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# The product of all the prime numbers less than 20 is closest

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The product of all the prime numbers less than 20 is closest  [#permalink]

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02 Jul 2012, 02:21
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Difficulty:

85% (hard)

Question Stats:

47% (01:11) correct 53% (01:15) wrong based on 2899 sessions

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The product of all the prime numbers less than 20 is closest to which of the following powers of 10 ?

(A) 10^9
(B) 10^8
(C) 10^7
(D) 10^6
(E) 10^5

Diagnostic Test
Question: 15
Page: 22
Difficulty: 650

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The product of all the prime numbers less than 20 is closest  [#permalink]

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02 Jul 2012, 02:22
19
1
27
SOLUTION

The product of all the prime numbers less than 20 is closest to which of the following powers of 10 ?

(A) 10^9
(B) 10^8
(C) 10^7
(D) 10^6
(E) 10^5

We should find the approximate value of 2*3*5*7*11*13*17*19 to some power of 10.

# of different approximations are possible.

Approach #1:

2*5=10;
3*7=~20 (actually more than 20);
11*19=~200 (actually more than 200);
13*17=~200 (actually more than 200);

$$2*3*5*7*11*13*17*19\approx{10*20*200*200=8*10^6}\approx{10^7}$$.

Approach #2:

2*5=10
3*17=~50 (actually more than 50);
7*13=~100 (actually less than 100);
11*19=~200 (actually more than 200)

$$2*3*5*7*11*13*17*19\approx{10*50*100*200}=10^7$$.

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Re: The product of all the prime numbers less than 20 is closest  [#permalink]

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25 Oct 2013, 04:45
9
1
3
I have approached in the following way:-

2 x 3 x 5 x 7 x 11 x 13 x 17 x 19 = 10 x 21x 11 x 13 x 17 x 19 = 10 x 10 x 2.1 x 10 x 1.1 x 10 x 1.3 x 10 x 1.7 x 10 x 1.9 = 10^6 x (2.1 x1.1 x 1.3 x 1.7 x 1.9)

The approximation of the (2.1 x 1.7 x 1.9 x 1.3 x 1.1) will be close to 10, so the answer is 10^7
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Re: The product of all the prime numbers less than 20 is closest  [#permalink]

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Updated on: 02 Jul 2012, 03:39
1
Hi,

Difficulty level: 650

Product of prime numbers less than 20 = 2*3*5*7*11*13*17*19
=(2*5)*3*(7*13)*(11*17)*19
=3*10*91*189*19
~(30)*(90*19)*190
~5700*1710
~5700*1700
~9690000 $$(=10^6)$$

Regards,

Originally posted by cyberjadugar on 02 Jul 2012, 03:17.
Last edited by cyberjadugar on 02 Jul 2012, 03:39, edited 2 times in total.
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Re: The product of all the prime numbers less than 20 is closest  [#permalink]

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02 Jul 2012, 03:25
2
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Bunuel wrote:
The Official Guide for GMAT® Review, 13th Edition - Quantitative Questions Project

The product of all the prime numbers less than 20 is closest to which of the following powers of 10 ?

(A) 10^9
(B) 10^8
(C) 10^7
(D) 10^6
(E) 10^5

Diagnostic Test
Question: 15
Page: 22
Difficulty: 650

GMAT Club is introducing a new project: The Official Guide for GMAT® Review, 13th Edition - Quantitative Questions Project

Each week we'll be posting several questions from The Official Guide for GMAT® Review, 13th Edition and then after couple of days we'll provide Official Answer (OA) to them along with a slution.

We'll be glad if you participate in development of this project:
2. Please vote for the best solutions by pressing Kudos button;
3. Please vote for the questions themselves by pressing Kudos button;
4. Please share your views on difficulty level of the questions, so that we have most precise evaluation.

Thank you!

The product is $$2*3*5*7*11*13*17*19 = 10 * 21 * 11*(13*17) * 19 \approx10*20*10*15^2*20=900*10^4\approx10^7$$.

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Re: The product of all the prime numbers less than 20 is closest  [#permalink]

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02 Jul 2012, 03:45
2
4
Hi,

The magnitude of error depends on the approximation done in the question.

For example: 51*671=34221
50*671=33550 (error = 671)
51*670=34170 (error = 51)

Thus, the when the larger number is approximated the error is less.

Regards,
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Re: The product of all the prime numbers less than 20 is closest  [#permalink]

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02 Jul 2012, 05:53
Hi,

The magnitude of error depends on the approximation done in the question.

For example: 51*671=34221
50*671=33550 (error = 671)
51*670=34170 (error = 51)

Thus, the when the larger number is approximated the error is less.

Regards,

then why D instead of C.
i got the same and chose D
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Re: The product of all the prime numbers less than 20 is closest  [#permalink]

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02 Jul 2012, 05:58
kashishh wrote:
Hi,

The magnitude of error depends on the approximation done in the question.

For example: 51*671=34221
50*671=33550 (error = 671)
51*670=34170 (error = 51)

Thus, the when the larger number is approximated the error is less.

Regards,

then why D instead of C.
i got the same and chose D

Hi,

I believe you are right and answer should be (C).

Regards,
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Re: The product of all the prime numbers less than 20 is closest  [#permalink]

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02 Jul 2012, 12:58
1
1
2*3*5*7*11*13*17*19 = 210*(10+1)(10+3)(10+7)(10+9)
210 ~ 10*10*2
So the original equation will be equivalent to 10*10*2*(10+1)(10+3)(10+7)(10+9)
The highest order of the equation will be 10^6, and other part of expression will not make it closer to 10^7.
I have solved it mostly on hunch, no sure how correct I am.
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Re: The product of all the prime numbers less than 20 is closest  [#permalink]

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05 Mar 2014, 11:48
1
Similar to Bunuel's first approach:
I thought it was easiest to just do :
2*3*5*7 = 210
11 roughly 10
13 roughly 10
17 roughly 20
19 roughly 20
so 210*10*10*20*20 = 8*10^6 = 10^7
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Re: The product of all the prime numbers less than 20 is closest  [#permalink]

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06 Mar 2014, 21:40
1
Product of Prime no between 1 & 20 =

(2*3*5*7)*(11*13)* (17*19)

210 * 143 * 323

$$(10^2+10) * (10^1+43) * (10^3+23)$$

$$(10^3+4300+10^2+430) * (10^3+23)$$

$$5830 * (10^3+23)$$

5830000 + 134090

= 6974090 this is near to 10^7

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Re: The product of all the prime numbers less than 20 is closest  [#permalink]

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10 Mar 2014, 10:15
2*19 =~40
3*17=~60
5*13=~50 (More than 50)
7*11=~70 (")

=40*60*50*70
=4*6(10^2)*5*7(10^2)
=2*(10^1*10^2)*3.5(10^1*10^2)
=7*10^6
=~10^7

Like this any prime number can be multiplied with.
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Re: The product of all the prime numbers less than 20 is closest  [#permalink]

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11 Apr 2014, 19:07
Can anyone comment on my method's effectiveness/efficiency:
Prime numbers < 20 are: 2, 3, 5, 7, 11, 13, 17, 19 (Count = 8)
Median = 9 (through (7+11)/2)
So product should be approx. close to 9^8 which in turn is closest to 10^7, thus answer is 10^7.

Thanks for reviewing.
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The product of all the prime numbers less than 20 is closest  [#permalink]

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21 Jun 2015, 03:36
1
Here's my solution
2*3*5*7*11*13*17*19

1) 2*3*5*19 ~ 30*20
2) 11*13 ~ 140
3) 7*17 ~ 120

SO we have 600*140*120 = 10^4*168*6 (168*6 ~ 1000) --> ~ 10^4*10^3 ~10^7

Method 2

2*3*5*7*11*13*17*19 = 2*5*11*19*(10-7)(10+7)(10-3)(10+3) ~ 2*10*10*10*50*80=100*10*10*10*10*8 ~$$10^7$$ Answer (D)
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The product of all the prime numbers less than 20 is closest  [#permalink]

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29 Aug 2016, 19:15
Bunuel wrote:
SOLUTION

The product of all the prime numbers less than 20 is closest to which of the following powers of 10 ?

(A) 10^9
(B) 10^8
(C) 10^7
(D) 10^6
(E) 10^5

We should find the approximate value of 2*3*5*7*11*13*17*19 to some power of 10.

# of different approximations are possible.

Approach #1:

2*5=10;
3*7=~20 (actually more than 20);
11*19=~200 (actually more than 200);
13*17=~200 (actually more than 200);

$$2*3*5*7*11*13*17*19\approx{10*20*200*200=8*10^6}\approx{10^7}$$.

Approach #2:

2*5=10
3*17=~50 (actually more than 50);
7*13=~100 (actually less than 100);
11*19=~200 (actually more than 200)

$$2*3*5*7*11*13*17*19\approx{10*50*100*200}=10^7$$.

Approach #1:

2*5=10;
3*7=~20 (actually more than 20);
11*19=~200 (actually more than 200);
13*17=~200 (actually more than 200);

$$2*3*5*7*11*13*17*19\approx{10*20*200*200=8*10^6}\approx{10^7}$$.

HELP WITH Approach #1:

10*20*200*200
$${10^1} {10^1} {10^2} {10^2} = {10^6}$$

Where does the 8 comes from?
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Re: The product of all the prime numbers less than 20 is closest  [#permalink]

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29 Aug 2016, 19:50
Bunuel wrote:
The product of all the prime numbers less than 20 is closest to which of the following powers of 10 ?

(A) 10^9
(B) 10^8
(C) 10^7
(D) 10^6
(E) 10^5

Diagnostic Test
Question: 15
Page: 22
Difficulty: 650

Prime nos. less than 20 are 2,3,5,7,11,13,17,19

(2*5)x(11*19)x(3*7) (13*17)
10x209x21x221
210*200*200
200 x 200 x 200
8x10^6 = 10^7
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The product of all the prime numbers less than 20 is closest  [#permalink]

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03 Dec 2016, 19:39
High Quality Question from the Official Set.
Its always good to remember the first 10 primes
2
3
5
7
11
13
17
19
23
29

Now the product of all primes <20 => 2*3*5*7*11*13*19
Product => 210*143*323
Approximating => Product
=>200*150*300 => 9000000 which is close to 10^7
Hence C
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The product of all prime numbers less than 20 is closest to which of  [#permalink]

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26 Dec 2016, 07:59
1
interceptor77 wrote:
The product of all prime numbers less than 20 is closest to which of the following powers of 10?

A. 10^9
B. 10^8
C. 10^7
D. 10^6
E. 10^5

Hi,

This has been discussed here: the-product-of-all-the-prime-numbers-less-than-20-is-closest-135192.html

(2*5)(3*7)(11*19)(13*17)=(10)(~20)(~200)(~200)=(~8)$$(10^6)=10(10^6)=10^7$$
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Re: The product of all the prime numbers less than 20 is closest  [#permalink]

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22 Jun 2017, 09:29
dansanso wrote:
Bunuel wrote:
SOLUTION

The product of all the prime numbers less than 20 is closest to which of the following powers of 10 ?

(A) 10^9
(B) 10^8
(C) 10^7
(D) 10^6
(E) 10^5

We should find the approximate value of 2*3*5*7*11*13*17*19 to some power of 10.

# of different approximations are possible.

Approach #1:

2*5=10;
3*7=~20 (actually more than 20);
11*19=~200 (actually more than 200);
13*17=~200 (actually more than 200);

$$2*3*5*7*11*13*17*19\approx{10*20*200*200=8*10^6}\approx{10^7}$$.

Approach #2:

2*5=10
3*17=~50 (actually more than 50);
7*13=~100 (actually less than 100);
11*19=~200 (actually more than 200)

$$2*3*5*7*11*13*17*19\approx{10*50*100*200}=10^7$$.

Approach #1:

2*5=10;
3*7=~20 (actually more than 20);
11*19=~200 (actually more than 200);
13*17=~200 (actually more than 200);

$$2*3*5*7*11*13*17*19\approx{10*20*200*200=8*10^6}\approx{10^7}$$.

HELP WITH Approach #1:

10*20*200*200
$${10^1} {10^1} {10^2} {10^2} = {10^6}$$

Where does the 8 comes from?

2*2*2*1=8. & There are six 0's hence the 8 * 10^6.
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Re: The product of all the prime numbers less than 20 is closest  [#permalink]

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28 Jun 2017, 04:09
2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 = ?
The questions tests the ability to approximate final results. Keeping this in mind, immediately simplify the task.
(2 * 5 take as 10) * (3 * 7 take as 20) * (11 take as 10) * (13 * 17 take as 220) * (19 take as 20) = 10 * 20 * 10 * 220 * 20 = 88 *10^5 = 0.88 * 10^7 -> quite close to 1 * 10^7 = 10^7. Answer C.
Re: The product of all the prime numbers less than 20 is closest &nbs [#permalink] 28 Jun 2017, 04:09

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Question 9

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## Friday, June 19, 2015

### The Prophet and the Dajjal

I saw Prophet Muhammad (peace be upon him). He was eating when the visitor arrived at his doorstep. It was a woman with a boy, small boy,… around 4 or 6 years of age. She offered his son to the Prophet (peace be upon him) saying: “please take care of my son. Adopt him. Raise and teach him.” He agreed and took the boy.

Years went by, the small boy lived in that household, raised by the Prophet. Whether the Prophet was traveling for da’wa or for business, the boy always tag along.

One day, they went to a far-distant place. On the way back home, they travelled beside a very long river. (yes a river amidst the desert.) On camel, the boy was seated on the lap of the Prophet. Suddenly, he jumped out, crossed the river, and on to the other side and on to the desert, making his path perpendicular to that of the Prophet’s. And it was said this boy would be the “Dajjal.” Then I woke up.

What is your interpretation of such dream? Did I really see the prophet? Based on the real history, such event did not occur but what or who could that boy symbolize? Please share your thoughts.

## Monday, January 27, 2014

### The Story of Hatim and the Messenger Sent to Kill Him

1.
Our being is like a mirror, vulnerable to the incoming dust. Every now and then, we are vulnerable to the dusts of feeling of anger, negative thoughts and so on. Yet, just like the ordinary mirror we can wipe the dust off. As to our being, we can wipe these negative dusts by taking dosage of tale with beautiful wisdom.

Today, I'm gonna give you a dose of that taken from one of the great masterpieces of all time, the book entitled "The Bustan of Saadi."

2.
One of the kings of Yemen was renowned for his liberality, yet the name of Hatim was never mentioned in his presence without his falling into a rage. “How long,” he would ask, “will you speak of that vain man, who possesses neither a kingdom, nor power, nor wealth?”

On one occasion he prepared a royal feast, which the people were invited to attend. Someone began to speak of Hatim, and another to praise him. Envious, the king dispatched a man to slay the Arabian chief, reflecting, “So long as Hatim lives, my name will never become famous.

The messenger departed, and traveled far seeking for Hatim that he might kill him. As he went along the road a youth came out to meet him. He was handsome and wise, and showed friendliness toward the messenger, whom he took to his house to pass the night. Such liberality did he shower upon his guest that the heart of the evil-minded one was turned to goodness.

In the morning the generous youth kissed his hand and said, “Remain with me for a few days.”

"I can not stay here any longer,” replied the messenger, “for urgent business is before me.”

“If you will entrust me with your secret,” said the youth, “I will not hesitate to help you.”

“O, generous man!” was the reply, “give ear to me, for I know that the generous are concealers of secrets. Perhaps in this country you know Hatim, who is of lofty mind and noble qualities. The king of Yemen desires his head, though I know not what enmity has arisen between them. Grateful shall I be if you will direct me to where he is. This hope from your kindness do I entertain, O friend!”

The youth laughed and said, “I am Hatim, see here my head! Strike it from my body with your sword. I would not wish that harm should befall you, or that you should fall in your endeavor.”

Throwing aside his sword, the man fell on the ground and kissed the dust of Hatim’s feet. “If I injured a hair on your body,” he cried, “I should no longer be a man.” So saying, he clasped Hatim to his breast and took his way back to Yemen.

“Come,” said the king as the man approached, “what news do you have for me?” Why did you not tie his head (Hatim's) to your saddle-straps? Perhaps that famous one attacked you and you were too weak to engage in combat.”

The messenger kissed the ground and said, “O, wise and just king! I found Hatim, and saw him to be generous and full of wisdom, and in courage superior to myself. My back was bent by the burden of his favors; with the sword of kindness and bounty he killed me.”

When he had related all that he had seen of Hatim’s generosity, the king uttered praises upon the family of the Arab chief and rewarded the messenger with gold.

3.
Look! the man was sent by the king to slay Hatim with a mighty sword, but returned weeping, slain by Hatim's overpowering kindness.
The hard hatred melted away by the heat of love.

## Sunday, December 1, 2013

### A Look at Ka Freddie's Wedding

Recently, news outlets and social media were filled with the headline about the wedding of Philippine music legend Freddie Aguilar. He married his 16-year-old-girlfriend under Islamic rites in Maguindanao on November 22, 2013.
 Freddie Aguilar

This turned into mixed reactions. While some affectionately congratulated them, others frowned at this that he and his partner converted into Islam for no other reason than just to get married. (yes, they are considered as converts and were even given muslim names.)
But wait a second, let's not point those hasty "bad" conclusions to the great man. Whether they'll be good muslims or not is God's business for no-one can exactly know what's in their hearts for that matter. Let's set aside that.

As of this moment, from where we see it, there are two certain things:
1. Islam offers the solution to human needs.
Unlike other religions wherein celibacy is advocated, In Islam marriage is encouraged. It is even said that when a man marries, he has completed half of the faith. Islam permits the marriage once the primary requirements are met: the consent of the groom, the bride and the bride's parents (or guardian) had consented, mahr or bridal gift, witnesses, and (possibly) pre-nuptial conditions set by both parties.
As to the age of the two, as long as they are "old enough." And there is no restriction on the age gap.
So, even if the man is relatively older than the woman or the reverse, as long as these requirements are met the marriage of the two are considered decent and honorable in Islamic point of view.

If a man, for example, failed to marry while he was young, became too much busy with other things in his life. And only at some later years in his life that he found the time for marriage. In Islam, he is still given the chance to marry not only to woman about his age but also to younger one. Whether he marries old or young woman, he is respected just like any other married man in Islam.

In western society, this is shameful. To avoid the sneer of the general public, an old man (if he couldn't find a woman about his age) would be forced to practice celibacy or worse, involved in rape, fornication or prostitution. Would you like to see sir Freddie get married legally or discover him involved in those immoral acts?

2. Their love is too great.
Sir Freddie is a legendary singer. No doubt. His tune and meaningful lyrics can never be forgotten. His reputation is already well built through the decades. His fans is almost everywhere.
Yet, he and his new wife submitted to Islam to make their love legal through matrimony. . . Even if that appeared to a betrayal to the christian community for some people, or even if that appeared to be abandoning their comrade and his fans, they could bravely do it. They don't really care about what other people might say. How could they? Its a proof of their great love for each other. Now, isn't that something?

May their marriage be a happy one. And if their conversion to Islam was only for their marriage, may the Almighty make this as a mean to find the absolute truth that they may become true muslims.
$latex \square$

References:
http://www.abs-cbnnews.com/entertainment/11/22/13/teen-explains-why-she-married-freddie-aguilar
http://www.pep.ph/news/41281/Freddie-Aguilar-after-marrying-16-year-old-girlfriend-under-Muslim-rites:-Sana-tumigil-na-ang-detractors-namin.
The photo above is from http://entertainment.inquirer.net/122665/lawyer-accusing-freddie-aguilar-gets-death-threats

## Tuesday, June 18, 2013

### Unplanned Visit To The Blue Mosque

We’ve arrived at the Ninoy Aquino International Airport (NAIA) terminal 3 five minutes passed 7 in the morning on Monday after approximately 1 hour travel by plane from Cagayan De Oro. After fetching my baggage, I hastened to walk towards the main hallway leading to the exit door. Before even reaching the hallway, my cell phone rang. I immediately pressed the answer button and put it right beside my ear right away.

“Hello! Are you already here?” asked a woman’s voice in my phone.

“Yes,” answered. I put the phone away from my ear to see who was calling. It was the woman I contacted a few days ago.

“I’m so sorry. I tried my best to get hold of your passport but my co-worker here has accidentally included it in the regular ones and had sent it already to Cagayan De Oro last Saturday.”

She was working in the Department of Foreign Affairs (DFA) in Pasay City. Three days ago, I asked her if she could help me release my passport the soonest possible since I was going to attend a two-day international conference in Thailand later that week. Official date of release of my passport given by DFA Cagayan De Oro was on Tuesday. One week before the release date I went to Cagayan De Oro to request for an earlier date but to my disappointment, I got denied. . . They even told me that my passport’s release would be delayed by one week from the official date which would disallow me to participate in the conference. Fortunately, someone from our university knows this woman. She suggested the only way to release it on time is that she should get hold of the passport in their office and then I would claim it there.

But now, things have gone out of control. Dark sadness looms in my head and sweat of anxiety started to flow from my glands. I already bought round trip ticket for travel between Cagayan De Oro City and Manila, in the anticipation that I would be in Thailand for the conference.

"Oh No! this can't be happening." I thought. While my professors, who were to attend the same conference, were starting to leave the airport, I took refuge in a corner behind a big pillar of the terminal and started finding an electric outlet to charge my cell phone which was already dying of charge. I called and texted several people, trying to find somebody who could claim my passport in Cagayan De Oro and send it to me in the shortest possible time. Hours went by, I could contact no-one to do what I needed. Then the feeling  of despair started to crumple my very being. While I scanned my phonebook on my phone to call another somebody who might know someone who could do the job, I uttered silent prayers entangled with a soundless cry.

And thank God, after four hours of continuously calling and texting while sitting in the same spot, I’ve found one person. She's is a sister-in-law of one of  my professors. She was working in Cagayan De Oro.

That afternoon, we were able to buy my plane ticket for Bangkok, Thailand. She successfully claimed my passport and it would arrive the following day.
 My very kind friend Bash. I shot this picture of him in NAIA after his duty.
While waiting for my flight for Thailand which was scheduled on Wednesday, I stayed in Taguig City. My beloved friend Bash, who was residing in the place, had arranged my accommodation at a condo unit. He told me that just a block from the building where I stayed, is the famed big masjid called the “Blue Mosque.” But he couldn’t show me around the place since he had to go to work.

By 3 in the afternoon I went out to find it. I saw this boy walking down the street. He was wearing a Pakistani dress so I was pretty sure he was a Muslim. He looked to the sides and also saw me walking from aisle of two buildings towards the street. I raised my right hand while we were looking at each other, signaling for him to stop.

On reaching him, I greeted him with salam and shook his right hand.

“Where is the nearest mosque here?” I immediately asked. Just after saying that, a loud call for prayer sounded from a speaker. And I suddenly observed that it came from behind him. Just a few steps from where we were standing was an entrance of a small masjid.

“Here it is,” He smiled while pointing to the entrance. I couldn’t help but smiled also. I continued my glance from the flip flops on the entrance, the roof, and then the small dome behind some branches of tree. I realized that behind him was a small masjid.

“I mean the Blue Mosque. I heard it is near from here.”

Then he accompanied me. It was located at an elevated ground, south of the condo building where I stayed. It was enclosed by surrounding fence with several metal gates. Beautifully painted with light yellow color mainly on its walls, white on its fence, and blue on its big dome and edges, its huge structure could attractively be seen even from a far distance. Upon entering the main gate, we climbed up a stair of few steps with green plants on both sides. A few steps more from the main entrance gate was a circle ground of several feet in radius where a flower plant was grown. The name of the Almighty “Allah” in Arabic were splendidly written on two sides of the outside wall of its porch where we entered the main building. After removing our shoes, we had finally reached the inside of the masid. The floor was totally covered with soft and thick maroon carpet. Long grey curtain was partially unfurled on our left which indicates the space for women and the pulpit (mimbar in Arabic) was on our right. Then I realized we have entered the mosque from the North.

 The Blue Mosque
The area for ablution was in South, side by side with the bathrooms. Soon, the congregational afternoon prayer started. The imam’s voice could be heard clearly through the sound system's speaker. There were electric fans here and there, either attached to a huge pillar or standing on its own. But since the three sides of the mosque, north, south and east were all open allowing free air to flow in and out, the fans needed not be in maximum to counter the mild hotness of the afternoon.

After the prayer, I sat in the south porch, my back against the wall of the mosque. I enjoyed the ambience: air,  space offered by the huge structure, little noise from adolescents playing nearby.

I also had the chance to talk with a guy in the da'wah (Islam propagation) there.

Most of the afternoon, I spent with the boy who had accompanied me. He said his name was Zaynuddin. He was among the students there who focus on the memorization of the entire Qur'an. He also told me some additional information regarding the masjid. There were two administrations involved there: one focused on the maintenance of the masjid and the other is focused on the madrasa or school.

There were several classrooms on the east side of the masjid, far behind the space for women, intended for madrasa classes. And behind these classrooms were the dormitory rooms for the students.

 A worshiper in retreat.
Some people stay there for the night often. Perhaps, homeless or others are known to be performing i'tikaf, a practice consisting of a period of spiritual retreat in a masjid for a certain number of days in accordance with the believer's own wish.

It is refreshing and inspiring to know that even in most populated and busiest place, there are still people who continue to uphold a discipline on spirituality, not forgetful of the remembrance to the Lord.

Now, I seem to understand why I got delayed in my trip to Thailand. Had my passport not delayed, probably I would have joined my professors to go to Thailand the same day I arrived to Manila and could not have visited this place. But I guess the Almighty just wanted to give me a chance to visit this beautiful masjid and His fellow sincere worshipers in this city.

“He knows what is before them And what is behind them: And to Allah go back All questions (for decision)” Qur’an : Surah Al Hajj 22:76

Let us not judge things or events by its immediate outcome. The Almighty is the All-Knowledgable, the All-Knower. He chooses to show us things but sometimes we are not shown the wisdom directly but behind some things or some events. Everything happens to us for a reason. $latex \square$

 The arabic word "Allah" engraved on the main entrance.

 The main gate as seen from the inside of the Blue Mosque.

 Zaynuddin - the boy who had accompanied me.

## Saturday, June 15, 2013

### Terminal - A book review

 The cover of the book Terminal written by Prof Almayrah Tiburon.
Terminal is a book containing collection of Short Stories. This book was originally written by a friend, Prof. Almayrah Tiburon, a Filipino professor of MSU-Main. It contains eleven beautiful short stories whose settings are in the parallel realm of Marawi City and featuring the Meranao people and culture, and some issues that have existed in the region. All of them are written in Filipino language with some phrases in Meranao dialect. A glossary is included where the readers may find the concise meanings of some hard Filipino and Meranao terms.

They mention Horror Stories, Meranao culture, Drug addiction, Disadvantages of working abroad, Betrayal to one's spouse, prostitution, witchcraft, ghosts, and marriage and divorce. The descriptions are clear and did not fail in capturing my full attention when I was reading them.

I may say that the author is very talented in writing... so talented that she can even make up stories about physical objects. In the book, two of the eleven stories were even entitled after two things: container and Dyaket. At just knowing these titles, you might wonder what are the plots of these stories?

I also like the story entitled "Lamber." It's about a young man who became a drug addict. In the plot, his hallucinations, and struggles of his loving family were seriously depicted. I could related to this kind of story for in the past I had also met some of people like these. Indeed, prohibited drugs can shatter one's life. The shattered dream and misery in this plot is too great that it was made into a short film. The film was shot right in the campus where the author is working as a college professor. It is yet to be submitted in a competition of short films.

The author also explores the story of married people in another story entitled "L'pad." In this story, while her husband went out of Philippines to work abroad, she was tempted, and had a love affair with a married man. The secret and forbidden love were enjoyed by both. Certainly, truth would be known. And the shameful acts of these two people were eventually discovered, and known by their family and extended relatives. The author exploits this conflict to introduce to the readers the law provided by the Meranao culture in dealing with these kinds of issues.

But if I were to decide which among the twelve have attracted me the most, it would be “Paktol.”

“Paktol” is story about witchcraft, love and revenge. In this story, Yusoph and Sahara were married. The marriage seemed to be perfect; Husband is a responsible man; wife is kind, very loving and a good daughter who came from a clan of good family background; man and wife is in good compatibility and in constant care and love for each other; their in-laws are in good terms. And this marriage became even happier when they were blessed with a son after a year in union.

But this colorful living came into gloomy atmosphere when Yusoph fell sick one morning. Each passing day, the sickness became worse. Instead of going to a doctor, he went up to a pamomolong, a Meranaw term for one who is good healer at sickness if were caused by witchcraft. But even the pamomolong admitted that this magic was all too strong. And this kind could have only been done by a person who had been wronged by Yusoph in the past. And the author at this point unfurls the past history of Yusoph using a flashback.

The culprit was a woman named Aklimah, someone who had fallen in love with Yusoph so madly when they were still in college. She didn’t feel love from her direct family and so she became too engrossed upon Yusoph’s comfort. She had totally believed that he was her only ticket to happiness in life. One day, they discovered that she was pregnant and she rejoiced at this. For Aklimah, this was assurance to finally be his legal wife. For him and his family this was shame and immorality. His parents disapproved her to be his wife. And he was a very obedient son and eventually he ended up with Sahara, the woman his family favored. This was way too painful for Aklimah. Her life even became bitter when she lost her unborn baby with Yusoph. The feeling of being betrayed and forsaken by her man and hated by his family resulted to despair and evil plot in bitter vengeance.

The conflict in arranged marriage in Meranao and other muslim cultures in Philippines is common. But the battle in this story became most unique since it is mixed with witchcraft. You’d be surprised at how the author vividly described the procedure in using witchcraft to take revenge, exploiting the blood of the dead unborn child, the miserable scene created where the faithful wife carrying the burden of feeling not able to find any medicine to her husband’s harsh unknown sickness. While I was reading this, I felt I was there at the scenes where it had all occurred.

Is there such a cure for even the most powerful black magic ever known? Will the bad spirit of vengeance prevail over the love for the good man she had cherished the most?

To find out the what's the ending of this story and to feel the thrill and suspense of stories like these, you’ll have to read this book.

Terminal by Prof. Almayrah Tiburon is available at the Filipino Department, College of Social Sciences and Humanities in MSU-Main Campus. $latex \square$

 I asked her to write something for me in my copy of her book since she was the author. And there they are, those inspiring words for me. :-)

## Monday, May 13, 2013

### Island Global Insitute, Inc. and My Visit to Malabang

Plan, Execute, Conquer – these are the three-word formula shared by one of today’s youngest millionaires, Kevin Abdulrahman, in order to succeed in life. It means that in order for one to be able to achieve his dream in life, the very first step is to plan. Next is executing the plan, putting the plan into action. And then eventually, the desired result will most likely be achieved – that is, conquered!

 A beautiful, and vast land of Malabang.
These days, I am witnessing a great dream starting to unfold. The dream of a very young, vibrant, intelligent and talented man, my beloved friend, Aleem Abdulnasser Garo. His beautiful dream is a great legacy to his hometown and that is to build a world class university in his homeland, Malabang. And his plan for this massive dream is being executed these days.

 The 29-year-old founder of Island Global Institute, Alim Abdulnasser Garo.
I first met him a few years ago in MSU-Main campus, when he was still a college student. His course was International Relation (IR) then. Among our circle of friends, he was known as very knowledgeable at Arabic language and Islam, hence we addressed him "Ustad." I was fond of talking to him about Arabic and religious practices. I once even requested him to teach me Arabic language. He was also so smart in other disciplines and he had this strong urge to learn almost everything. He even enrolled extra subjects, like Math, even if they were not included in the curriculum of his course. But with all these impressive characteristics, he remains humble, soft-spoken and kind. I knew it then that this was one hint of a great person in the future.

Just last week, he invited me to attend their Summer Advance Class 2013 Culmination program in Malabang. Malabang is a municipality under the region Lanao Del Sur, Philippines. It is located northwest off Parang. Its southwest side is oulined by sea of Illana Bay. Malabang comes from the maranao word "Mala" which means big and "Bang" or Adhan in Arabic which is a call to prayer. History has it that an Arab missionary name Sharif Kabunsuan arrived to the Sultanate of Tubok around 14th century to introduce Islam to the people in mainland Mindanao. Upon docking in the shore of Tubok which is known today as barangay Tambara, the Arab Missionary call a prayer with his loud voice. The First Adhan or "Bang" in Lanao. The vilagers heard it and called it a BIG ADHAN, or locally "Mala a Bang", as referring to the birth of Islam in the Island. The Islam spread through the mainland Mindanao. It reaches the northern settlers in northern Mindanao. Being marked as the historical foundation of the religion Islam in central and northern Mindanao, they name the town as Malabang.

On May 6, 2013, my friend, Engr. Mohaimin Samanoden fetched me from Marawi city. We followed the route along Lake lanao, passing through Marantao onto Pualas. Then, we passed onto side of Lake Dapao,  The scenery of travel was just amazing. There's a highway overlooking the edge of Lake Lanao at some height, flat vast farm with high mountains at the outskirts extending up towards the cool-looking clouds  above. No-one can deny that the nature's beauty is indeed a gift from the Almighty.

Right after dawn of May 7, we went to a natural spring to bathe in order to prepare early for that day's programme. At first, I expected the spring to be small... small that can be filled only by two to five persons I assumed. But when we got there, I was mistaken. The spring was wide, filled with cold water. It was so wide that you can't swim it at once from one side to the opposite. Since I grew up at an island, I am used to swim through waves of seawater. However, since this was a spring, there was no such waves. It looked magically so calm and undisturbed, that you could almost see totally the inverted image of the opposite edge's view. The surface was smooth especially in its deep middle region.

The Culmination program whose theme was Learn Deeply was to be held that morning. This Summer Advance Class is the very first operation of the future world-class university, Islands Global Institute, Inc. (IGI) which will formally open precisely in September 2014. Thoughts of this grand dream was first drafted 10 years ago (2003) by Alim Garo. With the help coming from his very supportive family and friends, he has made IGI formally and legally as a school at the age of just 29 -- By the way, yes, that's his age right at this very moment. His family and friends are very proud of this guy --The style of schedule and learning of this elegant institution will be based on the style of University of Cambridge (United Kingdom). The start of its academic school year will always start in September each year like other universities abroad. And the faculty will be composed of world-class professors, some will be from foreign countries. So will some of the students be.

 Guest, Administrators and Teachers of IGI.
The day's event was graced by the supportive parents of the students and some guests from a neighboring school Our Lady Peace High School of Malabang and from Silsila forum.

 Supportive Parents of IGI's first advance summer students.
After the culmination program, we went to the lot where the IGI will stand. The area was flat and there is even a defunct airport nearby. I was so astonished upon looking at the vast 16 hectares land area intended for the school’s campus. The architectural designs of the buildings are marvelous. The school's classroom, administrator's buildings, dormitories and even commercial center will be incorporated inside the compound. I jokingly said to him, “can you pinch me so that I’ll be pretty sure that I am not daydreaming right now.”
 Alim Garo, and Me. The fence of IGI campus can be seen on the left.

Who knows that in few years to come this place shall bloom because of IGI. Take a look at a pattern from history. If we roll back time by 50 years from now, we’d see boring and haunted cold forest at the location where MSU-Main Campus is. People flocked to Marawi when MSU-Main got established. Students coming from different places went there to study. Other people also took advantage of this and established good businesses in satisfying the needs of people there, especially the faculty and students.

 The first advance summer class students of IGI.
No doubt, Malabang is already an attractive place - the nature's beauty there is one of the country's wealth. In three days of visit, I've seen more than enough. If only peace and order shall be established well, it'll be a better place to live in. Then with the complete installation of IGI, it will ignite the place's massive development. We hope and pray that the same good faith will  happen to Malabang. If Marawi had flourished because of MSU, so will Malabang because of IGI.

 The defunct airport. When the tricycle speeds up while passing by this, it almost felt like starting to fly in an airplane.
 Part of the 16 hectares lot for IGI campus.

 Concrete fence of IGI campus, under construction.

 IGI's president Alim Garo, and vice-president Samanoden

## Thursday, March 7, 2013

### Determining the Distance between Lightning and an Observer

We can actually measure the distance between the location of lightning and an observer according to his observation. Let's illustrate that by treating the question in a book by Paul Hewitt.

What is the approximate distance of a thunderstorm when you note a 3-s delay between the flash of lightning and the sound of thunder?

To make things easier, let's assume that the speed of light is 300,000,000 m/s and that the speed of sound is about 340 m/s (speed of sound in air at 20 deg Celsius). Indeed, light travels faster than sound. Consequently, when light and sound came from the same source, light arrives to a certain destination first.

Variables and Relationships
Let
$latex \displaystyle t_{L}$ be the time for light to travel the distance between the origin of lighning and you.
and
$latex \displaystyle t_{S}$ be the time for sound to travel the same distance.
From the question, 3-s delay between the flash of lightning and the sound of thunder, we can translate this into a mathematical equation.
$latex \displaystyle t_{S}=t_{L}+ 3 \textup{ s} \longrightarrow (\textup{Eq.} 1)$
Let's call this as Eq. 1.
Both travel at constant speeds. So, the relationship among the speed, distance travelled and time can be expressed mathematically as
$latex v_{S}=\frac{d}{t_{S}} \longrightarrow (\textup{Eq.} 2)$
$latex v_{L}=\frac{d}{t_{L}} \longrightarrow (\textup{Eq.} 3)$
Solution
We substitute the expressions for the times described by Eq. 2 and Eq. 3 into Eq. 1 and we'll have.
$latex \frac{d}{v_{S}} = \frac{d}{v_{L}} + 3 \textup{ s}$
Now, we have one equation with only one unknown, that is, d.
$latex \frac{d}{v_{S}} - \frac{d}{v_{L}} = 3 \textup{ s}$
$latex d \Bigg (\frac{1}{v_{S}} - \frac{1}{v_{L}} \Bigg) = 3 \textup{ s}$
$latex d = \frac {3 \textup{ s}}{\cfrac{1}{v_{S}} - \cfrac{1}{v_{L}} }$
$latex d = \frac {3 \textup{ s}}{\cfrac{1}{340 \textup{ m/s}} - \cfrac{1}{300,000,000 \textup{ m/s}} }$
$latex d = 1,020 \textup{ m }$
And that's the distance we're looking for. Well, this is just an estimate. Here we've rounded off the value of speed of light. In reality it's not really exactly 300,000,000 m/s. Also, the sound may not really be 340 m/s but may depend on the temperature of the air in the surrounding. But anyway, the important thing here is the process in solving the question and not the values. :-)


text:
# Oar Shaft Stiffness

Oar Bending has been discussed in various venues before, but the descriptions often remain diffuse and imprecise. Without claiming this would be an exhaustive description of the mechanics, I want to reiterate the basic mechanics and discuss what effect oar shaft stiffness may have on rowing performance.

Fig. 1: Measuring shaft stiffness, according to Conecpt2.
Source: http://www.concept2.com/oars/oar-options/shafts/stiffness

The Oar represents a first class lever with the gate being the fulcrum (see [1] for discussion).  “Bend” describes the deflection of the oar under load applied to the blade with a “softer” shaft having more deflection at similar load than a “stiff” shaft. Various ways exist to measure shaft stiffness, for example this is the method used by Concept2 to describe their oar shafts (see Fig. 1).

Let’s for a second assume we have a theoretical shaft of infinite stiffness, i.e. a shaft that has no deflection under any load. In this case the ratio of blade and handle force (called “gearing ratio”) would be exactly that of inboard to outboard. Since outboard is in practice is always longer than inboard, the force at the blade is smaller than the force at the handle. A really accurate description though would need to consider that the point of force application on the handle is not the tip of the handle but rather somewhere near the center of the hands (see Fig. 2).

Fig. 2: The point of handle force application is not at the tip of the handle.

Furthermore, it moves throughout the stroke, especially in sweep rowing where the force applied by the inner and outer hand are not balanced. Similar thoughts need to be applied to the blade. See this paper by Dr. Kleshnev for more information. Leaving the issue of the actual gearing ratio aside, the transfer of mechanical energy is unaffected by a lever of perfect stiffness. All the work performed on the handle is transferred to the blade, even though the force on the handle is larger than on the blade. This is because the mechanic work on the handle $$W=F*s$$  is performed with a higer force on a shorter distance while the work on the blade has lower force but longer distance.

Shafts in the real world of course do not have infinite stiffness and deflect under load. What effect does this have? We need to look at this from two different perspectives: The energy balance and the force balance.

Let’s look at energy balance first. A bent oar stores mechanical energy, very much like a spring. It is however not a perfect storage for mechanical energy but exhibits elastic hysteresis due to internal friction in the shaft (Fig. 3). Some of the energy put into the shaft will thus dissipate as heat and cannot be recovered for propulsion.

Fig. 3: Elastic Hysteresis.
Source: https://en.wikipedia.org/wiki/Hysteresis

This phenomenon is one of the reasons for the often claimed “inefficiency” of softer shafts.

Second, we need to look at the change in forces introduced by shaft flexibility. When the blades enter the water, the shaft is fully relaxed. The subsequent force application on the handle by the rower now however has a different temporal loading profile. When the shaft flexes, the handle force profile will ramp up slower because the oar handle needs to travel a further distance until the shaft can produce a counter force for handle force at the same magnitude as if the shaft were stiffer. In Figure 5 below we can see the result of simulating how shaft stiffness affects handle force and blade force (called “blade propulsive force” in the figure), assuming an identical rotation profile of the gate between both simulation runs.

Fig 4: Source: “Modelling the effect of oar shaft bending during the rowing stroke”, A. Sliasas and S. Tullis, Proceedings of the Institution of Mechanical Engineers, Part P: Journal of Sports Engineering and Technology December 2011 vol. 225 no. 4 265-270.

We can observe multiple interesting results in this figure. First, we can see the effect of gearing at work: blade forces are much smaller than handle forces. Second, a more rigid oar requires a much more front-loaded handle force profile if the rower was to maintain a similar gate rotation profile. Third, handle force becomes negative for the simulated rigid oar towards the end of the drive. This is of course an unrealistic result and a consequence of the simulation constraint (identical gate rotation profile). Nonetheless, it is a consequence of the blade velocity falling below shell velocity towards the end of the drive.

This hints at a realistic problem though: as the shell accelerates throughout the drive and handle force decreases naturally after the leg drive is finished because the weaker trunk and arms muscles are at work, the crew must maintain a positive difference between blade and shell velocity in order not to stop the boat. A shaft that is still flexed at the end of the drive phase helps maintaining this positive difference as it ensures positive blade force is applied. Problematic however is a shaft that is still flexed during extraction of the blades. Not only will it make extraction of the blades more difficult, but also the potential elastic energy stored in the shaft can not be efficiently recovered for propulsion of the rower+boat system.

As the rower applies force at the relaxed handle at the catch, the shaft flexes and will store mechanical energy (called elastic potential energy). This portion of mechanical effort of the rower is thus not available for propulsion at this point in time. The total amount of mechanic energy stored in the shaft has been estimated to be between 2.5% (stiff) and 3.3% (soft) of total work per stroke.

What effect does storing mechanical energy in the shaft have for the propulsion of the rower boat system? There are a lot of factors at play here so that it is difficult to make a reasonable judgement about the advantages or disadvantages of using a softer or stiffer shaft. Let’s recap:

• Energy stored in the shaft during the beginning of the drive phase does not accelerate the boat, which would be advantageous to maximize fluid drag efficiency. Theoretically, using a stiffer shaft should result in a more front-loaded boat acceleration profile which would be easy to measure with an acceleration measurement system like Rowing in Motion
• Energy release of the shaft is subject to hysteresis (probably of marginal effect), energy may thus be lost
• A flexible shaft helps maintain positive blade force towards the end of the drive phase
• The change in handler force profile due to a more flexible shaft may be more suited to the physiological capabilities of rowers (working angles of muscles, isometric vs. isokinetic force, muscle contraction speeds)
• Flexion of the shaft has a considerable influence on the blade force profile, which in combination with the variation of blade propulsive efficiency at different angles may lead to a higher or lower total blade efficiency for the stroke
• Softer shafts may be lighter (decreasing oar inertia and the work required to accelerate the oar) and of smaller diameter (decreasing wind resistance)

The interplay of all these factors is complex.

A recent thesis by Brock Laschowski from the university of Western Ontario, Canada (supervised by Dr. Volker Nolte) has set out to empirically test the effect of varying oar shaft stiffness. The full text of the thesis is unfortunately still under embargo, but the available abstract indicates that no statistically significant differences could be found.

I hope this article gives a reasonable overview of the different arguments to consider when choosing oar shaft stiffness. Curious whether changing oars and inboard/outboard setting has any effect on your stroke? Give Rowing in Motion a try to measure it and see for yourself.

[1]: here has been some discussion about this statement in the comments. My current thinking is that from the perspective of the rower, the oar represents a first class lever with the gate as a fulcrum. From the perspective of the water through which the boat is propelled, the blade’s tip is the fulcrum (although this fulcrum moves quite a bit!) and the oar is a second class lever. Happy to discuss this in the comments!)

• Not so sure that an oar is a first class lever. We aren’t moving the water past the boat; we’re moving the boat past the water. The water is effectively stationary and the boat is moving. I’d suggest therefore that an oar is a second class lever; fulcrum at the blade in the water, load at the gate and the effort at the handle. Of course there’s some slip of the blade through the water at least before it’s loaded and maybe during the drive. Perhaps it exhibits attributes of both first and second class levers with he latter being the dominant one.

• Hi Demot, interesting point and thanks for pointing it out. The FISA rule book (http://www.worldrowing.com/uploads/files/FISA_rule_book_EN_2011_HR_sans_traitsCoupes(12.05)_.pdf) e.g. states that oars should be used as levers of the second order for propulsion of a rowing boat. I like your description that it exhibits characteristics of both.

If we look at it from the frame of reference “water” it is indeed a second class lever. If we look at it from the frame of reference “boat” however, it looks to me more like a first class lever. In any case, the classification is irrelevant with regards to the formula at work describing the lever.

• Recently using “skinnys” for sculling so find this analysis very interesting. Since differences in stiffness may be have marginal results, am wondering if affects on the body might be more important. Is a more flexible scull easier on the lower back? Does a sculler have to be stronger in order to keep the “stored energy” from dissipating near the finish after the legs are down? Is a lighter scull-like a “skinny” easier to use because of it’s lighter weight?

Other questions related more to Physics: Does a flexed shaft change or degrade the Bernoulli affect on the blade by allowing more “slippage”?

• Hi Tim, these are very interesting questions indeed. One thing to be said regarding the whole “easy on the back”, “harder to pull” descriptions is that the handle force profile may indeed change with oar shaft stiffness. But (and this is important)
– handle force is created by muscles pulling on the handle, not the other way round. The shaft doesn’t “overload” any muscles, a stiffer shaft may only allow you to ramp up handle force faster.
– all a rower has to do at the end of the drive to ensure the shaft dissatipates its energy into the water is to ensure the shaft doesn’t pull his hands away from the body.

Regarding the effect of shaft stiffness on blade slippage: A bend shaft allows gate angle and blade angle to differ, and quite significantly so. This is indeed interesting but would require detailed simulation.

• Please see the article entitled “The effects of oar-shaft stiffness and length on rowing biomechanics” published in the Journal of Sports Engineering and Technology for full details [http://pip.sagepub.com/content/early/2015/05/05/1754337115582121?papetoc]

• Hi Brock, thanks for including the link. I’ve read the article but many interested rowers (i.e. those without university or library access) won’t be able to access it behind Sage’s paywall… That’s why I had included the link to your thesis instead.

• If interested, the University of Western Ontario [Canada] recently published a short story about this research. Please click the following link [http://news.westernu.ca/2015/06/research-floats-revolutionary-idea-on-rowing/]

• Another manuscript entitled “Modeling the
deflection of rowing oar-shafts” is currently under peer-review and should
be published soon. This article will provide an unprecedented looked into the
mechanical properties of rowing oar-shafts of different stiffness’ and lengths.
While previous anecdotal work by Kleshnev and others have been satisfactory,
this upcoming publication will provide the scientific backing to finally put these
questions to rest once and for all.

• Great to have you comment here. This is absolutely fantastic! Is there any chance you can arrange for an open access policy for the article?

• A note on the energy balance. I’d question the assumption of internal friction in the shaft that would dissipate some energy via heat. Elastic hysteresis is important for rubber and maybe for stresses near the yield point, but this is not the case for a bent shaft.

You might instead want to assume perfect linear elasticity, i.e. the oar simply obeys Hooke’s law. Therefore, it can store energy, but it cannot dissipate energy and the statement about the “inefficiency” of softer shafts is misleading.

• I fully agree with you, the hysteresis of a shaft is probably marginal (and I say so in the summary as well). Sorry if I made it sound like I would agree with that in the first part, but I felt it was important to mention it as it’s a commonplace myth.


text:
Does this vector set span $\Bbb{R}^3$?

Determine whether $S = \Big\{(1, 0, −1),(2, 1, 0),(0, 1, 1)\Big\}$ spans $\Bbb{R}^3$.

So I'm going to let $(u_1, u_2, u_3)$ be a random vector.

$$(u_1, u_2, u_3) = c_1(1,0,-1) + c_2(2,1,0) + c_3(0,1,1) \\ \hspace{0.5cm}= (c_1 + 2c_2, c_2 + c_3, -c_1 + c_3)$$

So it leads to the equations:

$$c_1 + 2c_2 = u_1$$ $$c_2 + c_3 = u_2$$ $$-c_1 + c_3 = u_3$$

$\Rightarrow$

$$\begin{bmatrix} 1 & 2 & 0 \\ 0 & 1 & 1 \\ -1 & 0 & 1 \end{bmatrix} \sim \begin{bmatrix} 1 & 2 & 0 \\ 0 & 1 & 1 \\ 0 & 2 & 1 \end{bmatrix}$$

I know one way to check this: the determinant of this matrix is found by taking the determinant of the first column:

$$1 * 1 * -1 = -1$$

So the matrix has a unique solution so any vector in $\Bbb{R}^3$ can be written with those 3 vectors so these vectors span $\Bbb{R}^3$

Is this right?

• Yes, this is correct. You used quite a few more words than I would have in constructing the matrix., the columns are merely the vectors in question. – JMoravitz Sep 11 '18 at 22:17
• The "determinant of the first column" does not exist. – amsmath Sep 11 '18 at 22:18
• Can I not find the determinant in this way @amsmath? – Jwan622 Sep 11 '18 at 22:20
• @JMoravitz what words are redundant? – Jwan622 Sep 11 '18 at 22:20
• Well, at least I did not understand what you did there. – amsmath Sep 11 '18 at 22:20

$$\begin{bmatrix} 1 & 2 & 0 \\ 0 & 1 & 1 \\ -1 & 0 & 1 \end{bmatrix}\to \begin{bmatrix} 1 & 2 & 0 \\ 0 & 1 & 1 \\ 0 & 2 & 1 \end{bmatrix}\to \begin{bmatrix} 1 & 2 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & -1 \end{bmatrix}$$


text:
# Laplace Transform of an integral function of a convolution

Making suitable assumptions wherever necessary, what is the Laplace Transform $$\mathcal{L}(S(t))$$ where $$S(t)=\int_{0}^{t}\int_{0}^{t}f(t-s_1,t-s_2)g(s_1)h(s_2)ds_1ds_2$$.

I tried using the Double Laplace theorem but I am not sure if that is applicable here since $$S(t)$$ is a function of one argument.

Choosing a specific form of $$f(t_1,t_2)=f(\frac{t_1+t_2}{2})$$ , $$f(t)=0 \text{ }\forall t<0$$ and the Heaviside function $$H(t)$$
$$\hat{S}(p) =\mathcal{L}(S(t))=\int_{0}^{\infty}S(t)e^{-pt}dt\\ \Rightarrow \hat{S}(p) =\int_0^{\infty}e^{-pt}dt\int_0^{t}\int_0^{t}f(t-s_1,t-s_2)g(s_1)h(s_2)ds_1ds_2\\ \Rightarrow \hat{S}(p) = \int_0^{\infty}e^{-pt}dt\int_0^{\infty}\int_0^{\infty}f(t-\frac{s_1+s_2}{2})H(t-\frac{s_1+s_2}{2})g(s_1)h(s_2)ds_1ds_2\\ \Rightarrow \hat{S}(p) =\int_0^{\infty}g(s_1)ds_1\int_0^{\infty}h(s_2)ds_2\int_{\frac{s_1+s_2}{2}}^{\infty}f(t-\frac{s_1+s_2}{2})e^{-pt}dt\\ \text{with a change of variable } t-\frac{s_1+s_2}{2}=\tau\\ \Rightarrow \hat{S}(p) =\int_0^{\infty}g(s_1)e^{-s_1/2}ds_1\int_0^{\infty}h(s_2)e^{-s_2/2}ds_2\int_{0}^{\infty}f(\tau)e^{-p\tau}d\tau\\ \Rightarrow \hat{S}(p) = \hat{f}(p)\hat{g}(p/2)\hat h(p/2) \qed\\ \text{Please let me know if there is a better way to solve this without taking the specific form of }f(t_1,t_2)$$.


text:
# Inference with Fuzzy IF-THEN Rules

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This Demonstration illustrates the interpolation method for a system of fuzzy IF-THEN rules. In particular, it shows how to calculate the suitability of a house given the following three rules: - IF (House is Inexpensive OR Close-to-work), THEN Suitability is Good. - IF (House is Expensive OR Far-from-work), THEN Suitability is Low. - IF (House is Average-priced AND About-50-km-from-work), THEN Suitability is Regular.

[more]

These rules contain imprecise concepts -such as "Expensive" or "Close-to-work"- which are formalized as fuzzy sets. A fuzzy set F (e.g. denoting the concept "Expensive") is defined by a membership function that assigns to each possible element x within a certain universe of discourse X (e.g. all the houses in the world) a value between 0 and 1 that denotes the extent to which that element x (e.g. a particular house) belongs to the fuzzy set (i.e. the extent to which the particular house is expensive). In the example above, the extent to which a house of price=110 is expensive is (see the membership function of "Expensive" on the mid-left of the demonstration, plotted as a function of the price of the house).

[less]

Contributed by: Luis R. Izquierdo and Segismundo S. Izquierdo (February 2015)
Open content licensed under CC BY-NC-SA

## Details

This Demonstration illustrates the so-called interpolation method for multiconditional approximate reasoning systems of fuzzy IF-THEN rules [1], including fuzzification and defuzzification. A particular instance of this method is Mamdani inference (also called max-min inference), which is often used in fuzzy control. Another particular instance is max-prod inference. The method consists of the following four steps:

1. Calculate the degree of consistency between the inputs and the antecedent of each IF-THEN rule. The Demonstration lets you choose different functions for the logical operators (AND, OR). As an example, consider the first rule at the top row, with antecedent "Inexpensive OR Close-to-work". The computation for crisp inputs price=110 and distance=57, using the function Max (maximum) as logical OR, would be: OR(Inexpensive(110), Close-to-work(57)) ⟶ OR(0.45,0.16) ⟶ Max(0.45,0.16) ⟶ 0.45. The result of this step is a number for each rule (i.e. the degree of consistency between the inputs and each rule's antecedent).

2. Reshape the original consequent of each rule (which is a fuzzy set), given the degree of consistency between the inputs and the rule's antecedent. The result of this step is a (reshaped) fuzzy set for each rule. Possible operators for the reshaping method are Truncate (by default) and Multiplication. Consider, for example, the rule at the top. Its original consequent is the fuzzy set "Good suitability" with membership function . The computation would be: .

3. Aggregate all the reshaped consequents. The result of this step is one fuzzy set (labeled "Aggreg. Suitability"), which is the union of all the reshaped consequents. Possible operators for the aggregation are the Max, the probabilistic sum (ProbOr[a,b]:=a+b-ab) and the sum clipped at 1 (ClippedSum).

4. Defuzzify the aggregated fuzzy set. The defuzzification reduces the aggregated fuzzy set to one single number. Possible defuzzification methods are: center of gravity (COG), first of maxima (FOM), last of maxima (LOM), middle of maxima (MOM) and mean of maxima (MeOM).

If you want to use Mamdani (or max-min) inference, choose "Min" for logical AND, "Max" for logical OR, "Truncate" as "Reshaping method"," Max" as "Aggregation method", and "COG" as "Defuzzification method".

Reference

[1] G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Upper Saddle River, New Jersey: Prentice Hall PTR, 1995.

## Permanent Citation

Luis R. Izquierdo and Segismundo S. Izquierdo

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text:
# Who gave you infinitesimal epsilon?

As someone reputed among certain historians to have given you the epsilon Cauchy startled me by using $\varepsilon$ to denote an infinitely small number in his 1826 text on differential geometry; see page 98 here:

[S]i l'on désigne par $\varepsilon$ un nombre infiniment petit, on aura $$\frac{\sin\left(\frac{\pi}{2}\pm\varepsilon\right)}{r}= \frac{\sin(\pm\Delta\tau)}{\sqrt{\Delta x^2+\Delta y^2}}$$

Are there other instances of such nonweierstrassian usage by Cauchy?

• Maybe Resumé (1823), 4ème Lecon, page 13: "Soient ... $i$ une quantité infiniment petite, et $h$ une quantité finie. Si l'on pose $i=\alpha x$, $\alpha$ sera encore une quantité infiniment petite, et l'on aura...$\dfrac {f(x+ \alpha h)-f(x)} { \alpha}$" where $i$ and $\alpha$ are used for "infinitesimal small" quantities. – Mauro ALLEGRANZA Jul 12 '17 at 15:08
• Also on page 27 and 83 of this volume $\delta$ and $\epsilon$ appear as "nombres tres-petits" and as quantities differing "tres peu de zero". – Jan Peter Schäfermeyer Jul 17 '17 at 12:28


text:
# Astronomy:Planetary migration

Planetary migration occurs when a planet or other body in orbit around a star interacts with a disk of gas or planetesimals, resulting in the alteration of its orbital parameters, especially its semi-major axis. Planetary migration is the most likely explanation for hot Jupiters (exoplanets with Jovian masses but orbits of only a few days). The generally accepted theory of planet formation from a protoplanetary disk predicts that such planets cannot form so close to their stars, as there is insufficient mass at such small radii and the temperature is too high to allow the formation of rocky or icy planetesimals.

It has also become clear that terrestrial-mass planets may be subject to rapid inward migration if they form while the gas disk is still present. This may affect the formation of the cores of the giant planets (which have masses of the order of 10 to 1000 Earth masses), if those planets form via the core-accretion mechanism.

## Types of disk

### Gas disk

Observations suggest that gas in Protoplanetary disks orbiting young stars have lifetimes of a few to several million years.[1] If planets with masses of around an Earth mass or greater form while the gas is still present, the planets can exchange angular momentum with the surrounding gas in the protoplanetary disk so that their orbits change gradually. Although the sense of migration is typically inwards in locally isothermal disks, outward migration may occur in disks that possess entropy gradients.[2]

### Planetesimal disk

During the late phase of planetary system formation, massive protoplanets and planetesimals gravitationally interact in a chaotic manner causing many planetesimals to be thrown into new orbits. This results in angular-momentum exchange between the planets and the planetesimals, and leads to migration (either inward or outward). Outward migration of Neptune is believed to be responsible for the resonant capture of Pluto and other Plutinos into the 3:2 resonance with Neptune.

## Types of migration

There are many different mechanisms by which planets' orbits can migrate, which are described below as disk migration (Type I migration, Type II migration, or Type III migration), tidal migration, planetesimal-driven migration, gravitational scattering, and Kozai cycles and tidal friction. This list of types is not exhaustive or definitive: Depending on what is most convenient for any one type of study, different researchers will distinguish mechanisms in somewhat different ways.

Classification of any one mechanism is mainly based on the circumstances in the disk that enable the mechanism to efficiently transfer energy and / or angular momentum to and from planetary orbits. As the loss or relocation of material in the disk changes the circumstances, one migration mechanism will give way to another mechanism, or perhaps none. If there is no follow-on mechanism, migration (largely) stops and the stellar system becomes (mostly) stable.

### Disk migration

Disk migration arises from the gravitational force exerted by a sufficiently massive body embedded in a disk on the surrounding disk's gas, which perturbs its density distribution. By the reaction principle of classical mechanics, the gas exerts an equal and opposite gravitational force on the body, which can also be expressed as a torque. This torque alters the angular momentum of the planet's orbit, resulting in a variation of the semi-major axis and other orbital elements. An increase over time of the semi-major axis leads to outward migration, i.e., away from the star, whereas the opposite behavior leads to inward migration.

Three sub-types of disk migration are distinguished as Types I, II, and III. The numbering is not intended to suggest a sequence or stages.

#### Type I migration

Small planets undergo Type I disk migration driven by torques arising from Lindblad and co-rotation resonances. Lindblad resonances excite spiral density waves in the surrounding gas, both interior and exterior of the planet's orbit. In most cases, the outer spiral wave exerts a greater torque than does the inner wave, causing the planet to lose angular momentum, and hence migrate toward the star. The migration rate due to these torques is proportional to the mass of the planet and to the local gas density, and results in a migration timescale that tends to be short relative to the million-year lifetime of the gaseous disk.[3] Additional co-rotation torques are also exerted by gas orbiting with a period similar to that of the planet. In a reference frame attached to the planet, this gas follows horseshoe orbits, reversing direction when it approaches the planet from ahead or from behind. The gas reversing course ahead of the planet originates from a larger semi-major axis and may be cooler and denser than the gas reversing course behind the planet. This may result in a region of excess density ahead of the planet and of lesser density behind the planet, causing the planet to gain angular momentum.[4][5]

The planet mass for which migration can be approximated to Type I depends on the local gas pressure scale height and, to a lesser extent, the kinematic viscosity of the gas.[3][6] In warm and viscous disks, Type I migration may apply to larger mass planets. In locally isothermal disks and far from steep density and temperature gradients, co-rotation torques are generally overpowered by the Lindblad torques.Cite error: Closing </ref> missing for <ref> tag In more realistic situations, unless extreme thermal and viscosity conditions occur in a disk, there is an ongoing flux of gas through the gap.[7] As a consequence of this mass flux, torques acting on a planet can be susceptible to local disk properties, akin to torques at work during Type I migration. Therefore, in viscous disks, Type II migration can be typically described as a modified form of Type I migration, in a unified formalism.[8][6] The transition between Type I and Type II migration is generally smooth, but deviations from a smooth transition have also been found.[9][10] In some situations, when planets induce eccentric perturbation in the surrounding disk's gas, Type II migration may slow down, stall, or reverse.[11]

From a physical viewpoint, Type I and Type II migration are driven by the same type of torques (at Lindblad and co-rotation resonances). In fact, they can be interpreted and modeled as a single regime of migration, that of Type I appropriately modified by the perturbed gas surface density of the disk.[8][6]

#### Type III disk migration

Type III disk migration applies to fairly extreme disk / planet cases and is characterized by extremely short migration timescales.[12][13][8] Although sometimes referred to as "runaway migration", the migration rate does not necessarily increase over time.[12][13] Type III migration is driven by the co-orbital torques from gas trapped in the planet's libration regions and from an initial, relatively fast, planetary radial motion. The planet's radial motion displaces gas in its co-orbital region, creating a density asymmetry between the gas on the leading and the trailing side of the planet.[8][3] Type III migration applies to disks that are relatively massive and to planets that can only open partial gaps in the gas disk.[3][8][12] Previous interpretations linked Type III migration to gas streaming across the orbit of the planet in the opposite direction as the planet's radial motion, creating a positive feedback loop.[12] Fast outward migration may also occur temporarily, delivering giant planets to distant orbits, if later Type II migration is ineffective at driving the planets back.[14]

### Gravitational scattering

Another possible mechanism that may move planets over large orbital radii is gravitational scattering by larger planets or, in a protoplantetary disk, gravitational scattering by over-densities in the fluid of the disk.[15] In the case of the Solar System, Uranus and Neptune may have been gravitationally scattered onto larger orbits by close encounters with Jupiter and/or Saturn.[16][17] Systems of exoplanets can undergo similar dynamical instabilities following the dissipation of the gas disk that alter their orbits and in some cases result in planets being ejected or colliding with the star.

Planets scattered gravitationally can end on highly eccentric orbits with perihelia close to the star, enabling their orbits to be altered by the tides they raise on the star. The eccentricities and inclinations of these planets are also excited during these encounters, providing one possible explanation for the observed eccentricity distribution of the closely orbiting exoplanets.[18] The resulting systems are often near the limits of stability.[19] As in the Nice model, systems of exoplanets with an outer disk of planetesimals can also undergo dynamical instabilities following resonance crossings during planetesimal-driven migration. The eccentricities and inclinations of the planets on distant orbits can be damped by dynamical friction with the planetesimals with the final values depending on the relative masses of the disk and the planets that had gravitational encounters.[20]

### Tidal migration

Tides between the star and planet modify the semi-major axis and orbital eccentricity of the planet. If the planet is orbiting very near to its star, the tide of the planet raises a bulge on the star. If the star's rotational period is longer than the planet's orbital period the location of the bulge lags behind a line between the planet and the center of the star creating a torque between the planet and the star. As a result, the planet loses angular momentum and its semi-major axis decreases with time.

If the planet is in an eccentric orbit the strength of the tide is stronger when it is near perihelion. The planet is slowed the most when near perihelion, causing its aphelion to decrease faster than its perihelion, reducing its eccentricity. Unlike disk migration – which lasts a few million years until the gas dissipates – tidal migration continues for billions of years. Tidal evolution of close-in planets produces semi-major axes typically half as large as they were at the time that the gas nebula cleared.[21]

### Kozai cycles and tidal friction

A planetary orbit that is inclined relative to the plane of a binary star can shrink due to a combination of Kozai cycles and tidal friction. Interactions with the more distant star cause the planet's orbit to undergo an exchange of eccentricity and inclination due to the Kozai mechanism. This process can increase the planet's eccentricity and lower its perihelion enough to create strong tides between the planet on the star increases. When it is near the star the planet loses angular momentum causing its orbit to shrink.

The planet's eccentricity and inclination cycle repeatedly, slowing the evolution of the planets semi-major axis.[22] If the planet's orbit shrinks enough to remove it from the influence of the distant star the Kozai cycles end. Its orbit will then shrink more rapidly as it is tidally circularized. The orbit of the planet can also become retrograde due to this process. Kozai cycles can also occur in a system with two planets that have differing inclinations due to gravitational scattering between planets and can result in planets with retrograde orbits.[23][24]

### Planetesimal-driven migration

The orbit of a planet can change due to gravitational encounters with a large number of planetesimals. Planetesimal-driven migration is the result of the accumulation of the transfers of angular momentum during encounters between the planetesimals and a planet. For individual encounters the amount of angular momentum exchanged and the direction of the change in the planet's orbit depends on the geometry of the encounter. For a large number of encounters the direction of the planet's migration depends on the average angular momentum of the planetesimals relative to the planet. If it is higher, for example a disk outside the planet's orbit, the planet migrates outward, if it is lower the planet migrates inward. The migration of a planet beginning with a similar angular momentum as the disk depends on potential sinks and sources of the planetesimals.[25]

For a single planet system, planetesimals can only be lost (a sink) due to their ejection, which would cause the planet to migrate inward. In multiple planet systems the other planets can act as sinks or sources. Planetesimals can be removed from the planet's influence after encountering an adjacent planet or transferred to that planet's influence. These interactions cause the planet's orbits to diverge as the outer planet tends to remove planetesimals with larger momentum from the inner planet influence or add planetesimals with lower angular momentum, and vice versa. The planet's resonances, where the eccentricities of planetesimals are pumped up until they intersect with the planet, also act as a source. Finally, the planet's migration acts as both a sink and a source of new planetesimals creating a positive feedback that tends to continue its migration in the original direction.[25]

Planetesimal-driven migration can be damped if planetesimals are lost to various sinks faster than new ones are encountered due to its sources. It may be sustained if the new planetesimals enter its influence faster than they are lost. If sustained migration is due to its migration only, it is called runaway migration. If it is due to the loss of planetesimals to another planets influence, it is called forced migration[25] For a single planet orbiting in a planetesial disk the shorter timescales of the encounters with planetesimals with shorter period orbits results in more frequent encounters with the planetesimals with less angular momentum and the inward migration of the planet.[26] Planetesimal-driven migration in a gas disk, however, can be outward for a particular range of planetesimal sizes because of the removal of shorter period planetesimals due to gas drag.[27]

## Resonance capture

The migration of planets can lead to planets being captured in resonances and chains of resonances if their orbits converge. The orbits of the planets can converge if the migration of the inner planet is halted at the inner edge of the gas disk, resulting in a systems of tightly orbiting inner planets;[28] or if migration is halted in a convergence zone where the torques driving Type I migration cancel, for example near the ice line, in a chain of more distant planets.[29]

Gravitational encounters can also lead to the capture of planets with sizable eccentricities in resonances.[30] In the Grand tack hypothesis the migration of Jupiter is halted and reversed when it captured Saturn in an outer resonance.[31] The halting of Jupiter's and Saturn's migration and the capture of Uranus and Neptune in further resonances may have prevented the formation of a compact system of super-earths similar to many of those found by Kepler.[32] The outward migration of planets can also result in the capture of planetesimals in resonance with the outer planet; for example the resonant trans-Neptunian objects in the Kuiper belt.[33]

Although planetary migration is expected to lead to systems with chains of resonant planets most exoplanets are not in resonances. The resonance chains can be disrupted by gravitational instabilities once the gas disk dissipates.[34] Interactions with leftover planetesimals can break resonances of low mass planets leaving them in orbits slightly outside the resonance.[35] Tidal interactions with the star, turbulence in the disk, and interactions with the wake of another planet could also disrupt resonances.[36] Resonance capture might be avoided for planets smaller than Neptune with eccentric orbits.[37]

## In the Solar System

Main page: Astronomy:Nice model
Simulation showing outer planets and Kuiper belt: (a) Before Jupiter/Saturn 2:1 resonance. (b) Scattering of Kuiper belt objects into the Solar System after the orbital shift of Neptune. (c) After ejection of Kuiper belt bodies by Jupiter[17]

The migration of the outer planets is a scenario proposed to explain some of the orbital properties of the bodies in the Solar System's outermost regions.[38] Beyond Neptune, the Solar System continues into the Kuiper belt, the scattered disc, and the Oort cloud, three sparse populations of small icy bodies thought to be the points of origin for most observed comets. At their distance from the Sun, accretion was too slow to allow planets to form before the solar nebula dispersed, because the initial disc lacked enough mass density to consolidate into a planet. The Kuiper belt lies between 30 and 55 AU from the Sun, while the farther scattered disc extends to over 100 AU,[38] and the distant Oort cloud begins at about 50,000 AU.[39]

According to this scenario the Kuiper belt was originally much denser and closer to the Sun: it contained millions of planetesimals, and had an outer edge at approximately 30 AU, the present distance of Neptune. After the formation of the Solar System, the orbits of all the giant planets continued to change slowly, influenced by their interaction with the large number of remaining planetesimals. After 500–600 million years (about 4 billion years ago) Jupiter and Saturn divergently crossed the 2:1 orbital resonance, in which Saturn orbited the Sun once for every two Jupiter orbits.[38] This resonance crossing increased the eccentricities of Jupiter and Saturn and destabilized the orbits of Uranus and Neptune. Encounters between the planets followed causing Neptune to surge past Uranus and plough into the dense planetesimal belt. The planets scattered the majority of the small icy bodies inwards, while moving outwards themselves. These planetesimals then scattered off the next planet they encountered in a similar manner, moving the planets' orbits outwards while they moved inwards.[40] This process continued until the planetesimals interacted with Jupiter, whose immense gravity sent them into highly elliptical orbits or even ejected them outright from the Solar System. This caused Jupiter to move slightly inward. This scattering scenario explains the trans-Neptunian populations' present low mass. In contrast to the outer planets, the inner planets are not believed to have migrated significantly over the age of the Solar System, because their orbits have remained stable following the period of giant impacts.[41]

## Notes

1. Ercolano, B.; Pascucci, I. (2017). "The dispersal of planet-forming discs: theory confronts observations". Royal Society Open Science 4 (2): 170114. doi:10.1098/rsos.170114. Bibcode2017RSOS....470114E.
2. D'Angelo, G.; Lissauer, J. J. (2018). "Formation of Giant Planets". in Deeg H., Belmonte J.. Handbook of Exoplanets. Springer International Publishing AG, part of Springer Nature. pp. 2319–2343. doi:10.1007/978-3-319-55333-7_140. ISBN 978-3-319-55332-0. Bibcode2018haex.bookE.140D.
3. Lubow, S.H.; Ida, S. (2011). "Planet Migration". in Seager, S.. Exoplanets. University of Arizona Press, Tucson, AZ. pp. 347–371. Bibcode2011exop.book..347L.
4. Paardekooper, S.-J.; Mellema, G. (2006). "Halting type I planet migration in non-isothermal disks". Astronomy and Astrophysics 459 (1): L17–L20. doi:10.1051/0004-6361:20066304. Bibcode2006A&A...459L..17P.
5. Brasser, R.; Bitsch, B.; Matsumura, S. (2017). "Saving super-Earths: Interplay between pebble accretion and type I migration". The Astronomical Journal 153 (5): 222. doi:10.3847/1538-3881/aa6ba3. Bibcode2017AJ....153..222B.
6. Cite error: Invalid <ref> tag; no text was provided for refs named dangelo_lubow_2010
7. Lubow, S.; D'Angelo, G. (2006). "Gas flow across gaps in protoplanetary disks". The Astrophysical Journal 641 (1): 526–533. doi:10.1086/500356. Bibcode2006ApJ...641..526L.
8. D'Angelo, G.; Lubow, S. H. (2008). "Evolution of migrating planets undergoing gas accretion". The Astrophysical Journal 685 (1): 560–583. doi:10.1086/590904. Bibcode2008ApJ...685..560D.
9. D'Angelo, G.; Kley, W.; Henning T. (2003). "Orbital migration and mass accretion of protoplanets in three-dimensional global computations with nested grids". The Astrophysical Journal 586 (1): 540–561. doi:10.1086/367555. Bibcode2003ApJ...586..540D.
10. Masset, F.S.; D'Angelo, G.; Kley, W. (2006). "On the migration of protogiant solid cores". The Astrophysical Journal 652 (1): 730–745. doi:10.1086/507515. Bibcode2006ApJ...652..730M.
11. D'Angelo, Gennaro; Lubow, Stephen H.; Bate, Matthew R. (2006). "Evolution of Giant Planets in Eccentric Disks". The Astrophysical Journal 652 (2): 1698–1714. doi:10.1086/508451. Bibcode2006ApJ...652.1698D.
12. Masset, F.S.; Papaloizou, J.C.B. (2003). "Runaway migration and the formation of hot Jupiters". The Astrophysical Journal 588 (1): 494–508. doi:10.1086/373892. Bibcode2003ApJ...588..494M.
13. D'Angelo, G.; Bate, M.R.B.; Lubow, S.H. (2005). "The dependence of protoplanet migration rates on co-orbital torques". Monthly Notices of the Royal Astronomical Society 358 (2): 316–332. doi:10.1111/j.1365-2966.2005.08866.x. Bibcode2005MNRAS.358..316D.
14. Pierens, A.; Raymond, S.N. (2016). "Migration of accreting planets in radiative discs from dynamical torques". Monthly Notices of the Royal Astronomical Society 462 (4): 4130–4140. doi:10.1093/mnras/stw1904. Bibcode2016MNRAS.462.4130P.
15. R. Cloutier; M-K. Lin (2013). "Orbital migration of giant planets induced by gravitationally unstable gaps: the effect of planet mass". Monthly Notices of the Royal Astronomical Society 434 (1): 621–632. doi:10.1093/mnras/stt1047. Bibcode2013MNRAS.434..621C.
16. E. W. Thommes; M. J. Duncan; H. F. Levison (2002). "The Formation of Uranus and Neptune among Jupiter and Saturn". Astronomical Journal 123 (5): 2862. doi:10.1086/339975. Bibcode2002AJ....123.2862T.
17. Gomes, R.; Levison, H.F.; Tsiganis, K.; Morbidelli, A. (2005). "Origin of the cataclysmic Late Heavy Bombardment period of the terrestrial planets". Nature 435 (7041): 466–469. doi:10.1038/nature03676. PMID 15917802. Bibcode2005Natur.435..466G. Retrieved 2008-06-08.
18. Ford, Eric B.; Rasio, Frederic A. (2008). "Origins of Eccentric Extrasolar Planets: Testing the Planet-Planet Scattering Model". The Astrophysical Journal 686 (1): 621–636. doi:10.1086/590926. Bibcode2008ApJ...686..621F.
19. Raymond, Sean N.; Barnes, Rory; Veras, Dimitri; Armitage, Phillip J.; Gorelick, Noel; Greenberg, Richard (2009). "Planet-Planet Scattering Leads to Tightly Packed Planetary Systems". The Astrophysical Journal Letters 696 (1): L98–L101. doi:10.1088/0004-637X/696/1/L98. Bibcode2009ApJ...696L..98R.
20. Raymond, Sean N.; Armitage, Philip J.; Gorelick, Noel (2010). "Planet-Planet Scattering in Planetesimal Disks: II. Predictions for Outer Extrasolar Planetary Systems". The Astrophysical Journal 711 (2): 772–795. doi:10.1088/0004-637X/711/2/772. Bibcode2010ApJ...711..772R.
21. Jackson, Brian; Greenberg, Richard; Barnes, Rory (4 Jan 2008). "Tidal evolution of close-in extra-solar planets". arXiv:0801.0716 [astro-ph]. Submitted [for publication] on 4 Jan 2008
22. Fabrycky, Daniel; Tremaine, Scott (2007). "Shrinking Binary and Planetary Orbits by Kozai Cycles with Tidal Friction". The Astrophysical Journal 669 (2): 1298–1315. doi:10.1086/521702. Bibcode2007ApJ...669.1298F.
23. Naoz, Smadar; Farr, Will M.; Lithwick, Yoram; Rasio, Frederic A.; Teyssandier, Jean (2011). "Hot Jupiters from secular planet-planet interactions". Nature 473 (7346): 187–189. doi:10.1038/nature10076. PMID 21562558. Bibcode2011Natur.473..187N.
24. Nagasawa, M.; Ida, S.; Bessho, T. (2008). "Formation of Hot Planets by a Combination of Planet Scattering, Tidal Circularization, and the Kozai Mechanism". The Astrophysical Journal 678 (1): 498–508. doi:10.1086/529369. Bibcode2008ApJ...678..498N.
25. Levison, H.F.; Morbidelli, A.; Gomes, R.; Backman, D. (2007). "Planet Migration in Planetesimal Disks". Protostars and Planets V. University of Arizona Press. pp. 669–684. Retrieved 6 April 2017.
26. Kirsh, David R.; Duncan, Martin; Brasser, Ramon; Levison, Harold F. (2009). "Simulations of planet migration driven by planetesimal scattering". Icarus 199 (1): 197–209. doi:10.1016/j.icarus.2008.05.028. Bibcode2009Icar..199..197K.
27. Capobianco, Christopher C.; Duncan, Martin; Levison, Harold F. (2011). "Planetesimal-driven planet migration in the presence of a gas disk". Icarus 211 (1): 819–831. doi:10.1016/j.icarus.2010.09.001. Bibcode2011Icar..211..819C.
28. Cossou, Cchristophe; Raymond, Sean N.; Hersant, Franck; Pierens, Arnaud (2014). "Hot super-Earths and giant planet cores from different migration histories". Astronomy & Astrophysics 569: A56. doi:10.1051/0004-6361/201424157. Bibcode2014A&A...569A..56C.
29. Cossou, C.; Raymond, S.N.; Pierens, A. (2013). "Convergence zones for Type I migration: An inward shift for multiple planet systems". Astronomy & Astrophysics 553: L2. doi:10.1051/0004-6361/201220853. Bibcode2013A&A...553L...2C.
30. Raymond, Sean N.; Barnes, Rory; Armitage, Philip J.; Gorelick, Noel (2008). "Mean motion resonances from planet-planet scattering". The Astrophysical Journal Letters 687 (2): L107. doi:10.1086/593301. Bibcode2008ApJ...687L.107R.
31. Walsh, Kevin J.; Morbidelli, Alessandro; Raymond, Sean N.; O'Brien, David P.; Mandell, Avi M. (2011). "A low mass for Mars from Jupiter's early gas-driven migration". Nature 475 (7355): 206–209. doi:10.1038/nature10201. PMID 21642961. Bibcode2011Natur.475..206W.
32. Izidoro, André; Raymond, Sean N.; Morbidelli, Alessandro; Hersant, Franck; Pierens, Arnaud (2015). "Gas giant planets as dynamical barriers to inward-migrating super-Earths". Astrophysical Journal Letters 800 (2): L22. doi:10.1088/2041-8205/800/2/L22. Bibcode2015ApJ...800L..22I.
33. Malhotra, Renu (1995). "The origin of Pluto's orbit: Implications for the Solar System beyond Neptune". Astronomical Journal 110: 420. doi:10.1086/117532. Bibcode1995AJ....110..420M.
34. Izidoro, Andre; Ogihara, Masahiro; Raymond, Sean N.; Morbidelli, Alessaandro; Pierens, Arnaud; Bitsch, Bertram; Cossou, Christophe; Hersant, Franck (2017). "Breaking the Chains: Hot Super-Earth systems from migration and disruption of compact resonant chains". Monthly Notices of the Royal Astronomical Society 470 (2): 1750–1770. doi:10.1093/mnras/stx1232. Bibcode2017MNRAS.470.1750I.
35. Chatterjee, Sourav; Ford, Eric B. (2015). "Planetesimal interactions can explain the mysterious period ratios of small near-resonant planets". The Astrophysical Journal 803 (1): 33. doi:10.1088/0004-637X/803/1/33. Bibcode2015ApJ...803...33C.
36. Baruteau, C.; Crida, A.; Paardekooper, S.-M.; Masset, F.; Guilet, J.; Bitsch, B.; Nelson, R.; Kley, W. et al. (2014). "Planet-Disk Interactions and Early Evolution of Planetary Systems". Protostars and Planets VI. University of Arizona Press. pp. 667–689. doi:10.2458/azu_uapress_9780816531240-ch029. ISBN 9780816531240. Bibcode2014prpl.conf..667B.
37. Pan, Margaret; Schlichting, Hilke E. (2017). "Avoiding resonance capture in multi-planet extrasolar systems". arXiv:1704.07836 [astro-ph.EP].
38. Levison, Harold F. et al. (2007). "Origin of the structure of the Kuiper Belt during a dynamical instability in the orbits of Uranus and Neptune". Icarus 196 (1): 258. doi:10.1016/j.icarus.2007.11.035. Bibcode2008Icar..196..258L.
39. Alessandro Morbidelli (2005). "Origin and dynamical evolution of comets and their reservoirs". arXiv:astro-ph/0512256.
40. Taylor, G. Jeffrey (21 August 2001). "Uranus, Neptune, and the Mountains of the Moon". Planetary Science Research Discoveries. Hawaii Institute of Geophysics & Planetology.
41. Lin, Douglas N.C. (May 2008). "The Chaotic Genesis of Planets". Scientific American 298 (5): 50–59. PMID 18444325. Bibcode2008SciAm.298e..50C. Retrieved 2008-06-08.

## References

• Goldreich, P., and Tremaine, S. 1979, Astrophysical Journal, 233, 857
• Lin, D. N. C., and Papaloizou, J. 1979, Monthly Notices of the Royal Astronomical Society, 186, 799
• Ward, W. R. 1997, Icarus, 126, 261
• Tanaka, H., Takeuchi, T., and Ward, W. R. 2002, Astrophysical Journal, 565, 1257


text:
### A quick question about collide

There are no stupid questions, just stupid answers.

#### Re: A quick question about collide

Jason Campbell
Posts: 266
Joined: Fri Feb 13, 2015 2:51 am

This is strange, I added this to vehicleWheeled.cs in scripts/server to see what was happening and it only registers if I hit AIPlayer, Bouncy Rocks, AITurrets but not TSStatic shapes. I'll keep looking into this, I hope tonight.

Code: Select all

function WheeledVehicleData::onCollision(%this, %obj, %col, %vec, %speed) { // Collision with other objects, including items echo("car collision:",%col.getName()); }

#### Re: A quick question about collide

LukasPJ
Posts: 388
Joined: Tue Feb 03, 2015 7:25 pm

It would seem someone decided not to have collisions with TSStatic's as they aren't really meant to be interacted with I guess.
The culprit is here:
https://github.com/GarageGames/Torque3D ... pp#L50-L55

Would have to add TSStatic there.

#### Re: A quick question about collide

OTHGMars
Posts: 25
Joined: Fri Jul 14, 2017 2:19 am
TSStatic is a StaticShapeObjectType, so that mask should detect it.
https://github.com/GarageGames/Torque3D ... c.cpp#L108.

Edit: And it does, but LukasPJ is correct. Apparently it was decided that Vehicles should only generate collision callbacks for ShapeBase derived objects.
https://github.com/GarageGames/Torque3D ... 1415-L1419
https://github.com/GarageGames/Torque3D ... 1610-L1624
You could make your rocks StaticShape instead of TSStatic and get the callbacks.

#### Re: A quick question about collide

Jason Campbell
Posts: 266
Joined: Fri Feb 13, 2015 2:51 am

I just tried to get StaticObjectType added to the mask. I added it and included tsStatic.h and it compiled without errors but still did not detect TSStatic on collision.

Think it was removed because TSStatics are just supposed to sit there.

@
OTHGMars
: probably that is the answer, using StaticShapes. OP could also mess about with Rigid Shapes and applyImpulse to the collision for some simple movement stuff.

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