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# 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag

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Senior Manager
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5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag  [#permalink]

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16 Oct 2006, 11:42
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55% (hard)

Question Stats:

71% (02:47) correct 29% (02:28) wrong based on 300 sessions

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5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag. If 4 marbles are drawn without replacement, what is the probability that the result will not be 2 blue and 2 purple marbles?

A. $$\frac{4}{33}$$

B. $$(\frac{5}{36})^2$$

C. $$\frac{1}{2}$$

D. $$(\frac{31}{36})^2$$

E. $$\frac{29}{33}$$
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Re: 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag  [#permalink]

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11 Aug 2010, 08:45
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21
5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag. If 4 marbles are drawn WITHOUT replacement, what is the probability that the result will not be 2 blue and 2 purple marbles?

Given: 5B+3R+4P=12 marbles. Question: what is the probability of NOT having scenario BBPP.

Calculate the probability of an opposite event and subtract this value from 1.

Opposite event is that out of 4 marbles we will have 2 blue and 2 purple marbles (scenario BBPP):
$$P(opposite \ event)=\frac{4!}{2!2!}*\frac{5}{12}*\frac{4}{11}*\frac{4}{10}*\frac{3}{9}=\frac{4}{33}$$, we are multiplying by $$\frac{4!}{2!2!}$$ as scenario BBPP can occur in several ways: BBPP, PPBB, BPBP, ... So scenario BBPP can occur in $$\frac{4!}{2!2!}$$ # of ways (# of permuations of 4 letters BBPP out of which 2 B's and 2 P's are identical);

Or with combinatorics $$P(opposte \ event)=\frac{C^2_5*C^2_4}{C^4_{12}}=\frac{4}{33}$$.

So $$P=1-\frac{4}{33}=\frac{29}{33}$$.

5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag. If 4 marbles are drawn WITH replacement, what is the probability that the result will not be 2 blue and 2 purple marbles?

Again Calculate the probability of an opposite event and subtract this value from 1.

Opposite event is that out of 4 marbles we will have 2 blue and 2 purple marbles:
$$P(opposite \ event)=\frac{4!}{2!2!}*\frac{5}{12}*\frac{5}{12}*\frac{4}{12}*\frac{4}{12}=\frac{25}{216}$$, the same reason to multiply by $$\frac{4!}{2!2!}$$;

So $$P=1-\frac{25}{216}=\frac{191}{216}$$.

Similar problem: probability-colored-balls-55253.html

Hope it helps.
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Re: 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag  [#permalink]

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16 Oct 2006, 20:08
3
2

The probability of 2 blue and 2 purple marbles selected is 5C2.4C2/12C4 =4/33.

Subtracting the above from 1 we get 29/33
##### General Discussion
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Re: 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag  [#permalink]

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16 Oct 2006, 12:47
The probability that the result will be 2 blue and 2 purple marbles:
(5/12)*(5/12)*(4/12)*(4/12)=25/1296

The probability that it will not happen should be 1-(25/1296)=1271/1296.... and this is not an equivalent of D...

Where is my mistake?
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Joined: 30 Aug 2006
Posts: 355
Re: 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag  [#permalink]

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16 Oct 2006, 13:24
No cigar yet Simaq!

Its without replacement, and above u have done the equivalent of with replacement. E.g whats the probability of picking 2 blue marbles out of a box containing 5 blue, 2 red, 3 black without replacement?

5/10 * 4/9 = 20/90 = 2/9
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Re: 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag  [#permalink]

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16 Oct 2006, 13:34
Ok missed the replacement part....

Then the long way:

blue blue purple purple: 5/12*4/11*4/10*3/9
purple purple blue blue: 4/12*3/11*5/10*4/10
purple blue purple blue: 4/12*5/11*3/10*5/9
blue purple blue purple: 5/12*4/11*4/10*3/9

1-sum of all those values... must be a shortcut though
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Re: 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag  [#permalink]

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10 Aug 2010, 18:57
1
vijay2001 wrote:

The probability of 2 blue and 2 purple marbles selected is 5C2.4C2/12C4 =4/33.

Subtracting the above from 1 we get 29/33

How would this answer be different if it said "with replacement"? Can you provide the #of combinations in that case?
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Re: 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag  [#permalink]

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11 Aug 2010, 07:51
I'm thinking that with replacement you would have a total number of possible draws of 12^4 (could choose any of the 12 with any single draw). I think the numerator would look like 5*5*4*4 (5 ways to choose the first blue marble times 5 ways to choose the second blue marble times 4 ways to choose the first purple marble, etc.). Hence we would ultimately have

1 - 25*16/12^4 = 98% chance of not having this draw

Howver, after reading this post probability-colored-balls-55253.html#p637525 I'm confused. Can anyone chime in an clear up the correct way to compute the probability with replacement?
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Re: 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag  [#permalink]

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11 Aug 2010, 09:37
Excellent Bunuel. Now if you can clarify one more point. Same cases as 1 and 2 above, but in this case the question is - we pull balls 1by1 , what is the probability of seeing a blue one in the 9th draw (i) with replacement - i think this is easy 5/12 (ii) without replacement - i believe the answer here is also 5/12. My question how?
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Re: 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag  [#permalink]

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11 Aug 2010, 11:53
mainhoon wrote:
Excellent Bunuel. Now if you can clarify one more point. Same cases as 1 and 2 above, but in this case the question is - we pull balls 1by1 , what is the probability of seeing a blue one in the 9th draw (i) with replacement - i think this is easy 5/12 (ii) without replacement - i believe the answer here is also 5/12. My question how?

A jar contains 5 blue balls and 7 black balls. One by one, every ball is selected at random without replacement. What is the probability that the 9th ball selected is blue?

The probability of drawing blue ball is 5/12 and it will not change for ANY successive drawing: second, third, fourth...

Obviously for case with replacement the answer would be the same - every time we are drawing from 12 balls out of which 5 are blue, so every time probability of drawing blue ball is 5/12.

Similar problems:
probability-of-picking-last-97015.html?hilit=initial%20probability

Hope it helps.
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Re: 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag  [#permalink]

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11 Aug 2010, 14:25
thanks a lot Bunuel
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Re: 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag  [#permalink]

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11 Aug 2010, 14:28
Yes excellent. With or without replacement gives same answer. Can you give me an example where
The results are different?

Also while in the topic of probability can you answer the following please, my guesses are mentioned

Distribute 10 different rocks among 15 people ? 10^15
Distribute 10 identical rocks among 15 ? Not sure
Distribute 15 different rocks among 10 ? 15^10
Distribute 15 identical rocks among 10? 15+9 C 9
thanks

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Re: 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag  [#permalink]

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12 Aug 2010, 04:23
1
3
mainhoon wrote:
Yes excellent. With or without replacement gives same answer. Can you give me an example where
The results are different?

Also while in the topic of probability can you answer the following please, my guesses are mentioned

Distribute 10 different rocks among 15 people ? 10^15
Distribute 10 identical rocks among 15 ? Not sure
Distribute 15 different rocks among 10 ? 15^10
Distribute 15 identical rocks among 10? 15+9 C 9
thanks

Posted from my mobile device

Posted from my mobile device

Distribute 10 different rocks among 15 people? 10^15
Distribute 10 identical rocks among 15? (10+15-1)C(15-1)=24C14
Distribute 15 different rocks among 10? 15^10
Distribute 15 identical rocks among 10? (15+10-1)C(10-1)=24C9

Direct formulas for identical objects:
The total number of ways of dividing n identical items among r persons, each one of whom, can receive 0,1,2 or more items is $$n+r-1C_{r-1}$$.

The total number of ways of dividing n identical items among r persons, each one of whom receives at least one item is $$n-1C_{r-1}$$.

I doubt you'll see division of identical object involving such a big numbers on GMAT or division of different objects when each can get at least n objects (the question you've asked before) involving such a big numbers. With smaller numbers you can solve with different approaches depending on problem. So I would not worry about this questions too much. Better to concentrate on other areas.
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Re: 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag  [#permalink]

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12 Aug 2010, 04:40
Bunuel wrote:
mainhoon wrote:
Excellent Bunuel. Now if you can clarify one more point. Same cases as 1 and 2 above, but in this case the question is - we pull balls 1by1 , what is the probability of seeing a blue one in the 9th draw (i) with replacement - i think this is easy 5/12 (ii) without replacement - i believe the answer here is also 5/12. My question how?

A jar contains 5 blue balls and 7 black balls. One by one, every ball is selected at random without replacement. What is the probability that the 9th ball selected is blue?

The probability of drawing blue ball is 5/12 and it will not change for ANY successive drawing: second, third, fourth...
--> WHY??

If I draw 1 blue ball only 4 will be left (no replacement...) - next prob. of drawing blue should be 4/11 ??
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Re: 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag  [#permalink]

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12 Aug 2010, 05:05
1
AndreG wrote:
Bunuel wrote:
mainhoon wrote:
Excellent Bunuel. Now if you can clarify one more point. Same cases as 1 and 2 above, but in this case the question is - we pull balls 1by1 , what is the probability of seeing a blue one in the 9th draw (i) with replacement - i think this is easy 5/12 (ii) without replacement - i believe the answer here is also 5/12. My question how?

A jar contains 5 blue balls and 7 black balls. One by one, every ball is selected at random without replacement. What is the probability that the 9th ball selected is blue?

The probability of drawing blue ball is 5/12 and it will not change for ANY successive drawing: second, third, fourth...
--> WHY??

If I draw 1 blue ball only 4 will be left (no replacement...) - next prob. of drawing blue should be 4/11 ??

I think you misinterpreted the question (or I made it ambiguous). The question is: what is the probability that the 9th ball will be blue and it's meant that we don't know the results of first 8 draws (or drawing has not started yet and we are just curious what is the probability that 9th ball will be blue). If I pick one by one and every time I know the result then of course the probability of drawing of a blue ball will change depending on results of previous drawings.

Consider the following: these 12 balls were put in a line randomly. Now, what is the probability that the 9th ball will be blue? ANY ball in a line has equal chances to be blue and ANY ball has equal chances to be black. Why would the probability of say 4th and 7th balls of being blue be different? For any ball in a line p=5/12 to be blue and for any ball p=7/12 to be black: for any ball in a line the probability of being blue plus the probability of being black must be 1, as there are only black and blue balls in a line. This example is basically the same as our original question, so answer for original question is also 5/12.

Hope it helps.
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Re: 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag  [#permalink]

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12 Aug 2010, 05:24
Makes more sense now, thanks! Was just relatively hard to comprehend at first, what really was meant, but you made it clear!
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Re: 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag  [#permalink]

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13 Aug 2010, 05:44
Thanks Bunuel. I just chose the numbers at. Random, I see GMAT use smaller numbers, and then some start existing options, but the formula is more powerful. Also was curious to know how identical vs different changed the answers.

I wouldn't have known the replacement vs nonreplacement giving the same probability, thanks for the explanation to the prioranswer.

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Re: 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag  [#permalink]

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04 Jun 2012, 08:19
Bunuel wrote:
5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag. If 4 marbles are drawn WITHOUT replacement, what is the probability that the result will not be 2 blue and 2 purple marbles?

Given: 5B+3R+4P=12 marbles. Question: what is the probability of NOT having scenario BBPP.

Calculate the probability of an opposite event and subtract this value from 1.

Opposite event is that out of 4 marbles we will have 2 blue and 2 purple marbles (scenario BBPP):
$$P(opposite \ event)=\frac{4!}{2!2!}*\frac{5}{12}*\frac{4}{11}*\frac{4}{10}*\frac{3}{9}=\frac{4}{33}$$, we are multiplying by $$\frac{4!}{2!2!}$$ as scenario BBPP can occur in several ways: BBPP, PPBB, BPBP, ... So scenario BBPP can occur in $$\frac{4!}{2!2!}$$ # of ways (# of permuations of 4 letters BBPP out of which 2 B's and 2 P's are identical);

Or with combinatorics $$P(opposte \ event)=\frac{C^2_5*C^2_4}{C^4_{12}}=\frac{4}{33}$$.

So $$P=1-\frac{4}{33}=\frac{29}{33}$$.

5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag. If 4 marbles are drawn WITH replacement, what is the probability that the result will not be 2 blue and 2 purple marbles?

Again Calculate the probability of an opposite event and subtract this value from 1.

Opposite event is that out of 4 marbles we will have 2 blue and 2 purple marbles:
$$P(opposite \ event)=\frac{4!}{2!2!}*\frac{5}{12}*\frac{5}{12}*\frac{4}{12}*\frac{4}{12}=\frac{25}{216}$$, the same reason to multiply by $$\frac{4!}{2!2!}$$;

So $$P=1-\frac{25}{216}=\frac{191}{216}$$.

Similar problem: probability-colored-balls-55253.html

Hope it helps.

Is there a way to solve WITH replacement cases using combinatorics? I notice that all WITH replacement cases have only been solved using only one method.
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Re: 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag  [#permalink]

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11 Dec 2014, 06:28
Thank you Bunuel !
I wish you had been there in my school for maths

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Re: 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag  [#permalink]

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16 Apr 2018, 17:12
londonluddite wrote:
5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag. If 4 marbles are drawn without replacement, what is the probability that the result will not be 2 blue and 2 purple marbles?

A. 4/33
B. (5/36)^2
C. 1/2
D. (31/36)^2
E. 29/33

We can use the formula:

P(will not be 2 blue and 2 purple marbles) = 1 - P(2 blue and 2 purple marbles)

We can select two blue marbles in 5C2 = (5 x 4)/2! = 10 ways.

We can select 2 purple marbles in 4C2 = (4 x 3)/2! = 6 ways.

Thus, the total number of pairings of 2 blue and 2 purple marbles is 10 x 6 = 60 ways.

We can select 4 marbles from 12 in 12C4 = (12 x 11 x 10 x 9)/(4 x 3 x 2) = 11 x 5 x 9 = 495 ways.

Thus, P(2 blue and 2 purple marbles) = 60/495 = 20/165 = 4/33.

So, P(will not be 2 blue and 2 purple marbles) = 1 - 4/33 = 29/33.

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Re: 5 blue marbles, 3 red marbles and 4 purple marbles are placed in a bag   [#permalink] 16 Apr 2018, 17:12

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excel-vba,compare,range,msgbox , Compare two sets of Ranges and MsgBox amount and value of common cell values

## Question:

Tag: excel-vba,compare,range,msgbox

I have a problem with this code and I'm not able to resolve it.

I have two sets of ranges; Range("C14:C19") and Range("E14:E19") each containing 6 Numbers generated randomly (I already coded the random selection).

My goal is to, after generating the Numbers, compare both Ranges. If both Ranges contain identical Numbers, I want to MsgBox the amount of identical Numbers found in both Ranges and also the value of each Numbers.

Example: After generating the Numbers, if both Ranges have the values 9 and 7 in common, the MsgBox would look like:

There is/are 2 number(s) in the winning combination and it's/there value(s) is/are : 7, 9

Here Is what I was able to do up to now.

Private Sub Compare()

Dim rgnChoix As Range, rgnTirage As Range, i As Range, j As Range, iVal As Integer

Set rgnChoix = Range("C14:C19")
Set rgnTirage = Range("E14:E19")

For Each i In rgnChoix.Cells

For Each j In rgnTirage.Cells

If i.value = j.value Then

MsgBox "There is/are" & " " & iVal & " " & "number(s) in the winning combination and it's/there value(s) is/are :" & " " & i.value, , "Erika"
Exit Sub

End If
Next j
Next i

iVal = Application.WorksheetFunction.CountIf(Range("C14:C19"),"i.value=j.value")

End Sub

Thanks!! Erika

It may work as you wish. But if same number occurs more than once in a range it counts it as a seperate match. It counts all the matchs seperately. If one range includes two 4 and the other includes three 4 it counts six matching.

Private Sub Compare()
Dim rgnChoix As Range, rgnTirage As Range, i As Range, j As Range, iVal As Integer, sVal As String
Set rgnChoix = Range("C14:C19")
Set rgnTirage = Range("E14:E19")
For Each i In rgnChoix.Cells
For Each j In rgnTirage.Cells
If i.Value = j.Value Then
iVal = iVal + 1
sVal = sVal & "," & CStr(i.Value)
End If
Next j
Next i
sVal = Mid(sVal, 2)
MsgBox "There is/are" & " " & iVal & " " & "number(s) in the winning combination and it's/there value(s) is/are :" & " " & sVal, , "Erika"
End Sub

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## VBA - .find printing wrong value

excel,vba,excel-vba
I have two sections of code that basically do the same thing but with two different columns. The code finds the header "CUTTING TOOL" and "HOLDER" (looping through multiple files) and prints the information from those columns into one worksheet, masterfile. I was using a less efficient method of setting...

## Date formatting - making dd/m/yyyy into dd/mm/yyyy

excel-vba,date,format
I got a pretty simple question (but yet I've been stuck at it for some time now). Does anyone know how to make the date value from dd/m/yyyy into dd/mm/yyyy in a variable? dim lastdaylastmonth as date lastdaylastmonth = DateSerial(Year(Date), Month(Date), 0) So this code, as of now, would return...

## VBA how to initialize vCPath

vba,excel-vba,excel-2010
How do I initialize vCPath? VBA Run-Time Error 1004 @Garry's Student says I "must somehow use the info you get from Application.FileDialog(msoFileDialogOpen) to generate the full filespec of the file you wish to open". What's the simplest way to do this? I am a VBA beginner: I have been programming...

format: plain


text:
• ### Observational constraints on the sub-galactic matter-power spectrum from galaxy-galaxy strong gravitational lensing(1803.05952)

March 15, 2018 astro-ph.CO, astro-ph.GA
Constraining the sub-galactic matter-power spectrum on 1-10 kpc scales would make it possible to distinguish between the concordance $\Lambda$CDM model and various alternative dark-matter models due to the significantly different levels of predicted mass structure. Here, we demonstrate a novel approach to observationally constrain the population of overall law-mass density fluctuations in the inner regions of massive elliptical lens galaxies, based on the power spectrum of the associated surface-brightness perturbations observable in highly magnified galaxy-scale Einstein rings and gravitational arcs. The application of our method to the SLACS lens system SDSS J0252+0039 results in the following limits (at the 99 per cent confidence level) on the dimensionless convergence-power spectrum (and the associated standard deviation in aperture mass): $\Delta^{2}_{\delta\kappa}<1$ ($\sigma_{AM}< 0.8 \times 10^8 M_\odot$) on 0.5-kpc scale, $\Delta^{2}_{\delta\kappa}<0.1$ ($\sigma_{AM}< 1 \times 10^8 M_\odot$) on 1-kpc scale and $\Delta^{2}_{\delta\kappa}<0.01$ ($\sigma_{AM}< 3 \times 10^8 M_\odot$) on 3-kpc scale. The estimated effect of CDM sub-haloes lies considerably below these first observational upper-limit constraints on the level of inhomogeneities in the projected total mass distribution of galactic haloes. Future analysis for a larger sample of galaxy-galaxy strong lens systems will narrow down these constraints and rule out all cosmological models predicting a significantly larger level of clumpiness on these critical sub-galactic scales.
• ### Spin transport in two-layer-CVD-hBN/graphene/hBN heterostructures(1712.00815)

We study room temperature spin transport in graphene devices encapsulated between a layer-by-layer-stacked two-layer-thick chemical vapour deposition (CVD) grown hexagonal boron nitride (hBN) tunnel barrier, and a few-layer-thick exfoliated-hBN substrate. We find mobilities and spin-relaxation times comparable to that of SiO$_2$ substrate based graphene devices, and obtain a similar order of magnitude of spin relaxation rates for both the Elliott-Yafet and D'Yakonov-Perel' mechanisms. The behaviour of ferromagnet/two-layer-CVD-hBN/graphene/hBN contacts ranges from transparent to tunneling due to inhomogeneities in the CVD-hBN barriers. Surprisingly, we find both positive and negative spin polarizations for high-resistance two-layer-CVD-hBN barrier contacts with respect to the low-resistance contacts. Furthermore, we find that the differential spin injection polarization of the high-resistance contacts can be modulated by DC bias from -0.3 V to +0.3 V with no change in its sign, while its magnitude increases at higher negative bias. These features mark a distinctive spin injection nature of the two-layer-CVD-hBN compared to the bilayer-exfoliated-hBN tunnel barriers.
• The first observation of the rare decay$B^0_s \to \phi\pi^+\pi^-$ and evidence for $B^0 \to \phi\pi^+\pi^-$ are reported, using $pp$ collision data recorded by the LHCb detector at centre-of-mass energies $\sqrt{s} = 7$ and 8~TeV, corresponding to an integrated luminosity of $3{\mbox{\,fb}^{-1}}$. The branching fractions in the $\pi^+\pi^-$ invariant mass range $400<m(\pi^+\pi^-)<1600{\mathrm{\,Me\kern -0.1em V\!/}c^2}$ are $[3.48\pm 0.23\pm 0.17\pm 0.35]\times 10^{-6}$ and $[1.82\pm 0.25\pm 0.41\pm 0.14]\times 10^{-7}$ for $B^0_s \to \phi\pi^+\pi^-$ and $B^0 \to \phi\pi^+\pi^-$ respectively, where the uncertainties are statistical, systematic and from the normalisation mode $B^0_s \to \phi\phi$. A combined analysis of the $\pi^+\pi^-$ mass spectrum and the decay angles of the final-state particles identifies the exclusive decays $B^0_s \to \phi f_0(980)$, $B_s^0 \to \phi f_2(1270)$, and $B^0_s \to \phi\rho^0$ with branching fractions of $[1.12\pm 0.16^{+0.09}_{-0.08}\pm 0.11]\times 10^{-6}$, $[0.61\pm 0.13^{+0.12}_{-0.05}\pm 0.06]\times 10^{-6}$ and $[2.7\pm 0.7\pm 0.2\pm 0.2]\times 10^{-7}$, respectively.
• ### The age of the young bulge-like population in the stellar system Terzan5: linking the Galactic bulge to the high-z Universe(1609.01515)

Sept. 6, 2016 astro-ph.GA, astro-ph.SR
The Galactic bulge is dominated by an old, metal rich stellar population. The possible presence and the amount of a young (a few Gyr old) minor component is one of the major issues debated in the literature. Recently, the bulge stellar system Terzan 5 was found to harbor three sub-populations with iron content varying by more than one order of magnitude (from 0.2 up to 2 times the solar value), with chemical abundance patterns strikingly similar to those observed in bulge field stars. Here we report on the detection of two distinct main sequence turn-off points in Terzan 5, providing the age of the two main stellar populations: 12 Gyr for the (dominant) sub-solar component and 4.5 Gyr for the component at super-solar metallicity. This discovery classifies Terzan 5 as a site in the Galactic bulge where multiple bursts of star formation occurred, thus suggesting a quite massive progenitor possibly resembling the giant clumps observed in star forming galaxies at high redshifts. This connection opens a new route of investigation into the formation process and evolution of spheroids and their stellar content.
• ### Plasma Propulsion of a Metallic Micro-droplet and its Deformation upon Laser Impact(1604.00214)

The propulsion of a liquid indium-tin micro-droplet by nanosecond-pulse laser impact is experimentally investigated. We capture the physics of the droplet propulsion in a scaling law that accurately describes the plasma-imparted momentum transfer, enabling the optimization of the laser-droplet coupling. The subsequent deformation of the droplet is described by an analytical model that accounts for the droplet's propulsion velocity and the liquid properties. Comparing our findings to those from vaporization-accelerated mm-sized water droplets, we demonstrate that the hydrodynamic response of laser-impacted droplets is scalable and independent of the propulsion mechanism.
• ### A low-mass protostar's disk-envelope interface: disk-shadowing evidence from ALMA DCO+ observations of VLA1623(1505.07761)

May 28, 2015 astro-ph.SR
Due to instrumental limitations and a lack of disk detections, the structure between the envelope and the rotationally supported disk has been poorly studied. This is now possible with ALMA through observations of CO isotopologs and tracers of freezeout. Class 0 sources are ideal for such studies given their almost intact envelope and young disk. The structure of the disk-envelope interface of the prototypical Class 0 source, VLA1623A which has a confirmed Keplerian disk, is constrained from ALMA observations of DCO+ 3-2 and C18O 2-1. The physical structure of VLA1623 is obtained from the large-scale SED and continuum radiative transfer. An analytic model using a simple network coupled with radial density and temperature profiles is used as input for a 2D line radiative transfer calculation for comparison with the ALMA Cycle 0 12m array and Cycle 2 ACA observations of VLA1623. DCO+ emission shows a clumpy structure bordering VLA1623A's Keplerian disk, suggesting a cold ring-like structure at the disk-envelope interface. The radial position of the observed DCO+ peak is reproduced in our model only if the region's temperature is between 11-16K, lower than expected from models constrained by continuum and SED. Altering the density has little effect on the DCO+ position, but increased density is needed to reproduce the disk traced in C18O. The DCO+ emission around VLA1623A is the product of shadowing of the envelope by the disk. Disk-shadowing causes a drop in the gas temperature outside of the disk on >200AU scales, encouraging deuterated molecule production. This indicates that the physical structure of the disk-envelope interface differs from the rest of the envelope, highlighting the drastic impact that the disk has on the envelope and temperature structure. The results presented here show that DCO+ is an excellent cold temperature tracer.
• ### Evidence of Cluster Structure of $^9$Be from $^3$He+$^9$Be Reaction(1504.03942)

April 15, 2015 nucl-ex
The study of inelastic scattering and multi-nucleon transfer reactions was performed by bombarding a $^{9}$Be target with a $^3$He beam at an incident energy of 30 MeV. Angular distributions for $^9$Be($^3$He,$^3$He)$^{9}$Be, $^9$Be($^3$He,$^4$He)$^{8}$Be, $^9$Be($^3$He,$^5$He)$^{7}$Be, $^9$Be($^3$He,$^6$Li)$^6$Li and $^9$Be($^3$He,$^5$Li)$^7$Li reaction channels were measured. Experimental angular distributions for the corresponding ground states (g.s.) were analysed within the framework of the optical model, the coupled-channel approach and the distorted-wave Born approximation. Cross sections for channels leading to unbound $^5$He$_{g.s.}$, $^5$Li$_{g.s.}$ and $^8$Be systems were obtained from singles measurements where the relationship between the energy and the scattering angle of the observed stable ejectile is constrained by two-body kinematics. Information on the cluster structure of $^{9}$Be was obtained from the transfer channels. It was concluded that cluster transfer is an important mechanism in the investigated nuclear reactions. In the present work an attempt was made to estimate the relative strengths of the interesting $^8$Be+$n$ and $^5$He+$\alpha$ cluster configurations in $^9$Be. The branching ratios have been determined confirming that the $^5$He+$\alpha$ configuration plays an important role. The configuration of $^9$Be consisting of two bound helium clusters $^3$He+$^6$He is significantly suppressed, whereas the two-body configurations ${}^{8}$Be+$n$ and ${}^{5}$He+$\alpha$ including unbound $^8$Be and $^5$He are found more probable.
• ### Observation of anomalous Hanle spin precession lineshapes resulting from interaction with localized states(1411.1193)

Nov. 5, 2014 cond-mat.mes-hall
It has been shown recently that in spin precession experiments, the interaction of spins with localized states can change the response to a magnetic field, leading to a modified, effective spin relaxation time and precession frequency. Here, we show that also the shape of the Hanle curve can change, so that it cannot be fitted with the solutions of the conventional Bloch equation. We present experimental data that shows such an effect arising at low temperatures in epitaxial graphene on silicon carbide with localized states in the carbon buffer layer. We compare the strength of the effect between materials with different growth methods, epitaxial growth by sublimation and by chemical vapor deposition. The presented analysis gives information about the density of localized states and their coupling to the graphene states, which is inaccessible by charge transport measurements and can be applied to any spin transport channel that is coupled to localized states.
• ### Signatures of warm carbon monoxide in protoplanetary discs observed with Herschel SPIRE(1408.5432)

Aug. 22, 2014 astro-ph.GA, astro-ph.SR
Molecular gas constitutes the dominant mass component of protoplanetary discs. To date, these sources have not been studied comprehensively at the longest far-infrared and shortest submillimetre wavelengths. This paper presents Herschel SPIRE FTS spectroscopic observations toward 18 protoplanetary discs, covering the entire 450-1540 GHz (666-195 $\mu$m) range at R~400-1300. The spectra reveal clear detections of the dust continuum and, in six targets, a significant amount of spectral line emission primarily attributable to $^{12}$CO rotational lines. Other targets exhibit little to no detectable spectral lines. Low signal-to-noise detections also include signatures from $^{13}$CO, [CI] and HCN. For completeness, we present upper limits of non-detected lines in all targets, including low-energy transitions of H2O and CH$^+$ molecules. The ten $^{12}$CO lines that fall within the SPIRE FTS bands trace energy levels of ~50-500 K. Combined with lower and higher energy lines from the literature, we compare the CO rotational line energy distribution with detailed physical-chemical models, for sources where these are available and published. Our 13CO line detections in the disc around Herbig Be star HD 100546 exceed, by factors of ~10-30, the values predicted by a model that matches a wealth of other observational constraints, including the SPIRE $^{12}$CO ladder. To explain the observed $^{12}$CO/$^{13}$CO ratio, it may be necessary to consider the combined effects of optical depth and isotope selective (photo)chemical processes. Considering the full sample of 18 objects, we find that the strongest line emission is observed in discs around Herbig Ae/Be stars, although not all show line emission. In addition, two of the six T Tauri objects exhibit detectable $^{12}$CO lines in the SPIRE range.
• ### Constraining the structure of the transition disk HD 135344B (SAO 206462) by simultaneous modeling of multiwavelength gas and dust observations(1403.6193)

April 18, 2014 astro-ph.SR
HD 135344B is an accreting (pre-) transition disk that displays the emission of warm CO extending tens of AU inside its 30 AU dust cavity. We used the dust radiative transfer code MCFOST and the thermochemical code ProDiMo to derive the disk structure from the simultaneous modeling of the spectral energy distribution (SED), VLT/CRIRES CO P(10) 4.75 micron, Herschel/PACS [O I] 63 micron, Spitzer-IRS, and JCMT 12CO J=3-2 spectra, VLTI/PIONIER H-band visibilities, and constraints from (sub-)mm continuum interferometry and near-IR imaging. We found a disk model able to describe the current observations simultaneously. This disk has the following structure. (1) To reproduce the SED, the near-IR interferometry data, and the CO ro-vibrational emission, refractory grains (we suggest carbon) are present inside the silicate sublimation radius (0.08<R<0.2 AU). (2) The dust cavity (R<30 AU) is filled with gas, the surface density of this gas must increase with radius to fit the CO P(10) line profile, a small gap of a few AU in the gas is compatible with current data, and a large gap in the gas is not likely. (4) The gas/dust ratio inside the cavity is > 100 to account for the 870 micron continuum upper limit and the CO P(10) line flux. (5) The gas/dust ratio at 30<R<200 AU is < 10 to simultaneously describe the [O I] 63 micron line flux and the CO P(10) line profile. (6) In the outer disk, most of the mass should be located in the midplane, and a significant fraction of the dust is in large grains. Conclusions: Simultaneous modeling of the gas and dust is required to break the model degeneracies and constrain the disk structure. An increasing gas surface density with radius in the inner dust cavity echoes the effect of a migrating Jovian planet. The low gas mass (a few MJupiter) in the HD 135344B's disk suggests that it is an evolved disk that has already lost a large portion of its mass.
• ### Resolved Imaging of the HR 8799 Debris Disk with Herschel(1311.2977)

Nov. 12, 2013 astro-ph.SR, astro-ph.EP
We present Herschel far-infrared and submillimeter maps of the debris disk associated with the HR 8799 planetary system. We resolve the outer disk emission at 70, 100, 160 and 250 um and detect the disk at 350 and 500 um. A smooth model explains the observed disk emission well. We observe no obvious clumps or asymmetries associated with the trapping of planetesimals that is a potential consequence of planetary migration in the system. We estimate that the disk eccentricity must be <0.1. As in previous work by Su et al. (2009), we find a disk with three components: a warm inner component and two outer components, a planetesimal belt extending from 100 - 310 AU, with some flexibility (+/- 10 AU) on the inner edge, and the external halo which extends to ~2000 AU. We measure the disk inclination to be 26 +/- 3 deg from face-on at a position angle of 64 deg E of N, establishing that the disk is coplanar with the star and planets. The SED of the disk is well fit by blackbody grains whose semi-major axes lie within the planetesimal belt, suggesting an absence of small grains. The wavelength at which the spectrum steepens from blackbody, 47 +/- 30 um, however, is short compared to other A star debris disks, suggesting that there are atypically small grains likely populating the halo. The PACS longer wavelength data yield a lower disk color temperature than do MIPS data (24 and 70 um), implying two distinct halo dust grain populations.
• ### The HIFI spectral survey of AFGL2591 (CHESS). I. Highly excited linear rotor molecules in the high-mass protostellar envelope(1303.3339)

March 14, 2013 astro-ph.GA, astro-ph.SR
We aim to reveal the gas energetics in the circumstellar environment of the prototypical high-mass protostellar object AFGL2591 using space-based far-infrared observations of linear rotor molecules. Rotational spectral line signatures of CO, HCO+, CS, HCN and HNC from a 490-1240 GHz survey with Herschel/HIFI, complemented by ground-based JCMT and IRAM 30m spectra, cover transitions with E(up)/k between 5 and ~300 K (750K for 12C16O, using selected frequency settings up to 1850 GHz). The resolved spectral line profiles are used to separate and study various kinematic components. The line profiles show two emission components, the widest and bluest of which is attributed to an approaching outflow and the other to the envelope. We find evidence for progressively more redshifted and wider line profiles from the envelope gas with increasing energy level, qualitatively explained by residual outflow contribution picked up in the systematically decreasing beam size. Integrated line intensities for each species decrease as E(up)/k increases from <50 to 700K. We constrain the following: n(H2)~10^5-10^6 cm^-3 and T~60-200K for the outflow gas; T=9-17K and N(H2)~3x10^21 cm^-2 for a known foreground absorption cloud; N(H2)<10^19 cm^-2 for a second foreground component. Our spherical envelope radiative transfer model systematically underproduces observed line emission at E(up)/k > 150 K for all species. This indicates that warm gas should be added to the model and that the model's geometry should provide low optical depth pathways for line emission from this warm gas to escape, for example in the form of UV heated outflow cavity walls viewed at a favorable inclination angle. Physical and chemical conditions derived for the outflow gas are similar to those in the protostellar envelope, possibly indicating that the modest velocity (<10 km/s) outflow component consists of recently swept-up gas.
• Cassiopeia A was observed using the Low-Band Antennas of the LOw Frequency ARray (LOFAR) with high spectral resolution. This allowed a search for radio recombination lines (RRLs) along the line-of-sight to this source. Five carbon-alpha RRLs were detected in absorption between 40 and 50 MHz with a signal-to-noise ratio of > 5 from two independent LOFAR datasets. The derived line velocities (v_LSR ~ -50 km/s) and integrated optical depths (~ 13 s^-1) of the RRLs in our spectra, extracted over the whole supernova remnant, are consistent within each LOFAR dataset and with those previously reported. For the first time, we are able to extract spectra against the brightest hotspot of the remnant at frequencies below 330 MHz. These spectra show significantly higher (15-80 %) integrated optical depths, indicating that there is small-scale angular structure on the order of ~1 pc in the absorbing gas distribution over the face of the remnant. We also place an upper limit of 3 x 10^-4 on the peak optical depths of hydrogen and helium RRLs. These results demonstrate that LOFAR has the desired spectral stability and sensitivity to study faint recombination lines in the decameter band.
• ### The cooling phase of Type I X-ray bursts observed with RXTE in 4U 1820-30 does not follow the canonical F - T^4 relation(1301.1035)

Jan. 6, 2013 astro-ph.HE
We analysed the complete set of bursts from the neutron-star low-mass X-ray binary 4U 1820-30 detected with the Rossi X-ray Timing Explorer (RXTE). We found that all are photospheric radius expansion bursts, and have similar duration, peak flux and fluence. From the analysis of time-resolved spectra during the cooling phase of the bursts, we found that the relation between the bolometric flux and the temperature is very different from the canonical F - T^4 relation that is expected if the apparent emitting area on the surface of the neutron star remains constant. The flux-temperature relation can be fitted using a broken power law, with indices 2.0$\pm$0.3 and 5.72$\pm$0.06. The departure from the F - T^4 relation during the cooling phase of the X-ray bursts in 4U 1820-30 could be due to changes in the emitting area of the neutron star while the atmosphere cools-down, variations in the colour-correction factor due to chemical evolution, or the presence of a source of heat, e.g. residual hydrogen nuclear burning, playing an important role when the burst emission ceases.
• The decays $J/\psi\to p\bar{p}$ and $J/\psi\to n\bar{n}$ have been investigated with a sample of 225.2 million $J/\psi$ events collected with the BESIII detector at the BEPCII $e^+e^-$ collider. The branching fractions are determined to be $\mathcal{B}(J/\psi\to p\bar{p})=(2.112\pm0.004\pm0.031)\times10^{-3}$ and $\mathcal{B}(J/\psi\to n\bar{n})=(2.07\pm0.01\pm0.17)\times10^{-3}$. Distributions of the angle $\theta$ between the proton or anti-neutron and the beam direction are well described by the form $1+\alpha\cos^2\theta$, and we find $\alpha=0.595\pm0.012\pm0.015$ for $J/\psi\to p\bar{p}$ and $\alpha=0.50\pm0.04\pm0.21$ for $J/\psi\to n\bar{n}$. Our branching-fraction results suggest a large phase angle between the strong and electromagnetic amplitudes describing the $J/\psi\to N\bar{N}$ decay.
• Based on a data sample of 106 M $\psi^{\prime}$ events collected with the BESIII detector, the decays $\psi^{\prime}\ar\gamma\chi_{c0, 2}$,$\chi_{c0, 2}\ar\gamma\gamma$ are studied to determine the two-photon widths of the $\chi_{c0, 2}$ states. The two-photon decay branching fractions are determined to be ${\cal B}(\chi_{c0}\ar\gamma\gamma) = (2.24\pm 0.19\pm 0.12\pm 0.08)\times 10^{-4}$ and ${\cal B}(\chi_{c2}\ar\gamma\gamma) = (3.21\pm 0.18\pm 0.17\pm 0.13)\times 10^{-4}$. From these, the two-photon widths are determined to be $\Gamma_{\gamma \gamma}(\chi_{c0}) = (2.33\pm0.20\pm0.13\pm0.17)$ keV, $\Gamma_{\gamma \gamma}(\chi_{c2}) = (0.63\pm0.04\pm0.04\pm0.04)$ keV, and $\cal R$ $=\Gamma_{\gamma \gamma}(\chi_{c2})/\Gamma_{\gamma \gamma}(\chi_{c0})=0.271\pm 0.029\pm 0.013\pm 0.027$, where the uncertainties are statistical, systematic, and those from the PDG ${\cal B}(\psi^{\prime}\ar\gamma\chi_{c0,2})$ and $\Gamma(\chi_{c0,2})$ errors, respectively. The ratio of the two-photon widths for helicity $\lambda=0$ and helicity $\lambda=2$ components in the decay $\chi_{c2}\ar\gamma\gamma$ is measured for the first time to be $f_{0/2} =\Gamma^{\lambda=0}_{\gamma\gamma}(\chi_{c2})/\Gamma^{\lambda=2}_{\gamma\gamma}(\chi_{c2}) = 0.00\pm0.02\pm0.02$.
• ### The infant Milky Way(1204.1943)

April 9, 2012 astro-ph.CO
We investigate the physical properties of the progenitors of today living Milky Way-like galaxies that are visible as Damped Lya Absorption systems and Lya Emitters at higher redshifts (z ~ 2.3,5.7). To this aim we use a statistical merger-tree approach that follows the formation of the Galaxy and its dwarf satellites in a cosmological context, tracing the chemical evolution and stellar population history of the progenitor halos. The model accounts for the properties of the most metal-poor stars and local dwarf galaxies, providing insights on the early cosmic star-formation. Fruitful links between Galactic Archaeology and more distant galaxies are presented.
• ### Stellar archeology: a cosmological view of dwarf galaxies(1204.1946)

April 9, 2012 astro-ph.CO, astro-ph.GA
The origin of dwarf spheroidal galaxies (dSphs) is investigated in a global cosmological context by simultaneously following the evolution of the Milky Way Galaxy and its dwarf satellites. This approach enable to study the formation of dSphs in their proper birth environment and to reconstruct their own merging histories. The proposed picture simultaneously accounts for several dSph and Milky Way properties, including the Metallicity Distribution Functions of metal-poor stars. The observed features are interpreted in terms of physical processes acting at high redshifts.
• ### First stars in Damped Lyman Alpha systems(1111.6637)

Nov. 28, 2011 astro-ph.CO
In order to characterize Damped Lyman Alpha systems (DLAs) potentially hosting first stars, we present a novel approach to investigate DLAs in the context of Milky Way (MW) formation, along with their connection with the most metal-poor stars and local dwarf spheroidal (dSph) galaxies. The merger tree method previously developed is extended to include inhomogeneous reionization and metal mixing, and it is validated by matching both the Metallicity Distribution Function of Galactic halo stars and the Fe-Luminosity relation of dSph galaxies. The model explains the observed NHI-Fe relation of DLAs along with the chemical abundances of [Fe/H] < -2 systems. In this picture, the recently discovered z_abs ~ 2.34 C-enhanced DLA (Cooke et al. 2011a), pertains to a new class of absorbers hosting first stars along with second-generation long-living low-mass stars. These "PopIII DLAs" are the descendants of H2-cooling minihalos with Mh ~ 10^7 Msun, that virialize at z > 8 in neutral, primordial regions of the MW environment and passively evolve after a short initial period of star formation. The gas in these systems is warm Tg \sim (40-1000) K, and strongly C-enriched by long-living, extremely metal-poor stars of total mass M* \sim 10^{2-4} Msun.
• ### The JCMT Spectral Legacy Survey: physical structure of the molecular envelope of the high-mass protostar AFGL2591(1101.5529)

Jan. 28, 2011 astro-ph.GA
The understanding of the formation process of massive stars (>8 Msun) is limited, due to theoretical complications and observational challenges. We investigate the physical structure of the large-scale (~10^4-10^5 AU) molecular envelope of the high-mass protostar AFGL2591 using spectral imaging in the 330-373 GHz regime from the JCMT Spectral Legacy Survey. Out of ~160 spectral features, this paper uses the 35 that are spatially resolved. The observed spatial distributions of a selection of six species are compared with radiative transfer models based on a static spherically symmetric structure, a dynamic spherical structure, and a static flattened structure. The maps of CO and its isotopic variations exhibit elongated geometries on scales of ~100", and smaller scale substructure is found in maps of N2H+, o-H2CO, CS, SO2, CCH, and methanol lines. A velocity gradient is apparent in maps of all molecular lines presented here, except SO, SO2, and H2CO. We find two emission peaks in warm (Eup~200K) methanol separated by 12", indicative of a secondary heating source in the envelope. The spherical models are able to explain the distribution of emission for the optically thin H13CO+ and C34S, but not for the optically thick HCN, HCO+, and CS, nor for the optically thin C17O. The introduction of velocity structure mitigates the optical depth effects, but does not fully explain the observations, especially in the spectral dimension. A static flattened envelope viewed at a small inclination angle does slightly better. We conclude that a geometry of the envelope other than an isotropic static sphere is needed to circumvent line optical depth effects. We propose that this could be achieved in envelope models with an outflow cavity and/or inhomogeneous structure at scales smaller than ~10^4 AU. The picture of inhomogeneity is supported by observed substructure in at least six species.
• ### Parsec-Scale Bipolar X-ray Shocks Produced by Powerful Jets from the Neutron Star Circinus X-1(1008.0647)

Aug. 3, 2010 astro-ph.HE
We report the discovery of multi-scale X-ray jets from the accreting neutron star X-ray binary, Circinus X-1. The bipolar outflows show wide opening angles and are spatially coincident with the radio jets seen in new high-resolution radio images of the region. The morphology of the emission regions suggests that the jets from Circinus X-1 are running into a terminal shock with the interstellar medium, as is seen in powerful radio galaxies. This and other observations indicate that the jets have a wide opening angle, suggesting that the jets are either not very well collimated or precessing. We interpret the spectra from the shocks as cooled synchrotron emission and derive a cooling age of approximately 1600 yr. This allows us to constrain the jet power to be between 3e35 erg/s and 2e37 erg/s, making this one of a few microquasars with a direct measurement of its jet power and the only known microquasar that exhibits stationary large-scale X-ray emission.
• ### Fossil evidence for spin alignment of SDSS galaxies in filaments(1001.4479)

July 5, 2010 astro-ph.CO
We search for and find fossil evidence that the distribution of the spin axes of galaxies in cosmic web filaments relative to their host filaments are not randomly distributed. This would indicate that the action of large scale tidal torques effected the alignments of galaxies located in cosmic filaments. To this end, we constructed a catalogue of clean filaments containing edge-on galaxies. We started by applying the Multiscale Morphology Filter (MMF) technique to the galaxies in a redshift-distortion corrected version of the Sloan Digital Sky Survey DR5. From that sample we extracted those 426 filaments that contained edge-on galaxies (b/a < 0.2). These filaments were then visually classified relative to a variety of quality criteria. Statistical analysis using "feature measures" indicates that the distribution of orientations of these edge-on galaxies relative to their parent filament deviate significantly from what would be expected on the basis of a random distribution of orientations. The interpretation of this result may not be immediately apparent, but it is easy to identify a population of 14 objects whose spin axes are aligned perpendicular to the spine of the parent filament (\cos \theta < 0.2). The candidate objects are found in relatively less dense filaments. This might be expected since galaxies in such locations suffer less interaction with surrounding galaxies, and consequently better preserve their tidally induced orientations relative to the parent filament. The technique of searching for fossil evidence of alignment yields relatively few candidate objects, but it does not suffer from the dilution effects inherent in correlation analysis of large samples.
• ### Imaging of a Transitional Disk Gap in Reflected Light: Indications of Planet Formation Around the Young Solar Analog LkCa 15(1005.5162)

July 1, 2010 astro-ph.SR, astro-ph.EP
We present H- and Ks-band imaging data resolving the gap in the transitional disk around LkCa 15, revealing the surrounding nebulosity. We detect sharp elliptical contours delimiting the nebulosity on the inside as well as the outside, consistent with the shape, size, ellipticity, and orientation of starlight reflected from the far-side disk wall, whereas the near-side wall is shielded from view by the disk's optically thick bulk. We note that forward-scattering of starlight on the near-side disk surface could provide an alternate interpretation of the nebulosity. In either case, this discovery provides confirmation of the disk geometry that has been proposed to explain the spectral energy distributions (SED) of such systems, comprising an optically thick outer disk with an inner truncation radius of ~46 AU enclosing a largely evacuated gap. Our data show an offset of the nebulosity contours along the major axis, likely corresponding to a physical pericenter offset of the disk gap. This reinforces the leading theory that dynamical clearing by at least one orbiting body is the cause of the gap. Based on evolutionary models, our high-contrast imagery imposes an upper limit of 21 Jupiter masses on companions at separations outside of 0.1" and of 13 Jupiter masses outside of 0.2". Thus, we find that a planetary system around LkCa 15 is the most likely explanation for the disk architecture.
• ### High redshift Lya emitters: clues on the Milky Way infancy(1005.4422)

May 24, 2010 astro-ph.CO
With the aim of determining if Milky Way (MW) progenitors could be identified as high redshift Lyman Alpha Emitters (LAEs) we have derived the intrinsic properties of z ~ 5.7 MW progenitors, which are then used to compute their observed Lyman-alpha luminosity, L_alpha, and equivalent width, EW. MW progenitors visible as LAEs are selected according to the canonical observational criterion, L_alpha > 10^42 erg/s and EW > 20 A. Progenitors of MW-like galaxies have L_alpha = 10^(39-43.25) erg/s, making some of them visible as LAEs. In any single MW merger tree realization, typically only 1 (out of ~ 50) progenitor meets the LAE selection criterion, but the probability to have at least one LAE is very high, P = 68%. The identified LAE stars have ages, t_* ~ 150-400 Myr at z ~ 5.7 with the exception of five small progenitors with t_* < 5 Myr and large EW = 60-130 A. LAE MW progenitors provide > 10% of the halo very metal-poor stars [Fe/H] < -2, thus establishing a potentially fruitful link between high-z galaxies and the Local Universe.
• ### Chemical stratification in the Orion Bar: JCMT Spectral Legacy Survey observations(0902.1433)

Feb. 23, 2010 astro-ph.GA
Photon-dominated regions (PDRs) are expected to show a layered structure in molecular abundances and emerging line emission, which is sensitive to the physical structure of the region as well as the UV radiation illuminating it. We aim to study this layering in the Orion Bar, a prototypical nearby PDR with a favorable edge-on geometry. We present new maps of 2 by 2 arcminute fields at 14-23 arcsecond resolution toward the Orion Bar in the SO 8_8-9_9, H2CO 5_(1,5)-4_(1,4), 13CO 3-2, C2H 4_(9/2)-3_(7/2) and 4_(7/2)-3_(5/2), C18O 2-1 and HCN 3-2 transitions. The data reveal a clear chemical stratification pattern. The C2H emission peaks close to the ionization front, followed by H2CO and SO, while C18O, HCN and 13CO peak deeper into the cloud. A simple PDR model reproduces the observed stratification, although the SO emission is predicted to peak much deeper into the cloud than observed while H2CO is predicted to peak closer to the ionization front than observed. In addition, the predicted SO abundance is higher than observed while the H2CO abundance is lower than observed. The discrepancies between the models and observations indicate that more sophisticated models, including production of H2CO through grain surface chemistry, are needed to quantitatively match the observations of this region.


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# random sample simulator

The AQL system according to DIN 40 080 can be simulated using the random sample simulator. In a total population of n=1000 balls, 0.5%, 1%, 2%, 5%, 10% and 20% of the balls are of different colours. This symbolizes the quantity of “bad” parts. Only the appropriate colour is considered and evaluated, depending on the AQL value assumed. A random sample of n=100 can be taken and evaluated by simply tilting the simulator. The balls roll in ten channels. The random sample is thus shown clearly and the evaluation is simplified. Smaller random samples can also be simulated by evaluating only one part of the channel.

ordernumber 1075


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# Is energy conserved during the expansion of the universe? [duplicate]

Consider the following thought experiment. Suppose we initialize a parallel plate capacitor with the plates of area $$A$$ separated by a distance $$d$$ in empty space. The energy in the capacitor is approximated as $$E\approx\epsilon_0 Ad$$. Let a large amount of time pass by. Assume that the distance between the plates increases due to the expansion of the universe (and in that case $$A$$ increases as well.) Then it seems our capacitor has now gained some potential energy "from nowhere".

Now I have read that, though the spacetime metric is observed to be expanding on large scales, it does not follow that spacetime is expanding on small scales. But it might be.

Also, I know this is a silly example, as we should not expect laws of household capacitors to hold on cosmological scales. Perhaps someone could come up with a more sophisticated example. What I am getting at, is that in many physical systems, the potential energy of that system is related to the distance between the components of that system. If the distance between the components changed due to the expansion of the universe, it would seem the potential energy of the system also changed. Yet, no work was performed upon the system. What is the discrepancy here? Where did the difference in energy go? (I think this general way of looking of the problem should overcome the objection that spacetime might not be expanding on small scale. Just define our system to occur on some scale where expansion is observed.)

• Energy is not conserved in an expanding universe because it doesn't have time translation symmetry - the universe evolves in a preferred time direction. Feb 28 at 3:38
• Possible duplicates: physics.stackexchange.com/q/35431/2451 and links therein. Feb 28 at 4:31


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## Distribution of three kinds of objects

Hi. This problem sounds seemingly simple but I am not sure if it's as simple as I think so I've decided to post it here.

We have three kinds of objects and there are $2n$objects of each kind. How many ways of dividing those $6n$ objects between two people are there so that each of them gets $3n$ objects?


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# ACT Math : Sine

## Example Questions

← Previous 1 3 4 5

### Example Question #1 : Sine

What is the period of 2sin(4Θ)?

Possible Answers:

None of the answers are correct

Correct answer:

Explanation:

The period of sinΘ is 2Π, so we set the new angle equal to the base period of 2Π and solve for Θ.

So 4Θ = 2Π and Θ = Π/2.

### Example Question #2 : Sine

A function with period P will repeat on intervals of length P, and these intervals are referred to as periods.

Find the period of

.

Possible Answers:

Correct answer:

Explanation:

For the function

the period is equal to

in this case

which reduces to .

### Example Question #1 : Sine

A function with period P will repeat on intervals of length P, and these intervals are referred to as periods.

Find the period of the function

.

Possible Answers:

Correct answer:

Explanation:

For the function

the period is equal to

in this case

which reduces to .

### Example Question #1 : Sine

What is the period of the function ?

Possible Answers:

Correct answer:

Explanation:

To find the period of Sine and Cosine functions you use the formula:
where  comes from . Looking at our formula you see b is 4 so

### Example Question #1 : Sine

What is the period of the given trigonometric function:

. Leave your answer in terms of , simplify all fractions.

Possible Answers:

Correct answer:

Explanation:

To find the period of a sine, cosine, cosecant, or secant funciton use the formula:

where  comes from the general formula: . We see that for our equation  and so the period is  when you reduce the fraction.

### Example Question #6 : Sine

Find the period of the following formula:

Possible Answers:

Correct answer:

Explanation:

To find period, simply remember the following formula:

where B is the coefficient in front of x. Thus,

### Example Question #1 : Sine

Find the domain of the function:

Possible Answers:

Correct answer:

Explanation:

The function  is related to the parent function , which has a domain of .

The value of theta for  has no restriction and is valid for all real numbers.

The answer is .

### Example Question #8 : Sine

What is the domain of the given trigonometric function:

Possible Answers:

Correct answer:

Explanation:

For both Sine and Cosine, since there are no asymptotes like Tangent and Cotangent functions, the function can take in any value for . Thus the domain is:

### Example Question #1 : How To Find The Range Of The Sine

Which of the following statements is (are) true:

I. The domain of the tangent function is all real numbers

II. The range of the sine function is all real numbers

III. The periods of the tangent, sine, and cosine functions are the same.

Possible Answers:

none of the above

II and III only

II only

I and II only

I only

Correct answer:

II only

Explanation:

The domain of the tangent function does not include any values of x that are odd multiples of π/2 .

The range of the sine function is from [-1, 1].

The period of the tangent function is π, whereas the period for both sine and cosine is 2π.

### Example Question #1 : How To Find The Range Of The Sine

Which of the following represents a sine wave with a range of ?

Possible Answers:

Correct answer:

Explanation:

The range of a sine wave is altered by the coefficient placed in front of the base equation. So, if you have , this means that the highest point on the wave will be at  and the lowest at .  However, if you then begin to shift the equation vertically by adding values, as in, , then you need to account for said shift.  This would make the minimum value to be  and the maximum value to be . For our question, then, it is fine to use . The  for the function parameter only alters the period of the equation, making its waves "thinner."

← Previous 1 3 4 5


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# Math Insight

### Applet: Parametrized ellipse

The vector-valued function $\adllp(t)=(3\cos \frac{t^2}{2\pi})\vc{i} + (2 \sin \frac{t^2}{2\pi}) \vc{j}$ parametrizes an ellipse, shown in cyan. This ellipse is the image of the interval $[0,2\pi]$ (shown in red) under the mapping of $\adllp$. For each value of $t$, the blue vector is $\adllp(t)$. As you change $t$ by moving the blue point along the interval $[0,2\pi]$, the head of the arrow traces out the ellipse.

Can you show mathematically that $\adllp(t)$ traces out an ellipse? To do so, let $(x,y)$ be the point defined by $(x,y)=\adllp(t)$ for some value of $t$. Since $\adllp(t) = (3\cos \frac{t^2}{2\pi}, 2 \sin \frac{t^2}{2\pi})$, we conclude that $x = 3\cos \frac{t^2}{2\pi}$ and $y=2\sin \frac{t^2}{2\pi}$. Can you see that $x$ and $y$ satisfy \begin{align*} \frac{x^2}{3^2} + \frac{y^2}{2^2} = 1, \end{align*} which is the equation for the above ellipse? (You remember that $\cos^2s + \sin^2s = 1$ for any $s$. You can calculate expressions for $x^2/3^2$ and $y^2/2^2$, and add them together.)

Applet file: parametrized_ellipse_2.ggb

#### Applet links

This applet is found in the pages

List of all applets

#### General information about Geogebra Web applets

This applet was created using Geogebra. In most Geogebra applets, you can move objects by dragging them with the mouse. In some, you can enter values with the keyboard. To reset the applet to its original view, click the icon in the upper right hand corner.

You can download the applet onto your own computer so you can use it outside this web page or even modify it to improve it. You simply need to download the above applet file and download the Geogebra program onto your own computer.


text:
# On the continuum limit for discrete NLS with long-range lattice interactions

Kirkpatrick, Kay and Lenzmann, Enno and Staffilani, Gigliola. (2013) On the continuum limit for discrete NLS with long-range lattice interactions. Communications in mathematical physics, Vol. 317, H. 3. pp. 563-591.

Full text not available from this repository.

Official URL: http://edoc.unibas.ch/dok/A6194582


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## January 12, 2016

### A Compositional Framework for Markov Processes

#### Posted by John Baez

Last summer my students Brendan Fong and Blake Pollard visited me at the Centre for Quantum Technologies, and we figured out how to understand open continuous-time Markov chains! I think this is a nice step towards understanding the math of living systems.

Admittedly, it’s just a small first step. But I’m excited by this step, since Blake and I have been trying to get this stuff to work for a couple years, and it finally fell into place. And we think we know what to do next. Here’s our paper:

And here’s the basic idea….

### Open detailed balanced Markov processes

A continuous-time Markov chain is a way to specify the dynamics of a population which is spread across some finite set of states. Population can flow between the states. The larger the population of a state, the more rapidly population flows out of the state. Because of this property, under certain conditions the populations of the states tend toward an equilibrium where at any state the inflow of population is balanced by its outflow.

In applications to statistical mechanics, we are often interested in equilibria such that for any two states connected by an edge, say $i$ and $j,$ the flow from $i$ to $j$ equals the flow from $j$ to $i.$ A continuous-time Markov chain with a chosen equilibrium having this property is called ‘detailed balanced’.

I’m getting tired of saying ‘continuous-time Markov chain’, so from now on I’ll just say ‘Markov process’, just because it’s shorter. Okay? That will let me say the next sentence without running out of breath:

Our paper is about open detailed balanced Markov processes.

Here’s an example:

The detailed balanced Markov process itself consists of a finite set of states together with a finite set of edges between them, with each state $i$ labelled by an equilibrium population $q_i >0,$ and each edge $e$ labelled by a rate constant $r_e > 0.$

These populations and rate constants are required to obey an equation called the ‘detailed balance condition’. This equation means that in equilibrium, the flow from $i$ to $j$ equal the flow from $j$ to $i.$ Do you see how it works in this example?

To get an ‘open’ detailed balanced Markov process, some states are designated as inputs or outputs. In general each state may be specified as both an input and an output, or as inputs and outputs multiple times. See how that’s happening in this example? It may seem weird, but it makes things work better.

People usually say Markov processes are all about how probabilities flow from one state to another. But we work with un-normalized probabilities, which we call ‘populations’, rather than probabilities that must sum to 1. The reason is that in an open Markov process, probability is not conserved: it can flow in or out at the inputs and outputs. We allow it to flow both in and out at both the input states and the output states.

Our most fundamental result is that there’s a category ${DetBalMark}$ where a morphism is an open detailed balanced Markov process. We think of it as a morphism from its inputs to its outputs.

We compose morphisms in ${DetBalMark}$ by identifying the output states of one open detailed balanced Markov process with the input states of another. The populations of identified states must match. For example, we may compose this morphism $N$:

with the previously shown morphism $M$ to get this morphism $M \circ N$:

And here’s our second most fundamental result: the category ${DetBalMark}$ is actually a dagger compact category. This lets us do other stuff with open Markov processes. An important one is ‘tensoring’, which lets us take two open Markov processes like $M$ and $N$ above and set them side by side, giving $M \otimes N$:

The compactness is also important. This means we can take some inputs of an open Markov process and turn them into outputs, or vice versa. For example, using the compactness of ${DetBalMark}$ we can get this open Markov process from $M$:

In fact all the categories in our paper are dagger compact categories, and all our functors preserve this structure. Dagger compact categories are a well-known framework for describing systems with inputs and outputs, so this is good.

### The analogy to electrical circuits

In a detailed balanced Markov process, population can flow along edges. In the detailed balanced equilibrium, without any flow of population from outside, the flow along from state $i$ to state $j$ will be matched by the flow back from $j$ to $i.$ The populations need to take specific values for this to occur.

In an electrical circuit made of linear resistors, charge can flow along wires. In equilibrium, without any driving voltage from outside, the current along each wire will be zero. The potentials will be equal at every node.

This sets up an analogy between detailed balanced continuous-time Markov chains and electrical circuits made of linear resistors! I love analogy charts, so this makes me very happy:

 Circuits Detailed balanced Markov processes potential population current flow conductance rate constant power dissipation

This analogy is already well known. Schnakenberg used it in his book Thermodynamic Network Analysis of Biological Systems. So, our main goal is to formalize and exploit it. This analogy extends from systems in equilibrium to the more interesting case of nonequilibrium steady states, which are the main topic of our paper.

Earlier, Brendan and I introduced a way to ‘black box’ a circuit and define the relation it determines between potential-current pairs at the input and output terminals. This relation describes the circuit’s external behavior as seen by an observer who can only perform measurements at the terminals.

An important fact is that black boxing is ‘compositional’: if one builds a circuit from smaller pieces, the external behavior of the whole circuit can be determined from the external behaviors of the pieces. For category theorists, this means that black boxing is a functor!

Our new paper with Blake develops a similar ‘black box functor’ for detailed balanced Markov processes, and relates it to the earlier one for circuits.

When you black box a detailed balanced Markov process, you get the relation between population–flow pairs at the terminals. (By the ‘flow at a terminal’, we more precisely mean the net population outflow.) This relation holds not only in equilibrium, but also in any nonequilibrium steady state. Thus, black boxing an open detailed balanced Markov process gives its steady state dynamics as seen by an observer who can only measure populations and flows at the terminals.

### The principle of minimum dissipation

At least since the work of Prigogine, it’s been widely accepted that a large class of systems minimize entropy production in a nonequilibrium steady state. But people still fight about the the precise boundary of this class of systems, and even the meaning of this ‘principle of minimum entropy production’.

For detailed balanced open Markov processes, we show that a quantity we call the ‘dissipation’ is minimized in any steady state. This is a quadratic function of the populations and flows, analogous to the power dissipation of a circuit made of resistors. We make no claim that this quadratic function actually deserves to be called ‘entropy production’. Indeed, Schnakenberg has convincingly argued that they are only approximately equal.

But still, the ‘dissipation’ function is very natural and useful—and Prigogine’s so-called ‘entropy production’ is also a quadratic function.

### Black boxing

I’ve already mentioned the category ${DetBalMark},$ where a morphism is an open detailed balanced Markov process. But our paper needs two more categories to tell its story! There’s the category of circuits, and the category of linear relations.

A morphism in the category ${Circ}$ is an open electrical circuit made of resistors: that is, a graph with each edge labelled by a ‘conductance’ $c_e > 0,$ together with specified input and output nodes:

A morphism in the category ${LinRel}$ is a linear relation $L : U \rightsquigarrow V$ between finite-dimensional real vector spaces $U$ and $V.$ This is nothing but a linear subspace $L \subseteq U \oplus V.$ Just as relations generalize functions, linear relations generalize linear functions!

In our previous paper, Brendan and I introduced these two categories and a functor between them, the ‘black box functor’:

$\blacksquare \colon {Circ} \to {LinRel}$

The idea is that any circuit determines a linear relation between the potentials and net current flows at the inputs and outputs. This relation describes the behavior of a circuit of resistors as seen from outside.

Our new paper introduces a black box functor for detailed balanced Markov processes:

$\square \colon {DetBalMark} \to {LinRel}$

We draw this functor as a white box merely to distinguish it from the other black box functor. The functor $\square$ maps any detailed balanced Markov process to the linear relation obeyed by populations and flows at the inputs and outputs in a steady state. In short, it describes the steady state behavior of the Markov process ‘as seen from outside’.

How do we manage to black box detailed balanced Markov processes? We do it using the analogy with circuits!

### The analogy becomes a functor

Every analogy wants to be a functor. So, we make the analogy between detailed balanced Markov processes and circuits precise by turning it into a functor:

$K : {DetBalMark} \to {Circ}$

This functor converts any open detailed balanced Markov process into an open electrical circuit made of resistors. This circuit is carefully chosen to reflect the steady-state behavior of the Markov process. Its underlying graph is the same as that of the Markov process. So, the ‘states’ of the Markov process are the same as the ‘nodes’ of the circuit.

Both the equilibrium populations at states of the Markov process and the rate constants labelling edges of the Markov process are used to compute the conductances of edges of this circuit. In the simple case where the Markov process has exactly one edge from any state $i$ to any state $j,$ the rule is this:

$C_{i j} = H_{i j} q_j$

where:

• $q_j$ is the equilibrium population of the $j$th state of the Markov process,
• $H_{i j}$ is the rate constant for the edge from the $j$th state to the $i$th state of the Markov process, and
• $C_{i j}$ is the conductance (that is, the reciprocal of the resistance) of the wire from the $j$th node to the $i$th node of the resulting circuit.

The detailed balance condition for Markov processes says precisely that the matrix $C_{i j}$ is symmetric! This is just right for an electrical circuit made of resistors, since it means that the resistance of the wire from node $i$ to node $j$ equals the resistance of the same wire in the reverse direction, from node $j$ to node $i.$

### A triangle of functors

If you paid careful attention, you’ll have noticed that I’ve described a triangle of functors:

And if you’ve got the tao of category theory flowing in your veins, you’ll be wondering if this diagram commutes.

In fact, this triangle of functors does not commute! However, a general lesson of category theory is that we should only expect diagrams of functors to commute up to natural isomorphism, and this is what happens here:

The natural transformation $\alpha$ ‘corrects’ the black box functor for resistors to give the one for detailed balanced Markov processes.

The functors $\square$ and $\blacksquare \circ K$ are actually equal on objects. An object in ${DetBalMark}$ is a finite set $X$ with each element $i \in X$ labelled a positive populations $q_i.$ Both functors map this object to the vector space $\mathbb{R}^X \oplus \mathbb{R}^X.$ For the functor $\square,$ we think of this as a space of population-flow pairs. For the functor $\blacksquare \circ K,$ we think of it as a space of potential-current pairs. The natural transformation $\alpha$ then gives a linear relation

$\alpha_{X,q} : \mathbb{R}^X \oplus \mathbb{R}^X \rightsquigarrow \mathbb{R}^X \oplus \mathbb{R}^X$

in fact an isomorphism of vector spaces, which converts potential-current pairs into population-flow pairs in a manner that depends on the $q_i.$ I’ll skip the formula; it’s in the paper.

But here’s the key point. The naturality of $\alpha$ actually allows us to reduce the problem of computing the functor $\square$ to the problem of computing $\blacksquare.$ Suppose

$M \colon (X,q) \to (Y,r)$

is any morphism in ${DetBalMark}.$ The object $(X,q)$ is some finite set $X$ labelled by populations $q,$ and $(Y,r)$ is some finite set $Y$ labelled by populations $r.$ Then the naturality of $\alpha$ means that this square commutes:

Since $\alpha_{X,q}$ and $\alpha_{Y,r}$ are isomorphisms, we can solve for the functor $\square$ as follows:

$\square(M) = \alpha_Y \circ \blacksquare K(M) \circ \alpha_X^{-1}$

This equation has a clear intuitive meaning! It says that to compute the behavior of a detailed balanced Markov process, namely $\square(f),$ we convert it into a circuit made of resistors and compute the behavior of that, namely $\blacksquare K(f).$ This is not equal to the behavior of the Markov process, but we can compute that behavior by converting the input populations and flows into potentials and currents, feeding them into our circuit, and then converting the outputs back into populations and flows.

### What we really did

So that’s a sketch of what we did, and I hope you ask questions if it’s not clear. But I also hope you read our paper! Here’s what we actually do in there. After an introduction and summary of results:

• Section 3 defines open Markov processes and the open master equation.
• Section 4 introduces detailed balance for open Markov processes.
• Section 5 recalls the principle of minimum power for open circuits made of linear resistors, and explains how to black box them.
• Section 6 introduces the principle of minimum dissipation for open detailed balanced Markov processes, and describes how to black box these.
• Section 7 states the analogy between circuits and detailed balanced Markov processes in a formal way.
• Section 8 describes how to compose open Markov processes, making them into the morphisms of a category.
• Section 9 does the same for detailed balanced Markov processes.
• Section 10 describes the ‘black box functor’ that sends any open detailed balanced Markov process to the linear relation describing its external behavior, and recalls the similar functor for circuits.
• Section 11 makes the analogy between between open detailed balanced Markov processes and open circuits even more formal, by making it into a functor. We prove that together with the two black box functors, this forms a triangle that commutes up to natural isomorphism.
• Section 12 is about geometric aspects of this theory. We show that linear relations in the image of these black box functors are Lagrangian relations between symplectic vector spaces. We also show that the master equation can be seen as a gradient flow equation.
• Section 13 is a summary of what we have learned.

Finally, Appendix A is a quick tutorial on decorated cospans. This is a key mathematical tool in our work, developed by Brendan in an earlier paper.

Posted at January 12, 2016 12:28 PM UTC

TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2858

## 2 Comments & 0 Trackbacks

### Re: A Compositional Framework for Markov Processes

in an open Markov process, probability is not conserved: it can flow in or out at the inputs and outputs.

You lost me here. I’m looking at your example and I think I see how the detailed balance condition holds. On the right, for instance, we have $1\cdot 4$ flowing to the right and $2\cdot 2$ flowing to the left, which are equal.

Is it correct that the detailed balance condition implies that the populations are in equilibrium? Since at node $i$ there is the same amount flowing out to $j$ as flowing in from $j$, for every $j$ individually.

Now none of this seems to have anything to do with choosing some states to call “input” and some to call “output”. Okay, you can choose some nodes and give them labels. But what is this “probability” thing? The only thing I can think of for it to mean is to sum up the populations and divide each of them by the total, so that in your example the total is $6+1+2+\frac{1}{2} = \frac{19}{2}$, and the populations would get normalized to probabilities of $\frac{12}{19}, \frac{2}{19}, \frac{4}{19}, \frac{1}{19}$. But dividing all the populations by the same number doesn’t affect the detailed balance condition or the fact that they are in equilibrium, does it? So what does it mean to say that probability is not conserved?

(By the way, \leadsto is apparently not supported in iTeX, but I think you can use \rightsquigarrow.)

Posted by: Mike Shulman on January 13, 2016 5:32 AM | Permalink | Reply to this

### Re: A Compositional Framework for Markov Processes

Mike wrote:

John wrote:

in an open Markov process, probability is not conserved: it can flow in or out at the inputs and outputs.

You lost me here.

Sorry. I’ll say some stuff that may help.

Is it correct that the detailed balance condition implies that the populations are in equilibrium?

Yes. Detailed balance is a stronger condition, which turns out to be very commonly obeyed in physics, and has enormous consequences.

Since at node $i$ there is the same amount flowing out to $j$ as flowing in from $j$, for every $j$ individually.

Right.

Now none of this seems to have anything to do with choosing some states to call “input” and some to call “output”.

No, none of this depends on that.

But what is this “probability” thing?

I only mentioned the word ‘probability’ to explain why, unlike most people working on Markov processes, we don’t talk about probability distributions. I was trying to say that in an open Markov process, the total population is not conserved: it can flow in or out at the inputs and outputs. So, even if we normalize it to 1 at some time, it won’t stay normalized.

To clarify that, I think I should tell you what an open Markov process is. My introduction was not really trying to explain that in any detail.

An open Markov process consists of a finite set $X$ of states, a subset $B \subseteq X$ of boundary states, and an infinitesimal stochastic linear operator $H: \mathbb{R}^X \to \mathbb{R}^X,$ meaning one whose matrix obeys

$H_{i j} \geq 0$

for all $i \neq j$ and

$\sum_{i \in X} H_{i j} = 0$

for all $j \in X$.

I’ll explain these two conditions in a minute.

For each $i \in X$ we introduce a population $p_i \in [0,\infty),$ which may also depend on time, $t \in \mathbb{R}$. We call the resulting time-dependent function $p : X \to [0,\infty)$ the population distribution.

We demand that the population distribution evolves in time according to the open master equation. This says that the master equation

$\displaystyle{ \frac{d p_i(t)}{d t} = \sum_{j \in X} H_{i j}p_j}$

holds for all $i \in X-B$, while

$p_i(t) = b_i(t)$

for all $i \in B$.

So, the populations $p_i$ obey a linear differential equation at states $i$ that are not in the boundary, but they are specified ‘by the user’ to be chosen functions of time $b_i$ at the boundary states.

The off-diagonal entries, $H_{i j}$ with $i \neq j$, are the rates at which population hops from the $j$th to the $i$th state. This lets us understand the definition of an infinitesimal stochastic operator. The first condition,

$H_{i j} \geq 0$

for all $i \neq j$, says that the rate for population to transition from one state to another is non-negative. The second:

$\sum_{i \in X} H_{i j} = 0$

for all $j$, says that population is conserved, at least if there are no boundary states. Population can flow in or out at boundary states, since the master equation doesn’t hold there!

A steady state is a solution of the open master equation that does not change with time. A steady state is an equilibrium if it obeys the master equation

$\displaystyle{ \frac{d p_i(t)}{d t} = \sum_j H_{i j}p_j}$

for all states $i$, even boundary states.

So, an equilibrium obeys the master equation at all states, while for a steady state this may not be true at the boundary states. In the former, the total population is conserved! In the latter, we imagine that population can flow in or out at the boundary.

We say an equilibrium $q : X \to [0,\infty)$ of a Markov process is detailed balanced if the rate at which population flows from the $i$th state to the $j$th state is equal to the rate at which it flows from the $j$th state to the $i$th:

$H_{j i}q_i = H_{i j}q_j$

for all $i, j \in X$.

A detailed balanced open Markov process is an open Markov process equipped with a specific detailed balanced equilibrium $q$.

Now what was that business about ‘inputs’ and ‘outputs’? Well, if we go slightly further and choose a cospan of finite sets

$I \stackrel{s}{\rightarrow} X \stackrel{t}{\leftarrow} O$

with

 im(s) \cup im(t) = B$then we make our open detailed balanced Markov process into a morphism from$I$to$O$. This is a morphism in the category we call$DetBalMark\$.

Posted by: John Baez on January 17, 2016 3:59 AM | Permalink | Reply to this

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text:
# Fast Allocation and Deallocation with an Improved Buddy System

## Gerth Stølting Brodal, Erik D. Demaine, and J. Ian Munro

Technical Report, ALCOMFT-TR-03-3, ALCOM-FT, 15 pages, May 2003.

## Abstract

We propose several modifications to the binary buddy system for managing dynamic allocation of memory blocks whose sizes are powers of two. The standard buddy system allocates and deallocates blocks in Θ(lg n) time in the worst case (and on an amortized basis), where n is the size of the memory. We present three schemes that improve the running time to O(1) time, where the time bound for deallocation is amortized for the first two schemes. The first scheme uses just one more word of memory than the standard buddy system, but may result in greater fragmentation than necessary. The second and third schemes have essentially the same fragmentation as the standard buddy system, and use O(2(1 + \sqrt{lg n}) lg lg n) bits of auxiliary storage, which is ω(lgk n) but o(n^ε) for all k ≥ 1 and ε > 0. Finally, we present simulation results estimating the effect of the excess fragmentation in the first scheme.

Online version

alcomft-tr-03-3.pdf (193 Kb)

BIBTEX entry

@techreport{alcomft-tr-03-3,
author = "Gerth St{\o}lting Brodal and Erik D. Demaine and J. Ian Munro",
institution = "ALCOM-FT",
month = "May",
number = "ALCOMFT-TR-03-3",
pages = "15",
title = "Fast Allocation and Deallocation with an Improved Buddy System",
year = "2003"
}


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# Composition of composable 2-cells in a 2-category is unambiguously defined?

I believe the following nice statement is true, but I cannot find a reference or proof it myself.

## In a 2-category(i.e., bicategory), the composition of composable 2-cells is unambiguously defined.

Where 2-cells are 2-commutative triangles (other shape will not give more information), "composable" is not easy to say in simple words but it is easy to imagine.

Special cases of this statement include vertical composition of 2-morphisms are associative and the exchange law of vertical composition and horizontal composition.

Any comments or references would be appreciated. Thank you guys!

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A well-known reference:

John Power, A 2-categorical pasting theorem, Journal of Algebra Volume 129, Issue 2, March 1990, Pages 439-445.

See the nLab page on pasting diagrams for more.

Power's paper just deals with strict 2-categories, if I remember correctly.

Edit: Following Todd's comment, here's the reference to Verity's thesis, where the bicategorical case is covered. The original thesis came out in 1992, but it has recently been reprinted:

Dominic Verity, Enriched categories, internal categories and change of base, Reprints in Theory and Applications of Categories No. 20 (2011) pp 1-266.

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Yes, just strict 2-categories. There's also a bicategorical pasting theorem in Dominic Verity's thesis. –  Todd Trimble Nov 23 '11 at 12:23
I just read the paper of Power. The most part of the paper is graph theory, i.e. how to phrase the theory precisely. The actual proof is a very short induction, and the induction part which have to do is left to reader"It is routine to apply the axioms for 2-category and ...". So, I think it works for weak 2-category too. –  Ma Ming Nov 23 '11 at 15:13
Yes, I would imagine so. –  Tom Leinster Nov 23 '11 at 16:16
A Weak 2-category or bicategory (I guess you mean) is biequivalent (doing by a pseudofunctor that locally is a equivalence) to a 2-category (strictifiable) by a generalized Yoneda Lemma, and then diagram of 2-cells is commutative (or coherent) in a bicategory iff it is commutative in its 2-category strictifications, then the coherence theorem (for 2-cells diagrams) on 2-category is automaticaly generalizable to bicategory. See T. Leinster article: front.math.ucdavis.edu/9810.5017 –  Buschi Sergio Nov 23 '11 at 19:08
@Buschi Sergio: that's the rough idea, of course. It's another matter to nail down the precise details. See Verity's thesis for a full accounting. –  Todd Trimble Nov 24 '11 at 1:13

I think Tom's answer is what was being looked for, but I just thought I would mention that there is another way of stating and proving such a theorem, namely using string diagrams instead of pasting diagrams. The corresponding theorem for string diagrams in a monoidal category was proven by Joyal and Street; the one for bicategories is an easy extension (but I don't know of anywhere that it appears in print). There are some references at the nLab page.

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Se also "General associativity and general composition for double categories - Dawson, Robert; Pare, Robert"

Cahiers de Topologie et Géométrie Différentielle Catégoriques, 34 no. 1 (1993), p. 57-79

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For simplicial compositions see Richard Steiner's paper

The algebraic structure of the universal complicial sets arXiv:1009.3384v1

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