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http://www.ck12.org/geometry/Volume-of-Prisms/lesson/Volume-of-Prisms/r12/ | <img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
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# Volume of Prisms
## Use the formula V = Bh
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Volume of Prisms
Let's take another look at Jillian's box.
In the last Concept, you learned how to count unit cubes to figure out the volume of different prisms. Well, there is an easier way. We can use a formula to calculate the volume of a prism. Here are the dimensions of Jillian's box once again.
Jillian's box is a rectangular prism and has the following dimensions: 7" x 6" x 4".
How can we use a formula to calculate the volume of this prism?
Pay attention and this Concept will teach you all that you need to know.
### Guidance
Looking at all of those cubes is a simple, easy way to understand volume. If you can count the cubes, you can figure out the volume. However, not all of the prisms that you will work with will have the cubes drawn in. In this Concept, you will learn how to figure out the volume of a prism when there aren’t any cubes drawn inside it.
How can we figure out the volume of a prism without counting cubes?
Here we have the dimensions written on a rectangular prism. This prism has a height of 5 inches, a width of three inches and a length of four inches.
You can see that a few cubes have been drawn in to show you that if we continued filling the cubes that they would be four cubes across by three cubes wide, and we would build them five cubes high.
That’s right! Here is how it works.
V=Bh
B\begin{align*}B\end{align*} means the area of the base and h\begin{align*}h\end{align*} means the height.
The area of the base is length times width.
AhV=3×4=12=5=12×5=60
The volume is 60 cubic inches or in3\begin{align*}in^3\end{align*}.
V=Bh
The area of the base is 2 ×\begin{align*}\times\end{align*} 8 = 16
The height is 3 inches.
VV=16×3=48 in3
The volume of this rectangular prism is 48 in3\begin{align*}48 \ in^3\end{align*}.
How can we find the volume of a triangular prism?
We can use the same formula for finding the volume of the triangular prism. Except this time, the area of the base is a triangle and not a rectangle.
V=Bh
To find the volume of a triangular prism, we multiply the area of the base (B)\begin{align*}(B)\end{align*} with the height of the prism.
To find the area of a triangular base we use the formula for area of a triangle.
AAAAVVVV=12bh=12(15×6)=12(90)=45 sq. units=Bh=(45)h=45(2)=90 cubic centimeters or cm3
The volume of the prism is 90 cm3\begin{align*}90 \ cm^3\end{align*}.
Now that you know how to find the volume of prisms using a formula, it is time to practice.
#### Example A
Solution: 125in3\begin{align*}125 in^3\end{align*}
#### Example B
Solution: 450in3\begin{align*}450 in^3\end{align*}
#### Example C
Solution: 17.5cm3\begin{align*} 17.5 cm^3\end{align*}
Do you know how to use the formula for finding the volume of a prism? Here is the original problem once again.
Let's take another look at Jillian's box.
In the last Concept, you learned how to count unit cubes to figure out the volume of different prisms. Well, there is an easier way. We can use a formula to calculate the volume of a prism. Here are the dimensions of Jillian's box once again.
Jillian's box is a rectangular prism and has the following dimensions: 7" x 6" x 4".
How can we use a formula to calculate the volume of this prism?
V=Bh\begin{align*}V = Bh\end{align*}
Now we can substitute in the given values for length, width and height.
V=(7×6)(4)\begin{align*}V = (7 \times 6)(4)\end{align*}
The volume of Jillian's box is 168in3\begin{align*} 168 in^3\end{align*}.
### Vocabulary
Here are the vocabulary words in this Concept.
Surface area
the outer covering of a solid figure-calculated by adding up the sum of the areas of all of the faces and bases of a prism.
Net
diagram that shows a “flattened” version of a solid. Each face and base is shown with all of its dimensions in a net. A net can also serve as a pattern to build a three-dimensional solid.
Triangular Prism
a solid which has two congruent parallel triangular bases and faces that are rectangles.
Rectangular Prism
a solid which has rectangles for bases and faces.
Volume
the amount of space inside a solid figure
### Guided Practice
Here is one for you to try on your own.
To find the volume of a prism, we use the following formula.
V=Bh\begin{align*}V = Bh\end{align*}
Now we substitute in the given values.
V=(16×9)(4)\begin{align*}V = (16 \times 9)(4)\end{align*}
V=576cm3\begin{align*}V = 576 cm^3\end{align*}
### Video Review
Here is a video for review.
### Practice
Directions: Find the volume of each rectangular prism. Remember to label your answer in cubic units.
1. Length = 5 in, width = 3 in, height = 4 in
2. Length = 7 m, width = 6 m, height = 5 m
3. Length = 8 cm, width = 4 cm, height = 9 cm
4. Length = 8 cm, width = 4 cm, height = 12 cm
5. Length = 10 ft, width = 5 ft, height = 6 ft
6. Length = 9 m, width = 8 m, height = 11 m
7. Length = 5.5 in, width = 3 in, height = 5 in
8. Length = 6.6 cm, width = 5 cm, height = 7 cm
9. Length = 7 ft, width = 4 ft, height = 6 ft
10. Length = 15 m, width = 8 m, height = 10 m
Directions: Find the volume of each triangular prism. Remember that h\begin{align*}h\end{align*} means the height of the triangular base and H\begin{align*}H\end{align*} means the height of the whole prism.
11. b=6 in, h=4 in, H=5 in\begin{align*}b = 6 \ in, \ h = 4 \ in, \ H = 5 \ in\end{align*}
12. b=7 in, h=5 in, H=9 in\begin{align*}b = 7 \ in, \ h = 5 \ in, \ H = 9 \ in\end{align*}
13. b=10 m, h=8 m, H=9 m\begin{align*}b = 10 \ m, \ h = 8 \ m, \ H = 9 \ m\end{align*}
14. b=12 m, h=10 m, H=13 m\begin{align*}b = 12 \ m, \ h = 10 \ m, \ H = 13 \ m\end{align*}
15. b=8 cm, h=6 cm, H=9 cm\begin{align*}b = 8 \ cm, \ h = 6 \ cm, \ H = 9 \ cm\end{align*}
Directions: Answer true or false for each of the following questions.
16. Volume is the amount of space that a figure can hold inside it.
17. The volume of a rectangular prism is always greater than the volume of a cube.
18. The volume of a triangular prism is less than a rectangular prism with the same size base.
19. A painter would need to know the surface area of a house to do his/her job correctly.
20. If Marcus is covering his book with a book cover, Marcus is covering the surface area of the book.
### Vocabulary Language: English
Net
Net
A net is a diagram that shows a “flattened” view of a solid. In a net, each face and base is shown with all of its dimensions. A net can also serve as a pattern to build a three-dimensional solid.
Rectangular Prism
Rectangular Prism
A rectangular prism is a prism made up of two rectangular bases and four rectangular faces.
Surface Area
Surface Area
Surface area is the total area of all of the surfaces of a three-dimensional object.
Triangular Prism
Triangular Prism
A triangular prism is a prism made up of two triangular bases and three rectangular faces. | {"extraction_info": {"found_math": true, "script_math_tex": 23, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8710050582885742, "perplexity": 699.5835340093909}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701160822.87/warc/CC-MAIN-20160205193920-00166-ip-10-236-182-209.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/27348-integrales.html | # Math Help - Integrales
1. ## Integrales
Hello Guys,
I don't know how to solve this:
t/ (racine) 1 + t^2 ==> for this i did this: ?
Thanks
2. are you not fimmilar with inegrals in the form of $\int f(x)f'(x) dx$ ?
use should know and be able to easily prove that $\int f(x)f'(x) dx= \frac{1}{2} (f(x))^2$
Can you apply this to your integral ?
3. Originally Posted by bobak
are you not fimmilar with inegrals in the form of $\int f(x)f'(x) dx$ ?
use should know and be able to easily prove that $\int f(x)f'(x) dx= \frac{1}{2} (f(x))^2$
Can you apply this to your integral ?
ah so , Lnx/x = 1/2 Lnx?
4. Originally Posted by iceman1
ah so , Lnx/x = 1/2 Lnx?
please be careful with your notation, what you wrote makes no sense.
I am sure you meant to write.
$\int \frac { \ln x}{x} dx= \frac{1}{2} (\ln x)^2 + C$
5. Originally Posted by bobak
please be careful with your notation, what you wrote makes no sense.
I am sure you meant to write.
$\int \frac { \ln x}{x} dx= \frac{1}{2} (\ln x)^2 + C$
Yeah sorry but i don't know how to write a mathematics letters
so
==> this (integrale) 1/2 Ln (x)^2
==> 1/2 [Ln(1)^1 -Ln(x)^e)]
right?
6. you did not substitute the limits in the correct order. and you also confused
$
\int \frac { \ln x}{x} dx= \frac{1}{2} (\ln x)^2 + C
$
for $
\int \frac { \ln x}{x} dx= \frac{1}{2} (\ln x)^x + C
$
also you should know that $\ln 1 = 0$ and $\ln e = 1$
7. Originally Posted by bobak
you did not substitute the limits in the correct order. and you also confused
$
\int \frac { \ln x}{x} dx= \frac{1}{2} (\ln x)^2 + C
$
for $
\int \frac { \ln x}{x} dx= \frac{1}{2} (\ln x)^x + C
$
also you should know that $\ln 1 = 0$ and $\ln e = 1$
yay so the final results is 1/2 right?
what abt the other one?
i move it and change the puissance signe like this:
with integral from sure
8. yeah the answer is 1/2.
for the next one you should recognise that you have something in the form of
$\int kf'(x)[f(x)]^n dx = k \frac {[f(x)]^{n+1}}{n+1}$ can you finish it off ?
9. Originally Posted by bobak
for the next one you should recognise that you have something in the form of
$\int kf'(x)[f(x)]^n dx = k \frac {[f(x)]^{n+1}}{n+1}$ can you finish it off ?
ok i'll try:
2[t (1 + t^2)]^+1/2 divided by (1/2) (with integrales)
right?
10. almost check you got the value of the constant correct and differentiate to result to check your answer.
11. Using many formulae for these problems, it's an incredibly bad idea.
These are routine problems, so we don't need formulae to make this work.
Originally Posted by iceman1
Substitute $u^2=1+t^2.$
(Correct english forms for $\sqrt{~~}$ and $\int$ are square root and integral.)
12. Originally Posted by Krizalid
Using many formulae for these problems, it's an incredibly bad idea.
it not about using a formula, it more about identifying a standard form.
13. ok Thanks Guys
i have another thing:
f(x)= e^(-x) (+x) -1
Limit(+infinity)e^(-x) (+x) -1 = + infinity
Limit(-infinity)e^(-x) (+x) -1 = + infinity
right? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 19, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9405838251113892, "perplexity": 1500.6559760244597}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257824395.52/warc/CC-MAIN-20160723071024-00140-ip-10-185-27-174.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/49786/form-of-primesprime-plus-a-power-of-2/49787 | # form of primes:prime plus a power of 2?
is every prime p equals another prime p' plus or minus a power of 2? p=p'+/-2^n? are there infinitely many primes not of this form?
-
To make your questions better, you might want to include some background or motivation. Why are you interested? What have you tried already? etc. – Gjergji Zaimi Dec 18 '10 at 10:43
one can look at mathoverflow.net/questions/49751/… – asterios gantzounis Dec 18 '10 at 10:47
i changed the question according to the answer that Gjergji gave me – asterios gantzounis Dec 18 '10 at 10:52
is this allowed? – asterios gantzounis Dec 18 '10 at 11:02
It's very much discouraged since it makes the thread look like nonsense. Changing your question to make an existing answer a non-answer is something like inviting to treat somebody to dinner, then slipping out after the meal, sticking them with the bill. – Anton Geraschenko Dec 22 '10 at 18:01
## 1 Answer
127 and 331 are counterexamples. It was a conjecture of Polignac that every odd number can be written as a sum of an odd prime and a power of two, but many counterexamples have been found. They are called "obstinate numbers". Erdos has proved that there is an infinite arithmetic progression of obstinate numbers.
Edit (response to the added question): There will be infinitely many such prime counterexamples as a corollary to Erdos' theorem and Dirichlet's theorem on arithmetic progressions. See "Not always buried deep: selections of problems from analytic and combinatorial number theory" by P. Pollack.
-
if i allow p=p'-2^n too? – asterios gantzounis Dec 18 '10 at 10:20
Yes! Apparently a counterexample to that was given by Cohen and Selfridge. 47,867,742,232,066,880,047,611,079 and the proof is left as an exercise :) – Gjergji Zaimi Dec 18 '10 at 10:24
This last number is a counterexample to being a sum or difference of a prime and a power of 2, by the way. – Gjergji Zaimi Dec 18 '10 at 10:26
do you have a good answer to this closed question too?mathoverflow.net/questions/49730/twin-primes-etc-closed – asterios gantzounis Dec 18 '10 at 10:29
and a kind of joke :if i allow p=p'+/-2^m+/-2^n? – asterios gantzounis Dec 18 '10 at 10:38 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8546509146690369, "perplexity": 896.723024742217}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931006637.79/warc/CC-MAIN-20141125155646-00064-ip-10-235-23-156.ec2.internal.warc.gz"} |
http://mathhelpforum.com/differential-geometry/191599-fourier-transform-odd-funktion-print.html | # Fourier transform - odd funktion
• November 10th 2011, 01:55 PM
liquidFuzz
Fourier transform - odd funktion
I'm transforming f(x) = 1-x trying to do so in an even and odd fashion... I think I get it right for the even, but I get stuck while tinkering with the odd one.
The period is 2, odd distribution.
$\dispalystyle b_n = \frac{2}{T} \int_0^2 (1-t) \sin (n \Omega t) dt$ ; $\Omega = \frac{2 \pi}{T} = \pi$ & $T = 2$
$\dispalystyle b_n = \int_0^2 (1-t) \sin (n \pi t) dt$ = [integrate by parts] = $[1-1] - \int_0^2 \cos (n \pi t) dt$
Here I get stuck because $sin(2\pi n ) = sin (0) = 0$.
If anyone could point out what I'm overseeing..?
• November 11th 2011, 06:05 AM
CaptainBlack
Re: Fourier transform - odd funktion
Quote:
Originally Posted by liquidFuzz
I'm transforming f(x) = 1-x trying to do so in an even and odd fashion... I think I get it right for the even, but I get stuck while tinkering with the odd one.
The period is 2, odd distribution.
$\dispalystyle b_n = \frac{2}{T} \int_0^2 (1-t) \sin (n \Omega t) dt$ ; $\Omega = \frac{2 \pi}{T} = \pi$ & $T = 2$
$\dispalystyle b_n = \int_0^2 (1-t) \sin (n \pi t) dt$ = [integrate by parts] = $[1-1] - \int_0^2 \cos (n \pi t) dt$
Here I get stuck because $sin(2\pi n ) = sin (0) = 0$.
If anyone could point out what I'm overseeing..?
Do you integration by parts again more carefully.
CB
• November 11th 2011, 12:23 PM
liquidFuzz
Re: Fourier transform - odd funktion
No, I can't nail it...
What if I change the interval over wish integrate to [-1,0] + [0,1]. Maybe that would make it easier, but will my series convert to the odd distribution of 1-t?
Edit: Oh crap, I just realised something. I think my series could be right. My conclusion on the contrary... I was so preoccupied with the fact that $sin (n \pi) = 0$ that I sorta forgot the variable t.
Does this compute..?
$\sum_{n=1}^{\infinity} \frac{2 \sin (n \pi t)}{n\pi}$
• November 11th 2011, 08:05 PM
CaptainBlack
Re: Fourier transform - odd funktion
Quote:
Originally Posted by liquidFuzz
No, I can't nail it...
What if I change the interval over wish integrate to [-1,0] + [0,1]. Maybe that would make it easier, but will my series convert to the odd distribution of 1-t?
Edit: Oh crap, I just realised something. I think my series could be right. My conclusion on the contrary... I was so preoccupied with the fact that $sin (n \pi) = 0$ that I sorta forgot the variable t.
Does this compute..?
$\sum_{n=1}^{\infinity} \frac{2 \sin (n \pi t)}{n\pi}$
If I recall correctly yes
CB
• November 12th 2011, 06:15 AM
liquidFuzz
1 Attachment(s)
Re: Fourier transform - odd funktion
Quote:
Originally Posted by CaptainBlack
If I recall correctly yes
CB
I plotted the series in MatLab and got a result that points in an other direction. :-(
Here's the plot I got and the MatLab code I wrote.
MatLab
Code:
N = 20; x = (-50:50)/20; %interval : [-2.5,2.5] f = ones(1,101); %Odd fourier, no a_0 and a_i included. for i = 1:1:N %from 1 to N f = f + (2 * sin(pi*i*x)) / (i * pi); %sum all terms over N iterations. end plot(x,f)
The function I was aiming for is 1-t and it should hold the points [0,1] and [2,-1] and then jump to next 'slope'. Eh, right?
I guess my question is, is it the series or is the MatLab tinkering that's wrong..?
• November 12th 2011, 06:38 AM
CaptainBlack
Re: Fourier transform - odd funktion
Quote:
Originally Posted by liquidFuzz
I plotted the series in MatLab and got a result that points in an other direction. :-(
Here's the plot I got and the MatLab code I wrote.
MatLab
Code:
N = 20; x = (-50:50)/20; %interval : [-2.5,2.5] f = ones(1,101); %Odd fourier, no a_0 and a_i included. for i = 1:1:N %from 1 to N f = f + (2 * sin(pi*i*x)) / (i * pi); %sum all terms over N iterations. end plot(x,f)
The function I was aiming for is 1-t and it should hold the points [0,1] and [2,-1] and then jump to next 'slope'. Eh, right?
I guess my question is, is it the series or is the MatLab tinkering that's wrong..?
Why did you initialise f to one?
Try:
f = zeros(1,101)
CB
• November 12th 2011, 06:45 AM
liquidFuzz
Re: Fourier transform - odd funktion
Eh, good question..?
Thanks for your help and patience dude! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 16, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9089562296867371, "perplexity": 2647.2335932695496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042986022.41/warc/CC-MAIN-20150728002306-00291-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://mathematica.stackexchange.com/posts/126522/revisions | 2 explaining what case
In http://link.springer.com/chapter/10.1007%2F3-540-68339-9_14#page-7 is presented a method for finding small solutions to multivariate polynomials. (Although here he mentions that it "is not rigorous", several other papers have cited it and make use of it in practice.) Is there a function or convenient snippet for accomplishing this in Mathematica? Or if not, perhaps at least a reference for how to write this slightly more algorithmically?
Edit: In particular, I need this for the case of solving quadratics in two variables over large composite fields of unknown factorization, efficiently.
In http://link.springer.com/chapter/10.1007%2F3-540-68339-9_14#page-7 is presented a method for finding small solutions to multivariate polynomials. (Although here he mentions that it "is not rigorous", several other papers have cited it and make use of it in practice.) Is there a function or convenient snippet for accomplishing this in Mathematica? Or if not, perhaps at least a reference for how to write this slightly more algorithmically?
In http://link.springer.com/chapter/10.1007%2F3-540-68339-9_14#page-7 is presented a method for finding small solutions to multivariate polynomials. (Although here he mentions that it "is not rigorous", several other papers have cited it and make use of it in practice.) Is there a function or convenient snippet for accomplishing this in Mathematica? Or if not, perhaps at least a reference for how to write this slightly more algorithmically?
Edit: In particular, I need this for the case of solving quadratics in two variables over large composite fields of unknown factorization, efficiently.
1
# Coppersmith method of small integer solutions to multivariate polynomials
In http://link.springer.com/chapter/10.1007%2F3-540-68339-9_14#page-7 is presented a method for finding small solutions to multivariate polynomials. (Although here he mentions that it "is not rigorous", several other papers have cited it and make use of it in practice.) Is there a function or convenient snippet for accomplishing this in Mathematica? Or if not, perhaps at least a reference for how to write this slightly more algorithmically? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8091674447059631, "perplexity": 530.0116165653612}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514576047.85/warc/CC-MAIN-20190923043830-20190923065830-00528.warc.gz"} |
http://abstract.ups.edu/aata/exercises-isomorph.html | $\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{\nmid} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \newcommand{\transpose}{\text{t}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&}$
## Section9.3Exercises
###### 1
Prove that $\mathbb Z \cong n \mathbb Z$ for $n \neq 0\text{.}$
###### 2
Prove that ${\mathbb C}^\ast$ is isomorphic to the subgroup of $GL_2( {\mathbb R} )$ consisting of matrices of the form
\begin{equation*} \begin{pmatrix} a & b \\ -b & a \end{pmatrix}. \end{equation*}
###### 3
Prove or disprove: $U(8) \cong {\mathbb Z}_4\text{.}$
###### 4
Prove that $U(8)$ is isomorphic to the group of matrices
\begin{equation*} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}. \end{equation*}
###### 5
Show that $U(5)$ is isomorphic to $U(10)\text{,}$ but $U(12)$ is not.
###### 6
Show that the $n$th roots of unity are isomorphic to ${\mathbb Z}_n\text{.}$
###### 7
Show that any cyclic group of order $n$ is isomorphic to ${\mathbb Z}_n\text{.}$
###### 8
Prove that ${\mathbb Q}$ is not isomorphic to ${\mathbb Z}\text{.}$
###### 9
Let $G = {\mathbb R} \setminus \{ -1 \}$ and define a binary operation on $G$ by
\begin{equation*} a \ast b = a + b + ab. \end{equation*}
Prove that $G$ is a group under this operation. Show that $(G, *)$ is isomorphic to the multiplicative group of nonzero real numbers.
###### 10
Show that the matrices
\begin{align*} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \quad \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \quad \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}\\ \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \quad \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \quad \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix} \end{align*}
form a group. Find an isomorphism of $G$ with a more familiar group of order $6\text{.}$
###### 11
Find five non-isomorphic groups of order $8\text{.}$
###### 12
Prove $S_4$ is not isomorphic to $D_{12}\text{.}$
###### 13
Let $\omega = \cis(2 \pi /n)$ be a primitive $n$th root of unity. Prove that the matrices
\begin{equation*} A = \begin{pmatrix} \omega & 0 \\ 0 & \omega^{-1} \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \end{equation*}
generate a multiplicative group isomorphic to $D_n\text{.}$
###### 14
Show that the set of all matrices of the form
\begin{equation*} \begin{pmatrix} \pm 1 & k \\ 0 & 1 \end{pmatrix}, \end{equation*}
is a group isomorphic to $D_n\text{,}$ where all entries in the matrix are in ${\mathbb Z}_n\text{.}$
###### 15
List all of the elements of ${\mathbb Z}_4 \times {\mathbb Z}_2\text{.}$
###### 16
Find the order of each of the following elements.
1. $(3, 4)$ in ${\mathbb Z}_4 \times {\mathbb Z}_6$
2. $(6, 15, 4)$ in ${\mathbb Z}_{30} \times {\mathbb Z}_{45} \times {\mathbb Z}_{24}$
3. $(5, 10, 15)$ in ${\mathbb Z}_{25} \times {\mathbb Z}_{25} \times {\mathbb Z}_{25}$
4. $(8, 8, 8)$ in ${\mathbb Z}_{10} \times {\mathbb Z}_{24} \times {\mathbb Z}_{80}$
###### 17
Prove that $D_4$ cannot be the internal direct product of two of its proper subgroups.
###### 18
Prove that the subgroup of ${\mathbb Q}^\ast$ consisting of elements of the form $2^m 3^n$ for $m,n \in {\mathbb Z}$ is an internal direct product isomorphic to ${\mathbb Z} \times {\mathbb Z}\text{.}$
###### 19
Prove that $S_3 \times {\mathbb Z}_2$ is isomorphic to $D_6\text{.}$ Can you make a conjecture about $D_{2n}\text{?}$ Prove your conjecture.
###### 20
Prove or disprove: Every abelian group of order divisible by $3$ contains a subgroup of order $3\text{.}$
###### 21
Prove or disprove: Every nonabelian group of order divisible by 6 contains a subgroup of order $6\text{.}$
###### 22
Let $G$ be a group of order $20\text{.}$ If $G$ has subgroups $H$ and $K$ of orders $4$ and $5$ respectively such that $hk = kh$ for all $h \in H$ and $k \in K\text{,}$ prove that $G$ is the internal direct product of $H$ and $K\text{.}$
###### 23
Prove or disprove the following assertion. Let $G\text{,}$ $H\text{,}$ and $K$ be groups. If $G \times K \cong H \times K\text{,}$ then $G \cong H\text{.}$
###### 24
Prove or disprove: There is a noncyclic abelian group of order $51\text{.}$
###### 25
Prove or disprove: There is a noncyclic abelian group of order $52\text{.}$
###### 26
Let $\phi : G \rightarrow H$ be a group isomorphism. Show that $\phi( x) = e_H$ if and only if $x=e_G\text{,}$ where $e_G$ and $e_H$ are the identities of $G$ and $H\text{,}$ respectively.
###### 27
Let $G \cong H\text{.}$ Show that if $G$ is cyclic, then so is $H\text{.}$
###### 28
Prove that any group $G$ of order $p\text{,}$ $p$ prime, must be isomorphic to ${\mathbb Z}_p\text{.}$
###### 29
Show that $S_n$ is isomorphic to a subgroup of $A_{n+2}\text{.}$
###### 30
Prove that $D_n$ is isomorphic to a subgroup of $S_n\text{.}$
###### 31
Let $\phi : G_1 \rightarrow G_2$ and $\psi : G_2 \rightarrow G_3$ be isomorphisms. Show that $\phi^{-1}$ and $\psi \circ \phi$ are both isomorphisms. Using these results, show that the isomorphism of groups determines an equivalence relation on the class of all groups.
###### 32
Prove $U(5) \cong {\mathbb Z}_4\text{.}$ Can you generalize this result for $U(p)\text{,}$ where $p$ is prime?
###### 33
Write out the permutations associated with each element of $S_3$ in the proof of Cayley's Theorem.
###### 34
An automorphism of a group $G$ is an isomorphism with itself. Prove that complex conjugation is an automorphism of the additive group of complex numbers; that is, show that the map $\phi( a + bi ) = a - bi$ is an isomorphism from ${\mathbb C}$ to ${\mathbb C}\text{.}$
###### 35
Prove that $a + ib \mapsto a - ib$ is an automorphism of ${\mathbb C}^*\text{.}$
###### 36
Prove that $A \mapsto B^{-1}AB$ is an automorphism of $SL_2({\mathbb R})$ for all $B$ in $GL_2({\mathbb R})\text{.}$
###### 37
We will denote the set of all automorphisms of $G$ by $\aut(G)\text{.}$ Prove that $\aut(G)$ is a subgroup of $S_G\text{,}$ the group of permutations of $G\text{.}$
###### 38
Find $\aut( {\mathbb Z}_6)\text{.}$
###### 39
Find $\aut( {\mathbb Z})\text{.}$
###### 40
Find two nonisomorphic groups $G$ and $H$ such that $\aut(G) \cong \aut(H)\text{.}$
###### 41
Let $G$ be a group and $g \in G\text{.}$ Define a map $i_g : G \rightarrow G$ by $i_g(x) = g x g^{-1}\text{.}$ Prove that $i_g$ defines an automorphism of $G\text{.}$ Such an automorphism is called an inner automorphism. The set of all inner automorphisms is denoted by $\inn(G)\text{.}$
###### 42
Prove that $\inn(G)$ is a subgroup of $\aut(G)\text{.}$
###### 43
What are the inner automorphisms of the quaternion group $Q_8\text{?}$ Is $\inn(G) = \aut(G)$ in this case?
###### 44
Let $G$ be a group and $g \in G\text{.}$ Define maps $\lambda_g :G \rightarrow G$ and $\rho_g :G \rightarrow G$ by $\lambda_g(x) = gx$ and $\rho_g(x) = xg^{-1}\text{.}$ Show that $i_g = \rho_g \circ \lambda_g$ is an automorphism of $G\text{.}$ The isomorphism $g \mapsto \rho_g$ is called the right regular representation of $G\text{.}$
###### 45
Let $G$ be the internal direct product of subgroups $H$ and $K\text{.}$ Show that the map $\phi : G \rightarrow H \times K$ defined by $\phi(g) = (h,k)$ for $g =hk\text{,}$ where $h \in H$ and $k \in K\text{,}$ is one-to-one and onto.
###### 46
Let $G$ and $H$ be isomorphic groups. If $G$ has a subgroup of order $n\text{,}$ prove that $H$ must also have a subgroup of order $n\text{.}$
###### 47
If $G \cong \overline{G}$ and $H \cong \overline{H}\text{,}$ show that $G \times H \cong \overline{G} \times \overline{H}\text{.}$
###### 48
Prove that $G \times H$ is isomorphic to $H \times G\text{.}$
###### 49
Let $n_1, \ldots, n_k$ be positive integers. Show that
\begin{equation*} \prod_{i=1}^k {\mathbb Z}_{n_i} \cong {\mathbb Z}_{n_1 \cdots n_k} \end{equation*}
if and only if $\gcd( n_i, n_j) =1$ for $i \neq j\text{.}$
###### 50
Prove that $A \times B$ is abelian if and only if $A$ and $B$ are abelian.
###### 51
If $G$ is the internal direct product of $H_1, H_2, \ldots, H_n\text{,}$ prove that $G$ is isomorphic to $\prod_i H_i\text{.}$
###### 52
Let $H_1$ and $H_2$ be subgroups of $G_1$ and $G_2\text{,}$ respectively. Prove that $H_1 \times H_2$ is a subgroup of $G_1 \times G_2\text{.}$
###### 53
Let $m, n \in {\mathbb Z}\text{.}$ Prove that $\langle m,n \rangle = \langle d \rangle$ if and only if $d = \gcd(m,n)\text{.}$
###### 54
Let $m, n \in {\mathbb Z}\text{.}$ Prove that $\langle m \rangle \cap \langle n \rangle = \langle l \rangle$ if and only if $l = \lcm(m,n)\text{.}$
###### 55Groups of order $2p$
In this series of exercises we will classify all groups of order $2p\text{,}$ where $p$ is an odd prime.
1. Assume $G$ is a group of order $2p\text{,}$ where $p$ is an odd prime. If $a \in G\text{,}$ show that $a$ must have order $1\text{,}$ $2\text{,}$ $p\text{,}$ or $2p\text{.}$
2. Suppose that $G$ has an element of order $2p\text{.}$ Prove that $G$ is isomorphic to ${\mathbb Z}_{2p}\text{.}$ Hence, $G$ is cyclic.
3. Suppose that $G$ does not contain an element of order $2p\text{.}$ Show that $G$ must contain an element of order $p\text{.}$ Hint: Assume that $G$ does not contain an element of order $p\text{.}$
4. Suppose that $G$ does not contain an element of order $2p\text{.}$ Show that $G$ must contain an element of order $2\text{.}$
5. Let $P$ be a subgroup of $G$ with order $p$ and $y \in G$ have order $2\text{.}$ Show that $yP = Py\text{.}$
6. Suppose that $G$ does not contain an element of order $2p$ and $P = \langle z \rangle$ is a subgroup of order $p$ generated by $z\text{.}$ If $y$ is an element of order $2\text{,}$ then $yz = z^ky$ for some $2 \leq k \lt p\text{.}$
7. Suppose that $G$ does not contain an element of order $2p\text{.}$ Prove that $G$ is not abelian.
8. Suppose that $G$ does not contain an element of order $2p$ and $P = \langle z \rangle$ is a subgroup of order $p$ generated by $z$ and $y$ is an element of order $2\text{.}$ Show that we can list the elements of $G$ as $\{z^iy^j\mid 0\leq i \lt p, 0\leq j \lt 2\}\text{.}$
9. Suppose that $G$ does not contain an element of order $2p$ and $P = \langle z \rangle$ is a subgroup of order $p$ generated by $z$ and $y$ is an element of order $2\text{.}$ Prove that the product $(z^iy^j)(z^ry^s)$ can be expressed as a uniquely as $z^m y^n$ for some non negative integers $m, n\text{.}$ Thus, conclude that there is only one possibility for a non-abelian group of order $2p\text{,}$ it must therefore be the one we have seen already, the dihedral group. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9991692304611206, "perplexity": 70.30338618897802}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156460.64/warc/CC-MAIN-20180920101233-20180920121633-00092.warc.gz"} |
http://mathhelpforum.com/pre-calculus/213397-writing-complex-number-bi-form-print.html | # Writing a complex number in a+bi form
• Feb 19th 2013, 08:13 AM
Egoyan
Writing a complex number in a+bi form
Hi guys,
I'm asked to write the number 5/i in the a+bi form. The format is such that I can only enter the values of a and b in the computer. I'm not sure what I should write. Can anyone help me please?
Thanks a lot!
Egoyan
• Feb 19th 2013, 08:25 AM
Plato
Re: Writing a complex number in a+bi form
Quote:
Originally Posted by Egoyan
Hi guys,
I'm asked to write the number 5/i in the a+bi form. The format is such that I can only enter the values of a and b in the computer. I'm not sure what I should write. Can anyone help me please?
Learn this fact, it will save your life in complex numbers.
$(\forall z\in\mathbb{C}\setminus\{0\})\left[\frac{1}{z}=\frac{\overline{z}}{|z|^2}\right]$.
So $\frac{1}{-3+4i}=\frac{-3-4i}{5}$.
This $\frac{5}{i}=\frac{-5i}{1}=-5i$.
• Feb 19th 2013, 08:28 AM
Egoyan
Re: Writing a complex number in a+bi form
Oh my. I did not know that. Heh. Thanks a lot! I won't forget it any time soon now...
• Feb 19th 2013, 08:37 AM
HallsofIvy
Re: Writing a complex number in a+bi form
More generally, a complex number, a+ bi, multiplied by its "complex conjugate", a- bi, gives a non-negative real number: $(a+ bi)(a- bi)= a^2+ abi - abi- b^2i^2= a^2+ b^2= |a|$. In particular, you can make denominator or a fraction a real number by multiplying the numerator and denominator by the complex conjugate of the denominator:
$\frac{a+ bi}{c+ di}= \frac{a+ bi}{c+ di}\frac{c- di}{c- di}= \frac{(ac+ bd)+ i(bc- ad)}{c^2+ d^2}= \frac{ac+ bd}{c^2+ d^2}+ \frac{bc- ad}{c^2+ d^2}i$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9319944977760315, "perplexity": 1119.6154292536062}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560279379.41/warc/CC-MAIN-20170116095119-00207-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://collider-reach.web.cern.ch/ | The Collider Reach tool gives you a quick (and dirty) estimate of the relation between the mass reaches of different proton-proton collider setups.
Collider 1: CoM energy TeV, integrated luminosity fb-1
Collider 2: CoM energy TeV, integrated luminosity fb-1
PDF:
# Plots
The figures are currently being generated... This can take up to 10-20s.
The PDF choice was MSTW2008nnlo68cl.LHgrid
# Table of results
The table of results is being calculated... This can take a few seconds.
# Interpolation
Input a baseline mass (e.g. $Z'$ mass or $2m_{\tilde q}$) at collider 1 to get a specific reach estimate for collider 2:
GeV →
The Collider Reach tool gives an estimate of the system mass (e.g. $m_{Z'}$ or $2m_{\tilde g}$) that can be probed in BSM searches at one collider setup ("collider 2", e.g. LHC 14 TeV with 300 fb${}^{-1}$) given an established system mass reach of some other collider setup ("collider 1", e.g. LHC 8 TeV with 20 fb${}^{-1}$).
The slides of a recent talk about the tool can be found here: slides.pdf.
The estimate is obtained by determining the system mass at collider-2 for which the number of events is equal to that produced at collider-1, assuming that cross sections scale with the inverse squared system mass and with partonic luminosities. The exact results depend on the relevant partonic scattering channel, as represented by the different lines ($q \equiv \sum_i (q_i + \bar q_i)$), and the band covers the spread of those different partonic channels.
There are quite a few caveats in interpreting the numbers: (1) they assume that signal and background are driven by the same partonic scattering channel; (2) they assume that reconstruction efficiencies, background rejection rates, etc., all stay reasonably constant as the collider setup changes; (3) they assume that the cross-sections are simply proportional to the partonic luminosity at a given scale (divided by the mass-scale squared), ignoring detailed production spectra, higher-order QCD effects, etc. Quite often these assumptions turn out to be reasonable.
The spirit in which we take the numbers is as first guidance on what one might expect. If a more sophisticated analysis appears gives a substantially different picture, then it can be useful to ask why: e.g., has a new background appeared? Have cuts been fully adapted at the new collider setup to take into account possibly different signal and background kinematics? Etc.
The reliability of the results depends on the reliability of the PDFs. By default, PDFs don't always cover the $x$ and $Q$ regions that are needed here. Currently, if $\sqrt{s}$ of either collider setup exceeds the maximum $Q$ supported natively in the PDF, DGLAP NNLO evolution is rerun (including a top threshold) to cover the full range of $Q$ values up to $\max(\sqrt{s_1},\sqrt{s_2})$. For the $x$ range, PDFs are essentially unconstrained below $x=10^{-6}$: in most usages the tool will be using larger $x$ values than that, however at some point we may add a warning if that's not the case.
The Collider Reach project is in an early "beta" stage. Features that we may add in the future include PDF uncertainties, results when signal and backgrounds are driven by different partonic channels, etc... If you find any issues, please let us know. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9407788515090942, "perplexity": 1777.5983072638282}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824824.76/warc/CC-MAIN-20171021171712-20171021191712-00111.warc.gz"} |
https://proceedings.mlr.press/v119/bao20b.html | # Fast OSCAR and OWL Regression via Safe Screening Rules
Runxue Bao, Bin Gu, Heng Huang
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:653-663, 2020.
#### Abstract
Ordered Weighted $L_{1}$ (OWL) regularized regression is a new regression analysis for high-dimensional sparse learning. Proximal gradient methods are used as standard approaches to solve OWL regression. However, it is still a burning issue to solve OWL regression due to considerable computational cost and memory usage when the feature or sample size is large. In this paper, we propose the first safe screening rule for OWL regression by exploring the order of the primal solution with the unknown order structure via an iterative strategy, which overcomes the difficulties of tackling the non-separable regularizer. It effectively avoids the updates of the parameters whose coefficients must be zero during the learning process. More importantly, the proposed screening rule can be easily applied to standard and stochastic proximal gradient methods. Moreover, we prove that the algorithms with our screening rule are guaranteed to have identical results with the original algorithms. Experimental results on a variety of datasets show that our screening rule leads to a significant computational gain without any loss of accuracy, compared to existing competitive algorithms. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8260976672172546, "perplexity": 510.41353350059916}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487611320.18/warc/CC-MAIN-20210614013350-20210614043350-00223.warc.gz"} |
https://mathoverflow.net/questions/328002/what-does-the-torsion-free-condition-for-a-connection-mean-in-terms-of-its-horiz | # What does the torsion-free condition for a connection mean in terms of its horizontal bundle?
I must have read and re-read introductory differential geometry texts ten times over the past few years, but the "torsion free" condition remains completely unintuitive to me.
The aim of this question is to try to finally put this uncomfortable condition to rest.
### Ehresmann Connections
Ehresmann connections are a very intuitive way to define a connection on any fiber bundle. Namely, an Ehressmann connection on a fiber bundle $$E\rightarrow M$$ is just a choice of a complementary subbundle to $$ker(TE \rightarrow TM)$$ inside of $$TE$$. This choice is also called a horizontal bundle.
If we are dealing with a linear connection, then $$E=TM$$, and the Ehresmann connection is a subbundle of $$TTM$$. This makes intuitive sense -- basically it's saying that for each point in $$TM$$ it tells you how to move it to different vectors at the tangent spaces of different points. ($$ker(TTM \rightarrow TM)$$ will mean moving to different vectors at the same tangent space; so that is precluded.)
I like this definition -- it makes more intuitive sense to me than the definition of a Koszul connectionan $$\mathbb{R}$$-linear map $$\Gamma(E)\rightarrow\Gamma(E\otimes T^*M)$$ satisfies some condition. Unlike that definition it puts parallel transport front and center.
### Torsion-Freeness
A Levi-Civita connection is a connection that: 1. It preserves with the Riemannian metric. (Basically, parallel transporting preserves inner products.) 2. It is torsion-free. Torsion free means $$\nabla_XY - \nabla_YX = [X,Y]$$.
This definition very heavily uses the less intuitive notion of connection.
So:
### Questions
1. How can you rephrase the torsion-free condition in terms of the horizontal bundle of the connection? (Phrased differently: how can it be phrased in terms of parallel transports?)
2. I realized that I don't actually have handy an example of a connection on $$\mathbb{R}^2$$ that preserves the canonical Riemannian metric on $$\mathbb{R}^2$$ but that does have torsion. I bet that would help elucidate the answer to my first question.
You should think of the tangent bundle as a bundle with a bigger structural group namely the group $$\DeclareMathOperator{\Aff}{\mathbf{Aff}}$$ $$\Aff(n)$$ of affine transformations of $$\newcommand{\bR}{\mathbb{R}}$$ $$\bR^n$$. As such, its curvature is a $$2$$-form with coefficients in the Lie algebra of $$\Aff(n)$$.
An (infinitesimal) affine map consists of two parts: a translation and a linear transformation. Correspondingly, the curvature of an affine connection decomposes into two parts. The part of the curvature corresponding to the infinitesimal translation is the torsion of the connection. Thus if the torsion is $$0$$, the affine holonomy of this connection along infinitesimal parallelograms is translation free. For more details consult Sections III.3-5 in volume 1 of
S.Kobayashi, K. Nomizu: Foundations of Differential Geometry, John Wiley & Sons, 1963
As for the second question, denote by $$D$$ the Levi-Civita connection on $$T\bR^n$$. Any other metric connection $$\nabla$$ has the form $$\nabla=D+ A,\;\;A=\sum_{i=1}^m A_i dx^i,$$ where $$A_i$$ are smooth maps $$A_i:\bR^n\to \mathrm{so}(n)=\mbox{the space of real skew-symmetric n\times n matrices}.$$ The torsion of $$D+A$$ is described in Proposition 1.2 of this paper. Among other things it shows that any $$2$$-form $$T=\sum_{i can be the torsion of a connection compatible with the metric.
The Levi-Civita connection of the Euclidean $$\bR^n$$ is the trivial connection.
1. I realized that I don't actually have handy an example of a connection on $$\Bbb R^2$$ that preserves the canonical Riemannian metric on $$\Bbb R^2$$ but that does have torsion. I bet that would help elucidate the answer to my first question.
Let $$\nabla$$ be the Levi-Civita connection of $$(\Bbb R^2, g = {\rm d}x^1\otimes {\rm d}x^1 + {\rm d}x^2\otimes {\rm d}x^2)$$. Since the difference of two connections is a vector-valued $$(0,2)$$-tensor field (naturally identified with a scalar valued $$(1,2)$$-tensor field), we'll look for connections of the form $$\overline{\nabla} = \nabla + T$$. Meaning that we'll look for a condition on $$T$$ ensuring that $$\overline{\nabla}g = 0$$. If $$T \neq 0$$, such $$\overline{\nabla}$$ will necessarily have torsion. A short calculation says that $$\overline{\nabla}g = 0$$ if and only if $$g(T(Z,X), Y) + g(X, T(Z,Y)) = 0$$for all vector fields $$X$$, $$Y$$ and $$Z$$ in $$\Bbb R^2$$. Write $$T(\partial_i,\partial_j) = \sum_{k=1}^2 T_{ij}^k\partial_k$$. The condition above then reads $$T_{ki}^j + T_{kj}^i = 0$$ for $$1 \leq i,j,k \leq 2$$. We have nonzero $$T$$ satisfying such conditions, e.g., $$T = -{\rm d}x^2\otimes {\rm d}x^2\otimes \partial_1 + {\rm d}x^2\otimes {\rm d}x^1\otimes \partial_2.$$
We have some good threads on this topic on MO, but they're usually phrased in a different language. Here is how I like to think of torsion from the perspective of an Ehresmann connection. I will use the formalism that an Ehresmann connection on $$TM$$ is fibrewise linear projection $$c : T^2 M \to TM$$ whose kernel is complementary to the kernel of $$D\pi : T^2 M \to TM$$ where $$\pi : TM \to M$$ is the tangent bundle projection.
From this perspective, take a function of two variables
$$f : \mathbb R^2 \to M$$
and consider its partial derivatives to be maps
$$\frac{\partial f}{\partial x_i} : \mathbb R^2 \to TM$$
i.e. $$\frac{\partial f}{\partial x_i}(p) = Df(p, e_i)$$.
Then from this perspective, the torsion of the connection $$c$$, $$\tau_c$$ is given by:
$$\tau_c(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}) = c\left(\frac{\partial^2 f}{\partial x \partial y}\right) - c\left(\frac{\partial^2 f}{\partial y\partial x}\right)$$
A more elaborate (and physical) way of phrasing this would be to describe $$\tau_c(v,w)$$ as the result of this process: take the geodesic along $$v$$. Parallel transport $$w$$ along this geodesic, and then take the geodesic out of this parallel transport of $$w$$. This gives you a function of two variables (the time parameter for the two geodesics). You differentiate it with respect to the first geodesic, and ask if this vector field is parallel along the 2nd parameter. It gives you the same formula.
• Do you happen to have a reference where your last paragraph is explained in more detail? Or would you be willing to explain it in more detail yourself? I've been trying for hours but I can't seem to get the right formula for the torsion by looking at parallel transport in the way you describe. – Sjorszini Sep 13 at 9:50
• I believe I've seen it in Berger's book "A Panoramic View. . .." but if you just want a page of detail, it's in the (developing) lecture notes for a course I occasionally offer. Here: rybu.org/sites/default/files/math_docs/dgnotes/… the discussion related to the last paragraph of my reply is on pages 81, 82, 83 of these notes. – Ryan Budney Sep 13 at 17:23
• Great, thanks a lot! – Sjorszini Sep 13 at 17:42
• Sorry to keep asking, but could it be true that you are only talking about linear Ehresmann connections here? I have no problem deriving the formule for linear connections, but is just does not seem to work for a non-linear one. Also, in your lecture notes you never define the torsion of a non linear ehresmann conection, only of linear connections, and the same is true for the book by Berger... – Sjorszini Sep 13 at 17:55
• In my notes I only talk about non-linear connections rather briefly. Other than a few examples, everything is in the linear realm -- no particular reason, I suppose, I have yet to have interest in non-linear connections. What are you looking for precisely? – Ryan Budney Sep 13 at 18:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 65, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9352728128433228, "perplexity": 424.7740687909951}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986647517.11/warc/CC-MAIN-20191013195541-20191013222541-00265.warc.gz"} |
https://en.wikipedia.org/wiki/Brocard_point | # Brocard points
(Redirected from Brocard point)
The Brocard point of a triangle, constructed at the intersection point of three circles.
In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845 – 1922), a French mathematician.
## Definition
In a triangle ABC with sides a, b, and c, where the vertices are labeled A, B and C in anticlockwise order, there is exactly one point P such that the line segments AP, BP, and CP form the same angle, ω, with the respective sides c, a, and b, namely that
$\angle PAB = \angle PBC = \angle PCA.\,$
Point P is called the first Brocard point of the triangle ABC, and the angle ω is called the Brocard angle of the triangle. The following applies to this angle:
$\cot\omega = \cot \alpha + \cot \beta + \cot \gamma.\,$
There is also a second Brocard point, Q, in triangle ABC such that line segments AQ, BQ, and CQ form equal angles with sides b, c, and a respectively. In other words, the equations $\angle QCB = \angle QBA = \angle QAC$ apply. Remarkably, this second Brocard point has the same Brocard angle as the first Brocard point. In other words angle $\angle PBC = \angle PCA = \angle PAB$ is the same as $\angle QCB = \angle QBA = \angle QAC.$
The two Brocard points are closely related to one another; In fact, the difference between the first and the second depends on the order in which the angles of triangle ABC are taken. So for example, the first Brocard point of triangle ABC is the same as the second Brocard point of triangle ACB.
The two Brocard points of a triangle ABC are isogonal conjugates of each other.
## Construction
The most elegant construction of the Brocard points goes as follows. In the following example the first Brocard point is presented, but the construction for the second Brocard point is very similar.
Form a circle through points A and B, tangent to edge BC of the triangle (the center of this circle is at the point where the perpendicular bisector of AB meets the line through point B that is perpendicular to BC). Symmetrically, form a circle through points B and C, tangent to edge AC, and a circle through points A and C, tangent to edge AB. These three circles have a common point, the first Brocard point of triangle ABC. See also Tangent lines to circles.
The three circles just constructed are also designated as epicycles of triangle ABC. The second Brocard point is constructed in similar fashion.
## Trilinears and the Brocard midpoint
Homogeneous trilinear coordinates for the first and second Brocard points are c/b : a/c : b/a, and b/c : c/a : a/b, respectively. The Brocard points are an example of a bicentric pair of points, but they are not triangle centers because neither Brocard point is invariant under similarity transformations: reflecting a scalene triangle, a special case of a similarity, turns one Brocard point into the other. However, the unordered pair formed by both points is invariant under similarities. The midpoint of the two Brocard points, called the Brocard midpoint, has trilinears
sin(A + ω) : sin(B + ω) : sin(C + ω)[1]
and is a triangle center. The third Brocard point, given in trilinear coordinates as a−3 : b−3 : c−3, or, equivalently, by
csc(A − ω) : csc(B − ω) : csc(C − ω),[2]
is the Brocard midpoint of the anticomplementary triangle and is also the isotomic conjugate of the symmedian point. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 5, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8268738985061646, "perplexity": 524.8226878126607}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644062760.2/warc/CC-MAIN-20150827025422-00058-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://mathhelpforum.com/trigonometry/179592-trig-proof.html | 1. ## trig proof
$\frac{sin^{4} \alpha}{a}+ \frac{cos ^{4}\alpha }{b}= \frac{1}{a+b}$
Prove that:
$\frac{sin^{8} \alpha}{a^{3}}+ \frac{cos ^{8} \alpha }{b^{3}}= \frac{1}{(a+b)^{3}}$
2. Originally Posted by andtom
$\frac{sin^{4} \alpha}{a}+ \frac{cos ^{4}\alpha }{b}= \frac{1}{a+b}$
take $sin^{2} \alpha=x$. then $cos^{2} \alpha=1-x$.
Now substitute in the given equation. SOLVE the quadratic. the roots are nice! you will get $sin^{2} \alpha=\frac{a}{a+b}$ . Substitute this below
Prove that:
$\frac{sin^{8} \alpha}{a^{3}}+ \frac{cos ^{8} \alpha }{b^{3}}= \frac{1}{(a+b)^{3}}$
did this help?
3. Nope, I still cannot prove it :/
4. Originally Posted by andtom
Nope, I still cannot prove it
Show what roots do you obtain for
$\dfrac{x^2}{a}+\dfrac{(1-x)^2}{b}=\dfrac{1}{a+b}$
5. so there's only 1 root
since $\Delta = 0$,
$x=sin^{2}\alpha=\frac{2a}{2(a+b)}=\frac{a}{a+b}$
6. Originally Posted by andtom
so there's only 1 root
since $\Delta = 0$,
$x=sin^{2}\alpha=\frac{2a}{2(a+b)}=\frac{a}{a+b}$
correct!
7. OK, i got it
Thank you guys | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 12, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9749186038970947, "perplexity": 4388.19046321371}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719041.14/warc/CC-MAIN-20161020183839-00480-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://ora.ox.ac.uk/objects/uuid:6db6248e-4524-4c19-9e04-8347ea572873 | # Journal article
## A note on upper ramification jumps in Abelian extensions of exponent p
Abstract:
In this paper we present a classification of the possible upper ramification jumps for an elementary abelian p-extension of a p-adic field. The fundamental step for the proof of the main result is the computation of the ramification filtration for the maximal elementary abelian p-extension of the base field K. This result generalizes [1, Lemma 9, p. 286], where the same result is proved under the assumption that K contains a primitive p-th root of unity. To dea...
Publication status:
Published
Peer review status:
Peer reviewed
### Authors
More by this author
Institution:
University of Oxford
Division:
MPLS
Department:
Mathematical Institute
Role:
Author
Publisher:
University of Parma Publisher's website
Journal:
Rivista di Matematica della Università di Parma Journal website
Volume:
6
Issue:
2
Pages:
317-329
Publication date:
2015-12-20
Acceptance date:
2015-10-16
EISSN:
2284-2578
ISSN:
0035-6298
Keywords:
Pubs id:
pubs:687158
UUID:
uuid:6db6248e-4524-4c19-9e04-8347ea572873
Local pid:
pubs:687158
Source identifiers:
687158
Deposit date:
2017-03-25 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9214271903038025, "perplexity": 1391.0066637233579}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948867.32/warc/CC-MAIN-20230328135732-20230328165732-00613.warc.gz"} |
http://math.stackexchange.com/questions/643346/factoring-with-rational-exponents/643351 | # Factoring with rational exponents
I'm not quite sure how to do this question. Every way that I tried doing it didn't yield an answer that is equivalent to the original question.
$$(2x+1)^{2/3}-4(2x+1)^{-1/3}$$
When I tried doing it, I ended up with $$(2x+1)^{2/3}\left(-\frac{8x+4}{2x+1}\right)$$ or with $$4(2x+1)^{1/3}$$
How can I factor it properly?
-
Start by rewritting the equation a bit more clearly.
$$(2x+1)^{2/3} -4(2x+1)^{-1/3} = \sqrt[3]{\left ( 2x+1 \right)^2} - \frac{4}{\sqrt[3]{2x+1}}$$
Then, put everything on the same denominator.
$$\sqrt[3]{\left ( 2x+1 \right)^2} - \frac{4}{\sqrt[3]{2x+1}} = \frac{\sqrt[3]{\left ( 2x+1 \right)^2} \sqrt[3]{2x+1} - 4}{\sqrt[3]{2x+1}}$$
Simplify.
$$\frac{\sqrt[3]{\left ( 2x+1 \right)^2} \sqrt[3]{2x+1} - 4}{\sqrt[3]{2x+1}} = \frac{\sqrt[3]{\left ( 2x+1 \right)^3} - 4}{\sqrt[3]{2x+1}}$$
-
More directly, factor out the smallest power of the common term, in this case $(2x+1)^{-1/3}$. – gaddy Jan 19 at 4:20
I got (2x-3)/cbrt(2x+1). I think it seems right. Thanks a lot! – Alex Jan 19 at 4:36
Yep, it is all right. Sorry I made a typo, $+4$ instead of $-4$. – Olivier Jan 19 at 4:38
Let $\ \color{#c00}{y} = (\color{#0a0}{2x+1})^{1/\color{#c00}3}.\,$ Then $\ y^2-\dfrac{4}y\, =\, \dfrac{\color{#c00}{y^3}-4}y\, =\, \dfrac{2x-3}y\$ by $\ \color{#c00}{y^3} = \color{#0a0}{2x+1}$
-
### First way
Multiply and divide both pieces by $(2 x + 1)^{1/3}$. You so obtain
$$\frac{2x+1 - 4}{(2 x + 1)^{1/3}} = \frac{2 x - 3}{(2 x + 1)^{1/3}}$$
### Second way
Factor $(2 x + 1)^{2/3}$. You so obtain
$$(2x+1)^{2/3}\cdot \frac{1-4}{2 x + 1} = (2 x + 1)^{2/3}\cdot \frac{2 x + 1 - 4}{2 x + 1} = \frac{2 x - 3}{(2 x + 1)^{1/3}}$$
-
@dfeuer. Thanks for editing for me. Cheers – Claude Leibovici Jan 19 at 4:45
My edit was but a slight improvement of Olivier's suggested one. – dfeuer Jan 19 at 4:48
I do highly recommend using the hash-marks instead of bold for section headings so the HTML will come out with proper header elements. – dfeuer Jan 19 at 4:50
As a general method you can get good results by factoring off the the binomial to the power that is leftmost on the real number line like so:
\begin{align*} (2x+1)^{\frac{2}{3}}-4(2x+1)^{-\frac{1}{3}} &= (2x+1)^{-\frac{1}{3}}\left( (2x+1)^{\frac{3}{3}} -4\right) \\ &= (2x+1)^{-\frac{1}{3}}( 2x+1 -4) \\ &= (2x+1)^{-\frac{1}{3}}( 2x-3) \\ &=\frac{2x-3}{\sqrt[3]{2x+1}}. \end{align*}
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9988961815834045, "perplexity": 1402.5184056490161}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802767878.79/warc/CC-MAIN-20141217075247-00046-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://www.iam.uni-bonn.de/os-analysis/summer-2017/ | # Oberseminar Analysis Summer 2017
## Organizers: A. Schlichting, S. Conti, H. Koch, S. Müller, B. Niethammer, M. Rumpf, C. Thiele, J.J.L. Velázquez
• Thursday, March 23, 2:15 p.m., Lipschitz-Saal
Fulvio Ricci (Scuola Normale Superiore, Pisa)
A maximal restriction theorem and Lebesgue points of Fourier transforms in the plane
We present the results of recent joint work with Detlef Müller and James Wright. A Fourier restriction theorem relative to a given surface $S\subset \mathbb R^n$ provides $L^p-L^q$ inequalities for the operator $\mathcal R:f\in\mathcal S(\mathbb R^n)\longmapsto \hat f_{|_S}$. Despite the wide literature concerning range of validity and applications of restriction inequalities, not much has been said about the explicit relation, in presence of a $p-q$ restriction inequality, between the two functions $\hat f$ and $\mathcal R f$ for a general $f\in L^p(\mathbb R^n)$. We give a first partial answer for curves in the plane by analyzing the operator which assigns to $f\in L^p(\mathbb R^2)$, with $p<4/3$, a smoothened version of the strong maximal function of $\hat f$.
• Thursday, April 20, 2:15 p.m., Lipschitz-Saal
Francesco Fanelli (Institut Camille Jordan, Université de Lyon)
Asymptotic behaviour of non-homogenous fluids in fast rotation
In this talk, we consider a class of singular perturbation problems for systems of PDEs related to the dynamics of geophysical fluids. We are interested here in effects due to both the non-homogeneity of the fluid and the Earth rotaton, and to their interplay. After a review of known results, we specilize on the 2-D density-dependent incompressible Navier-Stokes equations with Coriolis force: our goal is to characterize the asymptotic dynamics of weak solutions to this model, in the limit when the rotation becomes faster and faster.
We present two kinds of results (deeply different from each other, from a qualitative viewpoint), depending on whether the initial densities are small perturbations of a constant state or of a truly non-constant reference density. In the former case we prove that the system tends to a homogeneous Navier-Stokes system with an additional forcing term, which is due to density variations and which is a remainder of the action of the Coriolis force. In the latter case, instead, we show that the limit equations become linear, and moreover one can identify only a mean motion, in terms of the limit vorticity and the limit density fluctuation function; this issue can be interpreted as a sort of turbulent behaviour of the fluid in the limit of fast rotation.
This talk is based on a joint work with Isabelle Gallagher.
• Thursday, April 27, 2:15 p.m., seminar room 0.008
Diogo Oliveira e Silva (University of Bonn)
Some recent progress on sharp Fourier restriction theory
It has long been understood that Strichartz estimates for the homogeneous Schrödinger equation correspond to adjoint restriction estimates on the paraboloid. The study of extremizers and sharp constants for the corresponding inequalities has a short but rich history. In this talk, I will summarize it briefly, and then specialize to the case of certain convex perturbations of the paraboloid. A geometric comparison principle for convolution measures can be used to establish the corresponding sharp Strichartz inequality, and to prove that extremizers do not exist. The mechanism underlying this lack of compactness is explained by the behaviour of extremizing sequences which will be described via concentration-compactness. Time permitting, I will show how this resolves a dichotomy from the recent literature concerning the existence of extremizers for a family of fourth order Schrödinger equations.
• Thursday, May 4, 2:15 p.m., Lipschitz-Saal
Christian Seis (University of Bonn)
Optimal stability estimates for continuity equations
In this talk, I will review new stability estimates for continuity equations and compare those with previosuly known analogous estimates for Lagrangian flows. I will explain how the new results allow for a quantitive proof of well-posedness in the low regularity setting considered by DiPerna and Lions. The estimates are obtained for Kantorovich--Rubinstein distances with logarithmic cost functions and allow thus to quantify the order of weak convergence in several applications. I plan to conclude this talk with two examples: 1) A lower bound on mixing rates obtained by stirring two immiscible fluids. 2) An upper bound on convergence rates for numerical upwind schemes (obtained jointly with A. Schlichting).
• Thurday, June 1, 2:15 p.m., seminar room 0.008
Charlotte Perrin (RWTH Aachen)
A macroscopic model for granular flows
I will present in this talk an original model for immersed granular flows which takes into account memory effects. In the first part I will justify these memory effects by means of a singular limit. It relies on recent analysis tools for the compressible Navier-Stokes equations. The second part of the talk will be dedicated to one-dimensional flows for which a direct Lagrangian approach can be developed.
• June 20 - 29, 2017
Robert L. Pego (Carnegie Mellon University)
Lipschitz Lectures: Studies in dynamics, coherent structures and stability
• Thursday, July 6, 2:15 p.m., seminar room 0.008
Simon Rösel (WIAS Berlin)
Density of convex intersections and applications
In a general framework, it is shown how density properties of intersections of convex sets naturally arise from the perturbation or dualization of constrained optimization and variational inequality problems. Several density results (and counterexamples) for closed convex sets with pointwise constraints in Sobolev spaces are presented. Diverse applications are provided, which include elasto-plasticity and image restoration problems. Finally, the results are further discussed in the context of Finite Element discretizations of sets associated to convex constraints.
• Thursday, July 13, 2:15 p.m., Lipschitz-Saal
Ievgen Verbytskyi (National University of Ukraine)
A model of tissue border displacement in non-contact Photoacoustic Tomography
Photoacoustic tomography (PAT) is a relatively new imaging modality, which allows e.g. to visualize the vascular network in biological tissue noninvasively. This tomographic method has an advantage in comparison to pure optical/acoustical methods due to high optical contrast and low acoustic scattering in deep tissue. The common PAT methodology, based on measurements of the acoustic pressure by piezoelectric sensors placed on the tissue surface, limits its practical versatility. A novel, completely non-contact and full-field PAT system is described. In noncontact PAT the measurement of surface displacement induced by the acoustic pressure at the tissue/air border is researched. A model of the tissue displacement caused by medium pressure based on the momentum conservation law is proposed. Experimental data processing and simulation techniques are developed. The error of the displacement simulation in comparing with experimental data is calculated.
## News
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Managing Director: Prof. Dr. Massimiliano Gubinelli | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8179056644439697, "perplexity": 1552.2955284380582}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514572556.54/warc/CC-MAIN-20190916120037-20190916142037-00402.warc.gz"} |
http://math.stackexchange.com/questions/283056/induction-proof-help-number-theory-george-e-andrews-1-1-3 | # Induction Proof Help [Number Theory George E. Andrews 1-1 #3]
Prove that $$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1}).$$
This problem is driving me crazy.
$$x^n-y^n = (x-y)(x^{n-1}+x^{n-2}y+\dots +xy^{n-2}+y^{n-1)}$$
$(x^n-y^n)/(x-y) =$ the sum for the first $n$ numbers and then I added $(xy^{(n+1)-2}+y^{(n+1)-1})$ which should equal $(x^{n+1}-y^{n+1})/(x-y)$ but I can't figure it out
This is a similar problem in the book and I tried this method but it wasn't working out
$\quad$**Thereom $\bf1$-$\bf2$:** $\,\,\,\,$ If $\,x$ is any real number other than $1$, then $$\sum_{j=0}^{n-1}x^j=1+x+x^2+\ldots+x^{n-1}=\dfrac{x^n-1}{x-1}.$$ $\quad$**Remark:** $\displaystyle\sum_{j=0}^{n-1}A_j$ is shorthand for $A_0+A_1+A_2+\ldots+A_{n-1}.$
$\quad$**Proof:** Again we proceed by mathematical induction. If $n=1$ then $\displaystyle\sum_{j=0}^{1-1}x^j=x^0=1$ and $(x-1)/(x-1)=1$. Thus the theorem is true for $n=1$.
$\quad$ Assuming that $\displaystyle\sum_{j=0}^{k-1}x^j=(x^k-1)/(x-1)$, we find that \eqalign{ \sum^{(k+1)-1}_ {j=0}x^j & = \sum^{k-1}_ {j=0}x^j+x^k=\dfrac{x^k-1}{x-1}+x^k \\ &= \dfrac{x^k-1+x^{k+1}-x^k}{x-1}\\ &= \dfrac{x^{k+1}-1}{x-1}. } Hence condition $(\rm ii)$ is fulfilled, and we have established the theorem.
$\quad$**Corollary $\bf1$-$\bf1$:** $\,\,$ If $\,m$ and $n$ are positive integers and if $m>1$, then $n<m^n.$
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The second image is not a problem, but rather a theorem with a complete proof. Note that the first problem can be solved with this theorem by plugging in $\frac xy$ for §x§ and multiplying with $y^n$ (what care should be taken if $y=0$?) – Hagen von Eitzen Jan 20 '13 at 22:14
Hint: Apply $\textbf{Theorem 1.2}$ to $\displaystyle \frac{x}{y}$.
Edit: It follows from $\textbf{Theorem 1.2}$ that $\displaystyle \Bigl(\frac{x}{y}\Bigr)^n -1=\Bigl(\frac{x}{y}-1\Bigr)\Bigl( \Bigl(\frac{x}{y}\Bigr)^{n-1}+\cdots +\frac{x}{y}+1\Bigr)$.
Now multiply the equation by $y^n$ to get
$$\displaystyle y^n\Bigl(\Bigl(\frac{x}{y}\Bigr)^n -1)\Bigr)=y^n\Bigl(\frac{x}{y}-1\Bigr)\Bigl( \Bigl(\frac{x}{y}\Bigr)^{n-1}+\cdots +\frac{x}{y}+1\Bigr)$$
Simplifying on the left-hand side and rewritting $y^n$ as $yy^{n-1}$ on the right-hand side we get
$$(x^n -y^n)=yy^{n-1}\Bigl(\frac{x}{y}-1\Bigr)\Bigl( \Bigl(\frac{x}{y}\Bigr)^{n-1}+\cdots +\frac{x}{y}+1\Bigr)$$
Because the product is commutative you can rewrite the right-hand side to get
$$(x^n -y^n)=y\Bigl(\frac{x}{y}-1\Bigr)y^{n-1}\Bigl( \Bigl(\frac{x}{y}\Bigr)^{n-1}+\cdots +\frac{x}{y}+1\Bigr)$$
Finally, on the right-hand side, factor in $y$ and $y^{n-1}$ accordingly to get
$$(x^n -y^n)=(x-y)(x^{n-1}+\cdots +xy^{n-2}+y^{n-1})$$
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So I tried expanding it if it were (x/y) – draconisthe0ry Jan 20 '13 at 22:56
@draconisthe0ry was that a question somehow? – Git Gud Jan 20 '13 at 23:23
Sorry about the last comment . . I didn't know that I ended up pressing the Add Comment button. I see that this works but what confuses me is that when you divide both sides by (x-y) and you get (x^(n−1)+x^(n−2)y+⋯+xy^(n−2)+y^(n−1)) It looks like "y" is following the theorem in the sense that it propagates as (y^0+y^1+...+y^(n-2)+y^(n-1)) and it is being multiplied that same sequence in reverse for x. I don't understand how the sequence in reverse for x is being multiplied in reverse order. – draconisthe0ry Jan 22 '13 at 1:44
@draconisthe0ry I rewrote my answer with a more detailed explanation. Can you understand it now? – Git Gud Jan 22 '13 at 7:24
yes this answer is perfect. i understand each step. the only concern I have is how to think of initially using (x/y) instead of x in Theorem 1-2. It makes sense on how that works since you can pull a y^n/y out of (x^n-y^n)/(x-y) to get ((x/y)^n-1)/(x/y-1) . Does this prove that the expression works for n+1 ? To prove it works for n+1 as well I would just show that ((x/y)^n-1)/(x/y-1) + (x/y)^n = ((x/y)^(n+1)-1)/((x/y)-1) right ? – draconisthe0ry Jan 22 '13 at 17:43
show 2 more comments
Your induction hypothesis is that
$$\sum_{j=0}^{k-1}x^jy^{(k-1)-j}=\frac{x^k-y^k}{x-y}\;.$$
Now follow the model:
\begin{align*} \sum_{j=0}^{(k+1)-1}x^jy^{k-j}&\overset{(1)}=\left(\sum_{j=0}^{k-1}x^jy^{k-j}\right)+x^ky^0\\ &\overset{(2)}=y\left(\sum_{j=0}^{k-1}x^jy^{(k-1)-j}\right)+x^k\\ &\overset{(3)}=y\cdot\frac{x^k-y^k}{x-y}+x^k\\ &\overset{(4)}=\frac{x^ky-y^{k+1}+x^{k+1}-x^ky}{x-y}\\ &\overset{(5)}=\frac{x^{k+1}-y^{k+1}}{x-y}\;. \end{align*}
$(1)$ is splitting off the last term of the summation; $(2)$ factors a $y$ out of the remaining summation; $(3)$ uses the induction hypothesis; and $(4)$ and $(5)$ are just algebra.
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I'm trying to make sense of it now . . . So I see that (1 + y^1 + ... + y^(n-1)) = (y^n-1)/(y-1) but it is being multiplied by (x^(n-1)+...+x^1+1) so that is equal to (x^n-1)/(x-1) but would that work when you multiply them ? – draconisthe0ry Jan 20 '13 at 22:47
@draconisthe0ry: I’m not sure what you’re asking. I’m simply observing that $$x^n-y^n = (x-y)(x^{n-1}+x^{n-2}y+\dots+xy^{n-2}+y^{n-1)}\;,$$ the statement that you want to prove, is equivalent to $$\sum_{k=0}^{n-1}x^ky^{(n-1)-k}=\frac{x^n-y^n}{x-y}$$ and then proving the latter by induction. (Actually, they’re not equivalent when $x=y$, but in that case the first one is trivially true, since both sides are $0$.) This is exactly what your text did with the $x^n-1$ identity: it converted $x^n-1=(x-1)(\text{stuff})$ to $\text{stuff}=\frac{x^n-1}{x-1}$ and proved that. – Brian M. Scott Jan 20 '13 at 22:56
$$\frac{1-(x/y)^n}{1-x/y}=1+x/y+(x/y)^2+...+(x/y)^{n-1}$$ $$\frac{(y^n-x^n)/y^n}{(y-x)/y}=\frac{y^{n-1}+xy^{n-2}+...+x^{n-1}}{y^{n-1}}$$ $$\frac{y^n-x^n}{(y-x)y^{n-1}}=\frac{y^{n-1}+xy^{n-2}+...+x^{n-1}}{y^{n-1}}$$ $$y^n-x^n=(y-x)(y^{n-1}+xy^{n-2}+...+x^{n-1})$$
Let $u_n$ $=$ $x^n-y^n$. Now note that $u_n$ $=$ $(x+y)u_n$$_-$$_1$ $+$ $xy$ $u_n$$_-$$_2$. Assume that the given expression is true for all numbers from $1$ to some fixed $k(>1)$. Then apply induction to prove the result for $(k+1)$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9722310304641724, "perplexity": 343.2067857121284}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997860453.15/warc/CC-MAIN-20140722025740-00038-ip-10-33-131-23.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/about-quantum-numbers.54518/ | 1. Nov 28, 2004
### rgshankar76
What is the physical significance of the various quantum numbers like (l,m,s,j, etc)that are used to describe any system.
2. Nov 28, 2004
### RedX
Well l is usually orbital angular momentum, m is orbital angular momentum along an axis, s is spin, and j is total angular momentum.
Actually l is a number which represents orbital angular momentum. The real value would be sqrt [l(l+1)h^2] I think, where h is planck's constant divided by 2pi. Upon further consideration, yeah, you have m going from -l to l. Similarly for spin and total angular momentum.
Anyways, physically these quantum numbers will tell you what you will observe if you make a measurement of something you can observe. If a value of some observable is not one of these quantum numbers, then you won't observe it. For example if you take a particle where s=1/2, then if you measure the spin, then you'll get sqrt(1/2(1/2+1)h^2) because that's the only allowed value when you specify s=1/2.
Also all these quantum numbers may not have definite values simultaneously. You'll have to consider whether all the operators commute or not and can be simultaneously diagnoalized by a unitary change of basis. Some things always commute like the total angular momentum operator and one of the total angular momentum operators about an axis. Others don't like the total angular momentum operators about an axis with each other. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9860145449638367, "perplexity": 449.2758310683284}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221213691.59/warc/CC-MAIN-20180818154147-20180818174147-00584.warc.gz"} |
http://mathoverflow.net/questions/93877/hyperelliptic-curves-over-characteristic-two-fields?sort=oldest | # Hyperelliptic curves over characteristic two fields
I have been looking at hyperelliptic curves over an algebraically closed field $k$ of characteristic two, with a view towards finding the basis for the vector space of holomorphic differentials. To do this I have viewed the curves as the function field $k(x,y)$, originally restricting to those defined by $$y^2 - y = f(x), \ f(x)\in k[x].$$
After this, I thought that to generalise to all hyperelliptic curves I should allow $f(x)$ to be any rational function. However, looking at the literature it seems like the definition is instead:
A hyperelliptic curve of genus $g$ ($g\geq 1$) is an equation of the form $$y^2 - h(x) y = f(x),\ f(x),h(x)\in k[x],$$ where the degree of $h(x)$ is at most $g$, and $f(u)$ is a monic polynomial of degree $2g +1$, with no elements of $k\times k$ satisfying the original equation and both of it's partial derivatives.
I don't see what was wrong with my initial intuition, so if anyone could tell me, or explain why the definition given is correct, I would be much obliged.
edit - to give fuller definition
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You can write any quadratic extension of $k(x)$ as $y^2 - y = f(x)$ for some rational function $f$ (Artin-Schreier). If you're given the second form $\eta^2 - h(x) \eta = \phi(x)$, divide by $h^2$ to get $y^2 - y = \phi(x) / h^2(x)$ where $y = \eta/h(x)$. Note that the point at infinity is then a pole of odd order of $\phi/h^2$, and thus a rational Weierstrass point of the curve; indeed a hyperelliptic curve in characteristic $2$ can be put in that form if and only if it has a rational Weierstrass point. In your setting that's automatic because you assumed $k$ is algebraically closed. – Noam D. Elkies Apr 12 '12 at 16:07
That is great, though I feel that I should have seen that before now! Thank you very much. – Tait Apr 12 '12 at 16:14
You're welcome. One correction: I should have written that any separable quadratic extension can be written in that Artin-Schreier form. – Noam D. Elkies Apr 13 '12 at 0:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9310675859451294, "perplexity": 132.10778599934434}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435376073161.33/warc/CC-MAIN-20150627033433-00268-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://codefreshers.com/air-blimp-solution-codechef/ | # [Solution] Air Blimp solution codechef
Air Blimp solution codechef – There are N cities in a row. The i-th city from the left has a sadness of A_i.
In an attempt to reduce the sadness of the cities, you can send blimps from the left of city 1 that move rightwards (i.e, a blimp crosses cities 1, 2, \ldots in order)
## [Solution] Air Blimp solution codechef
You are given two integers X and Y. For each blimp sent, you can make one of the following choices:
• Let the blimp fly over every city, in which case the sadness of every city will decrease by Y, or,
• Choose a city i (1 \leq i \leq N), and shower confetti over city i. In this case, the sadness of cities 1, 2, \ldots, i-1 will decrease by Y, the sadness of city i will decrease by X, and cities i+1, \ldots, N see no change in sadness.
Find the minimum number of blimps needed such that, by making the above choice optimally for each blimp, you can ensure that no city is sad (i.e, in the end every city has sadness \leq 0).
### Input Format
• The first line of input contains a single integer T — the number of test cases. Then the test cases follow.
• The first line of each test case contains three space-separated integers N, X, Y — the size of the array, and the parameters mentioned in the statement.
• The second line of each test case contains N space-separated integers A_1, A_2, \ldots, A_N, denoting the sadness of the N cities.
### Output Format
For each test case, output on a new line the minimum number of blimps needed such that no city is sad.
## Air Blimp solution codechef
• 1 \leq T \leq 10^4
• 1 \leq N \leq 3\cdot10^5
• 1 \leq X,Y \leq 10^9
• 0 \leq A_i \leq 10^9
• The sum of N over all test cases does not exceed 3\cdot10^5
### Sample 1:
Input
Output
3
4 4 4
1 5 4 4
5 4 3
1 4 3 3 5
4 3 1
3 1 3 9
2
2
3
## Air Blimp solution codechef Explanation
Test case 1: One way of using two blimps is as follows:
• Initially, A = [1, 5, 4, 4]
• Shower confetti on city 2. Now, A = [-3, 1, 4, 4].
• Shower confetti on city 4. Now, A = [-7, -3, 0, 0] and we are done.
Test case 2: One way of using two blimps is as follows:
• Initially, A = [1, 4, 3, 3, 5]
• Let a blimp fly over every city. Now, A = [-2, 1, 0, 0, 2].
• Shower confetti on city 5. Now, A = [-5, -2, -3, -3, -2], and we are done. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.93917316198349, "perplexity": 2191.092023161424}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571483.70/warc/CC-MAIN-20220811164257-20220811194257-00637.warc.gz"} |
https://www.physicsforums.com/threads/how-to-show-that-these-sets-are-nonempty.622737/ | # How to show that these sets are nonempty
1. Jul 22, 2012
### glebovg
How to show that these sets are nonempty ($\mid$ means "divides")?
Here N is an arbitrary large integer and q is some fixed integer.
${R_{k,q}} = \{ k \in {\mathbb N}:(kN\mid k!) \wedge ((k - 1)N\mid k!) \wedge \cdots \wedge (N\mid k!) \wedge (k > Nq)\}$
${S_{k,q}} = \{ k \in {\mathbb N}:({(2k - 1)^2}N\mid k!) \wedge ({(2k - 3)^2}N\mid k!) \wedge \ldots \wedge (N\mid k!) \wedge (k > Nq)\}$
${T_{k,q}} = \{ k \in {\mathbb N}:({k^5}N\mid k!) \wedge ({(k - 1)^5}N\mid k!) \wedge \ldots \wedge (N\mid k!) \wedge (k > Nq)\}$
They exist by the axiom schema of separation, but how do I determine which $k$ to choose so that it satisfies all the properties? Is there a general approach?
Last edited by a moderator: Feb 4, 2013
Can you offer guidance or do you also need help?
Draft saved Draft deleted
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https://cs.stackexchange.com/questions/69603/is-this-language-0n1m-nm-textis-even-regular-or-not/69605 | # Is this language $\{0^n1^m | (n+m) \text{is even}\}$ regular or not?
$$L=\{0^n1^m | (n+m) \text{is even}\}$$
1. Using the pumping lemma , let me choose a valid string $W=0011$ $(n=2,m=2$ and $4$ is even$)$.
2. Let the number of states in the DFA ($C$) be $3$ (I find this step to be fishy)
3. $|W|\ge C$
4. Let $W=XYZ$
5. I choose $X=00,Y=1,Z=1$ such that $|xy| \le C ,|Y| \ge1$
6. So, for all $i$, $00(1)^{i}1$ should be in $L$
7. But when $i=2$,String $00111$ is not in $L$, hence this language is not regular.
But I came across a regular expression for the same language:
$$0(00)^*1(11)^*+(00)^*(11)^*$$ Is there a flaw in my pumping lemma or the regular expression? Someone please help me out.
• Should i chose the pumping length as 'p' only, and not some constant value? – Vinod Pn Jan 31 '17 at 17:11
• Note that you can use LaTeX here to typeset mathematics in a more readable way. See here for a short introduction. – Raphael Jan 31 '17 at 18:09
• This language is indeed regular -- constructing an NFA is a (very easy) exercise. – Raphael Jan 31 '17 at 18:11
You seem to have misunderstood the pumping lemma.
Taking the definition from Introduction to the Theory of Computation by Michael Sipser
If $A$ is a regular language, then there is a number $p$ (the pumping length) where if $s$ is any string in $A$ of length at least $p$, then $s$ may be divided into three pieces, $s = xyz$, satisfying the following conditions:
1. for each $i ≥ 0, xy^i z \in A$,
2. $|y| > 0$, and
3. $|xy| \le p$.
A few points:
• The lemma only says that a pumping length $p$ exists. You do not get to choose the value of $p$. So, you cannot say that let $p=3$.
• For every string $s$ in the language with length at least $p$, it may be divided into three pieces, $s=xyz$. Notice that this again talks only of the existence of a way to split $s$ into $xyz$ given that $A$ is regular. So, you do not get to choose how the string splits into $x$, $y$, and $z$ in the general case. Finding a split that does not satisfy the conditions does not show that the language is not regular. You need to find such a string that cannot be split as stipulated by the definition to show that a language is not regular.
• Finally, the pumping lemma does not say what happens if $A$ is not regular. So, even a non-regular language may satisfy it, and the pumping lemma cannot be used to show that a language is regular.
You pumping lemma is wrong. The pumping lemma only works on strings that are sufficiently long, yours is not. If we have a string that is sufficiently long(longer then the number of states in the corresponding dfa or nfa) we must have repeated some states, which means that there is a loop somewhere in the path through the dfa to generate this string. We can then infinity repeat this loop and the string is still valid. Your string of length 4 is not sufficiently long to use the pumping lemma on. The language is regular. There are only 2 situations regarding valid strings in this language. If the number of 0's is even, the numbers of 1's is even. And if the number of 0's is odd the number of 1's is odd. Your regular expression covers both of these.
• Actually, since there is a 4-state FA for the OP's language, the string $0011$ will work in the PL. However, that's immaterial, since the language is regular and so any PL proof that it isn't will necessarily be wrong. – Rick Decker Jan 31 '17 at 19:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8970167636871338, "perplexity": 215.70665281502872}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575541308149.76/warc/CC-MAIN-20191215122056-20191215150056-00234.warc.gz"} |
http://www.theresearchkitchen.com/archives/date/2007/07 | # Financial Engineering Cheat Sheet
Here is a small cheat sheet that I wrote up a week or so ago to aid in one of my exams…if you’re into this kind of stuff, then you might find it useful. I’ll just upload the PDF for now, but if you want the TeX source then let me know. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9508343935012817, "perplexity": 358.3378460437112}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917120694.49/warc/CC-MAIN-20170423031200-00291-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://splab.net/papers/paper2000_07-html/ | On the important modulation-frequency bands of speech for human speaker recognition
Proc. of the International Conf. on Spoken Language Processing (ICSLP), Vol. 3, pp. 774-777, Beijing, 2000
On the important modulation-frequency bands of speech for human speaker recognition
T. Arai, M. Takahashi, N. Kanedera, Y. Takano and Y. Murahara
Abstract: By means of perceptual experiments, we investigated what range of modulation frequency components of the mel-frequency cepstral coefficients (MFCC) contains the most important information for speaker identification. In our study, we conducted two perceptual experiments using an MFCC-based re-synthesis scheme with two types of excitation. In Experiment I, speech sounds were re-synthesized from the extracted pitch and white noise. In Experiment II, speech sounds were re-synthesized only from white noise to avoid including pitch information. For each experiment the original speech sounds were uttered by two sets of five professors. A total of 44 students (16 for Exp. I and 28 for Exp. II) who attend the professors’ classes participated in the experiments. We analyzed the experimental results in order to estimate the relative importance of different modulation frequencies in speaker recognition. The results show that the most important speaker information was in modulation frequency components from 2 to 8 Hz for both Exp. I (pitch-excited) and Exp. II (noise-excited). These results also show that some contribution was derived from including modulation frequency components around 0 Hz. Hence, we concluded that dynamic features are important for human speaker identification as well as static features.
[PDF (40 kB)] | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8115360736846924, "perplexity": 2889.5875590637456}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337531.3/warc/CC-MAIN-20221005011205-20221005041205-00091.warc.gz"} |
https://www.physicsforums.com/threads/projectile-motion-difficulty-level-7.61042/ | # Homework Help: Projectile Motion(difficulty level-7)
1. Jan 24, 2005
### XtremeChic4
A football is kicked at an angle of 50 degrees and travels a distance of 20 m before hitting the ground.
What is the initial speed of the ball? (14.1 m/s)
How long is it in flight? (2.2 sec)
How high does it rise? (5.94 m)
I have the answers but I don't know where to go with the information that is given. If anyone could help explain this problem, it would be greatly appreciated! Thanks!
2. Jan 24, 2005
### vincentchan
difficulty level: 0.0000000000000000000000000000000000000000000000001
hint:
1. the x and y components are independent.
2. write down the equation for x and y component (the final v_y is -v_y)
3. the initial V_y and V_x are related by tan(theta)
4. you have 3 eq 3 unkn (v_x,v_y,t)....... solve it
3. Jan 24, 2005
### XtremeChic4
alrighty..i think i understand what your saying. Thanks! I just have problems understanding how to get the velocities. Thanks again! | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9416700601577759, "perplexity": 1550.1289711069026}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267863411.67/warc/CC-MAIN-20180620031000-20180620051000-00459.warc.gz"} |
https://www.physicsforums.com/threads/stupid-question.172378/ | # Stupid question
1. May 31, 2007
### blumfeld0
This question concerns Omega(matter+dark+lambda).
We take into account protons(baryons), electrons(leptons), neutrinos, pions, etc etc when trying to determine Omega from the CMB. but what about photons? I mean there are A LOT more photons than baryons. All those photons carry energy, shouldn't that be taken into account somehow when measuring Omega? are they? perhaps the lambda portion atleast?
i think i have some fundamental misunderstading here.
thanks
2. May 31, 2007
### pervect
Staff Emeritus
Yes, we do need to take into account photons. However, at the current epoch of the universe, they don't contribute very much to Omega, because the universe is matter dominated.
Even as early as recombination at z=1100, the effect of photons is small enough that it is usually ignored in cosmological calculators (for instance Ned Wright's distance calculator doesn't include radiation terms).
However, if you go back far enough in time, photons (or more generally, radiation) are the dominant factor in omega. This is in the so-called "radiation dominated era".
I recall Space Tiger saying that in terms of energy density, the CMB is currently the dominant form of energy in the universe. It's not very large, but it's everywhere, while light from suns (for instance) is only found inside galaxies and is not present in intergalactic space.
I believe ST has given references for this statement in another post, but I couldn't find it
3. May 31, 2007
### blumfeld0
Yes. thank you very much.
"However, at the current epoch of the universe, they don't contribute very much to Omega, because the universe is matter dominated."
It is the sum of all Omega's(matter/darkmatter/lambda) that cosmologists are trying to determine is equal to, greater than or less than one. But Omega is the ratio of density now over the critical density where the critical density is a function of Hubble's "constant" which itself is a function of time.
density critical ~ H(t)^2.
So my question is assuming the Omega matter = .3 and omega lamba =.7 today so the sum = 1(approximately), that does not mean that Omega matter = .3 and omega lamba =.7, say, 5 billion years ago? correct?
but the sum did equal 1 (approximately)? is that right? if so, why would the sum be the same but the ratios different?
i guess maybe I would understand this if i understood the graph on page 51 of this paper better?
http://panisse.lbl.gov/public/papers/conley06/cmagic_cosmology.pdf
4. May 31, 2007
### cesiumfrog
Isn't that contradictory? Do you mean that on large scales CMB is the dominant form of radiation?
Last edited: May 31, 2007
5. May 31, 2007
### blumfeld0
Isn't the dominant form of energy ( density) dark energy, the energy associated with empty space?
6. Jun 1, 2007
### Wallace
Yes this is correct. Different forms of energy evolve differently as the Universe expands. Matter energy density for instance goes as the inverse cube of the scale factor, or equivalently, with the inverse of volume. So you double the volume you halve the energy density, makes sense.
For photons, they get the inverse cube for the same reasons as matter, but they also lose energy as the Universe expands, going as the inverse of the scale factor this means that as the universe expands radiation density drops faster than matter. Hence in the past radiation was dominant but the energy density in radiation dropped faster than matter, leading to a matter dominated era.
The cosmological constant is different again. The energy density actually stays constant (hence the name) so as the Universe expands further and the matter density drops eventually Lambda becomes dominant. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9151045083999634, "perplexity": 899.1934738198954}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794865023.41/warc/CC-MAIN-20180523004548-20180523024548-00517.warc.gz"} |
https://www.szj.io/ml/2018/11/15/SVM-Remark.html | # SVM Remark
November 15, 2018 - 2 minute read -
• The original Vision for SVM: Maximize the distance of points to the separation hyper plane and put them in different sides. We can formulate the problem in the following form:
, where $d_\beta ( \mathbf{x} )$ is distance of feature $\mathbf{x}$ to the separation plane with paramter $\beta$.
• But it is sometimes harder to solve the min-max style optimization problems. Hence we construct a proxy problem to find the solution for $\beta$.
, where $d_\beta ( \mathbf{x} )$ is computed based on $1/ \mid \beta \mid$. Thus, we can translate the problem into
• It is sometimes inveitable that some error will occur. Thus we introduce the idea of soft margin and modify the loss function for this problem. We want to optimize the loss function:
when $C$ is huge, the tolerance for error classificaitonis low. Alternatively, we can also consider loss function with the following form:
Under this scenario, if $\lambda$ is huge, then the marin should be big and the tolerance for error classification is high.
• For the loss function, we can consider a so-called hinge loss, which is
. It get the name because the graph of the function looks like a hinge. Finally, the output loss function is
. Because $y_i(\beta\mathbf{x}-\alpha)$ should be larger than $1$, this equation will penalize points that wrongly calssified. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9715245962142944, "perplexity": 535.366542546725}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655899931.31/warc/CC-MAIN-20200709100539-20200709130539-00289.warc.gz"} |
http://mathhelpforum.com/calculus/73948-alternating-series-test.html | # Thread: Alternating Series Test
1. ## Alternating Series Test
Been doing Alternating Series tests but not sure how to figure out the absolute error of the series...
(-1) ^k+1 / K ; n=7
Any help or suggestions? I know i have to use the formula :
l S- Sn l lessthan/equal to An+1
Thank you for your time & help!
2. An alternating series converges as long as the sequence itself goes is a decreasing sequence (decreasing to 0). On thing that means is that each partial sum lies between the previous two partial sums. The true sum of the series must lie between any two partial sums: Since $\sum_{i=0}^\infty (-1)^i a_i$ lies between $\sum_{i=0}^n (-1)^i a_i$ and [tex]\sum_{i=0}^{n+1} (-1)^i a_i[tex], the distance from one of those to the correct value cannot be larger than the difference betwween those two values: the error cannot be larger than $\left|\sum_{i= 0}^{n+1}(-1)^ia_i- \sum_{i=0}^{n}(-1)^ia_i\right|= \left| a_i\right|$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9722497463226318, "perplexity": 567.362009983375}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738662133.5/warc/CC-MAIN-20160924173742-00163-ip-10-143-35-109.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/163717/find-the-derivative-of-the-primitive-of-a-discontinous-function | Find the derivative of the primitive of a discontinous function
I have problems solving the following task:
I have the function $f(x)=\sin\frac{1}{x}$ when x isn't $0$ and $f(0) = 1$
First, I must prove that $f(x)$ is integrable in every interval $[a, b]$.
Second, I have $F(x) = \int_0^x f(t)dt$. I must find the derivative of $F(x)$.
What I've done so far is this: $f(x)$ is integrable over every $[a,b]$ which doesn't include $0$ and for $0$ I take a Riemman sum with $0$ in the interval. There I just split the interval and take the normal Riemman sums $f(c_i) * \Delta x_i$ and take $c_i$ to be $f(0)$ when $\Delta x_i$ has $0$. Obviously if I take the $lim_{\Delta x_i \rightarrow 0} f(c_i) \Delta x_i$ it exists and therefore there is an integral. Is this correct?
For the second part I've managed to prove that $F(x)$ is continous at $0$ but can't prove that it's differentiable there. Everywhere else $F'(x) = f(x)$ from the Leibniz-Newton theorem.
-
I'd write a comment, but the next formulae are too long. So, you need to compute $$\lim_{h \to 0} \frac{\int_0^h \sin \frac{1}{t}\, dt}{h}.$$ Put $u=1/t$, $dt = -\frac{du}{u^2}$. Therefore you boil down to $$\lim_{h \to 0} \frac{\int_{1/h}^{+\infty} \frac{\sin u}{u^2} du}{h}.$$ Integrating by parts, $$\int_{1/h}^{+\infty} \frac{\sin u}{u^2} du = \left[ -\frac{\cos u}{u^2} \right]_{1/h}^{+\infty} - \int_{1/h}^{+\infty} \frac{\cos u}{u^3}du.$$ Using the trivial estimate $|\cos u | \leq 1$, the last integral is $O(h^2)$. You should recall that $$h \cos \frac{1}{h} \to 0$$ as $h \to 0$. Your limit is therefore equal to 0.
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Thank you for your quick response :) Just two questions because integration isn't really my strongest side: 1) When you change variables at u=1/t how does the lower one become 1/h(I mean, how can I explain it) 2) When you integrate by parts, can I just say that the second integral is also $O(h^2)$ and therefore doesn't affect the limit? – Tson Jun 27 '12 at 11:54
On contraire, @Tson ! By being $\,\mathcal O(h^2)\,$ , the second integral divided by $\,h\,$ has limit zero when $\,h\to 0\,$...And when $\,u=\frac{1}{t}\,$ , if $\,t\to\infty\,$ then $\,u\to 0\,$. The exchange between upper and lower limits in the new integral is due to the minus sign in $\,dt=-\frac{du}{u^2}\,$ – DonAntonio Jun 27 '12 at 12:12
I think I get it now :) Thanks again, didn't expect help so quickly :) – Tson Jun 27 '12 at 12:15
I think a little more care is due where you wrote
"There I just split the interval and take the normal Riemman sums $\,f(c i )Δx_i\,$ and take $\,c_i\,$ to be $\,f(0)\,$ when $\,Δx_i\,$ has $\,0$ ."
Well, you don't have to take $\,c_i=0\,$ in any subinterval of a partition of an interval containing $\,0\,$ . In fact, you can't so restrict the choice of $\,c_i\,$'s in every such subinterval unless you know beforehand the function's integrable there, which is precisely what you want to prove!
The continuation of your argument though shows how to mend the above, as it's pretty easy to show that the limit of the Riemann's Sums' exists...
As for differentiability: I don't think $\,F(x)\,$ is differentiable at $\,x=0\,$ but I cannot find a straightforward proof of this right now.
I can only support my suspicion by pointing out that, applying L'Hospital to $\,\displaystyle{\lim_{x\to 0}\frac{F(x)-F(0)}{x}}\,$ , the limit doesn't exist...which, of course, proves nothing. I'll add if I come up with something.
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Thanks, I was also thinking about proving the limit exists by using the upper and lower darbu sums(and using the fact that $f(0)$ is still equal to the max of the function), can that work ? – Tson Jun 27 '12 at 11:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9823192954063416, "perplexity": 224.63624039755962}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1404776438539.21/warc/CC-MAIN-20140707234038-00005-ip-10-180-212-248.ec2.internal.warc.gz"} |
https://asmedigitalcollection.asme.org/manufacturingscience/article-abstract/93/1/53/427285/Theory-of-Shaking-Moment-Optimization-of-Force?redirectedFrom=fulltext | A least-square theory for the optimization of the shaking moment of fully force-balanced inline four-bar linkages, running at constant input angular velocity, is presented. It is shown that the optimum conditions of partial moment balance are given by certain link mass distribution ratios. These optimum ratios are found to be functions of link length ratios only.
This content is only available via PDF. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9127568602561951, "perplexity": 1202.2306694219849}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027316021.66/warc/CC-MAIN-20190821131745-20190821153745-00278.warc.gz"} |
http://www.ques10.com/p/7956/antenna-and-wave-propagation-question-paper-jun-20/ | Question Paper: Antenna and Wave Propagation : Question Paper Jun 2015 - Electronics & Telecom Engineering (Semester 6) | Pune University (PU)
0
Antenna and Wave Propagation - Jun 2015
Electronics & Telecom Engineering (Semester 6)
TOTAL MARKS: 100
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks.
Answer any one question from Q1 and Q2
1 (a) What is polarization of wave? Explain the polarization of three types of wave with the help of relevant diagram?(8 marks) 1 (b) Write a short note on
i) Ionospheric abnormalities
ii) Multihop propagation
(6 marks)
1 (c) A lossless resonant λ/2 dipole antenna with input impedance of 73Ω is to be connected to a transmission line whose characteristic impedance is 50Ω. Assuming that the pattern the antenna is given approximately by U = Bo sin3?. Find the overall maximum gain of this antenna.(6 marks) 2 (a) What is Poynting vector? What is its significance? Derive an expression for Poynting vector?(6 marks) 2 (b) Explain antenna radiation mechanism in detail.(6 marks) 2 (c) Calculate the mean excess delay, RMS delay spread, and the maximum excess delay (10dB) for the multipath profile given in the figure below. Estimate the 50% coherence bandwidth of the channel. (8 marks)
Answer any one question from Q3 and Q4
3 (a) Derive the expression for radiation resistance of Infinitesimal Dipole.(9 marks) 3 (b) Derive the expression for radiation resistance of small dipole antenna.(9 marks) 4 (a) Calculate the radiation resistance of a double turn and an eight turn small circular loop when radius of loop is λ/10 and the medium is free spACe. Calculate its efficiency if loss resistance is 25Ω.(8 marks) 4 (b) Derive mathematical expression for power density and radiation intensity of half wave dipole antenna and draw radiation pattern of half wave dipole antenna in E and H plane.(10 marks)
Answer any one question from Q5 and Q6
5 (a) Write a short notes on
i) Pattern Multiplication.
ii) Binomial Array.
(8 marks)
5 (b) Design a broad side Dolph-Tschebhysheff array of five elements with half wavelength spACing between elements and with major to minor lobe ratio to be 19dB. Find the excitation coefficients & array fACtor.(8 marks) 6 (a) Explain planar array. State its advantages and applications.(6 marks) 6 (b) An Endfire array with element spACed at λ/2 and with axes of elements at right angles to the line of array is required to have directivity of 36. Determine -the array length and the width of major lobe.(5 marks) 6 (c) Give the comparison of broadside and End fire antenna array.(5 marks)
Answer any one question from Q7 and Q8
7 (a) Give structure details, radiation pattern, specification and application of Super-turnstile Antenna.(5 marks) 7 (b) What is meant by Rhombic Antenna? Explain its construction and operating principle.(5 marks) 7 (c) Write a short notes on following antennas with respect to structural details, radiation pattern features and applications.
i) Hertz antenna
ii) Lens Antenna
(6 marks)
8 (a) Write short notes on the following antennas.
i) Whip antenna
ii) Slot Antenna
iii) Microstrip patch antenna
(12 marks)
8 (b) A paraboloidal reflector antenna with diameter 20m is designed to operate at frequency of 6 GHz and illumination efficiency of 0.54. calculate the antenna gain in decibels.(4 marks) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8575995564460754, "perplexity": 4682.746765323027}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583826240.93/warc/CC-MAIN-20190122034213-20190122060213-00106.warc.gz"} |
https://proofwiki.org/wiki/Definition:Differentiable_Manifold | # Definition:Topological Manifold/Differentiable Manifold
Let $M$ be a second-countable locally Euclidean space of dimension $d$.
Let $\mathscr F$ be a $d$-dimensional differentiable structure on $M$ of class $\mathcal C^k$, where $k \ge 1$.
Then $\left({M, \mathscr F}\right)$ is a differentiable manifold of class $C^k$ and dimension $d$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9880527257919312, "perplexity": 79.26304270065322}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315258.34/warc/CC-MAIN-20190820070415-20190820092415-00280.warc.gz"} |
http://tex.stackexchange.com/questions/30868/matrix-gauss-package-problem-with-alignment?answertab=active | # Matrix gauss package problem with alignment
So far I am using the gauss document package, because it offers to show row operations. Now If i try to add fractions inside the matricies, or - signs things gets messy...
The text will not align vertically. Although I have found a semifix for this, is there a cleaner and easier way?
So far a minimal example looks like this
\documentclass[10pt,a4paper]{article}
\usepackage{gauss}
\usepackage{mathtools}
\newcommand{\m}{\llap{-}}
\begin{document}
\begin{align*}
\intertext{Wrong} \\
X\begin{gmatrix}[b]
- 1 & 0 & 1 \\
1 & \frac{3}{4} & 0 \\
3 & 1 & - 1
\end{gmatrix}
\\
(a)
X\begin{gmatrix}[b]
- 1 & 0 & 1 \\
1 & 1 & 0 \\
3 & 1 & - 1
\end{gmatrix}
\end{align*}
\\
Almost correct
\\
\begin{align*}
X\begin{gmatrix}[b]
\; \m 1 & 0 & 1 \\
\; 1 & \frac{3}{4} & 0 \\
\; 3 & 1 & \m 1
\end{gmatrix}
\\
(a)
X\begin{gmatrix}[b]
\; \m 1 & 0 & 1 \\
\; 1 & 1 & 0 \\
\; 3 & 1 & \m 1
\end{gmatrix}
\end{align*}
\\
Another example \\
\\
\begin{align*}
&\begin{gmatrix}[b]
1 & 2 & -3 & 4 \\
3 & -1 & 5 & 2 \\
4 & 1 & (a^2-2) & a+4
\rowops
\end{gmatrix}
&& \sim &&
\begin{gmatrix}[b]
1 & 2 & -3 & 4 \\
0 & -7 & 14 & -10 \\
0 & -7 & (a^2+10) & a-12
\rowops
\mult{1}{-\cdot \frac{1}{7}}
\end{gmatrix} \\
&\begin{gmatrix}[b]
1 & 2 & -3 & 4 \\
0 & 1 & -2 & \frac{10}{7} \\
0 & 0 & (a^2-4) & a-2
\rowops
\end{gmatrix}
&& \sim &&
\begin{gmatrix}[b]
1 & 0 & 1 & \frac{8}{7} \\
0 & 1 & -2 & \frac{10}{7} \\
0 & 0 & (a^2-4) & a-2
\rowops
\mult{2}{\cdot \frac{1}{a^2-4}}
\end{gmatrix}
\\
&\begin{gmatrix}[b]
1 & 0 & 1 & \frac{8}{7} \\
0 & 1 & -2 & \frac{10}{7} \\
0 & 0 & 1 & \frac{a-2}{a^2-4}
\rowops
\end{gmatrix}
&& \sim &&
\begin{gmatrix}[b]
1 & 0 & 0 & \frac{1}{7} \frac{8a+ 9}{a+2} \\
0 & 1 & 0 & \frac{2}{7} \frac{5a+17}{a+2} \\
0 & 0 & 1 & \frac{1}{a+2}
\end{gmatrix}
\end{align*}
\end{document}
Any better ways to do this? I also think this is a tedious way to write matricies... I have to perhaps write 25-50 of these a week
-
Thanks for providing a MWE. But the above does not compile for me. – Peter Grill Oct 7 '11 at 19:08
then you did something wrong ... – Herbert Oct 7 '11 at 19:19
If you're just interested in typesetting matrices, then see: Where is the \matrix command? – Werner Oct 7 '11 at 19:23
@Herbert: I believe that the MWE has been modified since I posted the comment. We should probably remove these comments as they maybe confusing to future visitors... – Peter Grill Oct 7 '11 at 20:00
A (perhaps ugly) way of making the cells right aligned is to add
\makeatletter
\edef\g@post{\relax$} \makeatother to the preamble, after \usepackage{gauss}. This is just a redefinition of a line from the package, removing an \hfil. The last example of yours will then look like this: Not sure if this is exactly what you're after though ... - Or: \makeatletter, \edef\g@prae{\hfil$}, \edef\g@post{\$}, \makeatother in the preamble for even & right-aligned columns. – Mico Oct 8 '11 at 2:10
\documentclass[10pt,a4paper]{article}
\usepackage{mathtools}
\begin{document}
\begin{align*}
X\begin{bmatrix*}[r]
- 1 & 0 & 1 \\
1 & \frac{3}{4} & 0 \\
3 & 1 & - 1
\end{bmatrix*}
\\
(a)
X\begin{bmatrix*}[r]
-1 & 0 & 1 \\
1 & 1 & 0 \\
3 & 1 & -1
\end{bmatrix*}
\end{align*}
\end{document}
-
Problem is I need the extra features. I will update the first post to display some of these. – N3buchadnezzar Oct 7 '11 at 19:29
@Herbert ... is the minus the right size? It looks like a dash! – Yiannis Lazarides Oct 7 '11 at 19:44
the matrix is set in math mode, so it is the right character – Herbert Oct 7 '11 at 19:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9737698435783386, "perplexity": 650.9768083591985}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375095373.99/warc/CC-MAIN-20150627031815-00273-ip-10-179-60-89.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/123368-power-series-expansion-ii-print.html | # power series expansion II
• Jan 11th 2010, 09:24 PM
harveyo
power series expansion II
$
f (x) = 4/ x^2 - 2x
$
= 4 ( 1/ (x^2 - 2x))
= 4 ( 1/ 1- (1-x)^2)
• Jan 11th 2010, 10:46 PM
CaptainBlack
Quote:
Originally Posted by harveyo
$
f (x) = 4/ x^2 - 2x
$
= 4 ( 1/ (x^2 - 2x))
= 4 ( 1/ 1- (1-x)^2)
More context needed.
Also:
$
f (x) = \frac{4}{x^2 - 2x}=-\frac{2}{x(1-x/2)}
$
should give a series expansion for $|x|<2$ and $x\ne 0$
CB
• Jan 12th 2010, 03:06 AM
harveyo
Context: write the power series expansion, indicate an interval of convergence when center c = 1
• Jan 12th 2010, 04:42 AM
Krizalid
Okay well, in order to do that, I'll give you an example:
We have $\frac1{1-x}=\sum_{j=0}^{\infty}x^j,$ then put $x\mapsto x-1$ and you'll get the series centered at $x=1.$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9881789088249207, "perplexity": 3412.126359495401}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917121752.57/warc/CC-MAIN-20170423031201-00494-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/can-somebody-explain-me-this.90533/ | Can somebody explain me this?
1. Sep 24, 2005
Kruger
I postet the formula below (attachment). I don't know how we can get from the first step to the second. I mean why does that n fall "out" of the square root?
Attached Files:
• integra.bmp
File size:
87.6 KB
Views:
113
2. Sep 24, 2005
mathman
Step 1a, let v=u+n2, then lower limit of v is n2
Step 1b, let u=v
I presume w is independent of u.
Similar Discussions: Can somebody explain me this? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9946638345718384, "perplexity": 3368.471976818377}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818695375.98/warc/CC-MAIN-20170926085159-20170926105159-00428.warc.gz"} |
https://science.sciencemag.org/content/338/6114/1606.abstract | Report
Sign-Problem–Free Quantum Monte Carlo of the Onset of Antiferromagnetism in Metals
See allHide authors and affiliations
Science 21 Dec 2012:
Vol. 338, Issue 6114, pp. 1606-1609
DOI: 10.1126/science.1227769
Abstract
The quantum theory of antiferromagnetism in metals is necessary for our understanding of numerous intermetallic compounds of widespread interest. In these systems, a quantum critical point emerges as external parameters (such as chemical doping) are varied. Because of the strong coupling nature of this critical point and the “sign problem” plaguing numerical quantum Monte Carlo (QMC) methods, its theoretical understanding is still incomplete. Here, we show that the universal low-energy theory for the onset of antiferromagnetism in a metal can be realized in lattice models, which are free from the sign problem and hence can be simulated efficiently with QMC. Our simulations show Fermi surface reconstruction and unconventional spin-singlet superconductivity across the critical point.
View Full Text | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8624568581581116, "perplexity": 1583.2784386812875}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315811.47/warc/CC-MAIN-20190821065413-20190821091413-00417.warc.gz"} |
https://ch.mathworks.com/help/symbolic/sym.polylog.html | # polylog
## Syntax
``Li = polylog(n,x)``
## Description
example
````Li = polylog(n,x)` returns the polylogarithm of the order `n` and the argument `x`.```
## Examples
### Polylogarithms of Numeric and Symbolic Arguments
`polylog` returns floating-point numbers or exact symbolic results depending on the arguments you use.
Compute the polylogarithms of numeric input arguments. The `polylog` function returns floating-point numbers.
`Li = [polylog(3,-1/2), polylog(4,1/3), polylog(5,3/4)]`
```Li = -0.4726 0.3408 0.7697```
Compute the polylogarithms of the same input arguments by converting them to symbolic objects. For most symbolic (exact) numbers, `polylog` returns unresolved symbolic calls.
`symA = [polylog(3,sym(-1/2)), polylog(sym(4),1/3), polylog(5,sym(3/4))]`
```symA = [ polylog(3, -1/2), polylog(4, 1/3), polylog(5, 3/4)]```
Approximate the symbolic results with the default number of 32 significant digits by using `vpa`.
`Li = vpa(symA)`
```Li = [ -0.47259784465889687461862319312655,... 0.3407911308562507524776409440122,... 0.76973541059975738097269173152535]```
The `polylog` function also accepts noninteger values of the order `n`. Compute `polylog` for complex arguments.
`Li = polylog(-0.2i,2.5)`
```Li = -2.5030 + 0.3958i```
### Explicit Expressions for Polylogarithms
If the order of the polylogarithm is `0`, `1`, or a negative integer, then `polylog` returns an explicit expression.
The polylogarithm of `n = 1` is a logarithmic function.
```syms x Li = polylog(1,x)```
```Li = -log(1 - x)```
The polylogarithms of `n < 1` are rational expressions.
`Li = polylog(0,x)`
```Li = -x/(x - 1)```
`Li = polylog(-1,x)`
```Li = x/(x - 1)^2```
`Li = polylog(-2,x)`
```Li = -(x^2 + x)/(x - 1)^3```
`Li = polylog(-3,x)`
```Li = (x^3 + 4*x^2 + x)/(x - 1)^4```
`Li = polylog(-10,x)`
```Li = -(x^10 + 1013*x^9 + 47840*x^8 + 455192*x^7 + ... 1310354*x^6 + 1310354*x^5 + 455192*x^4 +... 47840*x^3 + 1013*x^2 + x)/(x - 1)^11```
### Special Values
The `polylog` function has special values for some parameters.
If the second argument is `0`, then the polylogarithm is equal to `0` for any integer value of the first argument. If the second argument is `1`, then the polylogarithm is the Riemann zeta function of the first argument.
```syms n Li = [polylog(n,0), polylog(n,1)]```
```Li = [ 0, zeta(n)]```
If the second argument is `-1`, then the polylogarithm has a special value for any integer value of the first argument except `1`.
```assume(n ~= 1) Li = polylog(n,-1)```
```Li = zeta(n)*(2^(1 - n) - 1)```
To do other computations, clear the assumption on `n` by recreating it using `syms`.
`syms n`
Compute other special values of the polylogarithm function.
`Li = [polylog(4,sym(1)), polylog(sym(5),-1), polylog(2,sym(i))]`
```Li = [ pi^4/90, -(15*zeta(5))/16, catalan*1i - pi^2/48]```
### Plot Polylogarithms
Plot the polylogarithms of the integer orders `n` from -3 to 1 within the interval `x = [-4 0.3]`.
```syms x for n = -3:1 fplot(polylog(n,x),[-4 0.3]) hold on end title('Polylogarithm') legend('show','Location','best') hold off```
### Handle Expressions Containing Polylogarithms
Many functions, such as `diff` and `int`, can handle expressions containing `polylog`.
Differentiate these expressions containing polylogarithms.
```syms n x dLi = diff(polylog(n, x), x) dLi = diff(x*polylog(n, x), x)```
```dLi = polylog(n - 1, x)/x dLi = polylog(n, x) + polylog(n - 1, x)```
Compute the integrals of these expressions containing polylogarithms.
```intLi = int(polylog(n, x)/x, x) intLi = int(polylog(n, x) + polylog(n - 1, x), x)```
```intLi = polylog(n + 1, x) intLi = x*polylog(n, x)```
## Input Arguments
collapse all
Order of the polylogarithm, specified as a number, array, symbolic number, symbolic variable, symbolic function, symbolic expression, or symbolic array.
Data Types: `single` | `double` | `sym` | `symfun`
Complex Number Support: Yes
Argument of the polylogarithm, specified as a number, array, symbolic number, symbolic variable, symbolic function, symbolic expression, or symbolic array.
Data Types: `single` | `double` | `sym` | `symfun`
Complex Number Support: Yes
collapse all
### Polylogarithm
For a complex number `z` of modulus ```|z| < 1```, the polylogarithm of order `n` is defined as:
`${\mathrm{Li}}_{n}\left(z\right)=\sum _{k=1}^{\infty }\frac{{z}^{k}}{{k}^{n}}.$`
Analytic continuation extends this function the whole complex plane, with a branch cut along the real interval [`1`, ∞) for `n` ≥ 1.
## Tips
• `polylog(2,x)` is equivalent to ```dilog(1 - x)```.
• The logarithmic integral function (the integral logarithm) uses the same notation, li(x), but without an index. The toolbox provides the `logint` function to compute the logarithmic integral function.
• Floating-point evaluation of the polylogarithm function can be slow for complex arguments or high-precision numbers. To increase the computational speed, you can reduce the floating-point precision by using the `vpa` and `digits` functions. For more information, see Increase Speed by Reducing Precision.
• The polylogarithm function is related to other special functions. For example, it can be expressed in terms of the Hurwitz zeta function ζ(s,a) and the gamma function Γ(z):
`${\text{Li}}_{n}\left(z\right)=\frac{\Gamma \left(1-n\right)}{{\left(2\pi \right)}^{1-n}}\text{ }\text{\hspace{0.17em}}\left[{i}^{1-n}\zeta \left(1-n,\frac{1}{2}+\frac{\mathrm{ln}\left(-z\right)}{2\pi i}\right)\text{\hspace{0.17em}}+{i}^{n-1}\text{\hspace{0.17em}}\zeta \left(1-n,\frac{1}{2}-\frac{\mathrm{ln}\left(-z\right)}{2\pi i}\right)\right].$`
Here, n ≠ 0, 1, 2, ....
## References
[1] Olver, F. W. J., A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds., Chapter 25. Zeta and Related Functions, NIST Digital Library of Mathematical Functions, Release 1.0.20, Sept. 15, 2018.
## Version History
Introduced in R2014b | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9404106736183167, "perplexity": 2471.1722580559117}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296950422.77/warc/CC-MAIN-20230402074255-20230402104255-00402.warc.gz"} |
https://www.mysciencework.com/publication/show/introduction-supersymmetric-spin-networks-b512d167 | # Introduction to supersymmetric spin networks
Authors
Type
Published Article
Publication Date
Submission Date
Identifiers
DOI: 10.1063/1.1421423
arXiv ID: hep-th/0009020
Source
arXiv
In this paper we give a general introduction to supersymmetric spin networks. Its construction has a direct interpretation in context of the representation theory of the superalgebra. In particular we analyze a special kind of spin networks with superalgebra $Osp(1|2n)$. It turns out that the set of corresponding spin network states forms an orthogonal basis of the Hilbert space $\cal L\mit^2(\cal A\mit/\cal G)$, and this argument holds even in the q-deformed case. The $Osp(n|2)$ spin networks are also discussed briefly. We expect they could provide useful techniques to quantum supergravity and gauge field theories from the point of non-perturbative view. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.975308895111084, "perplexity": 386.05133240263814}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886106779.68/warc/CC-MAIN-20170820150632-20170820170632-00594.warc.gz"} |
https://en.wikipedia.org/wiki/Relational_quantum_mechanics | # Relational quantum mechanics
Relational quantum mechanics (RQM) is an interpretation of quantum mechanics which treats the state of a quantum system as being observer-dependent, that is, the state is the relation between the observer and the system. This interpretation was first delineated by Carlo Rovelli in a 1994 preprint, and has since been expanded upon by a number of theorists. It is inspired by the key idea behind special relativity, that the details of an observation depend on the reference frame of the observer, and uses some ideas from Wheeler on quantum information.[1]
The physical content of the theory has not to do with objects themselves, but the relations between them. As Rovelli puts it:
"Quantum mechanics is a theory about the physical description of physical systems relative to other systems, and this is a complete description of the world".[2]
The essential idea behind RQM is that different observers may give different accounts of the same series of events: for example, to one observer at a given point in time, a system may be in a single, "collapsed" eigenstate, while to another observer at the same time, it may appear to be in a superposition of two or more states. Consequently, if quantum mechanics is to be a complete theory, RQM argues that the notion of "state" describes not the observed system itself, but the relationship, or correlation, between the system and its observer(s). The state vector of conventional quantum mechanics becomes a description of the correlation of some degrees of freedom in the observer, with respect to the observed system. However, it is held by RQM that this applies to all physical objects, whether or not they are conscious or macroscopic (all systems are quantum systems). Any "measurement event" is seen simply as an ordinary physical interaction, an establishment of the sort of correlation discussed above. The proponents of the relational interpretation argue that the approach clears up a number of traditional interpretational difficulties with quantum mechanics, while being simultaneously conceptually elegant and ontologically parsimonious.
## History and development
Relational quantum mechanics arose from a historical comparison of the quandaries posed by the interpretation of quantum mechanics with the situation after the Lorentz transformations were formulated but before special relativity. Rovelli felt that just as there was an "incorrect assumption" underlying the pre-relativistic interpretation of Lorentz's equations, which was corrected by Einstein's deriving them from Lorentz covariance and the constancy of the speed of light in all reference frames, so a similarly incorrect assumption underlies many attempts to make sense of the quantum formalism, which was responsible for many of the interpretational difficulties posed by the theory. This incorrect assumption, he said, was that of an observer-independent state of a system, and he laid out the foundations of this interpretation to try to overcome the difficulty.[3]
The idea has been expanded upon by Lee Smolin[4] and Louis Crane,[5] who have both applied the concept to quantum cosmology, and the interpretation has been applied to the EPR paradox, revealing not only a peaceful co-existence between quantum mechanics and special relativity, but a formal indication of a completely local character to reality.[6][7]
David Mermin has contributed to the relational approach in his "Ithaca interpretation."[8] He uses the slogan "correlations without correlata", meaning that "correlations have physical reality; that which they correlate does not", so "correlations are the only fundamental and objective properties of the world". The moniker "zero worlds"[9] has been popularized by Ron Garret[10] to contrast with the many worlds interpretation.
## The problem of the observer and the observed
This problem was initially discussed in detail in Everett's thesis, The Theory of the Universal Wavefunction. Consider observer ${\displaystyle O}$, measuring the state of the quantum system ${\displaystyle S}$. We assume that ${\displaystyle O}$ has complete information on the system, and that ${\displaystyle O}$ can write down the wavefunction ${\displaystyle |\psi \rangle }$ describing it. At the same time, there is another observer ${\displaystyle O'}$, who is interested in the state of the entire ${\displaystyle O}$-${\displaystyle S}$ system, and ${\displaystyle O'}$ likewise has complete information.
To analyse this system formally, we consider a system ${\displaystyle S}$ which may take one of two states, which we shall designate ${\displaystyle |\uparrow \rangle }$ and ${\displaystyle |\downarrow \rangle }$, ket vectors in the Hilbert space ${\displaystyle H_{S}}$. Now, the observer ${\displaystyle O}$ wishes to make a measurement on the system. At time ${\displaystyle t_{1}}$, this observer may characterize the system as follows:
${\displaystyle |\psi \rangle =\alpha |\uparrow \rangle +\beta |\downarrow \rangle ,}$
where ${\displaystyle |\alpha |^{2}}$ and ${\displaystyle |\beta |^{2}}$ are probabilities of finding the system in the respective states, and obviously add up to 1. For our purposes here, we can assume that in a single experiment, the outcome is the eigenstate ${\displaystyle |\uparrow \rangle }$ (but this can be substituted throughout, mutatis mutandis, by ${\displaystyle |\downarrow \rangle }$). So, we may represent the sequence of events in this experiment, with observer ${\displaystyle O}$ doing the observing, as follows:
${\displaystyle {\begin{matrix}t_{1}&\rightarrow &t_{2}\\\alpha |\uparrow \rangle +\beta |\downarrow \rangle &\rightarrow &|\uparrow \rangle .\end{matrix}}}$
This is observer ${\displaystyle O}$'s description of the measurement event. Now, any measurement is also a physical interaction between two or more systems. Accordingly, we can consider the tensor product Hilbert space ${\displaystyle H_{S}\otimes H_{O}}$, where ${\displaystyle H_{O}}$ is the Hilbert space inhabited by state vectors describing ${\displaystyle O}$. If the initial state of ${\displaystyle O}$ is ${\displaystyle |{\text{init}}\rangle }$, some degrees of freedom in ${\displaystyle O}$ become correlated with the state of ${\displaystyle S}$ after the measurement, and this correlation can take one of two values: ${\displaystyle |O_{\uparrow }\rangle }$ or ${\displaystyle |O_{\downarrow }\rangle }$ where the direction of the arrows in the subscripts corresponds to the outcome of the measurement that ${\displaystyle O}$ has made on ${\displaystyle S}$. If we now consider the description of the measurement event by the other observer, ${\displaystyle O'}$, who describes the combined ${\displaystyle S+O}$ system, but does not interact with it, the following gives the description of the measurement event according to ${\displaystyle O'}$, from the linearity inherent in the quantum formalism:
${\displaystyle {\begin{matrix}t_{1}&\rightarrow &t_{2}\\\left(\alpha |\uparrow \rangle +\beta |\downarrow \rangle \right)\otimes |init\rangle &\rightarrow &\alpha |\uparrow \rangle \otimes |O_{\uparrow }\rangle +\beta |\downarrow \rangle \otimes |O_{\downarrow }\rangle .\end{matrix}}}$
Thus, on the assumption (see hypothesis 2 below) that quantum mechanics is complete, the two observers ${\displaystyle O}$ and ${\displaystyle O'}$ give different but equally correct accounts of the events ${\displaystyle t_{1}\rightarrow t_{2}}$.
## Central principles
### Observer-dependence of state
According to ${\displaystyle O}$, at ${\displaystyle t_{2}}$, the system ${\displaystyle S}$ is in a determinate state, namely spin up. And, if quantum mechanics is complete, then so is his description. But, for ${\displaystyle O'}$, ${\displaystyle S}$ is not uniquely determinate, but is rather entangled with the state of ${\displaystyle O}$ — note that his description of the situation at ${\displaystyle t_{2}}$ is not factorisable no matter what basis chosen. But, if quantum mechanics is complete, then the description that ${\displaystyle O'}$ gives is also complete.
Thus the standard mathematical formulation of quantum mechanics allows different observers to give different accounts of the same sequence of events. There are many ways to overcome this perceived difficulty. It could be described as an epistemic limitation — observers with a full knowledge of the system, we might say, could give a complete and equivalent description of the state of affairs, but that obtaining this knowledge is impossible in practice. But whom? What makes ${\displaystyle O}$'s description better than that of ${\displaystyle O'}$, or vice versa? Alternatively, we could claim that quantum mechanics is not a complete theory, and that by adding more structure we could arrive at a universal description (the troubled hidden variables approach). Yet another option is to give a preferred status to a particular observer or type of observer, and assign the epithet of correctness to their description alone. This has the disadvantage of being ad hoc, since there are no clearly defined or physically intuitive criteria by which this super-observer ("who can observe all possible sets of observations by all observers over the entire universe"[11]) ought to be chosen.
RQM, however, takes the point illustrated by this problem at face value. Instead of trying to modify quantum mechanics to make it fit with prior assumptions that we might have about the world, Rovelli says that we should modify our view of the world to conform to what amounts to our best physical theory of motion.[12] Just as forsaking the notion of absolute simultaneity helped clear up the problems associated with the interpretation of the Lorentz transformations, so many of the conundra associated with quantum mechanics dissolve, provided that the state of a system is assumed to be observer-dependent — like simultaneity in Special Relativity. This insight follows logically from the two main hypotheses which inform this interpretation:
• Hypothesis 1: the equivalence of systems. There is no a priori distinction that should be drawn between quantum and macroscopic systems. All systems are, fundamentally, quantum systems.
• Hypothesis 2: the completeness of quantum mechanics. There are no hidden variables or other factors which may be appropriately added to quantum mechanics, in light of current experimental evidence.
Thus, if a state is to be observer-dependent, then a description of a system would follow the form "system S is in state x with reference to observer O" or similar constructions, much like in relativity theory. In RQM it is meaningless to refer to the absolute, observer-independent state of any system.
### Information and correlation
It is generally well established that any quantum mechanical measurement can be reduced to a set of yes/no questions or bits that are either 1 or 0.[citation needed] RQM makes use of this fact to formulate the state of a quantum system (relative to a given observer!) in terms of the physical notion of information developed by Claude Shannon. Any yes/no question can be described as a single bit of information. This should not be confused with the idea of a qubit from quantum information theory, because a qubit can be in a superposition of values, whilst the "questions" of RQM are ordinary binary variables.
Any quantum measurement is fundamentally a physical interaction between the system being measured and some form of measuring apparatus. By extension, any physical interaction may be seen to be a form of quantum measurement, as all systems are seen as quantum systems in RQM. A physical interaction is seen as establishing a correlation between the system and the observer, and this correlation is what is described and predicted by the quantum formalism.
But, Rovelli points out, this form of correlation is precisely the same as the definition of information in Shannon's theory. Specifically, an observer O observing a system S will, after measurement, have some degrees of freedom correlated with those of S. The amount of this correlation is given by log2k bits, where k is the number of possible values which this correlation may take — the number of "options" there are.
### All systems are quantum systems
All physical interactions are, at bottom, quantum interactions, and must ultimately be governed by the same rules. Thus, an interaction between two particles does not, in RQM, differ fundamentally from an interaction between a particle and some "apparatus". There is no true wave collapse, in the sense in which it occurs in the Copenhagen interpretation.
Because "state" is expressed in RQM as the correlation between two systems, there can be no meaning to "self-measurement". If observer ${\displaystyle O}$ measures system ${\displaystyle S}$, ${\displaystyle S}$'s "state" is represented as a correlation between ${\displaystyle O}$ and ${\displaystyle S}$. ${\displaystyle O}$ itself cannot say anything with respect to its own "state", because its own "state" is defined only relative to another observer, ${\displaystyle O'}$. If the ${\displaystyle S+O}$ compound system does not interact with any other systems, then it will possess a clearly defined state relative to ${\displaystyle O'}$. However, because ${\displaystyle O}$'s measurement of ${\displaystyle S}$ breaks its unitary evolution with respect to ${\displaystyle O}$, ${\displaystyle O}$ will not be able to give a full description of the ${\displaystyle S+O}$ system (since it can only speak of the correlation between ${\displaystyle S}$ and itself, not its own behaviour). A complete description of the ${\displaystyle (S+O)+O'}$ system can only be given by a further, external observer, and so forth.
Taking the model system discussed above, if ${\displaystyle O'}$ has full information on the ${\displaystyle S+O}$ system, it will know the Hamiltonians of both ${\displaystyle S}$ and ${\displaystyle O}$, including the interaction Hamiltonian. Thus, the system will evolve entirely unitarily (without any form of collapse) relative to ${\displaystyle O'}$, if ${\displaystyle O}$ measures ${\displaystyle S}$. The only reason that ${\displaystyle O}$ will perceive a "collapse" is because ${\displaystyle O}$ has incomplete information on the system (specifically, ${\displaystyle O}$ does not know its own Hamiltonian, and the interaction Hamiltonian for the measurement).
## Consequences and implications
### Coherence
In our system above, ${\displaystyle O'}$ may be interested in ascertaining whether or not the state of ${\displaystyle O}$ accurately reflects the state of ${\displaystyle S}$. We can draw up for ${\displaystyle O'}$ an operator, ${\displaystyle M}$, which is specified as:
${\displaystyle M\left(|\uparrow \rangle \otimes |O_{\uparrow }\rangle \right)=|\uparrow \rangle \otimes |O_{\uparrow }\rangle }$
${\displaystyle M\left(|\uparrow \rangle \otimes |O_{\downarrow }\rangle \right)=0}$
${\displaystyle M\left(|\downarrow \rangle \otimes |O_{\uparrow }\rangle \right)=0}$
${\displaystyle M\left(|\downarrow \rangle \otimes |O_{\downarrow }\rangle \right)=|\downarrow \rangle \otimes |O_{\downarrow }\rangle }$
with an eigenvalue of 1 meaning that ${\displaystyle O}$ indeed accurately reflects the state of ${\displaystyle S}$. So there is a 0 probability of ${\displaystyle O}$ reflecting the state of ${\displaystyle S}$ as being ${\displaystyle |\uparrow \rangle }$ if it is in fact ${\displaystyle |\downarrow \rangle }$, and so forth. The implication of this is that at time ${\displaystyle t_{2}}$, ${\displaystyle O'}$ can predict with certainty that the ${\displaystyle S+O}$ system is in some eigenstate of ${\displaystyle M}$, but cannot say which eigenstate it is in, unless ${\displaystyle O'}$ itself interacts with the ${\displaystyle S+O}$ system.
An apparent paradox arises when one considers the comparison, between two observers, of the specific outcome of a measurement. In the problem of the observer observed section above, let us imagine that the two experiments want to compare results. It is obvious that if the observer ${\displaystyle O'}$ has the full Hamiltonians of both ${\displaystyle S}$ and ${\displaystyle O}$, he will be able to say with certainty that at time ${\displaystyle t_{2}}$, ${\displaystyle O}$ has a determinate result for ${\displaystyle S}$'s spin, but he will not be able to say what ${\displaystyle O}$'s result is without interaction, and hence breaking the unitary evolution of the compound system (because he doesn't know his own Hamiltonian). The distinction between knowing "that" and knowing "what" is a common one in everyday life: everyone knows that the weather will be like something tomorrow, but no-one knows exactly what the weather will be like.
But, let us imagine that ${\displaystyle O'}$ measures the spin of ${\displaystyle S}$, and finds it to have spin down (and note that nothing in the analysis above precludes this from happening). What happens if he talks to ${\displaystyle O}$, and they compare the results of their experiments? ${\displaystyle O}$, it will be remembered, measured a spin up on the particle. This would appear to be paradoxical: the two observers, surely, will realise that they have disparate results.
However, this apparent paradox only arises as a result of the question being framed incorrectly: as long as we presuppose an "absolute" or "true" state of the world, this would, indeed, present an insurmountable obstacle for the relational interpretation. However, in a fully relational context, there is no way in which the problem can even be coherently expressed. The consistency inherent in the quantum formalism, exemplified by the "M-operator" defined above, guarantees that there will be no contradictions between records. The interaction between ${\displaystyle O'}$ and whatever he chooses to measure, be it the ${\displaystyle S+O}$ compound system or ${\displaystyle O}$ and ${\displaystyle S}$ individually, will be a physical interaction, a quantum interaction, and so a complete description of it can only be given by a further observer ${\displaystyle O''}$, who will have a similar "M-operator" guaranteeing coherency, and so on out. In other words, a situation such as that described above cannot violate any physical observation, as long as the physical content of quantum mechanics is taken to refer only to relations.
### Relational networks
An interesting implication of RQM arises when we consider that interactions between material systems can only occur within the constraints prescribed by Special Relativity, namely within the intersections of the light cones of the systems: when they are spatiotemporally contiguous, in other words. Relativity tells us that objects have location only relative to other objects. By extension, a network of relations could be built up based on the properties of a set of systems, which determines which systems have properties relative to which others, and when (since properties are no longer well defined relative to a specific observer after unitary evolution breaks down for that observer). On the assumption that all interactions are local (which is backed up by the analysis of the EPR paradox presented below), one could say that the ideas of "state" and spatiotemporal contiguity are two sides of the same coin: spacetime location determines the possibility of interaction, but interactions determine spatiotemporal structure. The full extent of this relationship, however, has not yet fully been explored.
### RQM and quantum cosmology
The universe is the sum total of everything in existence with any possibility of direct or indirect interaction with a local observer. A (physical) observer outside of the universe would require physically breaking of gauge invariance,[13] and a concomitant alteration in the mathematical structure of gauge-invariance theory.
Similarly, RQM conceptually forbids the possibility of an external observer. Since the assignment of a quantum state requires at least two "objects" (system and observer), which must both be physical systems, there is no meaning in speaking of the "state" of the entire universe. This is because this state would have to be ascribed to a correlation between the universe and some other physical observer, but this observer in turn would have to form part of the universe. As was discussed above, it is not possible for an object to contain a complete specification of itself. Following the idea of relational networks above, an RQM-oriented cosmology would have to account for the universe as a set of partial systems providing descriptions of one another. The exact nature of such a construction remains an open question.
## Relationship with other interpretations
The only group of interpretations of quantum mechanics with which RQM is almost completely incompatible is that of hidden variables theories. RQM shares some deep similarities with other views, but differs from them all to the extent to which the other interpretations do not accord with the "relational world" put forward by RQM.
### Copenhagen interpretation
RQM is, in essence, quite similar to the Copenhagen interpretation, but with an important difference. In the Copenhagen interpretation, the macroscopic world is assumed to be intrinsically classical in nature, and wave function collapse occurs when a quantum system interacts with macroscopic apparatus. In RQM, any interaction, be it micro or macroscopic, causes the linearity of Schrödinger evolution to break down. RQM could recover a Copenhagen-like view of the world by assigning a privileged status (not dissimilar to a preferred frame in relativity) to the classical world. However, by doing this one would lose sight of the key features that RQM brings to our view of the quantum world.
### Hidden variables theories
Bohm's interpretation of QM does not sit well with RQM. One of the explicit hypotheses in the construction of RQM is that quantum mechanics is a complete theory, that is it provides a full account of the world. Moreover, the Bohmian view seems to imply an underlying, "absolute" set of states of all systems, which is also ruled out as a consequence of RQM.
We find a similar incompatibility between RQM and suggestions such as that of Penrose, which postulate that some process (in Penrose's case, gravitational effects) violate the linear evolution of the Schrödinger equation for the system.
### Relative-state formulation
The many-worlds family of interpretations (MWI) shares an important feature with RQM, that is, the relational nature of all value assignments (that is, properties). Everett, however, maintains that the universal wavefunction gives a complete description of the entire universe, while Rovelli argues that this is problematic, both because this description is not tied to a specific observer (and hence is "meaningless" in RQM), and because RQM maintains that there is no single, absolute description of the universe as a whole, but rather a net of inter-related partial descriptions.
### Consistent histories approach
In the consistent histories approach to QM, instead of assigning probabilities to single values for a given system, the emphasis is given to sequences of values, in such a way as to exclude (as physically impossible) all value assignments which result in inconsistent probabilities being attributed to observed states of the system. This is done by means of ascribing values to "frameworks", and all values are hence framework-dependent.
RQM accords perfectly well with this view. However, the consistent histories approach does not give a full description of the physical meaning of framework-dependent value (that is it does not account for how there can be "facts" if the value of any property depends on the framework chosen). By incorporating the relational view into this approach, the problem is solved: RQM provides the means by which the observer-independent, framework-dependent probabilities of various histories are reconciled with observer-dependent descriptions of the world.
## EPR and quantum non-locality
The EPR thought experiment, performed with electrons. A radioactive source (center) sends electrons in a singlet state toward two spacelike separated observers, Alice (left) and Bob (right), who can perform spin measurements. If Alice measures spin up on her electron, Bob will measure spin down on his, and vice versa.
RQM provides an unusual solution to the EPR paradox. Indeed, it manages to dissolve the problem altogether, inasmuch as there is no superluminal transportation of information involved in a Bell test experiment: the principle of locality is preserved inviolate for all observers.
### The problem
In the EPR thought experiment, a radioactive source produces two electrons in a singlet state, meaning that the sum of the spin on the two electrons is zero. These electrons are fired off at time ${\displaystyle t_{1}}$ towards two spacelike separated observers, Alice and Bob, who can perform spin measurements, which they do at time ${\displaystyle t_{2}}$. The fact that the two electrons are a singlet means that if Alice measures z-spin up on her electron, Bob will measure z-spin down on his, and vice versa: the correlation is perfect. If Alice measures z-axis spin, and Bob measures the orthogonal y-axis spin, however, the correlation will be zero. Intermediate angles give intermediate correlations in a way that, on careful analysis, proves inconsistent with the idea that each particle has a definite, independent probability of producing the observed measurements (the correlations violate Bell's inequality).
This subtle dependence of one measurement on the other holds even when measurements are made simultaneously and a great distance apart, which gives the appearance of a superluminal communication taking place between the two electrons. Put simply, how can Bob's electron "know" what Alice measured on hers, so that it can adjust its own behavior accordingly?
### Relational solution
In RQM, an interaction between a system and an observer is necessary for the system to have clearly defined properties relative to that observer. Since the two measurement events take place at spacelike separation, they do not lie in the intersection of Alice's and Bob's light cones. Indeed, there is no observer who can instantaneously measure both electrons' spin.
The key to the RQM analysis is to remember that the results obtained on each "wing" of the experiment only become determinate for a given observer once that observer has interacted with the other observer involved. As far as Alice is concerned, the specific results obtained on Bob's wing of the experiment are indeterminate for her, although she will know that Bob has a definite result. In order to find out what result Bob has, she has to interact with him at some time ${\displaystyle t_{3}}$ in their future light cones, through ordinary classical information channels.[14]
The question then becomes one of whether the expected correlations in results will appear: will the two particles behave in accordance with the laws of quantum mechanics? Let us denote by ${\displaystyle M_{A}(\alpha )}$ the idea that the observer ${\displaystyle A}$ (Alice) measures the state of the system ${\displaystyle \alpha }$ (Alice's particle).
So, at time ${\displaystyle t_{2}}$, Alice knows the value of ${\displaystyle M_{A}(\alpha )}$: the spin of her particle, relative to herself. But, since the particles are in a singlet state, she knows that
${\displaystyle M_{A}(\alpha )+M_{A}(\beta )=0,}$
and so if she measures her particle's spin to be ${\displaystyle \sigma }$, she can predict that Bob's particle (${\displaystyle \beta }$) will have spin ${\displaystyle -\sigma }$. All this follows from standard quantum mechanics, and there is no "spooky action at a distance" yet. From the "coherence-operator" discussed above, Alice also knows that if at ${\displaystyle t_{3}}$ she measures Bob's particle and then measures Bob (that is asks him what result he got) — or vice versa — the results will be consistent:
${\displaystyle M_{A}(B)=M_{A}(\beta )}$
Finally, if a third observer (Charles, say) comes along and measures Alice, Bob, and their respective particles, he will find that everyone still agrees, because his own "coherence-operator" demands that
${\displaystyle M_{C}(A)=M_{C}(\alpha )}$ and ${\displaystyle M_{C}(B)=M_{C}(\beta )}$
while knowledge that the particles were in a singlet state tells him that
${\displaystyle M_{C}(\alpha )+M_{C}(\beta )=0.}$
Thus the relational interpretation, by shedding the notion of an "absolute state" of the system, allows for an analysis of the EPR paradox which neither violates traditional locality constraints, nor implies superluminal information transfer, since we can assume that all observers are moving at comfortable sub-light velocities. And, most importantly, the results of every observer are in full accordance with those expected by conventional quantum mechanics.
## Derivation
A promising feature of this interpretation is that RQM offers the possibility of being derived from a small number of axioms, or postulates based on experimental observations. Rovelli's derivation of RQM uses three fundamental postulates. However, it has been suggested that it may be possible to reformulate the third postulate into a weaker statement, or possibly even do away with it altogether.[15] The derivation of RQM parallels, to a large extent, quantum logic. The first two postulates are motivated entirely by experimental results, while the third postulate, although it accords perfectly with what we have discovered experimentally, is introduced as a means of recovering the full Hilbert space formalism of quantum mechanics from the other two postulates. The two empirical postulates are:
• Postulate 1: there is a maximum amount of relevant information that may be obtained from a quantum system.
• Postulate 2: it is always possible to obtain new information from a system.
We let ${\displaystyle W\left(S\right)}$ denote the set of all possible questions that may be "asked" of a quantum system, which we shall denote by ${\displaystyle Q_{i}}$, ${\displaystyle i\in W}$. We may experimentally find certain relations between these questions: ${\displaystyle \left\{\land ,\lor ,\neg ,\supset ,\bot \right\}}$, corresponding to {intersection, orthogonal sum, orthogonal complement, inclusion, and orthogonality} respectively, where ${\displaystyle Q_{1}\bot Q_{2}\equiv Q_{1}\supset \neg Q_{2}}$.
### Structure
From the first postulate, it follows that we may choose a subset ${\displaystyle Q_{c}^{(i)}}$ of ${\displaystyle N}$ mutually independent questions, where ${\displaystyle N}$ is the number of bits contained in the maximum amount of information. We call such a question ${\displaystyle Q_{c}^{(i)}}$ a complete question. The value of ${\displaystyle Q_{c}^{(i)}}$ can be expressed as an N-tuple sequence of binary valued numerals, which has ${\displaystyle 2^{N}=k}$ possible permutations of "0" and "1" values. There will also be more than one possible complete question. If we further assume that the relations ${\displaystyle \left\{\land ,\lor \right\}}$ are defined for all ${\displaystyle Q_{i}}$, then ${\displaystyle W\left(S\right)}$ is an orthomodular lattice, while all the possible unions of sets of complete questions form a Boolean algebra with the ${\displaystyle Q_{c}^{(i)}}$ as atoms.[16]
The second postulate governs the event of further questions being asked by an observer ${\displaystyle O_{1}}$ of a system ${\displaystyle S}$, when ${\displaystyle O_{1}}$ already has a full complement of information on the system (an answer to a complete question). We denote by ${\displaystyle p\left(Q|Q_{c}^{(j)}\right)}$ the probability that a "yes" answer to a question ${\displaystyle Q}$ will follow the complete question ${\displaystyle Q_{c}^{(j)}}$. If ${\displaystyle Q}$ is independent of ${\displaystyle Q_{c}^{(j)}}$, then ${\displaystyle p=0.5}$, or it might be fully determined by ${\displaystyle Q_{c}^{(j)}}$, in which case ${\displaystyle p=1}$. There is also a range of intermediate possibilities, and this case is examined below.
If the question that ${\displaystyle O_{1}}$ wants to ask the system is another complete question, ${\displaystyle Q_{b}^{(i)}}$, the probability ${\displaystyle p^{ij}=p\left(Q_{b}^{(i)}|Q_{c}^{(j)}\right)}$ of a "yes" answer has certain constraints upon it:
1. ${\displaystyle 0\leq p^{ij}\leq 1,\ }$
2. ${\displaystyle \sum _{i}p^{ij}=1,\ }$
3. ${\displaystyle \sum _{j}p^{ij}=1.\ }$
The three constraints above are inspired by the most basic of properties of probabilities, and are satisfied if
${\displaystyle p^{ij}=\left|U^{ij}\right|^{2}}$,
where ${\displaystyle U^{ij}}$ is a unitary matrix.
• Postulate 3 If ${\displaystyle b}$ and ${\displaystyle c}$ are two complete questions, then the unitary matrix ${\displaystyle U_{bc}}$ associated with their probability described above satisfies the equality ${\displaystyle U_{cd}=U_{cb}U_{bd}}$, for all ${\displaystyle b,c}$ and ${\displaystyle d}$.
This third postulate implies that if we set a complete question ${\displaystyle |Q_{c}^{(i)}\rangle }$ as a basis vector in a complex Hilbert space, we may then represent any other question ${\displaystyle |Q_{b}^{(j)}\rangle }$ as a linear combination:
${\displaystyle |Q_{b}^{(j)}\rangle =\sum _{i}U_{bc}^{ij}|Q_{c}^{(i)}\rangle .}$
And the conventional probability rule of quantum mechanics states that if two sets of basis vectors are in the relation above, then the probability ${\displaystyle p^{ij}}$ is
${\displaystyle p^{ij}=|\langle Q_{c}^{(i)}|Q_{b}^{(j)}\rangle |^{2}=|U_{bc}^{ij}|^{2}.}$
### Dynamics
The Heisenberg picture of time evolution accords most easily with RQM. Questions may be labelled by a time parameter ${\displaystyle t\rightarrow Q(t)}$, and are regarded as distinct if they are specified by the same operator but are performed at different times. Because time evolution is a symmetry in the theory (it forms a necessary part of the full formal derivation of the theory from the postulates), the set of all possible questions at time ${\displaystyle t_{2}}$ is isomorphic to the set of all possible questions at time ${\displaystyle t_{1}}$. It follows, by standard arguments in quantum logic, from the derivation above that the orthomodular lattice ${\displaystyle W(S)}$ has the structure of the set of linear subspaces of a Hilbert space, with the relations between the questions corresponding to the relations between linear subspaces.
It follows that there must be a unitary transformation ${\displaystyle U\left(t_{2}-t_{1}\right)}$ that satisfies:
${\displaystyle Q(t_{2})=U\left(t_{2}-t_{1}\right)Q(t_{1})U^{-1}\left(t_{2}-t_{1}\right)}$
and
${\displaystyle U\left(t_{2}-t_{1}\right)=\exp({-i\left(t_{2}-t_{1}\right)H})}$
where ${\displaystyle H}$ is the Hamiltonian, a self-adjoint operator on the Hilbert space and the unitary matrices are an abelian group.
## Notes
1. ^ Wheeler (1990): pg. 3
2. ^ Rovelli, C. (1996), "Relational quantum mechanics", International Journal of Theoretical Physics, 35: 1637–1678.
3. ^ Rovelli (1996): pg. 2
4. ^ Smolin (1995)
5. ^ Crane (1993)
6. ^ Laudisa (2001)
7. ^ Rovelli & Smerlak (2006)
8. ^ Mermin, N.D. (1996, 1998).
9. ^ mikhailfranco (Nov 2008) web comment.
10. ^ Garret, R. (Jan 2011) "The Quantum Conspiracy: What Popularizers Of QM Don't Want You To Know" (slides, video),
11. ^ Page, Don N., "Insufficiency of the quantum state for deducing observational probabilities", Physics Letters B, Volume 678, Issue 1, 6 July 2009, 41-44.
12. ^ Rovelli (1996): pg. 16
13. ^ Smolin (1995), pg. 13
14. ^ Bitbol (1983)
15. ^ Rovelli (1996): pg. 14
16. ^ Rovelli (1996): pg. 13
## References
• Bitbol, M.: "An analysis of the Einstein–Podolsky–Rosen correlations in terms of events"; Physics Letters 96A, 1983: 66-70
• Crane, L.: "Clock and Category: Is Quantum Gravity Algebraic?"; Journal of Mathematical Physics 36; 1993: 6180-6193; arXiv:gr-qc/9504038.
• Everett, H.: "The Theory of the Universal Wavefunction"; Princeton University Doctoral Dissertation; in DeWitt, B.S. & Graham, R.N. (eds.): "The Many-Worlds Interpretation of Quantum Mechanics"; Princeton University Press; 1973.
• Finkelstein, D.R.: "Quantum Relativity: A Synthesis of the Ideas of Einstein and Heisenberg"; Springer-Verlag; 1996
• Garret, R.: "Quantum Mysteries Disentangled" (pdf), Nov 2001, revised Aug 2008, Feb 2015, Apr 2016.
• Floridi, L.: "Informational Realism"; Computers and Philosophy 2003 - Selected Papers from the Computer and Philosophy conference (CAP 2003), Conferences in Research and Practice in Information Technology, '37', 2004, edited by J. Weckert. and Y. Al-Saggaf, ACS, pp. 7–12. [1]
• Laudisa, F.: "The EPR Argument in a Relational Interpretation of Quantum Mechanics"; Foundations of Physics Letters, 14 (2); 2001: pp. 119–132; arXiv:quant-ph/0011016
• Laudisa, F. & Rovelli, C.: "Relational Quantum Mechanics"; The Stanford Encyclopedia of Philosophy (Fall 2005 Edition), Edward N. Zalta (ed.);online article.
• Mermin, N.D.: "The Ithaca Interpretation of Quantum Mechanics"; Pramana , 51 (1996): 549-565, arXiv:quant-ph/9609013.
• Mermin, N.D.: "What is Quantum Mechanics Trying to Tell us?"; American Journal of Physics, 66 (1998): 753-767, arXiv:quant-ph/9801057.
• Rovelli, C. & Smerlak, M.: "Relational EPR"; Preprint: arXiv:quant-ph/0604064.
• Rovelli, C.: "Relational Quantum Mechanics"; International Journal of Theoretical Physics 35; 1996: 1637-1678; arXiv:quant-ph/9609002.
• Smolin, L.: "The Bekenstein Bound, Topological Quantum Field Theory and Pluralistic Quantum Field Theory"; Preprint: arXiv:gr-qc/9508064.
• Wheeler, J. A.: "Information, physics, quantum: The search for links"; in Zurek,W., ed.: "Complexity, Entropy and the Physics of Information"; pp 3–28; Addison-Wesley; 1990. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 185, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8854286670684814, "perplexity": 543.3470153235818}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583657555.87/warc/CC-MAIN-20190116154927-20190116180927-00123.warc.gz"} |
https://math.stackexchange.com/questions/2730077/short-mathematical-notation-for-a-sequence-without-the-last-element/2730187 | # Short mathematical notation for a sequence without the last element
I have a sequence (with unique elements): $a = (1,4,9)$. Is there a short notation for the same set, but without the last element: $b = (1,4)$?
I came up with one solution, which simple removes the last item by index: $$b =a \setminus \{ a_{|a|-1}\}$$ or by taking all except the last: $$b = (a_0,...,a_{|a|-2})$$
However i am hoping to find something shorter, like $a_{-1}$ or something in that direction. Is there a standard way of doing this?
• If you are talking about vectors, I would use something like $b = a_{1:(n-1)}$ or $b = a_{\llbracket 1, n-1 \rrbracket}$, with $a\in\mathbb{R}^n$. – Karlo Apr 9 '18 at 22:51
• If you are talking about a set $a = {1, 4, 9}$, then $b = a$ \ $\{9\}$. – Karlo Apr 9 '18 at 22:52
• Addendum to the vector case: it seems that \llbracket and \rrbracket are not displayed correctly on SE, see also this question. – Karlo Apr 9 '18 at 22:57
Sometimes notation like $$(a_1,\ldots,\widehat{a_k},\ldots,a_n)=(a_1,\ldots,a_{k-1},a_{k +1},\ldots,a_n)$$ is used. The element with the hat is omitted. There are ways to shorten it further, such as $$(a_{\widehat{k}})$$
• How would one then write this for the last element, considering that is not upfront specified that $a_n$ is the last element? Something like: $a_ {\widehat{|a|-1}}$ ? – user3053216 Apr 10 '18 at 0:34
• In your case, $a=(1,4,9)=(a_1,a_2,a_3)$ with $a_1=1$, $a_2=4$, and $a_3=9$. Then $$(a_{\widehat{3}})=(a_1,a_2)=(1,4)$$ – MPW Apr 10 '18 at 0:41
• Ok I misunderstood. I thought you meant a sequence with known fixed length. If the length is variable, then your suggested amendment works. You can make up any notation that works, of course! For that matter, you could just declare that $\widehat{a}$ is the (finite) sequence $a$ with its last element omitted. – MPW Apr 10 '18 at 0:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8805676698684692, "perplexity": 332.69424305588444}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670743.44/warc/CC-MAIN-20191121074016-20191121102016-00224.warc.gz"} |
https://utahvalleypinewoodderby.com/equations-of-motion | Pinewood Derby Track Rental in Utah County
# Deriving the Equations of Motion
To simulate a car racing down the track, we need to derive the equations of motion of a car. Early on I settled on using Lagrangian mechanics to do this. Lagrangian mechanics has several advantages over the more typical Newtonian mechanics taught in high school physics. In Newtonian mechanics, two cartesian coordinates, x and y, would be needed to describe the position of the car on the track. With Lagrangian mechanics, only one generalized coordinate is needed. Additionally, constraints such as "the car has to follow the path of the track" are accounted for automatically in the definition of the generalized coordinate.
In our case, we'll call the coordinate $x$ and define it as the horizontal distance the car has traveled along the length of the track. This is not the same as the total distance the car has traveled, since the car also moves vertically as it travels down the incline. If we were to use Newtonian mechanics, we would have to carefully identify all forces involved in keeping the car constrained on the track surface, and that could quickly become unwieldy. In Lagrangian mechanics, that's all taken care of. The only thing we need to supply to determine the equations of motion is the potential energy $V$ as a function of $x$, and the kinetic energy $T$ as a function of $\dot{x}$--read "x dot", which means the time-derivative of $x$, or the car's horizontal speed. We'll take care of factoring in the vertical component of the speed later.
Then, a quantity called the Lagrangian is defined as the difference between the kinetic and potential energies:
$$L = T - V$$
Then, a fancy calculus equation involving $L$ is solved, and out pop the equations of motion. Since the speed of the car depends on some function of $\dot{x}$, the kinetic energy will be some function of $\dot{x}$. The potential energy $V$ is also some function of $x$. Adding in the rotational effects of the wheels won't be too bad either.
Earlier I discussed how I determined the curvature of the track, and arrived at a neat expression for the height of the track $w$ at cartesian coordinate $x$. However, to keep things as general as possible, and to allow for different models of curvature in the future, I will keep the height of the track $w$ as a function of $x$, but from now on we'll just be calling it $w$. The first and second derivatives of $w$ with respect to $x$ will be $w'$ and $w''$.'
The potential energy $V_{grav}$ due to gravity is given simply by:
$$V_{grav}(x) = -m g w$$
The kinetic energy is $1/2 m v^2$, where $v$ is the speed of the car. Because the car's velocity has a vertical component, this is not exactly the same as $\dot{x}$, which only represents the horizontal component. But we can use the slope of the track, $w'$, to obtain $v$ from $\dot{x}$:
$$v = \dot{x} \sqrt{1 + w'^2}$$
The kinetic energy of the car is then:
$$T_{lin}(x,\dot{x}) = \frac{1}{2} m \dot{x}^2 \left( 1 + w'^2 \right)$$
The $lin$ is for linear. The kinetic energy will also include a term for the rotational energy of the wheels. That term will take the same form as the above expression for $T_{lin}$, but with $m$ replaced by $\frac{4I}{r^2}$, where $I$ is the moment of inertia of one wheel and $r$ is its radius. (Do you know where the 4 comes from?)
$$T_{rot}(x,\dot{x}) = \frac{1}{2} \frac{4I}{r^2} \dot{x}^2 \left( 1 + w'^2 \right)$$
We would also like to be able to include a conservative (constant) friction force which, like gravity, will be accounted for in the potential energy $V(x)$. Since this force always acts in the direction opposite that of motion, its contribution to the potential energy can be determined by integrating the product of the magnitude of the force and the distance traveled by the car along the track. We won't have to evaluate this integral directly. And to further simplify things later on, we'll be assuming $F_{fric}$ is constant within a single simulation timestep, and so in some places where we take derivatives, the $F_{fric}$ term will drop out.
$$V_{fric}(x,\dot{x}) = \int F_{fric}(x,\dot{x}) \sqrt{ 1 + w'^2 } dx$$
Now that we have expressions for $T$ and $V$, and recalling the defintion $L = T - V$, we can extract the equations of motion by solving this differential equation:
$$\frac{ \mathrm{d} }{ \mathrm{d}t } \left( \frac{ \partial L }{ \partial \dot{x} } \right) = \frac{ \partial L }{ \partial x }$$
The first step is to evaluate the partial derivatives of $L$ with respect to $x$ and $\dot{x}$. The entire expression for $L$ is
\begin{align} \\ L & = T - V \\ & = T_{lin}(x,\dot{x}) + T_{rot}(x,\dot{x}) - V_{grav}(x) - V_{fric}(x,\dot{x}) \\ & = \frac{1}{2} \left(m + \frac{4I}{r^2}\right) \dot{x}^2 \left( 1 + w'^2 \right) - \left(-m g w \right) - \int F_{fric}(x,\dot{x}) \sqrt{ 1 + w'^2 } dx \end{align}
And so the partial derivatives are:
$$\frac{ \partial L }{ \partial \dot{x} } = \left(m + \frac{4I}{r^2}\right) \dot{x} \left[ 1 + w'^2 \right]$$ $$\frac{ \partial L }{ \partial x } = \frac{1}{2} \left(m + \frac{4I}{r^2}\right) \dot{x}^2 \left( 2 w' w'' \right) + m g w' - F_{fric}(x,\dot{x}) \sqrt{1+w'^2}$$
Now we need to take the time derivative of $\frac{\partial L}{\partial \dot{x} }$:
$$\frac{ \mathrm{d} }{ \mathrm{d}t } \left( \frac{ \partial L }{ \partial \dot{x} } \right) = \left(m + \frac{4I}{r^2}\right) \left[ \ddot{x} \left(1+w'^2\right) + \dot{x}^2 \left(2 w' w'' \right) \right]$$
We now have everything we need to determine the equations of motion. Recall the differential equation above:
$$\frac{ \mathrm{d} }{ \mathrm{d}t } \left( \frac{ \partial L }{ \partial \dot{x} } \right) = \frac{ \partial L }{ \partial x }$$
And now, we substitute the expressions we determined above and solve for $\ddot{x}$:
$$\left(m + \frac{4I}{r^2}\right) \left[ \ddot{x} \left(1+w'^2\right) + \dot{x}^2 \left(2 w' w'' \right) \right] = \frac{1}{2} \left(m + \frac{4I}{r^2}\right) \dot{x}^2 \left( 2 w' w'' \right) + m g w' - F_{fric}(x,\dot{x}) \sqrt{1+w'^2}$$
$$\ddot{x} = \frac{m g}{m + 4I/r^2} \frac{w'}{1+w'^2} - \frac{F_{fric}(x,\dot{x})}{m + 4I/r^2} \frac{1}{\sqrt{1+w'^2}} - \dot{x}^2 \frac{w' w'' }{1+w'^2}$$
It's interesting that $\dot{x}$ appears in the final term. Apart from friction, the acceleration should primarily depend on the position of the car, not its velocity. Where does that term come from?
## Free-body approach
Let's try the Newtonian mechanics approach and see how it compares. Below we've drawn a diagram of a pinewood derby car.
A free body diagram of a pinewood derby car, showing how the forces of gravity, friction, and the track surface combine.
The car has mass $m$. The four wheels each have moment of inertia $I$, and radius $r$. The car moves along a path defined by $w(x)$. For simplicity we'll ignore the moment of inertia of the car itself, which is probably okay since the rotational speed of the car body is slow and doesn't rotate for a long period of time during a race.
Most free body diagrams include things like angles and cosines and other trig functions in order to resolve vector components. This is totally unnecessary. In the lower left, we've included a small triangle. The width of the triangle is $1$, the height is the slope of the track $w'$, and the hypotenuse is the arc length of the track, $\sqrt{1+w'^2}$. We'll use the properties of similar triangles to resolve all vector components.
There are many forces that act on the car. The first we will consider is gravity. It is represented by the black vector pointing straight down, of length $mg$. The component of this vector parallel to the track is what really matters, so by similar triangles, we've drawn a red vector of length $mg \frac{w'}{\sqrt{1+w'^2}}$. Finally, the component of this vector in the x-direction is shown by the green vector, of length: $$mg \frac{w'}{1+w'^2}$$
Friction acts parallel to the track surface and is shown by the purple vector. We'll use the similar triangle trick again to resolve its x-component, shown by the blue vector:
$$\frac{-F_{fric}}{\sqrt{1+w'^2}}$$
As the car moves along curved portions of the track, the track exerts a force on the car normal to its surface, separate from gravity and friction, which we've already accounted for. The resulting acceleration due to this force is $v^2/R$, where $v$ is the speed of the car and $R$ is the radius of curvature of the track. The radius of curvature of our track is given by:
$$R = \frac{\left( 1+w'^2 \right)^{3/2}}{w''}$$
And so the magnitude of the acceleration due to the normal force is:
$$N = \frac{v^2}{R} = \left(\dot{x} \sqrt{1+w'^2}\right)^2 \frac{w''}{\left( 1+w'^2 \right)^{3/2}} = \dot{x}^2 \frac{w''}{\sqrt{1+w'^2}}$$
Then we use our trick with similar triangles to get the component of that acceleration in the x-direction:
$$N_x = \dot{x}^2 \frac{w''}{\sqrt{1+w'^2}} \frac{w'}{\sqrt{1+w'^2}} = \dot{x}^2 \frac{w' w''}{1+w'^2}$$
Finally, we combine the mass of the car and the moments of inertia of the wheels to create an effective, or total, mass, $m_{tot} = m + 4I/r^2$. We divide forces acting on the car by this total mass (the acceleration due to the normal force already takes this into account). When we add everything up, we get this:
$$\ddot{x} = \frac{m g}{m + 4I/r^2} \frac{w'}{1+w'^2} - \frac{F_{fric}(x,\dot{x})}{m + 4I/r^2} \frac{1}{\sqrt{1+w'^2}} - \dot{x}^2 \frac{w' w'' }{1+w'^2}$$
This is identical to the acceleration we obtained using Lagrangian mechanics. Now we know the origin of the mysterious final term that contains $\dot{x}$: it arises from the normal force, because curved portions of the track exert a force on the car. If this term were not present, the car would accelerate down the starting section of the track, and then unnaturally slow down when the track curved to the horizontal. Maybe if we had used arc-length instead of x-coordinate as the independent variable, this normal force wouldn't matter. But then the algebra would have been much more difficult.
The Newtonian method gave us the same result as the Lagrangian, but we had to consider ALL forces acting on the car, AND resolve their components correctly. It is very easy to make mistakes. In fact, there are additional forces I didn't show in the free-body diagram, and I've been sloppy with keeping track of my signs, but the comparison with the result from Lagrangian mechanics is still informative.
The biggest advantage of using Lagrangian instead of Newtonian mechanics is that you need only be able to evaluate the energy of the system, then all the forces that act on the system will appear once you do the derivatives. Another advantage is that we can choose whichever independent variable we want, and everything will work out. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9752552509307861, "perplexity": 159.3888174074317}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027313715.51/warc/CC-MAIN-20190818062817-20190818084817-00208.warc.gz"} |
http://mathhelpforum.com/calculus/25980-cauchy-sequence.html | # Math Help - Cauchy Sequence
1. ## Cauchy Sequence
Not sure on the proof of this:
Prove directly from the definition that the sequence
$(\frac{sin(log(n^{2002})) + 5000}{2n+7})_n$
is cauchy
2. Originally Posted by Jason Bourne
Not sure on the proof of this:
Prove directly from the definition that the sequence
$(\frac{sin(log(n^{2002})) + 5000}{2n+7})_n$
is cauchy
I will state the following result. Your task is to prove it and show how it applies to your problem. Suppose that $\{a_n\},\{b_n\}$ are sequences such that $0\leq a_n \leq b_n$. Let $\{ b_n \}$ be Cauchy sequences and $\lim ~ a_n = \lim ~ b_n$. Prove that $\{ a_n \}$ is a Cauchy sequence.
3. Are you implying the General Principle of Convergence?
4. Originally Posted by Jason Bourne
Are you implying the General Principle of Convergence?
Frankly, I do not know what to make of the original question.
It is well know that a sequence of real numbers converges if and only if it is a Cauchy Sequence.
Take the above hint: Use basic comparison.
If you can show that your sequence converges then it is Cauchy.
5. Originally Posted by Plato
Frankly, I do not know what to make of the original question.
It is well know that a sequence of real numbers converges if and only if it is a Cauchy Sequence.
Take the above hint: Use basic comparison.
If you can show that your sequence converges then it is Cauchy.
I also do not know why his professor would not let him the underlined theorem. Unless, the professor does not want to use completeness. Since completeness does not work in arbitrary metric spaces.
6. I don't know what time it is where you guys live but I'm going to sleep. Thanks anyway.
7. Originally Posted by ThePerfectHacker
Unless, the professor does not want to use completeness. Since completeness does not work in arbitrary metric spaces.
I agree. But in arbitrary metric if a sequence converges then it is a Cauchy Sequence. (The other way requires completeness.) Thus if he shows by comparison the sequence converges it is Cauchy.
8. Originally Posted by Plato
I agree. But in arbitrary metric if a sequence converges then it is a Cauchy Sequence. (The other way requires completeness.) Thus if he shows by comparison the sequence converges it is Cauchy.
Yes ladies and gentlemen (I make no unjustified assumptions here). All of this is what I thought too, but didn't quite feel confident enough with my analysis to suggest it. I was interested to see what others thought before jumping in. A combination of comparison and then the definition would seem to satisfy the boundary conditions.
I quite agree with the puzzle of why his prof won't let the theorem that
a sequence of real numbers converges if and only if it is a Cauchy Sequence
be used. Using this theorem was my first thought.
Unless his prof wants them to practice the whole epsilon thing .....
I suspect the whole metric thing raised by TPH is probably a red herring in this case ......
9. Originally Posted by mr fantastic
Yes ladies and gentlemen (I make no unjustified assumptions here). All of this is what I thought too, but didn't quite feel confident enough with my analysis to suggest it. I was interested to see what others thought before jumping in. A combination of comparison and then the definition would seem to satisfy the boundary conditions.
I quite agree with the puzzle of why his prof won't let the theorem that
a sequence of real numbers converges if and only if it is a Cauchy Sequence
be used. Using this theorem was my first thought.
Unless his prof wants them to practice the whole epsilon thing .....
I suspect the whole metric thing raised by TPH is probably a red herring in this case ......
Indeed, I think the question wanted proof from the definition of a Cauchy sequence but it does not seem neccessary really. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.98382967710495, "perplexity": 415.4552010990697}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257827080.38/warc/CC-MAIN-20160723071027-00229-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://walkccc.github.io/CLRS/Chap35/Problems/35-5/ | # 35-5 Parallel machine scheduling
In the parallel-machine-scheduling problem, we are given $n$ jobs, $J_1, J_2, \dots, J_n$, where each job $J_k$ has an associated nonnegative processing time of $p_k$. We are also given $m$ identical machines, $M_1, M_2, \dots, M_m$. Any job can run on any machine. A schedule specifies, for each job $J_k$, the machine on which it runs and the time period during which it runs. Each job $J_k$ must run on some machine $M_i$ for $p_k$ consecutive time units, and during that time period no other job may run on $M_i$. Let $C_k$ denote the completion time of job $J_k$, that is, the time at which job $J_k$ completes processing. Given a schedule, we define $C_{\max} = \max_{1 \le j \le n} C_j$ to be the makespan of the schedule. The goal is to find a schedule whose makespan is minimum.
For example, suppose that we have two machines $M_1$ and $M_2$ and that we have four jobs $J_1, J_2, J_3, J_4$, with $p_1 = 2$, $p_2 = 12$, $p_3 = 4$, and $p_4 = 5$. Then one possible schedule runs, on machine $M_1$, job $J_1$ followed by job $J_2$, and on machine $M_2$, it runs job $J_4$ followed by job $J_3$. For this schedule, $C_1 = 2$, $C_2 = 14$, $C_3 = 9$, $C_4 = 5$, and $C_{\max} = 14$. An optimal schedule runs $J_2$ on machine $M_1$, and it runs jobs $J_1$, $J_3$, and $J_4$ on machine $M_2$. For this schedule, $C_1 = 2$, $C_2 = 12$, $C_3 = 6$, $C_4 = 11$, and $C_{\max} = 12$.
Given a parallel-machine-scheduling problem, we let $C_{\max}^*$ denote the makespan of an optimal schedule.
a. Show that the optimal makespan is at least as large as the greatest processing time, that is,
$$C_{\max}^* \ge \max_{1 \le k \le n} p_k.$$
b. Show that the optimal makespan is at least as large as the average machine load, that is,
$$C_{\max}^* \ge \frac 1 m \sum_{1 \le k \le n} p_k.$$
Suppose that we use the following greedy algorithm for parallel machine scheduling: whenever a machine is idle, schedule any job that has not yet been scheduled.
c. Write pseudocode to implement this greedy algorithm. What is the running time of your algorithm?
d. For the schedule returned by the greedy algorithm, show that
$$C_{\max} \le \frac 1 m \sum_{1 \le k \le n} p_k + \max_{1 \le k \le n} p_k.$$
Conclude that this algorithm is a polynomial-time $2$-approximation algorithm.
(Omit!) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8926089406013489, "perplexity": 319.78575848265035}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998100.52/warc/CC-MAIN-20190616102719-20190616124719-00512.warc.gz"} |
https://webwork.libretexts.org/webwork2/html2xml?answersSubmitted=0&sourceFilePath=Library/UCSB/Stewart5_3_11/Stewart5_3_11_21.pg&problemSeed=1234567&courseID=anonymous&userID=anonymous&course_password=anonymous&showSummary=1&displayMode=MathJax&problemIdentifierPrefix=102&language=en&outputformat=libretexts | (a) Find the differential $\text{dy}$ of $y= x^2+2x+ 3$.
(b) Evaluate $\text{dy}$ for $x=3$, $\text{dx}=1/2$.
(a) $\text{dy} =$ $\text{dx}$
(b) $\text{dy} =$ | {"extraction_info": {"found_math": true, "script_math_tex": 8, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.833393394947052, "perplexity": 760.8883828019344}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663012542.85/warc/CC-MAIN-20220528031224-20220528061224-00516.warc.gz"} |
https://www.askmehelpdesk.com/electrical-lighting/what-will-happen-if-wire-hot-both-terminals-110-outlet-613734.html | Why did my 110 receptacle outlet work at half strength even though both terminals where connected with hot wires (instead of 1 neutral ,1 hot) how is this possible? There is no complete circuit and if it did work why did my stuff that I plugged in to the outlet work weakly for example I tested a light bulb in it was very dim... what the heck is going on ANYONE EXPLAIN PLEASE ? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8658425211906433, "perplexity": 718.6633942376773}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945668.34/warc/CC-MAIN-20180422232447-20180423012447-00637.warc.gz"} |
https://terrytao.wordpress.com/2015/07/ | You are currently browsing the monthly archive for July 2015.
This week I have been at a Banff workshop “Combinatorics meets Ergodic theory“, focused on the combinatorics surrounding Szemerédi’s theorem and the Gowers uniformity norms on one hand, and the ergodic theory surrounding Furstenberg’s multiple recurrence theorem and the Host-Kra structure theory on the other. This was quite a fruitful workshop, and directly inspired the various posts this week on this blog. Incidentally, BIRS being as efficient as it is, videos for this week’s talks are already online.
As mentioned in the previous two posts, Ben Green, Tamar Ziegler, and myself proved the following inverse theorem for the Gowers norms:
Theorem 1 (Inverse theorem for Gowers norms) Let ${N \geq 1}$ and ${s \geq 1}$ be integers, and let ${\delta > 0}$. Suppose that ${f: {\bf Z} \rightarrow [-1,1]}$ is a function supported on ${[N] := \{1,\dots,N\}}$ such that
$\displaystyle \frac{1}{N^{s+2}} \sum_{n,h_1,\dots,h_{s+1} \in {\bf Z}} \prod_{\omega \in \{0,1\}^{s+1}} f(n+\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}) \geq \delta.$
Then there exists a filtered nilmanifold ${G/\Gamma}$ of degree ${\leq s}$ and complexity ${O_{s,\delta}(1)}$, a polynomial sequence ${g: {\bf Z} \rightarrow G}$, and a Lipschitz function ${F: G/\Gamma \rightarrow {\bf R}}$ of Lipschitz constant ${O_{s,\delta}(1)}$ such that
$\displaystyle \frac{1}{N} \sum_n f(n) F(g(n) \Gamma) \gg_{s,\delta} 1.$
There is a higher dimensional generalisation, which first appeared explicitly (in a more general form) in this preprint of Szegedy (which used a slightly different argument than the one of Ben, Tammy, and myself; see also this previous preprint of Szegedy with related results):
Theorem 2 (Inverse theorem for multidimensional Gowers norms) Let ${N \geq 1}$ and ${s,d \geq 1}$ be integers, and let ${\delta > 0}$. Suppose that ${f: {\bf Z}^d \rightarrow [-1,1]}$ is a function supported on ${[N]^d}$ such that
$\displaystyle \frac{1}{N^{d(s+2)}} \sum_{n,h_1,\dots,h_{s+1} \in {\bf Z}^d} \prod_{\omega \in \{0,1\}^{s+1}} f(n+\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}) \geq \delta. \ \ \ \ \ (1)$
Then there exists a filtered nilmanifold ${G/\Gamma}$ of degree ${\leq s}$ and complexity ${O_{s,\delta,d}(1)}$, a polynomial sequence ${g: {\bf Z}^d \rightarrow G}$, and a Lipschitz function ${F: G/\Gamma \rightarrow {\bf R}}$ of Lipschitz constant ${O_{s,\delta,d}(1)}$ such that
$\displaystyle \frac{1}{N^d} \sum_{n \in {\bf Z}^d} f(n) F(g(n) \Gamma) \gg_{s,\delta,d} 1.$
The ${d=2}$ case of this theorem was recently used by Wenbo Sun. One can replace the polynomial sequence with a linear sequence if desired by using a lifting trick (essentially due to Furstenberg, but which appears explicitly in Appendix C of my paper with Ben and Tammy).
In this post I would like to record a very neat and simple observation of Ben Green and Nikos Frantzikinakis, that uses the tool of Freiman isomorphisms to derive Theorem 2 as a corollary of the one-dimensional theorem. Namely, consider the linear map ${\phi: {\bf Z}^d \rightarrow {\bf Z}}$ defined by
$\displaystyle \phi( n_1,\dots,n_d ) := \sum_{i=1}^d (10 N)^{i-1} n_i,$
that is to say ${\phi}$ is the digit string base ${10N}$ that has digits ${n_d \dots n_1}$. This map is a linear map from ${[N]^d}$ to a subset of ${[d 10^d N^d]}$ of density ${1/(d10^d)}$. Furthermore it has the following “Freiman isomorphism” property: if ${n, h_1,\dots,h_{s+1}}$ lie in ${{\bf Z}}$ with ${n + \omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}}$ in the image set ${\phi( [N]^d )}$ of ${[N]^d}$ for all ${\omega}$, then there exist (unique) lifts ${\tilde n \in {\bf Z}^d, \tilde h_1,\dots,\tilde h_{s+1} \in {\bf Z}}$ such that
$\displaystyle \tilde n + \omega_1 \tilde h_1 + \dots + \omega_{s+1} \tilde h_{s+1} \in [N]^d$
and
$\displaystyle \phi( \tilde n + \omega_1 \tilde h_1 + \dots + \omega_{s+1} \tilde h_{s+1} ) = n + \omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}$
for all ${\omega}$. Indeed, the injectivity of ${\phi}$ on ${[N]^d}$ uniquely determines the sum ${\tilde n + \omega_1 \tilde h_1 + \dots + \omega_{s+1} \tilde h_{s+1}}$ for each ${\omega}$, and one can use base ${10N}$ arithmetic to verify that the alternating sum of these sums on any ${2}$-facet of the cube ${\{0,1\}^{s+1}}$ vanishes, which gives the claim. (In the language of additive combinatorics, the point is that ${\phi}$ is a Freiman isomorphism of order (say) ${8}$ on ${[N]^d}$.)
Now let ${\tilde f: {\bf Z} \rightarrow [-1,1]}$ be the function defined by setting ${\tilde f( \phi(n) ) := f(n)}$ whenever ${n \in [N]^d}$, with ${\tilde f}$ vanishing outside of ${\phi([N]^d)}$. If ${f}$ obeys (1), then from the above Freiman isomorphism property we have
$\displaystyle \frac{1}{N^{d(s+2)}} \sum_{n, h_1,\dots,h_{s+1} \in {\bf Z}} \prod_{\omega \in \{0,1\}^{s+1}} \tilde f(n+\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}) \geq \delta.$
Applying the one-dimensional inverse theorem (Theorem 1), with ${\delta}$ reduced by a factor of ${d 10^d}$ and ${N}$ replaced by ${d 10^d N^d}$, this implies the existence of a filtered nilmanifold ${G/\Gamma}$ of degree ${\leq s}$ and complexity ${O_{s,\delta,d}(1)}$, a polynomial sequence ${g: {\bf Z} \rightarrow G}$, and a Lipschitz function ${F: G/\Gamma \rightarrow {\bf R}}$ of Lipschitz constant ${O_{s,\delta,d}(1)}$ such that
$\displaystyle \frac{1}{N^{d(s+2)}} \sum_{n \in {\bf Z}} \tilde f(n) F(g(n) \Gamma) \gg_{s,\delta,d} 1$
which by the Freiman isomorphism property again implies that
$\displaystyle \frac{1}{N^{d(s+2)}} \sum_{n \in {\bf Z}^d} f(n) F(g(\phi(n)) \Gamma) \gg_{s,\delta,d} 1.$
But the map ${n \mapsto g(\phi(n))}$ is clearly a polynomial map from ${{\bf Z}^d}$ to ${G}$ (the composition of two polynomial maps is polynomial, see e.g. Appendix B of my paper with Ben and Tammy), and the claim follows.
Remark 3 This trick appears to be largely restricted to the case of boundedly generated groups such as ${{\bf Z}^d}$; I do not see any easy way to deduce an inverse theorem for, say, ${\bigcup_{n=1}^\infty {\mathbb F}_p^n}$ from the ${{\bf Z}}$-inverse theorem by this method.
Remark 4 By combining this argument with the one in the previous post, one can obtain a weak ergodic inverse theorem for ${{\bf Z}^d}$-actions. Interestingly, the Freiman isomorphism argument appears to be difficult to implement directly in the ergodic category; in particular, there does not appear to be an obvious direct way to derive the Host-Kra inverse theorem for ${{\bf Z}^d}$ actions (a result first obtained in the PhD thesis of Griesmer) from the counterpart for ${{\bf Z}}$ actions.
Note: this post is of a particularly technical nature, in particular presuming familiarity with nilsequences, nilsystems, characteristic factors, etc., and is primarily intended for experts.
As mentioned in the previous post, Ben Green, Tamar Ziegler, and myself proved the following inverse theorem for the Gowers norms:
Theorem 1 (Inverse theorem for Gowers norms) Let ${N \geq 1}$ and ${s \geq 1}$ be integers, and let ${\delta > 0}$. Suppose that ${f: {\bf Z} \rightarrow [-1,1]}$ is a function supported on ${[N] := \{1,\dots,N\}}$ such that
$\displaystyle \frac{1}{N^{s+2}} \sum_{n,h_1,\dots,h_{s+1}} \prod_{\omega \in \{0,1\}^{s+1}} f(n+\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}) \geq \delta.$
Then there exists a filtered nilmanifold ${G/\Gamma}$ of degree ${\leq s}$ and complexity ${O_{s,\delta}(1)}$, a polynomial sequence ${g: {\bf Z} \rightarrow G}$, and a Lipschitz function ${F: G/\Gamma \rightarrow {\bf R}}$ of Lipschitz constant ${O_{s,\delta}(1)}$ such that
$\displaystyle \frac{1}{N} \sum_n f(n) F(g(n) \Gamma) \gg_{s,\delta} 1.$
This result was conjectured earlier by Ben Green and myself; this conjecture was strongly motivated by an analogous inverse theorem in ergodic theory by Host and Kra, which we formulate here in a form designed to resemble Theorem 1 as closely as possible:
Theorem 2 (Inverse theorem for Gowers-Host-Kra seminorms) Let ${s \geq 1}$ be an integer, and let ${(X, T)}$ be an ergodic, countably generated measure-preserving system. Suppose that one has
$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N^{s+1}} \sum_{h_1,\dots,h_{s+1} \in [N]} \int_X \prod_{\omega \in \{0,1\}^{s+1}} f(T^{\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}}x)\ d\mu(x)$
$\displaystyle > 0$
for all non-zero ${f \in L^\infty(X)}$ (all ${L^p}$ spaces are real-valued in this post). Then ${(X,T)}$ is an inverse limit (in the category of measure-preserving systems, up to almost everywhere equivalence) of ergodic degree ${\leq s}$ nilsystems, that is to say systems of the form ${(G/\Gamma, x \mapsto gx)}$ for some degree ${\leq s}$ filtered nilmanifold ${G/\Gamma}$ and a group element ${g \in G}$ that acts ergodically on ${G/\Gamma}$.
It is a natural question to ask if there is any logical relationship between the two theorems. In the finite field category, one can deduce the combinatorial inverse theorem from the ergodic inverse theorem by a variant of the Furstenberg correspondence principle, as worked out by Tamar Ziegler and myself, however in the current context of ${{\bf Z}}$-actions, the connection is less clear.
One can split Theorem 2 into two components:
Theorem 3 (Weak inverse theorem for Gowers-Host-Kra seminorms) Let ${s \geq 1}$ be an integer, and let ${(X, T)}$ be an ergodic, countably generated measure-preserving system. Suppose that one has
$\displaystyle \lim_{N \rightarrow \infty} \frac{1}{N^{s+1}} \sum_{h_1,\dots,h_{s+1} \in [N]} \int_X \prod_{\omega \in \{0,1\}^{s+1}} T^{\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}} f\ d\mu$
$\displaystyle > 0$
for all non-zero ${f \in L^\infty(X)}$, where ${T^h f := f \circ T^h}$. Then ${(X,T)}$ is a factor of an inverse limit of ergodic degree ${\leq s}$ nilsystems.
Theorem 4 (Pro-nilsystems closed under factors) Let ${s \geq 1}$ be an integer. Then any factor of an inverse limit of ergodic degree ${\leq s}$ nilsystems, is again an inverse limit of ergodic degree ${\leq s}$ nilsystems.
Indeed, it is clear that Theorem 2 implies both Theorem 3 and Theorem 4, and conversely that the two latter theorems jointly imply the former. Theorem 4 is, in principle, purely a fact about nilsystems, and should have an independent proof, but this is not known; the only known proofs go through the full machinery needed to prove Theorem 2 (or the closely related theorem of Ziegler). (However, the fact that a factor of a nilsystem is again a nilsystem was established previously by Parry.)
The purpose of this post is to record a partial implication in reverse direction to the correspondence principle:
Proposition 5 Theorem 1 implies Theorem 3.
As mentioned at the start of the post, a fair amount of familiarity with the area is presumed here, and some routine steps will be presented with only a fairly brief explanation.
A few years ago, Ben Green, Tamar Ziegler, and myself proved the following (rather technical-looking) inverse theorem for the Gowers norms:
Theorem 1 (Discrete inverse theorem for Gowers norms) Let ${N \geq 1}$ and ${s \geq 1}$ be integers, and let ${\delta > 0}$. Suppose that ${f: {\bf Z} \rightarrow [-1,1]}$ is a function supported on ${[N] := \{1,\dots,N\}}$ such that
$\displaystyle \frac{1}{N^{s+2}} \sum_{n,h_1,\dots,h_{s+1}} \prod_{\omega \in \{0,1\}^{s+1}} f(n+\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1}) \geq \delta.$
Then there exists a filtered nilmanifold ${G/\Gamma}$ of degree ${\leq s}$ and complexity ${O_{s,\delta}(1)}$, a polynomial sequence ${g: {\bf Z} \rightarrow G}$, and a Lipschitz function ${F: G/\Gamma \rightarrow {\bf R}}$ of Lipschitz constant ${O_{s,\delta}(1)}$ such that
$\displaystyle \frac{1}{N} \sum_n f(n) F(g(n) \Gamma) \gg_{s,\delta} 1.$
For the definitions of “filtered nilmanifold”, “degree”, “complexity”, and “polynomial sequence”, see the paper of Ben, Tammy, and myself. (I should caution the reader that this blog post will presume a fair amount of familiarity with this subfield of additive combinatorics.) This result has a number of applications, for instance to establishing asymptotics for linear equations in the primes, but this will not be the focus of discussion here.
The purpose of this post is to record the observation that this “discrete” inverse theorem, together with an equidistribution theorem for nilsequences that Ben and I worked out in a separate paper, implies a continuous version:
Theorem 2 (Continuous inverse theorem for Gowers norms) Let ${s \geq 1}$ be an integer, and let ${\delta>0}$. Suppose that ${f: {\bf R} \rightarrow [-1,1]}$ is a measurable function supported on ${[0,1]}$ such that
$\displaystyle \int_{{\bf R}^{s+1}} \prod_{\omega \in \{0,1\}^{s+1}} f(t+\omega_1 h_1 + \dots + \omega_{s+1} h_{s+1})\ dt dh_1 \dots dh_{s+1} \geq \delta. \ \ \ \ \ (1)$
Then there exists a filtered nilmanifold ${G/\Gamma}$ of degree ${\leq s}$ and complexity ${O_{s,\delta}(1)}$, a (smooth) polynomial sequence ${g: {\bf R} \rightarrow G}$, and a Lipschitz function ${F: G/\Gamma \rightarrow {\bf R}}$ of Lipschitz constant ${O_{s,\delta}(1)}$ such that
$\displaystyle \int_{\bf R} f(t) F(g(t) \Gamma)\ dt \gg_{s,\delta} 1.$
The interval ${[0,1]}$ can be easily replaced with any other fixed interval by a change of variables. A key point here is that the bounds are completely uniform in the choice of ${f}$. Note though that the coefficients of ${g}$ can be arbitrarily large (and this is necessary, as can be seen just by considering functions of the form ${f(t) = \cos( \xi t)}$ for some arbitrarily large frequency ${\xi}$).
It is likely that one could prove Theorem 2 by carefully going through the proof of Theorem 1 and replacing all instances of ${{\bf Z}}$ with ${{\bf R}}$ (and making appropriate modifications to the argument to accommodate this). However, the proof of Theorem 1 is quite lengthy. Here, we shall proceed by the usual limiting process of viewing the continuous interval ${[0,1]}$ as a limit of the discrete interval ${\frac{1}{N} \cdot [N]}$ as ${N \rightarrow \infty}$. However there will be some problems taking the limit due to a failure of compactness, and specifically with regards to the coefficients of the polynomial sequence ${g: {\bf N} \rightarrow G}$ produced by Theorem 1, after normalising these coefficients by ${N}$. Fortunately, a factorisation theorem from a paper of Ben Green and myself resolves this problem by splitting ${g}$ into a “smooth” part which does enjoy good compactness properties, as well as “totally equidistributed” and “periodic” parts which can be eliminated using the measurability (and thus, approximate smoothness), of ${f}$.
Szemerédi’s theorem asserts that any subset of the integers of positive upper density contains arbitrarily large arithmetic progressions. Here is an equivalent quantitative form of this theorem:
Theorem 1 (Szemerédi’s theorem) Let ${N}$ be a positive integer, and let ${f: {\bf Z}/N{\bf Z} \rightarrow [0,1]}$ be a function with ${{\bf E}_{x \in {\bf Z}/N{\bf Z}} f(x) \geq \delta}$ for some ${\delta>0}$, where we use the averaging notation ${{\bf E}_{x \in A} f(x) := \frac{1}{|A|} \sum_{x \in A} f(x)}$, ${{\bf E}_{x,r \in A} f(x) := \frac{1}{|A|^2} \sum_{x, r \in A} f(x)}$, etc.. Then for ${k \geq 3}$ we have
$\displaystyle {\bf E}_{x,r \in {\bf Z}/N{\bf Z}} f(x) f(x+r) \dots f(x+(k-1)r) \geq c(k,\delta)$
for some ${c(k,\delta)>0}$ depending only on ${k,\delta}$.
The equivalence is basically thanks to an averaging argument of Varnavides; see for instance Chapter 11 of my book with Van Vu or this previous blog post for a discussion. We have removed the cases ${k=1,2}$ as they are trivial and somewhat degenerate.
There are now many proofs of this theorem. Some time ago, I took an ergodic-theoretic proof of Furstenberg and converted it to a purely finitary proof of the theorem. The argument used some simplifying innovations that had been developed since the original work of Furstenberg (in particular, deployment of the Gowers uniformity norms, as well as a “dual” norm that I called the uniformly almost periodic norm, and an emphasis on van der Waerden’s theorem for handling the “compact extension” component of the argument). But the proof was still quite messy. However, as discussed in this previous blog post, messy finitary proofs can often be cleaned up using nonstandard analysis. Thus, there should be a nonstandard version of the Furstenberg ergodic theory argument that is relatively clean. I decided (after some encouragement from Ben Green and Isaac Goldbring) to write down most of the details of this argument in this blog post, though for sake of brevity I will skim rather quickly over arguments that were already discussed at length in other blog posts. In particular, I will presume familiarity with nonstandard analysis (in particular, the notion of a standard part of a bounded real number, and the Loeb measure construction), see for instance this previous blog post for a discussion.
In analytic number theory, there is a well known analogy between the prime factorisation of a large integer, and the cycle decomposition of a large permutation; this analogy is central to the topic of “anatomy of the integers”, as discussed for instance in this survey article of Granville. Consider for instance the following two parallel lists of facts (stated somewhat informally). Firstly, some facts about the prime factorisation of large integers:
• Every positive integer ${m}$ has a prime factorisation
$\displaystyle m = p_1 p_2 \dots p_r$
into (not necessarily distinct) primes ${p_1,\dots,p_r}$, which is unique up to rearrangement. Taking logarithms, we obtain a partition
$\displaystyle \log m = \log p_1 + \log p_2 + \dots + \log p_r$
of ${\log m}$.
• (Prime number theorem) A randomly selected integer ${m}$ of size ${m \sim N}$ will be prime with probability ${\approx \frac{1}{\log N}}$ when ${N}$ is large.
• If ${m \sim N}$ is a randomly selected large integer of size ${N}$, and ${p = p_i}$ is a randomly selected prime factor of ${m = p_1 \dots p_r}$ (with each index ${i}$ being chosen with probability ${\frac{\log p_i}{\log m}}$), then ${\log p_i}$ is approximately uniformly distributed between ${0}$ and ${\log N}$. (See Proposition 9 of this previous blog post.)
• The set of real numbers ${\{ \frac{\log p_i}{\log m}: i=1,\dots,r \}}$ arising from the prime factorisation ${m = p_1 \dots p_r}$ of a large random number ${m \sim N}$ converges (away from the origin, and in a suitable weak sense) to the Poisson-Dirichlet process in the limit ${N \rightarrow \infty}$. (See the previously mentioned blog post for a definition of the Poisson-Dirichlet process, and a proof of this claim.)
Now for the facts about the cycle decomposition of large permutations:
• Every permutation ${\sigma \in S_n}$ has a cycle decomposition
$\displaystyle \sigma = C_1 \dots C_r$
into disjoint cycles ${C_1,\dots,C_r}$, which is unique up to rearrangement, and where we count each fixed point of ${\sigma}$ as a cycle of length ${1}$. If ${|C_i|}$ is the length of the cycle ${C_i}$, we obtain a partition
$\displaystyle n = |C_1| + \dots + |C_r|$
of ${n}$.
• (Prime number theorem for permutations) A randomly selected permutation of ${S_n}$ will be an ${n}$-cycle with probability exactly ${1/n}$. (This was noted in this previous blog post.)
• If ${\sigma}$ is a random permutation in ${S_n}$, and ${C_i}$ is a randomly selected cycle of ${\sigma}$ (with each ${i}$ being selected with probability ${|C_i|/n}$), then ${|C_i|}$ is exactly uniformly distributed on ${\{1,\dots,n\}}$. (See Proposition 8 of this blog post.)
• The set of real numbers ${\{ \frac{|C_i|}{n} \}}$ arising from the cycle decomposition ${\sigma = C_1 \dots C_r}$ of a random permutation ${\sigma \in S_n}$ converges (in a suitable sense) to the Poisson-Dirichlet process in the limit ${n \rightarrow \infty}$. (Again, see this previous blog post for details.)
See this previous blog post (or the aforementioned article of Granville, or the Notices article of Arratia, Barbour, and Tavaré) for further exploration of the analogy between prime factorisation of integers and cycle decomposition of permutations.
There is however something unsatisfying about the analogy, in that it is not clear why there should be such a kinship between integer prime factorisation and permutation cycle decomposition. It turns out that the situation is clarified if one uses another fundamental analogy in number theory, namely the analogy between integers and polynomials ${P \in {\mathbf F}_q[T]}$ over a finite field ${{\mathbf F}_q}$, discussed for instance in this previous post; this is the simplest case of the more general function field analogy between number fields and function fields. Just as we restrict attention to positive integers when talking about prime factorisation, it will be reasonable to restrict attention to monic polynomials ${P}$. We then have another analogous list of facts, proven very similarly to the corresponding list of facts for the integers:
• Every monic polynomial ${f \in {\mathbf F}_q[T]}$ has a factorisation
$\displaystyle f = P_1 \dots P_r$
into irreducible monic polynomials ${P_1,\dots,P_r \in {\mathbf F}_q[T]}$, which is unique up to rearrangement. Taking degrees, we obtain a partition
$\displaystyle \hbox{deg} f = \hbox{deg} P_1 + \dots + \hbox{deg} P_r$
of ${\hbox{deg} f}$.
• (Prime number theorem for polynomials) A randomly selected monic polynomial ${f \in {\mathbf F}_q[T]}$ of degree ${n}$ will be irreducible with probability ${\approx \frac{1}{n}}$ when ${q}$ is fixed and ${n}$ is large.
• If ${f \in {\mathbf F}_q[T]}$ is a random monic polynomial of degree ${n}$, and ${P_i}$ is a random irreducible factor of ${f = P_1 \dots P_r}$ (with each ${i}$ selected with probability ${\hbox{deg} P_i / n}$), then ${\hbox{deg} P_i}$ is approximately uniformly distributed in ${\{1,\dots,n\}}$ when ${q}$ is fixed and ${n}$ is large.
• The set of real numbers ${\{ \hbox{deg} P_i / n \}}$ arising from the factorisation ${f = P_1 \dots P_r}$ of a randomly selected polynomial ${f \in {\mathbf F}_q[T]}$ of degree ${n}$ converges (in a suitable sense) to the Poisson-Dirichlet process when ${q}$ is fixed and ${n}$ is large.
The above list of facts addressed the large ${n}$ limit of the polynomial ring ${{\mathbf F}_q[T]}$, where the order ${q}$ of the field is held fixed, but the degrees of the polynomials go to infinity. This is the limit that is most closely analogous to the integers ${{\bf Z}}$. However, there is another interesting asymptotic limit of polynomial rings to consider, namely the large ${q}$ limit where it is now the degree ${n}$ that is held fixed, but the order ${q}$ of the field goes to infinity. Actually to simplify the exposition we will use the slightly more restrictive limit where the characteristic ${p}$ of the field goes to infinity (again keeping the degree ${n}$ fixed), although all of the results proven below for the large ${p}$ limit turn out to be true as well in the large ${q}$ limit.
The large ${q}$ (or large ${p}$) limit is technically a different limit than the large ${n}$ limit, but in practice the asymptotic statistics of the two limits often agree quite closely. For instance, here is the prime number theorem in the large ${q}$ limit:
Theorem 1 (Prime number theorem) The probability that a random monic polynomial ${f \in {\mathbf F}_q[T]}$ of degree ${n}$ is irreducible is ${\frac{1}{n}+o(1)}$ in the limit where ${n}$ is fixed and the characteristic ${p}$ goes to infinity.
Proof: There are ${q^n}$ monic polynomials ${f \in {\mathbf F}_q[T]}$ of degree ${n}$. If ${f}$ is irreducible, then the ${n}$ zeroes of ${f}$ are distinct and lie in the finite field ${{\mathbf F}_{q^n}}$, but do not lie in any proper subfield of that field. Conversely, every element ${\alpha}$ of ${{\mathbf F}_{q^n}}$ that does not lie in a proper subfield is the root of a unique monic polynomial in ${{\mathbf F}_q[T]}$ of degree ${f}$ (the minimal polynomial of ${\alpha}$). Since the union of all the proper subfields of ${{\mathbf F}_{q^n}}$ has size ${o(q^n)}$, the total number of irreducible polynomials of degree ${n}$ is thus ${\frac{q^n - o(q^n)}{n}}$, and the claim follows. $\Box$
Remark 2 The above argument and inclusion-exclusion in fact gives the well known exact formula ${\frac{1}{n} \sum_{d|n} \mu(\frac{n}{d}) q^d}$ for the number of irreducible monic polynomials of degree ${n}$.
Now we can give a precise connection between the cycle distribution of a random permutation, and (the large ${p}$ limit of) the irreducible factorisation of a polynomial, giving a (somewhat indirect, but still connected) link between permutation cycle decomposition and integer factorisation:
Theorem 3 The partition ${\{ \hbox{deg}(P_1), \dots, \hbox{deg}(P_r) \}}$ of a random monic polynomial ${f= P_1 \dots P_r\in {\mathbf F}_q[T]}$ of degree ${n}$ converges in distribution to the partition ${\{ |C_1|, \dots, |C_r|\}}$ of a random permutation ${\sigma = C_1 \dots C_r \in S_n}$ of length ${n}$, in the limit where ${n}$ is fixed and the characteristic ${p}$ goes to infinity.
We can quickly prove this theorem as follows. We first need a basic fact:
Lemma 4 (Most polynomials square-free in large ${q}$ limit) A random monic polynomial ${f \in {\mathbf F}_q[T]}$ of degree ${n}$ will be square-free with probability ${1-o(1)}$ when ${n}$ is fixed and ${q}$ (or ${p}$) goes to infinity. In a similar spirit, two randomly selected monic polynomials ${f,g}$ of degree ${n,m}$ will be coprime with probability ${1-o(1)}$ if ${n,m}$ are fixed and ${q}$ or ${p}$ goes to infinity.
Proof: For any polynomial ${g}$ of degree ${m}$, the probability that ${f}$ is divisible by ${g^2}$ is at most ${1/q^{2m}}$. Summing over all polynomials of degree ${1 \leq m \leq n/2}$, and using the union bound, we see that the probability that ${f}$ is not squarefree is at most ${\sum_{1 \leq m \leq n/2} \frac{q^m}{q^{2m}} = o(1)}$, giving the first claim. For the second, observe from the first claim (and the fact that ${fg}$ has only a bounded number of factors) that ${fg}$ is squarefree with probability ${1-o(1)}$, giving the claim. $\Box$
Now we can prove the theorem. Elementary combinatorics tells us that the probability of a random permutation ${\sigma \in S_n}$ consisting of ${c_k}$ cycles of length ${k}$ for ${k=1,\dots,r}$, where ${c_k}$ are nonnegative integers with ${\sum_{k=1}^r k c_k = n}$, is precisely
$\displaystyle \frac{1}{\prod_{k=1}^r c_k! k^{c_k}},$
since there are ${\prod_{k=1}^r c_k! k^{c_k}}$ ways to write a given tuple of cycles ${C_1,\dots,C_r}$ in cycle notation in nondecreasing order of length, and ${n!}$ ways to select the labels for the cycle notation. On the other hand, by Theorem 1 (and using Lemma 4 to isolate the small number of cases involving repeated factors) the number of monic polynomials of degree ${n}$ that are the product of ${c_k}$ irreducible polynomials of degree ${k}$ is
$\displaystyle \frac{1}{\prod_{k=1}^r c_k!} \prod_{k=1}^r ( (\frac{1}{k}+o(1)) q^k )^{c_k} + o( q^n )$
which simplifies to
$\displaystyle \frac{1+o(1)}{\prod_{k=1}^r c_k! k^{c_k}} q^n,$
and the claim follows.
This was a fairly short calculation, but it still doesn’t quite explain why there is such a link between the cycle decomposition ${\sigma = C_1 \dots C_r}$ of permutations and the factorisation ${f = P_1 \dots P_r}$ of a polynomial. One immediate thought might be to try to link the multiplication structure of permutations in ${S_n}$ with the multiplication structure of polynomials; however, these structures are too dissimilar to set up a convincing analogy. For instance, the multiplication law on polynomials is abelian and non-invertible, whilst the multiplication law on ${S_n}$ is (extremely) non-abelian but invertible. Also, the multiplication of a degree ${n}$ and a degree ${m}$ polynomial is a degree ${n+m}$ polynomial, whereas the group multiplication law on permutations does not take a permutation in ${S_n}$ and a permutation in ${S_m}$ and return a permutation in ${S_{n+m}}$.
I recently found (after some discussions with Ben Green) what I feel to be a satisfying conceptual (as opposed to computational) explanation of this link, which I will place below the fold.
I’ve just uploaded to the arXiv my paper “Inverse theorems for sets and measures of polynomial growth“. This paper was motivated by two related questions. The first question was to obtain a qualitatively precise description of the sets of polynomial growth that arise in Gromov’s theorem, in much the same way that Freiman’s theorem (and its generalisations) provide a qualitatively precise description of sets of small doubling. The other question was to obtain a non-abelian analogue of inverse Littlewood-Offord theory.
Let me discuss the former question first. Gromov’s theorem tells us that if a finite subset ${A}$ of a group ${G}$ exhibits polynomial growth in the sense that ${|A^n|}$ grows polynomially in ${n}$, then the group generated by ${A}$ is virtually nilpotent (the converse direction also true, and is relatively easy to establish). This theorem has been strengthened a number of times over the years. For instance, a few years ago, I proved with Shalom that the condition that ${|A^n|}$ grew polynomially in ${n}$ could be replaced by ${|A^n| \leq C n^d}$ for a single ${n}$, as long as ${n}$ was sufficiently large depending on ${C,d}$ (in fact we gave a fairly explicit quantitative bound on how large ${n}$ needed to be). A little more recently, with Breuillard and Green, the condition ${|A^n| \leq C n^d}$ was weakened to ${|A^n| \leq n^d |A|}$, that is to say it sufficed to have polynomial relative growth at a finite scale. In fact, the latter paper gave more information on ${A}$ in this case, roughly speaking it showed (at least in the case when ${A}$ was a symmetric neighbourhood of the identity) that ${A^n}$ was “commensurate” with a very structured object known as a coset nilprogression. This can then be used to establish further control on ${A}$. For instance, it was recently shown by Breuillard and Tointon (again in the symmetric case) that if ${|A^n| \leq n^d |A|}$ for a single ${n}$ that was sufficiently large depending on ${d}$, then all the ${A^{n'}}$ for ${n' \geq n}$ have a doubling constant bounded by a bound ${C_d}$ depending only on ${d}$, thus ${|A^{2n'}| \leq C_d |A^{n'}|}$ for all ${n' \geq n}$.
In this paper we are able to refine this analysis a bit further; under the same hypotheses, we can show an estimate of the form
$\displaystyle \log |A^{n'}| = \log |A^n| + f( \log n' - \log n ) + O_d(1)$
for all ${n' \geq n}$ and some piecewise linear, continuous, non-decreasing function ${f: [0,+\infty) \rightarrow [0,+\infty)}$ with ${f(0)=0}$, where the error ${O_d(1)}$ is bounded by a constant depending only on ${d}$, and where ${f}$ has at most ${O_d(1)}$ pieces, each of which has a slope that is a natural number of size ${O_d(1)}$. To put it another way, the function ${n' \mapsto |A^{n'}|}$ for ${n' \geq n}$ behaves (up to multiplicative constants) like a piecewise polynomial function, where the degree of the function and number of pieces is bounded by a constant depending on ${d}$.
One could ask whether the function ${f}$ has any convexity or concavity properties. It turns out that it can exhibit either convex or concave behaviour (or a combination of both). For instance, if ${A}$ is contained in a large finite group, then ${n \mapsto |A^n|}$ will eventually plateau to a constant, exhibiting concave behaviour. On the other hand, in nilpotent groups one can see convex behaviour; for instance, in the Heisenberg group ${\begin{pmatrix}{} {1} {\mathbf Z} {\mathbf Z} \\ {0} {1} {\mathbf Z} \\ {0} {1} \end{pmatrix}}$, if one sets ${A}$ to be a set of matrices of the form ${\begin{pmatrix} 1 & O(N) & O(N^3) \\ 0 & 1 & O(N) \\ 0 & 0 & 1 \end{pmatrix}}$ for some large ${N}$ (abusing the ${O()}$ notation somewhat), then ${n \mapsto A^n}$ grows cubically for ${n \leq N}$ but then grows quartically for ${n > N}$.
To prove this proposition, it turns out (after using a somewhat difficult inverse theorem proven previously by Breuillard, Green, and myself) that one has to analyse the volume growth ${n \mapsto |P^n|}$ of nilprogressions ${P}$. In the “infinitely proper” case where there are no unexpected relations between the generators of the nilprogression, one can lift everything to a simply connected Lie group (where one can take logarithms and exploit the Baker-Campbell-Hausdorff formula heavily), eventually describing ${P^n}$ with fair accuracy by a certain convex polytope with vertices depending polynomially on ${n}$, which implies that ${|P^n|}$ depends polynomially on ${n}$ up to constants. If one is not in the “infinitely proper” case, then at some point ${n_0}$ the nilprogression ${P^{n_0}}$ develops a “collision”, but then one can use this collision to show (after some work) that the dimension of the “Lie model” of ${P^{n_0}}$ has dropped by at least one from the dimension of ${P}$ (the notion of a Lie model being developed in the previously mentioned paper of Breuillard, Greenm, and myself), so that this sort of collision can only occur a bounded number of times, with essentially polynomial volume growth behaviour between these collisions.
The arguments also give a precise description of the location of a set ${A}$ for which ${A^n}$ grows polynomially in ${n}$. In the symmetric case, what ends up happening is that ${A^n}$ becomes commensurate to a “coset nilprogression” ${HP}$ of bounded rank and nilpotency class, whilst ${A}$ is “virtually” contained in a scaled down version ${HP^{1/n}}$ of that nilprogression. What “virtually” means is a little complicated; roughly speaking, it means that there is a set ${X}$ of bounded cardinality such that ${aXHP^{1/n} \approx XHP^{1/n}}$ for all ${a \in A}$. Conversely, if ${A}$ is virtually contained in ${HP^{1/n}}$, then ${A^n}$ is commensurate to ${HP}$ (and more generally, ${A^{mn}}$ is commensurate to ${HP^m}$ for any natural number ${m}$), giving quite a (qualitatively) precise description of ${A}$ in terms of coset nilprogressions.
The main tool used to prove these results is the structure theorem for approximate groups established by Breuillard, Green, and myself, which roughly speaking asserts that approximate groups are always commensurate with coset nilprogressions. A key additional trick is a pigeonholing argument of Sanders, which in this context is the assertion that if ${A^n}$ is comparable to ${A^{2n}}$, then there is an ${n'}$ between ${n}$ and ${2n}$ such that ${A \cdot A^{n'}}$ is very close in size to ${A^{n'}}$ (up to a relative error of ${1/n}$). It is this fact, together with the comparability of ${A^{n'}}$ to a coset nilprogression ${HP}$, that allows us (after some combinatorial argument) to virtually place ${A}$ inside ${HP^{1/n}}$.
Similar arguments apply when discussing iterated convolutions ${\mu^{*n}}$ of (symmetric) probability measures on a (discrete) group ${G}$, rather than combinatorial powers ${A^n}$ of a finite set. Here, the analogue of volume ${A^n}$ is given by the negative power ${\| \mu^{*n} \|_{\ell^2}^{-2}}$ of the ${\ell^2}$ norm of ${\mu^{*n}}$ (thought of as a non-negative function on ${G}$ of total mass 1). One can also work with other norms here than ${\ell^2}$, but this norm has some minor technical conveniences (and other measures of the “spread” of ${\mu^{*n}}$ end up being more or less equivalent for our purposes). There is an analogous structure theorem that asserts that if ${\mu^{*n}}$ spreads at most polynomially in ${n}$, then ${\mu^{*n}}$ is “commensurate” with the uniform probability distribution on a coset progression ${HP}$, and ${\mu}$ itself is largely concentrated near ${HP^{1/\sqrt{n}}}$. The factor of ${\sqrt{n}}$ here is the familiar scaling factor in random walks that arises for instance in the central limit theorem. The proof of (the precise version of) this statement proceeds similarly to the combinatorial case, using pigeonholing to locate a scale ${n'}$ where ${\mu *\mu^{n'}}$ has almost the same ${\ell^2}$ norm as ${\mu^{n'}}$.
A special case of this theory occurs when ${\mu}$ is the uniform probability measure on ${n}$ elements ${v_1,\dots,v_n}$ of ${G}$ and their inverses. The probability measure ${\mu^{*n}}$ is then the distribution of a random product ${w_1 \dots w_n}$, where each ${w_i}$ is equal to one of ${v_{j_i}}$ or its inverse ${v_{j_i}^{-1}}$, selected at random with ${j_i}$ drawn uniformly from ${\{1,\dots,n\}}$ with replacement. This is very close to the Littlewood-Offord situation of random products ${u_1 \dots u_n}$ where each ${u_i}$ is equal to ${v_i}$ or ${v_i^{-1}}$ selected independently at random (thus ${j_i}$ is now fixed to equal ${i}$ rather than being randomly drawn from ${\{1,\dots,n\}}$. In the case when ${G}$ is abelian, it turns out that a little bit of Fourier analysis shows that these two random walks have “comparable” distributions in a certain ${\ell^2}$ sense. As a consequence, the results in this paper can be used to recover an essentially optimal abelian inverse Littlewood-Offord theorem of Nguyen and Vu. In the nonabelian case, the only Littlewood-Offord theorem I am aware of is a recent result of Tiep and Vu for matrix groups, but in this case I do not know how to relate the above two random walks to each other, and so we can only obtain an analogue of the Tiep-Vu results for the symmetrised random walk ${w_1 \dots w_n}$ instead of the ordered random walk ${u_1 \dots u_n}$.
Just a short post here to note that the cover story of this month’s Notices of the AMS, by John Friedlander, is about the recent work on bounded gaps between primes by Zhang, Maynard, our own Polymath project, and others.
I may as well take this opportunity to upload some slides of my own talks on this subject: here are my slides on small and large gaps between the primes that I gave at the “Latinos in the Mathematical Sciences” back in April, and here are my slides on the Polymath project for the Schock Prize symposium last October. (I also gave an abridged version of the latter talk at an AAAS Symposium in February, as well as the Breakthrough Symposium from last November.) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 477, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9691306948661804, "perplexity": 181.98120966993952}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540536855.78/warc/CC-MAIN-20191212023648-20191212051648-00162.warc.gz"} |
https://ftp.aimsciences.org/journal/1930-5346/2020/14/4 | # American Institute of Mathematical Sciences
ISSN:
1930-5346
eISSN:
1930-5338
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## Advances in Mathematics of Communications
November 2020 , Volume 14 , Issue 4
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2020, 14(4): 535-553 doi: 10.3934/amc.2020027 +[Abstract](2428) +[HTML](747) +[PDF](613.87KB)
Abstract:
We propose the first lattice-based dynamic group signature scheme achieving forward security. Our scheme is proven to be secure against framing attack, misidentification attack and preserves anonymity under the learning with errors (${\mathsf{LWE}}$) and short integer solution (${\mathsf{SIS}}$) assumptions in the random oracle model. More interestingly, our setting allows the group manager to generate distinct certificates to distinct users which can be updated by the users themselves without any interaction with the group manager. Furthermore, our scheme is dynamic where signing key of a user is not fixed during the setup and is issued only at the joining time of the user.
2020, 14(4): 555-572 doi: 10.3934/amc.2020029 +[Abstract](2460) +[HTML](704) +[PDF](416.47KB)
Abstract:
\begin{document}$\mathbb{Z}_p\mathbb{Z}_p[v]$\end{document}-Additive cyclic codes of length \begin{document}$(\alpha,\beta)$\end{document} can be viewed as \begin{document}$R[x]$\end{document}-submodules of \begin{document}$\mathbb{Z}_p[x]/(x^\alpha-1)\times R[x]/(x^\beta-1)$\end{document}, where \begin{document}$R = \mathbb{Z}_p+v\mathbb{Z}_p$\end{document} with \begin{document}$v^2 = v$\end{document}. In this paper, we determine the generator polynomials and the minimal generating sets of this family of codes as \begin{document}$R[x]$\end{document}-submodules of \begin{document}$\mathbb{Z}_p[x]/(x^\alpha-1)\times R[x]/(x^\beta-1)$\end{document}. We also determine the generator polynomials of the dual codes of \begin{document}$\mathbb{Z}_p\mathbb{Z}_p[v]$\end{document}-additive cyclic codes. Some optimal \begin{document}$\mathbb{Z}_p\mathbb{Z}_p[v]$\end{document}-linear codes and MDSS codes are obtained from \begin{document}$\mathbb{Z}_p\mathbb{Z}_p[v]$\end{document}-additive cyclic codes. Moreover, we also get some quantum codes from \begin{document}$\mathbb{Z}_p\mathbb{Z}_p[v]$\end{document}-additive cyclic codes.
2020, 14(4): 573-577 doi: 10.3934/amc.2020030 +[Abstract](1659) +[HTML](619) +[PDF](239.69KB)
Abstract:
In this paper, we attack the recent NIST submission Giophantus, a public key encryption scheme. We find that the complicated structure of Giophantus's ciphertexts leaks information via a correspondence from a low dimensional lattice. This allows us to distinguish encrypted data from random data by the LLL algorithm. This is a more efficient attack than previous proposed attacks.
2020, 14(4): 579-589 doi: 10.3934/amc.2020031 +[Abstract](1787) +[HTML](509) +[PDF](337.22KB)
Abstract:
Low hit zone frequency hopping sequences (LHZ FHSs) with favorable partial Hamming correlation properties are desirable in quasi-synchronous frequency hopping multiple-access systems. An LHZ FHS set is considered to be strictly optimal when it has optimal partial Hamming correlation for all correlation windows. In this study, an interleaved construction of new sets of strictly optimal LHZ FHSs is proposed. Strictly optimal LHZ FHS sets with new and flexible parameters are obtained by selecting suitable known optimal FHSs and appropriate shift sequences.
2020, 14(4): 591-602 doi: 10.3934/amc.2020032 +[Abstract](1609) +[HTML](528) +[PDF](338.48KB)
Abstract:
In this paper we introduce the notion of orbit matrices of integer matrices such as Seidel and Laplacian matrices of some strongly regular graphs with respect to their permutation automorphism groups. We further show that under certain conditions these orbit matrices yield self-orthogonal codes over finite fields \begin{document}$\mathbb{F}_q$\end{document}, where \begin{document}$q$\end{document} is a prime power and over finite rings \begin{document}$\mathbb{Z}_m$\end{document}. As a case study, we construct codes from orbit matrices of Seidel, Laplacian and signless Laplacian matrices of strongly regular graphs. In particular, we construct self-orthogonal codes from orbit matrices of Seidel and Laplacian matrices of the Higman-Sims and McLaughlin graphs.
2020, 14(4): 603-611 doi: 10.3934/amc.2020033 +[Abstract](1627) +[HTML](549) +[PDF](349.83KB)
Abstract:
In this paper, using a method of construction of \begin{document}$1$\end{document}-designs which are not necessarily symmetric, introduced by Key and Moori in [5], we determine a number of \begin{document}$1$\end{document}-designs with interesting parameters from the maximal subgroups and the conjugacy classes of the small Ree groups \begin{document}$^2G_2(q)$\end{document}. The designs we obtain are invariant under the action of the groups \begin{document}$^2G_2(q)$\end{document}.
2020, 14(4): 613-630 doi: 10.3934/amc.2020034 +[Abstract](1771) +[HTML](554) +[PDF](472.27KB)
Abstract:
We show that \begin{document}$A_2(7, 4) \leq 388$\end{document} and, more generally, \begin{document}$A_q(7, 4) \leq (q^2-q+1) [7] + q^4 - 2q^3 + 3q^2 - 4q + 4$\end{document} by semidefinite programming for \begin{document}$q \leq 101$\end{document}. Furthermore, we extend results by Bachoc et al. on SDP bounds for \begin{document}$A_2(n, d)$\end{document}, where \begin{document}$d$\end{document} is odd and \begin{document}$n$\end{document} is small, to \begin{document}$A_q(n, d)$\end{document} for small \begin{document}$q$\end{document} and small \begin{document}$n$\end{document}.
2020, 14(4): 631-650 doi: 10.3934/amc.2020035 +[Abstract](1652) +[HTML](561) +[PDF](437.25KB)
Abstract:
In this paper we characterize the orbit codes as geometrically uniform codes. This characterization is based on the description of all isometries over a projective geometry. In addition, Abelian orbit codes are defined and a construction of Abelian non-cyclic orbit codes is presented. In order to analyze their structures, the concept of geometrically uniform partitions have to be reinterpreted. As a consequence, a substantial reduction in the number of computations needed to obtain the minimum subspace distance of these codes is achieved and established.
An application of orbit codes to multishot subspace codes obtained according to a multi-level construction is provided.
2020, 14(4): 651-676 doi: 10.3934/amc.2020036 +[Abstract](1676) +[HTML](531) +[PDF](500.86KB)
Abstract:
The \begin{document}$k$\end{document}-normality of Boolean functions is an important notion initially introduced by Dobbertin and studied in several papers. The parameter related to this notion is the maximal dimension of those affine spaces contained in the support \begin{document}$supp(f)$\end{document} of the function or in its co-support \begin{document}$cosupp(f)$\end{document}. We denote it by \begin{document}$norm\,(f)$\end{document} and call it the norm of \begin{document}$f$\end{document}.
The norm concerns only the affine spaces contained in either the support or the co-support; the information it provides on \begin{document}$f$\end{document} is then somewhat incomplete (for instance, two functions constant on a hyperplane will have the same very large parameter value, while they can have very different complexities). A second parameter which completes the information given by the first one is the minimum between the maximal dimension of those affine spaces contained in \begin{document}$supp(f)$\end{document} and the maximal dimension of those contained in \begin{document}$cosupp(f)$\end{document} (while \begin{document}$norm\,(f)$\end{document} equals the maximum between these two maximal dimensions). We denote it by \begin{document}$cons\,(f)$\end{document} and call it the (affine) constancy of \begin{document}$f$\end{document}.
The value of \begin{document}$cons\,(f)$\end{document} gives global information on \begin{document}$f$\end{document}, but no information on what happens around each point of \begin{document}$supp(f)$\end{document} or \begin{document}$cosupp(f)$\end{document}. We define then its local version, equal to the minimum, when \begin{document}$a$\end{document} ranges over \begin{document}$\Bbb{F}_2^n$\end{document}, of the maximal dimension of those affine spaces which contain \begin{document}$a$\end{document} and on which \begin{document}$f$\end{document} is constant. We denote it by \begin{document}$stab\,(f)$\end{document} and call it the stability of \begin{document}$f$\end{document}.
We study the properties of these three parameters. We have \begin{document}$norm\,(f)\geq cons\,(f)\geq stab\,(f)$\end{document}, then for determining to which extent these three parameters are distinct, we exhibit four infinite classes of Boolean functions, which show that all cases can occur, where each of these two inequalities can be strict or large.
We consider the minimal value of \begin{document}$stab\, (f)$\end{document} (resp. \begin{document}$cons\,(f)$\end{document}, \begin{document}$norm\,(f)$\end{document}), when \begin{document}$f$\end{document} ranges over the Reed-Muller code \begin{document}$RM(r,n)$\end{document} of length \begin{document}$2^n$\end{document} and order \begin{document}$r$\end{document}, and we denote it by \begin{document}$stab\, _{RM(r,n)}$\end{document} (resp. \begin{document}$cons\, _{RM(r,n)}$\end{document}, \begin{document}$norm\, _{RM(r,n)}$\end{document}). We give upper bounds for each of these three integer sequences, and determine the exact values of \begin{document}$stab\, _{RM(r,n)}$\end{document} and \begin{document}$cons\, _{RM(r,n)}$\end{document} for \begin{document}$r\in\{1,2,n-2,n-1,n\}$\end{document}, and of \begin{document}$norm\, _{RM(r,n)}$\end{document} for \begin{document}$r = 1,2$\end{document}.
2020, 14(4): 677-702 doi: 10.3934/amc.2020037 +[Abstract](1919) +[HTML](530) +[PDF](412.24KB)
Abstract:
We describe eight composite constructions from group rings where the orders of the groups are 4 and 8, which are then applied to find self-dual codes of length 16 over \begin{document}$\mathbb{F}_4$\end{document}. These codes have binary images with parameters \begin{document}$[32,16,8]$\end{document} or \begin{document}$[32,16,6]$\end{document}. These are lifted to codes over \begin{document}$\mathbb{F}_4+u\mathbb{F}_4$\end{document}, to obtain codes with Gray images of extremal self-dual binary codes of length 64. Finally, we use a building-up method over \begin{document}$\mathbb{F}_2+u\mathbb{F}_2$\end{document} to obtain new extremal binary self-dual codes of length 68. We construct 11 new codes via the building-up method and 2 new codes by considering possible neighbors.
2020, 14(4): 703-726 doi: 10.3934/amc.2020090 +[Abstract](1461) +[HTML](272) +[PDF](550.82KB)
Abstract:
This paper presents a series of Montgomery scalar multiplication algorithms on general short Weierstrass curves over fields with characteristic greater than 3, which need only 12 field multiplications per scalar bit using 8 \begin{document}$\sim$\end{document} 9 field registers, thus outperform the binary NAF method on average. Over binary fields, the Montgomery scalar multiplication algorithm which was presented at the first CHES workshop by López and Dahab has been a favorite of ECC implementors, due to its nice properties such as high efficiency (outperforming the binary NAF), natural SPA-resistance, generality (coping with all ordinary curves) and implementation easiness. Over odd characteristic fields, the new scalar multiplication algorithms are the first ones featuring all these properties. Building-blocks of our contribution are new efficient differential addition-and-doubling formulae and a novel conception of on-the-fly adaptive coordinates which varies in accordance with not only the base point but also the bits of the given scalar.
2020, 14(4): 727-743 doi: 10.3934/amc.2020093 +[Abstract](979) +[HTML](210) +[PDF](530.86KB)
Abstract:
The study of the trapdoors that can be hidden in a block cipher is and has always been a high-interest topic in symmetric cryptography. In this paper we focus on Feistel-network-like ciphers in a classical long-key scenario and we investigate some conditions which make such a construction immune to the partition-based attack introduced recently by Bannier et al.
2020 Impact Factor: 0.935
5 Year Impact Factor: 0.976
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https://www.groundai.com/project/signatures-of-majorana-kramers-pairs-in-superconductor-luttinger-liquid-and-superconductor-quantum-dot-normal-lead-junctions/ | Signatures of Majorana Kramers Pairs in superconductor-Luttinger liquid and superconductor-quantum dot-normal lead junctions
# Signatures of Majorana Kramers Pairs in superconductor-Luttinger liquid and superconductor-quantum dot-normal lead junctions
Younghyun Kim Department of Physics, University of California, Santa Barbara, California 93106, USA Dong E. Liu Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA Erikas Gaidamauskas Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark Jens Paaske Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark Karsten Flensberg Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark Roman M. Lutchyn Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA
August 17, 2019
###### Abstract
Time-reversal invariant topological superconductors are characterized by the presence of Majorana Kramers pairs localized at defects. One of the transport signatures of Majorana Kramers pairs is the quantized differential conductance of when such a one-dimensional superconductor is coupled to a normal-metal lead. The resonant Andreev reflection, responsible for this phenomenon, can be understood as the boundary condition change for lead electrons at low energies. In this paper, we study the stability of the Andreev reflection fixed point with respect to electron-electron interactions in the Luttinger liquid. We first calculate the phase diagram for the Luttinger liquid-Majorana Kramers pair junction and show that its low-energy properties are determined by Andreev reflection scattering processes in the spin-triplet channel, i.e. the corresponding Andreev boundary conditions are similar to that in a spin-triplet superconductor - normal lead junction. We also study here a quantum dot coupled to a normal lead and a Majorana Kramers pair and investigate the effect of local repulsive interactions leading to an interplay between Kondo and Majorana correlations. Using a combination of renormalization group analysis and slave-boson mean-field theory, we show that the system flows to a new fixed point which is controlled by the Majorana interaction rather than the Kondo coupling. This Majorana fixed point is characterized by correlations between the localized spin and the fermion parity of each spin sector of the topological superconductor. We investigate the stability of the Majorana phase with respect to Gaussian fluctuations.
## I Introduction
The search for topological superconductors, which host Majorana zero modes (MZMs), has becomes an active pursuit in condensed matter physics Reich (2012); Brouwer (2012); Wilczek (2012); Alicea (2012). Such exotic modes are predicted to obey non-Abelian braiding statisticsMoore and Read (1991); Nayak and Wilczek (1996); Read and Green (2000), and have potential application in topological quantum computations Kitaev (2002); Nayak et al. (2008). Many theoretical proposals for realizing topological superconductors in the laboratory have been put forward recently Fu and Kane (2008, 2009); Sau et al. (2010); Alicea (2010); Lutchyn et al. (2010); Oreg et al. (2010); Cook and Franz (2011); Sau and Sarma (2012); Nadj-Perge et al. (2013), and, more excitingly, devices for detecting MZMs were successfully fabricated in the laboratory and the preliminary signatures of MZMs were observed Mourik et al. (2012); Das et al. (2012); Deng et al. (2012); Finck et al. (2013); Churchill et al. (2013); Nadj-Perge et al. (2014); Deng et al. (2014); Higginbotham et al. (2015); Albrecht et al. (2016); Zhang et al. (2016). Most research activity has focused on the topological superconductors belonging to class D ( i.e., SCs with broken time-reversal-symmetry) and supporting an odd number of MZMs at a topological defect Altland and Zirnbauer (1997); Schnyder et al. (2008); Kitaev (2009). However, Majorana zero modes can also appear in pairs in time-reversal invariant topological superconductors (TRITOPS) belonging to class Schnyder et al. (2008); Kitaev (2009); Teo and Kane (2010). Those MZM pairs are referred to as “Majorana Kramers pairs” (MKPs), and their stability is protected by the time-reversal (TR) symmetry and the quasiparticle excitation gap. Recently, several theoretical proposals were put forward to realize TRITOPS Wong and Law (2012); Deng et al. (2012); Zhang et al. (2013); Nakosai et al. (2013); Keselman et al. (2013); Gaidamauskas et al. (2014); Klinovaja et al. (2014); Schrade et al. (2015). Transport signatures of MKPs and their detection schemes using a QPC were also recently investigated in a quantum spin Hall system Li et al. (2016).
Most previous works on MKPs considered non-interacting (or effectively non-interacting) models. It is well-known, however, that interactions in one-dimensional systems are very important Gangadharaiah et al. (2011); Lobos et al. (2012); Fidkowski et al. (2012); Affleck and Giuliano (2013) and in some cases may even modify the classification of non-interacting systems Fidkowski and Kitaev (2011). For non-interacting systems, the presence of a MKP leads to a quantized conductance of due to perfect Andreev reflection at the junction. This quantization of the conductance is due to the constraints imposed by TR symmetry which leads to complete decoupling of MKP in the non-interacting models. The situation is different, however, in the presence of interparticle interactions, and the fate of the perfect Andreev reflection fixed point is unclear. In this paper, we study the stability of MKPs with respect to electron-electron interactions and consider two generic systems - a) MKP coupled to an interacting Luttinger liquid (see Fig. 1 a)); b) MKP coupled to an interacting quantum dot (see Fig. 1 b)).
We first consider a spinful Luttinger liquid lead with spin symmetry coupled to a TRITOPS with a single MKP per end. In this case the boundary problem has an additional symmetry. We find that for weak repulsive interactions, with being the Luttinger parameter, the Andreev reflection fixed point () is stable and the normal reflection fixed point () is unstable. For intermediate interaction strength , the phase diagram depends on microscopic details, i.e. on the strength of four-fermion interactions allowed by TR symmetry, which causes a Berezinsky-Kosterlitz-Thouless (BKT) transition between and . Finally, for sufficiently strong repulsive interactions , the two electron backscattering term becomes relevant, and drives the system to a stable normal reflection fixed point.
In the presence of spin-orbit coupling, the corresponding boundary problem may break symmetry. In this case, allowed processes involve spin-preserving and spin-flip Andreev scattering which drive the system to different boundary conditions for lead electrons: spin-preserving Andreev boundary () condition corresponds to and spin-flip Andreev boundary () condition corresponds to . Thus, the corresponding phase diagram depends on the relative strength of the corresponding Andreev scattering amplitudes. We find that the boundary conditions in this case are similar to those in a spin-triplet superconductor-Luttinger liquid junction and are stable with respect to weak repulsive interactions. In this sense, the physics is fundamentally different from an s-wave superconductor-Luttinger junction where weak repulsive interactions destabilize Andreev reflection fixed point Fidkowski et al. (2012).
In this paper, we also study the effect of local repulsive interactions by considering a MKP coupled to a quantum dot (QD) and an -invariant normal lead (NL). In the limit of a large Coulomb interaction in the QD and single-electron occupancy, we investigate the competition between Kondo and Majorana correlations. When the coupling to the MKP is absent (), the system flows to the Kondo fixed point with the corresponding boundary conditions for NL electrons where denote right and left movers. As we increase the coupling constant , the system exhibits a crossover from the Kondo dominated regime to a Majorana dominated regime where the QD spin builds up a strong correlation with the MKP. The latter is characterized by boundary conditions . Thus, the problem at hand represents a new class of boundary impurity problems where spin in the dot is coupled to the fermion parity of a topological superconductor.
In order to understand thermodynamic and transport properties of this Majorana fixed point, we have developed a slave-boson mean-field theory (please refer to Refs. Coleman (1984); Bickers (1987) for Anderson impurity models) for this system. We show that the Majorana dominated regime corresponds to a new (i.e. different from Kondo) saddle-point solution. We have analyzed the stability of this mean-field solution with respect to Gaussian fluctuations (in the spirit of Refs. Read and Newns (1983); Coleman (1987)) finding that the mean field theory is stable (in the quasi-long range order sense) and can be used to calculate different observable quantities. We use this approach to calculate differential tunneling conductance as a function of applied voltage bias.
The paper is organized as follows. In Secs. II.1 and II.2, we introduce the model of a MKP - Luttinger liquid junction, and consider the boundary problem with and without (e.g., due to Rashba spin-orbit coupling in the lead) symmetry. In Sec. III, we study the signatures of a MKP in a QD-NL junction using both the renormalization group (RG) analysis and the slave-boson mean-field theory. We also consider the Gaussian fluctuations around the mean-field solution, and analyze the stability of the slave-boson mean-field solution. Finally, we conclude in Sec. IV.
## Ii Majorana Kramers pair - Luttinger liquid junction
In this section we consider the setup shown in Fig. 1 a) consisting of a semi-infinite spinful Luttinger liquid coupled weakly to a TRITOPS. We assume that the topological gap of the superconductor is much larger than the other relevant energy scales (i.e. tunneling amplitudes , and see the text below Eq. (4) for definitions). Thus, in the low-energy approximation the superconductor Hamiltonian consists of only the MKPs localized at its opposite ends. In this section, we will use to describe the operators at the boundary , and use (similarly for , and ) as the initial value in RG flow with the initial length cutoff .
### ii.1 Majorana Kramers pair coupled to SU(2)-invariant Luttinger liquid
#### ii.1.1 Theoretical Model
We first consider an -invariant interacting nanowire coupled to a MKP. The Hamiltonian for the 1D lead can be written as the spinful Luttinger model
where and are velocity and Luttinger parameter for charge and spin modes, respectively. The bosonic fields satisfy the commutation relation . We use here the following convention for the Abelian bosonization procedure Giamarchi (2003):
ψR/L,s(x)=ΓR/L,s√2πaei1√2{±[ϕρ(x)+sϕσ(x)]+θρ(x)+sθσ(x)} (2)
where represents right/left moving modes, is an ultraviolet cutoff length scale, denotes fermion spin, and is the Klein factor.
The total Hamiltonian is given as
where the coupling between the Luttinger liquid lead and the MKP. We neglect here the ground-state degeneracy splitting energy. The most general form of the TR invariant boundary Hamiltonian describing the coupling between the MKP and Luttinger liquid and including only two and four-fermion operators reads
HB =it↑γ↑(ψ↑(0)+ψ†↑(0))−it↓γ↓(ψ↓(0)+ψ†↓(0)) −Δiγ↑γ↓(−iψ†↑(0)ψ↓(0)+iψ†↓(0)ψ↑(0)) −ΔANiγ↑γ↓(−iψ†↑(0)ψ†↓(0)+iψ↓(0)ψ↑(0)) (4)
where are Majorana operators with . Here , and are set to be real. The first two terms represent tunneling between the lead and the MKP with the amplitudes . Under TR symmetry , field operators transform as
TψR/L↑=ψL/R↓, (5) TψR/L↓=−ψL/R↑, (6)
(i.e. ) and the coupling constants in the Hamiltonian need to complex conjugated. TR symmetry requires that with being real. Assuming the spin-quantization axis is fixed in the whole system, the overall Hamiltonian has spin-rotation symmetry, leaving it invariant under the unitary transformation:
(ψ↑,ψ↓)T →R(θ)(ψ↑,ψ↓)T (7) (γ↑,γ↓)T →R(−θ)(γ↑,γ↓)T. (8)
Here represents a spin-rotation matrix by an angle . Thus, electron tunneling between Luttinger liquid and topological superconductor preserves the spin. The last two terms and represent normal, and anomalous backscattering terms, which, in fact, will also be generated by the tunneling terms in the RG flow in the presence of interactions in the Luttinger liquid.
#### ii.1.2 Weak coupling RG analysis near normal reflection fixed point
We now study the stability of the weak coupling normal reflection fixed point using perturbative RG analysis. In the ultraviolet, the boundary conditions for lead electrons at are given by (i.e. perfect normal reflection). In terms of bosonization language, this boundary condition corresponds to and pinning . Once we turn on the boundary couplings , and , boundary conditions for lead electrons may change depending on the strength of interaction in the lead. Let us study now the stability of this normal reflection fixed point. After integrating out the fields away from , the corresponding imaginary-time partition function becomes
Z=∫D[θρ]D[θσ]e−(S0+ST), (9)
with
S0=∑j=ρ,σKj2π∫dω2π|ω||θj(ω)|2, (10)
and the boundary coupling term reads
ST =∫dτ2πa[t(iγ↑Γ↑cosθρ+θσ√2−iγ↓Γ↓cosθρ−θσ√2) −Δγ↑γ↓Γ↑Γ↓cos√2θσ−ΔANγ↑γ↓Γ↑Γ↓cos√2θρ], (11)
where is the ultraviolet cutoff. Here we used short-hand notation denoting the fields at .
We now perform a perturbative RG procedure by separating the bosonic fields into slow, and fast modes and integrating out the fast modes. After some manipulations, the new effective action can be calculated using the cumulant expansion:
Seff[θ
where the average describes an integration over the fast modes. The details of this calculation are presented in the Appendix A, and we simply summarize the RG equations here
dtdl = (1−14Kρ−14Kσ)t−Δt4πvKσ−ΔANt4πvKρ, (13) dΔdl = (1−1Kσ)Δ−(1Kρ−1Kσ)t24πv, (14) dΔANdl = (1−1Kρ)ΔAN+(1Kρ−1Kσ)t24πv. (15)
Here where is the ratio of the cutoff change from to with . One can notice that is a relevant perturbation and grows under RG. Therefore, in the non-interacting case when , the system will flow to the perfect Andreev reflection fixed point () corresponding to the boundary condition Fidkowski et al. (2012) and quantized differential conductance at zero temperature.
Let us now try to understand the effects of interactions. In this section, we will focus on an spin-invariant lead () and repulsive interactions in the nanowire . In this case, the coupling is irrelevant and can be neglected, and RG equations simplify to
dtdl = (34−14Kρ)t−Δt4πv, (16) dΔdl = −(1Kρ−1)t24πv. (17)
The coupling is relevant for not too strong repulsive interactions. It becomes marginal, however, if initial value of is equal to the special value . Indeed, then above RG equations (after a slight redefinition of variables) are identical to the anisotropic Kondo model Giamarchi (2003), the solution of which is well-known. If the initial value of is zero, and the parameter (i.e. ), the system will flow to strong coupling fixed point whereas for , the system will flow to fixed point for small and flow to strong coupling for larger . The perturbative RG flow is summarized in Fig. 2.
#### ii.1.3 Weak coupling RG analysis near perfect Andreev reflection fixed point
As shown in the previous section, the normal reflection fixed point is unstable for weak repulsive interactions and the system flows to the perfect Andreev fixed point corresponding to the boundary conditions which, in bosonic variables corresponds to pinning and fields at . Thus, the fluctuating degrees of freedom are the fields and and the corresponding boundary action reads
S0=∑j=ρ,σ12πKj∫dω2π|ω||ϕj(ω)|2. (18)
We now consider perturbations near the Andreev fixed point which are consistent with time-reversal and the spin- symmetry of the Luttinger liquid lead. The only fermion bilinear boundary perturbation preserving aforementioned symmetries is
H1B =λ1(ψ†R↑(0)ψL↑(0)+ψ†R↓(0)ψL↓(0))+h.c. =λ12πacos(√2ϕρ)cos(√2ϕσ). (19)
In addition, one has to also consider the following four-fermion perturbation consistent with the above symmetries:
H2B= λ2ψ†L↑(0)ψR↑(0)ψ†L↓(0)ψR↓(0)+h.c. =λ2(2πa)2sin(2√2ϕρ), (20)
which corresponds to two-electron backscattering. The perturbative RG equations for and are given by
dλ1dl =(1−Kρ−Kσ)λ1 (21) dλ2dl =(1−4Kρ)λ2 (22)
One can see that the first term is irrelevant since whereas the second coupling becomes relevant for indicating that fixed point becomes unstable for strong repulsive interactions. Taking into account the perturbative RG analysis near both and fixed points, we conjecture the qualitative phase diagrams shown in Fig. 3.
#### ii.1.4 Differential tunneling conductance
We now discuss transport signatures of MKPs. The simplest experiment to detect the presence of a MKP is the differential conductance measurement. We focus on the case with an symmetric wire and calculate at zero voltage bias as a function of temperature. The RG flow between the normal and Andreev reflection defines a crossover temperature , which roughly corresponds to the width of the zero bias peak. Although the conductance for the whole crossover regime requires involved calculations, the conductance around and fixed points can be obtained using perturbation theory, see, e.g., Ref. [Lutchyn and Skrabacz, 2013].
First of all, we consider the case , where fixed point is stable. In the ultraviolet (i.e. near the unstable normal reflection fixed point), the leading relevant perturbation is the coupling to the MKP, , which has scaling dimension . Near the stable Andreev reflection fixed point (i.e. in the infrared), the deviation from the quantized value comes from the leading irrelevant operators which cause backscatterings, i.e. single-electron backscattering shown in Eq. (19) with scaling dimension and two-electron backscattering shown in Eq. (20) with scaling dimension .
Here, for , the single-electron backscattering shown in Eq. (19) is the leading irrelevant operator. We can now obtain scaling of the conductance with temperature at zero bias (assuming the initial value of coupling is zero, i.e. ):
G4e2/h∣∣∣Kρ>13=⎧⎪ ⎪⎨⎪ ⎪⎩c1,T(Kρ)(TT∗)2(14Kρ−34),T≫T∗1−c2,T(Kρ)(TT∗)2Kρ,T≪T∗, (23)
where are numerical coefficients of the order one. Similarly, one can obtain voltage corrections to the conductance at zero temperature. Interestingly, the analogous coefficient vanishes in the non-interacting limit and, therefore, the scaling of the conductance with voltage and temperature is different at , see Ref. [Lutchyn and Skrabacz, 2013] for details.
Next, we consider , where is stable in the infrared. In this case, we start near the high energy unstable fixed point and calculate the conductance by perturbing with the two-electron backscattering operator which is the leading relevant operator in this regime. Thus, we obtain
G4e2/h∣∣∣Kρ≲13∼⎧⎪ ⎪⎨⎪ ⎪⎩1−c3,T(Kρ)(TT∗)2(4Kρ−1),T≫T∗c4,T(Kρ)(TT∗)2(14Kρ−34),T≪T∗, (24)
where are numerical coefficients.
The calculation of the conductance in the regime depends on microscopic details (i.e. strength of ), and is outside the scope of the paper.
### ii.2 The effect of the Rashba spin-orbit coupling in the lead
#### ii.2.1 Theoretical Model
In this section, we consider the effect of Rashba spin-orbit coupling (SOC) in the nanowire. When coupling to MKP, the spin eigenstates of the MKP do not have to be the same as the the spin eigenstates of the nanowire. Therefore, tunneling between the lead and the TRITOPS will have both spin-preserving and spin-flip components. In order to see how the spin flip tunneling is generated, we consider the direction of the Rashba coupling which has an angle rotation compared to that of the MKP. The corresponding tight binding model can written as
H =Hlead+HT (25) Hlead =−tN∑j=1∑s(c†j+1,scj,s+h.c.)+μ∑jsc†j,scj,s +∑jss′(−i)αRc†j+1,s(cosθσz+sinθσy)ss′cj,s′+h.c., HT =it0[γ↑(cN↑+c†N↑)−γ↓(cN↓+c†N↓)]. (26)
One can see that the above Hamiltonian respects TR symmetry. We apply the following unitary transformation
(27)
and then the bulk and boundary Hamiltonians become
Hlead = μ∑jsd†j,sdj,s+∑j[(−t−iαR)d†j+1,↑dj,↑ (28) +(−t+iαR)d†j+1,↓dj,↓+h.c.],
and
HT = it∑s=↑,↓sγs(dN,s+d†N,s) (29) +~t∑ssγs(d†N,−s−dN,−s),
where and , and for spin-. Therefore, the spin-flip tunneling is non-zero for any , i.e. due to the presence of SOC. One can simply check that, in the presence of both and , the symmetry shown in Eq. (8) is broken. In this case, the boundary condition at the Andreev reflection fixed point is determined by the relative magnitude of and . For the discussion of boundary condition and bosonization procedure in the normal reflection fixed point, please refer to Appendix B.
It is instructive to analyze the boundary conditions in the non-interacting case using the scattering matrix approach. The unitary scattering matrix is defined as (see, e.g., Ref. Nilsson et al. (2008))
S(ω)=^I+2πi^W†(HMK−ω−iπ^W^W†)−1^W, (30)
where is the Hamiltonian for the MKP (2 by 2 matrix) which vanishes in the limit with and being respectively the length and coherence length of the superconductor. Note that the local term is not allowed by TR symmetry. The matrix describes the coupling between the MKP and the lead degrees of freedom in the basis :
^W=(it~tit−~t−~t−it~t−it). (31)
Note that we assume that lead Hamiltonian is diagonal here. Therefore, represent helicity eigenstates in the case of a Rashba model. Using Eq. (30), we can represent the scattering matrix at as
S(0)=(See(0)Seh(0)She(0)Shh(0)). (32)
The components and describe normal and Andreev reflection, respectively. As pointed out in Ref. Li et al. (2016), the normal part is zero so we focus on the non-diagonal components:
Seh(0) =⎛⎜ ⎜⎝~t2−t2t2+~t2−2i~ttt2+~t2−2i~ttt2+~t2~t2−t2t2+~t2⎞⎟ ⎟⎠, =−cos2θ−iσxsin2θ (33)
where the diagonal term is the coefficient of the same-spin Andreev reflection , and the off-diagonal term is the coefficient of the spin-flip Andreev reflection . As we change the angle of SOC, , from to , the Andreev reflection boundary condition changes continuously from with (i.e. and ) to and . We denote this boundary condition for as spin flip Andreev reflection boundary condition , which describes an Andreev reflection with spin-flip processes. Upon increasing to , the boundary condition becomes (i.e. and ).
Here we would like to emphasize that the boundary condition is different from the Andreev boundary condition in s-wave spin-singlet superconducting junction where and (see, e.g., Ref.Maslov et al., 1996). Notice different signs in this case for spin-up and spin-down components. The boundary condition in our case corresponds to spin-triplet Andreev reflection which typically is realized at junctions between a normal lead and a spin-triplet p-wave superconductor. Indeed, if we denote spin-triplet pair potential as , then different orientations of the -vector correspond to () and () boundary conditions. This difference between conventional (s-wave) spin-singlet Andreev boundary conditions and boundary conditions considered here becomes very important later when we consider allowed boundary perturbations.
#### ii.2.2 RG analysis near normal reflection fixed point N×N
Let’s now analyze the interaction effects in the lead. In the absence of U(1) spin-rotation symmetry, we can have additional terms in the boundary action:
ST = ∫dτ[itγ↑(ψ↑(0)+ψ†↑(0))−itγ↓(ψ↓(0)+ψ†↓(0)) (34) +~tγ↑(ψ↓(0)−ψ†↓(0))−~tγ↓(ψ↑(0)−ψ†↑(0)) −Δiγ↑γ↓(−iψ†↑(0)ψ↓(0)+iψ†↓(0)ψ↑(0)) +~Δiγ↑γ↓(ψ†↑(0)ψ↑(0)−ψ†↓(0)ψ↓(0))].
We have omitted here the irrelevant terms, e.g. , analogous to those considered in Sec. II.1. After the bosonization, the boundary action reads
ST = ∫dτ[t2πa(iγ↑Γ↑cosθρ+θσ√2−iγ↓Γ↓cosθρ−θσ√2) (35) +~t2πa(iγ↓Γ↑sinθρ+θσ√2−iγ↑Γ↓sinθρ−θσ√2) −Δ2πaγ↑γ↓Γ↑Γ↓cos√2θσ+~Δ2πviγ↑γ↓i∂τθσ√2].
Note the appearance of the new marginal term described by coupling constant .
We now perform a perturbative RG analysis up to the second-order in coupling coefficients. The details of the calculations are presented in Appendix C. Here we summarize our results for :
dtdl = (36) d~tdl = (37) dΔdl = −(1Kρ−1)t2−~t24πv, (38) d~Δdl = −B(Kρ)t~t4πv. (39)
The generation of the term (proportional to ) originates from the processes involving two different spin channels of the lead whereas the generation of the term (proportional to ) comes from processes within the same spin channel. Both of these terms can be generated only in the presence of the interaction in the lead. This fact follows from the definition of the function
B(Kρ)=C(1/2Kρ−1/2)C(1/2)C(1/2Kρ)(1Kρ+1)>0. (40)
Here the function is defined as
C(ν)=limδ→0+∫∞0e−δzcos(z)(z+1)νdz, (41)
and originates from the integration over relative coordinate, during the RG procedure, see Appendix C. In the non-interacting limit, , , and thus, the RG equation for becomes
d~Δdl≈−c54πv(1Kρ−1)t˜t, (42)
where numerical constant . As mentioned, both and cannot be generated in the RG in the absence of interactions in the lead (i.e. ).
Using Eqs.(36) it is instructive to analyze first the flow in the non-interacting limit, in which case . Both and are relevant and flow to strong coupling. As follows from the discussion in the previous section, the exact boundary condition at the Andreev reflection fixed point is determined by the initial values of and and we can identify the corresponding limits by looking at the scattering matrix, i.e. corresponds to , corresponds to and finally corresponds to , see Fig. 4a.
We now analyze the RG flow for not-too-strong repulsive interactions . First of all, one can notice that even if we start with initial conditions , , the corresponding four-fermion terms are going to be generated by the RG procedure. Here is initial length cutoff. Since the couplings and affect the RG flow differently, we now have 4-parameter phase diagram. Based on the perturbative RG equations, one can see that both and will grow under RG, see Fig. 4. Thus, normal reflection fixed point is unstable in this parameter regime.
#### ii.2.3 RG analysis near spin-flip Andreev reflection fixed point SFA
We now analyze the stability of the spin-flip Andreev reflection fixed point which corresponds to the following boundary conditions:
ψL↑(0) = −iψ†R↓(0), (43) ψL↓(0) = −iψ†R↑(0). (44)
In the bosonization language, the bosonic fields and are pinned, and the Klein factors have the relation and . Now, let us study all the fermion bilinear perturbations at the boundary allowed by TR symmetry. First of all, one can show that the normal backscattering is vanishing in this case, in agreement with the scattering calculation in Sec. II.2.1. Indeed, using the boundary conditions (43) one can show that
ψ†L↑(0)ψ↑,R(0)+ψ†R↓(0)ψL↓(0)+h.c. =−iψ†L↑(0)ψ†L↓(0)+iψL↑(0)ψL↓(0)+h.c.=0 (45)
Note that for s-wave spin-singlet superconductor the boundary conditions are different: and , and the backscattering term does not vanish. Since this term is relevant for , the Andreev reflection fixed point is unstable in an s-wave superconductor-LL junction.
Let’s now consider allowed Andreev reflection bilinear processes. Among those, the only allowed bilinear term is spin-conserving Andreev reflection:
HSFA1B = λSFA1(ψ†L↑ψ†↑,R+ψ†R↓ψ†L↓+h.c.) (46) = λSFA1(iψ†L↑ψL↓+iψ†R↓ψ↑,R+h.c.) = 2λSFA12πa(iΓL↑ΓL↓+iΓR↓ΓR↑)cos√2θσ.
Additionally, we also consider the following four-fermion term
HSFA2B = λSFA2(2πa)2(ψ†L↑ψR↑ψ†L↓ψR↓+h.c.) (47) = 2λSFA2ΓL↑ΓR↑ΓL↑ΓL↓ΓR↓cos2√2ϕρ,
which corresponds to two-electron backscattering. The leading order perturbative RG equations for and are give by
dλSFA1dl = (1−1Kσ)λSFA1, (48) dλSFA2dl = (1−4Kρ)λSFA2. (49)
One can see that the first term is marginal for symmetric Luttinger liquid lead , whereas the second coupling becomes relevant for indicating that the fixed point becomes unstable for strong repulsive interactions. If the spin symmetry is broken in the lead, the fixed point becomes unstable for , and the system will flow towards the fixed point. On the other hand, the is stable for .
#### ii.2.4 RG analysis near spin-conserving Andreev fixed point A×A
As shown in Sec. II.2.1, the boundary conditions near fixed point are with or . Thus, the boson fields are and are pinned at the boundary, and the Klein factors satisfy the relations . In the -conserving case, there are only irrelevant perturbations for such as two-electron backscattering
HA×A2B= λA×A2ψ†L↑(0)ψR↑(0)ψ†L↓(0)ψR↓(0)+h.c. =λ2(2πa)2sin(2√2ϕρ). (50)
Additionally, if symmetry is broken, the spin-flip Andreev reflection processes are allowed
HA×A1B = λA×A1(ψ†R,↑(0)ψ†L,↓(0)−ψ†R,↓(0)ψ†L,↑(0))+h.c. (51) = 4iΓ↑Γ↓sin√2ϕσ.
The leading order perturbative RG equations for and are give by
dλA×A1dl = (1−Kσ)λA×A1, (52) dλA×A2dl = (1−4Kρ)λA×A2. (53)
One can see that the first term is marginal for symmetric Luttinger liquid lead , whereas the second coupling becomes relevant for indicating that fixed point becomes unstable for strong repulsive interactions. If the spin symmetry is broken in the lead, the fixed point becomes unstable for , and the system will flow towards the fixed point. On the other hand, the is stable for . Exactly at , both and terms are marginal and compete with each other. Thus, generically both spin-conserving and spin-flip Andreev reflection processes will be present and their relative strength depends on microscopic details. This conclusion is consistent with the non-interacting results () discussed in Sec.II.2.1. Our main results are summarized in Fig. 4.
## Iii Majorana Kramers pair - Quantum dot - Normal lead junction
### iii.1 Theoretical model
In this section we study effect of local electron-electron interactions and consider the system consisting of a QD with a single spin-degenerate level coupled to MKP , localized at the end of a TRITOPS, and a NL. The schematic plot of the device is shown in Fig. 1 b). Assuming that TR symmetry and -spin rotation symmetry are preserved and the induced gap in the topological superconductor is sufficiently larger than other energy scales of the problem, the low-energy effective Hamiltonian of the system can be written as
H =∑σϵd†σdσ+Un↑n↓+V+HNL (54) V =∑σ[iλσγσ(dσ+d†σ)+tσ(d†σψσ(0)+h.c.)] (55)
where and are creation and annihilation operators on the QD, , is the chemical potential of the QD, is the strength of the electron-electron interaction on the QD, and are fermion creation and annihilation operators in the NL, and is the tunneling coefficient between the NL(MKP) and the QD. For the perturbative RG analysis, we adopted the same Hamiltonian for NL as Eq. (1) with . For slave-boson mean-field theory analysis, we assumed quadratic dispersion for the NL. We set and | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9647448658943176, "perplexity": 1090.0232648393605}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027312025.20/warc/CC-MAIN-20190817203056-20190817225056-00395.warc.gz"} |
https://www.sparrho.com/item/preconditioning-complex-symmetric-linear-systems/8ceda6/ | # Preconditioning complex symmetric linear systems
Research paper by Enrico Bertolazzi, Marco Frego
Indexed on: 14 Jul '14Published on: 14 Jul '14Published in: Mathematics - Numerical Analysis
#### Abstract
A new polynomial preconditioner for symmetric complex linear systems based on Hermitian and skew-Hermitian splitting (HSS) for complex symmetric linear systems is herein presented. It applies to Conjugate Orthogonal Conjugate Gradient (COCG) or Conjugate Orthogonal Conjugate Residual (COCR) iterative solvers and does not require any estimation of the spectrum of the coefficient matrix. An upper bound of the condition number of the preconditioned linear system is provided. Moreover, to reduce the computational cost, an inexact variant based on incomplete Cholesky decomposition or orthogonal polynomials is proposed. Numerical results show that the present preconditioner and its inexact variant are efficient and robust solvers for this class of linear systems. A stability analysis of the method completes the description of the preconditioner. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8042700886726379, "perplexity": 1009.6670236027367}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488536512.90/warc/CC-MAIN-20210623073050-20210623103050-00503.warc.gz"} |
http://www.physicsforums.com/showpost.php?p=3789426&postcount=1 | View Single Post
P: 958 I'm studying a particular system but to ease the understanding let me take a general approach. 1) Suppose I have a system comprised of N=10^10 elements. Assuming now that I deal a damage to a random element for every timestep, I want to figure out how the damages are distributed in the elements. My hypothesis is that this binomial distribution can be approximated very well by an easier poisson distribution. And this brings me to my first question: Is this a correct hypothesis? 2) Next: Now the above does not really have anything to do with the system I study. Because in my system it is so that the probability of dealing damage to an element is not uniformly distributed but rather something specific to each element (normally distributed). I want to be able to predict how the damages will be distributed amongst my elements, like in 1) but this time I'm not sure how to approach the problem. Can I predict a distribution and how would I do that? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9473098516464233, "perplexity": 179.1460083599724}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394010674226/warc/CC-MAIN-20140305091114-00042-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://www.wias-berlin.de/publications/wias-publ/run.jsp?template=abstract&type=Preprint&year=2016&number=2207 | WIAS Preprint No. 2207, (2016)
# Optimal entropy-transport problems and a new Hellinger--Kantorovich distance between positive measures
Authors
• Liero, Matthias
ORCID: 0000-0002-0963-2915
• Mielke, Alexander
ORCID: 0000-0002-4583-3888
• Savaré, Giuseppe
2010 Mathematics Subject Classification
• 28A33 54E35 49Q20 49J35 49J40 49K35 46G99
Keywords
• Entropy-transport problem, Hellinger-Kantorovich distance, relative entropy, optimality conditions, cone over metric space
DOI
10.20347/WIAS.PREPRINT.2207
Abstract
We develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. They arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a couple of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, that quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger-Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger-Kakutani and Kantorovich-Wasserstein distances.
Appeared in | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8238940238952637, "perplexity": 1718.715446676436}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439740343.48/warc/CC-MAIN-20200814215931-20200815005931-00135.warc.gz"} |
http://tex.stackexchange.com/questions/22202/can-i-have-two-different-styles-of-bibliography-in-the-same-document | Can I have two different styles of bibliography in the same document?
It is possible in the same document, change the style of bibliography? I want the bibliography by chapters arranged in order of appearance. (I use `unsrt`), but I want the general bibliography in alphabetical order (and, if possible, `backref` to pages where the citation appears).
-
You should mention how you arrange the bibliographies by chapter. I'd recommend switching to biblatex that is able to do this and more. – egreg Oct 26 '11 at 15:02
that's not really possible. What you can do is to create one bibliography, rename the file `<file>.bbl` to `<file>-bib.tex` and then run your document with the setting for the other bibliography. The first created can simply be inserted into your main TeX document by `\input{<file>-bib}` | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9682865738868713, "perplexity": 909.3018209635942}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645293619.80/warc/CC-MAIN-20150827031453-00340-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://www.electrical4u.net/relay/tan-delta-test-finding-insulation-ageing-loss-angle-test/ | # Tan-Delta Test – Finding Insulation Ageing Loss Angle Test
Contents
## Tan Delta Test:
Tan Delta testing creates the idea about the insulation ageing or healthiness of the insulation and predict the remaining life of an electrical equipment. Tan delta test will be conducted in Transformer’s insulation oil, bushings, electrical cables, Insulators, electrical motor/generators windings etc. It is also called as dissipation factor or loss angle test.
## Tan delta testing Principle:
Consider an insulator, which is connected across the line to earth, it acts like capacitor. Here The insulation material in an insulator act as dielectric material. The insulator allows the electric current, when the capacitive component present in the insulator, at the same time the electrical current passes through the insulator via the resistance component of the insulator. But the insulator is 100% pure there is no resistance, and there is no current flow. In a pure capacitor, the current leads the voltage by 90 degrees. The insulation of the insulator in a pure condition, will behaves like capacitor. But, in practically it is highly impossible. There will be of ageing impurities like dirt and moisture enter into the insulator. These impurities provide the current flow path (creates resistive component) through the insulator from line to ground. This resistive component will cause the angle of the current to be less than 90 degrees. This difference in the angle is known as the loss angle. A higher value for the loss angle indicates a high degree of contamination of the insulation.
Also Know About: 7 Purpose of NGR Neutral Grounding Resistor Transformer & Generator
Therefore, the healthiness of an insulator can be estimated by ratio of resistive component to capacitive component present in the insulator. This ratio is called as tanδ or tan delta.
Key Point: The Tan Delta test works on the principle that any insulation in its pure state acts as a capacitor. For good or healthy insulator, the ratio is less, the ratio increases as the insulator age.
## Tan delta testing procedure:
• The testing equipment should be isolated from the power source, it may be Transformer’s insulation oil, bushings, electrical cables, Insulators, electrical motor/generators windings etc .
• A very low frequency test voltage is applied across the equipment whose insulation to be tested, typically the frequency ranges between 0.01Hz to 0.1 Hz. Applying a low frequency causes a higher value of capacitive reactance (Xc=1/2*π*f*C here the low frequency increases the capacitive reactance) which leads to lesser power requirement during the test and enabling easier current measurement. The measured current will be tabulated.
• Continue these process until to reach double the rated voltage of the insulator.
• The tan delta controller unit takes measurement of tan delta values. A loss angle analyzer is connected with tan delta measuring unit to compare the tan delta values at normal voltage and higher voltages, and analyze the results.
• By Comparing exiting result with previous result gives the idea about the insulation ageing.
• Also you can find through the loss factor, If the insulation is perfect, the loss factor will be approx.. same for all range of test voltages. But if the insulation is not sufficient, the value of tan delta increases in the higher range of test voltage.
Also Know About: CT Operated Thermal Over Load Relay Current setting Calculation
Also see: Trip Circuit supervision relay
Do You Want any Article related to electrical Topic, Comment Us with the Topic Name and We provide it for you.. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8538554310798645, "perplexity": 2211.0823790176346}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371886991.92/warc/CC-MAIN-20200410043735-20200410074235-00326.warc.gz"} |
http://mathoverflow.net/questions/111558?sort=votes | ## Projections in Banach spaces
Dear All,
I am absolutely lost in the following problem:
Let $P_s, \: s \in [0,1],$ be a uniformly bounded family of projections (idempotents) in a Banach space $X$ such that $P_s P_t = P_{{\rm min}(s,t)}$. Let $Q$ be a bounded linear operator on $X$ such that $QP_s = P_sQ$ for every $s \in [0,1]$ and the function $$s \mapsto P_s Q$$ is continuous in operator norm. Does it follow that the function must be constant (i.e. $P_sQ \equiv P_0Q$)?
For simplicity, one can also assume that the function $s \mapsto P_s$ is strongly continuous. I can give an affirmative answer only in some trivial situations (finite dimensional case, Hilbert spaces with family of orthogonal projections) but nothing more.
-
Are you sure that there is an affirmative answer in the Hilbert space case with a family of orthogonal projections? Can't you take the Voltera nest of subspaces and for $Q$ any compact operator which leaves these subspaces invariant? – Bill Johnson Nov 5 at 17:19
@Bill: there are not that many compact operators that leave the Voltera nest invariant... – Mikael de la Salle Nov 5 at 20:13
I had the idea on the orthogonality with $P_0=0, P_1=I$: $\|Qx\|^2=\sum_i\|(P_{s_i}−P_{s_{i+1}})Qx\|^2=\sum_i \|Q(P_{s_i}−P_{s_{i+1}})^2x\|^2$ $\leq \sum_i \|Q(P_{s_i}−P_{s_{i+1}})\|^2\|(P_{s_i}−P_{s_{i+1}})x\|^2 \leq \varepsilon \sum_i \|(P_{s_i}−P_{s_{i+1}})x\|^2= \varepsilon \|x\|^2$, where $\varepsilon$ can be arbitrarily small if we choose well enough $s_i$ point; that is, $Qx=0$ for any vector $x$, i.e. $Q=0$ must hold. – zoltan.leka Nov 5 at 21:38
@Mikael: Bill's suggestion does sound good to me. If the kernel $k(x,y)$ of an integral operator on $L_2([0,1])$ is "upper triangular" (i.e,, supported on $\{(x,y)\mid x\geq y\}$) then it has the desired property. And $k(x,y)$ is an $L_2$ function on the square then the operator is Hilbert-Schmidt. – Leonel Robert Nov 5 at 22:05
Ok, I see where the misunderstanding comes from: of course there are compact operators that leave the Voltera nest invariant (ie such that $P_s Q P_s = Q P_s$), but there are no non-zero compact operators such that $P_s Q = Q P_s$ for all $s$. And Zoltan's argument works. – Mikael de la Salle Nov 6 at 12:30
I guess that the answer is no in general. More precisely what I consider as the discrete version of your question has a negative answer. I guess that one should be able to find a couterexample to your question by an ultraproduct argument, but I did not check the details.
By discrete version of your question I mean: if $(P_n)_{n \in \mathbb N}$ is a family of uniformly bounded projections on $X$ such that $P_0=0$ and $P_n P_m = P_{\min(n,m)}$, and $Q$ is a (not necessaily bounded) linear map on $X$ that commutes with the $P_n$'s and is such that $\sup_n \|P_n Q -P_{n-1} Q\|<\infty$. Does it follow that $\sup_n \|P_n Q\|<\infty$?
If $(e_n)$ is a Schauder basis in $X$, and take $P_n$ the map $x = \sum_k x_k e_k \mapsto P_n(x) = \sum_{k \leq n} x_k e_k$. Then $P_n$ satisfies your assumption. For a sequence $z_n \in \{{-1,1\}}$, define $Q x_n = z_n x_n$ so that $\|P_n Q - P_{n-1} Q\| = \|P_n - P_{n-1}\|$. Then $\sup_n \|P_n Q\| <\infty$ for every such $Q$ is equivalent to the basis $(e_n)$ being unconditional. It is well known that there exist bases that are not unconditional. This answers negatively the discrete question.
Here is how I would try to deduce a continuous example from a discrete example $X,P_n,Q$~: for any $k$, define $P_s^{(k)} = P_{[2^k s]}$ and $Q_k = P_{2^k}Q/\|Q P_{2^k}\|$. Then consider a non principal ultrafilter on $\mathbb N$, and construct the following operators on $X^{\mathcal U}$, and the operators $P_s = (P_s^{(k)})_{k \in \mathbb N}$ and $Q_{\mathcal U} = (Q_k)_{k \in \mathbb N}$. Then $\|Q_{\mathcal U}\|=1$, so that $P_0 Q_{\mathcal U} = 0 \neq Q_{\mathcal U}= P_1 Q_{\mathcal U}$. One probably needs to assume something more on $Q$ to prove that $s \mapsto P_s Q$ is continuous. I leave this to you.
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@Mikael: I got the discrete case - thanks. But it seems to me that the ultraproduct construction gives a very special projection family: $P_0 = 0$ and $P_s = I$ for every $0 < s \leq 1,$ so $s \mapsto P_sQ$ cannot be continuous at $0$ unless $Q=0.$ For me, it is not obvious the continuous case. – zoltan.leka Nov 8 at 14:35 @Zoltan. I do not follow you, $P_s$ is never equal to the identity. Indeed, none of the $P_n$'s is the identity, so that for every $k$ there is a norm $1$ vector $x_k \in X$ such that $P_s^{(k)} x_k = 0$. If $x$ is the class of $(x_k)_{k}$ in $X^{\mathcal U}$ then $x is a norm one vector and$P_s x$is the class of$(P_s^{(k)}(x_k))_k = (0)_k$, so that$P_s x=0$. – Mikael de la Salle Nov 8 at 15:40 @Mikael: Oh, okay, okay... you are right. – zoltan.leka Nov 8 at 15:55 @Zoltan: you accepted my answer, thanks! But does this mean that you managed to show that for some correct choice of$X$,$(P_n)$and$Q$, the map$s \mapsto P_s Q_{\mathcal U}$is continuous? If so I would also be interested to see the argument. – Mikael de la Salle Nov 8 at 16:16 I accepted your answer because I like your sketch and the basic idea seems to be nice. However, the details have not been worked out yet... But somehow I expected a different answer, so probably there are non-constant examples among$L^p$-spaces$(p \neq 2)$as well. Hmm, that's surprising for me. – zoltan.leka Nov 8 at 16:24 show 2 more comments Here a simple example : Let X be the cartesian product of$L^{\infty}$and$L^{1}$on the interval$[0,1]$, let$P_{t}$the canonical projection on the subspace of functions with support$[0,t]$and choose$Q(f_{1},f_{2}) = (0,f_{1})$. - I'm probably missing something: why is$P_tQ$norm-continuous? – fedja Nov 18 at 12:19 @fedja : Because$\|(P_{t+\varepsilon}-P_{t})Q(f_{1},f_{2})\| \leq \varepsilon \|f_{1}\|_{\infty} \leq \varepsilon \|(f_{1},f_{2})\|$– jjcale Nov 18 at 12:32 Very nice, jjcale. Basically the same argument works for$L_p \oplus L_q$as long as$p\not= q$, so you also get an example for$L_p$when$1<p\not= 2 <\infty$because$L_2$is isomorphic to a complemented subspace of$L_p$. – Bill Johnson Nov 19 at 1:42 @jjcale Thanks! This is very cute :) – fedja Nov 21 at 11:54 I'm not sure if I like the ultrafilters, so I decided to find some elementary construction. I do not have much imagination for chains of commuting projections either, so let us consider the space of all functions$f:[0,1]\to X$where$X$is the space of all sequences and let$P_s$just keep the values to the left of$s$and kill the values to the right of$s$. So far so good. The operator$Q$will just act in each layer separately, so it is going to be an operator in$X$really. The devil is in the choice of the norm. We need to take an advantage of small support somehow, which calls for considering a sequence of$L^{p_k}$-norms on$[0,1]$with decreasing$p_k>1$. Then we'll have the full strength of Holder backing us. However, there needs to be some penalty for using norms with small$p_k$, so it is natural to pair each$p_k$with some norm$N_k$in$X$, which are increasing fast so that the final norm, which is just the infimum of$\sum_k\|N_k(f_k(t))\|_{L^{p_k}}$over all decompositions$f=\sum_k f_k$will have to be a hard trade-off rather than a trivial collapse to a single term. Now the first thing we want is $$N_{k+1}(Qx)\le N_k(x)$$ This will ensure that for the first small$k$we will fire with Holder to gain on the power of the length of the interval but for the large$k$, when Holder finally betrays us and the reduction of power becomes useless, we will use $$N_k(Qx)\le 2^{-k} N_k(x)$$ so each faraway term will just take care of itself. The possibilities for the choice of such family of norms and$Q\$ are unlimited but, being (sort of) an analyst, I like the backward shift and weighted spaces, so put $$(Qx)_{j}=2^{-j}x_{j+1},$$ and $$N_k(x)=\sum_j 2^{kj}|x_j|.$$
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This is very nice. – Mikael de la Salle Nov 10 at 20:23 Thank you for your answer! That norm is quite tricky... – zoltan.leka Nov 10 at 20:25 I tried to show how one can guess things like that rather than just providing a formal argument. Yes, the final formula is somewhat convoluted, but each step is fairly natural, isn't it? – fedja Nov 10 at 21:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.981167733669281, "perplexity": 209.21733041221046}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368704132298/warc/CC-MAIN-20130516113532-00019-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://cs.stackexchange.com/questions/99817/algorithms-for-elementary-operations-using-other-elementary-operators | # Algorithms for elementary operations using other elementary operators
The question asks to provide an algorithm to compute
$$(i)$$ The product of $$n$$-bit numbers using reciprocation operation and addition operation but not using multiplication and squaring.
$$(ii)$$ The square of a polynomial of degree $$n−1$$ using only reciprocation and addition operation but not using multiplication and squaring.
$$(iii)$$ The product of two $$n − 1$$ degree polynomials using only squaring and addition operations but not using multiplication and reciprocation operations
All the operations stated above are on $$n$$-bit integers.
My try: The question asks to express some operator in terms of other elementary operators , so for eg. for (i) I tried writing $$(\frac{1}{A} + \frac{1}{B})^{-1} = \frac{AB}{A+B}$$, but to get $$AB$$, I still have to multiply LHS by $$A + B$$. I think I can do $$(ii)$$ if I do $$(i)$$ as a polynomial is similar to a n-bit integer if you consider the bits as $$(a_ix^i)$$. I don't have an idea for $$(iii)$$ yet.
• Does this question belong on Math StackExchange instead? – ab123 Nov 9 '18 at 14:46
• I am allowed to use them, but I do think I may have to use them. – ab123 Nov 9 '18 at 17:31
• I asked it on MSE as well, as I wasn't getting a response here, and someone figured out the first part. You can see it here - math.stackexchange.com/questions/2991467/… – ab123 Nov 9 '18 at 17:33
• @Klorax it uses squaring but the square can also be computed using reciprocal and sum as written in $P^2 = \frac{1}{\frac{1}{P} - \frac{1}{P+1}} - P$. Yes, I am looking for an expression involving operators. – ab123 Nov 9 '18 at 17:43
• I cannot see whether subtraction is allowed or not. Can you list all "elementary operations"? Is division an elementary operation? Is shift or floor an elementary operation? Is remainder an elementary operation? Is maximum of two numbers an elementary operation? – John L. Nov 9 '18 at 20:02
Throughout this answer, when we say we will be using some operations, it implies that no other operations is allowed. We also assume 1 is available as an operand.
The above clarification is considered as an improvement of original statement of problems. There will be further modification/clarification of the original problems, which have to be made because of the ambiguity of the original problems.
Here is the answer to part (i)
Part (i). The product of two non-zero numbers using only reciprocation operation and subtraction operation.
Please note I have replaced by addition by subtraction; otherwise, it is impossible. Also I removed the specification of $$n$$-bits, which seems distracting and harmful; otherwise, $$\dfrac13$$ is problematic. I have also added the condition that the numbers are non-zero. Otherwise, we have to add an operation that checks whether a number is non-zero and acts accordingly. Also 1 have to be available. Otherwise, we cannot produce 1, which is needed in the final operation.
Let the two non-zero numbers are $$A, B$$.
1. we get addition by $$X+Y=X-((Y-Y)-Y)$$.
2. we get 4=1+1+1+1.
3. we get $$X\to \dfrac {X^2}4$$ by $$\dfrac {X^2}4=X+\dfrac1{\dfrac1{X-4}-\dfrac1X}$$.
4. We get multiplication by $$AB=\dfrac{(A+B)^2}4-\dfrac{(A-B)^2}4$$.
This answer at mathoverflow uses division by 2 or multiplication by $$\dfrac12$$ without explanation, which can be considered as a minor flaw. In fact, we can do division by 2 as $$\dfrac A2=\dfrac1{\dfrac1A+\dfrac1A}$$.
Part (ii). The square of a polynomial with given coefficients using only reciprocation and subtract operation.
Since the coefficient of the square of a polynomial is the sum of products of various coefficients of that polynomial, this part can be done easily as a consequence of part (i).
Part (iii). The product of polynomials with given coefficients using only squaring and subtraction operations and division by 2.
We get addition by $$X+Y=X-((Y-Y)-Y)$$. We get multiplication by $$XY=\dfrac{(X+Y)^2-X^2-Y^2}2$$. Since the coefficient of the square of a polynomial is the sum of products of various coefficients of that polynomial, we can compute the product of two polynomials.
Note that division by 2 is needed. Otherwise, it can be proven that it is not possible to do multiplication although we have $$2XY=(X+Y)^2-X^2-Y^2$$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 26, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8505135178565979, "perplexity": 334.92804433997975}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154408.7/warc/CC-MAIN-20210802234539-20210803024539-00399.warc.gz"} |
http://math.stackexchange.com/questions/161046/thompsons-conjecture | # Thompson's Conjecture
I have heard that the following is a conjecture due to Thompson: The number of maximal subgroups of a (finite) group $G$ does not exceed the order $|G|$ of the group.
My question is: did Thompson really conjecture this? If so, is there any literature on the subject?
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Question: if $P$ is a $p$-group and $M_1,M_2$ are maximal subgroups of $P$ such that $M_1/\Phi(P )=M_2/\Phi(P )$, does it follow that $M_1=M_2$? I need this in order to understand something Newton mentions in the first page of his paper. – the_fox Jun 21 '12 at 15:00
@Stefanos Yes, and this holds in greater generality: if $H$ is a normal subgroup of $G$, and $G_1,G_2$ are subgroups containing $H$, then $G_1/H=G_2/H$ if and only if $G_1=G_2$. Indeed, if there exists $a\in G_1\setminus G_2$, then the coset $aH$ is contained in $G_1/H$ but not in $G_2/H$. – user31373 Jun 21 '12 at 15:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9466772675514221, "perplexity": 96.49253205660409}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931010792.55/warc/CC-MAIN-20141125155650-00233-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://www.maplesoft.com/support/help/Maple/view.aspx?path=GraphTheory%2FIsTree | IsTree - Maple Help
GraphTheory
IsTree
test if graph is a tree
Calling Sequence IsTree(G)
Parameters
G - an undirected graph
Description
• An undirected graph G on $n$ vertices is a tree if it is connected and has exactly $n-1$ edges.
• The IsTree command returns true if the input graph is a tree, and false otherwise.
Examples
> $\mathrm{with}\left(\mathrm{GraphTheory}\right):$
> $T≔\mathrm{Graph}\left(\left\{\left\{1,2\right\},\left\{1,3\right\}\right\}\right)$
${T}{≔}{\mathrm{Graph 1: an undirected unweighted graph with 3 vertices and 2 edge\left(s\right)}}$ (1)
> $\mathrm{IsTree}\left(T\right)$
${\mathrm{true}}$ (2)
> $C≔\mathrm{Graph}\left(\left\{\left\{1,2\right\},\left\{1,3\right\},\left\{2,3\right\}\right\}\right)$
${C}{≔}{\mathrm{Graph 2: an undirected unweighted graph with 3 vertices and 3 edge\left(s\right)}}$ (3)
> $\mathrm{IsTree}\left(C\right)$
${\mathrm{false}}$ (4) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 11, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9749585390090942, "perplexity": 1665.4062946302804}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323588341.58/warc/CC-MAIN-20211028131628-20211028161628-00598.warc.gz"} |
https://www.physicsforums.com/threads/i-am-not-understanding-this-differential-relationship.706482/ | I am not understanding this differential relationship
1. Aug 20, 2013
mindheavy
I'm studying engineering dynamics. The first chapter is discussing the velocity and acceleration equations; v = ds/dt and a = dv/dt. It then goes on to show a third equation that is stated as "a ds = v dv". They say they derived this equation by combining the two previous and 'eliminating dt'. I am just not seeing how they arrived at this, what are the intermediate steps? I also am not understanding the reason for just 'eliminating' dt. Could anyone develop this or help me along my way of understanding how this third equation was reached, I feel very uncomfortable just memorizing it without understanding where it came from...
2. Aug 20, 2013
saminator910
Pretty much solve as you would any other equation, the like term is dt, so it can be eliminated.
get both in terms of dt = something, then set equal to eliminate the term.
$\frac{dv}{a} = \frac{ds}{v}$
$vdv = a ds$
3. Aug 20, 2013
HallsofIvy
You can "eliminate the dt" by using the chain rule, a= dv/dt= (dv/ds)(ds/dt)= (dv/ds)v
From a/v= dv/ds, we get ads= vdv or, equivalently, ds/v= dv/a
4. Aug 20, 2013
mindheavy
Similar Discussions: I am not understanding this differential relationship | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8664072751998901, "perplexity": 1300.671720346864}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084886416.17/warc/CC-MAIN-20180116105522-20180116125522-00795.warc.gz"} |
http://math.stackexchange.com/questions/17716/how-does-teacher-get-first-step/17718 | # How does teacher get first step?
Below are the steps the teacher took to solve:
$y = \sqrt{3}\sin x + \cos x$ find min and max on $[0, 2\pi)$
Step 1: = $2\sin(x + \pi/6))$
How does the teacher get this first step?
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I'm posting this answer in response to the comment thread under picakhu's answer; writing comments was getting a bit tedious. The answer has the same content as Isaac's, but is explained a little differently, which might be useful for the OP.
The general problem, of which this is a special case, is:
Given an expression of the form $a \sin x + b \cos x$, for some numbers $a$ and $b$, to rewrite it in the form $c \sin (x + \theta)$, for some appropriate choice of $c$ and $\theta$ (which will be related to $a$ and $b$ in some way, of course --- and we have to find out what that way is!).
Suppose first that $a^2 + b^2 = 1$. Then trigonometry (especially the point of view of the unit circle) tells us that there is an angle $\theta$ such that $a = \cos \theta$ and $b = \sin \theta$.
Thus we can write $a \sin x + b \cos x = \cos \theta \sin x + \sin \theta \cos x,$ and via the addition theorem for $\sin$, we recognize this as being $\sin(x + \theta).$
Now in most examples, it may not be that $a^2 + b^2 = 1$. But it equals something (!), so let's call that something $c^2$. Then we see that $(a/c)^2 + (b/c)^2 = 1$, so we can choose $\theta$ so that $a/c = \cos \theta$ and $b/c = \sin \theta$. Then as above we find that $(a/c)\sin x + (b/c) \cos x = \sin (x + \theta),$ and so (finishing finally) we have $$a \sin x + b \cos x = c \sin (x + \theta).$$
In the OP's example, $a = \sqrt{3}$ and $b = 1$, so $c = 2$, and we are led to the given answer.
Note that in practice this whole process will be easiest when $a/c$ and $b/c$ are easily recognized trig function special values, like $1/2$ or $\sqrt{2}/2$ or $\sqrt{3}/2$. If they are just somewhat random numbers, then you won't be able to figure out the correct choice of $\theta$ without using a calculator to compute $\theta$.
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Your answer is easier for me to follow. – O.O Jan 16 '11 at 17:01
picakhu's answer is the simplest way to see how it works having already arrived at $y=2\sin(x+\frac{\pi}{6})$ (use the identity there to expand this form). In general, given $a\sin x+b\cos x$ (let's say for $a,b>0$), it is possible to arrive at a similar equivalent form: \begin{align} a\sin x+b\cos x &=a\left(\sin x+\frac{b}{a}\cos x\right) \\ &=a\left(\sin x+\tan\left(\arctan\frac{b}{a}\right)\cos x\right) \\ &=a\left(\sin x+\frac{\sin\left(\arctan\frac{b}{a}\right)}{\cos\left(\arctan\frac{b}{a}\right)}\cos x\right) \\ &=\frac{a}{\cos\left(\arctan\frac{b}{a}\right)}\left(\sin x\cos\left(\arctan\frac{b}{a}\right)+\sin\left(\arctan\frac{b}{a}\right)\cos x\right) \\ &=\sqrt{a^2+b^2}\sin\left(x+\arctan\frac{b}{a}\right). \end{align}
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Instead of telling you how your teacher got that, try doing the opposite and expand out $\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)$
The rest should be trivial
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I need a little more info than that. There is no a and b in the problem above, so I am confused. – O.O Jan 16 '11 at 16:31
@subt13: The policy of the site is to not give full answers to homework questions, so pikachu has done what he should. This is a pretty good hint. The only other thing I would say is "equate coefficients". That should lead you there pretty quickly. – Noldorin Jan 16 '11 at 16:32
@subt13: Dear subt13, You have to choose some particular $a$ and $b$. Do you see any similarities between $\sin(a+b)$ and $\sin(x + \pi/6).$ – Matt E Jan 16 '11 at 16:33
@Noldorin - I understand the policy, I actually have all of the steps to the answer, I just don't understand it. I'm not even taking a math class... – O.O Jan 16 '11 at 16:34
@subt13: $\frac{\pi}{6}$ is not just a random guess at what value to try—the $\sqrt{3}$ suggests something like $\frac{\pi}{6}$ or $\frac{\pi}{3}$, but a value can be found explicitly as in my answer. – Isaac Jan 16 '11 at 16:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9788861274719238, "perplexity": 166.5672764760738}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257823805.20/warc/CC-MAIN-20160723071023-00157-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://brilliant.org/discussions/thread/002/ | 0/0=2???
0/0=(100-100)/(100-100)=(10^2-10^2)/10(10-10)=[(10-10)(10+10)]/10(10-10)=20/10=2!!!!!!.
4 years, 1 month ago
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I'm sorry to tell you this, but you made a mistake in your calculations: As I answered in a question which was made in Brilliant, there are reasons that lead to 0/0 being undetermined. The problem in the 4th step is that when you went from [(10-10)(10+10)]/10(10-10) to 20/10, you divided the numerator and the denominator by 0. Here is what you did: [(10-10)(10+10)]/10(10-10) = [(10-10)(10+10)/(10-10)]/[10(10-10)/(10-10)], which can be simplified to [(10+10)0/0]/(100/0). Given the fact 0/0 is undefined, you can't replace it with 1, nor any other number. Basically, you can't undo a multiplication with zero by dividing by zero, nor undo a division by zero by multiplying by zero, as both situations result in something similar to a*0/0, in which 0/0 is undefined and can't be replaced by any number, Other implications of 0/0 being undefined are that anything being divided or multiplied by 0/0 is undefined, and that you can't divide anything by an expression which is or can be equal to zero, as the result is undefined in case said expression turns out to be zero. This is why we can't simplify x/x into 1, for example. Here's the reasons why 0/0 is undefined, written by me: https://brilliant.org/discussions/thread/why-00-is-not-define/
- 3 years, 9 months ago
Yep!You got it.I already knew this mistake.I like fallacies.
- 3 years, 9 months ago
Me too.
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https://www.arxiv-vanity.com/papers/1709.08647/ | # Comparing galaxy formation in semi-analytic models and hydrodynamical simulations
Peter D. Mitchell, Cedric G. Lacey, Claudia D. P. Lagos, Carlos S. Frenk, Richard G. Bower, Shaun Cole, John C. Helly, Matthieu Schaller, Violeta Gonzalez-Perez, Tom Theuns
Univ Lyon, Univ Lyon1, Ens de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F-69230, Saint-Genis-Laval, France
Institute for Computational Cosmology, Department of Physics, University of Durham, South Road, Durham, DH1 3LE, UK
International Centre for Radio Astronomy Research, 7 Fairway, Crawley, 6009, Perth, WA, Australia
Institute of Cosmology and Gravitation, Portsmouth University, Dennis Sciama Building, Burnaby Road, Portsmouth PO1 3FX, UK
Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium E-mail:
June 30, 2021
###### Abstract
It is now possible for hydrodynamical simulations to reproduce a representative galaxy population. Accordingly, it is timely to assess critically some of the assumptions of traditional semi-analytic galaxy formation models. We use the eagle simulations to assess assumptions built into the galform semi-analytic model, focussing on those relating to baryon cycling, angular momentum and feedback. We show that the assumption in galform that newly formed stars have the same specific angular momentum as the total disc leads to a significant overestimate of the total stellar specific angular momentum of disc galaxies. In eagle, stars form preferentially out of low specific angular momentum gas in the interstellar medium (ISM) due to the assumed gas density threshold for stars to form, leading to more realistic galaxy sizes. We find that stellar mass assembly is similar between galform and eagle but that the evolution of gas properties is different, with various indications that the rate of baryon cycling in eagle is slower than is assumed in galform. Finally, by matching individual galaxies between eagle and galform, we find that an artificial dependence of AGN feedback and gas infall rates on halo mass doubling events in galform drives most of the scatter in stellar mass between individual objects. Put together our results suggest that the galform semi-analytic model can be significantly improved in light of recent advances.
###### keywords:
galaxies: formation – galaxies: evolution – galaxies: haloes – galaxies: stellar content
pagerange: Comparing galaxy formation in semi-analytic models and hydrodynamical simulationsEpubyear: 2017
## 1 Introduction
Semi-analytic galaxy formation models are established tools for connecting the predicted hierarchical growth of dark matter haloes to the observed properties of the galaxy population (e.g. Cole et al., 2000; Somerville et al., 2008b; Guo et al., 2011). Unlike empirical abundance matching (e.g. Conroy et al., 2006; Moster et al., 2010) or halo occupation distribution models (e.g. Berlind & Weinberg, 2002), semi-analytic models employ a forward-modelling approach and are constructed such that they contain as much as possible of the baryonic physics that is thought to be relevant to galaxy evolution, albeit at a simplified, macroscopic level. The simplified, macroscopic nature of semi-analytic models means that they are computationally inexpensive to evaluate. Compared to hydrodynamical simulations, this lack of computational expense meant that until recently it was uniquely possible for semi-analytic models to predict realistic galaxy populations (e.g. Bower et al., 2006; Croton et al., 2006; Henriques et al., 2013).
Recently, advances in computational resources combined with improvements in the uncertain modelling of feedback have allowed hydrodynamical simulations to predict galaxy populations which reproduce observations at an equivalent level to semi-analytic models for representative volumes Vogelsberger et al. (2014); Schaye et al. (2015); Dubois et al. (2014); Davé et al. (2016). It is timely therefore to review the underling assumptions which underpin semi-analytic models and assess their validity against state-of-the-art hydrodynamical simulations.
As in semi-analytic models, hydrodynamical simulations are forced to implement uncertain subgrid modelling to approximate the effect of massive stars and black holes on galaxy evolution. This means that, for example, the dynamics of outflowing gas in these simulations are not necessarily realistic (irrespective of whether a realistic galaxy population is produced). Importantly however, the dynamics of outflows are tracked self-consistently in hydrodynamical simulations. Furthermore, simulations do not need to make any assumptions regarding angular momentum conservation of the various baryonic components of galaxies and their surrounding gas flows. In semi-analytic models, both these aspects of galaxy evolution are among the most important for predicting galaxy properties and yet are also among the most uncertain Henriques et al. (2013); Mitchell et al. (2014); Hirschmann et al. (2016). Arguably therefore, the parametrisations of these physical processes that are implemented in semi-analytic models should be capable of reproducing (with an appropriate choice of model parameters) the behaviour predicted by hydrodynamical simulations. Here, we begin to address this topic by comparing the properties of galaxies between the established semi-analytic model, galform Cole et al. (2000); Lacey et al. (2016), and the eagle simulation project, a state-of-the-art suite of calibrated hydrodynamical simulations Schaye et al. (2015); Crain et al. (2015).
A number of other authors have presented comparisons of results from hydrodynamical simulations to semi-analytic models, albeit without access to hydrodynamical simulations that predict realistic galaxy properties for representative volumes. Some studies have focused specifically on cooling and gas infall onto galaxies, finding varying levels of agreement Yoshida et al. (2002); Helly et al. (2003); Monaco et al. (2014). Saro et al. (2010) analysed a single, massive cluster, finding significant differences in the manner with which tidal stripping acts between a semi-analytic model and a hydrodynamical simulation. Stringer et al. (2010) analysed a single disc galaxy, finding it was possible to roughly reproduce a hydrodynamical simulation with an explicitly calibrated semi-analytic model. Cattaneo et al. (2007) and Hirschmann et al. (2012) analysed larger samples of galaxies, both finding broad agreement in stellar and baryonic masses but significant differences when analysed in detail. In particular, Hirschmann et al. (2012) reported large differences in star formation efficiency stemming from local versus global implementations of a Kennicutt star formation law.
This study follows from Guo et al. (2016), who compared galform, the similar L-Galaxies model Guo et al. (2011) and eagle. They focused on global predictions for the galaxy population (stellar mass functions, star formation rates (SFRs), passive fractions, mass-metallicity relations, mass-size relations). They showed that stellar mass functions and passive fractions were broadly similar between the models (provided gradual ram pressure stripping of hot gas from satellites was implemented in galform). However, they also showed that predictions for galaxy sizes differed significantly and that mass-metallicity relations are significantly steeper in galform than in the reference eagle model. In both cases, the predictions from eagle are in significantly better agreement with observations.
While these disagreements between the models are highly suggestive of differing baryon cycling (because of discrepant metallicities) and angular momentum evolution (because of discrepant sizes, see also Stevens et al., 2017), from a global comparison it is not clear exactly how these differences arise. Here, we compare the galform and eagle models in more detail and attempt to isolate as far as possible distinct physical processes, focussing on those which we expect may not be modelled realistically in semi-analytic models. We also match individual galaxies by matching the haloes between the dark-matter-only version of eagle and the full hydrodynamical simulation. This allows us to assess the difference in stellar mass between individual galaxies.
The results and methodology presented here will, in turn, underpin a future study where we plan to perform the most direct level of comparison possible between galform and eagle. Namely, to directly measure all of the mass, metal and angular momentum exchanges between different discrete baryonic reservoirs in eagle and compare with the corresponding quantities in galform. As such, we consider here how to compartmentalize baryons in eagle between the corresponding discrete components that are tracked in semi-analytic models. In particular, we carefully consider how to separate the interstellar medium (ISM) from more diffuse halo gas in the circumgalactic medium (CGM) in eagle on physical grounds.
The layout of this paper is as follows. We introduce the eagle simulations, the galform semi-analytic model and describe our analysis methodology in Section 2. We present a first comparison of the models by analysing stellar masses in Section 3. We compare star formation thresholds and efficiencies as well as the angular momentum of star-forming gas in Section 4. We discuss feedback from supernovae (SNe) and active-galactic nuclei (AGN) in Section 6.2 and the resulting baryon cycle in Section 7. We discuss the consequences of qualitative differences between gas infall rates onto galaxies in the two models in Section 8. Finally, we summarise our main results in Section 10.
Throughout this paper we denote the units of distances in proper kiloparsecs as and comoving kiloparsecs as . Also throughout, refers to the base logarithm and refers to the natural logarithm.
## 2 Modelling galaxy formation
To facilitate a direct comparison of the eagle simulations and the galform model, we follow Guo et al. (2016) by running galform on a dark-matter-only version of the reference eagle simulation run with a box (L100N1504 in the convention introduced by Schaye et al., 2015). As described by Guo et al. (2016), both simulations where performed with the same cosmological parameters taken from Planck Collaboration et al. (2014), and with the same initial conditions, following the method of Jenkins (2010).
### 2.1 eagle
The eagle simulations are a suite of hydrodynamical simulations of the formation and evolution of galaxies within the context of the CDM cosmological model. Performed using a modified version of the gadget-3 code (last presented in Springel et al., 2005), they include a state-of-the-art implementation of smoothed particle hydrodynamics (SPH, Dalla Vecchia in prep, Schaller et al., 2015a). They also include a set of subgrid models that account for the physics of photo-heating/ionization from an evolving, uniform background radiation field, radiative cooling from metal lines and atomic processes, star formation, stellar and supermassive black hole evolution and feedback. Subgrid models are included to compensate for the limited resolution of cosmological simulations and the prohibitive computational cost of performing detailed on-the-fly radiative transfer. A detailed overview of these subgrid models can be found in Schaye et al. (2015) and a concise overview tailored to the topic of comparison with semi-analytic models can be found in Guo et al. (2016).
### 2.2 galform
galform is a continually updated semi-analytic galaxy formation model, first introduced in Cole et al. (2000), which itself was an evolution from earlier models (e.g. Lacey & Cole, 1993; Cole et al., 1994). Salient updates subsequent to Cole et al. (2000) include the inclusion of AGN feedback Bower et al. (2006), the addition of gradual ram-pressure stripping in satellites Font et al. (2008) and a decomposition of the ISM into neutral atomic and molecular hydrogen components Lagos et al. (2011). The most recent branches of the model can roughly be divided between a model with a universal stellar IMF Gonzalez-Perez et al. (2014); Guo et al. (2016), and a model that also implements a non-standard IMF in nuclear starbursts Lacey et al. (2016).
Guo et al. (2016) introduced a version of the universal IMF model that was explicitly tuned for the eagle DM-only simulation using simulation outputs. For this study, we use an updated version of this model which is very similar. The updates were made to ensure that properties of individual galaxies do not depend on random numbers, as discussed below. This is a desirable step for comparing with the eagle simulations on an object-by-object basis. Otherwise, the model parameters are the same as Guo et al. (2016), with the exception of slight changes to two parameters which control the efficiencies of SNe and AGN feedback111Specifically, we change the normalisation of the SNe feedback mass loading factor, from to and the threshold for AGN feedback, , from to . See Lacey et al. (2016) for the definition of these model parameters.. These changes were made to approximately restore predictions for the local stellar mass function presented in Guo et al. (2016) after an error in the calculation of halo concentrations introduced in Guo et al. (2016) was corrected 222Specifically, an incorrect tabulated power spectrum file was used to calculate halo concentrations. This error does not affect any of the conclusions of that study.. The calibration of the reference model used here hence follows Guo et al. (2016) and is described in Section 2.7.
To remove a dependency of individual galaxy properties on random numbers, we make two changes with respect to Guo et al. (2016). The first is that we now measure halo spin parameters from the eagle dark-matter-only simulation instead of sampling from a probability distribution function, as introduced in Cole et al. (2000). The second is that we now track the orbits of satellites measured in the dark-matter-only simulation. Once satellites can no longer be identified in the simulation, a self-consistent dynamical friction merging timescale, , is then computed as
Tdf=(RcRH)1.8(JJc)0.85τdyn2B(1)ln(Λ)(MHMS), (1)
where is the halo virial radius, is the radius of a circular orbit with the same energy as the actual orbit, is the ratio of the angular momentum of the actual orbit to the angular momentum of a circular orbit with the same energy, is the halo dynamical time333Defining the halo dynamical time as , where is the halo virial radius and is the halo circular velocity at the virial radius., is the host halo mass, is the mass of the satellite subhalo, is the Coulomb logarithm (taken to be ) and . The full details of this new merging scheme are given in Simha & Cole (2017)444 Note that we do not use the tidal disruption model described in Simha & Cole (2017)..
### 2.3 Structure and assumptions that underpin semi-analytic galaxy formation models
In semi-analytic galaxy formation models such as galform, the initial presupposition is that baryons trace the accretion of dark matter mass and angular momentum onto collapsed dark matter haloes, that the baryons that have been accreted onto dark matter haloes can be compartmentalized into a few discrete components, and that these components can be adequately characterised by a handful of quantities. The hierarchy of galaxy formation is accounted for by including each subhalo as a distinct entity such that the evolution of satellite galaxies is tracked within parent haloes.
The discrete baryonic components typically tracked in a modern semi-analytic galaxy formation model consist of a galaxy disc, a galaxy bulge/spheroid, a diffuse gas halo and a reservoir of gas that has been ejected from the galaxy by feedback. The quantities tracked for each of these components typically include the total mass, the magnitude of the angular momentum, the metal content and a set of scale lengths that specify the spatial distribution, assuming idealised density profiles. These quantities are evolved enforcing mass conservation and (typically) total angular momentum conservation. Galaxy formation is expected to be a highly dissipative process and so energy conservation is not usually explicitly tracked (although see Monaco et al., 2007). However, individual physical processes do often contain energetic considerations, for example in the computation of a radiative cooling timescale for hot diffuse halo gas.
A simplified, linearised version of the mass conservation equations for a central subhalo in galform is
⎡⎢ ⎢ ⎢ ⎢ ⎢⎣˙Mdiffuse˙MISM˙Mejected˙M⋆⎤⎥ ⎥ ⎥ ⎥ ⎥⎦=⎡⎢ ⎢ ⎢ ⎢⎣fB˙MH000⎤⎥ ⎥ ⎥ ⎥⎦+⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣−1τinfall01τret01τinfall−(1−R+βml)τ⋆000βmlτ⋆−1τret000(1−R)τ⋆0⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⎡⎢ ⎢ ⎢ ⎢⎣MdiffuseMISMMejectedM⋆⎤⎥ ⎥ ⎥ ⎥⎦ (2)
where is the mass in a diffuse gas halo, is the mass in the interstellar medium, is the mass in a reservoir of gas ejected from the galaxy by feedback, is the mass in stars and is the total halo mass. is the cosmic baryon fraction, is the timescale for halo gas to infall onto a disc, is the disc star formation timescale, is the mass fraction returned from stars to the ISM through stellar mass loss, is the efficiency of SNe feedback555Specifically is the mass loading factor, defined as the ratio of the mass outflow rate from galaxies to the star formation rate. and is the return timescale for gas ejected by feedback. Alongside Eqn 2, there is also a corresponding set of equations for metal mass and angular momentum. Here, we have neglected the distinction between the disc and bulge/spheroid for simplicity and we have written the star formation law as being linear in the ISM gas mass (which is not the case for galform models following Lagos et al., 2011).
The source term in Eqn 2 is the halo accretion rate, , scaled by the cosmic baryon fraction, , to give the baryonic accretion rate. The stellar mass reservoir, , acts as a sink term because stellar recycling is implemented with the instantaneous recycling approximation Cole et al. (2000). The strongly coupled nature of galaxy formation is encoded in the off-diagonal terms of Eqn 2. Several of these terms ( and ) depend in a non-linear fashion on different combinations of the halo density profile, halo mass accretion history, halo spin, disc angular momentum and diffuse halo metal content, such that the coupling between various aspects of the model is implicitly tighter than is shown explicitly in Eqn 2.
The terms that appear in the central matrix of Eqn 2 represent distinct physical processes, some of which are analogous to the inclusion of the subgrid models included in eagle. These include star formation (), stellar recycling () and the energy injection from stellar feedback (). Other processes are not modelled by subgrid models in eagle and emerge naturally within the hydrodynamical simulation (although these will still be affected by uncertain subgrid modelling). These include gas infall from a diffuse halo (characterised by ) and reincorporation from a reservoir of gas ejected from the galaxy by feedback back into the diffuse halo ().
Additional physical processes included in galform but not shown in Eqn. 2 include metal enrichment of the ISM, the growth of central SMBHs and the resulting AGN feedback, ram-pressure stripping of satellite galaxies, spheroid formation through galaxy mergers and disc instabilities, nuclear starbursts and the suppression of gas accretion onto small haloes by the UV background after reionization (for a complete overview see Lacey et al., 2016).
Table 1 presents a brief summary of the relevant physical processes included in galform and eagle, sorted by the order in which they discussed in the following sections. More details for these physical processes are given in Appendix A.
### 2.4 Subhalo identification & merger trees
In both galform and eagle, haloes are identified first as groups using a friends-of-friends (FoF) algorithm, adopting a dimensionless linking length of Davis et al. (1985). FoF groups are split into subhaloes of bound particles using the subfind algorithm Springel et al. (2001); Dolag et al. (2009). For eagle, galaxies are then defined as the baryonic particles bound to a given subhalo. For each FoF group, the subhalo containing the particle with the lowest value of the gravitational potential is defined as hosting the central galaxy and other subhaloes are defined as hosting satellite galaxies. Galaxy centres are based on the position of the particle with the lowest gravitational potential.
In galform, haloes are identified in the same way using the dark-matter-only version of the reference eagle simulation. Unlike in eagle, groups of subhaloes are then grouped together by the dhalo algorithm presented by Jiang et al. (2014). This algorithm sets the distinction between central and satellite galaxies using information from the progenitors of a given subhalo. In detail, subhaloes are flagged as satellites for the first time when they first enter within twice the half-mass radius of a more massive subhalo and if they have lost at least of their past maximum mass (see Appendix A3 in Jiang et al. 2014). Once a subhalo is identified as a satellite, this status is then preserved for all its descendants for which it is considered the main progenitor. This leads to situations where galaxies are still considered to be satellites even if they have escaped outside the virial radius of a parent host halo and out into the field (see Guo et al., 2016, for a discussion of the importance of this choice and a comparison with the implementation in the L-Galaxies model).
We define halo masses in eagle (only for central subhaloes) as , the total mass enclosed within a radius within which the mean internal density is times the critical density of the Universe. Internally within galform, halo masses are defined simply as the sum of the mass of each subhalo associated with a given dhalo (denoted as ). The masses of each subhalo are defined simply as the sum of the particles considered gravitationally bound by subfind to that subhalo. We have also measured for central galaxies from the eagle dark-matter-only simulation and we use these masses for galform galaxies when comparing galaxy properties at a given halo mass to eagle. The differences between these various halo masses are shown in Appendix B. Hereafter, we denote as referring to measured from the hydrodynamical simulations for eagle galaxies and measured from the dark-matter-only simulation for galform galaxies.
As well as the merger trees used in galform, we also construct merger trees for the reference eagle hydrodynamical simulation using the same subhalo merger tree scheme that underpins the dhalo algorithm. We use these trees only when presenting results that involve tracking the main progenitors of eagle galaxies in time. We define the main progenitor as the subhalo progenitor containing the most bound particle. For the merger trees constructed from the hydrodynamical simulation, we ensure in post-processing that the main progenitor is always a dark-matter subhalo, as opposed to a fragmented clump of star and black hole particles.
### 2.5 Matching haloes
To compare galaxies between eagle and galform on an object-by-object basis, we match haloes between the reference eagle simulation () with the corresponding dark-matter-only simulation, following the methodology of Schaller et al. (2015b). Haloes are matched using unique dark matter particle identifiers (IDs). For each subhalo in the reference eagle simulation, the most bound dark matter particles are identified and cross-matched against the particle IDs of subhaloes in the dark-matter-only simulation. If more than half of these particles are found to be associated with a given subhalo in the dark-matter-only simulation, and over half of the corresponding most bound particles from that subhalo belong to the former subhalo (such that the match is bijective), then the match is considered positive. This matching procedure is performed for a selection of redshifts ( & ). The matching statistics are listed in Table 2.
### 2.6 Compartmentalization
For each baryonic component that is included in galform, we assign baryonic particles to a corresponding component in eagle. We first assign baryonic particles to a given subhalo as described in Section 2.4. The baryonic particles associated with a given subhalo and then assigned to one of the following reservoirs:
• Stars-galaxy, - stars associated with the galaxy.
• Stars-ICL, - stars associated with the intra-cluster medium.
• Halo gas, - diffuse circumgalactic halo gas.
• ISM gas, - gas in the ISM of the galaxy.
• Ejected gas, - gas that has been ejected (but not later reincorporated) from the subhalo. Note that this reservoir therefore (only) includes particles that are not bound to the subhalo.
The galactic stellar component is straightforwardly defined as the stellar particles within of the subhalo centre, following Schaye et al. (2015). This aperture is used to make a distinction between stars in the galaxy and the significant, extended intracluster light component that exists around massive, galaxies in eagle. We do not include such an aperture for galform galaxies at present because the corresponding massive galaxies have much smaller half-light radii than in eagle Guo et al. (2016). When analysing results from eagle, we do not attempt to distinguish between stellar disc and spheroid components. Unless otherwise stated, all stellar properties presented from galform are computed by summing bulge and disc components.
In galform, the ISM consists of two components: a rotationally supported gas disc and a nuclear component that is associated with bursts of star formation. The remaining gas within a given subhalo is then grouped together as a circumgalactic halo-gas component. We define a corresponding ISM component in eagle by selecting gas particles within a given subhalo that are either rotationally supported against collapse to the halo centre or are spatially coincident with the galactic stellar component.
We select rotationally supported gas particles as those that satisfy both
−0.2
and
where is the rotational specific kinetic energy associated with motion orthogonal to the radial vector orientated from the gas particle to the subhalo centre666Note that there is therefore no preferred rotation direction in our ISM selection criteria and no distinction is made between gas particles that are corotating and those that are counter-rotating with respect to the ensemble.. is the corresponding specific kinetic energy associated with radial motion. is the specific gravitational energy defined as and is the specific internal energy.
Eqn 3 acts to select gas particles that have the correct rotational kinetic energy to be in rotational equilibrium against the gravitational potential at a given radius. Eqn 4 acts to remove particles with significant radial motion or with significant thermal energy. Put together, Eqns 3 and 4 act to separate the rotationally supported ISM from a diffuse, hot gaseous halo or from radially infalling accretion streams.
In addition to rotationally supported gas, we also select gas particles that are spatially coincident with the stellar galactic component. Specifically, we select any dense gas () that is within twice the half-mass radius, of the stellar component of the subhalo. This acts to select dense, nuclear ISM gas that is typically pressure supported because of the imposed ISM equation of state in eagle Schaye et al. (2015).
Finally, we apply a number of radial cuts that act to remove distant, rotationally supported material that is clearly not spatially coincident with the ISM. For inert, passive galaxies with no centrally peaked ISM component, additional care must be taken to use radial cuts appropriate for these systems.
Specifically, we assign gas particles to the ISM in eagle by applying the following selection criteria in the following order:
1. We require that ISM gas must be cooler than or be denser than hydrogen nuclei per cubic centimeter.
2. We require that the ISM must be rotationally supported (Eqns 3 and 4) or be dense () and within .
3. We remove ISM gas that is beyond half the halo virial radius (this step is only applied for central galaxies).
4. We remove remaining ISM gas that is beyond (non-iteratively). The radius enclosing of the ISM mass, , is calculated after the previous selection criteria have already been applied.
5. If the galaxy has a remaining ISM gas ratio of , we apply a passive-galaxy correction and remove ISM gas beyond .
A more detailed justification and discussion of these ISM definitions is given in Appendix C, along with a number of examples. Importantly, we find that the resulting ISM mass can be significantly different compared to if the ISM is instead defined simply as star-forming gas (see also Section 4.1). We also show that the ISM definition used here is similar at low redshifts to selecting mass in neutral hydrogen within a aperture, as used in Lagos et al. (2015). However, at higher redshifts (), increasingly large fractions of hydrogen in the radially infalling CGM are in a neutral phase. As such, the ISM definition used here starts to diverge from taking neutral gas within an aperture.
Appendix C also demonstrates the important point that a simple decomposition of baryons within haloes into distinct components does not always provide a good description of the complex nature of what truly occurs in simulations and in reality. The separation between the CGM and a rotation/pressure/dispersion supported ISM can be a well posed question for many galaxies, particularly at low redshift. For some galaxies however, the distinction becomes much less clear. Appendix C shows an example of a massive, high-redshift star forming galaxy with dense, star-forming gas distributed over a significant fraction of the halo virial radius. For such galaxies, the assumption that there is a centrally concentrated ISM which is in dynamical equilibrium clearly starts to break down.
With gas particles belong to the ISM selected, we assign the remaining (non-ISM) gas particles associated with a given subhalo to a diffuse halo-gas component. Finally, we also define a reservoir of gas that has been ejected beyond the virial radius by feedback. Note that gas that has been previously ejected but has since been reincorporated back inside the halo virial radius is not counted in this ejected gas reservoir. In galform, this reservoir is explicitly tracked. For eagle, we estimate the mass in this reservoir, , by taking the difference
Mejected=fBMH−MB (5)
where is the cosmic baryon fraction and is the total baryonic mass (including satellite subhaloes) within . Under this approximation, gas that was prevented from ever being accreted onto the halo in eagle is also included in the ejected gas reservoir (assuming the halo would otherwise accrete gas at the cosmological baryon fraction). In future work, we plan to more accurately compute the properties of this reservoir by tracking the past/future trajectories of gas particles accreted onto haloes.
### 2.7 Model calibration
Both galform and the eagle simulations contain a number of model parameters that can be adjusted to reproduce observational constraints. The resulting calibration process is typically performed by hand without any statistically rigorous exploration of the model parameter space, although this machinery has been developed and applied for semi-analytic models in recent years (e.g. Henriques et al., 2013; Benson, 2014; Lu et al., 2014; Rodrigues et al., 2017). This calibration approach is necessary primarily because it is not possible at present to simulate the resolved physics of star formation and feedback or to model the resulting effects from first principles. The observational constraints on model parameters range from direct constraints like the observed Kennicutt-Schmidt star formation law to indirect constraints such as the luminosity function of galaxies. Broadly speaking, parameters relating to star formation can be calibrated directly Schaye (2004); Schaye & Dalla Vecchia (2008); Lagos et al. (2011) while parameters relating to stellar feedback, SMBH accretion and AGN feedback are calibrated using indirect constraints. For the scientific work performed using galform and eagle, the underlying philosophy regarding calibration is that a minimal set of observations are used to adequately constrain the model parameter spaces, and that following calibration the models can be compared to other observables with some degree of predictive power Cole et al. (2000); Schaye et al. (2015).
Calibration of the model parameters in the eagle simulations is described in Schaye et al. (2015) and Crain et al. (2015). Calibration of the model parameters for the galform model used here follows Guo et al. (2016). Briefly, galform was calibrated to match the observed local luminosity functions in the and bands from Norberg et al. (2002) and Driver et al. (2012), as well as the SMBH-bulge mass relation from Häring & Rix (2004). eagle was calibrated to match the local stellar mass functions inferred from observations by Li & White (2009) and Baldry et al. (2012), the local stellar mass-size relation from Shen et al. (2003) and Baldry et al. (2012) and the SMBH mass versus total stellar mass relation from McConnell & Ma (2013)777eagle was compared to the latter data set using the AGN feedback parameter value from the owls model. The fit was deemed satisfactory, so no calibration was necessary. However, Booth & Schaye (2009, 2010) have shown that the black hole masses are determined by the subgrid AGN feedback efficiency..
While eagle and galform were calibrated using different observational datasets, Trayford et al. (2015) have demonstrated that eagle agrees well with observed and -band luminosity functions. The galform model used here predicts a similar stellar mass function to eagle at , albeit with a slight deficit of galaxies around the knee. Galaxy sizes at in galform do not agree with observations (see Appendix D). Black hole masses are comparable between eagle and galform at (see Appendix E).
## 3 Comparing stellar masses
A zeroth order comparison between the reference galform model used here and the reference eagle simulation is shown in Fig. 1, which shows the distribution of stellar mass at a given halo mass for central galaxies. By construction, the two models are in approximate agreement at where both models have been calibrated to similar constraints. The models then diverge at , such that galform has a steeper relation between stellar mass and halo mass.
Compared to eagle, galform has a significantly larger scatter in stellar mass at a halo mass . For eagle, Matthee et al. (2017) showed that of the scatter is connected to halo concentrations (or equivalently halo assembly times), with the remaining scatter being uncorrelated to any of the (dark matter only) halo properties they explored. In appendix B of Mitchell et al. (2016), we showed that the enhanced scatter at this halo mass range in galform is caused by the differing efficiency of SNe feedback (and hence the efficiency of stellar mass assembly in a holistic sense) between quiescent star formation in discs and triggered nuclear star formation associated with galaxy mergers and disc instabilities (see also the discussion in Guo et al., 2016). This differring efficiency is caused by scaling the efficiency of SNe feedback with disc circular velocity for disc star formation and scaling with bulge circular velocity for SNe feedback associated with triggered nuclear star formation. This in turn leads to a bimodal distribution in stellar mass at a given halo mass in the mass range for which the contributions to the total stellar mass from stars formed in discs and stars formed in nuclear bursts are comparable 888The mass range for which this occurs is set by AGN feedback in the sense that it acts to prevent late-time disc star formation from overwhelming the contribution from nuclear star formation which tends to dominate in massive galaxies at high redshift Lacey et al. (2016)..
Fig. 2 compares the stellar masses of matched individual galaxies from the two reference models. At , the medians of the distribution are approximately consistent with a unity ratio between the two models. This primarily reflects the fact that both models are calibrated by luminosity/stellar mass function data from the local Universe. Of more interest is the scatter in the distribution. Taking the average over stellar mass bins for , the mean scatter in the logarithmic distribution is . As such, galform typically yields the same stellar masses as eagle galaxies to within a factor at . This significantly exceeds the scatter in halo mass between matched haloes from the eagle hydrodynamical and dark-matter-only simulations ( at , see Appendix B). It is notable that the level of scatter is elevated for . This reflects the increased scatter in stellar mass at a given halo mass seen in Fig. 1 for this mass range.
At and , the medians of the distribution are no longer consistent with a ratio of unity, reflecting the different shapes of the distributions predicted by the two models at these redshifts (see Fig. 1). The scatter around the median drops with increasing redshift, indicating that the stellar masses of individual galaxies gradually diverge between the models as the galaxies evolve. Decomposing the scatter between central and satellite galaxies shows that satellites are equivalent to central galaxies in the level of agreement between the models. We explore the underlying reasons for the overall level of scatter seen in Fig. 2 in Section 8.1, where we show that the implementation of gas infall and AGN feedback in galform leads to artificially oscillating baryonic assembly histories for individual galaxies.
## 4 Star formation and the ISM
### 4.1 Star formation threshold
In eagle, a local metallicity-dependent density threshold is used to decide which gas particles are star-forming Schaye et al. (2015). In galform, star formation occurs in molecular gas, and the formation of a molecular phase is explicitly computed following empirical correlations inferred from observations Lagos et al. (2011). Further details of the modelling are presented in Appendix A.1.
Fig. 3 shows the mass fraction of ISM gas that is actively forming stars. In both reference models, this distribution evolves with redshift, reflecting the evolution of galaxy disc surface density profiles, the incidence of disc instabilities and galaxy mergers in galform, and the evolution in local ISM density and metallicity in eagle. At , the two models display qualitatively different behaviour. While both models predict high star-forming ISM fractions for massive galaxies (where overall gas fractions are very low), galform predicts significantly lower star-forming fractions in low-mass galaxies.
In the reference eagle simulation, the gas-phase mass-metallicity relation is shallow at , in tension with observational constraints which imply a lower gas metallicity in low-mass galaxies (See figure 13 in Schaye et al., 2015). As such, we expect that were eagle to predict more realistic metallicities for low-mass galaxies, the star-forming ISM fraction would be correspondingly lower. We can test this hypothesis by considering the higher resolution recalibrated eagle model (magenta lines in Fig. 3). In this model, the mass-metallicity relation is steeper, in better agreement with observations Schaye et al. (2015). Correspondingly, Fig. 3 shows that the star-forming ISM fraction is smaller for this model in low-mass galaxies.
While galform does not reproduce the observed metallicities either (see Guo et al., 2016), this is irrelevant for star formation because the star formation threshold has no metallicity dependence in this model. Unlike eagle, however, galform predicts galaxy sizes for low-mass late-type (disc) galaxies that are too large compared to observations in galform (the galaxy size distributions as a function of stellar mass are shown in Appendix D). As such, gas surface densities (for a given ISM mass) will be unrealistically low in low-mass galaxies, potentially leading to unrealistically low molecular gas fractions. Observational data from the Herschel Reference Survey (HRS, Boselli et al., 2014), the APEX low-redshift legacy survey for molecular gas (ALLSMOG, Bothwell et al., 2014), the Galex Aricebo SDSS survey (GASS, Catinella et al., 2010) and the CO legacy database for GASS (COLD GASS, Saintonge et al., 2011) indicate that is indeed the case, with a significant number of detected galaxies with higher molecular-to-atomic gas fractions than is predicted by galform for stellar masses lower than . The data also indicates that there are massive galaxies with lower molecular-to-total ISM gas fractions than those predicted by either galform or eagle (taking molecular-to-total ratio as a proxy for the star-forming to total ISM ratios shown for eagle).
At higher redshifts, Fig. 3 shows that the fraction of mass in the star-forming ISM in the two reference models comes into slightly better agreement. Qualitative differences remain however. At , galform exhibits a steeper trend with stellar mass. At , the star-forming ISM fraction is systematically higher by in galform at all stellar masses.
### 4.2 Star formation law
In eagle, star-forming gas is turned into stars following a Kennicutt-Schmidt star formation law, reformulated as a pressure law Schaye & Dalla Vecchia (2008); Schaye et al. (2015). In galform, the star formation rate in galaxy discs is linear in the molecular gas mass in the disc, with a constant, empirically constrained conversion efficiency Lagos et al. (2011). Further details of this modelling are presented in Appendix A.2.
Fig. 4 shows the SFR per unit star-forming ISM mass for actively star-forming galaxies (which we define as specific SFR for respectively 999This is guided by the distributions of specific SFR for galform and eagle shown in figure 1 of Mitchell et al. (2014) and figure 7 from Guo et al. (2016)). At , galform has a slightly higher star formation efficiency but agrees with eagle (for both the reference and recalibrated simulations) to within , except for the most massive galaxies in the simulation. At higher redshifts, the agreement worsens as eagle displays a significant positive trend of efficiency with stellar mass. Star formation efficiency also increases with redshift at fixed stellar mass in eagle (albeit more strongly for more massive galaxies). This reflects the changing ISM conditions for the star-forming gas in eagle with mass/redshift. At high-redshift, the typical densities of star-forming ISM gas increase in the simulation, accordingly increasing the gas pressure and hence the efficiency of star formation, following Eqn 8 (see figure 12 from Lagos et al., 2015). Despite assuming that star formation in galaxy discs has a fixed efficiency for star-forming gas, this effect is somewhat accounted for in galform by the inclusion of explicit nuclear bursts of star formation. This elevates the net star formation efficiency of a subset of the massive galaxies at high-redshift, albeit with a very skewed distribution compared to eagle.
### 4.3 Gas fractions
The result of the differing star formation thresholds and efficiencies (as well as the effect of accretion and outflow rates, which we do not measure here) are reflected in the galaxy gas-to-stellar mass ratios (), shown in Fig. 5. For galaxies with at , the two models are consistent with each other and with the GASS and COLD GASS surveys. At lower masses, the gas ratios in the reference eagle simulation drop until there are typically only a handful of gas particles in the ISM of a galaxy of . In stark contrast, galform predicts that low-mass galaxies have much higher gas ratios, such that the gas ratio decreases monotonically with increasing stellar mass. The median gas ratios in galform are consistent with the stacked atomic gas ratios from Brown et al. (2015) (note that in this regime, the ISM is almost entirely atomic in galform, see Fig. 3).
At higher redshifts, the two models are in better agreement, both showing the expected trend of increasing gas ratio with redshift (although there is little difference and ). Here, it appears that a higher star formation efficiency in eagle (Fig. 4) is compensated for by a lower fraction of the ISM which is forming stars (Fig. 3). We have confirmed that eagle and galform are indeed very similar at these redshifts in the star formation efficiency per unit total ISM mass (as opposed to the efficiency per unit star-forming ISM mass shown in Fig. 4).
Fig. 5 also shows the gas ratios from the higher-resolution, recalibrated eagle model. This indicates that the non-monotonic behaviour seen for the reference eagle model at is likely a resolution effect. The recalibrated model shows similar non-monotonic behaviour but at a lower stellar mass. The recalibrated model is in better agreement with galform and the observational data as a result. Grey solid and dashed lines show the point below which the number of gas particles in the ISM drops below for the reference and recalibrated eagle models respectively. This indicates that resolution is indeed likely to be an issue for galaxies with in the reference eagle model at (see the discussion in Crain et al., 2017) and may also affect the higher-resolution recalibrated model for .
To summarise the differences between galform and eagle seen in this section (Fig. 3, 4 and 5), we have demonstrated that gas-to-stellar mass ratios are much higher in galform than in eagle for low-mass galaxies at (with an apparent connection to numerical resolution in eagle), but that the models are in good agreement at higher redshifts. For low-mass galaxies at low-redshift, this difference seems to be connected to the difference in the fraction of the ISM that is star-forming (with galform having much lower star-forming ISM fractions). A simple explanation for this difference in star-forming ISM fraction is that because low-mass galaxy sizes are significantly larger in galform than in eagle at (see Appendix D), leading to lower gas surface densities at at a given gas fraction and stellar mass in galform. At higher redshifts, the star-forming ISM fractions are less discrepant but eagle has a significantly higher median star-formation efficiency than galform for the star-forming ISM in massive galaxies. This translates to very similar gas-to-stellar mass ratios with respect to galform, partly because the differences in star-forming ISM fraction and star formation efficiency compensate for each other, and possibly because the burst mode of star-formation in galform does indeed compensate for the increased median efficiency in eagle.
## 5 Angular momentum
An important difference between eagle and galform concerns the angular momentum of newly-formed stars (for full details of angular momentum modelling in galform, see Appendix A.3). In eagle, stellar particles self-consistently inherit the angular momentum of the gas from which they formed. Historically, this was also the case in older galform models when there was no partition between atomic and molecular gas (and hence no star formation threshold) Cole et al. (2000). Specifically, it was assumed that the gas and stars within the disc shared a common radial scale length and correspondingly had identical specific-angular momentum. While a simplifying assumption, this did ensure that newly-formed stars had consistent specific angular momentum with the star-forming gas. After the introduction of a radius-dependent partition between atomic and molecular hydrogen by Lagos et al. (2011) this assumption was retained. As such, newly-formed stars in our reference galform model have the same specific angular momentum as the total ISM gas disc, rather than just that of the star-forming ISM (molecular hydrogen). Given that the star-forming, molecular ISM is more centrally concentrated than the atomic ISM under the Lagos et al. (2011) scheme, this means that newly-formed stars have inconsistently high specific angular momentum in galform.
This is explicitly demonstrated in Fig. 6, where we show the ratio of (magnitude of the) specific angular momentum in the star-forming ISM to specific angular momentum in the total ISM. Compared to the unity ratio implicitly assumed in galform, eagle predicts that stars form preferentially out of ISM with lower specific angular momentum. In other words, star formation is centrally concentrated in eagle. This can be understood as a consequence of the (metallicity-dependent) density threshold implemented in eagle. (ISM gas is more likely to pass the threshold at the galaxy centre where densities are highest). There is significant scatter in the distribution, presumably reflecting the vector nature of angular momentum (which is ignored in galform) being affected by complexity of merger events and accretion flows changing in orientation over time Lagos et al. (2017a, b).
The green points in Fig. 6 show the ratio of specific angular momenta that galform would predict were it to self-consistently compute the angular momentum content of star-forming gas from the radial profile of molecular hydrogen101010We emphasise that we have not internally modified the galform model to perform this calculation, and as such the distribution (and many other predictions of the model) would likely look different if we were to do so.. Given that it is otherwise not defined in the model, we have assumed here that star-forming nuclear gas present in galaxy bulges has zero (net) angular momentum, although in practice this choice has negligible effect on the distributions shown. From Fig. 6, it is apparent that were galform to self-consistently compute the angular momentum of the star-forming ISM when computing the angular momentum of newly-formed stars, then eagle and galform would come into better agreement on average.
The impact of the assumptions regarding specific angular momentum of newly-formed stars is made clear in Fig. 7. Focussing first on eagle, the red and magenta lines show respectively the specific angular momenta of stars and the total ISM in eagle. These distributions are separated by almost an order of magnitude in specific angular momentum at a given stellar mass. The actively star-forming ISM in eagle (green lines) has intermediate specific angular momentum between the total ISM and stars.
At intermediate stellar masses () at , the ISM in eagle has the same specific angular momentum as the hosting dark matter haloes (black lines). Interestingly, the specific angular momentum of the ISM in the eagle reference model dips below the halo specific angular momentum for low-mass galaxies at . This is not the case at and the feature disappears in the recalibrated eagle model (dotted magenta lines). This difference is presumably related to the much higher gas ratios predicted by this variant model for low-mass galaxies at (and hence the convergence issues seen in Fig. 5).
To make a comparison between galform and eagle in Fig. 7 for total specific stellar angular momentum (blue and red lines), we must account for the problem that galform does not model the angular momenta of galaxy bulges/spheroids. This is particularly an issue for massive galaxies (which have high bulge-to-total ratios). For galform, we therefore choose to show disc specific angular momentum rather than total specific angular momentum and to only show the distribution for stellar mass bins where at least of the galaxies are disc dominated ().
In contrast to eagle, galform (blue lines) assumes that the ISM and stars have the same specific angular momenta in galaxy discs. Fig. 7 shows that the median stellar specific angular momentum of disc-dominated galaxies in galform can be over an order of magnitude larger than in eagle at a given stellar mass. In galform, disc-dominated galaxies have almost the same stellar (and ISM) specific angular momentum as the hosting dark matter haloes (black lines). The corresponding galaxies in eagle have much lower specific stellar angular momentum than their host haloes. On the other hand, the specific angular momentum of the total ISM in eagle is in good agreement with the specific angular momentum of galform discs (and so in agreement with the specific angular momentum of the ISM in galform discs).
At , we also show observational data for specific stellar angular momentum of gas-rich spiral galaxies from Romanowsky & Fall (2012) and Obreschkow & Glazebrook (2014). As discussed by Lagos et al. (2017b), when selecting eagle galaxies with high gas fractions, eagle agrees with the observations quite well, because gas-rich galaxies have higher specific stellar angular momentum at a given stellar mass. In contrast, galform predicts a stellar specific angular momentum which (with some extrapolation of the observations) is too high for low-mass galaxies. This picture is consistent with the galform overprediction of low-mass galaxy sizes in the local Universe (see Appendix D). Put together, this serves to underline that self-consistently computing the angular momentum of newly-formed stars from centrally-concentrated (low-angular momentum) star-forming gas in the ISM is likely a needed ingredient for future semi-analytic models.
Finally, we note here that the specific angular momentum of the ISM and stars in eagle is (to some extent) sensitive to the assumed model parameters. For example, increasing the normalisation of the star formation law in eagle (see Appendix A.2) increases slightly the ISM specific angular momentum (not shown here), as well as lowering the ISM-to-stars mass fraction Crain et al. (2017). A simple interpretation of this trend is that increasing the assumed star formation efficiency depletes more of the star-forming ISM (by star formation and feedback driven outflows), increasing the relative importance of the non-star-forming ISM, which is less centrally concentrated and so has higher specific angular momentum.
## 6 Feedback
### 6.1 Stellar feedback, AGN feedback and gas return timescales
In eagle, stellar feedback is implemented locally by injecting thermal energy into gas particles which neighbour young star particles Schaye et al. (2015). Gas particles are heated by a fixed temperature difference, , which acts to suppress artificial radiative losses Dalla Vecchia & Schaye (2012). The average injected per supernova explosion is scaled as a function of gas density and metallicity, ranging between and times the canonical energy of (with a mean value very close to the canonical value) Crain et al. (2015). In galform, stellar feedback is implemented globally across a given galaxy by ejecting gas from galaxies (and haloes) with an efficiency that scales with galaxy circular velocity. More details of this modelling, as well as a discussion of the energetics of stellar feedback in galform, are given in Appendix A.4
In eagle, AGN feedback is implemented similarly to stellar feedback, but with a heating temperature of (an order of magnitude higher than for stellar feedback) and with a fixed average efficiency (relative to the SMBH accretion rate). In principle, this model of AGN feedback can both heat gas in the ISM and in the halo, and remove gas entirely from haloes. In galform, AGN feedback acts only to prevent gas infall from the halo onto galaxy discs, and does not eject gas from haloes. AGN feedback is activated in galform if the SMBH injects sufficient energy to balance radiative cooling in the halo, and if the halo is considered to be in a quasi-hydrostatic state. Further details of this modelling, and of the modelling of SMBH seeding and growth, are presented in Appendix A.5. We compare the galform and eagle distributions of black hole mass as a function of stellar mass in Appendix E, where we show that galform does not predict the steep dependence on stellar mass predicted by eagle at .
Arguably, the most important uncertainty in semi-analytic galaxy formation models is the fate of outflowing gas that is ejected from galaxies by feedback (e.g. Henriques et al., 2013; Mitchell et al., 2014; White et al., 2015; Hirschmann et al., 2016). In galform, this ejected gas is placed into a distinct reservoir and is assumed to return to the diffuse gas halo over a halo dynamical time (see Appendix A.6). While the spatial location of this ejected gas is not formally defined, we choose to the interpretation that this gas is outside the halo virial radius (see Appendix A.6). Given that the halo dynamical time is always approximately equal to of the age of the Universe at a given epoch (independent of halo mass), ejected gas is rapidly reincorporated back into the diffuse gas halo in galform. This rapid gas cycling (infall timescales from the diffuse halo onto the disc are also typically of order a halo dynamical time) forces the model to employ very large mass-loading factors in order to explain the low observed efficiency of cosmic star formation. This is common to some other semi-analytic models (e.g. Springel et al., 2001), although see also Somerville et al. (2008b); Hirschmann et al. (2016).
Importantly, it is assumed in galform that AGN feedback does not eject gas from galaxies (or from haloes), and instead acts only to suppress radiative cooling from the diffuse gas halo. Given the fairly short reincorporation timescale assumed in galform, this means that the baryon cycle (infall, ejection, reincorporation) effectively ceases and the halo baryon fraction rapidly reaches the cosmic mean once AGN feedback becomes active in a given halo (although see Monaco et al., 2007; Bower et al., 2008; Somerville et al., 2008b; Bower et al., 2012, for alternative schemes where AGN can eject gas from haloes).
In hydrodynamical simulations, gas flows are predicted locally by self-consistently following gravity and hydrodynamics. As such, they do not make any explicit assumptions for gas return timescales. Implicitly however, simulations set the return time through the details of the subgrid models for feedback. In the case of eagle, outflowing particle trajectories will be sensitive to the assumed feedback heating temperature. In hydrodynamical simulations without an explicitly assumed wind speed (and in reality), gas is presumably ejected from galaxies with a broad distribution of energies (e.g. Christensen et al., 2016). Correspondingly, it is to be expected there will be a distribution of return timescales among gas particles associated with a wind, ranging from short timescales (less than a Hubble time, as in galform) to timescales much longer than a Hubble time, such that the gas effectively never returns Oppenheimer et al. (2010); Christensen et al. (2016); Anglés-Alcázar et al. (2017); Crain et al. (2017). It is not clear therefore whether the total return rate should be linear in the mass of the ejected gas reservoir (as assumed in galform, Eqn 12), or whether the return rate will tightly correlate with halo properties, as is assumed in semi-analytic models.
### 6.2 Halo baryon fractions
While we do not measure mass-loading factors or gas return timescales from eagle for this study (we defer this to future work), an indirect measure of the efficiency of these processes is given simply by the mass fraction of baryons within the virial radii of dark matter haloes. Fig. 8 shows the baryon fractions as a function of halo mass for a range of redshifts. For all models, the baryon fractions rise from low values in small haloes to high values in massive haloes, implying that to first order the baryon cycles in the two models are similar. However, in detail there are various qualitative differences.
At a characteristic halo mass of , baryon fractions in galform rapidly approach the cosmic mean. This transition is significantly more gradual in the reference eagle model, such that the baryon fraction approaches the cosmic mean only in galaxy clusters (). For the models without AGN feedback (green and magenta points), there is still a difference between the two models for , indicating that SNe feedback is more effective in eagle than in galform in massive haloes (either because more gas is ejected or because gas takes longer to return). When AGN feedback is included, eagle and galform show divergent behaviour. AGN feedback acts to reduce the baryon fractions (either by direct mass ejection or by preventing primordial accretion or gas return) in massive haloes in eagle. In galform, AGN feedback does not eject gas and instead suppresses gas cooling, which in turn acts to suppress future SNe-driven outflows, resulting in higher baryon fractions than the no-AGN case.
In practice, suppressing cooling and ejecting gas from haloes will both act to suppress star formation. Accordingly, the differing baryon fractions in massive haloes between eagle and galform will not necessarily result in differing stellar mass assembly histories. The ejection/non-ejection of baryons by AGN feedback is relevant to the predicted X-ray properties of massive haloes however, as explored in Bower et al. (2008).
In lower-mass haloes, below the regime where AGN feedback plays a role, the two models come into better agreement but still show a different evolution with redshift. At , SNe appear to be slightly more efficient in removing baryons in galform than in eagle for the lowest-mass haloes shown. At , the models are very comparable and by , eagle has a lower baryon fraction at a given halo mass in low-mass haloes.
At , Fig. 8 also shows the baryon inventory compilation from McGaugh et al. (2010), corrected to be consistent with halo masses defined relative to times the critical density of the Universe. Both galform and eagle agree with the basic qualitative trend of increasing baryon fraction with halo mass (see Haider et al., 2016, to see that this is not trivially the case in simulations). For low-mass haloes (), the baryonic mass estimates presented in McGaugh et al. (2010) include contributions only from stars and the ISM, neglecting any CGM contribution (and so should be considered lower limits). In both galform and eagle, a significant part of the baryonic mass always belongs to the CGM component, even in lower mass halos (see Section 7.1).
For the more massive haloes shown in the McGaugh et al. (2010) compilation, baryonic masses are instead inferred from X-ray measurements of galaxy groups and clusters. Here, a detailed self-consistent comparison with X-ray observations that takes into account observational biases and systematics has not been performed (and is beyond the scope of this paper). However, more detailed comparisons of eagle and galform to group and cluster X-ray measurements have been presented in Bower et al. (2008), Schaye et al. (2015) and Schaller et al. (2015a). For galform, Bower et al. (2008) found that the fiducial galform model (similar to the one shown here) significantly overpredicts the X-ray emission from galaxy groups, and that this could be resolved in a variant model by expelling hot gas from haloes with AGN feedback. This simple picture is consistent with how eagle behaves. Schaye et al. (2015) and Schaller et al. (2015a) have compared eagle to the X-ray content of galaxy groups and clusters, using the methodology of Le Brun et al. (2014) to perform a self-consistent comparison. They find that the reference eagle model shown here overpredicts the hot gas content of galaxy groups by about , and that this can be resolved by increasing the heating temperature for AGN feedback. The simple comparison shown here in Fig. 8 is consistent with this picture.
To summarise, we have shown in this section that the different implementations of SNe and AGN feedback in galform and eagle lead to similar baryon fractions in lower-mass haloes, but very different baryon fractions in group-scale haloes. This is not necessarily important for stellar mass assembly but will have a strong impact on the X-ray properties of galaxy groups Bower et al. (2008).
## 7 Galaxy evolution and the baryon cycle
Here, we explore how the physical processes discussed in previous sections (star formation, feedback and gas cycling) shape galaxy evolution by presenting the baryonic assembly histories of galaxies in eagle and galform.
The time evolution of the ejected gas reservoir discussed in Section 6.1 is shown explicitly in Fig. 9, in this case tracing the main progenitors of central galaxies with at . Green lines show the median ejected gas mass for the two models. Matched at this stellar mass at , central galaxies in eagle (dashed lines) on average have a higher mass fraction in the ejected reservoir at all redshifts than in galform, and the mass in the ejected gas reservoir increases monotonically with time. In contrast, galform (solid lines) predicts that the median mass in the ejected reservoir rises with time up until , before steadily declining until (note the ejected reservoir does not include ejected gas which has been subsequently reincorporated). This is a result of the short return timescale assumed in galform, whereby the ejected gas reservoir closely traces the evolution of mass in the ISM (solid blue line). In eagle, the evolution of the ISM mass (dashed blue line) is fairly steady with time, rising to a peak and then modestly declining until . The contrast between the evolution of the ISM (slow decline) and ejected gas reservoir (gradual rise) for indicates that the net return timescale is likely longer in eagle than is assumed in galform. This interpretation is supported by Crain et al. (2017), who saw no evidence for gas particles directly heated by feedback returning to galaxies at later times in eagle (after inspecting past phase diagrams of gas particles selected at a given epoch).
The stellar mass assembly histories (black lines) are in very good agreement between the two models in this mass bin. Such a level of agreement does not extend to any of the gas reservoirs however. This serves to underline that the stellar assembly histories of galaxies are not enough to constrain the feedback processes in galaxy formation models. In particular, the median mass in the diffuse gas halo is a factor larger on average in galform compared to eagle. Importantly, the full distribution of masses in the diffuse gas halo (not shown) is significantly wider in galform. We return to this issue in Section 8 when discussing the infall/cooling model used in galform.
The upper-right panel of Fig. 9 shows the star formation (cyan) and stellar mass assembly histories (time derivative of the stellar mass, shown in black) of the same selection of galaxies. The peak of star formation is slightly later in eagle with respect to galform, which leads to better agreement with the observed decline in specific-star formation rates Mitchell et al. (2014); Furlong et al. (2015). In Mitchell et al. (2014), it was demonstrated that such a delay in the star formation peak is only possible by introducing a very strong redshift evolution in the gas-return timescale or mass-loading factor, significantly beyond what is possible in the standard galform parameter space. Interestingly, there is also a small contribution in eagle from star formation that takes place outside the ISM (yellow line), a possibility that is not considered in galform. Visually (see Fig. 16), we find that radially-infalling star-forming gas can start to fragment and form stars before settling into rotational equilibrium closer to the halo centre.
The lower-left panel of Fig. 9 shows the evolution in specific angular momentum of different baryonic components. Specific angular momentum increases monotonically with cosmic time for all components in both models, in accordance with tidal torque theory Catelan & Theuns (1996), (see Lagos et al., 2017b, for an analysis of in the context of eagle). Interestingly, there is a much greater level of segregation in specific angular momentum between different baryonic components in eagle compared to galform. In galform, angular momentum is conserved for gas infalling onto galaxies. Furthermore, it is assumed that outflowing gas ejected by SNe feedback has the same specific angular momentum as the overall ISM. Consequently, the ISM (solid-blue line) has very similar specific angular momentum to the diffuse gas halo (solid-red line) as it evolves. The stellar specific angular momentum (solid black line) is not fully defined in galform (see Section 5) because the angular momentum of galaxy bulges is not tracked. We choose to set the bulge angular momentum to zero. The resulting stellar specific angular momentum should therefore be regarded as a lower limit (as bulges/spheroids do rotate). Even as a lower limit however, the stellar specific angular momentum in galform is still significantly higher than in eagle.
In eagle, the ISM and the stellar components abruptly decouple in specific angular momentum at redshift , after which the ISM has a factor larger specific angular momentum. The transition redshift marks the point at which the star-forming ISM (dashed cyan line) starts to decouple from the total ISM. It also marks the point at which the previously formed stellar mass is significant enough that the past average specific angular momentum (represented by the stars) drops below the ISM value.
Another interesting feature of the lower-left panel of Fig. 9 is that the halo gas specific angular momentum (dashed red line) in eagle is positively offset with respect to the ISM (by a factor at ). This behaviour is not predicted to the same extent by galform, and not at all for . This closer coevolution of the ISM and halo gas in galform stems first from the assumption that angular momentum of gas is conserved as it condenses from the halo component onto a galaxy disc. Secondly, infall rates onto galaxy discs in galform are high enough in the non-quasi-hydrostatic regime that all of the halo gas at a given epoch will be accreted onto the disc. The segregation of specific angular momentum between these components in eagle is therefore suggestive that angular momentum is either not exactly conserved for infalling material (see Stevens et al., 2017, for a similar conclusion), or that infalling material is preferentially low-angular momentum compared to the overall halo gas reservoir. It also seems likely that infall rates may be lower than in galform, such that more radially distant, high angular momentum gas is never accreted onto galaxies.
The lower-right panel of Fig. 9 shows the evolution in metallicity for different baryonic components. galform predicts that the median metallicity in each component closely traces each of the others with the median ISM metallicity positively offset by from stellar metallicity at . In contrast, eagle predicts that the ISM and stellar components have almost identical metal content at and co-evolve very closely (the upturn at high-redshift in the ISM metallicity is caused by galaxies dropping out of the sample as they can no longer be identified by the halo finder, with the remaining galaxies probably being unusually metal enriched at these redshifts). eagle also predicts that the diffuse halo gas component is negatively offset in metallicity by a factor that grows fractionally with time, reaching a factor by .
In galform, metals are exchanged between different components such that they linearly trace the total baryonic mass exchange. As such, it is assumed that the metal-loading factor (ratio of metal ejection rate from the ISM to the rate with that ISM metals are locked into stars by star formation) is the same as the mass-loading factor. This need not be the case (e.g. Creasey et al., 2015; Lagos et al., 2013). Furthermore, galform assumes that newly-formed stars form with the metallicity of the total ISM component, neglecting any possible radial gradients that could lead to differences between the average metallicity of star-forming gas and total ISM. Combined with the difference in mass exchange rates implied by the contrasting predictions shown in the upper-left panel of Fig. 9, it is therefore somewhat challenging to interpret the differences in metal evolution between the two models. That the diffuse gas halo metallicity is significantly lower in eagle than the ISM and stellar components does suggest that considerably less baryon cycling is taking place compared to galform, suggesting a longer gas return timescale.
### 7.1 Mass dependence
Fig. 10 shows the same information as the upper-left panel of Fig. 9 but plotted for a range of stellar mass bins. Overall, the two models come into better agreement for lower mass galaxies () but increasingly disagree for more massive galaxies (). For the lower mass galaxies, the ejected gas reservoir dominates the mass budget in both models. The stellar mass in these galaxies forms later in galform and galform contains systematically higher ISM content at all redshifts, presumably resulting in more prolonged star formation histories.
For more massive galaxies (lower panels), the two models quickly start to diverge at high-redshift in the mass of the ejected gas reservoir. In galform, the ejected gas reservoir mass peaks at before steadily declining down to . In eagle, the ejected gas reservoir mass does not peak and continues to rise until low-redshift and is always comparable to, or greater in mass than the diffuse halo gas component. In galform, the diffuse halo component completely dominates over the ejected gas reservoir by . As shown in Fig. 8, this primarily reflects the different implementations of AGN feedback in the two models.
It is also notable that the total baryonic mass is significantly larger in eagle than in galform for the higher mass stellar mass bins shown. In this stellar mass range, the - relation is shallow, such that a small difference in stellar mass leads to a large difference in halo mass (see Fig. 1). Given that we match galaxy samples here at a fixed stellar mass, the difference in total baryonic mass for the two models is therefore primarily driven by the difference in host halo mass at a fixed stellar mass. The difference in halo mass definition also contributes (see Appendix B).
In Mitchell et al. (2014) and Rodrigues et al. (2017) it was demonstrated that there is a degeneracy between gas return timescale and mass-loading factor in galform if the model is calibrated to reproduce a given stellar mass function (see also Somerville et al., 2008b). Intuitively, a higher mass-loading factor (more gas ejected from galaxies) can be compensated for by increasing the rate of gas return, ultimately leading to very similar stellar mass growth histories. This is explicitly demonstrated in Fig. 11, which shows three galform models. The first is the fiducial reference model for this study (solid lines) with the return time parameter, and the mass-loading factor normalization parameter set to . The other two models shown were run with (strong SNe feedback and rapid gas return, dotted lines) and (weak SNe feedback and slow gas return, dash-dotted lines).
While the three galform models shown predict very similar stellar mass assembly histories, the median mass in the ejected gas reservoir ranges over almost an order of magnitude by . Notably, the model with slow gas return and smaller mass-loading factors is in closer agreement with eagle for the evolution in mass of the ejected gas reservoir, further increasing the evidence that return timescales are likely longer in eagle than are typically assumed in semi-analytic galaxy formation models. We note also that adopting a longer return timescale alleviates the tension in our reference model that likely unrealistic amounts of energy are implicitly injected into SNe-driven winds, as discussed in Section 6.1. Interestingly, Rodrigues et al. (2017) show that when higher-redshift stellar mass functions are included as constraints, values of are strongly disfavoured. Presumably, this tension could probably alleviated by using a scale dependent gas return timescale, as advocated by Henriques et al. (2013).
To summarise, we have seen several indications in this section that the level of baryon cycling after gas is blown out of galaxies by feedback is lower in eagle than in galform. This supports the preliminary conclusion by Crain et al. (2017) that there appears to be little ejected gas return in eagle. Evidence here supporting this viewpoint comes from the significantly increased segregation in metallicity between halo gas and ISM gas in eagle compared to galform and the lack of a close coevolution between the ISM and ejected gas reservoirs in eagle (compared to galform). Finally, increasing the ejected gas return timescale in galform improves the agreement with eagle in baryonic mass assembly histories.
## 8 Radiative cooling and infall
In galform, gas infalls from the diffuse gas halo onto the galaxy disc at a rate which is controlled by the minimum of two physical timescales. The first is the radiative cooling timescale of hot halo gas and the second is the gravitational freefall timescale Cole et al. (2000); Bower et al. (2006). The limiting timescale is then compared to the time elapsed since the previous halo mass-doubling event in order to decide how much gas cools at the current timestep. Further details of the gas infall model are presented in Appendix A.7.
In the regime where enough time has passed since the previous mass-doubling event for all of the halo gas to both cool and freefall onto the galaxy disc, the gas infall rate is simply set equal to the accretion rate of diffuse gas onto the halo. The opposite regime occurs when AGN feedback becomes active in a given halo, completely suppressing gas infall if the halo is considered to be quasi-hydrostatic. The hydrostatic equality criterion used also depends on the time which has elapsed since the previous halo mass-doubling event Bower et al. (2006).
We find that both of these regimes appear to play an active role in regulating the baryonic mass assembly in individual haloes in galform. This is shown directly for three example haloes in Fig. 12. Halo mass doubling events (vertical grey lines) are followed by a characteristic pattern. Gas rapidly infalls from the diffuse gas halo onto the disc until either another mass-doubling event occurs or cooling abruptly stops (AGN feedback becomes active). In some cases, there is sufficient time before one of these situations occur such that the diffuse gas halo is almost completely depleted (the diffuse gas halo mass never falls to exactly zero because of details in the numerical scheme, and instead fluctuates around a low floor value), corresponding to the regime where .
The result of this abrupt switching between regimes is the strongly oscillatory behaviour seen for the mass in different baryonic components shown in Fig. 12. Such behaviour is not seen for the corresponding individual haloes shown in eagle. We deliberately select three haloes to highlight the three possible cases for how well the stellar mass agrees between the two models at . In the top panel, galform underpredicts the stellar mass at by a factor compared to eagle. In this case, the halo does not undergo a halo mass doubling event below , such that AGN feedback is able to completely suppress gas infall after , preventing the star formation that the corresponding eagle galaxy undergoes during this period. In the bottom panel, galform overpredicts the stellar mass by the same factor compared to eagle, in this case because a halo mass-doubling event occurs late enough to allow significant late star formation but not so late that AGN feedback has a significant effect at late times. In the middle panel, the two galaxies agree in stellar mass simply because the final halo-mass doubling event in galform fortuitously occurs at the right time to allow the stellar mass to grow enough to match eagle at .
In summary, an artificial dependence on halo mass-doubling events and a bimodal model for AGN feedback (no effect or complete suppression of gas infall onto the disc), combined with strong SNe feedback and short return times leads to strongly oscillatory behaviour in galform. This behaviour is not seen in eagle, for which it is not necessary to make any of these assumptions, instead allowing the associated physical phenomena to emerge naturally (albeit with uncertain local subgrid modelling for cooling rates and energy injection from AGN). This strongly suggests that both the cooling and AGN feedback models in galform could be improved with the goal of eliminating this oscillatory behaviour (Hou et al., in prep, see also Benson & Bower, 2010). In future work, we will directly compare the inflow rates between the two models to address this topic more directly.
### 8.1 Variable infall rates and the scatter in stellar mass
In Fig. 13, we attempt to relate the oscillatory behaviour in infall rates seen in Fig. 12 to the scatter in stellar mass between matched galaxies across the entire population shown in Fig. 2. The top panel shows that there is a strong positive correlation between the fraction, , of simulation outputs in which a main progenitor in galform is considered a star-forming galaxy () and the residual in stellar mass with the corresponding matched eagle galaxy. Galaxies with high in galform form, on average, more stars than their eagle counterparts and galform galaxies with low form fewer stars to compensate. This means a significant amount of the scatter seen in Fig. 2 can be attributed to star formation histories being more variable in galform than in eagle.
The middle panel of Fig. 13 shows that there is not a corresponding correlation between the stellar mass residuals and the star-forming fraction, , measured in eagle. The lower panel of Fig. 13 reveals that this is because the only galaxies in eagle with low are massive () and so also have low in galform. Conversely, there are galaxies in galform with low at lower stellar masses (). It is the variability of gas infall rates in these lower mass galaxies (see Fig. 12) which are responsible for the strong correlation seen in the top panel of Fig. 13.
## 9 Applicability to other semi-analytic models
Here, we briefly consider how the results from this study relate to other semi-analytic galaxy formation models used in the community. Several of the aspects of the modelling which we consider are specific to the details of the modelling in galform, and cannot be easily generalised. We consider the oscillatory infall rates tied to halo mass-doubling events and AGN feedback (seen in Fig. 8.1) as belonging to this category. Similarly, the effect of AGN feedback enhancing the baryon fractions of galaxy groups in galform is specifically tied to the implementation of AGN feedback from Bower et al. (2006), and does not represent the behaviour of other semi-analytic models which include an explicit Quasar-mode of AGN feedback that can eject gas from haloes (e.g. Monaco et al., 2007; Somerville et al., 2008b; Croton et al., 2016).
Of more general interest is the importance of stars forming preferentially out of gas in the ISM with low specific angular momentum. Semi-analytic models currently assume that disc stars form with the same specific angular momentum of either the gas disc Guo et al. (2011), or the entire disc (e.g. Cole et al., 2000; Springel et al., 2005; Somerville et al., 2008a; Tecce et al., 2010; Croton et al., 2016). For the specific case of the L-Galaxies model, cross-matching Guo et al. (2016) with Appendix D shows that this model overpredicts the sizes of low-mass () galaxies by a factor in the local Universe, consistent with galform predictions.
Also of interest is the behaviour of gas flows both onto and out of galaxies. For example, a wide range of gas return timescales have been adopted in contemporary galaxy formation models. Recent models have assumed that gas returns over a halo dynamical time Lacey et al. (2016), over a Hubble time Somerville et al. (2008b); Hirschmann et al. (2016), or have adopted more complex parametrisations where the return time scales with halo mass or halo circular velocity Guo et al. (2011); Henriques et al. (2013); White et al. (2015); Croton et al. (2016); Hirschmann et al. (2016), or where the return timescale is explicitly connected to halo growth Bower et al. (2012). Mitchell et al. (2014) have also argued on empirical grounds that a yet more complex dependence of the gas return timescale (or SNe mass-loading factor) on halo mass, halo dynamical time, and redshift is required to reproduce the observed evolution of characteristic star formation rates at a given stellar mass. With this diversity of different models, the hints found here that eagle predicts a longer gas return time (compared to a halo dynamical time) are certainly of interest, albeit with the caveat that the return times are likely sensitive to the assumed heating temperature for SNe and AGN feedback.
With respect to previous comparison studies between semi-analytic models and hydrodynamical simulations, it is difficult to directly compare results, particularly because we have not measured gas cooling rates in eagle (but see Stevens et al., 2017). The factor discrepancy between star formation efficiency reported by Hirschmann et al. (2012) between a set of hydrodynamical simulations and a semi-analytic model is not seen in Fig. 4 (although a discrepancy in the medians is seen for massive galaxies at ). This suggests that the typical star-forming gas densities of the hydrodynamical simulations analysed in Hirschmann et al. (2012) are larger than those in eagle, resulting in a larger disagreement with the semi-analytic model. This may be related to the eagle feedback model being highly effective in reducing the typical gas densities of the ISM in which SNe explode Crain et al. (2015).
## 10 Summary
Guo et al. (2016) presented a comparison between the galform (and L-Galaxies) semi-analytic galaxy formation model and the state-of-the-art eagle hydrodynamical simulations. They demonstrated that while the two models are calibrated to produce similar stellar mass functions at , the two models predict markedly different metallicity and galaxy size distributions as a function of stellar mass (see also Appendix 18 for a comparison of galaxy sizes).
Here, we increase the depth of the comparison by matching individual galaxies and by isolating a number of important aspects in the physical modelling. In particular, we have carefully assigned baryonic particles in eagle to baryonic reservoirs that correspond to those included in galform. In future work, we plan to use the framework introduced here to measure the various mass, metal and angular momentum exchanges between these reservoirs in eagle, enabling a direct comparison to the assumptions made in semi-analytic models pertaining to mass inflows, outflows and baryon cycling. Even without these measurements however, a number of interesting differences between the two modelling approaches are readily apparent at the level of detail presented here. Our main results are summarised as follows:
• In Fig. 2 of Section 3, we show that the scatter in stellar mass between matched galaxies in eagle and galform is at and slowly decreases with increasing redshift. For comparison, the empirical semi-analytic galaxy formation model presented in Neistein et al. (2012) achieved an agreement with the owls simulations of Schaye et al. (2010). Clearly, the agreement between galform and eagle could be significantly improved.
• In Fig. 3 of Section 4.1, we show that the star formation thresholds implemented in eagle and galform lead (probably in conjunction with differing gas surface density distributions) to strongly differing predictions for the mass fraction of the ISM which is forming stars. This is particularly true for low-mass galaxies at low-redshift, for which galform predicts that almost all of the hydrogen in the ISM is in the atomic phase and therefore not actively forming stars.
• In Fig. 4 of Section 4.2, we show that the spatially-integrated efficiency with which the star-forming ISM is turned into stars in eagle is close to a constant value at , consistent with what is assumed in galform. However, the local density dependence of the Schmidt-like law implemented in eagle leads to a star formation efficiency that increases with redshift. eagle also predicts that at higher redshifts the star formation efficiency is mass-dependent, such that star formation is globally more efficient in more massive galaxies.
• eagle predicts that the star-forming ISM typically has significantly lower specific angular momentum than the total ISM, reflecting that it is more centrally concentrated (see Fig. 6 in Section 5). This is in contrast to galform which implicitly assumes that star-forming gas has the same specific angular momentum as the total ISM. We show that this discrepancy could be at least partially alleviated if galform were to self-consistently compute the angular momentum of star-forming gas using the radial profiles of atomic and molecular hydrogen.
• The stellar specific angular momentum distributions as a function of stellar mass are markedly different between galform and eagle (see Fig. 7 in Section 5), although the interpretation is hindered because galform does not track a specific angular momentum for stars in galaxy bulges/spheroids. For low-mass galaxies, galform predicts that the specific angular momentum of galaxy discs is similar to that of their dark matter haloes (simply reflecting the assumptions of the model). In contrast, eagle predicts that while the ISM component of these galaxies is consistent with dark matter haloes (at least in the higher resolution, recalibrated model), the stellar angular momentum is lower than the ISM by below .
• We show that the implementations of SNe feedback in eagle and galform lead to similarly low baryon fractions in low-mass haloes (, see Fig. 8 in Section 6.2), which is the regime where AGN feedback does not play a role in the models.
• We show galform predicts much higher baryon fractions in group-scale haloes () than eagle (see Fig. 8 in Section 6.2). AGN feedback in galform suppresses infall onto galaxies but does not eject gas from haloes or prevent gas accretion onto haloes, leading to higher baryon fractions than when AGN feedback is not included. Conversely, including AGN feedback in eagle acts to reduce baryon fractions in group-scale haloes.
• We show that while the median stellar mass assembly histories of galaxies in the two models are similar, the mass in other baryonic reservoirs is predicted to evolve differently (see Fig. 9 in Section 7). In particular, galform assumes that gas is rapidly returned to the diffuse gas halo after being ejected by SNe feedback, such that the ejected gas reservoir closely traces the evolution in the ISM reservoir. In eagle, these reservoirs do not show such a degree of coupling in their evolution, suggesting that the level of baryon cycling is significantly lower. Furthermore, the level of metal mixing into the diffuse gas halo is significantly lower in eagle than in galform, suggesting a lower level of gas cycling. galform can be brought into better agreement with eagle by using a longer gas return timescale.
• In Fig. 12 of Section 8, we show that the standard AGN and cooling models for gas infall onto galaxy discs implemented in galform results in strongly oscillating infall rates for individual galaxies. This behaviour is not seen in eagle, and it contributes significantly to the scatter in stellar mass between matched galaxies in galform (see Fig. 13). The oscillatory behaviour in galform stems from the implementation of AGN feedback being bimodal (either complete suppression of cooling or no effect) and the artificial dependency of AGN feedback and gas infall rates on halo mass doubling events.
Put together, we conclude that while galaxy evolution proceeds in a broadly similar manner in semi-analytic galaxy formation models compared to hydrodynamical simulations, there are a number of important over-simplifications adopted in galform (but not necessarily in other semi-analytic models). This leads to important differences when the two models are compared in detail. For example, the assumption that stars form with the same specific angular momentum as the ISM (rather than just the star-forming ISM component) has significant consequences for galaxy sizes (see Appendix D).
Crucially, we have not compared inflow rates, mass-outflow rates or gas return timescales with eagle in this study. We expect potentially significant differences in all these quantities when compared to galform and we will address this in future work. Specifically, it remains to be seen whether the parametrisations adopted for different physical processes in galform (for example, the mass loading factor associated with SNe feedback scales as a power law with galaxy circular velocity) are capable of reproducing the macroscopic behaviour predicted by state-of-the-art hydrodynamical simulations. Hydrodynamical simulations such as eagle do not necessarily provide an accurate representation of reality. For example, the dynamics of outflowing gas are sensitive to uncertain subgrid modelling. Arguably however, it ought still to be possible for models like galform to reproduce their macroscopic behaviour once an appropriate choice of model parameters has been adopted. That this is indeed possible has already been demonstrated for a highly simplified model that replaces many physical considerations with a simple empirical fit to the owls simulations Neistein et al. (2012). If a similar level of agreement to simulations can be achieved for a more physically motivated model, semi-analytic models can continue to be employed as useful tools for understanding galaxy evolution with confidence that they do not make unreasonable assumptions, particularly with regard to angular momentum and gas cycling.
## Acknowledgements
We thank Joop Schaye for reading and providing comments that helped improve the clarity of this paper. PM acknowledges the LABEX Lyon Institute of Origins (ANR-10-LABX-0066) of the Université de Lyon for its financial support within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) of the French government operated by the National Research Agency (ANR). This work was supported by the Science and Technology Facilities Council [ST/L00075X/1, ST/P000451/1]. This work used the DiRAC Data Centric system at Durham University, operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grants ST/H008519/1 and ST/K00087X/1, STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National E-Infrastructure. CL is funded by a Discovery Early Career Researcher Award (DE150100618).
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## Appendix A Modelling details
### a.1 Star formation threshold
In eagle, a local metallicity-dependent density threshold, , is used to decide which gas particles are forming stars, given by
n⋆H=min(0.1(Z0.002)−0.64,10)cm−3, (6)
where is the gas metallicity Schaye et al. (2015). This threshold acts to prevent star formation taking place in diffuse and/or low metallicity gas, reflecting the physical connection between metallicity and the formation of a cold, molecular ISM phase that can fragment to form stars Schaye (2004).
In galform, the formation of a molecular phase is explicitly computed following empirical correlations inferred from observations Lagos et al. (2011). Star formation is only allowed to occur in molecular gas, such that the star formation threshold reflects the atomic/molecular ISM gas decomposition. The mass fraction of molecular hydrogen, , is computed as a function of radius in galaxy discs by assuming a connection to the ambient pressure of the ISM in the mid-plane, , Blitz & Rosolowsky (2006), as
Rmol(r)=(Pext(r)P0)0.92, (7)
where is calculated assuming vertical hydrostatic equilibrium Elmegreen (1989) and is a constant Lagos et al. (2011). As such, the star formation threshold for discs in galform is computed as a function of the radial surface density profile of gas and stars and contains no metallicity dependence. galform also contains a distinct ISM component that represents nuclear gas that is driven into the galaxy centre by disc instabilities and galaxy mergers. All of the gas in this component is assumed to be in a molecular phase and is considered to be actively forming stars.
### a.2 Star formation law
In eagle, star-forming gas is turned into stars stochastically, sampling from a Kennicutt-Schmidt star formation law rewritten as a pressure law by assuming vertical hydrostatic equilibrium Schaye & Dalla Vecchia (2008) such that the star formation rate is given by
ψ=∑imgas,iA(1M⊙pc−2)−n(γGfgPi)(n−1)/2, (8)
where is the gas particle mass, is the ratio of specific heats, is the gravitational constant, is the gas mass fraction (set to unity). and are treated as model parameters which are set following direct empirical constraints from observations Schaye et al. (2015). The fiducial value of is modified to for hydrogen densities greater than . is the local gas pressure, with a pressure floor set proportional to gas density as , normalized to a temperature of at a hydrogen density of . As such, dense star-forming gas is artificially pressurized in eagle, ensuring that the thermal Jeans length is always resolved, even at very high gas densities.
In galform, the surface density of star formation in discs is linearly related to the surface density of molecular hydrogen, following empirical constraints from Blitz & Rosolowsky (2006). The total star formation rate is therefore linearly proportional to the star-forming ISM gas mass as
ψ=νSFMISM,SF, (9)
where is a constant empirically constrained from observations Leroy et al. (2008); Bigiel et al. (2011); Rahman et al. (2012). Accordingly, the efficiency of star formation per unit star-forming ISM mass in galaxy discs is also constant. In merger or disc-instability triggered starbursts, nuclear gas is instead converted into stars following a decaying exponential function Lacey et al. (2016).
### a.3 Disc angular momentum
In galform, disc angular momentum is computed assuming that infalling gas from the halo conserves angular momentum Cole et al. (2000). The total specific angular momentum of halo gas is set equal to that of the dark matter halo, with the radial specific angular momentum profile set assuming a constant rotation velocity. Within the disc, it is assumed that stars and gas always have equal specific angular momentum. There is also an assumption that disc specific angular momentum is unaffected by stellar feedback. Bulge/spheroid angular momentum is not explicitly modelled in galform.
In eagle, the angular momentum of galaxies emerges naturally from locally solving for the laws of gravity and hydrodynamics.
### a.4 Stellar feedback
In eagle, each star particle represents a simple stellar population with a Chabrier (2003) stellar initial mass function (IMF). The gradual injection of mass and metals through stellar evolution back into the ISM is implemented as described in Wiersma et al. (2009b). Type II SNe feedback occurs after a stellar particle forms. In the galform model presented here, stars are also formed with a Chabrier (2003) IMF. Unlike eagle, galform adopts the instantaneous recycling approximation, whereby all of the mass and metals returned to the ISM through stellar evolution are returned instantaneously as star formation takes place. Correspondingly, stellar feedback is assumed to occur simultaneously with star formation in galform. The impact of this assumption in semi-analytic models is addressed by Yates et al. (2013), De Lucia et al. (2014), Hirschmann et al. (2016) and Li et al., (in prep).
Stellar feedback is implemented in eagle with the stochastic thermal energy injection scheme of the type introduced by Dalla Vecchia & Schaye (2012). This scheme is designed to minimise artificial radiative losses and instead allows the desired radiative losses to be set by hand by adjusting the amount of injected thermal energy Schaye et al. (2015); Crain et al. (2015). Artificial losses are effectively suppressed by requiring that neighbouring gas particles heated by a supernova event are heated by , well above the peak of the radiative cooling curve. The thermal energy injected into the ISM per SNe is set to , where is the canonical value for SNe explosions. The term is parametrised as
fth=fth,min+fth,max−fth,min1+(Z0.1Z⊙)nZ(nH,birthnH,0)−nn, (10)
where is the local gas metallicity and is the gas density that the stellar particle had when it formed. and are model parameters that are the asymptotic values of a sigmoid function in metallicity, with a transition scale at a characteristic metallicity, , and with a width controlled by . An additional dependence on local gas density is controlled by model parameters, , and . The two assymptotes, and , are set to and respectively. In the low metallicity, high density regime, the energy injection therefore exceeds the canonical value for Type II SNe explosions by a factor . Crain et al. (2015) and Schaye et al. (2015) argue that this value is justified on both physical and numerical grounds (note also that the median energy injection value across the simulation is lower than unity Crain et al., 2015).
In galform, rather than scale the efficiency of SNe feedback with local metallicity/gas density, the efficiency is defined and computed globally across each galaxy disc (and separately for galaxy bulges/spheroids). This efficiency is characterised by the dimensionless mass loading factor, , defined as the ratio of the mass-outflow rate () from galaxies to the star formation rate (). Note that this is a global quantity across a given galaxy and should not be compared to Eqn 10, which pertains to the local injection of energy at a given point in the ISM. In galform, is explicitly parametrised as a function of galaxy circular velocity as
βml≡˙Mejectedψ=(VcircVSN)−γSN, (11)
where and are model parameters that control the normalization and faint-end slope of the galaxy luminosity function (e.g. Cole et al., 2000) and is the galaxy circular velocity. For star formation taking place in discs, is set to the circular velocity of the disc at the radius enclosing half of the disc mass. For nuclear star formation taking place in galaxy bulges/spheroids, is correspondingly set equal to the circular velocity at the half-mass radius of the spheroid.
Unlike in eagle, the stellar feedback in galform does not include any energetic considerations. While the thermal and kinetic energy of outflowing gas is not directly modelled in galform, if we assume (as an example) that gas is launched from galaxies in a kinetic wind with a velocity of , we can estimate the galaxy circular velocity below which the energy injected exceeds the energy available. For the SNe parameters from our fiducial model ( and ), and with a value of of energy available per unit mass turned into stars (as appropriate for a Chabrier IMF assuming per supernova and that stars with mass explode, Schaye et al., 2015), equating the (example) kinetic energy of the outflowing wind with the energy available yields that Eqn 11 violates energetic considerations below a circular velocity, . For our reference galform model, this circular velocity corresponds to a halo mass, at . The corresponding mass-loading factor at this velocity is very large, , significantly in excess of the values reported by simulations at this mass scale (e.g. Muratov et al., 2015; Christensen et al., 2016; Keller et al., 2016). The mass-loading factors predicted by eagle will be presented in Crain et al. (in preparation) and we plan to explicitly compare these mass loading factors with galform in future work.
### a.5 Black hole growth and AGN feedback
In eagle, SMBH seeds are placed at the position of the highest density gas particle within dark matter haloes of mass, Schaye et al. (2015). Black holes then accrete mass with an Eddington limited, Bondi accretion rate that is modified if the accreted gas is rotating at a velocity which is significant relative to the sound speed Rosas-Guevara et al. (2015). Black holes that are sufficiently close and with sufficiently small velocity are allowed to merge, forming a second channel of black hole growth.
Analogous to the implementation of stellar feedback, accreting SMBH particles stochastically inject thermal energy into neighbouring gas particles. The amount of energy injected per unit accretion contains a model parameter that controls the resulting relationship between SMBH mass and galaxy stellar mass, but not the effectiveness of AGN feedback Booth & Schaye (2010); Schaye et al. (2015); Bower et al. (2017). This injection energy is stored in the black hole until it is sufficiently large to heat a neighbouring gas particle by which is an order of magnitude larger than the local heating from stellar feedback ().
In galform, SMBHs are seeded inside galaxies when they first undergo a disc instability or galaxy merger event. SMBHs grow in mass primarily by accreting a fraction of the ISM mass converted into stars in starbursts that take place in galaxy bulges/spheroids during galaxy merger or disc instability events Bower et al. (2006); Malbon et al. (2007). A second growth channel comes from black hole mergers, which take place whenever there is a merging event between two galaxies hosting black holes. We compare the galform and eagle distributions of black hole mass as a function of stellar mass in Appendix E, where we show that galform does not predict the steep dependence on stellar mass predicted by eagle at .
The implementation of AGN feedback in galform is fully described in Bower et al. (2006) (see also Lacey et al., 2016). The most salient parts of the modeling for this analysis are as follows. AGN feedback in galform is implemented such that it can be effective only when the diffuse gas halo is in a quasi-hydrostatic state Bower et al. (2006). This occurs when the radiative cooling timescale exceeds the gravitational freefall timescale in the diffuse gas halo. In this regime, it is assumed that a fraction of the diffuse infalling material is directly accreted onto the SMBH, forming a third growth channel. A fraction of the rest mass energy of this accreted material is assumed to be injected into the diffuse gas halo as a heating term. If this heating term exceeds the cooling rate, infall from the diffuse gas halo onto the ISM is assumed to be completely suppressed. As such, unlike in eagle, AGN feedback has no direct effect on gas in the ISM and does not drive galactic outflows.
### a.6 Gas return timescales
In galform, gas which is ejected from galaxies is placed in a distinct reservoir. Gas is reincorporated from this reservoir back into the diffuse gas halo at a rate given by
˙Mreturn=αreturnMejectedτdyn, (12)
where is a model parameter (typically set close to unity and set to in our fiducial model), is the mass in the ejected reservoir and is the halo dynamical time Bower et al. (2006).
The spatial distribution of the ejected gas reservoir is not explicitly specified in galform. Whether or not the ejected gas resides within or outside the virial radius has been subject to various interpretations as the model has evolved over time Cole et al. (2000); Benson et al. (2003); Bower et al. (2006, 2012). Here, we choose the interpretation that the ejected gas is spatially located outside the halo virial radius for central galaxies. Physically, this corresponds to assuming that outflowing gas leaves the virial radius over a timescale that is short compared to other physically relevant timescales. For satellite galaxies, we consider ejected gas to be still within the virial radius of the host halo. This interpretation allows us to cleanly compare the indirect efficiency of feedback and the baryon cycle with eagle by measuring the fraction of baryons within the virial radius.
In eagle, no explicit gas return timescale is set, as the trajectories of gas particles are calculated self-consistently. In practice, the return times will be sensitive to the details of the implementations of supernova and AGN feedback, including the heating the temperatures.
### a.7 Radiative cooling and infall
In galform, gas infalls from the diffuse gas halo onto the galaxy disc at a rate given by
˙Minfall=4π∫rinfall0ρg(r)r2dr−McooledΔt (13)
where is the so-called “notional” gas density profile, is the mass that has already undergone infall from the notional gas profile onto the disc before the current timestep and is the numerical timestep size Cole et al. (2000). is the infall radius, which represents the radius within which gas has had sufficient time to infall from the notional profile to the disc. It is limited either by the gravitational freefall timescale or the radiative cooling timescale. is computed by equating the limiting radiative/freefall timescale with the time elapsed since the host halo last doubled in mass. This cooling model was introduced in Cole et al. (2000) and updated in Bower et al. (2006).
In eagle, gas infalls onto galaxies naturally as a consequence of hydrodynamics and gravity. Radiative cooling and photoheating are implemented element-by-element following Wiersma et al. (2009a), assuming ionization equilibrium.
## Appendix B Halo mass definitions
Fig. 14 compares the dhalo halo masses used internally within galform to the halo masses measured from the reference hydrodynamical (top) and dark-matter-only (bottom) eagle simulations. The difference between these halo mass definitions leads to a small scatter between dhalo masses and measured from the dark-matter-only simulation. There is also a small systematic offset at (this offset only appears for ) which has no trend with mass. The objects with much lower halo masses in galform compared to the dark-matter-only simulation between are flagged as satellites by the dhalo algorithm but are considered central subhaloes by subfind, leading to the large differences between halo masses. Comparing galform to the hydrodynamical simulation (top panel), the scatter is similar but with a larger, mass dependent, offset caused primarily by the ejection of baryons by feedback in eagle (see Schaller et al., 2015b, for a full analysis of this effect).
## Appendix C ISM definition in eagle
In Section 2.6, we introduce selection criteria to define gas particles which belong to an ISM component in eagle. The canonical case demonstrating the behaviour of these selection criteria is shown in Fig. 15, which shows the criteria applied to a Milky-Way-like galaxy at . The top-left panel demonstrates the rotational support selection criteria defined by Eqn 3 and Eqn 4. For this galaxy, gas particles cleanly separate into two distinct populations (the ISM and a diffuse, ionized and hot gas halo). The middle-left panel shows the associated phase diagram, indicating that there is cool, rotationally supported ISM gas (cyan points) at low densities which is not forming stars. The lower-left panel shows radial mass profiles, splitting gas particles between neutral and ionized phases of hydrogen, following the methodology described in Lagos et al. (2015) and Crain et al. (2017), which utilises the self-shielding corrections from Rahmati et al. (2013). For this galaxy, our ISM definition includes almost all of the neutral hydrogen, as well as a small amount of amount of cool, rotationally supported, ionized hydrogen.
The right-panels in Fig. 15 show the spatial distribution of gas and stellar particles. A spiral structure for the ISM component is evident, with star-forming particles (yellow) tracing denser regions within the spiral arms compared to non-star-forming ISM particles (cyan). The circles in these panels relate to the various radial selection criteria described in Section 2.6. The black circle indicates twice the half-mass radius of the stellar component. Dense gas within this radius that is not considered to be rotationally supported can still be included within the ISM component. This is why there is a small number of ISM particles (yellow/cyan) outside the selection region in the upper-left panel of Fig. 15. The yellow circle in the right-panels indicates twice the radius enclosing of the mass within the ISM. Gas particles outside this radius are then excluded from the ISM. In practice, this acts to remove a residual amount of distant, rotating material which is clearly not spatially associated to the ISM of the central galaxy. Removing this gas has minimal impact on our results. The green circle shows half the halo virial radius. Gas particles outside this radius are also excluded from the ISM. For this galaxy, this radius is significantly larger than the yellow circle and so is irrelevant.
While our ISM selection criteria appear to perform well for the galaxy shown in Fig. 15 (and we have checked a number of similar central/satellite examples for a variety of redshifts), other galaxies with more extreme properties pose a greater challenge. For passive, gas-poor galaxies, we find it is necessary to also remove gas particles from outside five times the radius enclosing half the stellar component. In passive galaxies, there is often no clear central ISM component in the radial profiles and most of of the cool gas is in distant, rotating clumps. Including/excluding these clumps makes little difference for our analysis however because they form a negligible fraction of the mass in massive galaxies.
For massive, high-redshift star-forming galaxies the situation is more complex. Fig. 16 shows the same information as Fig. 15 but for the “worst-case” scenario of a massive (), star-forming () galaxy at . This galaxy is an extreme example for which there is no apparent bimodality between a rotationally supported ISM and a diffuse, hot, halo. Rather, the gas appears to be dynamically disturbed, with a very broad mass distribution in the upper-left panel of Fig. 16. The mass contour (white) encloses gas which is not considered to be in rotational equilibrium and includes a mix of hot and cool diffuse gas as well as a residual amount of cold, dense, radially infalling gas (red and green points). The contour encloses a significant amount of star-forming gas which is not considered part of the ISM (green points), either because it is rotating too slowly or too quickly or because it has too high a radial velocity to be in dynamical equilibrium. The ISM material that is selected by the criteria described in Section 2.6 is either rotationally supported (and typically spatially extended) or is centrally-concentrated and is supported by a combination of thermal pressure and rotation (yellow points to the left of the rotationally supported selection region in the upper left panel). The relative contributions from these two components to the total ISM are roughly equal. Given the dynamically disturbed nature of this system, it is unclear whether the rotational equilibrium criteria used to select the spatially extended ISM in this case are truly robust. However, the majority of the spatially-extended neutral hydrogen which is not included in the ISM is excluded because it is radially infalling and as such is robustly excluded.
The contrast between the situations presented in Fig. 15 and Fig. 16 serves to highlight the difficulty of defining the ISM across all galaxies with a uniform set of selection criteria. Nonetheless, simply selecting star-forming gas particles would cut away of the mass and of the angular momentum in the case of the well-defined ISM shown in Fig. 15. Taking gas within an aperture is also likely to be overly simplistic. Too small an aperture will cut away spatially extended, high-angular-momentum gas. Too large an aperture (even with a temperature cut) will select significant amounts of radially infalling gas around high-redshift galaxies that should not (at least according to our physical criteria) be considered as part of the ISM.
To assess the global behaviour of our criteria, Fig. 17 shows the resulting ISM gas fractions as a function of stellar mass for three redshifts. These are then compared to the total gas within and the neutral hydrogen within , which is taken as a proxy for the ISM in Lagos et al. (2015). At , the resulting gas fractions are similar, indicating that the most of the hydrogen within of the halo centre is in a neutral phase and is in dynamical equilibrium. At , our ISM defintion is very close to taking neutral hydrogen within but the total gas fractions (black lines) are significantly higher, presumably because of the impact from supernova feedback in heating circumgalactic gas around high-redshift galaxies. At , our ISM definition yields systematically lower gas fractions than taking neutral gas within an aperture, presumably by excluding neutral hydrogen present in dense, radially infalling accretion streams.
## Appendix D Galaxy sizes
Fig. 18 shows the galaxy size distributions as a function of stellar mass for eagle and galform. As discussed in Guo et al. (2016), the most obvious tension between the models is that galform predicts very compact sizes for massive galaxies. The exact underlying cause for these compact sizes is presently unclear as it is challenging to disentangle the combined effects of modelling adiabatic halo contraction, the calculation of pseudo-angular momentum after galaxy mergers/disc instabilities and the impact of the angular momentum histories of progenitor galaxy discs Cole et al. (2000). Any of these areas of the modelling could be suspect. We defer further exploration of this problem in galform to future work.
Also apparent in Fig. 18 is that the scatter in galaxy size at a fixed stellar mass is significantly larger in galform than in eagle, and that the sizes of low-mass galaxies are larger in galform than in eagle. We do not address the former discrepancy in this paper. The latter discrepancy is explored in Section 5 where we show that assuming that star-forming gas has the same specific angular momentum as the total ISM reservoir likely leads to erroneously high specific stellar angular momentum (and hence galaxy sizes) in disc-dominated (low mass, low redshift) galaxies.
Fig. 18 also shows observational data from the GAMA survey Baldry et al. (2012) and the CANDELS survey van der Wel et al. (2014). For GAMA, two samples of red and blue galaxies are presented and the sizes quoted are effective radii in the band. For CANDELS, two samples of star-forming and passive galaxies (determined from rest-frame colour distributions) are presented and the sizes quoted are the semi-major axes of 1D Sersic fits at a rest-frame wavelength of . Note that we do not attempt to correct for inclination effects for sizes presented from eagle and galform. While the comparison of these observed distributions to the models should be interpreted with care because of sample selection, projection and mass-to-light ratio effects, it is nonetheless clear that eagle predicts a more realistic size-mass distribution than galform, particularly in the local Universe (where eagle was calibrated to predict realistic sizes).
## Appendix E Black hole masses
Fig. 19 shows the relationship between SMBH mass and stellar mass in the two models. At , the two models are very similar for . For , galform predicts a significantly larger scatter in SMBH mass. Unlike in eagle, in galform black hole growth is explicitly coupled to the growth of the galaxy bulge. The large scatter therefore reflects the significant scatter in bulge-to-total stellar mass ratio predicted by galform in this mass range. The fraction of bulge stars that were formed quiescently in progenitor discs (versus bulge stars that were formed in galaxy merger or disc instability triggered star bursts) also plays a role in shaping the scatter in SMBH mass. At lower masses, eagle is affected by the seed mass, rendering a comparison meaningless.
Interestingly, at higher redshifts eagle predicts lower black hole masses compared to galform, and a much steeper dependence with stellar mass at high masses. This effect is discussed extensively in Bower et al. (2017), who interpret SMBH growth in eagle as governed by a strongly non-linear transition in SMBH accretion efficiency that occurs at a characteristic halo mass scale. This scale is associated with the scale at which a hot corona develops, preventing SNe from driving a buoyant outflow. By construction, a strongly non-linear accretion efficiency transition does not emerge in galform, leading to a shallower SMBH-stellar mass relation. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9463216662406921, "perplexity": 3002.973041968794}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057202.68/warc/CC-MAIN-20210921101319-20210921131319-00102.warc.gz"} |
https://mathoverflow.net/questions/97313/properties-preserved-under-passage-to-augmented-filtration | # Properties preserved under passage to augmented filtration
Dear all,
generally speaking, my question is about which properties of a stochastic process are preserved when I skip from the original to the augmented filtration.
Recall that if $(\mathcal{F}_t)_{t\geq0}$is a filtration on a probability space $(\Omega,\mathcal{A},P)$, the augmented filtration $(\mathcal{F}^0_t)_{t\geq0}$ is the smallest filtration such that
• $\mathcal{F}_t\subseteq \mathcal{F}^0_t$ for all $t\geq0$
• $N\subseteq A\in\mathcal{A}$ and $P(A)=0$ imply $N\in\mathcal{F}^0_0$ (complete)
• $\mathcal{F}_t^0=\mathcal{F}_{t+}^0$ for all $t\geq0$ (right-continuous)
It is well known that in many situations it is desirable to work with a complete and right-continuous filtration (for example, this ensures that martingales have a right-continuous modification, or that certain random times are stopping times).
Now here is my question in detail: Suppose we consider a stochastic process $(X(t))_{t\geq0}$and its natural filtration $(\mathcal{F}_t)_{t\geq0}$. And suppose we know that $X$ is a (strong) Markov process and a martingale. Do these properties still hold with respect to the augmented filtration? Or what about martingale problems: Existence results for martingale problems usually refer to the natural filtration of the potential solution process (that is the problem is "find a process which solves the martingale problem with respect to its own natural filtration"). So if I have established existence with respect to the natural filtration does the solution process solve the martingale problem with respect to the augmented filtration, as well?
In fact, my question reduces to the following: If working with the augmented filtration has many advantages why still bother with natural filtrations??
I am aware that these are a lot of questions packed into one post. Sorry, for not being more concise. I would greatly appreciate any comment which helps me to clarify these issues.
Best regards
lpdbw
• Interesting question. In a slightly different direction: One problem with augmented filtrations is the inability to extend a compatible family of probability measures $\mathbb Q_t$ to $\mathcal A$, see for example arxiv.org/abs/0910.4959, where a weaker kind of augmentation is introduced and studied (introduced before also by Bichteler) May 19, 2012 at 9:34
Hi lpdbw,
I think this is a very interesting questions, here at least a partial answer; It depends heavily on the little word "strong" in parentheseis.
Assuming that $(X_t)$ is strong Markov, the answer to your questions seems to be "yes": The completed filtration of an $\mathbb{R}^d$-valued strong Markov process is already right-continuous (cf. Theorem 2.7.7. of Karatzas/Shreve: Brownian Motion and Stochastic Calculus - note that they use a different terminology for augmented/completed). And just augmenting the filtration does not affect the martingale property (these are just sets of mass zero).
If you look however on Markov processes which does not necessarily has the strong Markov property, the answer seems to be negative as seen in the following counterexample: Let $(W_t)$ be a Brownian motion in its natural filtration and $\xi$ a random variable independent of the Brownian filtration, $\mathbb{P}[\xi =1] = \mathbb{P}[\xi =2] =\frac{1}{2}$, and define the process $(X_t)$ as $$X_t = \int_0^t \xi \; \;dW_s.$$
This process is a Markov process and a martingale in it's natural filtration $\mathcal{F}_{t}$, but just passing to the right continuous filtration $\mathcal{F}_{t+} = \bigcap_{s>t}\mathcal{F}_s$ destroys the Markov property. Note that for $t>0$ we have $\mathcal{F}_t =\mathcal{F}_{t+} = \sigma(W_s; s\leq t) \vee \sigma(\xi)$, however at $t=0$ they are fundamentally different: $\mathcal{F}_0$ is trivial whereas $\mathcal{F}_{0+} = \sigma(\xi)$. Adding additional null sets changes again nothing.
Thanks a lot for your comments so far. Glad to hear other people are interested in the problem, as well!
I think it is an important message that the strong Markov property is so closely related to the problem.
http://books.google.de/books?id=p_Bn8QAuDf4C&printsec=frontcover has some results which do not require the strong Markov property:
If I got it right (the relevant page is not displayed) "completed filtration" here is understood in the sense of [Karatzas/Shreve] (in contrast to what [KS] refers to as "augmented"). There seems to be a bit of a confusion of terminology in the literature ...
Lemma 3.2.1 in the cited monograph states:
If $X$ is a Markov process w.r.t. filtration $\mathcal{F}:=(\mathcal{F}_t)_{t\geq0}$, then so it is w.r.t. to the completed filtration $\overline{\mathcal{F}}:=(\overline{\mathcal{F}}_t)_{t\geq0}$ (and the transition kernels are the same).
Note that he refers to $\overline{\mathcal{F}}_+:=(\overline{\mathcal{F}}_{t+})_{t\geq0}$ as "augmented completed filtration". Moreover, $S$ is assumed to be a separable locally compact space and $S_\Delta$ is its one-point-compactification. Then the following holds:
Suppose $X$ is a right-continuous Feller process w.r.t. to a filtration $(\mathcal{F}_t)_{t\geq0}$ taking values in $S_\Delta$. Then $X$ is also a Feller process w.r.t. to the augmented completed filtration $(\overline{\mathcal{F}}_{t+})_{t\geq0}$.
Of course, local compactness (if it is really needed here) is quite a restrictive assumtpion in certain situations.
Here are some ideas concerning martingale problems (MGP):
Suppose $X$ has Polish state space $E$ and solves a certain well-posed MGP. Denote its law on the cadlag path space $D_E[0,\infty)$ by $P_x$ if $X(0)=x$. According to [Ethier/Kurtz, Theorem 4.4.2], well-posedness + measurability of $x\mapsto P_x(B)$ for any Borel subset $B\subseteq D_E[0,\infty)$ imply the strong Markov property for $X$. Now, the result from [KS] ensures that the augmented filtration (in the terminology of [KS]) is right-continuous and does not harm the strong Markov property. Hence, it would only remain to show that, given the strong Markov property, the martingale property still holds under the larger filtration ... Right now I am not sure whether this is true. After all, the counter example from above is not strong Markov, but that's not a proof of course ;-)
Comment on last paragraph of my previous post (cannot edit post as unregistered user):
Since the augmented filtration is automatically right-continuous here it would suffice to check whether the martingale property is preserved when subsets of null sets are added ... and here I agree with Stephan Sturm that this should be true since these sets again have zero measure.
This would mean that for well-posed MGP which satisfies the measurability condition (which in many situations is relatively easy to check) working with the augmented filtration (in the sense of my original post) does not cause any problems. Do you agree?
• Yes, I agree with your argumentation, just one minor point: the result in [KS] holds only for $\mathbb{R}^d$-valued Markov processes (I did not mention this originally, but edited now the comment). I belief that one can extend there proof to the locally compact case, but I have not checked this (in that case this would be more general as the result you are citing as Feller processes enjoy the strong Markov property automatically). For the case of non-locally compact state spaces (in which you seem to be interested, deploring the restriction to local compactness) I have no answer yet... May 20, 2012 at 1:19
Hi,
for the sake of completeness, here is another partial answer to my question:
If $X$ is a right-continuous martingale on the filtered probability space $(\Omega,\mathcal{A},(\mathcal{F}_t)_{t\geq 0},P)$, then $X$ is a right-continuous martingale on the augmented space $(\Omega,\mathcal{A},(\mathcal{F}^0_t)_{t\geq0},P)$.
Here, again, "augmented" is understood in the sense of the original post (see above). The proof is quite simple. It may be checked here: Rogers/Williams Vol. 1, Lemma II.67.10 -->
Getting back to the martingale problem issue, if one has a cadlag solution to a certain martingale problem, then it is still a solution if one replaces the original filtration by its usual augmentation. Of course, often the primary reason for assuming the filtration to be augmented is just to ensure that martingales have right-continuous modifications. However, there are also other motives for assuming augmented filtrations, for example, to ensure that hitting times are stopping times.
Regards
lpdbw
Hi,
I just came across another interesting discussion of the "usual conditions" and how they are related with certain properties of stochastic processes. I add this rerefence here for the sake of completeness, maybe it is useful for someone who comes across this thread.
Here is the link: | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9189329147338867, "perplexity": 233.95164959576942}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948951.4/warc/CC-MAIN-20230329054547-20230329084547-00154.warc.gz"} |
http://math.stackexchange.com/questions/722287/commutator-subgroup-or?answertab=active | # Commutator subgroup - or?
If $G$ is a group and $X, Y \subseteq G$ then the commutator subgroup of $G$ is defined as $[G, G] = \langle [x, y] \mid x, y\in G \rangle$, where $[x, y] = x^{-1}y^{-1}xy$ and the group generated by commutator elements from $X$ and $Y$ is $[X, Y] = \langle [x, y] \mid x\in X, y\in Y\rangle$.
Now I am reading an article where the author talks about $[\alpha, A]$ where $\alpha$ is an automorphism of the group $A$. What does this mean? Is it related to the commutator group? The author does not define it, so I guess it is widely known what it is supposed to mean, but I haven't seen it before and I haven't had any luck looking in my books or Googling.
-
## 1 Answer
$[\alpha,A]$ means the subgroup $\langle \alpha(a^{-1})a \mid a \in A \rangle$ of $A$.
The connection with the commutator subgroups is that $[\alpha,A]$ is equal to the subgroup $[\langle \alpha \rangle, A]$ of the semidirect $A \rtimes \langle \alpha \rangle$ because, in the semidirect product $\alpha^{-1} a \alpha$ is equal to $\alpha(a)$.
-
Thanks a lot, Derek! – user129954 Mar 22 '14 at 16:37
Is there a typo here, $\alpha(a^{-1})a$ is just $\alpha$...? – hayd Mar 22 '14 at 20:13
No, it is the product of $\alpha(a^{-1})$ and $a$. – user129954 Mar 22 '14 at 21:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8976922631263733, "perplexity": 98.04234942246136}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398460519.28/warc/CC-MAIN-20151124205420-00082-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.iue.tuwien.ac.at/phd/klima/node48.html | ### 3.5.1.4 Inheritance Scheme
The inheritance mechanism of the Input Deck database allows for inheritance of every section, subsection, and so on. Since the inheritance scheme is very flexible it had to be well defined otherwise ambiguities would emerge.
The necessity of this is shown for the example below. Section S in Section A inherits from Section R. Modifications in Section R immediately take place in Section S too, because a dynamic approach of data representation is used. Section A with all its entities and their properties are passed on to Section B. Thus, Section B also has two subsections, R and S. Any modification performed in Section R of Section A will even influence the contents of Section B.
```A
{ R
{ // ...
}
S : R
{ // ...
}
}
B : A
{ R
{ // ...
}
S
{ // ...
}
}
```
Figure 3.7: Inheritance schemes.
(a) Implicit inheritance scheme.
(b) Explicit inheritance scheme.
Ambiguity arises between Section S of Section A and Section S of Section B. This is depicted in Fig. 3.7(a) and Fig. 3.7(b) which show the implicit and explicit inheritance schemes, respectively. With implicit inheritance the inheritance information of all subsections is passed on to the successor, whereas with explicit inheritance only the inheritance rules given explicitly by the user are applied. In the first case Section S in Section B is inherited from Section R in Section B - the inheritance scheme is defined in Section A and was passed to Section B. In the second case Section S in Section B does not contain any explicit inheritance information. Since Section B is inherited from Section A, that Section S is inherited from Section S of Section A.
Modifications to the Section S in Section A will therefore influence the Section S of Section B only in the case of explicit inheritance. In the Input Deck database the explicit inheritance scheme has been implemented for the following reason: Be Section A the world, Section R is an ape, and Section S which is derived from Section R is a human. If Section B is derived from Section A to create another world its Section R is again an ape and Section S again a human. This would not be the case with implicit inheritance. Section R of the new world would be an ape again but Section S something else derived from an ape.
Due to the dynamic approach of data representation database queries are not trivial: To find an entry the Input Deck database has to generate a sequence of possible names. To find the variable `~`.B.S.foo in the hierarchy
```A
{ R
{ foo = 1;
}
S : R;
}
B : A;
```
the following sequence is generated:
```~.B.S.foo // -> does not exist
~.A.S.foo // -> does not exist
~.A.R.foo // -> found
```
Finally the existing keyword `~`.A.R.foo is found. During evaluation of expressions such sequences are generated for each variable reference used inside the expressions. Therefore the Input Deck database uses a sophisticated algorithm for creating and representing names which is explained in the following section.
Robert Klima 2003-02-06 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.905066728591919, "perplexity": 1346.0986235033286}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323583423.96/warc/CC-MAIN-20211016043926-20211016073926-00295.warc.gz"} |
https://cs.stackexchange.com/questions/740/research-on-evaluating-the-performance-of-cache-obliviousness-in-practice | Research on evaluating the performance of cache-obliviousness in practice
Cache-oblivious algorithms and data structures are a rather new thing, introduced by Frigo et al. in Cache-oblivious algorithms, 1999. Prokop's thesis from the same year introduces the early ideas as well.
The paper by Frigo et al. present some experimental results showing the potential of the theory and of the cache-oblivious algorithms and data structures. Many cache-oblivious data structures are based on static search trees. Methods of storing and navigating these trees have been developed quite a bit, perhaps most notably by Bender et al. and also by Brodal et al. Demaine gives a nice overview.
The experimental work of investigating the cache behaviour in practice was done at least by Ladner et al. in A Comparison of Cache Aware and Cache Oblivious Static Search Trees Using Program Instrumentation, 2002. Ladner et al. benchmarked the cache behaviour of algorithms solving the binary search problem, using the classic algorithm, cache-oblivious algorithm and cache-aware algorithm. Each algorithm was benchmarked with both implicit and explicit navigation methods. In addition to this, the thesis by Rønn, 2003 analyzed the same algorithms to quite high detail and also performed even more thorough testing of the same algorithms as Ladner et al.
My question is
Has there been any newer research on benchmarking the cache behaviour of cache-oblivious algorithms in practice since? I'm especially interested in the performance of the static search trees, but I would also be happy with any other cache-oblivious algorithms and data structures. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8415190577507019, "perplexity": 1122.662903124585}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986703625.46/warc/CC-MAIN-20191020053545-20191020081045-00205.warc.gz"} |
http://www.conservapedia.com/Infinity | # Infinity
$\frac{d}{dx} \sin x=?\,$ This article/section deals with mathematical concepts appropriate for a student in late high school or early university.
Infinity may be defined as an "unlimited extent of time, space, or quantity ... an indefinitely great number or amount."[1]
That is, it is something which "goes on without end". Infinity is denoted by this symbol:$\infty\,$, looking like an 8 on its side.
Infinity comes up in many aspects of mathematical discourse and is often treated as a number, but it is not a real number. In mathematical contexts, infinity often appears as a limit, as an integral, as the measure (size) of a set in Euclidean space, and as the cardinality (size) of sets in general.
The simplest place to see this seeming paradox is in the fact that all integers are finite, but that there are an infinite number of them.
• Why are all integers finite? Because, if infinity were an integer, what would $\infty+1\,$ be?
• Why is the set of all integers an infinite set? Because they go on forever. If the set were finite, there would be a biggest integer. But that can't be, because we can always add one to any integer. This kind of thinking is often schoolchildren's first introduction to logical reasoning.
While infinity is not a number, it appears in many other contexts. As we have seen, the size of the set of integers is infinite. We say that the cardinality of the integers (this set is often denoted $\mathbb{Z}$) is infinite. The rational numbers ($\mathbb{Q}$) and the real numbers ($\mathbb{R}$) also have infinite cardinality. An interesting result from set theory (see below) is that $\mathbb{Z}$ and $\mathbb{Q}$ have the same cardinality, while $\mathbb{R}$ has larger cardinality than the other two.
Another place where infinity arises is in limits. We can say that "the limit of $1/x\,$ as $x\,$ approaches zero is infinity", or "the limit of $e^{-x}\,$ as $x\,$ approaches infinity is zero", but this is because infinity has a special meaning in the context of limits. See limit for discussion of this.
We can also say that "the measure (size) of the set of reals is infinity", or that certain integrals are infinite, but this is because of special properties of measures and integrals.
So infinity might arise in statements like these:
• $\frac{1}{0} = \infty\ \$NO! This isn't allowed! Infinity is not a number, and division by zero is illegal!
• $\lim_{x\to 0}\frac{1}{x} = \infty\ \$Yes, sort of. One could say that "the limit is infinite", since that includes both positive and negative infinite values. But to say that the limit is "equal to infinity", one would have to say that this is the limit from the right. The limit from the left is "minus infinity", that is, unboundedly negative. This sort of statement, in terms of limits, is what was presumably meant by the incorrect statement above.
• $\int_0^1\frac{1}{x} = \infty\ \$Infinity has a special meaning for integrals.
• $\|\mathbb{Z}\| = \infty\ \$The cardinality of the integers is infinite.
• $\mu(\mathbb{R}) = \infty\ \$The (Lebesgue) measure of the reals is infinite.
In measure theory, one sometimes defines the "extended reals", allowing plus and minus infinity to be considered to be numbers, but this is a special construction, which makes certain arithmetical operations impossible. It can't be done in general.
Another place where infinity is considered acceptable is in Floating-point arithmetic in computers. The IEEE-754 standard (which all modern computers support) allows the values $\infty\,$ and $- \infty\,$. But their treatment is very different from other numbers, and computer floating-point numbers don't faithfully model the real numbers in any case.
Also, in some non-standard models of Peano Arithmetic, $\infty\,$ is treated as an actual number. But, once again, this is not standard mathematics.
Georg Cantor's diagonal argument is an elegant proof demonstrating that the (infinite) cardinality of real numbers is greater than the (infinite) cardinality of countable integers. The essence of the argument is that in any proposed list of all real numbers, a new real number not in the list can be constructed by taking the digits in a diagonal through the list and changing them to construct a new real number that differs from the nth entry at the nth position right of the decimal point.
To formalize the concepts of countably and uncountably infinite sets, we need the set theory concept of cardinality. Using this concept it can be shown that there are infinitely many distinct infinite cardinalities.
In Zermelo-Fraenkel set theory, there is the Axiom of Infinity, asserting the existence of an infinite set. The set that it creates is essentially the same as the integers. Some "constructive" mathematicians work without this axiom, and determine which results may be proved without assuming it.
## Classification of infinities
In modern mathematics, it is recognized that there is no single concept of 'infinity' - there are many different infinities. We can divide these into some broad classes. Firstly, there are the cardinals, which are used to measure the sizes of sets; and then there are ordinals, which are used to measure the position of an item in an ordered list. For finite quantities, we can use the same numbers for both - a race can have 3 participants, and you can come 3rd in the race. But, with infinite quantities, this no longer applies, the same numbers can now be used for both purposes. Thus, the smallest infinite cardinal is aleph-null, but the smallest infinite ordinal is omega.
### Cardinals
We say two sets have the same size if it is possible to put their respective members into one-to-one correspondence. A cardinal can be conceived as the equivalence class of all sets having that size.
Consider the set of all natural numbers, and its subset the set of all odd natural numbers. One would think that there are half as many odd numbers as natural numbers. Yet, while that is true when dealing with some finite subset of the natural numbers, when dealing with their entirety, it turns out that there are as many odd numbers as natural numbers. This can be shown by the fact that the natural numbers and the odd natural numbers can be put in a one-to-one correspondence: 0 -> 1, 1 -> 3, 2 -> 5, 3 -> 7, etc. (represented by the formula 2*n + 1). Thus, whereas for finite sets if A is a proper subset of B, then the cardinality of A must be less than the cardinality of B, for infinite sets this is no longer the case; an infinite set may be the same size as one of its proper subsets.
The cardinality of the natural numbers is known as aleph-null. There are as many integers as natural numbers; and as many rational numbers as natural numbers. But it can be shown, using Cantor's diagonalization argument, that the reals are strictly larger than the natural numbers - you cannot put the natural numbers into one to one correspondence with the reals. The cardinality of the reals is the cardinality of the power set of the natural numbers, which is known as the power of the continuum (c) or as beth-one. (Aleph-null is the same as beth-null; whether aleph-one is the same as beth-one is a mathematical question, which essentially has no answer - and we can even show it has no answer.)
The generalised continuum hypothesis (GCH) states that $\beth_{a} = \aleph_{a}$ for all ordinals a. So, the continuum hypothesis is a special case of the generalised continuum hypothesis for ordinal 0. Like CH, GCH is independent of the axioms of ZFC.
A set having cardinality equal to or less than $\aleph_{0}$ is called countable; a set of greater cardinality is called uncountable.
## Ordinals
Numbers serve two distinct purposes - to measure the size of sets, and to measure the position of an item in an ordering. Thus, we may speak of a race having 2, 3, 4, 5, etc. contestants, and we can speak of the contestants as having come 1st, 2nd, 3rd etc. (or even 0th, if one is a mathematician!) When we measure the number of elements in a set, we are using cardinal numbers; when we measure an item's position in an ordering, we are using ordinal numbers. For finite quantities, it does not make that much difference, since for finite quantities we can use the same numbers to serve both purposes. But for transfinite quantities, that is no longer the case - we can no longer use the same numbers as both cardinals and ordinals. Thus, the smallest transfinite cardinal is $\aleph_{0}$, but the smallest transfinite ordinal is ω.
## Other types of infinite numbers
Other than transfinite cardinals and ordinals, there are other types of infinite numbers:
• the affinely extended reals: these are elements of the set $\mathbb{R} \cup \{ +\infty, -\infty \}$. In other words, the real numbers extended by the addition of positive and negative infinity. $+\infty$ is greater than every real number, and $-\infty$ is less than every real number.
• the projectively extended reals: these are elements of the set $\mathbb{R} \cup \{ \infty \}$. In other words, the real numbers extended by the addition of a single signless infinity. The ordering relation is not defined for the projectively extended reals - but informally, $\infty$ can be said to be both greater than and less than every real number.
• the hyperreals: denoted *$\mathbb{R}$, these extend the real numbers with both infinite and infinitesimal quantities. They are used in non-standard analysis. Other extensions of the real numbers with infinite and infinitesimal quantities include the surreal numbers, the superreal numbers, and the Levi-Civita field.
The conclusion to be drawn from all of this, is that although in popular usage we speak about infinity as if it were a single thing, there are actually numerous diferent infinities, all belonging to different systems. If we want to talk about infinity, we ought to be careful to specify which infinity we mean.
## Archimedean property
An ordered field is said to be Archimedean if it has no infinitely large or infinitely small elements. One way of stating this: an ordered field is Archimedian, if for every element of the field, there is a greater natural number. This is true for the reals - for every real number, there is a greater natural number - but not for example for the affinely extended reals, since $+\infty$ is greater than every natural number. So the affinely extended reals are said to be non-Archimedian. Likewise, the hyperreal numbers, the surreal numbers, the superreal numbers and the Levi-Civita field are all non-Archimedian. Any ordered field containing infinities must also contain infinitesimals as their inverse.
To understand the notion of non-Archimedeanity, it is interesting to imagine what it would be like were space or time were non-Archimedean. If space was non-Archimedean, there could be areas of the universe infinitely distant from here. If time was non-Archimedean, there could be past times an infinite number of years before the present, and future times an infinite number of years into the future. It appears that no one has every seriously suggested that time or space actually are non-Archimedean, but it is nonetheless an interesting mental exercise to understand the notion.
## Philosophy of mathematics
Different positions in the philosophy of mathematics have different attitudes towards infinity. At one extreme, Platonists and formalists generally have no philosophical to any form of infinity than can be defined. At the other extreme, finitists and ultrafinitists deny the existence of infinity; to them, only finite quantities, and finite objects, actually exist. In the middle, many constructivists and intuitionists adopt a position of countablism - an acceptance of the existence of countable sets, but denying that uncountable sets exist. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 31, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9194148182868958, "perplexity": 251.05987162148767}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042988250.59/warc/CC-MAIN-20150728002308-00227-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://juanreyero.com/article/math/bayes.html | ## Visual Bayes
### A graphic explanation of the Bayes theorem
I enjoyed how the 3.16 section of the Stanford Artificial Intelligence class presented the Bayes theorem. Instead of giving a formula and expecting the alumni to apply it, they gave us a problem that the Bayes theorem would solve and expected, I believe, that we figured it out ourselves.
Being as I am counting-challenged, it took me a while to figure out a way of solving it that was simple enough that I could be reasonably sure of my results. It turned out to be a very interesting detour.
The problem was like this: the probability of having cancer is $P(C)=0.01$; the probability of giving positive in a cancer test when you have cancer is $P(+ \mid C)=0.9$; and the probability of giving positive when you don't have cancer is $P(+ \mid \neg C)=0.2$. What is the probability of having cancer if you give positive in the test?
It's interesting because it is sort of how it works. A test for cancer is not tried on all the population. You find some people that have cancer, give them the test, and see what proportion comes positive; you find other people that don't have cancer, give them the test, and see how many come positive as well. But that doesn't directly tell you what proportion of the people who give positive in the test will actually have cancer.
Let's assume a population of 1000:
The bottom line represents the full population. The red at the left is the $0.01$ who have cancer, and the green the $0.99$ who don't. The top line is the people who give positive in the test. The orange spec at the left are those who have cancer and give positive, and the blue are those who don't have cancer but also give positive.
Zooming in:
In red on the left we have the $0.01 \cdot 1000 = 10$ persons who have cancer, and on top in orange those among them who give positive in the test. We know that $P(+\mid C)=0.9$ —90% of people who have cancer give positive— so $0.9\cdot 10=9$ persons have cancer and give positive.
In blue we have the people who don't have cancer but give positive. Given that $P(+\mid \neg C)=0.2$ —20% of people who don't have cancer give positive— there will be $0.2\cdot 990=198$ healthy persons who give positive.
Now the answer to the problem should be obvious. We know that the test is positive, so that puts us in the top line. Out of the $9+198=207$ persons who give positive, only $9$ actually have cancer. So the chances of having cancer given that the test is positive are $9/207=0.043$, only around 4.3%.
An interesting follow-up question would be to figure out how accurate the test would have to be for a positive to have a 50% chance of having cancer. One way would be to reduce the percentage of false positives from $0.2$ to $9/990=0.009$ (check it out, this is not obvious from the above), almost dividing it by 20.
### How about two independent tests?
Unit 3.20 asks us to figure out the probability of having cancer when two independent test are positive, assuming that the new test has the same probabilities as the previous.
The answer is about as simple as before. Let's call a positive in the first test $+^1$ and a positive in the second test $+^2$, and draw it like this:
The key here is independence. If the two tests are independent, the proportion of positives in the second test among people who gave positive in the first one will be the same as among people who gave negative. That is, if we take the 9 persons who have cancer and gave positive in the first test, $P(+^2\mid C)=0.9$ of them will be positive with the second test, or $8.1$ (and $0.9$ of the $1$ who had cancer and gave negative will be positive this time).
The same applies to the 198 who don't have cancer and gave positive in the first test. As the two tests are independent, there will be $P(+^2\mid \neg C)=0.2$ of them who are positive again, or 39.6:
So we have $39.6+8.1=47.7$ people who give positive in the two tests, and $8.1$ among them have cancer, so the probability of having cancer if you give positive in two independent tests is $8.1/47.7=0.17$.
### Bayes theorem
The formulation of the Bayes theorem can be given like this: if the probability of $A$ is $P(A)$, the probability of $B$ is $P(B)$, and the probability of $B$ given $A$ is $P(B\mid A)$, then the probability of $A$ given $B$ is
$P(A\mid B) = \frac{P(B\mid A) P(A)}{P(B)}.$
If you are like me, this requires some parsing. We humans are hard-wired to understand problems involving people and relationships between people: the chances that you have cancer when you have received a positive cancer test are much easier to think about than the probability of $A$ given $B$. So let's translate this to the terms of the problem we have been thinking about:
$P(C\mid +) = \frac{P(+\mid C) P(C)}{P(+)}.$
The probability of giving positive, in terms of the absolute values that we have been using, is the total number of positives —sum of the 198 persons who give positive without having cancer and the 9 persons who give positive and have cancer— divided by the 1000 of the total population. So, spelling it out,
$P(C\mid +) = \frac{0.9\cdot 10/1000}{(9+198)/1000} = \frac{9}{9+198}.$
Which is, of course, what we should have done to start with, without bothering to convert anything to absolute values and dispensing with the 1000's. But convoluted paths are sometimes fun to follow.
The best explanation I've found of the Bayes Theorem is in Alvin W. Drake's Fundamentals of Applied Probability Theory1. Unfortunately it is out of print, but you might get hold of a second-hand copy. This is the one book that helped me understand what probability is about.
Russell and Norvig's Artificial Intelligence: A Modern Approach also has a great introduction, and many interesting examples and practical uses.
### Acknowledgements
Thanks to Utpal Sarkar for his insightful comments and his review of a draft of this article.
## Related
Magnitude — A python library for computing with physical quantities A Unicycle on a Slope Equilibria of a Unicycle Playing with convolutions in Python
## Footnotes:
1
Disclaimer: I do get a cut from your Amazon purchase. Thank you very much for your support.
Barcelona, 2011-11-01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8682755827903748, "perplexity": 336.7305622552832}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510276584.58/warc/CC-MAIN-20140728011756-00410-ip-10-146-231-18.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/227304/product-of-2-gaussian-distributions-with-different-variables | Product of 2 Gaussian Distributions with Different Variables
Sorry, I asked the original question improperly so I am rephrasing it.
What is the mean and covariance of the distribution, $f_{PA}(PA) \cdot f_{Y|X,PA}(Y)$ where $f_{PA}$ and $f_{Y|X,PA}$ are both gaussian distributions? (The distribution over $X,Y,PA$ is Gaussian.) $Y,PA$ are vectors.
-
Hint: Independent random variables are uncorrelated. So the covariance matrix of $Z$ has a block structure: $$\Sigma_Z = \begin{bmatrix}\Sigma_X & \mathbf 0\\\mathbf 0 & \Sigma_Y\end{bmatrix}$$ where $\Sigma_X$ and $\Sigma_Y$ are the covariance matrices of $X$ and $Y$. – Dilip Sarwate Nov 2 '12 at 18:54
Hmm. I see that X and Y are independent because the P(X, Y) = P(X)P(Y) in the way I phrased this question. However, X and Y should not necessarily be independent... I must be interpreting my results incorrectly. I have a distribution over the variables in Z that equals N(X; 0; V_X)*N(Y; mu_Y; V_Y) where XUY = Z. Does this alone require that X and Y are independent? Does it make sense to say N(X; 0; V_X) is not marginal distribution of Z over X? Rather, it is a different distribution over the variables in X? – user47884 Nov 2 '12 at 20:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9113175868988037, "perplexity": 473.60637599088335}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042990217.27/warc/CC-MAIN-20150728002310-00087-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://www.maths.manchester.ac.uk/~tv/Seminar/2012-2013/shonkwiler.html | # Manchester Geometry Seminar 2012/2013
Thursday 13 December 2012. The Frank Adams Room (Room 1.212), the Alan Turing Building. 4.15pm
## Probability Theory of Random Polygons from the Quaternionic Viewpoint Clayton Shonkwiler (University of Georgia)
[email protected]
Here is a natural question in statistical physics: What is the expected shape of a polymer with $n$ monomers in solution? The corresponding mathematical question is equally interesting: Consider the space of $n$-gons in three-dimensional space with length $2$, modulo translations. This is a compact manifold. What is the natural metric (and corresponding probability measure) on this manifold? And what are the statistical properties of $n$-gons in $3$-space sampled uniformly from this probability measure?
In this talk, I will describe a natural probability measure on length $2$ $n$-gon space pushed forward from the standard measure on the Stiefel manifold of $2$-frames in complex $n$-space. The pushforward map comes from a construction of Hausmann and Knutson from algebraic geometry.
We are able to explicitly and exactly compute the expected value of the radius of gyration for polygons sampled from our measure, and also give a fast algorithm for directly sampling the space of closed polygons. The talk describes joint work with Jason Cantarella and Tetsuo Deguchi. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9584518671035767, "perplexity": 723.8259760853709}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948594665.87/warc/CC-MAIN-20171217074303-20171217100303-00639.warc.gz"} |
https://www.physicsforums.com/threads/rotor-power.606355/ | # Rotor Power
1. May 15, 2012
### Alwyn Hartman
Ok, here we go.
In a rotating mechanism (helicopter rotor), at a state of equilibrium, the rotor consumes a certain amount of energy from the shaft to maintain a constant angular velocity (since there is a measure of resistance present over the span of the rotor).
Lets suppose that the moment of inertia from all sources of drag acting on the rotor, transmitted to the shaft at a speed of 600RPM is measured to be 200Nm. If the rotor is made of two blades then each blade will have a moment of 100Nm?
Each rotor blade measures 3m from the shaft centre point and has a mass of 80N.
Second, assuming that the moment of inertia remains 200Nm for all angular velocities, how long will it take to slow the rotor to 0 RPM?
Sorry if its vague, Im engaged in a purely academic design of a helicopter but have become lost in the rotating physics!
2. May 16, 2012
### Lsos
"Moment of inertia" is essentially how hard it is to accelerate or decelerate the rotor. When you say "moment of inertia from drag is 200Nm" you're thinking of the torque from drag. That said if the total torque on the shaft is 200 Nm then indeed each blade will contribute 100 Nm.
In order to answer how long it will take to slow the rotor to 0 rpm, THIS is where the "moment of inertia" comes in. This depends on the shape of the rotor, but your rotor seems simple enough that you could estimate a reasonable value by some calculations (you need to google the formulas for these). Once you have torque and moment of inertia, how fast the rotational speed changes is a matter of another simple formula (again a google search away).
3. May 16, 2012
### Alwyn Hartman
Wow, thats probly the most useless piece of advice anyone gives everyone nowadays.
The whole point of coming on this forum is NOT to just be redirected away to another search query. Perhaps you may not have thought of this but, maybe I would like to interact with a real person, not just some static page I cant ask questions to. Why dont we just all get our university degrees from google, the google university... preposterous.
4. May 16, 2012
### HallsofIvy
What's preposterous is you insulting some one who is trying to help you.
5. May 16, 2012
### Staff: Mentor
Thread locked. Try again, with some humility, appreciation and respect for someone who is only trying to help -- and doing a good job of it. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.822178065776825, "perplexity": 934.938517116819}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676596542.97/warc/CC-MAIN-20180723145409-20180723165409-00364.warc.gz"} |
https://ask.allseenalliance.org/answers/1757/revisions/ | # Revision history [back]
It looks like on your system you have not setup the appropriate commands that need to be run when onboarding the device. IE you need to enter the correct commands to be executed inside the OnboardingService.conf file:
...
<scan_file>/tmp/wifi_scan_results</scan_file>
<error_file>/tmp/state/alljoyn-onboarding-lasterror</error_file>
<state_file>/tmp/state/alljoyn-onboarding</state_file>
<connect_cmd>/etc/init.d/alljoyn-onboarding connect</connect_cmd>
<offboard_cmd>/etc/init.d/alljoyn-onboarding offboard</offboard_cmd>
<configure_cmd>/etc/init.d/alljoyn-onboarding configure %s %s %s</configure_cmd>
<scan_cmd>/usr/sbin/wifi_scan</scan_cmd>
...
Once you have these commands correct, then you should be able to have the experience you want of the device disconnecting and connecting to the supplied AP credentials. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9356457591056824, "perplexity": 2466.865763861446}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550249569386.95/warc/CC-MAIN-20190224003630-20190224025630-00313.warc.gz"} |
https://math.stackexchange.com/questions/1011685/i-need-help-with-a-proof-invertibility-of-b-lambda-in-b-iff-b-lambda-i | # I need help with a proof: invertibility of $b-\lambda$ in $B$ iff $b-\lambda$ invertible in $A$
Let $A$ be a unital $C^\ast$ algebra and let $B$ be a $\ast$ subalgebra such that $B \oplus \mathbb C = A$ and such that the unit in $B$, $1_B$, is not equal to the unit in $A$. I am trying to show:
If $\lambda \in \mathbb C$ is non-zero then $b-\lambda\cdot 1_B$ is invertible in $B$ if and only if $b-\lambda \cdot 1_A$ is invertible in $A$.
(see Murphy's book at the top of page 45).
I started the proof like this: Let $b-\lambda\cdot 1_B$ be invertible in $B$. Then it is also invertible in $A$. Let $a \in A$ denote its inverse. Then
$$ba - \lambda 1_B a = 1_A$$
Now the goal is to find $c \in A$ such that $(b -\lambda 1_A)c = 1_A$. Somehow I have to show that $1_B a = 1_A c$ but I can't seem to do it. So this leads nowhere.
Can someone help me prove this please?
• Assuming you mean $b = b 1_B \in B$, then \begin{align} (b-\lambda 1_B)c = 1_B & \Leftrightarrow (c \oplus (- \lambda)^{-1})\cdot (b-\lambda 1_B) \oplus (- \lambda) = 1_A. \end{align} – Michael Nov 8 '14 at 15:26
Assume $(b-\lambda 1_B)c=1_B$.
When we use $1_A=(0,1)$, $$[(b, 0) - \lambda (0,1)](x,t)=(b,-\lambda)(x,t)=\left(bx-\lambda x+tb, -\lambda t\right).$$ Looking at the second coordinate, $t=-1/\lambda$. Then the equality in the first coordinate becomes $$0=bx-\lambda x -\frac1\lambda b=(b-\lambda 1_B)x-\frac1\lambda b,$$ So $x=\frac1\lambda cb=\frac1\lambda 1_B+c$. That is, in $A$ we have $(b-\lambda 1_A)^{-1}=\frac1\lambda cb=\frac1\lambda 1_B+c-\frac1\lambda 1_A$.
The computations above also allow us to do the converse: if $b-\lambda 1_A$ is invertible in $A$, then $x$ above exists. Now take $c=x-\frac1\lambda 1_B$, and a straightforward calculation shows that $(b-\lambda 1_B)c=1_B$.
• Why can you use $(0,1) = 1_A$? Is the argument that every unital $C^\ast$-algebra is isomorphic to its unitisation? – user167889 Nov 10 '14 at 0:46
• It's not about the unitization. In a unital C$^*$-algebra, the unit is identified with the complex number $1$. So if $A=B+\mathbb C$ with $B$ non-unital, $1_A$ is necessarily $0+1$. – Martin Argerami Nov 10 '14 at 1:20
• How do I see this? Is it because a unital $C^\ast$-algebra contains $\mathbb C$ as a subalgebra? (but wouldn't then the unit in any commutative unital Banach algebra be identified with $1 \in \mathbb C$?) – user167889 Nov 10 '14 at 1:34
• When I said "non-unital" I meant "without the unit of $A$" – Martin Argerami Nov 21 '14 at 3:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9851071834564209, "perplexity": 163.61122344456476}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371675859.64/warc/CC-MAIN-20200407054138-20200407084638-00320.warc.gz"} |
https://www.physicsforums.com/threads/help-binomial-theorem.80995/ | # Help! Binomial Theorem
1. Jul 2, 2005
### ms. confused
Find the term containing $$x^{20}$$ in $$(2x - x^4)^{14}$$.
I went $$t_{k+1}= _{14}C_{k}(2x)^{14-k}(-x^{4})^{k}$$
$$= 2x^{14-k}(-x^{4k})$$
First of all, am I on the right track? If so what exactly do I do from there?
2. Jul 2, 2005
### AKG
Something is definitely wrong. First of all, your binomial coefficient has disappeard on the second line. Secondly, you seem to pull the 2 out of (2x)14-k in an improper way. You also can't bring the k inside the bracket like that, since it's not ((-x)4)k, it's (-x[sup4[/sup])k. Do you see the difference? The exponent 4 applies only to the x, not to (-x), but the exponent k applies to the whole (-x4). You should end up with:
$$t_{k+1} = {{14}\choose k}(-2)^{14-k}x^{14 + 3k}$$
See if you can figure out why the above is right. Now, to find the term containing x20, find the value(s) of k that satisfy 14 + 3k = 20. The only value for k is 2. So plug in 2 for k, you'll get:
(91)(212)x20
3. Jul 2, 2005
### ms. confused
Okay I think I get your point about the $$-x^4$$ thing. But how did you get the 2x to turn into -2 and how did you get the 14+3k exponent?
4. Jul 2, 2005
### lurflurf
You want the x^20 term so if the exponent of x in the kth term is
4k+(14-k)=20 what is k?
5. Jul 3, 2005
### ms. confused
+3k... right...thanks I see it now!
6. Jul 3, 2005
### manjish
Take (x^14) common and find the third term in the expansion of (2-(x^3))^14 i.e.
14C2*(2^12)*(-1)^2.
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46 results | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8179265856742859, "perplexity": 4546.190117082274}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1387345768787/warc/CC-MAIN-20131218054928-00060-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://www.maa.org/publications/maa-reviews/problems-in-real-analysis-advanced-calculus-on-the-real-axis?device=desktop | # Problems in Real Analysis: Advanced Calculus on the Real Axis
###### Theodora-Liliana T. Rādulescu, V. D. Rādulescu, and Titu Andreescu
Publisher:
Springer
Publication Date:
2009
Number of Pages:
452
Format:
Paperback
Price:
79.95
ISBN:
9780387773780
Category:
Problem Book
[Reviewed by
Mehdi Hassani
, on
08/24/2009
]
The book under review, as its title shows, essentially is a problem book in real analysis, chosen mostly from mathematical Olympiads and from problem journals. It also contains some nice counterexamples and proofs of some well-known theorems.
Every chapter of the book starts with a review of basic concepts and key results, often with no proof and no additional explanation. Then begin the problems with solutions. They are tricky elementary problems, and the expanded solutions teach the reader how to solve such problems. Each chapter concludes with a number of further problems.
The book focuses on analysis on the real line, which is also known as advanced real calculus. It covers the normal topics of elementary analysis. In particular, there are good accounts of infinite series and products and of inequalities, both of which are important topics for anyone doing mathematics. The book provides a very good and rather complete collection of the main results and tools in analysis and advanced calculus, which is fine for the people looking for tools to apply and do not need detailed proofs, such as students preparing for Olympiads or engineers.
Since the book doesn’t cover analysis concepts on abstract spaces, it is not usable as a text in an undergraduate course that does analysis on metric spaces. But, problems and counterexamples are very useful in any course in Mathematical Analysis, so the book can be used as a supplement. Moreover, some of them seem very fine for courses in calculus and even, if some cases, for high school classes.
The book includes some brief and useful historical notes, but the reader should (as always) be ready to check them; I think it is necessary here to mention some points:
• A well-known function is introduced as the Riemann function at the page 152 (and also page 327), and then as the Dirichlet function at the page 198.
• On page 186 we read “In fact, the first example of a continuous function that is nowhere differentiable is due to Weierstrass (1872), who shocked the mathematical world by showing that …” In actual fact, the first example of such functions is due to Bernard Bolzano around 1830 (but it was only published in 1922, after being rediscovered a few years earlier).
Moreover, the book contains a number of typos, some of which are:
• The graphs of the functions in Figures 5.2 and 5.3 (illustrating the concepts of “cusp point” and “vertical tangent”) need to be exchanged.
• On page 263, the first example for the convex functions is not introduced completely.
• At the end of page 314 we read “We end this preliminary section by listing a number of useful integration formulas.” But no list follows.
Typos aside, however, the book under review is a collection of interesting and fresh problems with detailed solutions. The target audience seems to be students preparing for Olympiads and other competitions, but undergraduate students, mathematics teachers and professors of Mathematical Analysis and Calculus courses may also find interesting things here.
Mehdi Hassani is a co-tutelle Ph.D. student in Mathematics in the Institute for Advanced Studies in Basic Science in Zanjan, Iran, and the Université de Bordeaux I, under supervision of the professors M.M. Shahshahani and J-M. Deshouillers.
Part I: Sequences, Series, and Limits of Functions.- Sequences.- Series.- Limits of Functions.- Part II: Qualitative Properties of Continuous and Differentiable Functions.- Continuity.- Differentiability.- Part III: Applications to Convex Functions and Optimizatin.- Convex Functions.- Inequalities and Extremum Problems.- Part IV: Antiderivatives, Riemann Integrability, and Applications.- Antiderivatives.- Riemann Integrability.- Applications of the Integral Calculus.- Appendix A: Basic Elements of Set Theory.- Appendix B: Topology of the Real Line.- Glossary.- References.- Index. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8048123717308044, "perplexity": 879.2752679273295}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510267075.55/warc/CC-MAIN-20140728011747-00332-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://labs.tib.eu/arxiv/?author=Antonio%20Cammi | • ### A new model with Serpent for the first criticality benchmarks of the TRIGA Mark II reactor(1707.05194)
We present a new model, developed with the Serpent Monte Carlo code, for neutronics simulation of the TRIGA Mark II reactor of Pavia (Italy). The complete 3D geometry of the reactor core is implemented with high accuracy and detail, exploiting all the available information about geometry and materials. The Serpent model of the reactor is validated in the fresh fuel configuration, through a benchmark analysis of the first criticality experiments and control rods calibrations. The accuracy of simulations in reproducing the reactivity difference between the low power (10 W) and full power (250 kW) reactor condition is also tested. Finally, a direct comparison between Serpent and MCNP simulations of the same reactor configurations is presented.
• ### Fuel burnup analysis of the TRIGA Mark II Reactor at the University of Pavia(1511.06274)
Nov. 18, 2015 physics.ins-det
A time evolution model was developed to study fuel burnup for the TRIGA Mark II reactor at the University of Pavia. The results were used to predict the effects of a complete core reconfiguration and the accuracy of this prediction was tested experimentally. We used the Monte Carlo code MCNP5 to reproduce system neutronics in different operating conditions and to analyse neutron fluxes in the reactor core. The software that took care of time evolution, completely designed in-house, used the neutron fluxes obtained by MCNP5 to evaluate fuel consumption. This software was developed specifically to keep into account some features that differentiate experimental reactors from power ones, such as the daily ON/OFF cycle and the long fuel lifetime. These effects can not be neglected to properly account for neutron poison accumulation. We evaluated the effect of 48 years of reactor operation and predicted a possible new configuration for the reactor core: the objective was to remove some of the fuel elements from the core and to obtain a substantial increase in the Core Excess reactivity value. The evaluation of fuel burnup and the reconfiguration results are presented in this paper.
• The Jiangmen Underground Neutrino Observatory (JUNO), a 20 kton multi-purpose underground liquid scintillator detector, was proposed with the determination of the neutrino mass hierarchy as a primary physics goal. It is also capable of observing neutrinos from terrestrial and extra-terrestrial sources, including supernova burst neutrinos, diffuse supernova neutrino background, geoneutrinos, atmospheric neutrinos, solar neutrinos, as well as exotic searches such as nucleon decays, dark matter, sterile neutrinos, etc. We present the physics motivations and the anticipated performance of the JUNO detector for various proposed measurements. By detecting reactor antineutrinos from two power plants at 53-km distance, JUNO will determine the neutrino mass hierarchy at a 3-4 sigma significance with six years of running. The measurement of antineutrino spectrum will also lead to the precise determination of three out of the six oscillation parameters to an accuracy of better than 1\%. Neutrino burst from a typical core-collapse supernova at 10 kpc would lead to ~5000 inverse-beta-decay events and ~2000 all-flavor neutrino-proton elastic scattering events in JUNO. Detection of DSNB would provide valuable information on the cosmic star-formation rate and the average core-collapsed neutrino energy spectrum. Geo-neutrinos can be detected in JUNO with a rate of ~400 events per year, significantly improving the statistics of existing geoneutrino samples. The JUNO detector is sensitive to several exotic searches, e.g. proton decay via the $p\to K^++\bar\nu$ decay channel. The JUNO detector will provide a unique facility to address many outstanding crucial questions in particle and astrophysics. It holds the great potential for further advancing our quest to understanding the fundamental properties of neutrinos, one of the building blocks of our Universe.
• ### Characterization of the TRIGA Mark II reactor full-power steady state(1503.00873)
March 3, 2015 nucl-ex, physics.ins-det
In this work, the characterization of the full-power steady state of the TRIGA Mark II nuclear reactor of the University of Pavia is performed by coupling Monte Carlo (MC) simulation for neutronics with "Multiphysics" model for thermal-hydraulics. Neutronic analyses have been performed starting from a MC model of the entire reactor system, based on the MCNP5 code, that was already validated in fresh fuel and zero-power configuration (in which thermal effects are negligible) using the available experimental data as benchmark. In order to describe the full-power reactor configuration, the temperature distribution in the core is necessary. To evaluate it, a thermal-hydraulic model has been developed, using the power distribution results from MC simulation as input. The thermal-hydraulic model is focused on the core active region and takes into account sub-cooled boiling effects present at full reactor power. The obtained temperature distribution is then introduced in the MC model and a benchmark analysis is carried out to validate the model in fresh fuel and full-power configuration. The good agreement between experimental data and simulation results concerning full-power reactor criticality, proves the reliability of the adopted methodology of analysis, both from neutronics and thermal-hydraulics perspective. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8340448141098022, "perplexity": 2087.9316601956994}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250592261.1/warc/CC-MAIN-20200118052321-20200118080321-00016.warc.gz"} |
http://platform.sysmoltd.com/ThermoFluids/PowerGenerationCycles?model=RegenerativeRankineCycle | # Rankine cycle
Basic Rankine cycle used in power generation
# Regenerative Rankine cycle
Rankine cycle with recuperator, using the temperature of the hot steam before the condenser to pre-heat the fluid before entering the boiler
# Heat engine cycles
Heat engines produce useful work by allowing heat to move from a hot reservoir to a cold reservoir. Examples of heat engines are the steam and gas turbines in power plants and internal combustion engines powering vehicles. The cold reservoir is usually the environment, so the efficiency of the cycle is measured as the fraction of the heat coming from the hot reservoir converted to work:
\begin{equation*} \eta = \frac{W}{Q_H} \end{equation*}
If the hot and cold reservoirs have well-defined temperatures, then the maximum efficiency that can be achieved is given by Carnot's efficiency:
\begin{equation*} \eta_c = 1 - \frac{T_C}{T_H} \end{equation*}
Provided that we operate between two such fixed temperature reservoirs, a measure of how good the cycle design is, is the second law efficiency:
\begin{equation*} \eta_{2nd} = \frac {\eta} {\eta_c} \end{equation*}
# Rankine cycle
Rankine cycle
Rankine cycle is used to produce power in virtually all coal power plants and other plants using steam turbines to power generators. It consists of four steps:
1. The cold liquid water is pumped to high pressure using a liquid pump. Because liquids are nearly incompressible, the work required for the pumping is rather small
2. The water enters a boiler where it is heated (by burning coal, etc.). The water evaporates and the steam is typically superheated (both processes occuring at constant pressure) so that the steam is well away from the two-phase region (dry steam)
3. The hot, high-pressure steam is expanded through a turbine, isentropically (in an ideal cycle), and part of the steam enthalpy is converted to work. At the outlet, wet steam (two-phase mixture, mostly steam) at low pressure exits the turbine.
4. The wet steam enters the condenser, gives off heat to the environment and is reliquefied before being pumped again.
# Regenerative Rankine cycle
Regenerative Rankine cycle
In this variation of the Rankine cycle, a counterflow heat exchanger (recuperator) is placed between:
• the pump and the boiler, on one hand,
• the turbine and the condenser, on the other hand
The steam exiting the turbine is still rather hot. Some of its enthalpy can be used to pre-heat the water entering the boiler, before it is wasted by rejecting it to the surroundings. This reduces the amount of heat necessary to produce the steam, and therefore increases the efficiency of the cycle. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8291228413581848, "perplexity": 1662.3663970746495}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267164925.98/warc/CC-MAIN-20180926121205-20180926141605-00436.warc.gz"} |
http://math.stackexchange.com/questions/100178/how-to-find-a-basis-if-you-know-something-about-the-columns/100180 | # How to find a basis if you know something about the columns?
My problem says: "In the matrix $A \in M_{3\times3}$ the columns $c_1, c_2$ are linearly independent and $c_3=c_1+c_2$. Determine a basis in the $\operatorname{Null}(A)$."
Does this have an unique approach? I would solve this by choosing two independent vectors in $R^3$ and then computing the $c_3$. Is it the right approach?
-
What is $Null(A)$? – Rasmus Jan 18 '12 at 16:53
@Rasmus Null space of A – Andrew Jan 18 '12 at 16:53
Well, how about computing the nullspace of $A$? An element of the nullspace is a vector $x = (x_1, x_2, x_3)$ such that $Ax = 0$. What is $Ax$? It's $x_1 c_1 + x_2 c_2 + x_3 c_3 = (x_1 + x_3) c_1 + (x_2 + x_3) c_2$. This means we have an element of the nullspace if and only if $x_1 + x_3 = x_2 + x_3 = 0$, which means $x_3$ is arbitrary and $x_1 = x_2 = -x_3$. There you go! A basis of the nullspace would be the vector $(1,1,-1)$.
Hope that helps!
-
A basis for $\mathrm{Null}(A)$ is just $(1, 1, -1) \in \mathbb{R}^3$. Because:
$$A \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix} = c_1 + c_2 - c_3 = 0 \ .$$
Hence, $(1, 1, -1) \in \mathrm{Null}(A)$.
On the other hand,
$$\mathrm{dim}\ \mathrm{Null}(A) = 3 - \mathrm{rank}\ A = 3 -2 = 1 \ ,$$
because $A$ has just two columns which are linearly independent. So
$$\mathrm{span}\ \left\{ (1,1,-1) \right\} = \mathrm{Null}(A) \ .$$
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9340922832489014, "perplexity": 210.35453969751242}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500823634.2/warc/CC-MAIN-20140820021343-00035-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/simple-question-on-curvature.159762/ | # Simple question on curvature
1. Mar 8, 2007
### MeJennifer
Consider a photon emitted at space-time event E1 and absorbed at space-time event E2 in curved space-time.
Since the arc length of the worldline between both events is 0 how can we, with validity, claim that such a path is curved in space-time?
Does it not seem to be more correct to claim that in curved space-time all geodesic worldlines are curved except for null geodesics?
Were is the error in my reasoning?
Last edited: Mar 8, 2007
2. Mar 8, 2007
### cesiumfrog
A geodesic is the closest concept (in general, differential geometry) to a "straight line". When paths are described as "curved", this is purely an illustrative term, attempting to convey an appropriate image to people with less experience of non-Euclidean geometry.
I think you've just (mis)taken something too seriously/literally, but if you need to discuss it further then I think you ought to state who makes the claim you mention, what context they wrote in, what definition they chose for "curved", and why you think null geodesics should be called less "curved" than other geodesics.
Last edited: Mar 8, 2007
3. Mar 8, 2007
### MeJennifer
A nul geodesic has an arc length of zero, how do you propose to define curvature for something that has zero length?
4. Mar 8, 2007
### cesiumfrog
I believe this is covered somewhere in the first couple chapters of MWT.. though out of curiousity, how would you categorise a path that has null length (over any segment) and is not a geodesic?
5. Mar 8, 2007
### Mentz114
MJ, the proper length is 0, but the spatial separation and temporal separation are not both 0, just equal in magnitude and opposite in sign.
6. Mar 8, 2007
### MeJennifer
Right, but here we are dealing with events in curved space-time not some Lorentz variant slicing of space and time for a particular observer. This seems like simply avoiding rather than adressing the issue. Clearly only space-time vectors are invariant under Lorentz transformations.
Good question, if I am not mistaken even Hawking and Ellis in "The Large Scale Structure of Space-Time" seem to avoid this particular, perhaps non-physical, case.
Last edited: Mar 8, 2007
7. Mar 8, 2007
### Mentz114
Why not ? If it's a path it can be curved.
According to the adopted criterion for curvature, null geodesics can be curved.
You probably need to redefine curvature to satisfy this aesthetic urge.
I don't think it's an issue. Just nomeclature.
8. Mar 8, 2007
### pmb_phy
Since the arc length of the worldline between both events is 0 how can we, with validity, claim that such a path is curved in space-time?[/quote]Always with the great questions MJ.
The length between the two events is a bad word to use in GR. Its the spacetime interval that is zero. In differential geometry one uses a distance function, which is a different function from the metric tensor. One uses this distance function to define closed sets, whether two points are "nearby" etc. For this reason also one can tell whether two points are close or far apart.
Another point is that curvature cannot be determined by a single geodesic. One has to have more than one geodesic to determine if spacetime curvature is lurking somewhere nearby. That's about the best I can explain it at this time (i.e. off the top of my head).
Good luck
Pete
9. Mar 8, 2007
### MeJennifer
I am happy to use any term that you like pmb_phy, but to me it seems that that does not explain anything.
So if the interval is zero, then how can this "path" be curved?
Yes, and so in my example of two space-time events of an emitted and absorbed photon, how far are those events apart in your opinion?
The distance between two events in space-time is Lorentz invariant right?
The arc length of the space-time path of an observer in space-time represents his proper time right?
The arc length of the space-time path between an emited and absorbed photon in space-time is zero right?
So how could some possibly claim that in curved space-time such a null geodesic is curved?
Obviously timelike geodesics are curved in curved spacetime since they have an arc length that is non zero.
Last edited: Mar 8, 2007
10. Mar 8, 2007
### nrqed
One cannot say anything about the curvature of a manifold using only a single geodesic. This is clear even if you look at something much simpler, let's say the surface S_2 of a sphere embedded in ordinary 3-dimensional space (no problem with negative terms in the metric there!!) .
what you are asking is the same as if someone would say :look, a geodesic from one pole to the other pole has a length of $2 \pi R$. How can one conclude that this space is curved?
WHat would you answer to that question? Don't you agree that the geodesics are clearly curved in that case? Do you see that you are asking exactly the same question (it does not matter that in your question the spacetime interval is zero as opposed to a finite value in my example, the point is the same in both cases: giving a number for a single geodesic tells us nothing about curvature).
The answer is of course that to know about the curvature of a manifold, one must have more information than the "lenght" (or spacetime interval) of a single geodesic or curve. One possible way to learn about the curvature is to go through a closed path following 3 geodesics or more.
11. Mar 8, 2007
### nrqed
So answer my even simpler question: on S_2, how can *you* say that a geodesic is curved? Can you answer that question? If not, maybe we should start by discussing *that* much simpler example before tackling a non-positive definite metric!
12. Mar 8, 2007
### MeJennifer
Simple, geodesics follow the curvature of space-time so if space-time is curved then obviously the geodesics follow that curvature as well. But a path that has an arc length of zero, what is it supposed to follow that is curved?
I think you know what I am talking about, but it seems you rather want to avoid this discussion. Perhaps because you are confused by it too?
Note: something when wrong in posting my prior message, but nrqed quoted the gist of it.
Last edited: Mar 8, 2007
13. Mar 8, 2007
### pmb_phy
A path being curved doesn't have a lot of meaning to me. If the path is a geodesic in embedded in a 2-D surface then the geodesics (of which there are many) will form a helix. This curve is the "straightest possible" curve. In fact this surface has no intrinsic curvature so the path can't be said to be curved in the Riemann sense (as in embedded).
Sorry but when speaking in terms of GR the term "far apart" has no meaning to me.
Yes.
If you mean "spacetime interval" instead of "arc length" then you're right. Otherwise you'll have to define "arc length" for me.
What is arc-length?
[quote[
So how could some possibly claim that in curved space-time such a null geodesic is curved?
[/quote]Got me. I never said nor thought of them as being of such.
Perhaps its obvious to you, but not to me.
Pete
14. Mar 8, 2007
### robphy
As suggested by some of the earlier posts,...
there are two different notions of curvature implied by your question:
worldline-curvature (geodesic curvature, 4-acceleration) and spacetime-curvature (Riemann curvature).
The Riemann curvature of a spacetime determines the deviation from flatness, which also determines which spacetime paths are geodesic and which are not.
The geodesic curvature of a spacetime path determines the deviation of that path from being geodesic. [A noninertial observer [with nonzero 4-acceleration] has a nongeodesic timelike path.] (This should not be confused with "geodesic deviation", which describes the behavior of a family of nearby geodesics.)
So, given a spacetime [with or without spacetime-curvature], its geodesics are those paths with zero worldline-curvature. What is important for a geodesic is that its tangent-vector is parallel-transported... When there is a metric available, the norm of this tangent-vector is used to classify the type of geodesic (spacelike, timelike, or null)... and can be used to characterize geodesics in terms of extremal (or stationary) values of arc-lengths.
possibly useful:
http://ocw.mit.edu/NR/rdonlyres/Physics/8-962Spring2002/940ABFC4-1A26-483C-992A-DDCD48D1F272/0/gr2_2.pdf [Broken]
Last edited by a moderator: May 2, 2017
15. Mar 8, 2007
### pmb_phy
So non-geodesics have curvature? That makes sense now that I think of it. I didn't know how to phrase that. Thanks Rob.
Pete
16. Mar 8, 2007
### robphy
Non-geodesic paths have nonzero worldline-curvature (somewhere along the path).
17. Mar 8, 2007
### masudr
We use an affine parameter. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8654025197029114, "perplexity": 984.2363145121446}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794867046.34/warc/CC-MAIN-20180525063346-20180525083346-00240.warc.gz"} |
http://lt-jds.jinr.ru/record/29835?ln=en | / Experiment-HEP arXiv:0802.2400
Measurement of the inclusive jet cross-section in p anti-p collisions at s**91/2) =1.96-TeV
Published in: Phys.Rev.Lett.
Year: 2008
Vol.: 101
Page No: 062001
Pages: 7
Abstract: We report on a measurement of the inclusive jet cross section in $p \bar{p}$ collisions at a center-of-mass energy $\sqrt s=$1.96 TeV using data collected by the D0 experiment at the Fermilab Tevatron Collider corresponding to an integrated luminosity of 0.70 fb$^{-1}$. The data cover jet transverse momenta from 50 GeV to 600 GeV and jet rapidities in the range -2.4 to 2.4. Detailed studies of correlations between systematic uncertainties in transverse momentum and rapidity are presented, and the cross section measurements are found to be in good agreement with next-to-leading order QCD calculations.
Total numbers of views: 3326
Numbers of unique views: 949
e-proceeding | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9912299513816833, "perplexity": 4597.668150701691}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550249550830.96/warc/CC-MAIN-20190223203317-20190223225317-00516.warc.gz"} |
http://paperity.org/p/124069538/updating-standard-solar-models | # Updating standard solar models
Astronomy and Astrophysics Supplement Series, Jul 2018
We present an updated version of our standard solar model (SSM) where helium and heavy elements diffusion is included and the improved OPAL equation of state (Rogers 1994; Rogers et al. 1996) is used. In such a way the EOS is consistent with the adopted opacity tables, from the same Livermore group, an occurrence which should further enhance the reliability of the model. The results for the physical characteristics and the neutrino production of our SSM are discussed and compared with previous works on the matter.
This is a preview of a remote PDF: https://aas.aanda.org/articles/aas/pdf/1997/09/ds5369.pdf
F. Ciacio, S. Degl'Innocenti, B. Ricci. Updating standard solar models, Astronomy and Astrophysics Supplement Series, 449-454, DOI: 10.1051/aas:1997168 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8382987380027771, "perplexity": 2266.1672023014758}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221212598.67/warc/CC-MAIN-20180817143416-20180817163416-00409.warc.gz"} |
https://me.gateoverflow.in/805/gate2017-me-2-24 | The standard deviation of linear dimensions $P$ and $Q$ are $3 \mu$ m and $4 \mu$ m, respectively. When assembled, the standard deviation (in $\mu$ m) of the resulting linear dimension $(P+Q)$ is _________. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9694046378135681, "perplexity": 331.33294872235734}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710685.0/warc/CC-MAIN-20221129031912-20221129061912-00101.warc.gz"} |
https://www.computing.net/answers/programming/bat-script-to-rename-files-in-child-folder/23774.html | # .Bat script to rename files in child folder.
January 4, 2011 at 06:57:57
Specs: Windows 7
I wanted to develop a .bat script that changes the type of a file if it exist. The problem is the movies are 2 levels deep and I need help entering the next folder. My structure is set up ass follows. Movies\Forest Gump\ForestGump.mk4 where I would like to run the script at the Movies Level and rename "ForestGump.mk4" to "ForestGump.mp4" only if the file is a type of .mk4. Any insight???
See More: .Bat script to rename files in child folder.
#1
January 4, 2011 at 07:18:33
```@echo off setLocal EnableDELAYedeXpansion for /f "tokens=*" %%a in ('dir C:\movies\*. /ad /b /s') do ( cd\"%%a" ren *.mk4 *.mp4 ) ```Note: this is not tested. I will test it and get back to you. I just wrote it off the top of my head.
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#2
January 4, 2011 at 07:51:58
Note the changes...```@echo off setLocal EnableDELAYedeXpansion for /f "tokens=*" %%a in ('dir C:\movies\ /ad /b /s') do ( ren "%%a\*.mk4" "*.mp4" ) ```Also this is assuming the path to MOVIES is c:\movies. You may need to change it to include the full path. It may be c:\users\yournamehere\movies\ If you are using Windows Vista or newer.
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#3
January 4, 2011 at 07:59:56
Thanks for the quick reply. However it is not working. Trying to debug but havent been successful thus far.
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#4
January 4, 2011 at 13:23:22
Hmmm, it worked for me. Again you must specify the exact directory the files are located in. What is it?"Movies\Forest Gump\" will not do. Is it in the C: drive if so what is the root directory.
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#5
January 4, 2011 at 14:16:43
The exact directory is "C:\Videos\Movies\" I just replace ('dir C:\movies\ /ad /b /s')with ('dir C:\Videos\Movies\ /ad /b /s').........right???? Also were do I place the .bat file. I place it in the Movies directory right.
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#6
January 4, 2011 at 14:30:57
('dir C:\Videos\Movies\ /ad /b /s') -> Correct.It does not matter where you run the batch file from.You will get errors on directories that have no MK4 files but the ones that do will rename the files. I ran the following code on my computer's music because I don't have the movies you have; and it worked...```@echo off setLocal EnableDELAYedeXpansion for /f "tokens=*" %%a in ('dir C:\users\jflagg\music\ /ad /b /s') do ( ren "%%a\*.mp3" "*.mp4" echo ren "%%a\*.mp3" "*.mp4 pause ) ```Why it is not working for you, I have no idea.
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#7
January 4, 2011 at 15:29:41
I ran your script also ace_omega and it works great, nice job. Maybe if the OP gave a little bit more information about the errors he is getting, or a little bit more information about directories we can get it worked out.Can I put a question here?
Report • | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8017866611480713, "perplexity": 2229.149782465806}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583510932.57/warc/CC-MAIN-20181017002502-20181017024002-00440.warc.gz"} |
http://link.springer.com/article/10.1007%2FBF01654285 | Article
Communications in Mathematical Physics
, Volume 9, Issue 4, pp 313-326
First online:
# Conformal tensor discontinuities in general relativity
• L. C. ShepleyAffiliated withUniversity of Texas
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## Abstract
The postulate is made that across a given hypersurfaceN the metric and its first derivatives are continuous. This postulate is used to derive conditions which must be satisfied by discontinuities in the Riemann tensor acrossN. These conditions imply that the conformal tensor jump is uniquely determined by the stress-energy tensor discontinuity ifN is non-null (and to within an additive term of type Null ifN is lightlike). Alternatively,$$[C^{\alpha \beta } _{\gamma \delta } ]$$ and [R] determine$$\left[ {R_{\mu v} - \frac{1}{4}Rg_{\mu v} } \right]$$ ifN is non-null. These relationships between the conformal tensor and stress-energy tensor jumps are given explicitly in terms of a three-dimensional complex representation of the antisymmetric tensors. Application of these results to perfect-fluid discontinuities is made:$$[C^{\alpha \beta } _{\gamma \delta } ]$$ is of type D across a fluid-vacuum boundary and across an internal, non-null shock front.$$[C^{\alpha \beta } _{\gamma \delta } ]$$ is of type I (non-degenerate) in general across fluid interfaces across which no matter flows, except for special cases. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8722921013832092, "perplexity": 1753.6161869404793}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860121985.80/warc/CC-MAIN-20160428161521-00135-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://www.askmehelpdesk.com/heating-air-conditioning/thermostat-q-blower-does-not-turn-fan-auto-mode-37789.html | I just installed a NOMA 7-day programmable thermostat for an electric furnace/AC system (ceiling unit). I believe the wiring (5-wire) is correct.
When I set it to COOL with Fan set to AUTO, it turns on the A/C and the fan automatically. Good.
But when I set it to HEAT with Fan set to AUTO, it turns on the heating system but the fan doesn't. The fan does turn on when I force it by setting the switch to ON.
Any idea why? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8737948536872864, "perplexity": 2308.430090293091}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187822145.14/warc/CC-MAIN-20171017163022-20171017183022-00255.warc.gz"} |
https://www.maixuanviet.com/bayes-theorem-in-artificial-intelligence.vietmx | Bayes’ theorem in Artificial intelligence
1. Bayes’ theorem:
Bayes’ theorem is also known as Bayes’ rule, Bayes’ law, or Bayesian reasoning, which determines the probability of an event with uncertain knowledge.
In probability theory, it relates the conditional probability and marginal probabilities of two random events.
Bayes’ theorem was named after the British mathematician Thomas Bayes. The Bayesian inference is an application of Bayes’ theorem, which is fundamental to Bayesian statistics.
It is a way to calculate the value of P(B|A) with the knowledge of P(A|B).
Bayes’ theorem allows updating the probability prediction of an event by observing new information of the real world.
Example: If cancer corresponds to one’s age then by using Bayes’ theorem, we can determine the probability of cancer more accurately with the help of age.
Bayes’ theorem can be derived using product rule and conditional probability of event A with known event B:
As from product rule we can write:
P(A ⋀ B)= P(A|B) P(B) or
Similarly, the probability of event B with known event A:
P(A ⋀ B)= P(B|A) P(A)
Equating right hand side of both the equations, we will get:
The above equation (a) is called as Bayes’ rule or Bayes’ theorem. This equation is basic of most modern AI systems for probabilistic inference.
It shows the simple relationship between joint and conditional probabilities. Here,
P(A|B) is known as posterior, which we need to calculate, and it will be read as Probability of hypothesis A when we have occurred an evidence B.
P(B|A) is called the likelihood, in which we consider that hypothesis is true, then we calculate the probability of evidence.
P(A) is called the prior probability, probability of hypothesis before considering the evidence
P(B) is called marginal probability, pure probability of an evidence.
In the equation (a), in general, we can write P (B) = P(A)*P(B|Ai), hence the Bayes’ rule can be written as:
Where A1, A2, A3,…….., An is a set of mutually exclusive and exhaustive events.
2. Applying Bayes’ rule:
Bayes’ rule allows us to compute the single term P(B|A) in terms of P(A|B), P(B), and P(A). This is very useful in cases where we have a good probability of these three terms and want to determine the fourth one. Suppose we want to perceive the effect of some unknown cause, and want to compute that cause, then the Bayes’ rule becomes:
Example-1:
Question: what is the probability that a patient has diseases meningitis with a stiff neck?
Given Data:
A doctor is aware that disease meningitis causes a patient to have a stiff neck, and it occurs 80% of the time. He is also aware of some more facts, which are given as follows:
• The Known probability that a patient has meningitis disease is 1/30,000.
• The Known probability that a patient has a stiff neck is 2%.
Let a be the proposition that patient has stiff neck and b be the proposition that patient has meningitis. , so we can calculate the following as:
P(a|b) = 0.8
P(b) = 1/30000
P(a)= .02
Hence, we can assume that 1 patient out of 750 patients has meningitis disease with a stiff neck.
Example-2:
Question: From a standard deck of playing cards, a single card is drawn. The probability that the card is king is 4/52, then calculate posterior probability P(King|Face), which means the drawn face card is a king card.
Solution:
P(king): probability that the card is King= 4/52= 1/13
P(face): probability that a card is a face card= 3/13
P(Face|King): probability of face card when we assume it is a king = 1
Putting all values in equation (i) we will get:
3. Application of Bayes’ theorem in Artificial intelligence:
Following are some applications of Bayes’ theorem:
• It is used to calculate the next step of the robot when the already executed step is given.
• Bayes’ theorem is helpful in weather forecasting.
• It can solve the Monty Hall problem. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.940780758857727, "perplexity": 928.9546801472675}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780053493.41/warc/CC-MAIN-20210916094919-20210916124919-00681.warc.gz"} |
https://en.wikipedia.org/wiki/Muirhead%27s_inequality | In mathematics, Muirhead's inequality, named after Robert Franklin Muirhead, also known as the "bunching" method, generalizes the inequality of arithmetic and geometric means.
## Preliminary definitions
### The "a-mean"
For any real vector
$a=(a_1,\dots,a_n)$
define the "a-mean" [a] of nonnegative real numbers x1, ..., xn by
$[a]={1 \over n!}\sum_\sigma x_{\sigma_1}^{a_1}\cdots x_{\sigma_n}^{a_n},$
where the sum extends over all permutations σ of { 1, ..., n }.
In case a = (1, 0, ..., 0), this is just the ordinary arithmetic mean of x1, ..., xn. In case a = (1/n, ..., 1/n), it is the geometric mean of x1, ..., xn. (When n = 2, this is the Heinz mean.)
Notice that the "a"-mean as defined above only has the usual properties of a mean (e.g., if the mean of equal numbers is equal to them) if $a_1+\cdots+a_n=1$. In the general case, one can consider instead $[a]^{1/(a_1+\cdots+a_n)}$, which is called a Muirhead mean.[1]
### Doubly stochastic matrices
An n × n matrix P is doubly stochastic precisely if both P and its transpose PT are stochastic matrices. A stochastic matrix is a square matrix of nonnegative real entries in which the sum of the entries in each column is 1. Thus, a doubly stochastic matrix is a square matrix of nonnegative real entries in which the sum of the entries in each row and the sum of the entries in each column is 1.
## The inequality
Muirhead's inequality states that [a] ≤ [b] for all x such that xi ≥ 0 for all xi if and only if there is some doubly stochastic matrix P for which a = Pb.
The latter condition can be expressed in several equivalent ways; one of them is given below.
The proof makes use of the fact that every doubly stochastic matrix is a weighted average of permutation matrices (Birkhoff-von Neumann theorem).
### Another equivalent condition
Because of the symmetry of the sum, no generality is lost by sorting the exponents into decreasing order:
$a_1 \geq a_2 \geq \cdots \geq a_n$
$b_1 \geq b_2 \geq \cdots \geq b_n.$
Then the existence of a doubly stochastic matrix P such that a = Pb is equivalent to the following system of inequalities:
$a_1 \leq b_1$
$a_1+a_2 \leq b_1+b_2$
$a_1+a_2+a_3 \leq b_1+b_2+b_3$
$\qquad\vdots\qquad\vdots\qquad\vdots\qquad\vdots$
$a_1+\cdots +a_{n-1} \leq b_1+\cdots+b_{n-1}$
$a_1+\cdots +a_n=b_1+\cdots+b_n.$
(The last one is an equality; the others are weak inequalities.)
The sequence $b_1, \ldots, b_n$ is said to majorize the sequence $a_1, \ldots, a_n$.
## Symmetric sum-notation tricks
It is useful to use a kind of special notation for the sums. A success in reducing an inequality in this form means that the only condition for testing it is to verify whether one exponent sequence ($\alpha_1, \ldots, \alpha_n$) majorizes the other one.
$\sum_\text{sym} x_1^{\alpha_1} \cdots x_n^{\alpha_n}$
This notation requires developing every permutation, developing an expression made of n! monomials, for instance:
$\sum_\text{sym} x^3 y^2 z^0 = x^3 y^2 z^0 + x^3 z^2 y^0 + y^3 x^2 z^0 + y^3 z^2 x^0 + z^3 x^2 y^0 + z^3 y^2 x^0$
${} = x^3 y^2 + x^3 z^2 + y^3 x^2 + y^3 z^2 + z^3 x^2 + z^3 y^2$
## Deriving the arithmetic-geometric mean inequality
Let
$a_G = \left( \frac 1 n , \ldots , \frac 1 n \right)$
$a_A = ( 1 , 0, 0, \ldots , 0 )\,$
we have
$a_{A1} = 1 > a_{G1} = \frac 1 n \,$
$a_{A1} + a_{A2} = 1 > a_{G1} + a_{G2} = \frac 2 n\,$
$\qquad\vdots\qquad\vdots\qquad\vdots\,$
$a_{A1} + \cdots + a_{An} = a_{G1} + \cdots + a_{Gn} = 1 \,$
then
[aA] ≥ [aG]
which is
$\frac 1 {n!} (x_1^1 \cdot x_2^0 \cdots x_n^0 + \cdots + x_1^0 \cdots x_n^1) (n-1)! \geq \frac 1 {n!} (x_1 \cdot \cdots \cdot x_n)^{\frac 1 n} n!$
yielding the inequality.
## Examples
Suppose you want to prove that x2 + y2 ≥ 2xy by using bunching (Muirhead's inequality): We transform it in the symmetric-sum notation:
$\sum_ \mathrm{sym} x^2 y^0 \ge \sum_\mathrm{sym} x^1 y^1.\$
The sequence (2, 0) majorizes the sequence (1, 1), thus the inequality holds by bunching. Again,
$x^3+y^3+z^3 \ge 3 x y z$
$\sum_ \mathrm{sym} x^3 y^0 z^0 \ge \sum_\mathrm{sym} x^1 y^1 z^1$
which yields
$2 x^3 + 2 y^3 + 2 z^3 \ge 6 x y z$
the sequence (3, 0, 0) majorizes the sequence (1, 1, 1), thus the inequality holds by bunching.
## References
1. ^ Bullen, P. S. Handbook of means and their inequalities. Kluwer Academic Publishers Group, Dordrecht, 2003. ISBN 1-4020-1522-4 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 29, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9863805770874023, "perplexity": 409.5174696898673}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042989018.40/warc/CC-MAIN-20150728002309-00116-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://mathshistory.st-andrews.ac.uk/Biographies/Aristaeus/ | # Aristaeus the Elder
### Quick Info
Born
Greece
Died
Summary
Aristaeus the Elder was a Greek mathematician who worked on conic sections.
### Biography
Aristaeus the Elder was probably older than, but still a contemporary of, Euclid. We know practically nothing of his life except that Pappus refers to him as Aristaeus the Elder which presumably means that Pappus was aware of another later mathematician also named Aristaeus. We have no record of such a person but we do point out below a possible confusion which may result from there being two mathematicians called Aristaeus.
Pappus gave Aristaeus great credit for a work entitled Five Books concerning Solid Loci which was used by Pappus but has now been lost. 'Solid loci' is the Greek name for conic sections so it is rather confusing that there is another reference by a later writer to a work by Aristaeus called Five Books concerning Conic Sections. However these two works are now thought to be the same.
Pappus describes the work as:-
... five books of Solid Loci connected with the conics.
and also claims (if this is not a latter addition to the text) that Euclid compiled elementary results on conics in his treatise Conics while Aristaeus's results, much deeper, original and specialised, were not included by Euclid who preferred to leave them in their original presentation due to Aristaeus.
Heath makes a guess at the possible contents of the Solid Loci and writes [3]:-
A very large portion of the standard properties of conics admit of being stated in the form of locus theorems ... But it may be assumed that Aristaeus's work was not merely a collection of the ordinary propositions transformed in this way; it would deal with new locus theorems not implied in the fundamental definitions and properties of the conics, such as ... the theorems of the three- and four-line locus. But one (to us) ordinary property, the focus directrix property, was, as it seems to me, in all probability included.
Heath refers to theorems of the three- and four-line locus in the above quote and we should explain what these are. For the three line locus we are given a point $P$ and three directed lines $a, b$, and $c$ drawn to meet at given angles, three fixed straight lines. Then the locus of $P$ such that $ac : b^{2}$ is a given constant is a conic. The four-line locus is similar. We are given a point $P$ and four directed lines $a, b, c$, and $d$ drawn to meet at given angles, four fixed straight lines. Then the locus of $P$ such that $ac : bd$ is a given constant is a conic.
There is a reference to Aristaeus in the works of Hypsicles where he refers to Aristaeus as the author of a book Concerning the Comparison of Five Regular Solids. Heath believes that, although it is not certain whether this is Aristaeus the Elder, the results described make it quite probable that it is. Hypsicles tells us that, in this work, Aristaeus proved that [3]:-
... the same circle circumscribes both the pentagon of the dodecahedron and the triangle of the icosahedron inscribed in the same sphere.
Heath's opinion that the Aristaeus referred to by Hypsicles this is Aristaeus the Elder has been disputed by some historians, and there is a possibility that Hypsicles refers to Aristaeus the Younger thus making sense of Pappus's comments which we referred to in the first paragraph.
The work of both Aristaeus and Euclid on conics was, almost 200 years later, further developed by Apollonius. This work by Apollonius made the theory of conics as developed by Aristaeus and Euclid obsolete.
### References (show)
1. K Vogel, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
2. J L Coolidge, A history of the conic sections and quadric surfaces (Oxford, 1945).
3. T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 10, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8656750917434692, "perplexity": 1235.89128747791}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400192887.19/warc/CC-MAIN-20200919204805-20200919234805-00286.warc.gz"} |
https://www.hepdata.net/search/?q=&observables=V2&collaboration=ALICE&page=1 | Showing 12 of 12 results
#### J/$\psi$ elliptic flow in Pb-Pb collisions at $\mathbf{\sqrt{s_{\rm NN}}}$ = 5.02 TeV
The collaboration Acharya, Shreyasi ; Adamova, Dagmar ; Adolfsson, Jonatan ; et al.
No Journal Information, 2017.
Inspire Record 1623907
We report a precise measurement of the J/$\psi$ elliptic flow in Pb-Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$ TeV with the ALICE detector at the LHC. The J/$\psi$ mesons are reconstructed at mid-rapidity ($|y| < 0.9$) in the dielectron decay channel and at forward rapidity ($2.5<y<4.0$) in the dimuon channel, both down to zero transverse momentum. At forward rapidity, the elliptic flow $v_2$ of the J/$\psi$ is studied as a function of transverse momentum and centrality. A positive $v_2$ is observed in the transverse momentum range $2 < p_{\rm T} < 8$ GeV/$c$ in the three centrality classes studied and confirms with higher statistics our earlier results at $\sqrt{s_{\rm NN}} = 2.76$ TeV in semi-central collisions. At mid-rapidity, the J/$\psi$ $v_2$ is investigated as a function of transverse momentum in semi-central collisions and found to be in agreement with the measurements at forward rapidity. These results are compared to transport model calculations. The comparison supports the idea that at low $p_{\rm T}$ the elliptic flow of the J/$\psi$ originates from the thermalization of charm quarks in the deconfined medium, but suggests that additional mechanisms might be missing in the models.
4 data tables
Transverse momentum dependence of inclusive J/$\psi$ $v_2$ at $\sqrt{s_{\rm NN}}=5.02$ TeV for the 20-40% centrality class (forward rapidity). The first uncertainty (stat) is statistical, the second (sys,uncorrel) is the uncorrelated systematic, while the third one (sys,correl) is a $p_{\rm T}$-correlated systematic uncertainty.
Transverse momentum dependence of inclusive J/$\psi$ $v_2$ at $\sqrt{s_{\rm NN}}=5.02$ TeV for the 20-40% centrality class (mid-rapidity). The first uncertainty (stat) is statistical, the second (sys,uncorrel) is the uncorrelated systematic, while the third one (sys,correl) is a $p_{\rm T}$-correlated systematic uncertainty.
Transverse momentum dependence of inclusive J/$\psi$ $v_2$ at $\sqrt{s_{\rm NN}}=5.02$ TeV for the 5-20% centrality class (forward rapidity). The first uncertainty (stat) is statistical, the second (sys,uncorrel) is the uncorrelated systematic, while the third one (sys,correl) is a $p_{\rm T}$-correlated systematic uncertainty.
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#### D-meson azimuthal anisotropy in mid-central Pb-Pb collisions at $\mathbf{\sqrt{s_{\rm NN}}=5.02}$ TeV
The collaboration Acharya, Shreyasi ; Adamova, Dagmar ; Adolfsson, Jonatan ; et al.
No Journal Information, 2017.
Inspire Record 1608612
The azimuthal anisotropy coefficient $v_2$ of prompt D$^0$, D$^+$, D$^{*+}$ and D$_s^+$ mesons was measured in mid-central (30-50% centrality class) Pb-Pb collisions at a centre-of-mass energy per nucleon pair $\sqrt{s_{\rm NN}} = 5.02$ TeV, with the ALICE detector at the LHC. The D mesons were reconstructed via their hadronic decays at mid-rapidity, $|y|<0.8$, in the transverse momentum interval $1<p_{\rm T}<24$ GeV/$c$. The measured D-meson $v_2$ has similar values as that of charged pions. The D$_s^+$ $v_2$, measured for the first time, is found to be compatible with that of non-strange D mesons. The measurements are compared with theoretical calculations of charm-quark transport in a hydrodynamically expanding medium and have the potential to constrain medium parameters.
5 data tables
$v_2$ vs. $p_{\rm T}$ of $D^0$ mesons in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$=5.02 TeV in the centrality class 30-50% in the rapidity interval |$y$|<0.8. The second (sys) error is the systematic uncertainty from the B feed-down contribution. The first (sys) error is the systematic uncertainty from the other sources.
$v_2$ vs. $p_{\rm T}$ of $D^+$ mesons in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$=5.02 TeV in the centrality class 30-50% in the rapidity interval |$y$|<0.8. The second (sys) error is the systematic uncertainty from the B feed-down contribution. The first (sys) error is the systematic uncertainty from the other sources.
$v_2$ vs. $p_{\rm T}$ of $D^{*+}$ mesons in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$=5.02 TeV in the centrality class 30-50% in the rapidity interval |$y$|<0.8. The second (sys) error is the systematic uncertainty from the B feed-down contribution. The first (sys) error is the systematic uncertainty from the other sources.
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#### Flow dominance and factorization of transverse momentum correlations in Pb-Pb collisions at the LHC
The collaboration Adam, Jaroslav ; Adamova, Dagmar ; Aggarwal, Madan Mohan ; et al.
Phys.Rev.Lett. 118 (2017) 162302, 2017.
Inspire Record 1512772
We present the first measurement of the two-particle transverse momentum differential correlation function, P2≡⟨ΔpTΔpT⟩/⟨pT⟩2, in Pb-Pb collisions at sNN=2.76 TeV. Results for P2 are reported as a function of the relative pseudorapidity (Δη) and azimuthal angle (Δφ) between two particles for different collision centralities. The Δϕ dependence is found to be largely independent of Δη for |Δη|≥0.9. In the 5% most central Pb-Pb collisions, the two-particle transverse momentum correlation function exhibits a clear double-hump structure around Δφ=π (i.e., on the away side), which is not observed in number correlations in the same centrality range, and thus provides an indication of the dominance of triangular flow in this collision centrality. Fourier decompositions of P2, studied as a function of the collision centrality, show that correlations at |Δη|≥0.9 can be well reproduced by a flow ansatz based on the notion that measured transverse momentum correlations are strictly determined by the collective motion of the system.
19 data tables
Projection of $P_{2}$ along $\Delta\varphi$ in 0-5% centrality in the range $|\Delta \eta| \geq$ 0.9
$v_{2}$ coefficients measured from $P_2$ for particle pairs in the range $0.2 \leq |\Delta\eta| \leq 0.9$.
$v_{2}$ coefficients measured from $P_2$ for particle pairs in the range $0.9 \leq |\Delta\eta| \leq 1.9$.
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#### Pseudorapidity dependence of the anisotropic flow of charged particles in Pb-Pb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV
The collaboration Adam, Jaroslav ; Adamova, Dagmar ; Aggarwal, Madan Mohan ; et al.
Phys.Lett. B762 (2016) 376-388, 2016.
Inspire Record 1456145
We present measurements of the elliptic ($\mathrm{v}_2$), triangular ($\mathrm{v}_3$) and quadrangular ($\mathrm{v}_4$) anisotropic azimuthal flow over a wide range of pseudorapidities ($-3.5< \eta < 5$). The measurements are performed with Pb-Pb collisions at $\sqrt{s_{\text{NN}}} = 2.76$ TeV using the ALICE detector at the Large Hadron Collider (LHC). The flow harmonics are obtained using two- and four-particle correlations from nine different centrality intervals covering central to peripheral collisions. We find that the shape of $\mathrm{v}_n(\eta)$ is largely independent of centrality for the flow harmonics $n=2-4$, however the higher harmonics fall off more steeply with increasing $|\eta|$. We assess the validity of extended longitudinal scaling of $\mathrm{v}_2$ by comparing to lower energy measurements, and find that the higher harmonic flow coefficients are proportional to the charged particle densities at larger pseudorapidities. Finally, we compare our measurements to both hydrodynamical and transport models, and find they both have challenges when it comes to describing our data.
9 data tables
No description provided.
No description provided.
No description provided.
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#### Version 2 Anisotropic flow of charged particles in Pb-Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV
The collaboration Adam, Jaroslav ; Adamova, Dagmar ; Aggarwal, Madan Mohan ; et al.
Phys.Rev.Lett. 116 (2016) 132302, 2016.
Inspire Record 1419244
We report the first results of elliptic (v2), triangular (v3), and quadrangular (v4) flow of charged particles in Pb-Pb collisions at a center-of-mass energy per nucleon pair of sNN=5.02 TeV with the ALICE detector at the CERN Large Hadron Collider. The measurements are performed in the central pseudorapidity region |η|<0.8 and for the transverse momentum range 0.2<pT<5 GeV/c. The anisotropic flow is measured using two-particle correlations with a pseudorapidity gap greater than one unit and with the multiparticle cumulant method. Compared to results from Pb-Pb collisions at sNN=2.76 TeV, the anisotropic flow coefficients v2, v3, and v4 are found to increase by (3.0±0.6)%, (4.3±1.4)%, and (10.2±3.8)%, respectively, in the centrality range 0%–50%. This increase can be attributed mostly to an increase of the average transverse momentum between the two energies. The measurements are found to be compatible with hydrodynamic model calculations. This comparison provides a unique opportunity to test the validity of the hydrodynamic picture and the power to further discriminate between various possibilities for the temperature dependence of shear viscosity to entropy density ratio of the produced matter in heavy-ion collisions at the highest energies.
11 data tables
Centrality dependence of $v_2$, with two- and multi-particle correlations, integrated over the $p_{\rm T}$ range 0.2 < $p_{\rm T}$ < 5.0 GeV/$c$, at $\sqrt{s_{\rm NN}}$ = 5.02 TeV.
Centrality dependence of $v_3$ and $v_4$, with two-particle correlations, integrated over the $p_{\rm T}$ range 0.2 < $p_{\rm T}$ < 5.0 GeV/c, at $\sqrt{s_{\rm NN}}$ = 5.02 TeV.
Centrality dependence of $v_2$, with two- and multi-particle correlations, integrated over the $p_{\rm T}$ range 0.2 < $p_{\rm T}$ < 5.0 GeV/c, at $\sqrt{s_{\rm NN}}$ = 2.76 TeV.
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#### Azimuthal anisotropy of charged jet production in $\sqrt{s_{\rm NN}}$ = 2.76 TeV Pb-Pb collisions
The collaboration Adam, Jaroslav ; Adamova, Dagmar ; Aggarwal, Madan Mohan ; et al.
Phys.Lett. B753 (2016) 511-525, 2016.
Inspire Record 1394678
We present measurements of the azimuthal dependence of charged jet production in central and semi-central sNN=2.76 TeV Pb–Pb collisions with respect to the second harmonic event plane, quantified as v2ch jet . Jet finding is performed employing the anti- kT algorithm with a resolution parameter R=0.2 using charged tracks from the ALICE tracking system. The contribution of the azimuthal anisotropy of the underlying event is taken into account event-by-event. The remaining (statistical) region-to-region fluctuations are removed on an ensemble basis by unfolding the jet spectra for different event plane orientations independently. Significant non-zero v2ch jet is observed in semi-central collisions (30–50% centrality) for 20<pTch jet<90 GeV/c . The azimuthal dependence of the charged jet production is similar to the dependence observed for jets comprising both charged and neutral fragments, and compatible with measurements of the v2 of single charged particles at high pT . Good agreement between the data and predictions from JEWEL, an event generator simulating parton shower evolution in the presence of a dense QCD medium, is found in semi-central collisions.
2 data tables
Second-order harmonic coefficient $v_2^{ch~jet}$ as function a of $p_{T}^{ch~jet}$ for 0--5% collision centrality.
Second-order harmonic coefficient $v_2^{ch~jet}$ as function a of $p_{T}^{ch~jet}$ for 30--50% collision centrality.
#### Forward-central two-particle correlations in p-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV
The collaboration Adam, Jaroslav ; Adamova, Dagmar ; Aggarwal, Madan Mohan ; et al.
Phys.Lett. B753 (2016) 126-139, 2016.
Inspire Record 1379977
Two-particle angular correlations between trigger particles in the forward pseudorapidity range ( 2.5<|η|<4.0 ) and associated particles in the central range ( |η|<1.0 ) are measured with the ALICE detector in p–Pb collisions at a nucleon–nucleon centre-of-mass energy of 5.02 TeV. The trigger particles are reconstructed using the muon spectrometer, and the associated particles by the central barrel tracking detectors. In high-multiplicity events, the double-ridge structure, previously discovered in two-particle angular correlations at midrapidity, is found to persist to the pseudorapidity ranges studied in this Letter. The second-order Fourier coefficients for muons in high-multiplicity events are extracted after jet-like correlations from low-multiplicity events have been subtracted. The coefficients are found to have a similar transverse momentum ( pT ) dependence in p-going (p–Pb) and Pb-going (Pb–p) configurations, with the Pb-going coefficients larger by about 16±6% , rather independent of pT within the uncertainties of the measurement. The data are compared with calculations using the AMPT model, which predicts a different pT and η dependence than observed in the data. The results are sensitive to the parent particle v2 and composition of reconstructed muon tracks, where the contribution from heavy flavour decays is expected to dominate at pT>2 GeV/c .
4 data tables
$v_{2}^{\mu}\{\rm{2PC,sub}\}$ extracted from muon-track correlations.
$v_{2}^{\mu}\{\rm{2PC,sub}\}$ coefficients from muon-tracklet correlations in p-going direction.
$v_{2}^{\mu}\{\rm{2PC,sub}\}$ coefficients from muon-tracklet correlations in Pb-going direction.
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#### Elliptic flow of identified hadrons in Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV
The collaboration Abelev, Betty Bezverkhny ; Adam, Jaroslav ; Adamova, Dagmar ; et al.
JHEP 1506 (2015) 190, 2015.
Inspire Record 1297103
66 data tables
Charged pions v2 as a function of pT for centrality: 0-5%.
Charged pions v2 as a function of pT for centrality: 5-10%.
Charged pions v2 as a function of pT for centrality: 10-20%.
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#### Azimuthal anisotropy of D meson production in Pb-Pb collisions at $\sqrt{s_{\rm NN}} = 2.76$ TeV
The collaboration Abelev, Betty Bezverkhny ; Adam, Jaroslav ; Adamova, Dagmar ; et al.
Phys.Rev. C90 (2014) 034904, 2014.
Inspire Record 1294938
4 data tables
Prompt D^0 meson v2 as a function of pT for centrality 0-10%. The first systematic uncertainty is from the data and the second from the B feed-down.
Prompt D^0 meson v2 as a function of pT for centrality 10-30%. The first systematic uncertainty is from the data and the second from the B feed-down.
Prompt D^0 meson v2 as a function of pT for centrality 30-50%. The first systematic uncertainty is from the data and the second from the B feed-down.
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#### D meson elliptic flow in non-central Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76TeV
The collaboration Abelev, B. ; Adam, J. ; Adamova, D. ; et al.
Phys.Rev.Lett. 111 (2013) 102301, 2013.
Inspire Record 1233087
4 data tables
v2 vs. pt for D0. The first systematic (sys) error is that from the data analysis and the second is from the B feed-down subtraction, as explained in the paper.
v2 vs. pt for D+. The first systematic (sys) error is that from the data analysis and the second is from the B feed-down subtraction, as explained in the paper.
v2 vs. pt for D*+. The first systematic (sys) error is that from the data analysis and the second is from the B feed-down subtraction, as explained in the paper.
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#### J/Psi Elliptic Flow in Pb-Pb Collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV
The collaboration Abbas, Ehab ; Abelev, Betty ; Adam, Jaroslav ; et al.
Phys.Rev.Lett. 111 (2013) 162301, 2013.
Inspire Record 1225273
4 data tables
Measurements of V2 for inclusive J/PSI production for semi-central (20%-40%) collisions as a function of transverse momentum.
Measurements of V2 for inclusive J/PSI production in the transverse momentum range 1.5-10 GeV/c as a function of the number of participating nucleons.
Measurements of the uncorrected mean transverse momentum for inclusive J/PSI production in the transverse momentum range 1.5-10 GeV/c as a function of the number of participating nucleons.
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#### Elliptic flow of charged particles in Pb-Pb collisions at 2.76 TeV
The collaboration
Phys.Rev.Lett. 105 (2010) 252302, 2010.
Inspire Record 877822
We report the first measurement of charged particle elliptic flow in Pb-Pb collisions at 2.76 TeV with the ALICE detector at the CERN Large Hadron Collider. The measurement is performed in the central pseudorapidity region (|eta|<0.8) and transverse momentum range 0.2< p_t< 5.0 GeV/c. The elliptic flow signal v_2, measured using the 4-particle correlation method, averaged over transverse momentum and pseudorapidity is 0.087 +/- 0.002 (stat) +/- 0.004 (syst) in the 40-50% centrality class. The differential elliptic flow v_2(p_t) reaches a maximum of 0.2 near p_t = 3 GeV/c. Compared to RHIC Au-Au collisions at 200 GeV, the elliptic flow increases by about 30%. Some hydrodynamic model predictions which include viscous corrections are in agreement with the observed increase.
5 data tables
Transverse momentum dependence of v2 for centrality 40-50% from the 2- and 4-particle cumulant methods.
Transverse momentum dependence of v2{4} for centralities 10-20%, 20-30% and 30-40%.
Centrality dependence of elliptic flow, integrated over the pT range 0.2 < pT < 5.0 GeV, estimated with two- and multi-particle correlation techniques.
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https://infoglitz.com/alien-civilization-can-use-black-holes-to-generate-energy/ | Breaking News
Home https://server7.kproxy.com/servlet/redirect.srv/sruj/smyrwpoii/p2/ Science https://server7.kproxy.com/servlet/redirect.srv/sruj/smyrwpoii/p2/ Alien civilization can use black holes to generate energy
# Alien civilization can use black holes to generate energy
The artist’s impression of the accumulated flow inside and the jet from the supermassive black hole when it actively feeds, such as a recently torn star. Image: ESO/L. Calsada
A 50-year-old theory begins with how to use alien civilization Black hole The energy generation has been verified by the first experiment in the Glasgow Research Laboratory.
In 1
969, the British physicist Roger Penrose proposed that energy can be generated by placing an object in the active layer of the black hole (outer layer of the black hole event horizon). In this layer, an object’s The speed of movement must be faster than the speed of light to remain still.
Penrose predicts that objects will gain negative energy in this unusual region of space. By dropping the object and dividing it into two halves so that half falls into the black hole and the other half is recovered, the recoil will measure the loss of negative energy – effectively, the other half of the recovery will obtain energy from the rotation of the black hole. The scale of the engineering challenge required by this process is so large, but Penrose believes that only very advanced, perhaps foreign civilizations can complete the mission.
Two years later, another physicist named Yakov Zel’dovich suggested that a more practical and practical experiment could be used to verify the theory. He proposed that the “twisted” light wave hits the surface of a rotating metal cylinder rotating at the correct speed. Due to the weirdness of the rotating Doppler effect, it will eventually be reflected by the extra energy extracted from the rotation of the cylinder.
But Zel’dovich’s idea has only been in the theoretical field since 1971, because to conduct experiments, the metal cylinder he proposed needs to rotate at least one billion times per second, which is another insurmountable current ergonomic limit Challenge.
Now, from University of GlasgowThe School of Physics and Astronomy finally found a way to prove the effect proposed by Penrose and Zel’dovich through experiments, that is, by distorting sound instead of light (a signal source with a much lower frequency), in the laboratory To demonstrate in practice.
In a new paper published on June 22, 2020, Natural physicsThe team described how they constructed a system that uses a small circle of speakers to produce sound wave distortions similar to those of Zel’dovich’s.
Image source: University of Glasgow
These distorted sound waves are directed to a rotating sound absorber made of foam disk. A group of microphones on the back of the disc picked up the sound when passing the disc from the speaker, thereby steadily increasing the rotation speed.
In order to know that Penrose and Zel’dovich’s theory is correct, the team hopes to hear a significant change in the frequency and amplitude of sound waves as they propagate through the disc. This is caused by the Doppler effect.
The main author of the paper is Marion Cromb, a doctoral student in the College of Physics and Astronomy. Marion said: “The linear form of the Doppler effect is familiar to most people, because this phenomenon occurs as the pitch of the ambulance siren rises as it approaches the listener and decreases as it approaches. Because sound waves reach the listener more frequently when the ambulance approaches, but less often when the ambulance passes, it seems to increase the sound wave.
The rotating Doppler effect is similar, but the effect is limited to circular space. When measured from the angle of a rotating surface, the distorted sound wave changes its pitch. If the surface rotates fast enough, the sound frequency may do something very strange-it can change from a positive frequency to a negative frequency, and doing so will steal some energy from the surface rotation. “
During the researchers’ experiments, as the turntable speed increased, the pitch of the sound from the speakers decreased until it became too low to be heard. Then the tone rises again until it reaches its previous tone-but louder, its amplitude is 30% greater than the original sound from the speaker.
Marion added: “What we heard in the experiment was extraordinary. What happened was that as the rotation speed increased, the frequency of the sound wave was positive Doppler shifted to zero. When the sound recovered again, it was because the wave It has been converted from a positive frequency to a negative frequency. These negative frequency waves can absorb some energy from the rotating foam disk and become larger in the process-just as Zeldovic proposed in 1971.”
Professor Daniele Faccio, who is also the School of Physics and Astronomy at the University of Glasgow, is the co-author of the paper. Professor Faccio added: “We are very pleased to be able to verify some very strange physics through experiments half a century after the theory was proposed. Strangely, we have been able to confirm that half of the The theory of the origin of the universe in the history of the century, but we think it will open up many new avenues for scientific exploration. We are eager to see how we can investigate the impact on different sources, such as electromagnetic waves, in the near future.”
References: Marion Cromb, Graham M. Gibson, Ermes Toninelli, Miles J . Padgett), Ewan M. Wright and Daniele Faccio, “Amplification of Rotating Waves”, June 22, 2020, Natural physics.
DOI: 10.1038/s41567-020-0944-3
The research group’s paper titled “Amplification of Waves from Rotating Bodies” was published on Natural physics. This research was supported by the Engineering and Physical Sciences Research Council (EPSRC) and the European Union’s Horizon 2020 program. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8181303143501282, "perplexity": 727.661317581387}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655879532.0/warc/CC-MAIN-20200702142549-20200702172549-00440.warc.gz"} |
http://crypto.stackexchange.com/questions/3090/why-does-it-matter-for-a-signature-scheme-to-be-without-random-oracles?answertab=votes | # Why does it matter for a signature scheme to be without random oracles?
There is a profusion of articles proposing signature schemes without random oracles (see for yourself). What does that mean, and why does it matter?
- | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9298525452613831, "perplexity": 1282.5146412549504}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510273381.44/warc/CC-MAIN-20140728011753-00494-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/forces-that-depend-only-on-distance.329081/ | # Forces that depend only on distance
1. Aug 5, 2009
### fisico30
It is said that if a force depends only on distance (And not on time), the equations that involve those forces are invariant to a change of the frame of reference.....is this true?
Coulomb's force law is only dependent on distance. At a certain space location the force has a value that does not change with time. But after the object is subjected to that force, it moves to a new spatial position and the force on it changes. So there is a change with time....
A time changing source generates a time changing field and therefore a time changing force on an object.... what is special about these type of forces? Why don't they fit in the frame of reference transformations? after all, in classical mechanics time is absolute and distance differences are invariant...
thanks,
fisico30
2. Aug 6, 2009
### Jano L.
Hi,
in classical mechanics, motion equations are invariant to changes of inertial reference frame.
Imagine the force acting on a particle given as a function of coordinates and time. Consider two mutually moving frames, with nocoinciding origins O, O'. The coordinates in these frames are related by Galilei transformation, which in the simplest case of two equally oriented coordinated systems and motion od S' along x with velocity v is
$$x' = x - vt, y' = y, z' = z, t' = t.$$
(In general, x,y,z would enter in some linear combinatinons, but t' would be always t). Then in S and S' frames the equations of motion are
$$\mathbf F(x,y,z, t) = m\mathbf a(t)~~,~~\mathbf F'(x',y',z', t) = m\mathbf a'(t).$$
The equations looks the same - every frame use its coordinates to describe the particle, but the form of law is the same. We say that Newton's law of motion is covariant with respect to Galilean transformations coordinates. Furthermore, because frames are inertial, forces are the same
$$\mathbf F(x,y,z, t) = \mathbf F' (x',y',z', t).$$
There is no need to have the time independent force; equations are covariant in general.
In relativistic description (which is closer to reality), the equations are again covariant in respect to, but Lorentz transformation of coordinates, which in our special case is
$$x' = \gamma(x - vt), y' = y, z' = z, t' = \gamma(t-vx/c^2).$$
For example, the law of motion of a particle in EM field is in S
$$qF^\mu_{~\nu}(x,y,z,t) u^\nu(t) = m\frac{d^2 x^\mu}{d\tau^2}(t)$$
and in S'
$$qF'^\mu_{~\nu}(x',y',z',t') u^\nu(t') = m\frac{d^2 x'^\mu}{d\tau^2}(t').$$
So there is no need to have the time independent force - formulae are invariant with respect to spacetime coordinate transformations in general. Best,
Jano | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9083511233329773, "perplexity": 587.9496253848818}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257645943.23/warc/CC-MAIN-20180318184945-20180318204945-00548.warc.gz"} |
https://artofproblemsolving.com/wiki/index.php?title=2020_AIME_I_Problems/Problem_12&diff=119369&oldid=119362 | # Difference between revisions of "2020 AIME I Problems/Problem 12"
## Problem
Let be the least positive integer for which is divisible by Find the number of positive integer divisors of
## Solution 1
Lifting the Exponent shows that so thus, divides . It also shows that so thus, divides .
Now, multiplying by , we see and since and then meaning that we have that by LTE, divides .
Since , and all divide , the smallest value of working is their LCM, also . Thus the number of divisors is .
~kevinmathz
## Solution 2 (Simpler, just basic mods and Fermat's theorem)
Note that for all n, is divisible by 1 because that is a factor. That is , so now we can clearly see that the smallest to make the expression divisible by is just . Similarly, we can reason that the smallest n to make the expression divisible by is just .
Finally, for , take mod and mod of each quantity (They happen to both be and respectively, so you only need to compute once). One knows from Fermat's theorem that the maximum possible minimum for divisibility by is , and other values are factors of . Testing all of them (just using mods-not too bad), is indeed the smallest value to make the expression divisible by , and this clearly is NOT divisible by . Therefore, the smallest to make this expression divisible by is .
Calculating the LCM of all these, one gets . Using the factor counting formula, the answer is = .
-Solution by thanosaops | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.905636191368103, "perplexity": 659.5806076532401}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703531335.42/warc/CC-MAIN-20210122175527-20210122205527-00752.warc.gz"} |
http://math.stackexchange.com/questions/287370/isomorphism-with-cyclotomic-extension | # Isomorphism with cyclotomic extension
Let $K$ be a field with $\mbox{char}(K)=0$. I know that if $\xi$ is a primitive $n$-th root of the unity and $K(\xi)\big/K$ is a cyclotomic extension, then $\mbox{Gal}\left(K(\xi)\big/K\right)$ is isomorphic to a subgroup of $\mathbb Z_n^*$ (the multiplicative group).
However, I've found in some examples that they use that $\mbox{Gal}\left(K(\xi)\big/K\right)$ is isomorphic to $\mathbb Z_n^*$. I wonder when this is possible.
-
I'm not sure what you meant with "the multiplicative group without the zero", but $\,\left(\Bbb Z/n\Bbb Z\right)^*\,$ is not the original group (ring, in fact) without the zero, but the multiplicative group of all units modulo $\,n\,$. Unless $\,n\,$ is a prime, this is less than the non-zero elements. – DonAntonio Jan 26 '13 at 15:53
+1 Well, that helped indeed. Could you provide an example, please? – synack Jan 26 '13 at 16:04
An example is furnished by any $n$ that’s not a prime. Like $n=4$. – Lubin Jan 26 '13 at 17:25
So, it would be $\mathbb (Z/4\mathbb Z)^* = \left\{1,\xi, \xi^2\right\}$, right? – synack Jan 26 '13 at 17:32
No, it is $\,\left(\Bbb Z/4\Bbb Z\right)^*=\{1,3\}\,$ , multiplicatively. A group of order two. – DonAntonio Jan 26 '13 at 17:56
$\mathrm{Gal}(K(\xi)/K)\cong (\mathbb{Z}/n\mathbb{Z})^*$ is true when $K=\mathbb{Q}$. More generally, it is true if the $n$-th cyclotomic polynomial is irreducible over $K$.
For $n$ a prime, and $K = \mathbf{Q}$, you will indeed get $\mathrm{Gal}(\mathbf{Q}(\xi)/\mathbf{Q}) \cong \mathbf{Z}_n \setminus \{ 0 \}$.
But then if you understand $\mathbf{Z}_n^{\star}$ to be the group of invertible elements of (the multiplicative monoid of) $\mathbf{Z}_n$, then $\mathrm{Gal}(\mathbf{Q}(\xi)/\mathbf{Q}) \cong \mathbf{Z}_n^{\star}$ for all $n$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9571253657341003, "perplexity": 182.54601457115595}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049276304.88/warc/CC-MAIN-20160524002116-00102-ip-10-185-217-139.ec2.internal.warc.gz"} |
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I'm new to TikZ and pgfplots but would like to know how to plot the square as well as how to plot the arrows + delta x as well as arrow + delta A: I'd be most grateful if some of you had some ... | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9222937822341919, "perplexity": 2111.401130573621}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250614880.58/warc/CC-MAIN-20200124011048-20200124040048-00445.warc.gz"} |
https://webstrategies.net.au/linalyl-acetate-axc/8ca57b-brackett-series-wavelengths | electron jumps from any state n, It is called energy of first excited state of The wave number is, v = R( 1/4 2 - 1/n 2 2) = R( 1/16 - 1/n 2 2) (v) Pfund series . The third line of Brackett series is formed when electron drops from n=7 to n=4. The lines appear in emission when hydrogen atoms' electrons descend to the fourth energy level from a higher level, and they appear in absorption when the electrons ascends from the fourth energy level to higher levels. series. n = 4 â λ = (4)2/ (1.096776 x107 m-1) = 1458.9 nm. Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail, Spectral series of hydrogen atom and Energy level diagram. Copyright © 2018-2021 BrainKart.com; All Rights Reserved. Solution not clear? The wave number of the Lyman series is given by, When the electron jumps from any of the outer 68 0 obj <> endobj Given RH = 1.094 x 107 m-1. calculate the wavelength of the second line in the brackett series for hydrogen? The series obtained by the transition of the electron from n 2 = 5, 6... to n 1 = 4 is called Brackett series. Brackett series is displayed when electron transition takes place from higher energy states (nh=5,6,7,8,9â¦) to nl=4 energy state. The wavelengths of these lines are in the infrared region. Whenever an electron in a hydrogen atom jumps 3.3k VIEWS. 3.Calculate the 4 largest wavelengths for the Brackett and Pfund series for Hydrogen. spectra. The wave Determine the values for the quantum number n for the two energy levels involved in the transition. what series of wavelengths will be emitted? orbits to the first orbit, the spectral lines emitted are in the ultraviolet Using the ⦠Here n, The series obtained by the transition of the are emitted when the electron jumps from outer most orbits to the third orbit. The energy of second, third, fourth,
excited states of the The shortest wavelength of the Brackett series of hydrogen like atom (atomic number = Z) is the same as the shortest wavelength of the Balmer series of hydrogen atom. Solution for The Brackett series in the hydrogen spectrum corresponds to transitions that have a final state of m = 4. All the wavelength of Brackett series falls in Infrared region of the electromagnetic spectrum. Calculate the ratio of ionization energies of H and D. chemistry. (a) Calculate the wavelengths of the first three lines in this series. Taking these energies on a linear scale, horizontal lines are drawn Name * Email * Website. OR . Sodium vapour Table 6.1. The two lamps work on the principle of hot cathode positive column. This series consists of all wavelengths which as energy level diagram. Balmer Series; Lyman Series; Paschen Series; Brackett Series; Pfund Series; Further, letâs look at the Balmer series in detail. The series obtained by the transition of the endstream endobj 69 0 obj <. for the first member of the series, n2 = 5Therefore,λ1 =R[421 â 521 ]λ1 =R[161 â 251 ]λ1 =R[4009 ]λ = 9R400 λ = 9×10.97×106400 λ = 98. This series of the hydrogen emission spectrum is known as the Balmer series. This is the only series of lines in the electromagnetic spectrum that lies in the visible region. Solution. When the electron jumps from any of the outer Lyman α emissions are weakly absorbed by the major components of the atmosphereâO, O2, and N2âbut they are absorbed readily by NO and ⦠5890Å. The Brackett series in the hydrogen spectrum corresponds to transitions that have a final state of m=4. Express your answers in micrometers to three significant figures. Table 6.1. These lines lie in the infrared with wavelengths from 4.05 microns (Brackett-alpha) to 1.46 microns (the series limit), and are named after the American physicist Frederick Brackett (1896–1980). Their formulas are similar to Balmer’s except that the constant term is the reciprocal of the square of 1, 3, 4, or 5, instead of 2, and the… Read More Expert Answer. (iv) Brackett series . two orbits (energy levels) between which the transition of electron takes composite light consisting of all colours in the visible spectrum. The Brackett series of lines, first observed by Frederick Sumner Brackett in 1922, results when an excited electron falls from a higher energy level (n ⥠5) to the n=4 energy level. The following are the spectral series of hydrogen atom. Wavelength of spectral lines emitted by mercury. Answer : D Solution : Related Video. For shortest wavelength in Paschen Series n 1 =2 and n 2 =. 9); the shortest wavelength (highest energy) corresponds to the largest value of n. For nââ, ãSolã While the kinetic energy of any particle is positive, the potent ial energy of any pair of particles that are mutually attracted is negative. So I think we have to do the math for each possible power level in the series then match the wavelengths we get to the corresponding wavelength region. Know: The first line of the Paschen series occurs at 18,751.1A with an energy of E n =-13.6/(3) 2. View More Questions. The Brackett series in the hydrogen spectrum corresponds to transitions that have a final state of m=4 . These lines are called sodium D1 and D2 lines. The wave number is, v = R (1/42 - ⦠R = 1.09737x 10^7 m-1. the hydrogen atom. 1 2 2 H k 1 n 1 R 1 − λ= − 2a) four largest λ for Bracket series: n = 4. k = 5 → λ = {R H*(1/4 2 – 1/52)}-1 = 4.05 µm . The different series of lines falling on the picture are each named after the person who discovered them. The first line in this series (n2 = -Find wavelengths fron 20 hz: -Find wavelengths for 20,000 hz: Chemistry. View All. 3. The value of z is. How to solve: Calculate the wavelength of the second line in the Brackett series (nf = 4) of the hydrogen emission spectrum. endstream endobj startxref Q. Text Solution. Refer to the table below for various wavelengths associated with spectral lines. The Brackett Series? The maximum wavelength of Brackett series of hydrogen atom will be _____ 8.7k LIKES. Where m = 4, R = 1.097 * 10^-2 nm^-1, and n is an integer greater than 4. laboratory as a source of monochromatic (single colour) light. Complicating everything - frequency and wavelength. The Brackett series of lines, first observed by Frederick Sumner Brackett in 1922, results when an excited electron falls from a higher energy level (n ≥ 5) to the n=4 energy level. orbits to the second orbit, we get a spectral series called the Balmer series. Here n, This series consists of all wavelengths which The value, 109,677 cm-1, is called the Rydberg constant for hydrogen. Calculate the wavelengths in \mathrm{nm} of the first two lines of this series. The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. The lines of the series are obtained when the electron jumps from any state n 2 = 6, 7... to n 1 =5. The energy of the electron in the nth orbit of the spectral line series. In 1885, when Johann Balmer observed a spectral series in the visible spectrum of hydrogen, he made the following observations: The longest wavelength is 656.3 nm Calculate the mass of the deuteron given that the first line in the Lyman series of H lies at 82259.08 cm-1 whereas that of D lies at 82281.476 cm-1. Then at one particular point, known as the series limit, the series stops. are emitted when the electron jumps from outer most orbits to the third orbit. What Is The Wavelength (in Nm) Of This Emission From The Excited State Of N = 9? ) = R( 1/16 - 1/n22 wavelength of prominent lines emitted by the mercury source is presented in a) What are the wavelengths of the first three lines in this series? Q:- Since, sodium and mercury atoms are in the vapour state, they emit line Convert the wavelength to meters and use the Rydberg wavelength equation to determine the initial energy level: λ = (1280 nm) x (1 m / 1.0 x 10^9 nm) = 1.28 x 10-6 m. Rydberg wavelength equation. place, various spectral lines are obtained. We get the Brackett series ⦠Brackett Series: If the transition of electron takes place from any higher orbit (principal quantum number = 5, 6, 7, â¦) to the fourth orbit (principal quantum number = 4). region. The wavelengths of these lines are in the infrared region. Know: The first line of the Paschen series occurs at 18,751.1A with an energy of En=-13.6/(3)2. of the two levels is emitted as a radiation of particular wavelength. 4. The Paschen lines all lie in the infrared band. n_i = In what region of the electromagnetic spectrum is this line observed? The released wavelength lies in the Infra Red region of the spectrum. By how much do the wavelengths differ? region of the spectrum and they are said to form a series called Lyman series ãSolã The wavelengths in the Brackett series are given in Equation (4. hydrogen atom are, E, Therefore, it is seen from the above values, . The classification of the series by the Rydberg formula was important in the development of quantum mechanics. When the electron jumps from any of the outer associated with the second orbit is given by. Given RH = 1.094 X 107 M-1. This formula gives a wavelength of lines in the Paschen series of the hydrogen spectrum. Balmer Series. It is one of the hydrogen line series, such as the Lyman series and Balmer series and is named after Frederick Sumner Brackett. The What are the wavelengths of the first three lines in t… The two lamps work on the principle of hot cathode positive column. A hydrogen atom consists of an electron orbiting its nucleus. SUBMIT TRY MORE QUESTIONS. Calculate the wavelengths (in nanometers) and energies (in kJ/mole) of the first two lines in the Brackett series. ò?Ó 8Óm Where λvac is the wavelength of the light emitted in vacuum (λ); R is the Rydberg const. Brackett series is displayed when electron transition takes place from higher energy states(n h =5,6,7,8,9…) to n l =4 energy state. Balmer n1=2 , n2=3,4,5,…. Thousands of Experts/Students are active. In the below diagram we can see the three of these series laymen, Balmer, and Paschen series. Calculate the longest wavelength that a line in the Balmer series could have. The wavelengths of some of the emitted photons during these electron transitions are shown below: One of the lines has a wavelength of 2625 nm. n2=5,6,7,….. Pfund n1=5 , n2=6,7,8,….. In spectral line series. The shortest wavelength in Paschen Series is therefore 818 nm. The sodium vapour lamp is commonly used in the from higher energy level to the lower energy level, the difference in energies The n = 3 to n = 1 emission line for atomic hydrogen occurs in the UV region (it is a member of the Lyman series). 3), is called the Hα-line, the second (n2=4), the Hβ-line and so on. Its free . Enter your answers in descending order separated by commas. lamps and mercury lamps have been used for street lighting, as the two lamps / inf2 = 0. This series overlaps with the next (Brackett) series, i.e. An electron of wavelength 1.74*10-10m strikes an atom of ionized helium (He+). Chemistry electron jumps from any state n2 = 6, 7... to n1=5. spectra. The Bohr model was later replaced by quantum mechanics in which the electron occupies an atomic orbital rather than an orbit, but the allowed energy levels of the hydrogen atom remained the same as in the earlier theory. The Brackett series of emission lines from atomic hydrogen occurs in the far infrared region. ★★★ Correct answer to the question: Aline in the brackett series of hydrogen has a wavelength of 1945 nm. I think we have to use the rydberg equation, look in your textbook page 315 for a decent example. When n = 3, Balmerâs formula gives λ = 656.21 nanometres (1 nanometre = 10 â9 metre), the wavelength of the line designated H α, the first member of the series (in the red region of the spectrum), and when n = â, λ = 4/ R, the series limit (in the ultraviolet). These states were visualized by the Bohr model of the hydrogen atom as being distinct orbits around the nucleus. spectral series.docx - Free download as Word Doc (.doc / .docx), PDF File (.pdf), Text File (.txt) or read online for free. The shortest wavelength of visible light, at the violet end of the spectrum, is about 390 nanometers. ). hÞbcÃÀÂÀ±AXânø00leàm ±±' give a more intense light at comparatively low cost. It is called ground state energy of the hydrogen atom. 2 to the orbit n' = 2. The range of human hearing extends from approximately 20 Hz to 20,000 Hz. them. lamps and mercury lamps have been used for street lighting, as the two lamps The lines of the series are obtained when the The wave number is, v = R( 1/52 - 1/n22 Using the Rydberg equation . The electromagnetic force between the electron and the nuclear proton leads to a set of quantum states for the electron, each with its own energy. (BS) Developed by Therithal info, Chennai. Therefore, it is seen from the above values, closer and closer to the maximum value zero corresponding to n = ∞inf. For Brackett series n1 = 4, n2 = 5, 6, 7 1λ = R1n1 2 - 1n2 2For maximum wavelength n2 = 5 1λmax = 1.09687 × 107 142 - 152 λmax = 40519 Ao %%EOF The wavelengths of the Paschen series for hydrogen are given by {eq}1/\lambda = R_H (1/3^2 - 1/n^2) {/eq}, n = 4, 5, 6, . radio gamma rays visible X rays microwaves ultraviolet infrared. To give meaningful results n2 Question 33 4 pts The wavelengths of the Brackett series for hydrogen are emission of electron relaxation from higher excited states to a state of n = 4. What is the wavelength (in nm) of this emission from the excited state of n = 9? This series was observed by Friedrich Paschen during the years 1908. The Brackett series is the set of hydrogen spectral lines emitted when an electron descends from an electron shell number n greater than 4 down to n = 4, or the analogous absorption lines when absorbed electromagnetic radiation makes the electron do the opposite. Search for: Recent Posts. These lines are called sodium D, Millikan's oil drop experiment - Determination of charge of an electron, Rutherford's α - particle scattering experiment, Excitation and ionization potential of an atom, Production of X-rays - Modern Coolidge tube, Detection, Diffraction and Absorption of X-rays. Each energy state, or orbit, is designated by an integer, n as shown in the figure. Other articles where Lyman series is discussed: ionosphere and magnetosphere: Photon absorption: (The Lyman series is a related sequence of wavelengths that describe electromagnetic energy given off by energized atoms in the ultraviolet region.) The wavelength of a spectral line is given as . The The mercury light is a Paschen series (Bohr series, n′ = 3) Named after the German physicist Friedrich Paschen who first observed them in 1908. In the Brackett Series for the emission spectra of hydrogen the final destination of a dropping electron from a higher orbit is n=4 . The series obtained by the transition of the electron from n2 = 5, 6... to n1 = 4 is called Brackett series. The Brackett series is a series of absorption lines or emission lines due to electron jumps between the fourth and higher energy levels of the hydrogen atom. Calculate the de Broglie wavelength (in pm) of a hydrogen atom traveling 450 m/s . where R is Rydberg’s constant (1.097 10 7 m −1) and Z is atomic number (Z = 1 for hydrogen atom). Calculate the energy (in J) of a photon emitted during a transition corresponding to the first line in the Brackett series (nf = 4) of the hydrogen emission spectrum. At what point(s) on the line joining the two charges is the electric potential zero? from what state did the electron originate? The Balmer series in a hydrogen atom relates the possible electron transitions down to the n = 2 position to the wavelength of the emission that scientists observe.In quantum physics, when electrons transition between different energy levels around the atom (described by the principal quantum number, n ) they either release or absorb a photon. The sodium vapour lamp emits yellow light of wavelength 5896Å and closer and closer to the maximum value zero corresponding to n =, The sodium vapour lamp is commonly used in the (Jim Clark). The Balmer series in a hydrogen atom relates the possible electron transitions down to the n = 2 position to the wavelength of the emission that scientists observe.In quantum physics, when electrons transition between different energy levels around the atom (described by the principal quantum number, n ) they either release or absorb a photon. The wavelengths of these lines are in the infrared region. All the wavelength of Brackett series falls in Infrared region of the electromagnetic spectrum. If you now look at the Balmer series or the Paschen series, you will see that the pattern is just the same, but the series have become more compact. Brackett series has the shortest wavelength and it overlaps with the Paschen series. The wavelengths of some of the emitted photons during these electron transitions are shown below: What are the wavelengths of the first… electron from n, The lines of the series are obtained when the All the lines of this series in hydrogen have their wavelength in the visible 91 0 obj <>/Filter/FlateDecode/ID[<78E3343784E52844ACA917082EF29769><3D6C0B969DA361499A612943D3E64AC2>]/Index[68 49]/Info 67 0 R/Length 111/Prev 107142/Root 69 0 R/Size 117/Type/XRef/W[1 3 1]>>stream These lines lie in the infrared part of the electromagnetic spectrum, with wavelengths ranging from 4.05 micrometres (Brackett-alpha) to 1.46 micrometres (the series limit). Check Answer and What is the transition? Q:-Two charges 5 x 10-8 C and -3 x 10-8 C are located 16 cm apart. hydrogen atom is given by. The different wavelengths Energy H±X$ñLø ×ó3°,b¯aãZ]©PäÀ üIZoWÌî ¨ ½Ú@Ùn/7ã^(£$@ pï Brackett Series . These wavelengths fall within the infrared region of the electromagnetic spectrum. ). Show your calculations. Here n2 = 4,5,6
and n1 = 3. the hydrogen atom. Five spectral series identified in hydrogen are. List : Brackett Series . All the lines of this series in hydrogen have their wavelength in the visible Your Comment. Brackett series corresponds transitions to and from n = 4 level [1] So the first transition/emission is n = 4 ↔ n = 5 given by. Balmer Series: 383.5384 : 5 : 9 -> 2 : Violet: 388.9049 : 6 : 8 -> 2 : Violet: 397.0072 : 8 : 7 -> 2 : Violet: 410.174 : 15 : 6 -> 2 : Violet: 434.047 : 30 : 5 -> 2 : Violet: 486.133 : 80 : 4 -> 2 : Bluegreen (cyan) 656.272 : 120 : 3 -> 2 : Red: 656.2852 : 180 : 3 -> 2 : Red: Paschen Series: 954.62 ... 8 -> 3 : IR: 1004.98 ... 7 -> 3 : IR: 1093.8 ... 6 -> 3 : IR: 1281.81 ... 5 -> 3 : IR What series of wavelengths will be emitted? What is the wavelength (m) of the light corresponding to the line in the emission spectrum with the smallest energy transition? 121.6 \text{nm} 1/lambda = \text{R}(1/(n_1)^2 - 1/(n_2)^2) * \text{Z}^2 where, R = Rydbergs constant (Also written is \text{R}_\text{H}) Z = atomic number Since the question is asking for 1^(st) line of Lyman series therefore n_1 = 1 n_2 = 2 since the electron is de-exited from 1(\text{st}) exited state (i.e \text{n} = 2) to ground state (i.e text{n} = 1) for first line of Lyman series. that, the energy associated with a state becomes less negative and approaches The Lyman series lies in the ultraviolet, whereas the Paschen, Brackett, and Pfund series lie in the infrared. The various colors correspond to light of definite wavelengths, and the series of lines is called a ... the Balmer series (in which all the lines are in the visible region) corresponds to n=2, the Paschen series to n=3, the Brackett series to n=4, and the Pfund series to n=5. laboratory as a source of monochromatic (single colour) light. For Brackett series, n1 = 4 and n2 =5.6,7,.........,âHence, the wavelengths of Brackett series are given by the formula:λ1 =R[421 â n22 1 ]For maximum wavelength i.e. wavelength of prominent lines emitted by the mercury source is presented in number is, v = R( 1/42 - 1/n22 . From what state did the electron originate? The Brackett series is a series of absorption lines or emission lines due to electron jumps between the fourth and higher energy levels of the hydrogen atom. Brackett series with $$n_1 = 4$$ Pfund series with $$n_1 = 5$$ Humphreys series with $$n_1 = 6$$ The spectral series of hydrogen based of the Rydberg Equation (on a logarithmic scale). The Rydberg's formula for the hydrogen atom is. Pfund series (n l =5) 3.3k SHARES. ) = R( 1/25 - 1/n22 region. The energy of second, third, fourth,
excited states of the 5. It is called energy of first excited state of These observed spectral lines are due to the electron making transitions between two energy levels in an atom. What series of wavelengths will be emitted? At what point(s) on the line joining the two charges is the electric potential zero? called a spectral line. Definition of hydrogen spectrum in the Definitions.net dictionary. 1/ λ = R[1/n1^2 - 1/n2^2] = R[1/16 -1/25] Solve for λ. The shortest wavelength of the Brackett series of a hydrogen like atom (atomic number =Z) is the same as the shortest wavelength of the Balmar series of hydrogen atom. Which of the spectral lines of the Brackett series is closest in wavelength to the first spectral ine of the Paschen series? The Paschen series arises from hydrogen electron transitions ending at energy level n=3. English Energy associated with the first orbit of the hydrogen atom is. The sodium vapour lamp emits yellow light of wavelength 5896Å and It is shown as the transition from the higher energy states to the energy state of n=3 happen. that, the energy associated with a state becomes less negative and approaches the shortest line in the Brackett series has a wavelength that falls among the Paschen series. hydrogen atom are, E3 = -1.51 eV, E4 = -0.85 eV, E5 orbits to the second orbit, we get a spectral series called the Balmer series. In the Balmer series, notice the position of the three visible lines from the photograph further up the page. Find the wavelengths of these extremes at a temperature of 26°C. Sodium vapour Answer. infrared region with the wave number given by, v = R( 1/32 - 1/n22 Q:- In a parallel plate capacitor with air between the plates, each plate has an area of 6 x 10-3 m 2 and the distance between the plates is 3 mm. Since, sodium and mercury atoms are in the vapour state, they emit line Question: > Question 33 4 Pts The Wavelengths Of The Brackett Series For Hydrogen Are Emission Of Electron Relaxation From Higher Excited States To A State Of N = 4. OR . give a more intense light at comparatively low cost. Which of the spectral lines of the Brackett series is closest in wavelength to the first spectral ine of the Paschen series? %PDF-1.5 %âãÏÓ By how much do the wavelengths differ? Where R is Rydberg constant for the Hydrogen atom and equals to 1.1 10 7 m-1. Comment Cancel reply. Q:-Two charges 5 x 10-8 C and -3 x 10-8 C are located 16 cm apart. hÞbbdbº"§É@ÉQ"cÀ¤*ì@\$!XXiXܬW D2tÈX-°,dH"N`éÃf¯>@òÿÙ& {.Õ30 electron from n2 = 5, 6... to n1 = 4 is called Brackett = -0.54eV ... when n =infinity ∞, Einf = -13.6 Looking closely at the above image of the spectrum, we see various hydrogen emission spectrum wavelengths. 3.Calculate the 4 largest wavelengths for the Brackett and Pfund series for Hydrogen. ). The ratio of the largest to shortest wavelengths in Balmer series of hydrogen spectra is: (A) (25/9) (B) (17/6) (C) (9/5) (D) (5/4). constitute spectral series which are the characteristic of the atoms emitting Lyman n1= 1 ,n2=2 ,3,4,5,6,…. The Lyman series is in the ultraviolet while the Balmer series is in the visible and the Paschen, Brackett, Pfund, and Humphreys series are in the infrared. 116 0 obj <>stream Spectrum where n_ { 1 } \ ): the first spectral ine of the by. Given in Equation ( 4 ) 2/ ( 1.096776 x107 m-1 ) 1458.9... 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The nucleus an energy of the outer orbits to the second line in the visible region the table below various! Wavelengths associated with the Paschen series wavelength lies in the development of mechanics... Quantum number n for the hydrogen spectrum where n_ { 1 } \ ): the Lyman series Balmer. Hydrogen electron transitions ending at energy level n=3 1 } \ ): the first three lines the! Broglie wavelength ( in nm ) of a spectral series identified in hydrogen are emitted... Therefore 818 nm series lie in the electromagnetic spectrum following are the spectral lines of emission. The wavelength of spectral series, notice the position of the electromagnetic spectrum lies! And wavelength ( 3 ) named after the person who discovered them of quantum brackett series wavelengths a linear scale, lines! 5 x 10-8 C are located 16 cm apart of quantum mechanics ratio of ionization energies of and! Forum can get you clear solutions for any problem and is named the. Prominent lines emitted by the Rydberg const orbits around the nucleus and it overlaps with the brackett series wavelengths ( Brackett series! Of a hydrogen atom is -1/25 ] Solve for λ which of the light emitted in vacuum ( )! Electron jumps from outer most orbits to the table below for various associated. Here n, this series of hydrogen the final destination of a hydrogen atom be... Wavelength ( m ) of a spectral line is given by the transition of the series stops hydrogen! The second orbit is n=4, n as shown in the Brackett series has the shortest wavelength of series! Are the wavelengths of these lines are in the far infrared region of the first spectral ine of the jumps... Line joining the two charges is the electric potential zero of this emission from the excited state of n 4...: wavelength of 2625 nm nth orbit of the outer orbits to the third orbit lamp is used. Of Brackett series in hydrogen have their wavelength in the visible spectrum get. And is named after the person who discovered them hydrogen atom consists of all colours in the series. Atom and equals to 1.1 10 7 m-1 shortest wavelength in the visible region = 4 the value 109,677! These lines are in the visible spectrum n_i = in what region of the hydrogen corresponds... _____ 8.7k LIKES i 'm super confused how they arrived at this.... A dropping electron from n2 = 6 → λ = { R H (... 2625 nm and n is an integer, n as shown in the transition at a temperature of.! And wavelength Balmer series could have significant figures monochromatic ( single colour ) light emits yellow of. Potential zero the ratio of ionization energies of H and D. Chemistry drops from to! An electron of wavelength 5896Å and 5890Å know: the Lyman series lies in the development quantum... Electron orbiting its nucleus wavelengths fron 20 hz: -find wavelengths for the lamps... From the higher energy states ( n l =4 energy brackett series wavelengths, or orbit, is designated an! Wavelengths in the Brackett and Pfund series for hydrogen ( He+ ) in infrared region, horizontal lines in... Hydrogen spectrum where n_ { 1 } \ ): the first two lines of the second is... Levels involved in the Brackett series for hydrogen as a source of monochromatic single! Series consists of all colours in the emission spectrum with the smallest energy transition of... Of Brackett series has the shortest wavelength in Paschen series series is in! Nm ) of this series of lines in the infrared region is n=4 three of these lines are the... Electric potential zero the wave number is, v = R [ 1/n1^2 - ]... ( 1.096776 x107 m-1 ) = R [ 1/n1^2 - 1/n2^2 ] = R [ 1/16 ]! Series is formed when electron drops from n=7 to n=4 Brackett and Pfund series lie in the.... Occurs at 18,751.1A with an energy of the hydrogen atom ( Fig.! Wavelengths in \mathrm { nm } of the hydrogen atom ( Fig ) of... From the excited state of m=4 at energy level n=3 two charges is the electric potential zero we... The first line of Brackett series in the Brackett series in the infrared region series identified in are... In the below diagram brackett series wavelengths can see the three visible lines from atomic has... Lines from the excited state of n=3 happen separated by commas called sodium D1 and D2 lines (! { nm } of the light corresponding to the table below for wavelengths... ( Brackett ) series, with wavelengths given by the Rydberg formula the atoms emitting them called series! Light corresponding to the table below for various wavelengths associated with spectral lines of the spectrum, see... Is called the Rydberg formula the value, 109,677 cm-1, is about 390 nanometers vapour. Electron 13.0 ev is used to bombard gaseous hydrogen observed by Friedrich Paschen during the years 1908 the Red. The Bohr model of the Paschen series is closest in wavelength to the orbit! Answer is n=9 with a sepperation of 577A but i 'm super confused how they arrived at answer... Visualized by the transition and Pfund series for the two lamps work on the joining... Second orbit is n=4 R = 1.097 * 10^-2 nm^-1, and Paschen series atom... The following are the wavelengths in \mathrm { nm } of the hydrogen consists... First three lines in the visible region in the transition from the photograph further up the page and (... The German physicist Friedrich Paschen during the years 1908 different series of atom! And energies ( in pm ) of this series = { R H (... Lamp is commonly used in the Brackett series has a wavelength that a line in the development of mechanics... Energies ( in pm ) of the hydrogen spectrum corresponds to transitions that a... First excited state of m=4 the smallest energy transition looking closely at the end. Equation ( 4 ) 2/ ( 1.096776 x107 m-1 ) = R ( 1/25 - 1/n22 ) = R 1/16. Atom is given by = 3 ) named after the person who discovered them x 10-8 C are 16... End of the three visible lines from the photograph further up the page -find wavelengths for hydrogen. Could have values for the Brackett series is formed when electron drops from n=7 to n=4 = 4 λ. Traveling 450 m/s x107 m-1 ) = R ( 1/25 - 1/n22 ), i.e - edu-answer.com where λvac the... We see various hydrogen emission spectrum is this line observed at this answer traveling! Located 16 cm apart largest wavelengths for 20,000 hz: Chemistry to n1=5 the. This emission from the excited state of n = 4 is called the Rydberg formula was important in the series... In hydrogen are and Balmer series and is named after Frederick Sumner Brackett three lines in the spectrum... Spectrum wavelengths =5,6,7,8,9… ) to n l =5 ) Five spectral series called the Balmer series, n′ 3! Three significant figures involved in the Brackett series is closest in wavelength the... List: wavelength of Brackett series is therefore 818 nm called ground state energy the! 10 7 m-1 energy level n=3 is used to bombard gaseous hydrogen the Infra Red region of the hydrogen.! At one particular point, known as the transition the third line of the obtained... Of an electron orbiting its nucleus wavelengths fron 20 hz: -find wavelengths 20! Bohr series, i.e wavelengths constitute spectral series which are emitted when the electron in the series., notice the position of the electromagnetic spectrum is this line observed =2 and n 2 = ( ). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8944512009620667, "perplexity": 1265.5370266023294}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487607143.30/warc/CC-MAIN-20210613071347-20210613101347-00442.warc.gz"} |
http://physics.stackexchange.com/questions/62487/infinite-energy-of-point-charges-in-the-context-of-classical-field-theories | # Infinite Energy of Point Charges (in the context of classical field theories)
In the context of classical physics,is there any renormalization method to avoid infinite energy of point charges?
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– Ben Crowell May 2 '13 at 1:04 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9581891298294067, "perplexity": 1087.753247921282}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936462316.51/warc/CC-MAIN-20150226074102-00104-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://physics.stackexchange.com/questions/506833/quantization-of-field-with-other-complete-orthogonal-system | # Quantization of field with other complete orthogonal system
I've learned the quantization of Klein-Gordon field using Fourier expansion. I understand that this process is kind of exchanging complex fourier coefficients to operator and makes it satisfying the canonical relation.
The result of usual process is given as $$\Phi(x) = \int (d^3k)a(k)e^{-ikx} + a(k)^\dagger e^{ikx}\bigr|_{k^0 = \omega_k}$$
I wonder if I can do a quantization with other complete orthonormal systems, rather than $$e^{ikx}$$'s. (For example, radial fourier expansion or legendre polynomials)
If not, what makes it difficult? If we can, can you suggests the result with detail, the proper choice of commutation relation and Hamiltonian? Are they also determined by the quantization of classical Hamiltonian equation of motion as usual result be?
• The exponentials are chosen as the complete orthogonal basis because they satisfy the classical equations of motion. – Sounak Sinha Oct 7 at 7:04
• You also put $k^0=\omega_k$. That imples the definition of $\Phi$ is not a simple Fourier transform. – Sounak Sinha Oct 7 at 7:24
• @SounakSinha I think that is not critical reason because that form is just a result of elliminating dirac delta function($\delta(k^2 - m^2)$) from original form. – ChoMedit Oct 7 at 7:28
• @SounakSinha Do you mean that there is no other complete orthogonal basis which satisfies canonical quantization condition? – ChoMedit Oct 7 at 7:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9541029334068298, "perplexity": 756.0409091427673}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496665976.26/warc/CC-MAIN-20191113012959-20191113040959-00398.warc.gz"} |
https://mathoverflow.net/questions/348745/some-simple-algebra-of-rational-functions-by-andr%C3%A9-weil | # Some simple algebra of rational functions by André Weil
In André Weil's dissertation, he considers two meromorphic functions $$x,y$$ on a complex curve. He assumes every pole of $$y$$ is a pole of $$x$$, and its multiplicity as a pole of $$y$$ is no greater than its multiplicity as a pole of $$x$$. Then he says there is some natural number $$k$$ and some complex $$a\neq 0$$ such that $$ay^k+xP(x,y)+Q(y)=0$$ where $$P(x,y)$$ and $$Q(y)$$ are polynomials in $$x,y$$ with degree $$
I see the proof for genus 0 curves. There the field of meromorphic functions is $$\mathbb{C}(z)$$ and one-variable polynomial algebra suffices (unless I've made a mistake). But I do not see it for other curves. Can someone tell me how it is done?
• I assume that the curve is compact, then the field of meromorphic functions on it is finitely generated over $\mathbb{C}$ with transcendence degree $1$. Therefore any $x,y$ would satisfy a polynomial relation. It can be rewritten in the above form. (I'm guessing the symbol between $a$ and $0$ is $\not=$.) – Donu Arapura Dec 20 '19 at 3:05
• @DonuArapura Yes he surely means compact, though writing in 1927 the closest he comes to saying so is to say he is using the "birational viewpoint." I expect you are right about the strategy he had in mind, and i will try to work through what he says using that idea. – Colin McLarty Dec 20 '19 at 4:34
Let us write the irreducible equation relating $$x$$ and $$y$$ as $$P_k(x)y^k+P_{k-1}(x)y^{k-1}+\ldots+P_0(x)=0.$$ Consider the Newton polygon (the graph of the smallest concave function $$\phi$$ with $$\phi(j)\geq \deg P_j,\; 0\leq j\leq k$$. Condition on the poles of $$x$$ and $$y$$ tells us that $$P_k=\mathrm{const}$$, and all slopes of this graph are $$\geq -1.$$ This implies that $$\deg P_j\leq m-j,\quad 0\leq j\leq k-1.$$ Now unite all constant terms of $$P_j$$ times powers of $$y$$ into the polynomial $$Q(y)$$, $$\deg Q, and the rest is $$xP(x,y)$$, where degree of $$P$$ with respect to $$x$$ and $$y$$ is at most $$k-1$$.
• This is likely the right route. But I do not see how the condition on poles shows $P_k$ has no poles. The condition allows shared poles of $x$ and $y$ to have arbitrarily higher multiplicity in $x$ than $y$. I suspect the proof may need a step of multiplying through by some high enough power of $y$ to make all multiplicities of poles in $y^k$ surpass their multiplicity in $x$. – Colin McLarty Dec 20 '19 at 20:03
• The proof is complete. If all poles of $y$ are poles of $x$ then $P_k$ is constant. Look at the properties of the Newton polygon. – Alexandre Eremenko Dec 20 '19 at 21:05
• Algebraic equation wrt $y$ has $k$ solutions for generic $x$. If $P_k$ is not constant, it has a root at $x_0$ and as $x\to x_0$ some of these roots tend to $\infty$. Which means that at some point of the Riemann surface $x$ equals $x_0$ while $y$ has a pole. See any book on algebraic functions for the details. – Alexandre Eremenko Dec 20 '19 at 22:46
• I see that argument. But the the conclusion on degrees in $x$ and $y$ seems to follow just by counting the multiplicity of any one pole. I do not see the role of the Newton polygon. – Colin McLarty Dec 21 '19 at 16:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 32, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9573867917060852, "perplexity": 237.92124464745302}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400214347.17/warc/CC-MAIN-20200924065412-20200924095412-00589.warc.gz"} |
https://www.physicsforums.com/threads/is-it-watt-per-second-or-watt-per-hour.512361/ | # Is it watt per second or watt per hour
1. Jul 6, 2011
### bionic6manuel
Say I have a device (Thermoelectric generator) that "generates power" at 4 watts, as per watt meter reading, is it producing 4 watts a second or 4 watts an hour. I say it is producing 4 watts a second but my brother in law says 4 watts an hour so the watt meter can tell the future. Please bear in mind I am not consuming energy but generating energy.
2. Jul 6, 2011
Hello .Watts is a unit of power(Joules per second) so the 4 Watt device is producing 4 Joules per second.
To clarify this further the Joule is a unit of energy so the device is producing 4 Joules of energy every second.
Last edited: Jul 6, 2011
3. Jul 6, 2011
### cjl
Watts are a unit of power, so it doesn't make sense to talk about watts per hour (or per second). They already have the "per unit time" built in.
4. Jul 6, 2011
### stevenb
It is generating 4 Watts for as long as you leave it on.
See Dadface's answer above. Power is a measure of how fast you generate energy (or consume it). Joules is a measure of total generated energy.
Your brother may be getting confused by the units WHr which is also a measure of energy (that's Watt hours, not Watts per hour). If you leave a 4 W device running for one hour, you generate 4 WHr of energy, which is 4 JHr/s x 3600 sec / 1 Hr, which is 14400 Joules.
5. Jul 7, 2011
### bionic6manuel
I thank you all very much for the explanation
6. Jul 7, 2011
### Pythagorean
Watts per time is the change in power over time :)
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https://mathemerize.com/tag/domain-range-and-periodicity-of-trigonometric-functions/ | ## Graph of Trigonometric Functions – Domain & Range
Here, you will learn graph of trigonometric functions and domain & range of trigonometric functions. Graph of Trigonometric Functions : y = sinx y = cosx y = tanx y = cotx y = secx y = cosecx Values of T-Ratio of some standard angles AnglesT-Ratio 0 $$\pi\over 6$$ $$\pi\over 4$$ $$\pi\over 3$$ $$\pi\over 2$$ … | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8148468136787415, "perplexity": 1692.3470579483214}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499768.15/warc/CC-MAIN-20230129211612-20230130001612-00788.warc.gz"} |
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