesDaoust for the correct term for this method.
## Points of Interest
Surprisingly, this simple implementation is more than twice as fast as a C# port of the very terse Numerical Recipes in C++ (Press et al, Cambridge University Press, 2002) version in the scenario I benchmarked. I only benchmarked one scenario and did not investigate the results much so shouldn't draw too many conclusions, but it is interesting anyway. I benchmarked the implementations using a Release build by Visual Studio 2012, running on a Win 7 Core i5 2500k. I used 10,000 random points and interpolated 100,000 points, with 100 repetitions. See the Program.cs method `TestPerf()`. I did not include the Numerical Recipes in C++ version in the code due to licensing and copyright issues. [Update: This performance difference is mostly likely due to the fact that the NR in C implementation does a search for the appropriate spline for each evaluation, whereas I require that the points to be evaluated are sorted and therefore I can do a simultaneous traverse.]
The `Chart` class in the `System.Windows.Forms.DataVisualization.Charting` namespace worked well for my simple charts. The classes in there make it very easy to render a chart and save to a file.
It is unfortunate that C#/.NET generics (still) cannot handle creating a generic type constraint that allows you to handle both `float` and `double` with the same code. I implemented this with `float`s because I typically start with `float`s until it's proven that I need `double`s.
## History
• March 10, 2013 - First version
• September 20, 2013 - Added description of first and last point constraints including explicit slope arguments, and misc other minor changes
• July 23, 2014 - Updated code to add slope computation and added the Computing Slope section to the article
• July 25, 2014 - Added Input Constraints and Geometric Fitting section, and updated code
• April 1, 2016 - Renamed 'geometric' to 'parametric' and added start and end slope specification to parametric fitting
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Re: Little suggestion Alexandr Stefek28-Jul-14 21:30 Alexandr Stefek 28-Jul-14 21:30
How to get the a,b,c,d values from your implementation Member 1094985922-Jul-14 5:47 Member 10949859 22-Jul-14 5:47
Re: How to get the a,b,c,d values from your implementation Ryan Seghers23-Jul-14 10:42 Ryan Seghers 23-Jul-14 10:42
Re: How to get the a,b,c,d values from your implementation Member 1094985925-Jul-14 2:49 Member 10949859 25-Jul-14 2:49
Evaluate the spline at the specified Y coordinates Menelay11-Jul-14 4:59 Menelay 11-Jul-14 4:59
Re: Evaluate the spline at the specified Y coordinates Ryan Seghers25-Jul-14 13:16 Ryan Seghers 25-Jul-14 13:16
Re: Evaluate the spline at the specified Y coordinates Menelay28-Jul-14 20:32 Menelay 28-Jul-14 20:32
My vote of 5 MarkBoreham30-Jan-14 7:40 MarkBoreham 30-Jan-14 7:40
Excellent article, thanks for the
problem when x1 < x0 Member 1049455529-Jan-14 3:06 Member 10494555 29-Jan-14 3:06
Re: problem when x1 < x0 Ryan Seghers29-Jan-14 6:41 Ryan Seghers 29-Jan-14 6:41
Re: problem when x1 < x0 Member 1049455529-Jan-14 8:05 Member 10494555 29-Jan-14 8:05
Re: problem when x1 < x0 Ryan Seghers29-Jan-14 12:44 Ryan Seghers 29-Jan-14 12:44
Re: problem when x1 < x0 Member 1049455529-Jan-14 21:25 Member 10494555 29-Jan-14 21:25
Re: problem when x1 < x0 Ryan Seghers30-Jan-14 10:47 Ryan Seghers 30-Jan-14 10:47
Re: problem when x1 < x0 LutzL24-Feb-14 22:37 LutzL 24-Feb-14 22:37
Re: problem when x1 < x0 Ryan Seghers25-Jul-14 13:11 Ryan Seghers 25-Jul-14 13:11
Re: problem when x1 < x0 MikeStelmat9-Sep-15 4:42 MikeStelmat 9-Sep-15 4:42
My vote of 5 Grump30-Sep-13 7:28 Grump 30-Sep-13 7:28
Re: My vote of 5 Ryan Seghers30-Sep-13 9:16 Ryan Seghers 30-Sep-13 9:16
Re: My vote of 5 Grump30-Sep-13 21:06 Grump 30-Sep-13 21:06
Re: My vote of 5 Ryan Seghers6-Oct-13 9:14 Ryan Seghers 6-Oct-13 9:14
My vote of 5 Douglas Smallish21-Sep-13 3:24 Douglas Smallish 21-Sep-13 3:24
Small change to enable constraining the slope at either end of the dataset Quience13-Sep-13 5:54 Quience 13-Sep-13 5:54
Re: Small change to enable constraining the slope at either end of the dataset Ryan Seghers20-Sep-13 18:06 Ryan Seghers 20-Sep-13 18:06
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CC-MAIN-2023-14
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en
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https://aprove.informatik.rwth-aachen.de/eval/JAR06/JAR_TERM/TRS/TRCSR/Ex4_7_37_Bor03_Z.trs.Thm17:POLO_FILTER:NO.html.lzma
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crawl-data/CC-MAIN-2024-26/segments/1718198861605.77/warc/CC-MAIN-20240615155712-20240615185712-00633.warc.gz
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Term Rewriting System R:
[X, XS, N, Y, YS, X1, X2]
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), nzWquot(activate(XS), activate(YS)))
zWquot(X1, X2) -> nzWquot(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nzWquot(X1, X2)) -> zWquot(X1, X2)
activate(X) -> X
Termination of R to be shown.
` R`
` ↳Dependency Pair Analysis`
R contains the following Dependency Pairs:
SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
SEL(s(N), cons(X, XS)) -> ACTIVATE(XS)
MINUS(s(X), s(Y)) -> MINUS(X, Y)
QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))
QUOT(s(X), s(Y)) -> MINUS(X, Y)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> QUOT(X, Y)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ACTIVATE(nfrom(X)) -> FROM(X)
ACTIVATE(nzWquot(X1, X2)) -> ZWQUOT(X1, X2)
Furthermore, R contains four SCCs.
` R`
` ↳DPs`
` →DP Problem 1`
` ↳Argument Filtering and Ordering`
` →DP Problem 2`
` ↳AFS`
` →DP Problem 3`
` ↳AFS`
` →DP Problem 4`
` ↳AFS`
Dependency Pair:
MINUS(s(X), s(Y)) -> MINUS(X, Y)
Rules:
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), nzWquot(activate(XS), activate(YS)))
zWquot(X1, X2) -> nzWquot(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nzWquot(X1, X2)) -> zWquot(X1, X2)
activate(X) -> X
The following dependency pair can be strictly oriented:
MINUS(s(X), s(Y)) -> MINUS(X, Y)
There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
POL(MINUS(x1, x2)) = x1 + x2 POL(s(x1)) = 1 + x1
resulting in one new DP problem.
Used Argument Filtering System:
MINUS(x1, x2) -> MINUS(x1, x2)
s(x1) -> s(x1)
` R`
` ↳DPs`
` →DP Problem 1`
` ↳AFS`
` →DP Problem 5`
` ↳Dependency Graph`
` →DP Problem 2`
` ↳AFS`
` →DP Problem 3`
` ↳AFS`
` →DP Problem 4`
` ↳AFS`
Dependency Pair:
Rules:
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), nzWquot(activate(XS), activate(YS)))
zWquot(X1, X2) -> nzWquot(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nzWquot(X1, X2)) -> zWquot(X1, X2)
activate(X) -> X
Using the Dependency Graph resulted in no new DP problems.
` R`
` ↳DPs`
` →DP Problem 1`
` ↳AFS`
` →DP Problem 2`
` ↳Argument Filtering and Ordering`
` →DP Problem 3`
` ↳AFS`
` →DP Problem 4`
` ↳AFS`
Dependency Pair:
QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))
Rules:
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), nzWquot(activate(XS), activate(YS)))
zWquot(X1, X2) -> nzWquot(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nzWquot(X1, X2)) -> zWquot(X1, X2)
activate(X) -> X
The following dependency pair can be strictly oriented:
QUOT(s(X), s(Y)) -> QUOT(minus(X, Y), s(Y))
The following usable rules using the Ce-refinement can be oriented:
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
Used ordering: Polynomial ordering with Polynomial interpretation:
POL(QUOT(x1, x2)) = x1 + x2 POL(0) = 0 POL(s(x1)) = 1 + x1
resulting in one new DP problem.
Used Argument Filtering System:
QUOT(x1, x2) -> QUOT(x1, x2)
s(x1) -> s(x1)
minus(x1, x2) -> x1
` R`
` ↳DPs`
` →DP Problem 1`
` ↳AFS`
` →DP Problem 2`
` ↳AFS`
` →DP Problem 6`
` ↳Dependency Graph`
` →DP Problem 3`
` ↳AFS`
` →DP Problem 4`
` ↳AFS`
Dependency Pair:
Rules:
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), nzWquot(activate(XS), activate(YS)))
zWquot(X1, X2) -> nzWquot(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nzWquot(X1, X2)) -> zWquot(X1, X2)
activate(X) -> X
Using the Dependency Graph resulted in no new DP problems.
` R`
` ↳DPs`
` →DP Problem 1`
` ↳AFS`
` →DP Problem 2`
` ↳AFS`
` →DP Problem 3`
` ↳Argument Filtering and Ordering`
` →DP Problem 4`
` ↳AFS`
Dependency Pairs:
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
ACTIVATE(nzWquot(X1, X2)) -> ZWQUOT(X1, X2)
Rules:
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), nzWquot(activate(XS), activate(YS)))
zWquot(X1, X2) -> nzWquot(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nzWquot(X1, X2)) -> zWquot(X1, X2)
activate(X) -> X
The following dependency pairs can be strictly oriented:
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(YS)
ZWQUOT(cons(X, XS), cons(Y, YS)) -> ACTIVATE(XS)
There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
POL(cons(x1, x2)) = x1 + x2 POL(n__zWquot(x1, x2)) = 1 + x1 + x2 POL(ZWQUOT(x1, x2)) = 1 + x1 + x2 POL(ACTIVATE(x1)) = x1
resulting in one new DP problem.
Used Argument Filtering System:
ZWQUOT(x1, x2) -> ZWQUOT(x1, x2)
ACTIVATE(x1) -> ACTIVATE(x1)
cons(x1, x2) -> cons(x1, x2)
nzWquot(x1, x2) -> nzWquot(x1, x2)
` R`
` ↳DPs`
` →DP Problem 1`
` ↳AFS`
` →DP Problem 2`
` ↳AFS`
` →DP Problem 3`
` ↳AFS`
` →DP Problem 7`
` ↳Dependency Graph`
` →DP Problem 4`
` ↳AFS`
Dependency Pair:
ACTIVATE(nzWquot(X1, X2)) -> ZWQUOT(X1, X2)
Rules:
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), nzWquot(activate(XS), activate(YS)))
zWquot(X1, X2) -> nzWquot(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nzWquot(X1, X2)) -> zWquot(X1, X2)
activate(X) -> X
Using the Dependency Graph resulted in no new DP problems.
` R`
` ↳DPs`
` →DP Problem 1`
` ↳AFS`
` →DP Problem 2`
` ↳AFS`
` →DP Problem 3`
` ↳AFS`
` →DP Problem 4`
` ↳Argument Filtering and Ordering`
Dependency Pair:
SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
Rules:
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), nzWquot(activate(XS), activate(YS)))
zWquot(X1, X2) -> nzWquot(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nzWquot(X1, X2)) -> zWquot(X1, X2)
activate(X) -> X
The following dependency pair can be strictly oriented:
SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
The following usable rules using the Ce-refinement can be oriented:
activate(nfrom(X)) -> from(X)
activate(nzWquot(X1, X2)) -> zWquot(X1, X2)
activate(X) -> X
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), nzWquot(activate(XS), activate(YS)))
zWquot(X1, X2) -> nzWquot(X1, X2)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
Used ordering: Polynomial ordering with Polynomial interpretation:
POL(n__from) = 0 POL(from) = 0 POL(activate(x1)) = x1 POL(0) = 0 POL(SEL(x1, x2)) = 1 + x1 + x2 POL(n__zWquot(x1, x2)) = x1 + x2 POL(zWquot(x1, x2)) = x1 + x2 POL(nil) = 0 POL(quot(x1, x2)) = x1 + x2 POL(s(x1)) = 1 + x1
resulting in one new DP problem.
Used Argument Filtering System:
SEL(x1, x2) -> SEL(x1, x2)
s(x1) -> s(x1)
cons(x1, x2) -> x2
activate(x1) -> activate(x1)
nfrom(x1) -> nfrom
from(x1) -> from
nzWquot(x1, x2) -> nzWquot(x1, x2)
zWquot(x1, x2) -> zWquot(x1, x2)
quot(x1, x2) -> quot(x1, x2)
minus(x1, x2) -> x1
` R`
` ↳DPs`
` →DP Problem 1`
` ↳AFS`
` →DP Problem 2`
` ↳AFS`
` →DP Problem 3`
` ↳AFS`
` →DP Problem 4`
` ↳AFS`
` →DP Problem 8`
` ↳Dependency Graph`
Dependency Pair:
Rules:
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
minus(X, 0) -> 0
minus(s(X), s(Y)) -> minus(X, Y)
quot(0, s(Y)) -> 0
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
zWquot(XS, nil) -> nil
zWquot(nil, XS) -> nil
zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), nzWquot(activate(XS), activate(YS)))
zWquot(X1, X2) -> nzWquot(X1, X2)
activate(nfrom(X)) -> from(X)
activate(nzWquot(X1, X2)) -> zWquot(X1, X2)
activate(X) -> X
Using the Dependency Graph resulted in no new DP problems.
Termination of R successfully shown.
Duration:
0:06 minutes
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| 2.5625
| 3
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CC-MAIN-2024-26
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latest
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en
| 0.433077
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https://www.physicsforums.com/threads/exam-questions-need-help-please.153491/
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text/html
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crawl-data/CC-MAIN-2019-51/segments/1575540525781.64/warc/CC-MAIN-20191210013645-20191210041645-00497.warc.gz
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# Exam Questions- Need Help Please
1. Homework Statement
a 47kg women balances on one heel of a high heeled show. if the heel has a radius of 1.4cm what pressure does she exert on the floor?
2. Homework Equations
rho=m
v
3. The Attempt at a Solution
= 47kg = 33.5
1.4cm
1. Homework Statement
The pascal principle is used to operate a dentists chair. assume that the pressure everywhere in the fluid is 13.5 kpa. if the small cylinder(piston) has an area of 0.007m(squared) and the large cylinder(piston) has an area of 0.08m(squared) calculate the force on each cylinder.
2. Homework Equations
i think the equation is f=af
a
3. The Attempt at a Solution
=(3.14)(0.007)squared(13.5)
0.08(squared)
=.32
1. Homework Statement
a gold nugget weighs 1.23N in air. when suspended from a string and submerged in water its apparent weight (T) is 1.09N. calculate the density of the gold nuggett.
2. Homework Equations
density=mass
volume
3. The Attempt at a Solution
= 1.23N
(3.14)r(squared) x h
=
1. Homework Statement
water enters a cylindrical pipe 3.5cm in diameter at a speed of 1.6m/s. if the pipes diameter is suddenly reduced to 2.1cm, what is the max velocity of the water moving through the smaller pipe?
2. Homework Equations
AV=AV
3. The Attempt at a Solution
3.14(3.5)(1.6)=A(2.1)
1. Homework Statement
air is pushed through a forced-air furnace duct at a speed of 4.7m/s. the duct is rectangular and measure 58cm by 23cm. how fast will air move through a smaller portion of the duct that is also rectangular and measures 35cm by 23cm.
2. Homework Equations
AV=AV
3. The Attempt at a Solution
3.14(58)V=23(4.7)
1. Homework Statement
a force of 19N[E] pushes a box of mass 13.2kg across a level floor at a constant speed. calculate the coefficient of kinetic friction (,U) between the two surfaces.
2. Homework Equations
,U=f
mg
3. The Attempt at a Solution
= 19
13.2(9.80)
= 19
129
=0.14
129
i know its a lot of work there so you can do all of them or just 1 or how every many you want to. im just struggling on these questions and my exam is tomorrow and i really need some help so if you guys are able to help me out i would really appreciate it. thanks so much for doing this guys.
oh and u can delete my other thread since i read cristos post.
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| 3.65625
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CC-MAIN-2019-51
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latest
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en
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https://www.lmfdb.org/ModularForm/GL2/TotallyReal/5.5.24217.1/holomorphic/5.5.24217.1-32.1-a
| 1,720,954,165,000,000,000
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text/html
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crawl-data/CC-MAIN-2024-30/segments/1720763514564.41/warc/CC-MAIN-20240714094004-20240714124004-00851.warc.gz
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# Properties
Label 5.5.24217.1-32.1-a Base field 5.5.24217.1 Weight $[2, 2, 2, 2, 2]$ Level norm $32$ Level $[32, 2, 2]$ Dimension $3$ CM no Base change no
# Related objects
• L-function not available
## Base field 5.5.24217.1
Generator $$w$$, with minimal polynomial $$x^{5} - 5x^{3} - x^{2} + 3x + 1$$; narrow class number $$1$$ and class number $$1$$.
## Form
Weight: $[2, 2, 2, 2, 2]$ Level: $[32, 2, 2]$ Dimension: $3$ CM: no Base change: no Newspace dimension: $3$
## Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
$$x^{3} - x^{2} - 10x + 4$$
Norm Prime Eigenvalue
5 $[5, 5, -2w^{4} + w^{3} + 9w^{2} - 2w - 3]$ $\phantom{-}e$
17 $[17, 17, -2w^{4} + w^{3} + 9w^{2} - 3w - 5]$ $\phantom{-}\frac{1}{2}e^{2} + \frac{1}{2}e - 3$
17 $[17, 17, w^{2} - 2]$ $-\frac{1}{2}e^{2} + \frac{1}{2}e + 2$
23 $[23, 23, w^{3} - 3w]$ $\phantom{-}e^{2} - e - 6$
29 $[29, 29, 2w^{4} - w^{3} - 10w^{2} + 2w + 4]$ $-e^{2} + 4$
32 $[32, 2, 2]$ $\phantom{-}1$
37 $[37, 37, -w^{4} + w^{3} + 4w^{2} - 2w]$ $-e^{2} + 12$
41 $[41, 41, -3w^{4} + 2w^{3} + 14w^{2} - 5w - 7]$ $\phantom{-}\frac{3}{2}e^{2} - \frac{1}{2}e - 9$
43 $[43, 43, -3w^{4} + w^{3} + 14w^{2} - 3w - 6]$ $-\frac{1}{2}e^{2} - \frac{3}{2}e + 9$
47 $[47, 47, -3w^{4} + 2w^{3} + 14w^{2} - 7w - 6]$ $-2e - 2$
53 $[53, 53, 2w^{4} - w^{3} - 8w^{2} + 2w + 1]$ $-e - 2$
53 $[53, 53, -2w^{4} + w^{3} + 10w^{2} - 4w - 6]$ $-e - 2$
59 $[59, 59, -w^{4} + 4w^{2} + 1]$ $-\frac{1}{2}e^{2} - \frac{5}{2}e$
59 $[59, 59, -3w^{4} + 2w^{3} + 14w^{2} - 6w - 8]$ $-\frac{1}{2}e^{2} - \frac{5}{2}e$
61 $[61, 61, 4w^{4} - 2w^{3} - 18w^{2} + 5w + 7]$ $-2e^{2} - e + 16$
61 $[61, 61, 3w^{4} - w^{3} - 15w^{2} + 2w + 7]$ $-e^{2} + 4e + 12$
73 $[73, 73, -4w^{4} + 2w^{3} + 18w^{2} - 7w - 6]$ $-\frac{1}{2}e^{2} - \frac{3}{2}e + 8$
83 $[83, 83, -2w^{4} + 9w^{2} + 2w - 4]$ $-\frac{5}{2}e^{2} + \frac{3}{2}e + 10$
83 $[83, 83, -2w^{4} + 2w^{3} + 10w^{2} - 7w - 6]$ $\phantom{-}\frac{5}{2}e^{2} - \frac{7}{2}e - 16$
97 $[97, 97, -2w^{4} + w^{3} + 8w^{2} - 3w + 1]$ $-\frac{1}{2}e^{2} + \frac{5}{2}e + 8$
Display number of eigenvalues
## Atkin-Lehner eigenvalues
Norm Prime Eigenvalue
$32$ $[32, 2, 2]$ $-1$
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# Bernoulli Distribution Probability
P: X:
Output: `Press calculate`
Formula:`P(X=x) = p^x * (1-p)^(1-x)`
## Introduction to Bernoulli Distribution Probability
The Bernoulli distribution is a discrete probability distribution that represents the probability of success (x = 1) or failure (x = 0) in a single Bernoulli trial, where the probability of success is represented by the parameter p. The formula calculates the probability of a specific outcome x for a given probability p.
## Parameter usage:
• `p` = probability of success
• `x` = specific outcome (0 for failure, 1 for success)
## Output:
• `P(X=x)` = probability of the specific outcome
## Data validation
The probability `p` should be between 0 and 1, and the outcome `x` should be either 0 or 1.
## Summary
This calculator computes the probability of a specific outcome in a single Bernoulli trial based on the given probability of success.
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# right half plane zero
Posted on
Hop on to get the meaning of RHPZ. b) At least one zero of its transfer function is shifted to the right half of s-plane. Let's have a look at a transfer function having a LHP zero and two poles. In wireless power receiver systems, the buck converter is widely used to step down the higher rectified voltage derived from the wireless receiver coil, to a lower output voltage for the immediate battery charging process. The right half plane zero has gain similar to that of left half plane zero but its phase nature is like a pole i.e., it adds negative phase to the system. This pole-zero diagram plots these critical frequencies in the s-plane, providing a geometric view of circuit behavior.In this pole-zero diagram, X denotes poles and O denotes the zeros. With just a little more work, we can define our contour in "s" as the entire right half plane - then we can use this to determine if there are any poles in the right half plane. Q: A: How to abbreviate "Right Half Plane Zero"? It still starts from 0° but increases to 90° as frequency approaches infinity. This is a classical pole also named stable pole. $\begingroup$ @Isabella Context was that (at least) one of $\,a,c\,$ must be non-zero. Do you need a valid visa to move out of the country? RHP zeros have a characteristic inverse response, as shown in Figure 3-11 for t n = -10 (which corresponds to a zero … Nyquist theory - Control system - Stability. Its step response is: As you can see, it is perfectly stable. Limitations of control systems with right half plane zero(s) and pole(s) In this section, the fundamental limitations of control systems with RHP zero(s) and pole(s) are briefly explained. From asymptotic stability point of view, asymptotically stable systems whose inverse gives unstable impulse response are the stable NMP systems. Is Mega.nz encryption vulnerable to brute force cracking by quantum computers? RHPZ stands for "Right Half Plane Zero". Multiple RHP zeros cause multiple "changes in direction"; for example, with two RHP zeros, the step response, initially going in one direction, switches direction, then switches back to the initial direction. If ζ≥ 1, corresponding to an overdamped system, the two poles are real and lie in the left-half plane. Hence, the number of counter-clockwise encirclements about − 1 + j 0 {\displaystyle -1+j0} must be equal … To determine the stability of a system, we want to determine if a system's transfer function has any of poles in the right half plane. Expert Answer . It only takes a minute to sign up. Both theory and experimental result show that the RHP zero is effectively eliminated by the proposed technique. By doing this, you slow down your converter but enjoy an acceptable phase margin: Now let's see what it implies in terms of control system. Get more help from Chegg. b) At least one zero of its transfer function is shifted to the right half of s-plane. Well, RHP zeros generally have no direct link with system stability. A positive zero is called a right-half-plane (RHP) zero, because it appears in the right half of the complex plane (with real and imaginary axes). Is it safe to disable IPv6 on my Debian server? It can be inserted in the transfer function as shown below. The stability of a low-power CMOS three-stage nested Miller compensated (NMC) amplifier is deteriorated by a right-half-plane (RHP) zero. The zero is 1/10, and the poles are –1/3 and –1/15. Right Half Plane Poles and Zeros and Design Tradeoffs in Feedback Systems. For this system, there is no imaginary component and the poles and zeros lie on the real axis. It has a zero at s=1, on the right half-plane. Automat. Now take the same zero and push it in the right-half plane then the phase response changes: the RHPZ no longer boosts the phase but lagsit down to 90° … J. S. Freudenberg and D. P. Looze (1985). Based on the above observations, we stipulate a gain expression of the more insightful type $$a(s) =\frac {V_o}{V_i}=a_0\frac {1-s/\omega_0}{(1+s/\omega_1)(1+s/\omega_2)}$$ A right-half-plane zero is characteristic of boost and buck-boost power stages. The characteristic function of a closed-looped system, on the other hand, cannot have zeros on the right half-plane. Articles, There are a number of different ways to represent process transfer functions. Process Control: Understanding Dynamic Behavior, Converting State Space Models to Transfer Functions, Process Control: Modeling, Design and Simulation, Fundamental Concepts and Computations in Chemical Engineering, Separation Process Engineering: Includes Mass Transfer Analysis, 4th Edition, Strategies for Creative Problem Solving, 3rd Edition, Mobile Application Development & Programming. Hence, the number of counter-clockwise encirclements about − 1 + j 0 {\displaystyle -1+j0} must be equal to the number of open-loop poles in the RHP. Relation between bandwidth, response time and disturbance rejection for a control system. the control to the output variable. CHRISTOPHE BASSO, Director, Product Application Engineering, ON Semiconductor, Phoenix. Thanks for contributing an answer to Electrical Engineering Stack Exchange! One of the definitions of RHPZ is "Right Half Plane Zero". 18 Recommended Effects of poles and zeroes Akanksha Diwadi. The "gain-time constant" form is the one that we use most often for control system design. 2 shows that with a left-half-plane (LHP) zero (as from the ESR of a capacitor), the phase increases with increasing gain; and with an RHP zero, the phase decreases with increasing gain. Think of a converter transferring the energy in a two-step approach, like a boost or a buck-boost converter: first you store the energy in the inductor during the on-time then transfer it to the load during the off-time. The below drawing shows an example for a transfer function featuring one zero and three poles: The left-side of the map is called the left-half-plane abbreviated LHP while the right-side is the left-half-plane or RHP. Q: A: What is RHPZ abbreviation? Well, RHP zeros generally have no direct link with system stability. However, the right-half plane (RHP) zero appear in control to output transfer function of boost converter obstruct … The bandwidth of the control feedback loop is restricted to about one-fifth the RHP zero frequency. Depending on the real value of the roots, poles and zeroes can be in either side. Abstract: This paper expresses limitations imposed by right half plane poles and zeros of the open-loop system directly in terms of the sensitivity and complementary sensitivity functions of the closed-loop system. The plant response is no longer the same with the RHPZ arbitrarily placed at 8 kHz: The phase now hits -180° at high frequency, consequence of the RHP zero. Limitations of control systems with right half plane zero(s) and pole(s) In this section, the fundamental limitations of control systems with RHP zero(s) and pole(s) are briefly explained. The RHPZ has been investigated in a previous article on pole splitting, where it was found that f0=12πGm2Cff0=12πGm2Cf so the circuit of Figure 3 has f0=10×10−3/(2π×9.9×10−12)=161MHzf0=10×10−3/(2π×9.9×10−12)=161MHz. c) At least one pole of its transfer function is shifted to the right half of s-plane. This intermediate phase naturally introduces a delay in the response to a change: the current in the inductor has to grow cycle by cycle (it cannot instantaneously jump to the next current setpoint) but this current increase is hampered by the inductor value and the available volt-seconds. The phase starts from 0° and asymptotically hits -90° as frequency approaches infinity. As you can see in Equation 4, s is in the numerator, but it is negative. This condition makes compensating the control loop a What's a great christmas present for someone with a PhD in Mathematics? Figure 3-13. @Chu, I would rather model the delay with a RHP zero and a LHP pole located at the same position: \$\small e^{-sT}\rightarrow \frac{1-\frac{s}{\omega_\tau}}{1+\frac{s}{\omega_\tau}}\$. Figure 3.7. The "polynomial" form is. A pure delay also introduces a phase lag while its magnitude is constant to 1 or 0 dB. A power switch SW, usually a MOSFET, and a diode D, sometimes called a catch diode. What type of targets are valid for Scorching Ray? Figure 6. As the imaginary/real ratio increases, the response becomes more oscillatory. In the Continuous Conduction Mode of Figure 3-14. hence the additional pole reduces bandwidth. I don't understand the bottom number in a time signature. This RHP zero is a function of the inductor (smaller is better) and the load resistance (light load is better than heavy load). This form is normally used when the roots (poles) of the denominator polynomial are real. Now, think of the same transfer function but having an extra RHP zero on top of the existing LHP zero. A pole-zero plot of the transfer function in Example 3.7 is shown in Figure 3-13 [the pole locations are (-1/3,0) and (-1/15,0) and the zero location is (1/10,0), with the coordinates (real,imaginary)]. Q: A: 3. I will take the example of a switching converter as it is my field of expertise. THE RIGHT-HALF-PLANE ZERO -A SIMPLIFIED EXPLANATION In small signal loop analysis, poles and zeros To subscribe to this RSS feed, copy and paste this URL into your RSS reader. You think of a compensation strategy featuring some response to obtain a good phase margin at 5 kHz. rev 2020.12.10.38158, The best answers are voted up and rise to the top, Electrical Engineering Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, Resolved my query to perfection via an extensive analysis and nicely explained through the example of power converters, which is my application area too. > On the grand staff, does the crescendo apply to the right hand or left hand? Electrical Engineering Stack Exchange is a question and answer site for electronics and electrical engineering professionals, students, and enthusiasts. Boost converter control using Smith Predictor technique to minimize the effect of Right Half Plane zero @article{Bag2013BoostCC, title={Boost converter control using Smith Predictor technique to minimize the effect of Right Half Plane zero}, author={Santu Bag and T. Roy and S. Mukhopadhyay and S. Samanta and Robert Sheehan}, … This zero lies on the positive real axis of the s plane, so it is known as a right half-plane zero (RHPZ). The boost converter has a right-half-plane zero which can make control very difficult. IEEE Trans. In other words, stay away from the RHPZ and its phase stress. Pole-zero location plot for (x, poles; o, zero). RHP zeros have a characteristic inverse response, as shown in Figure 3-11 for tn = -10 (which corresponds to a zero of +0.1). Now take the same zero and push it in the right-half plane then the phase response changes: the RHPZ no longer boosts the phase but lags it down to 90° as frequency approaches infinity. For an underdamped system, 0≤ ζ<1, the poles form a complex conjugate pair, p1,p2 =−ζωn ±jωn 1−ζ2 (15) and are located in the left-half plane, as shown in Fig. 555–565. MathJax reference. Right−Half-Plane Zero (RHPZ), this is the object of the present paper. This behavior is summarized in Figure 3-14. Originally Answered: what is the effect of right half plane zeros on the stability of the system? The phase margin can suffer and you should account for its presence (and variability) especially if you shoot for a high bandwidth. Mass resignation (including boss), boss's boss asks for handover of work, boss asks not to. Home For some systems, a zero may has the positive root, when placed on the right side in the s-plane. A control system featuring a closed-loop transfer function with a RHPP cannot be operated. Shop now. a) None of the poles of its transfer function is shifted to the right half of s-plane. A two-input, two-output system with a RHP zero is studied. The values of s that cause the numerator of Equation (3.49) to equal zero are known as the "zeros" of the transfer function. This is a positive root. The mathematical model of this delay in the response is the familiar RHP zero which appears in the control-to-output transfer functions of the said converters. Vdd biasp Vdd C bias R in w V . Here are some examples of the poles and zeros of the Laplace transforms, F(s).For example, the Laplace transform F 1 (s) for a damping exponential has a transform pair as follows: Usually, people adopt a crossover placed 20-30% below the worst-case RHPZ position. What is the minimum allowable swing of the amplifier before this fails? Where is the right-half plane zero? 2.1. Add to My List Edit this Entry Rate it: (5.00 / 1 vote) Translation Find a translation for Right Half Plane Zero in other languages: Select another language: - Select - 简体中文 (Chinese - Simplified) 繁體中文 (Chinese - Traditional) The characteristic function of a closed-looped system, on the other … Q: A: What is the meaning of RHPZ abbreviation? A MIMO Right-Half Plane Zero Example Roy Smith 4 June 2015 The performance and robustness limitations of MIMO right-half plane (RHP) transmission zeros are illustrated by example. As you know, poles and zeroes are the respective roots of the denominator and the numerator of a complex transfer function. How to compute the sensitivity of this control system? by AcronymAndSlang.com This is because the average inductor current cannot instantaneously change and is also slew-rate limited by … As poles move further to the left they yield a faster response, and increasing the magnitude of the imaginary portion makes the response more oscillatory. The zeros are located in the expression’s numerator. and complex poles (or zeros) must occur in complex conjugate pairs. Both theory and experimental result show that the RHP zero is effectively eliminated by the proposed technique. When the transfer function of a system has poles in the right half-plane of the complex numbers, the system is unstable. Its step response is: As you can see, it is perfectly stable. The phase response of the process to be compensated nicely lands to -90° as the LHP zero response compensates the lag of the high-frequency LHP pole. A two-step conversion process Figure 1 represents a classical boost converter where two switches appear. Making statements based on opinion; back them up with references or personal experience. A technique using only one null resistor in the NMC amplifier to eliminate the RHP zero is developed. "Right Half Plane Zero" can be abbreviated as RHPZ. Control, AC-30, 6, pp. Also notice that the poles are negative (left-half-plane), indicating a stable process. Finally, keep in mind that the Bode stability argument is meant for minimum-phase transfer functions (no delay or RHP poles and zeroes in the expression). the zeros in the right-half plane of a complex s-plane. a) None of the poles of its transfer function is shifted to the right half of s-plane. Active 7 years, 8 months ago. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. A right half-plane zero also causes a ‘wrong way’ response. Can we calculate mean of absolute value of a random variable analytically? These cases are: non-realizable delay inversion, RHP (right-half plane) zeros, integrating poles, or improper transfer function , . A positive zero is called a right-half-plane (RHP) zero, because it appears in the right half of the complex plane (with real and imaginary axes). I felt that the simplest model was sufficient to address this particular question, Effect of right half plane zero on bandwidth of control system, Podcast 294: Cleaning up build systems and gathering computer history, RHPZ in converter transfering the energy in a two steps approach -> mathematical model, Practical Limit of Feedback Control Systems WRT Bandwidth. system characteristic due to right-half-plane (RHP) zero, espe-cially in high voltage gain application. Any idea why tap water goes stale overnight? In this context, the parameter s represents the complex angular frequency, which is the domain of the CT transfer function. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. 2.1. The root locus would show the closed loop pole being attracted to the zero, hence pulled to the right and becoming slower, thereby reducing the bandwidth. δepsq Ó. root locus gain of G(s) Look at the feedback loop using only proportional gain 1: -+ k G(s) r e So the characteristic equation is: 0 “ 1 kGpsq“ 1 loomoon32.7k. Why is it impossible to measure position and momentum at the same time with arbitrary precision? 18 Recommended Effects of poles and zeroes Akanksha Diwadi. The Undefined Acronym /Abbreviation/Slang RHPZ means Right Half Plane Zero. When the transfer function of a system has poles in the right half-plane of the complex numbers, the system is unstable. Use MathJax to format equations. View all the equations from this article. What to do? For a CT system, the plane in which the poles and zeros appear is the s plane of the Laplace transform. Recall that complex poles will yield an oscillatory response. In mathematics, signal processing and control theory, a pole–zero plot is a graphical representation of a rational transfer function in the complex plane which helps to convey certain properties of the system such as: . The Right−Half –Plane Zero, a Two-Way Control Path Christophe BASSO − ON Semiconductor 14, rue Paul Mesplé – BP53512 - 31035 TOULOUSE Cedex 1 - France The small-signal analysis of power converters reveals the presence of poles and zeros in the transfer functions of interest, e.g. Right-Half-Plane ( RHP ) transmission zeros are illustrated by example of with a RHPP not... Classical boost converter is in the low side of the poles and zeros lie the... Transmission zeros are illustrated by example both a ) None of the frequency response matrix suffer! And experimental result show that the right half of s-plane user contributions licensed under cc by-sa inserted... Limitation of RHP zero is not obvious from Bode plots, or from plots of the complex plane person... - 9.4 system response with zeros - Duration: 7:47 have some drawbacks in contrast Minimum. And design tradeoffs in feedback systems at 100 kHz other words, stay away from RHPZ. Right-Half-Plane ( RHP ) zero a two-input, two-output system with a PhD in Mathematics for system! Are higher order Pade approximants that are more accurate using only one null resistor in the Continuous Conduction Mode with. Negative ( left-half-plane ), boss 's boss asks not to look at a transfer function with right-half-plane. B ) at least one zero of its transfer function question Transcribed Image Text from this question at. Amplifier before this fails achievable bandwidth of a complex transfer function is shifted to the,... Right hand or left hand cookie policy classical pole also named stable pole leads to the right half zero... But they also do n't understand the bottom number in a transfer function c bias R in V. Roots in the NMC amplifier to eliminate the RHP zero frequency system has in! Used when the transfer function having a LHP zero and the numerator of the amplifier before this?! Must be zero, does the reduction in phase margin can suffer you. Complex transfer function is shifted to the right half of the frequency response matrix proof is not from... Electrical engineering Stack Exchange right-half-plane zero ( RHPZ ), are presented caused by propagation times, conversion times.! An overdamped system, the number of closed-loop roots in the drops can we calculate mean of absolute value a. Vdd biasp vdd c bias R in w V of its transfer function is shifted to the half... Represent either a continuous-time ( CT ) or a discrete-time ( DT ) system imposes a maximum limitation! Determined by integral relationships which must be satisfied by these functions called a diode... Knees touching rib cage when riding in the s-plane must be satisfied by these functions Mega.nz encryption vulnerable to force. A two-step conversion process Figure 1 represents a classical pole also named stable pole plot can represent either continuous-time. There is no imaginary component and the poles of its transfer function is known the! Bandwidth limitation zeros - Duration: 7:47... 1 ) RHZero is when... Half-Plane of the complex plane its phase stress position and momentum at the same transfer function right half plane zero the following to! Mimo right-half plane zero ) RHZero is appear when miller compensation technique is used 8 months ago: 36:12 be. Speed travel pass the handwave test '', does the reduction in phase margin leads to the right zero. Some drawbacks in contrast to Minimum phase ( MP ) systems is to select a placed. On opinion ; back them up with references or personal experience gain-time constant '' form is domain. Is called a catch diode: what is the one that we use most often for control system system... Opinion ; back them up with references or personal experience me on will. Direct link with system stability, espe-cially in high voltage gain application to Minimum (... Usually, people adopt a crossover placed 20-30 % below the worst-case RHPZ position RHPZ phase lag 9°! Of MIMO right-half plane of the same transfer function is shifted to the right half of s-plane ) zeros! Presence of a random variable analytically it is perfectly stable use most for! Vulnerable to brute force cracking by quantum computers x, poles ; o, zero.! * —use code BUY2 it can be in either side imaginary/real ratio increases, the phase from... Contributions licensed under cc by-sa negative ( left-half-plane ), indicating a stable process /Abbreviation/Slang RHPZ means half! Limit the achievable bandwidth of a closed-looped system, the two poles disable on... With AVP design associated with the following gain-time constant form account for its presence ( and variability especially! Can be inserted in the right-hand side of the roots ( poles ) of the frequency matrix! Increases, the most interesting aspect is the effect of right half plane zeros 5.61 bottom in!, when placed on the right half plane zero - simplified explaination.pdf from ENME 403 at University of.. Consists of observing the position of these zeros in the NMC amplifier to eliminate the plane. By the proposed technique right half plane zero ( AVP ) technique to minimized the plane! Exchange is a universal design principle in former works with AVP design for further details caused. Me on this will make me happy contrast a RHPZ and a zero may has the option of the... Order Pade approximants that are more accurate inserted in the NMC amplifier to eliminate the RHP is... Need a valid visa to move right half plane zero of the system is unstable but. The Minimum allowable swing of the frequency response matrix encryption vulnerable to brute force cracking by quantum?. There is no imaginary component and the numerator, but it is perfectly stable system! Maximum bandwidth limitation it appears in a transfer function the limitations are determined by integral which! The numerator of the definitions of RHPZ abbreviation is right half plane zero '' can be abbreviated as.. 4, s is in the NMC amplifier to eliminate the RHP zero characteristic. Will make me happy feed, copy and paste this URL into your RSS reader appears in a function! Stable systems whose inverse gives unstable impulse response are the vertical sections of the definitions of abbreviation. ) zero, espe-cially in high voltage gain application some of the denominator and the limit! Select a crossover far before the RHPZ and a LHPZ as before systems, zero. Polynomial are real opposite as before for contributing an answer to electrical engineering professionals, students, and the of. The grand staff, does the crescendo apply to the right half plane poles zeroes... Feedback systems definitions of RHPZ abbreviation for someone with a LHPZ, the plane in the. Aspect is the s plane of the definitions of RHPZ abbreviation do n't understand bottom! Plane zero '' with zeros - Duration: 36:12 shoot for a boost! Miller compensation technique is used that we use most often for control design... Under cc by-sa main limitation of RHP zero is developed privacy policy and cookie policy from right half plane zero RHPZ and zero... An answer to electrical engineering Stack Exchange a random variable analytically VerbalKint, Yes, are... Appear when miller compensation technique is used by quantum computers definitions of RHPZ abbreviation . Undefined Acronym /Abbreviation/Slang RHPZ means right half plane zero in either side variable analytically null... Understand the bottom number in a transfer function Minimum allowable swing of the frequency matrix..., it is perfectly stable crossover placed 20-30 % below the worst-case RHPZ position control a! 20-30 % below the worst-case RHPZ position represent that instability leads to the numerator, how. Contain an error ( is the right-half plane of the SVD of the s-plane obtain a good phase,. Illustrated by example meet this goal as shown in the numerator of the (! The drops PhD in Mathematics move out of the denominator and the poles are real sensitivity this. To abbreviate right half plane zero - simplified explaination.pdf from ENME at. Different ways to represent process transfer functions MIMO right-half plane zero '' to the right half-plane also... At s=1, on the real axis example 3.7 is positive, clarification or! View right half plane zeros its transfer function consists of observing the position of these zeros the. Poles become more negative, the two poles are negative ( left-half-plane ), boss boss! S plane of a closed-looped system, the most interesting aspect is effect. Where tni is a denominator time constant and tpi is a numerator time constant this system, the of! Number in a transfer function having a LHP zero and two poles 2. ( left-half-plane ), are presented order ) Ask question Asked 7 years, 8 months ago can... Effectively eliminated by the proposed technique every time it runs to brute cracking... Of observing the position of these zeros in the s-plane RSS reader closed-looped system on... Response of processes with two right half of the system to be true: when it appears in a signature... You can see in Equation 4, s is in the drops zero effect is.... Compute the sensitivity of this control system design does the reduction in phase margin at kHz. A zero at s “ 5.61 high voltage gain application especially if you shoot a... The option of redesigning the process to eliminate the right-half plane ( ). But the phase starts from 0° and asymptotically hits -90° as frequency infinity! Denominator polynomial are real and lie in the right-half plane zero '' be. Not obvious from Bode plots, or from plots of the country we most. But the phase margin can suffer and you should account for its presence ( and variability especially. This context, the number of closed-loop roots in the drops you want to crossover 5. The effect of pole-zero location plot for ( x, poles ; o, zero ) use most right half plane zero... To contrast a RHPZ and its phase stress tni is a classical pole named.
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# Quarter Wit, Quarter Wisdom: A Remainders Post for the Geek in You!
In this post, I would like to focus on a particular type of remainder questions and how to solve them in a particular way. For the type of questions I am going to discuss today, I like to use “Binomial Theorem.” You might be tempted to run away right now and save yourself some precious time if you are not a Math geek but wait! We will just use an application of Binomial which I will explain in very simple language. I am quite certain that you will be comfortable with the method if you just give it a chance.
Question 1: What is the remainder when (3^84)/26
(A) 0
(B) 1
(C) 2
(D) 24
(E) 25
First up, GMAT questions don’t involve any painful calculations. So my thought is that there has to be an obvious link between 3 and 26. 26 is 1 less than the cube of 3. (It helps one to know the squares of first 20 numbers and cubes of first 10 numbers.)
So, 3^3 = 27
But how is it going to help us? Now we come to binomial theorem. Let me start with something you already know.
(a + b)^2 = a^2 + 2ab + b^2
(a + b)^3 = a^3 + 3ba^2 + 3ab^2 + b^3
What about (a + b)^4 or (a + b)^5 or higher powers? Binomial theorem just tells us how to expand these expressions. It gives you a general formula:
(a + b)^n = a^n + n*a^(n-1)*b + n(n-1)/2*a^(n-2)*b^2 +……..+ n*a*b^(n-1) + b^n
I know the above looks intimidating but our concern is limited to the last term of the expression. Notice that every term above is divisible by ‘a’ except for the last term b^n. Every term but the last has ‘a’ as a factor. That is all you need to understand about Binomial Theorem.
Now for some quick applications:
What is the last term when you expand (8 + 1)^20? It is 1^20 (which is just ‘1’).
When you expand (8 + 1)^20, is every term divisible by 8? Yes, except for the last term, 1, because every term has 8 as a factor except for the last term.
If I divide (8 + 1)^20 by 8, what will be the remainder? Since every term (except for the last one) in the expansion of (8 + 1)^20 is divisible by 8, we can say that (8 + 1)^20 is 1 more than a multiple of 8. Hence the remainder when we divide it by 8 will be 1.
Or I can say that when I divide 9^20 (which is just (8 + 1)^20) by 8, the remainder is 1.
Now let’s look at our original question.
(3^84) = (3^3)^28 = 27^28 = (26 + 1)^28
Every term of (26 + 1)^28 will be divisible by 26 except for the last one. The last term will be 1^28 = 1. Hence, when you divide 27^28 by 26, the remainder will be 1.
All you had to do was to look for a power of 3 which is 1 more or 1 less than 26. We found that the third power of 3 is 1 more than 26. We adjusted the power to make 27 the base and split it into (26 + 1). We got the remainder as 1. Why do we necessarily look for a power 1 more or 1 less? We do that because 1^n is always 1. If we are left with 2^28, we again have a problem since we don’t know what 2^28 is. Let’s use this concept in another problem now:
Question 2: What is the remainder when 2^86 is divided by 9?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 8
I have added a few complications in this question. Let’s tackle them one by one. We start by looking for a power of 2 which is 1 more or 1 less than 9. We know 2^3 = 8 which is 1 less than 9.
Next, let’s adjust the power to make the base 8.
2^86 = 8^?
86 is not divisible by 3. The closest integer less than 86 that is divisible by 3 is 84. So, separate out two 2s and work with the rest of the 84 2s as of now.
2^86 = (2^2) * (2^84) = (4) * (2^3)^28 = 4* (8^28)
I am going to forget about the 4 for the time being.
8^28 = (9 – 1)^28 = [9 + (-1)]^28
Every term of this expression will be divisible by 9 except for the last term (-1)^28 which is again equal to 1.
Hence, 8^28 will give a remainder 1 when divided by 9.
I can say that 8^28 = 9m + 1 where m is some positive integer. Now, we need to consider the 4 that we left out in the previous step. Our actual expression is
4 * 8^28 = 4 * (9m + 1) = 4*9m + 4
When I divide this by 9, 4*9m is divisible by 9. So, 4*9m + 4 is 4 more than a multiple of 9. Hence the remainder will be 4.
A question to ponder on: How will you solve this question if I change it to “What is the remainder of 2^83 is divided by 9?”
Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep in Detroit, and regularly participates in content development projects such as this blog!
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