url
stringlengths 15
1.13k
| text
stringlengths 100
1.04M
| metadata
stringlengths 1.06k
1.1k
|
|---|---|---|
https://arxiv.org/abs/1204.1896 | nucl-th
(what is this?)
# Title: (Oscillating) non-exponential decays of unstable states
Abstract: We discuss deviations from the exponential decay law which originate when going beyond the Breit-Wigner distribution for an unstable state. In particular, we concentrate on an oscillating behavior, reminiscent of the Rabi-oscillations, in the short-time region. We propose that these oscillations can explain the GSI experiment, which measured superimposed oscillations on top of the exponential law for hydrogen-like nuclides decaying via electron-capture (the so-called GSI anomaly). Moreover, we discuss the possibility, that the deviations from the Breit-Wigner distribution in the case of the GSI anomaly are (predominantly) caused by the interaction of the unstable state with the measurement apparatus. The consequences of this scenario, such as the non-existence of oscillations in an analogous experiment performed at the Berkeley Lab, are investigated.
Comments: 10 pages, 2 figures, revised version in print in the Proceedings of the "50th International Winter Meeting on Nuclear Physics", 23-27 January 2012, Bormio, Italy Subjects: Nuclear Theory (nucl-th); High Energy Physics - Phenomenology (hep-ph); Quantum Physics (quant-ph) Cite as: arXiv:1204.1896 [nucl-th] (or arXiv:1204.1896v2 [nucl-th] for this version)
## Submission history
From: Pagliara Giuseppe [view email]
[v1] Mon, 9 Apr 2012 15:14:08 GMT (24kb)
[v2] Wed, 11 Jul 2012 10:46:15 GMT (24kb) | {"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": 0.8528832793235779, "perplexity": 3932.8495817459616}, "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/1539583510893.26/warc/CC-MAIN-20181016221847-20181017003347-00187.warc.gz"} |
http://asterisk.apod.com/viewtopic.php?t=39968 | ## APOD: Spiral Galaxies Spinning Super-Fast (2019 Nov 05)
APOD Robot
Otto Posterman
Posts: 4474
Joined: Fri Dec 04, 2009 3:27 am
### APOD: Spiral Galaxies Spinning Super-Fast (2019 Nov 05)
Spiral Galaxies Spinning Super-Fast
Explanation: Why are these galaxies spinning so fast? If you estimated each spiral's mass by how much light it emits, their fast rotations should break them apart. The leading hypothesis as to why these galaxies don't break apart is dark matter -- mass so dark we can't see it. But these galaxies are even out-spinning this break-up limit -- they are the fastest rotating disk galaxies known. It is therefore further hypothesized that their dark matter halos are so massive -- and their spins so fast -- that it is harder for them to form stars than regular spirals. If so, then these galaxies may be among the most massive spirals possible. Further study of surprising super-spirals like these will continue, likely including observations taken by NASA's James Webb Space Telescope scheduled for launch in 2021.
<< Previous APOD This Day in APOD Next APOD >>
Ann
4725 Å
Posts: 11641
Joined: Sat May 29, 2010 5:33 am
### Re: APOD: Spiral Galaxies Spinning Super-Fast (2019 Nov 05)
Tully-Fisher relation. The more luminous a galaxy is,
the faster it spins. Illustration: Astro533Fa10
These super-fast spinning galaxies are so fascinating!
I'm strongly reminded of the Tully-Fischer relation:
Wikipedia wrote:
In astronomy, the Tully–Fisher relation (TFR) is an empirical relationship between the mass or intrinsic luminosity of a spiral galaxy and its asymptotic rotation velocity or emission line width. It was first published in 1977 by astronomers R. Brent Tully and J. Richard Fisher.[1] The luminosity is calculated by multiplying the galaxy's apparent brightness by {\displaystyle 4\pi d^{2}}{\displaystyle 4\pi d^{2}}, where {\displaystyle d}d is its distance from us, and the spectral-line width is measured using long-slit spectroscopy.
So the more luminous a galaxy is, the faster it spins, which results in broad and smeared spectral lines. Presumably, too, a luminous galaxy is also a massive galaxy. So we can assume that fast-spinning galaxies are massive.
Super fast spinning galaxies.
APOD Robot wrote:
It is therefore further hypothesized that their dark matter halos are so massive -- and their spins so fast -- that it is harder for them to form stars than regular spirals.
I get the idea of the massive dark matter halos, and I agree that the most massive galaxies rarely form a lot of stars. All right, but take a look at the middle galaxy in the bottom row, OGC 1304! This galaxy is among the fastest spinning galaxies known, and presumably it is very very massive, so it is not "allowed" to form a lot of stars. Well, OGC 1304 couldn't care less. Look at all the blue stuff all around it!
Click to play embedded YouTube video.
The middle galaxy in the top row is also relatively blue, but I agree that the other four galaxies are not churning out too many new stars. The two "end galaxies in the top row" are extremely elegant and beautiful. Why are they black and white? Were they photographed in (near) infrared light?
(By the way, I think the crazy kids in the carousel video are Danish.)
Ann
Color Commentator
orin stepanek
Plutopian
Posts: 6981
Joined: Wed Jul 27, 2005 3:41 pm
### Re: APOD: Spiral Galaxies Spinning Super-Fast (2019 Nov 05)
When a skater pulls their arms in they spin spins faster! So to slow down their arms must come out! So; If the galaxies staro to separate their arms; they will probably slow down! IMHO
Orin
Smile today; tomorrow's another day!
sillyworm 2
### Re: APOD: Spiral Galaxies Spinning Super-Fast (2019 Nov 05)
I'm fascinated by OGC 1403..the cosmic sunnyside up space egg.The central bulge appears massive.(unless that is a foreground star).Would love to see a clearer picture some day.
bystander
Apathetic Retiree
Posts: 20806
Joined: Mon Aug 28, 2006 2:06 pm
Location: Oklahoma
### Re: APOD: Spiral Galaxies Spinning Super-Fast (2019 Nov 05)
Know the quiet place within your heart and touch the rainbow of possibility; be
alive to the gentle breeze of communication, and please stop being such a jerk.
— Garrison Keillor
Chris Peterson
Abominable Snowman
Posts: 16210
Joined: Wed Jan 31, 2007 11:13 pm
### Re: APOD: Spiral Galaxies Spinning Super-Fast (2019 Nov 05)
orin stepanek wrote:
Tue Nov 05, 2019 12:08 pm
When a skater pulls their arms in they spin spins faster! So to slow down their arms must come out! So; If the galaxies staro to separate their arms; they will probably slow down! :mrgreen: IMHO
Well, these galaxies aren't expanding, so we're not going to see that. Also, keep in mind that a skater is a rigid body and a galaxy is not. While both are subject to conservation of momentum, the observed dynamics that result from it are somewhat different for the two.
Chris
*****************************************
Chris L Peterson
Cloudbait Observatory
http://www.cloudbait.com
TheOtherBruce
Science Officer
Posts: 102
Joined: Wed Jul 17, 2019 6:07 pm
### Re: APOD: Spiral Galaxies Spinning Super-Fast (2019 Nov 05)
sillyworm 2 wrote:
Tue Nov 05, 2019 1:23 pm
I'm fascinated by OGC 1403..the cosmic sunnyside up space egg.The central bulge appears massive.(unless that is a foreground star).Would love to see a clearer picture some day.
I dunno, I can see another couple of possibilities here. It could be two similar spiral-ish galaxies in mid face-to-face collision, with the two nuclei still visibly separate. Or maybe one of them is something way in the background? Alternatively, it could be a stretched-out nearly edge on spiral, and the blue patch at lower left would be the end of a spiral arm; the speculative bit in between won't be visible unless we get a better picture. Lots of weird-looking stuff out there.
Do we have any guesstimates how far away they are? That might skew other figures one way or another.
This universe shipped by weight, not by volume.
Some expansion of the contents may have occurred during shipment.
tomatoherd
### Re: APOD: Spiral Galaxies Spinning Super-Fast (2019 Nov 05)
And it looks like, in the first two galaxies at least, there is some gravitational lensing around them. Meaning more mass. Something you don't notice on typical galaxy shots, only clusters. Am i wrong?
dlw
Ensign
Posts: 88
Joined: Fri Jul 22, 2005 4:43 pm
Location: California
### Re: APOD: Spiral Galaxies Spinning Super-Fast (2019 Nov 05)
Is it possible to estimate a maximum angular velocity within these galaxies? Stating that "UGC 12591 spins at about 480 km/sec" doesn't give me a sense of how fast it is spinning.
NHcycler
### Re: APOD: Spiral Galaxies Spinning Super-Fast (2019 Nov 05)
Re: STScI: Super Spirals Spin Super Fast
neufer wrote:
Fri Oct 18, 2019 1:20 am
Ann wrote:
Thu Oct 17, 2019 8:02 pm
That's the old Tully-Fisher relation, isn't it?
• OLD
Hey... Tully & Fisher were fellow Astronomy grad students with me at the University of Maryland!
• (They both got PhDs & I got a draft notice from Nixon. )
Art Neuendorffer
Thank you for serving, Art!!
Last edited by bystander on Tue Nov 05, 2019 5:04 pm, edited 1 time in total.
Reason: reformatted
Chris Peterson
Abominable Snowman
Posts: 16210
Joined: Wed Jan 31, 2007 11:13 pm
### Re: APOD: Spiral Galaxies Spinning Super-Fast (2019 Nov 05)
dlw wrote:
Tue Nov 05, 2019 4:40 pm
Is it possible to estimate a maximum angular velocity within these galaxies? Stating that "UGC 12591 spins at about 480 km/sec" doesn't give me a sense of how fast it is spinning.
Well, the rotation speed changes with radial distance from the center. If we take the stated speeds to be the asymptotic speeds, that would be close to the speed at the outside edge of a galaxy. The paper places the size of most of these galaxies at around 200,000 ly diameter. So that means that they take about 300 million years to make a single rotation. Say an angular velocity of around one degree per million years.
Chris
*****************************************
Chris L Peterson
Cloudbait Observatory
http://www.cloudbait.com
neufer
Posts: 18555
Joined: Mon Jan 21, 2008 1:57 pm
Location: Alexandria, Virginia
### Re: APOD: Spiral Galaxies Spinning Super-Fast (2019 Nov 05)
Chris Peterson wrote:
Tue Nov 05, 2019 5:02 pm
dlw wrote:
Tue Nov 05, 2019 4:40 pm
Is it possible to estimate a maximum angular velocity within these galaxies?
Stating that "UGC 12591 spins at about 480 km/sec" doesn't give me a sense of how fast it is spinning.
Well, the rotation speed changes with radial distance from the center. If we take the stated speeds to be the asymptotic speeds, that would be close to the speed at the outside edge of a galaxy. The paper places the size of most of these galaxies at around 200,000 ly diameter. So that means that they take about 300 million years to make a single rotation. Say an angular velocity of around one degree per million years.
Noting that:
a more meaningful answer for a galaxy 4 times the mass of the Milky Way
might be to simply half the rotation periods stipulated for the Milky Way:
Milky Way:
Code: Select all
Sun's Galactic rotation period 240 Myr
Spiral pattern rotation period 220–360 Myr
Bar pattern rotation period 100–120 Myr
UGC 12591:
Code: Select all
Typical star's rotation period 120 Myr
Spiral pattern rotation period 110–180 Myr
Bar pattern rotation period 50–60 Myr
Say a maximal angular velocity of around 6 degrees per million years...to keep Danish kids happy.
Art Neuendorffer
Ann
4725 Å
Posts: 11641
Joined: Sat May 29, 2010 5:33 am
### Re: APOD: Spiral Galaxies Spinning Super-Fast (2019 Nov 05)
neufer wrote:
Tue Nov 05, 2019 9:22 pm
Chris Peterson wrote:
Tue Nov 05, 2019 5:02 pm
dlw wrote:
Tue Nov 05, 2019 4:40 pm
Is it possible to estimate a maximum angular velocity within these galaxies?
Stating that "UGC 12591 spins at about 480 km/sec" doesn't give me a sense of how fast it is spinning.
Well, the rotation speed changes with radial distance from the center. If we take the stated speeds to be the asymptotic speeds, that would be close to the speed at the outside edge of a galaxy. The paper places the size of most of these galaxies at around 200,000 ly diameter. So that means that they take about 300 million years to make a single rotation. Say an angular velocity of around one degree per million years.
Noting that:
a more meaningful answer for a galaxy 4 times the mass of the Milky Way
might be to simply half the rotation periods stipulated for the Milky Way:
Milky Way:
Code: Select all
Sun's Galactic rotation period 240 Myr
Spiral pattern rotation period 220–360 Myr
Bar pattern rotation period 100–120 Myr
UGC 12591:
Code: Select all
Typical star's rotation period 120 Myr
Spiral pattern rotation period 110–180 Myr
Bar pattern rotation period 50–60 Myr
Say a maximal angular velocity of around 6 degrees per million years...to keep Danish kids happy.
Very good point, Art. I, too, thought that a diameter of 200 000 light-years didn't sound like all that much for a humongous-sized galaxy.
Perhaps the luminous disk of the Milky Way really is about 100 000 light-years in diameter, but a low-luminosity halo or even outer disk makes our galaxy twice as wide? And possibly those super-large spiral galaxies also have low-luminosity outer disks that make them almost twice as big as they appear to be in our photographs of them?
Ann
Color Commentator | {"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": 0.8046239614486694, "perplexity": 4618.470517377414}, "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-2021-49/segments/1637964358842.4/warc/CC-MAIN-20211129194957-20211129224957-00308.warc.gz"} |
https://en.wikibooks.org/wiki/Circuit_Theory/TF_Examples/Example34/io | # Circuit Theory/TF Examples/Example34/io
## Starting Point
Starting point of io looks good. Take integral to visit Vc initial condition. Then get expression for Vtotal, then take integral of that to visit IL initial condition. Lots of integrals.
## Transfer Function
${\displaystyle H(s)={\frac {i_{o}}{i_{s}}}={\frac {\frac {1}{R_{2}+{\frac {1}{sC}}}}{{\frac {1}{sL}}+{\frac {1}{R_{1}}}+{\frac {1}{R_{2}+{\frac {1}{sC}}}}}}}$
L :=1;
R1 := 1/2;
C := 1/2;
R2 := 1.5;
simplify((1/(R2 + 1/(s*C))/(1/(s*L) + 1/R1 + 1/(R2 + 1/(s*C))))
${\displaystyle {\frac {i_{o}}{i_{s}}}={\frac {2s^{2}}{8s^{2}+11s+4}}}$
## Homogeneous Solution
Set the denominator of the transfer function to 0 and solve for s:
solve(8*s^2 + 11*s + 4)
${\displaystyle s_{1,2}={\frac {-11\pm {\sqrt {7}}i}{16}}}$
So the solution is going to have the form:
${\displaystyle i_{o_{h}}=e^{\frac {11t}{16}}(A\cos {\frac {7}{16}}+B\sin {\frac {7}{16}})}$
## Particular Solution
After a very long time the inductor shorts, all the current flows through it so:
${\displaystyle i_{o_{p}}=0}$
## Initial Conditions
MuPad screen shot finding the first equation associated with the constants
Adding the particular and homogeneous solutions, get:
${\displaystyle i_{o}(t)=i_{o_{p}}+i_{o_{h}}+C=e^{-{\frac {11t}{16}}}(A\cos {\frac {7t}{16}}+B\sin {\frac {7t}{16}})+C}$
Doing the final condition again, get:
${\displaystyle i_{o}(\infty )=0=C\Rightarrow C=0}$
Let's try for Vc first. From the terminal relation for a capacitor:
${\displaystyle V_{c}(t)={\frac {1}{C}}\int i_{o}dt}$
io := exp(-11*t/16)*(A*cos(7*t/16) + B*sin(7*t/16))
VC := 2* int(io,t)
We know that initially Vc = 1.5 so at t=0 can find equation for A and B:
t :=0;
solve(1.5 = VC)
At this point mupad goes numeric and get this equation:
${\displaystyle A=-0.6363636364B-0.7244318182}$
MuPad code for finding the second equation associated with the constants
Need another equation. Can find Vt by adding Vr and Vc. Then from Vt can find expression for the current through the inductor and visit it's initial condition. Need to start over in MuPad because t=0 has ruined the current session. So repeating the setup of VC:
io := exp(-11*t/16)*(A*cos(7*t/16) + B*sin(7*t/16))
VC := 2*int(io,t)
The integration constant is going to be zero because after a long time VC is zero (the inductor shorts).
VT := VC + io*1.5
From the terminal equation for an inductor:
${\displaystyle I_{L}(t)={\frac {1}{L}}\int V_{T}dt}$
IL := int(VT,t)
At this point have to figure out the integration constant. After a long time, the inductor's current is going to be 1 because it shorts the current source. Looking at IL in the mupad window can see that every term is multiplied by e-0.6875t which is going to zero as t goes to ∞. This means the integration constant is 1.
So add 1 to IL, then set t=0 and IL = 0.5 and again solve for A and B:
t :=0;
solve (IL + 1 = 0.5)
Get this equation:
${\displaystyle A=6.273453094B+1.802644711}$
Now need to solve the two equations and two unknowns:
solve([A = - 0.6363636364*B - 0.7244318182,A = 6.273453094*B + 1.802644711],[A,B])
Get:
${\displaystyle A=-0.4916992187,B=-0.3657226563}$
So now have time domain expression for io step response:
${\displaystyle i_{o}(t)\mu (t)=e^{-{\frac {11t}{16}}}(-0.492\cos {\frac {7t}{16}}-0.366\sin {\frac {7t}{16}})}$
## Impulse Response
MuPad code for finding the impulse response and using the convolution integral
The impulse response is the derivative of the step response:
i_u := exp(-11*t/16)*(-0.4916992187 * cos(7*t/16) - 0.3657226563*sin(7*t/16))
i_s := diff(i_u,t)
## Convolution Integral
The first step is to substitute into i_s for t:
i_sub := subs(i_s, t = y-x):
Now form the convolution integral:
f := i_sub*(1 + 3*cos(2*x)):
io := int(f,x = 0..y)
Replacing y with t:
i_o :=subs(io, y=t)
There is going to be an integration constant. This value can not be determined because the driving function oscillates. The initial conditions of the inductor and capacitor have already been visited. More information (like a specific value at a future time) is needed in order to compute an integration constant.
${\displaystyle i_{o}=0.335sin(2t)-0.0177cos(2t)-0.474e^{-0.6875t}cos(0.438t)-0.648e^{-0.6875t}sin(0.438t)+0.492}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 14, "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": 0.8092532157897949, "perplexity": 2521.1453096334267}, "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-2017-26/segments/1498128320489.26/warc/CC-MAIN-20170625101427-20170625121427-00428.warc.gz"} |
http://math.stackexchange.com/users/22446/thomas-e?tab=activity&sort=all&page=46 | Thomas E.
less info
reputation
2927
bio website location age 23 member for 2 years, 7 months seen yesterday profile views 528
Doing math.
1,143 Actions
May22 revised 'Nested Intervals Theorem' in $\mathbb{R}^2$ Added comment to the last question. May22 revised 'Nested Intervals Theorem' in $\mathbb{R}^2$ some grammar corrections. May22 answered 'Nested Intervals Theorem' in $\mathbb{R}^2$ May20 comment Has Euler's Constant $\gamma$ been proven to be irrational? @AD.: These people must be the smartest MS word users on this planet. :-) May20 comment Has Euler's Constant $\gamma$ been proven to be irrational? If you look at the other uploads of the same author, you see that he has also proven the Riemann hypothesis among many others. It's pretty obvious now. May20 comment Has Euler's Constant $\gamma$ been proven to be irrational? Looks like a word-document and not even latex. That's usually a sign to be extra careful. May20 revised Showing the lebesgue measure is sigma finite on sets added LaTeX May20 suggested suggested edit on Showing the lebesgue measure is sigma finite on sets May20 comment Prove Continuous functions are borel functions Hint: You may want to use the fact that the pre-image of an open set is open under a continuous function. May20 revised Prove Continuous functions are borel functions Added LaTeX May20 suggested suggested edit on Prove Continuous functions are borel functions May20 revised If $M$ is complete and $f : (M,d)\to(N,p)$ is continuous, then $f(M)$ is complete? added 4 characters in body May20 revised If $M$ is complete and $f : (M,d)\to(N,p)$ is continuous, then $f(M)$ is complete? added 689 characters in body May20 answered If $M$ is complete and $f : (M,d)\to(N,p)$ is continuous, then $f(M)$ is complete? May20 comment Sieves and topology @DaveL.Renfro: Thanks. I'll try to get as many of those in my hands as possible. May20 comment Sieves and topology Sure, I'll take a look on both of them. Thanks once again, this was really helpful. May20 accepted Sieves and topology May20 comment Sieves and topology It's amazing how clear this is now. Thanks. And about reading the book further: I can't wait :-) May20 comment Sieves and topology Alright, I think I got the idea now. Thanks for all the help. @t.b.: if you want to post it as an answer I will accept it. May20 comment Sieves and topology @SimonMarkett: Nope, sorry. This topic is very new for me. | {"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": 0.9201318025588989, "perplexity": 1460.6346875261809}, "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-2014-35/segments/1408500830834.3/warc/CC-MAIN-20140820021350-00024-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/what-is-the-resulting-increase-in-the-speed-of-the-flatcar.158476/ | What is the resulting increase in the speed of the flatcar?
1. Feb 27, 2007
norcal
1. The problem statement, all variables and given/known data
A man (weighing 915 N) stands on a long railroad flatcar (weighing 2805 N) as it rolls at 18.0 m/s in the positive direction of an x axis, with negligible friction. Then the man runs along the flatcar in the negative x direction at 40.00 m/s relative to the flatcar. What is the resulting increase in the speed of the flatcar?
2. Relevant equations
Pi=Pf
m1v1=m2v2
3. The attempt at a solution
(915+2805)(18)=(915)(-40+18)+(2805)(v+18)
v=13.05 m/s
Well this is the wrong answer and I have no clue why. I know that the fact that the weights are given and not mass is not an issue since W=ma and a would cancel out on both sides so I cannot figure out why this answer is wrong. Anything else that I can try???
2. Feb 27, 2007
pakmingki
Hm
well
my physics teacher always says when you are dealing with speed or velocity, you always use energy.
So, since the net vertical change is 0, change in potential energy is negligible.
PEi + KEi = PEf + KEf
You can rearrange to get -(deltaPE) = deltaKE. We know that deltaPE = 0, so we have this formula
deltaKE=0
KEf - KEi = 0
KEf = summation of final kinetic energy
KEi = summartion of initial kinetic energy
you can find m because w = mg and you are given w and g is known.
Once you find m, use the above KE equations and solve for v final of flatcar, and that should give you the right answer.
3. Feb 27, 2007
robb_
why do you have (v+18)?
4. Feb 27, 2007
robb_
pakmingki- be careful here. Total mechanical energy is not conserved here.
5. Feb 27, 2007
pakmingki
Why wouldn't it be? The only force that could possibly act against the system is friction, which is negligible.
6. Feb 27, 2007
norcal
I have v+18 because its asking for the resulting increase in speed. do you have a suggestion of doing it another way???
7. Feb 27, 2007
norcal
KE is not involved in this problem, only conservation of momentum.
8. Feb 27, 2007
robb_
Looks inelastic to me....
but it is late cause I dont see anything wrong with the work shown above. :zzz:
9. Feb 27, 2007
pakmingki
well,
using KE could be a different approach to the problem, and if there is mass and velocity involved, it could very well be a KE problem as much as a P problem.
10. Feb 27, 2007
robb_
KE is not conserved in inelastic collisions.
11. Feb 27, 2007
norcal
maybe so but in the section of this book it says not to use KE as KE is conserved. Thus, using KE for this problem would be ridiculous.
12. Feb 27, 2007
pakmingki
if its an inelastic collision, you can use the inelastic collision equation.
Vfinal = sum P / sum m
where Vfinal is the velocity of the system, which is the man and the train. | {"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": 0.8667131066322327, "perplexity": 875.6090845530288}, "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-2016-44/segments/1476988721392.72/warc/CC-MAIN-20161020183841-00004-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://algassert.com/math/2015/09/17/Misconceptions-Unintegrable.html | # My Misconceptions: The Unintegrable Function
17 Sep 2015
Sometimes, I have ideas. Dumb ideas. Usually I catch them, but sometimes I don't.
In "My Misconceptions" posts, I poke fun at myself while discussing wrong things I've thought.
# From Undifferentiable to Unintegrable
Historically, there was a time when people thought that every continuous function was differentiable. People realized their error when pathological counter-examples, such as the Weierstrass_function, were discovered.
For example, the function $f(x) = \Sum{n=0}{\infty} 2^{-n} \sin(4^n x)$ is continuous (due to higher terms in the sum being bounded between $\pm 2^{-n}$). But, if you differentiate $f$, then the $2^{-n}$ term gets multiplied by the $4^n$ factor inside the $\sin$ and you end up with $f'(x) = \Sum{n=0}{\infty} 2^n \cos(4^n x)$. That sum fails to converge, because later terms are multiplied by $2^n$ instead of $2^{-n}$ and so can be arbitrarily large. The derivative does not exist.
When I learned about the above trick for breaking differentation, I had an idea: why not break integration with the same idea in reverse? Just replace $\sin(4^n x)$ with $\sin(4^{-n} x)$, since integrating will divide by the internal $\sin$ factor instead of multiplying by it:
$g(x) = \Sum{n=0}{\infty} 2^{-n} \sin(4^{-n} x)$
$(\int g)(x)$ $= \int \Sum{n=0}{\infty} 2^{-n} \sin(4^{-n} x) dx$
$= \Sum{n=0}{\infty} \int 2^{-n} \sin(4^{-n} x) dx$
$= -\Sum{n=0}{\infty} 2^n \cos(4^{-n} x)$
Notice that the infinite sum in the resulting expression for $\int g$ fails to converge (it always diverges towards infinity). And suppose for the moment that swapping the order of the infinite sum and the integral was justified. Clearly $\int g$ doesn't exist.
At least, that's what I thought for an embarassingly long time.
# Fundamentally and Constantly Wrong
What's especially bone-headed about this mistake is how obvious it is, at least in hindsight.
How obvious? Well, my conclusion violates a somewhat well-known theorem. A theorem that says that every continuous function (e.g. $g$) has an integral. You may have heard of it; it's called "The Fundamental Theorem of Calculus".
Eventually someone took pitty on me (or rather, got angry at me for being wrong on the internet) and pointed something out:
$h(x) = -\Sum{n=0}{\infty} 2^n (\cos(4^{-n} x) - 1)$
$h'(x) = g(x)$
Oh.
Well.
That's definitely the worst forgetting-the-integration-constant mistake I've ever made.
# Summary
If you're going to play fast and loose with the order of integrals and infinite sums, the consequences of forgetting your constant(s) of integation might get infinitely worse. | {"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": 0.9652265310287476, "perplexity": 757.261470636718}, "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-2017-13/segments/1490218189472.3/warc/CC-MAIN-20170322212949-00282-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://quantumcomputing.stackexchange.com/questions/17997/what-is-the-observable-to-measure-spin-along-some-direction-for-a-2-qubit-system | # What is the observable to measure spin along some direction for a 2 qubit system?
Let $$M$$ be the $$2 \times 2$$ matrix corresponding to the observable to measure spin along some arbitrary axis $$\vec{v}$$. This matrix is given by following formula:
$$$$M = v_x X + v_y Y + v_z Z$$$$
where $$X, Y, Z$$ are $$2 \times 2$$ Pauli matrices and $$\vec{v} = (v_x, v_y, v_z)$$. E.g., see this
Now if I have a 2 qubit system, then what is the $$4 \times 4$$ matrix to measure the spin of the first qubit along $$\vec{v}$$?
• It should be $M\otimes I$. Jun 17 at 1:59
As @narip pointed out in a comment, it should be $$M \otimes I$$. Moreover, if you want to measure the second qubit along some axis $$\vec{n}$$ with corresponding matrix $$N$$, you can do $$M \otimes N$$.
Expanding this, we get
\begin{align} M \otimes N &= (v_x X_1 + v_y Y_1 + v_z Z_1) \otimes (n_x X_2 + n_y Y_2 + n_z Z_2) \\ &= v_xn_x X_1 X_2 + v_xn_yX_1Y_2 + v_xn_zX_1Z_2 \\ & \hspace{0.2in} + v_yn_xY_1X_2 + v_yn_yY_1Y_2 + v_yn_zY_1Z_2 \\ & \hspace{0.2in} + v_zn_xZ_1X_2 + v_zn_yZ_1Y_2 + v_zn_zZ_1Z_2 \end{align}
where $$X_1X_2 = X\otimes X$$. (Wrote it like this to avoid writing a lot of $$\otimes$$s.) As @DaftWullie pointed out in his comment, this way you will get an observable with two eigenvalues and therefore your measurement will have two possible values.
Update
Following with the discussion on the comments, if you want are expecting to get four possible outcomes out of the measurement, you can do something like $$M_1 + 2N_2$$ given that $$M$$ and $$N$$ have eigenvalues $$\pm 1$$. Since both of these are of the form
\begin{align} \sigma_i &= \begin{pmatrix} \cos\theta & \sin\theta\cos\phi - i\sin\theta\sin\phi \\ \sin\theta\cos\phi + i\sin\theta\sin\phi & -\cos\theta \end{pmatrix} \\ &= \begin{pmatrix} \cos\theta & \sin\theta(\cos\phi - i\sin\phi) \\ \sin\theta(\cos\phi + i\sin\phi) & -\cos\theta \end{pmatrix} \\ &= \begin{pmatrix} \cos\theta & e^{-i\phi}\sin\theta \\ e^{i\phi}\sin\theta & -\cos\theta \end{pmatrix}, \end{align}
and therefore their characteristic equation is
\begin{align} \text{det}(\sigma_i - \lambda I) &= -(\cos\theta - \lambda)(\cos\theta + \lambda) - \sin^2\theta \\ &= -\cos^2\theta + \lambda^2 - \sin^2\theta \\ &= \lambda^2 - 1 = 0, \end{align}
which gives us $$\lambda = \pm 1$$, we are safe using $$M_1 + 2N_2$$.
• $X\otimes X$ as an observable has two eigenvalues and hence the measurement has two outcomes. If you want to measure both qubits, you're probably expecting 4 outcomes (or, at least, you need a more precise statement of what you're expecting). In which case, you might be better with something like $X_1+2X_2$. Jun 17 at 8:05
• @DaftWullie you're right, thanks for your observation! Just some clarifications. Why is the coefficient of $2$ introduced? Would a observable like $M_1 + 2N_2$ with $M,N$ as defined above also work? Jun 17 at 10:50
• That depends on the eigenvalues of your operators. If $M$ and $N$ both have eignevalues $\pm 1$, it'll work fine. The trick is just to make sure that all the eigenvalues are distinct, i.e. $\pm1\pm 2$ doesn't have any repetitions, whereas if I'd just done $X_1+X_2$, the eigenvalues would have been $\pm1\pm 1$, and the 0 eigenvalue would have been repeated. Jun 17 at 12:00
• @DaftWullie I see, thank you! I didn’t know about this, could you point me to some theorem/concept/whatever where this is introduced? Jun 17 at 13:35
• I don't know if there is one. This is just the understanding I've come to over time. Jun 17 at 14:25 | {"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": 24, "wp-katex-eq": 0, "align": 3, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9999377727508545, "perplexity": 605.1680808570691}, "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/1634323585265.67/warc/CC-MAIN-20211019105138-20211019135138-00029.warc.gz"} |
https://stats.stackexchange.com/questions/114553/pymc-confusion-are-observed-nodes-fixed-or-stochastic | # PYMC Confusion: are observed nodes fixed or stochastic?
I've been trying to gain a better understanding of factor potentials in PYMC. In reading this article by Cam Davidson-Pilon on Yhat, I got confused about how observed nodes are understood by PYMC.
Are they fixed in PYMC when observed=True? Or are they random like a stochastic node?
The factor potential is specified as:
@mc.potential
def censorfactor(obs=obs):
return -100000
else:
return 0
The variable birth and lifetime_ are fixed numpy arrays. Hence, it seems, the only changeable variable is obs. But I thought that was fixed in PYMC because it is defined by:
obs = mc.Weibull( 'obs', alpha, beta, value = lifetime_, observed = True )
If obs is fixed then the potential function will return a log probability of -100000 every time.
I suspect I'm missing a fundamental point of MCMC here. Appreciate if someone can help me understand.
UPDATE
I did some additional research and it appears that observed nodes can indeed be (partly) stochastic. If you pass a masked array as observed data to a stochastic node, each of the masked values will be individually represented as a stochastic node.
Chris Fonnesbeck's tutorial from this year's Scipy conference covers this topic at the end of this IPython notebook. Here he describes how the posterior predictive distribution is used to estimate each of the missing data points:
$$p(\tilde{y}|y) = \int p(\tilde{y}|\theta) f(\theta|y) d\theta$$
The ability to impute missing data conditional on the data and the model parameters (simultaneously) a powerful technique. Plus we get traces, distributions, HDI etc for each missing value. | {"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": 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": 0.9026162028312683, "perplexity": 1978.4458313575406}, "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-2020-45/segments/1603107905965.68/warc/CC-MAIN-20201029214439-20201030004439-00265.warc.gz"} |
http://en.citizendium.org/wiki/Complex_number | Citizendium - building a quality free general knowledge encyclopedia. Click here to join and contribute—free Many thanks December donors; special to Darren Duncan. January donations open; need minimum total \$100. Let's exceed that. Donate here. By donating you gift yourself and CZ.
# Complex number
Main Article
Talk
Related Articles [?]
Bibliography [?]
Citable Version [?]
This editable Main Article has an approved citable version (see its Citable Version subpage). While we have done conscientious work, we cannot guarantee that this Main Article, or its citable version, is wholly free of mistakes. By helping to improve this editable Main Article, you will help the process of generating a new, improved citable version.
Complex numbers are numbers of the form $\scriptstyle a+b\cdot i$, where $a$ and $b$ are real numbers and $i$ denotes a number satisfying $\scriptstyle i^{2}=-1$.[1] Of course, since the square of any real number is nonnegative, $i$ cannot be a real number. At first glance, it is not even clear whether such an object exists and can be reasonably called a number; for example, can we sensibly associate with $i$ natural operations such as addition and multiplication? As it happens, we can define mathematical operations for these "complex numbers" in a consistent and sensible way and, perhaps more importantly, using complex numbers provides mathematicians, physicists, and engineers with an extremely powerful approach to expressing parts of these sciences in a convenient and natural way.
## Historical Development
A common complaint among math students is why they must bother with complex numbers when real numbers almost always seem sufficient for applications. Indeed, many familiar real world quantities, such as distance, temperature, and time are best described using real numbers. In contrast, although there are many physical situations which are mostly aptly described with complex numbers, most of these situations require some relatively sophisticated knowledge of physics, and the phenomena which they describe are more abstruse than distance, temperature, or time.
Many modern students first encounter complex numbers when solving quadratic equations, which can have complex number solutions. This presentation is historically misleading — the quadratic formula precedes recognition of the utility of complex numbers by many centuries. The ancients could simply dismiss quadratic equations whose solutions involve complex numbers as having no real roots.
The need for complex numbers might have appeared for the first time during the sixteenth century, when Italian mathematicians like Scipione del Ferro, Niccolò Fontana Tartaglia, Gerolamo Cardano and Rafael Bombelli tried to solve cubic equations. Even for equations with three real solutions, the method they used sometimes required calculations with numbers whose squares are negative. A historical example of this can be found on the "advanced" subpage for this article.
In modernity, complex numbers form the basis for the mathematical models of many physical phenomena, including electro-magnetism and quantum mechanics. Even so, it is still often difficult to find examples where an equivalent mathematical model cannot be formulated using pairs of real numbers instead.
## Working with complex numbers
As a first step in giving some legitimacy to the "number" $\scriptstyle \sqrt{-1}$, we will explain how to compute with it. How do you add, multiply and divide expressions with this number? It turns out that this is not that difficult; the main rule to keep in mind is that the square of $\scriptstyle \sqrt{-1}$ equals $\scriptstyle -1$.
In the remainder of the article, we will use the letter $i$ to denote one solution of the equation $\scriptstyle i^2 = -1$, where we previously used $\scriptstyle \sqrt{-1}$.[2] With this convention, all complex numbers can be written as $a + bi$, where $a$ and $b$ are real numbers. We call $a$ the real part of the complex number and $b$ the imaginary part. The complex number $a + 0\cdot i$ whose imaginary part is zero is considered to be the same thing as the real number $a$.
### Basic operations
Addition of complex numbers is straightforward, $\scriptstyle (a + b\cdot i) + (c + d\cdot i) = (a + c) + (b + d) \cdot i.$ The result is again a complex number.
Multiplication is more interesting. Suppose we want to compute $\scriptstyle (a+b \cdot i)\cdot(c+d\cdot i)$. Using $\scriptstyle i^2 = -1$, we can rewrite this product in a form which clearly shows it to be another complex number:
$(a + b\cdot i)\cdot (c + d\cdot i) = a\cdot c + a\cdot d\cdot i + b\cdot c\cdot i + b\cdot d\cdot i^2 = (a\cdot c - b\cdot d) + (b\cdot c + a\cdot d)i. \$
To handle division, we simply note that $(c + d\cdot i)\cdot (c - d\cdot i) = c^2 +d^2$, so, provided that c and d are not simultaneously zero,
$\frac{1}{c + d\cdot i} = \frac{c - d\cdot i}{c^2 + d^2},$
from which it follows that
$\frac{a + b\cdot i}{c + d\cdot i} = \frac{(a\cdot c + b\cdot d) + (b\cdot c - a\cdot d)i}{c^2 + d^2}.$
If $c = d = 0$ then division by $c+d\cdot i$ is not defined.
Going a bit further, we can introduce the important operation of complex conjugation. Given an arbitrary complex number $z = x + y\cdot i$, we define its complex conjugate to be $\scriptstyle \bar{z} = x - y \cdot i$. Using the identity $(a + b)\cdot(a - b) = a^2 - b^2$ we derive the important formula
$z \bar{z} = x^2 + y^2$
and we define the modulus of a complex number z to be
$|z| = \sqrt{z \bar{z}}$
Note that the modulus of a complex number is always a nonnegative real number. The modulus (also called absolute value) satisfies three important properties that are completely analogous to the properties of the absolute value of real numbers
• $|z| \ge 0$; furthermore, $|z| = 0$ if and only if $z = 0$
• $|z_1 z_2| = |z_1| |z_2| \$
• $|z_1 + z_2 | \le |z_1| + |z_2|$
The last inequality is known as the triangle inequality.
### The complex exponential
Recall that in real analysis, the ordinary exponential function may be defined as
$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$
The same series may be used to define the complex exponential function
$e^z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots$
(where, of course, convergence is defined in terms of the complex modulus, instead of the real absolute value).
The complex exponential has the same multiplicative property that holds for real numbers, namely
$e^{z_1 + z_2} = e^{z_1} e^{z_2} \$
The complex exponential function has the important property that
$e^{i\theta} = \cos \theta + i \sin \theta \$
as may be seen immediately by substituting $z = i\theta$ and comparing terms with the usual power series expansions of $\sin \theta$ and $\cos \theta$.
The familiar trigonometric identity
$\sin^2 \theta + \cos^2 \theta = 1 \$
immediately implies the important formula
$|e^{i\theta}| = 1$, for any $\theta \in \mathbb{R}.$
Another way to establish this identity is to note that $\scriptstyle \overline{e^{i\theta}} = e^{-i\theta}$, so
$|e^{i\theta}|^2 = e^{i\theta}e^{-i\theta} = e^0 = 1. \$
### Geometric interpretation
Graphical representation of a complex number and its conjugate
Since a complex number $z = x + iy$ is specified by two real numbers, namely $x$ and $y$, it can be interpreted as the point $(x,y)$ in the plane. When complex numbers are represented as points in the plane, the resulting diagrams are known as Argand diagrams, after Robert Argand. The geometric representation of complex numbers turns out to be very useful, both as an aid to understanding the properties of complex numbers and as a tool in applying complex numbers to geometrical and physical problems.
There are no real surprises when we look at addition and subtraction in isolation: addition of complex numbers is not essentially different from addition of vectors in $\scriptstyle \mathbb{R}^2$. Similarly, if $\scriptstyle \alpha \in \mathbb{R}$ is real, multiplication by $\alpha$ is just scalar multiplication. In $\scriptstyle \mathbb{C}$ we have
$z_1 + z_2 = (x_1 + iy_1) + (x_2 + iy_2) = (x_1 + x_2) + i(y_1 + y_2) \$
and
$\alpha z = \alpha(x + i \cdot y) = \alpha x + i\alpha y. \,$
To put it succinctly, $\scriptstyle \mathbb{C}$ is a 2-dimensional real vector space with respect to the usual operations of addition of complex numbers and multiplication by a real number. There doesn't seem to be much more to say. But there is more to say, and that is that the multiplication of complex numbers has geometric significance. This is most easily seen if we take advantage of the complex exponential, and write complex numbers in polar form
$z = r\cdot e^{i\theta}.$
Here, r is simply the modulus $\scriptstyle |z| = \sqrt{x^2 + y^2}$ or vector length. The number $\theta$ is just the angle formed with the $x$-axis, and is called the argument satisfying the condition $\rm tan \theta = y/x$. Now, when complex numbers are written in polar form, multiplication is very interesting
$z_1 z_2 = (r_1 e^{i\theta_1}) (r_2 e^{i\theta_2}) = r_1 r_2 e^{i(\theta_1 + \theta_2)}.$
Multiplication by $i$ amounts to rotation by 90 degrees
In other words, multiplication by a complex number $z$ has the effect of simultaneously scaling by the number's modulus and rotating by its argument. This is really astounding. For example, to multiply a given complex number $z$ by $i$ we need only to rotate $z$ by $\pi/2$ (that is, 90 degrees). Translation corresponds to complex addition, scaling to multiplication by a real number, and rotation to multiplication by a complex number of unit modulus. The one type of coordinate transformation that is missing from this list is reflection. On the other hand, there is an arithmetic operation we have not considered, and that is division. Recall that for non-zero $z$
$\frac{1}{z} = \frac{\bar{z}}{|z|^2}.$
Division of a complex number $z_1$ by a non-zero complex number $z_2$ can then be interpreted as multiplication of $z_1$ by $\frac{1}{z_2}$. This in turn corresponds to scaling of the modulus of $z_1$ by the inverse of the modulus of $z_2$ and a rotation of its argument by the negative of the argument of $z_2$. That is,
$\frac{z_1}{z_2}=z_1 (\frac{1}{z_2})=\frac{1}{|z_2|^2}z_1 \overline{z_2}=\frac{|z_1|}{|z_2|}e^{i(\theta_1-\theta_2)},$
where $\theta_1,\theta_2$ are the arguments of $z_1,z_2$, respectively.
Returning to the representation of complex numbers in rectangular form, we note that complex conjugation is just the transformation (or map) $\scriptstyle x + iy \;\mapsto\; x - iy$ or, in vector notation, $\scriptstyle (x, y)\; \mapsto \;(x, -y)$. This is nothing other than reflection in the $x$-axis, and any other reflection may be obtained by combining that transformation with rotations and translations.
Historically, this observation was very important and led to the search for higher dimensional algebras that could "arithmetize" Euclidean geometry. It turns out that there are such generalizations in dimensions 4 and 8, known as the quaternions and octonions (also known as Cayley numbers). At that point, the process stops, but the ideas developed in this process have played an important role in the development of modern differential geometry and mathematical physics).
## Algebraic closure
An important property of the set of complex numbers is that it is algebraically closed. This means that any non-constant polynomial with complex coefficients has a complex root. This result is known as the Fundamental Theorem of Algebra.
This is actually quite remarkable. We started out with the real numbers. There are many polynomials with real coefficients that do not have a real root. We took just one of these, the polynomial $x^2+1$, and we introduced a new number, $i$, which is defined to be a root of the polynomial. Suddenly, all non-constant polynomials have a root in this new setting where we allow complex numbers.
There are many proofs of the Fundamental Theorem of Algebra. Many of the simplest depend crucially on complex analysis. But it is by no means necessary to rely on complex analysis here. A proof using field theory is alluded to at the very end of this article.
## Formal definition
We have been treating complex numbers very much like real numbers and found that they can be very useful, but we have not yet proven that they exist or that they can be used without running into contradictions. In fact, it is quite easy to go wrong when using complex numbers. Consider for instance the following computation:
$-1=\sqrt{-1}\times\sqrt{-1}=\sqrt{(-1)\times(-1)}=\sqrt{1}=1.$
This computation seems to show that $-1$ equals $1$, which is nonsense. The point is that the second equality can not be applied. Positive real numbers satisfy the identity
$\sqrt{a}\times\sqrt{b} = \sqrt{a \times b},$
but this identity does not hold for negative real numbers, whose square roots are not real, because the square root symbol denotes only the positive solution to $x^2 = a$.
One possibility to feel more secure when using complex numbers is to define them in terms of constructs which are better understood. This approach was taken by Hamilton, who defined complex numbers as ordered pairs of real numbers, that is,
$\mathbb{C}= \{ (a,b) \colon a,b\in \mathbb{R} \}.$
Addition and multiplication of such pairs can be defined as follows:
• addition: $(a, b) + (c, d) = (a + c, b + d)$
• multiplication: $(a, b)(c, d) = (ac - bd, bc + ad)$
The multiplication may look artificial, but it is inspired by the formula
$(a + b\cdot i)(c + d\cdot i) = (a\cdot c - b\cdot d) + (b\cdot c + a\cdot d)\cdot i. \$
which we derived before.
These definitions satisfy most of the basic properties of addition and multiplication of real numbers, and we can employ many formulas from the elementary algebra we are accustomed to. More specifically, it can easily be shown that addition and multiplication as defined above are commutative and associative, and that multiplication is distributive over addition; in other words, the sum (or the product) of two numbers does not depend on the order of terms;[3] the sum (product) of three or more elements does not depend on order of operations ('we can suppress the parentheses');[4] the product of a complex number with a sum of two other numbers expands in the usual way.[5] In mathematical language this means that with addition and multiplication defined this way, $\mathbb{C}$ satisfies the axioms for a field and is called the field of complex numbers.
Now we are ready to understand the 'real' meaning of $i$. Observe that the pairs of type ($a$,0) are identical[6] to the set of reals, so we write $(a,0)=a$. Observe also that by definition $(0,1)(0,1) = (-1,0)=-1$. In other words, we can define $i$, the symbol we've been using, as the pair (0,1). In this way we have a way of indicating which one we mean of the two solutions of the equation $i^2=-1$; the other is now denoted (0,-1).
Another way to define the complex numbers comes from field theory. Because $x^2+1$ is irreducible in the polynomial ring $\mathbb{R}[x]$, the ideal generated by $x^2+1$ is a maximal ideal.[7] Therefore, the quotient ring $\mathbb{C}=\mathbb{R}[x]/\left(x^2+1\right)$ is a field. We can choose the polynomials of degree at most 1 as the representatives for the equivalence classes in this quotient ring. So in a sense, we can imagine that the dummy variable $x$ is the imaginary number $i$, and the elements of the quotient ring behave exactly the way we expect the complex numbers to behave. For example, $x^2$ is in the same equivalence class as $-1$, and so $x^2=-1$ in this quotient ring. (As a final comment in this analysis, we could next show that $\mathbb{C}$ has no finite extension and must therefore be algebraically closed.)
## Notes and references
1. This article follows the usual convention in mathematics and physics of using $i$ as the imaginary unit. Complex numbers are frequently used in electrical engineering, but in that discipline it is usual to use $j$ instead, reserving $i$ for electrical current. This usage is found in some programming languages too, notably Python.
2. Part of the reason for not using $\sqrt{-1}$ is that the symbol $\sqrt{a}$ (or $\sqrt[n]{a}$) with $a\in\mathbb{C}$ is sometimes used to denote the set of complex roots of $a$, i.e., the set of the solutions of the equation $x^2=a$ ($x^n=a$ respectively). The set contains 2 ($n$, respectively) "equally important" elements and there is no canonical way to distinguish a "representative". Consequently, no computations are performed using this symbol.
3. that is, the addition (multiplication) is commutative
4. This is called associativity
5. In other words, multiplication is distributive over addition
6. i.e., isomorphic, which basically means that the mapping $\mathbb{C}\ni (a,0)\mapsto a\in\mathbb{R},$ preserves the addition and multiplication.
7. An ideal $I = \left(f(x)\right)$ in a polynomial ring over a field is maximal if and only if $f(x)$ is irreducible over the field. | {"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": 131, "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": 0.9472900629043579, "perplexity": 175.4948569993724}, "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/1406510261771.50/warc/CC-MAIN-20140728011741-00025-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://mathoverflow.net/questions/89857/contour-integral-with-gamma-functions-and-2f1 | # Contour Integral with Gamma functions and 2F1
Given the following contour integral
$$\frac{1}{2\pi j}\int^{c+j\infty}_{c-j\infty} \frac{\Gamma(-1+a+s)\Gamma(b+s)}{\Gamma(3+a-s)}\cos(-1+a+s)\, {}_2F_1\Big(-1-a+s,-1+a+s;\frac{1}{2};z\Big) y^s\: \mathrm{d}s ,$$
where $a,b,z,y \in \mathbb{R}$, as noticed there are two poles given by
$$P^{(1)}_k = 1-a-k \quad P^{(2)}_k=-b-k \quad\text{ where }\quad k=0,1,2,\dotsc,\infty.$$
The questions are:
1. What is the suitable contour to sum the residues and solve the integral?
2. Does the zero caused by $\Gamma(3+a-s)$ cancel any of the poles?
1) I imagine this integral came from a Mellin transform approach to compute another integral. You would come up with an appropriate 'c' lying in the overlap region of definition of your two original Mellin transforms (e. g. one might be defined for $c > 1/2$ and the other for $0 < c < 1$, in which case you would choose $1/2 < c < 1$.
Then you need to estimate the behavior of your integrand for large $\Im(s)$, to determine which direction you could move the integration contour and pick up poles to develop a series for your integral.
2) Yes, the poles of the $\Gamma$ function in the denominator can cancel poles in the numerator, you just need to determine those $s$ where that will occur. This all depends on the relative values of $a$ and $b$. Also, the $\cos$ function could cancel poles.
I hope this is not too vague to be of help!
Regards, Tom
• yes, @Tom, you are definitely right. This has come as a result from applying the mellin transform method for integral evaluation. I see....the cos function will cancel poles. I get what you mean. As I understood from what you said, I need to remove the removable poles (those cancelled by the zeros) and then based on what is left I have to choose either the right or the left part of the plane for the contour. – Remy Mar 1 '12 at 17:40
• You will need to remove the poles, and you will also need to determine which direction(s) you can push the contour. This involves using asymptotic estimates for the functions in your integrand (e. g. Stirling's formula) to derive the large $t$ behavior of the integrand, where $s =\sigma + i t$. See the book Asymptotics and Mellin-Barnes Integrals. For example, if you find that the behavior is $e^{-\pi |t|}$, you can move the contour either direction since the integrand will decay to zero for any value of $\sigma$, but if you find $|t|^{\sigma-3}$ you would not be able to move past $\sigma=3$. – Tom Dickens Mar 1 '12 at 18:45
• @Remy, I've noticed your series of questions about integrals and series, and it seems like they've been related to a common goal. How is your progress? Reading between the lines, I would guess that you are working on a problem I would find interesting! – Tom Dickens Feb 9 '14 at 4:21 | {"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": 0.8428704142570496, "perplexity": 176.75688443245033}, "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-2020-29/segments/1593657134758.80/warc/CC-MAIN-20200712082512-20200712112512-00184.warc.gz"} |
https://www.aanda.org/articles/aa/full/2001/44/aa1323/aa1323.orig.html | ## Contents
A&A 379, 482-495 (2001)
DOI: 10.1051/0004-6361:20011280
## Optical spectra of five UX Orionis-type stars,
V. P. Grinin1,2 - O. V. Kozlova1 - A. Natta3 - I. Ilyin4 - I. Tuominen4 - A. N. Rostopchina1 - D. N. Shakhovskoy1
1 - Crimean Astrophysical Observatory, Crimea, 334413 Nauchny, Ukraine
2 - Pulkovo Astronomical Observatory, 196140 St. Petersburgh, Russia
3 - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy
4 - Astronomy Division, PO Box 3000, 90014, University of Oulu, Finland
Received 28 Mars 2001 / Accepted 7 September 2001
Abstract
We present in this paper several high resolution ( ) spectra of five UXORs (UX Ori, CQ Tau, BF Ori, RR Tau, WW Vul), which cover the entire visual range, from 3900 to about 8700 Å. There are between 4 and 7 spectra per star, obtained over a time interval of two years. Simultaneous or quasi-simultaneous photometric observations were also obtained at the Crimean Astrophysical Observatory. The complete, reduced and normalized spectra are available in electronic form. We show for each star a selection of the most interesting lines, and the full spectrum of UX Ori computed by averaging the spectra obtained when the star was at maximum light. For UX Ori we show also the synthetic spectrum and provide an identification of most of the lines. The spectra are too sparse to form true time sequences; however, they provide an extremely useful database for studies of UX Ori-type stars. We discuss briefly the main features of the spectra. We show that they contain many time-stable photospheric lines that can be described to a good approximation by the synthetic spectra of normal A stars with and we derive for each star effective temperature, gravity and rotational velocity. We examine the time variability of selected lines and study their connection with the photometric activity of the stars. Two different types of spectral variability are identified. One is common to all stars with circumstellar (CS) gas and is caused by perturbations of the physical and kinematic conditions of the emitting region. There is no correlation between this type of activity and the brightness variations of the star. On the contrary, a second type of spectral variability correlates well with the brightness variations and is very likely connected with the screening effect of an opaque dust cloud which sporadically intersects the line of sight. This type of variability has been observed in its simplest form in one of the RR Tau spectra, where the equivalent width of the forbidden line [O I] 6364 Å increased by a factor of about three as the star faded by approximately the same amount.
Key words: stars: individual: UX Ori, CQ Tau, BF Ori, RR Tau, WW Vul - stars: general
### 1 Introduction
This paper presents the results of high resolution spectroscopic observations of five pre-main-sequence stars of intermediate mass belonging to the UXORs group. UXORs are pre-main-sequence stars of intermediate mass, mostly found among Herbig Ae stars (HAe in the following), characterized by their typical photometric and polarimetric variability (see reviews by Thé 1994; Grinin 1994 and references therein). They all show sporadic deep minima at visual wavelengths (typically of 2-3 mag), during which the fraction of polarized light increases from typical interstellar values to a few percent. As the star fades, the radiation first becomes redder, then bluer again. The best interpretation of these phenomena requires that the star is surrounded by a disk, which scatters a fraction of the stellar radiation toward the observer (Grinin 1988). The minima are caused by a much larger than average column of intervening dust, which occasionally occults the star. At that point, the fraction of stellar radiation received directly by the observer drops, while the fraction seen after being scattered by the disk remains practically unchanged and represents a significant contribution to the total. It is likely that variable extinction is present in most HAe stars, but that we see its effects only when the circumstellar disks are seen under small inclination angles to the line of sight (Grinin et al. 1991; Natta & Whitney 2000).
The spectra of UXORs show a number of lines of photospheric and circumstellar origin. Several of them (D Na I, He I 5876, H are highly variable (Graham 1992; Grinin et al. 1994; Grady et al. 1996; de Winter 1999; Kozlova et al. 2000), showing evidence for complex motions of the circumstellar gas in the immediate vicinity of the star. Red-shifted absorption components are frequently seen in the Na I resonance lines and in some other metallic lines, as well as in the hydrogen Balmer lines. These components seem to appear and disappear on timescales of days or shorter, indicating infall of gas clumps onto the star, similar, apparently, to infalling events observed in the spectrum of Pictoris. The origin of such accretion events in UXORs is matter of debate (Sorelli et al. 1996; Grinin et al. 1996; Pérez & Grady 1997; de Winter et al. 1998; Grady et al. 2000; Natta et al. 2000a).
It is clear from all previous works that the understanding of the spectral variability of UXORs requires spectra which include many lines of different species over a wide range of wavelengths. Moreover, it is important to have simultaneous information on the brightness of the star, since the comparison of spectra taken when the star is in a deep minimum to those obtained at maximum light provides invaluable clues to the physical processes which take place in the circumstellar environment.
Many UXORs (including UX Ori itself) are among Herbig Ae/Be stars which are not associated with bright nebulosities, so that most of them were not included either in the original list of Herbig Ae/Be stars (Herbig 1960), in those of Finkenzeller & Mundt (1984) and Hamann & Persson (1992), but only in the most recent catalogue of Thé et al. (1994). This is the main reason why detailed spectroscopic studies of UXORs have begun to appear only recently (see, for example, de Winter et al. 1999; Eiroa et al. 2000).
The main purpose of this paper is to make available to the community a number of high resolution spectra of 5 of the photometrically most active stars of this type: UX Ori, CQ Tau, WW Vul, RR Tau, and BF Ori. For each star, we have between 4 and 7 spectra taken over a time interval of about two years. Each spectrum covers, with some exceptions, the whole optical region from 3900 to about 8700 Å. The amount of information they contain is huge. Although some of it has already been used by us in studies of UXORs properties (Natta et al. 2000b; Tambovtseva et al. 2001), and more work will follow, we think that the spectra may be of value for other groups as well.
Observations and data reduction are described in Sect. 2. We provide in Sects. 3 and 4 a brief discussion of the main features observed in the spectra. In Sect. 5 we re-determine the stellar parameters by comparing the most stable absorption features to the prediction of synthetic spectra. Some implications are briefly discussed in Sect. 6. The properties of the stars, as known from the literature, are summarized in Table 1.
### 2 Observations and data reduction
High-resolution spectra were collected with the SOFIN échelle spectrograph (Tuominen et al. 1999) at the 2.56 m Nordic Optical Telescope (NOT) during two observing runs in 1995 and 1996 (we thank P. Petrov for giving us time to obtain the first assessment spectrum in 1994).
We used the 3rd camera, which provides a spectral resolution per 2 CCD pixels of about 12 kms-1 with the entrance slit width of 1 7 ( ). The EEV P88200 pixels CCD was used as a detector, whose format allows coverage of all 43 spectral orders in one exposure. Given the length of each spectral order (about 140Å around H), two different spectral settings are necessary to cover the whole spectrum from 3700 to 10700Å in two exposures.
The CCD images of the échelle spectra were obtained and reduced with the 4A software package (Ilyin 2000). The standard procedure involves bias subtraction, correction for the master flat field, scattered light subtraction with aid of 2D-smoothing splines, definition of spectral orders, and weighted integration of the intensity along with elimination of cosmic spikes.
Name Sp D L Age V Max. Pol. (pc) () (Myr) (mag) (kms-1) UX Ori A3III 430 39 4 3.9 6.7 70 CQ Tau F2IV 100 6.6 10 2.2 7.9 110 BF Ori A5III 460 39 2.3 3.0 5.7 100 RR Tau A0III-IV 380 37 3 3.7 6.8 - WW Vul A3III 550 29 4 2.2 5.3 -
The wavelength calibration was done with a ThAr comparison spectrum. To reduce the effect of line drifts during long integrations caused by temporal changes in a Cassegrain mounted spectrograph, the long exposures were subdivided into 2-3 shorter ones with the ThAr spectrum taken at the beginning of the series. For each image, we used the Gaussian centered positions and wavelengths of all detected ThAr lines from all spectral orders, obtaining an accuracy of the global fit of about 50-80m in the image center.
Temporal variations in the spectrograph during integration due to change of its orientation, ambient temperature, CCD dewar nitrogen evaporation, etc., as well as the slit effect caused by variations of the collimator illumination due to change of the seeing and the differences in the placement of the star on the slit from one observation to another, all result in a small but significant spurious shift of the stellar spectral lines which cannot be completely eliminated with the ThAr spectrum. With the aid of the telluric lines, numerous in the red orders, the wavelength scale was corrected for the spurious shift, where the offset was determined by cross-correlation of the telluric orders with respect to an artificial telluric spectrum (see Ilyin & Dümmler 1999 for details). Finally, the wavelength scale of the spectra was corrected for the orbital and diurnal motions of the Earth, and the radial velocity of the star.
The bell-like shape of the stellar spectra caused by the vignetting in the 3rd short focus optical camera was corrected with use of the flat field échelle spectra taken after each object exposure. Although the shape of the vignetting function in the stellar spectrum and flat field are alike, a small discrepancy arise from the difference in the optical path of the seeing image and flat field source. Therefore, the correction for the vignetting function corrects the continuum, but may introduce some large scale deviations which are then eliminated with a continuum fitting.
We have removed the numerous telluric H2O lines that contaminate the spectra especially in the red using the spectrum of the reference stars Eri (A3III) and 59 Tau (B9V) observed during the same nights with about the same zenith angles. We used also for this purpose the telluric lines of the solar spectrum (Wallace et al. 1998). In this case the water lines were broadened by the instrumental profile and normalized to the same air mass.
The spectra are not flux-calibrated. We have normalized each spectral order to the observed continuum. This is generally easy, with the exception of those wavelength intervals dominated by the broad photospheric wings of the Balmer lines. In these cases we have used the synthetic spectra (see Sect. 5) for control.
A logbook of the observations is given in Table 2, which lists in Col. 1 the name of the star, in Col. 2 the epoch of the observation, in Col. 3 an estimate of the V magnitude of the star at that epoch, in Col. 4 the spectral setting (1 or 2) used in the observations, in Col. 5 the exposure time, in Col. 6 the signal-to-noise at Å, and in Col. 7 the equivalent width of H (see Sect. 3.1). Note that S/N decreases toward the blue part of the spectrum and at the end of each order, so that the quality of the data may vary from region to region.
The NOT spectral observations were accompanied by photometric observations with the Crimean Observatory 1.25 m telescope equipped with a five-channel photometer-polarimeter (Piirola 1975). The procedure and data reduction of these observations have been described in Grinin et al. (1988). The stellar magnitudes of the program stars in the V band are given in Table 2. In the majority of cases the photometric observations were carried out on the same night as the spectroscopic ones, in some cases the night before or after. In a few cases the time interval between the spectral and photometric observations was more then one day, and they are marked by a semicolumn in Table 2. Three stars showed significant photometric variability during the nights covered by the spectral observations, although no spectrum has been obtained for stars in deep minima. They are BF Ori ( = 0.98), CQ Tau ( = 0.80) and RR Tau ( 1.6). All the spectra of UX Ori and WW Vul were obtained when the stars were close to their bright state.
Figure 1: a) Selected lines of UX Ori. Each spectrum has been normalized to the continuum and shifted vertically for easier display. Each curve is labelled with the date of the observation. The thin lines show the synthetic photospheric components.
Figure 1: b) Selected lines of UX Ori.
### 3 General description of the spectra
The spectra are available in ascii form (wavelength and normalized fluxes for each of the nights) upon request.
To illustrate the potential of these (and similar) observations, we have produced a first group of figures which show for each star a selection of the most interesting circumstellar lines that display variability. For each line we plot the normalized profile as function of velocity shift at different epochs. For several of the lines we also plot the synthetic photospheric spectrum (see Sect. 5). The figures show the results for UX Ori (Figs. 1a-e), CQ Tau (Figs. 2a-e), BF Ori (Figs. 3a-e), RR Tau (Figs. 4a-e) and WW Vul (Figs. 5a-d). Only Fig. 1 (all the variable lines in the spectrum of UX Ori) and Fig. 4d (displaying the [OI] 6363 Å at various epochs in RR Tau) are shown in print, the others are available only electronically; nevertheless, we will refer to them in the following sections of this paper.
Figure 1: c) Selected lines of UX Ori. Note that when more than one line is shown in a panel the horizontal scale is in wavelengths.
Figure 1: d) Selected lines of UX Ori.
Figure 1: e) Selected lines of UX Ori.
Name Date V T Exp S/N W(H) mag min Å UX Ori 19.11.94 9.70 : 1 30 115 7.6 02.12.95 9.74 : 1 45 140 10.6 04.12.95 9.68 1 45 170 10.5 06.12.95 9.78 1 90 230 10.1 25.11.96 10.08 1 60 60 8.4 28.11.96 9.95 1 20 90 5.5 28.11.96 9.95 2 25 95 5.5 02.12.96 9.86 2 40 95 5.2 CQ Tau 03.12.95 9.80 : 1 60 120 8.4 05.12.95 10.10 1 60 170 5.1 21.11.96 10.50 1 60 120 12.0 26.11.96 10.48 1 60 130 8.6 29.11.96 10.33 1 30 100 7.5 29.11.96 10.33 2 40 80 7.6 02.12.96 10.29 2 40 50 6.1 BF Ori 01.12.95 10.83 1 90 120 12.9 04.12.95 10.94 1 45 100 14.3 25.11.96 9.93 1 53 130 13.5 28.11.96 9.97 1 20 130 15.5 28.11.96 9.97 2 25 87 15.5 RR Tau 03.12.95 10.86 : 1 60 120 30.0 05.12.95 11.03 1 60 130 29.5 24.11.96 12.35 : 1 85 90 56.5 27.11.96 10.86 1 90 105 25.8 01.12.96 10.89 2 90 110 31.3 WW Vul 20.11.96 11.22 : 1 45 90 20.7 25.11.96 10.83 2 90 100 21.5 29.11.96 10.94 2 80 90 21.0 01.12.96 10.94 2 56 50 23.9
In Fig. 6 we show the complete spectrum of UX Ori, computed by averaging the spectra of the three 1996 nights when the star was at maximum brightness. The full spectrum includes 27 panels ordered from "a'' to "zz'' in order of increasing wavelength. We show in print two selected wavelength spectral intervals; the rest of the figures is available electronically (in colors). Figure 6 shows also the complete synthetic spectrum ( = 9500 K and = 4.0), before and after convolution with the projected rotation velocity (= 140 kms-1). Most of the lines are identified, and one can clearly see the photospheric spectrum of the star, as well as a large number of lines of CS origin.
As a general comment, we note that the spectra of these stars contain numerous absorption lines of neutral and ionized metals, as well as hydrogen lines, most prominently those of the Balmer series. Most of these lines are typical for normal stars of similar spectral types, but some, such as He I 5876, Na I D or the O I 7774 triplet, are not observed with the same intensity in normal A stars. We detect in the spectra numerous absorption lines of circumstellar origin, so-called shell or CS components. The presence of these components has to be taken into account in the analysis of the spectra and in estimating the stellar parameters (see Sect. 5).
#### 3.1 The H line
In all the stars the H line shows the two-component profile that is typical of UX Ori-type stars (Grinin & Rostopchina 1996). H has broad, almost symmetric wings in emission extending up to 400 kms-1. The typical radial velocities at half intensity are 200-300 kms-1.
The relative intensity of the two emission peaks varies from star to star and, for the same star, with time. It seems, however, that in spite of much time variation there is for each star a general pattern that characterizes the H profile. Namely, in RR Tau the red and blue peaks tend to have similar intensity, in UX Ori and CQ Tau the blue peak is systematically stronger than the red one (inverse P Cygni type III profile), while in WW Vul the red peak is stronger than the blue one (P Cygni type III profile). This behaviour is confirmed by earlier observations (see, for example, for UX Ori Kolotilov 1977, Grinin et al. 1994, de Winter 1996; for WW Vul Reipurth et al. 1996; for CQ Tau Kozlova et al. 2000).
According to the interpretation given in Tambovtseva et al. (1999), these kinds of profiles expected if H forms in an accretion disk seen almost edge-on. The relative intensity of the blue and red peaks is determined by the combination of rotation and infall or outflow motions. If so, the symmetric profile of H in RR Tau and BF Ori suggests a situation where rotation dominates, the inverse P Cygni III profile of UX Ori and CQ Tau suggest a situation where infall motions are important and the P Cygni III profile of WW Vul a situation where outflow motions dominate the H formation region. Note, however, that at times the spectrum of WW Vul shows in some lines, such as He I 5876 and D Na I, red-shifted absorption components at high velocity, which are clear signatures of accreting gas (Fig. 5b and Grinin et al. 1996).
The equivalent width of H is given in Table 2. It has been computed as the emission in excess of the photospheric one, estimated from the synthetic spectra discussed in the following. With the exception of RR Tau, the values of EW(H) are quite weak in comparison with other HAe stars. For example, among the 15 HAe stars listed by Bhöm and Catala (1995) only 7 have W(H)20 Å, and in the sample of 22 studied by Corcoran and Ray (1997) only 4. This last result is particularly significant because they did not subtract the photospheric spectrum, so that their values of W(H) are in fact lower limits. In AB Aur, one of the best studied HAe star with a disk seen close to face-on (Grady et al. 1999), W(H Å (Böhm & Catala 1993; Pogodin 1992). The non-LTE models by Tambovtseva et al. (1999), who computed the H emission due to a circumstellar disk, show that this difference can be caused at least partially by a different disk orientation (more edge-on in UXORs, more face-on in AB Aur).
Figure 4: d) [OI] 6363Å line at various epochs in RR Tau. The corresponding V magnitude is given for each night.
#### 3.2 He I 5876 and O I 7774 lines
These lines are seen in all spectra, and they are clearly of CS origin, since they are absent (He I 5876) or weak (O I 7774) in the photospheric spectra. They appear as broad (up to 200-300 kms-1) and highly variable absorption components, generally redshifted by less than 100 kms-1. In some spectra one sees weak blue-shifted emission. In a few spectra there is indication of blue-shifted absorption at low velocity (e.g., BF Ori, 4.10.95, 25.11.96) The absorption lines are broadened by differential motions of emitting gas (combination of infall and rotation) and formed in the nearest vicinity of the stars (see Tambovtseva et al. 1999 for a study of the He I 5876 line). Strong night-to-night variations of their profiles indicate strong fluctuations of the temperature and density along the line of sight in this region of the CS gas envelope.
#### 3.3 Sodium resonance lines
The sodium D Na I resonance doublet is one of the most interesting spectral features of the UXORs spectra. Due to the low ionization energy (5.17 eV) the sodium atoms are fully ionized in the atmospheres of A stars (see synthetic the spectra in Fig. 6) and in their immediate surroundings. Nevertheless the observations show that variable CS components are present in the spectra of UXORs (Grinin et al. 1994, 1996; Grady et al. 1996; de Winter et al. 1999; Kozlova et al. 2000). In most cases one sees red-shifted absorption components with radial velocities up to 300 kms-1. These components can appear and disappear on time scale of days and even less without any visible brightness change of the stars. Sometimes the profiles show instead blue-shifted components indicating matter outflow. Blue-shifted absorption is frequently seen in the spectrum of WW Vul (Fig. 5c). In all stars, at times blue and red-shifted components are seen simultaneously (BF Ori, 1.12.95, Fig. 3c; RR Tau, 1.12.96, Fig. 4c). In all these cases the radial velocity of the blue-shifted components does not exceed velocities of roughly 100 kms-1.
The variable CS components of the D Na I lines are blended with narrow interstellar (IS) lines. According to Finkenzeller & Jankovics (1984) the radial velocity of the IS lines coincides within a few kilometers per second with the radial velocity of the HAe star, and we have used it to estimate the values of the heliocentric radial velocity given in Table 4.
The sodium resonance lines provide the clearest evidence that the gas motions in UXORs are very complex, with infall and outflow occurring in the same objects, sometime simultaneously.
Figure 6: Complete spectrum of UX Ori (selected regions). The spectrum is the average of the three 1996 nights when the star was at maximum brightness. The gray line shows the synthetic spectrum for =9500 K and =4.0. The black thin line shows the same spectrum after convolution with a projected rotation velocity =140 kms-1. The strongest lines are identified. The complete spectrum is available electronically.
#### 3.4 Pashen 14 line
The CS component of this hydrogen line is usually weak in the spectra of UXORs (see Fig. 1e). Nevertheless, it is important for diagnostics since the line is usually optically thin, and the upper level of this transition is known with high accuracy in LTE. Therefore the emission of the line (after subtraction of the photospheric component) is proportional to the emission measure of the emitting region (Tambovtseva et al. 2001).
### 4 Spectral variability of UXORs
In the case of UXORs we deal with two different sources of spectral variability. One of them is typical of many stars with CS gas envelopes and is connected with variations of temperature and density in the emitting region and along the line of sight. The observations show that such variations can be quite strong and lead to strong variability of the spectral lines. In particular, we want to call attention to the He I 5876 line (Figs. 1b-5b), which has the highest excitation energy (about 21 eV) of its lower level of all the lines in our spectra and is, therefore, the most sensitive to temperature fluctuations.
The other mechanism of spectral variability is quite specific for UX Ori-type stars. It is connected to the coronagraphic effect produced by the screens of dust that sporadically obscure the star and cause the deep minima that characterize UXORs. For convenience, in the following we will call such screens "clumps", although their geometry and origin are at present uncertain. When it intersects the line-of-sight and screens the star from the observer, a CS clump screens at the same time the central part of the gas envelope where the CS lines form. The outer part of the CS envelope, on the other hand, is not screened. This causes an increase of the strength of the emission component with respect to the stellar continuum, whose degree depends on the extension of the line emitting region with respect to the extension of the obscuring clump. We expect that this effect will be maximum for strong emission lines, such as H and minimum for the weaker lines that are seen mostly as absorption against the stellar photosphere.
The high resolution spectra of UX Ori around H obtained at the moment of a very deep minimum in August 1992 ( = 2.5 mag, Grinin et al. 1994) have shown that not only the equivalent width of H changes when the star fades but also the line profile, which turns from its usual double-peaked shape to single peak. Such changes are the consequence of the coronagraphic effect caused by CS clumps: when screening the star from the observer, they screen also that part of the CS gas envelope in front of the star which is responsible for the absorption component of the line profile. As a result the central absorption decreases and can disappear completely (see Grinin & Tambovtseva 1995).
We have among our spectra of RR Tau the most convincing case of a minimum, when the star fades by about 1.6 mag (24.11.96). One can see the strong relative increase of H whose equivalengt width increases by almost a factor of two (Table 2). Similar changes have been found by Kolotilov (1977) on the basis of low resolution spectra and by Herbst et al. (1983) from narrow-band photometry. However, the 24.11.96 minimum of RR Tau was not a very deep one for this star (see i.e. Rostopchina et al. 1997), and the transformation of the H line profile from double to single peaked, observed in the 1992 minimum of UX Ori, was not detected.
It is interesting to note that a strong increase relative to the stellar continuum is seen in RR Tau also in the forbidden line [O I] 6363, whose equivalent width increased in the 24.11.96 night by about a factor of three, from 0.11 to 0.33 Å, as the star faded by a similar amount (Fig. 4d). A similar increase of the equivalent width of the [O I] 6363 line was also observed in CQ Tau in 2.12.96 (Fig. 2d), when the star, however, was close to its maximum brightness.
The behaviour of CS lines during the photometric minima of UXORs supports the interpretation of the observed anti-correlation between linear polarization of UXORs and their brightness (Grinin et al. 1991) and gives us additional and independent evidence that the deep minima of UXORs are caused by variable obscuration of the star by CS dust clouds.
### 5 Stellar parameters
For all the five stars we have computed a typical spectrum by averaging all the spectra obtained when the star was in its bright state. These average spectra have then been compared to synthetic spectra calculated from the Kurucz (1993) models of stellar atmospheres (for solar chemical composition) and the list of spectral lines from the data base VALD (Kupka et al. 1998). We have varied effective temperature and gravity and convolved the result with different values of the projected rotation velocity until a good fit to the most stable photospheric features was achieved. The values of given in Table 3 have been adopted for each star. We have relied in particular on the photospheric wings of the Balmer lines, and supplemented our data with the low resolution spectra by Kozlova et al. (1995). All calculations were made with the codes SYNT and ROTATE by Piskunov (1992). The best values of , and are shown in Table 3. Note that our values of are systematically higher than those determined by Kovalchuk & Pugach (1997), based on lower resolution spectra.
The full average spectrum of UX Ori is shown in Fig. 6. We show in the same figures the best synthetic spectrum before and after convolution with the projected rotation velocity. One can see the good agreement between observed and synthetic spectra in the Balmer wings as well as in the numerous weak metallic lines. On the contrary, strong metallic lines like Fe II or Mg II tend to be deeper then predicted. Inspection of the individual spectra shows that their shape changes with time, indicating a strong contribution of a CS component. As we have already discussed, some spectral lines like He I 5876 have a purely CS origin and originate probably in the inner part of an accretion disk (Tambovtseva et al. 1999).
We did not find any signature of veiling of the photospheric lines by a non-stellar continuum emission. This is also typical of other non-embedded HAEBE stars (Corcoran & Ray 1994; Ghandour et al. 1994).
Name (K) (kms-1) (kms-1) UX Ori 9500 4.0 +18 140 CQ Tau 7000 3.5 +20 90 BF Ori 8750 4.0 +18 40 RR Tau 9750 3.5 +11 140 WW Vul 8500 3.5 -12 150
### 6 Discussion
The optical spectra of these UXORs, which are among the most active in the group, can be described in important detail by the synthetic spectra of normal A stars. This is the case, for example, for the broad wings of the Balmer lines as well as for numerous weak metallic lines. The strong metallic lines are usually blended with shell components. The comparison with the high resolution spectrum of the photometrically non-active Herbig Ae star AB Aur (Böhm & Catala 1993) shows that the shell components are more numerous in the spectra of UXORs. We think that this is a consequence of the edge-on orientation of the disks of UXORs relatively to the observer, and we suggest that these lines can be used to further study the structure of the disk in the vicinity of the star. It should be noted that the presence of many CS metal lines causes serious problems in the determination of the stellar parameters. A preliminary study of the time stability of each feature is necessary to define the "true" photospheric spectrum one wants to fit.
In some stars the spectral signatures of accretion and outflow can be observed simultaneously. For example in the 20.11.96 spectrum of WW Vul (Fig. 5a) H has a direct P Cygni type III profile, while H and H have a characteristic type III inverse P Cygni profile with a weaker blue-shifted emission component. The same combination of the line profiles was observed on the night 1.12.96. Simultaneous infall and outflow motions are commonly observed in UXORs, as well as in many T Tauri stars (see, for example, Edwards et al. 1984; Johns & Basri 1995; Petrov et al. 1996), where they are both interpreted as phenomena associated with disk accretion. The analysis of the spectra of UXORs will show if the same interpretation can be applied to higher mass stars as well.
#### 6.1 The gas accretion rate
The observations of the Balmer lines shown in this paper, together with the Pashen 14 and Brackett gamma lines, were used recently by Tambovtseva et al. (2001) to estimate the gas accretion rate . These authors have assumed that the lines form in the inner region of an accretion disk illuminated by the central star and performed a non-LTE analysis of the physical conditions of the gas. They derive a typical value of the gas accretion rate for UX Ori of about (2-5) per year. This is significantly less then the value per year suggested recently in the papers by Herbst & Shevchenko (1999) and Bertout (2000).
Small accretion rates agree with the absence of veiling in the spectra of UXORs (see Sect. 5). Also, the low in UXORs are in agreement with the mass of their CS disks based on observations at millimeter wavelenghts (Natta et al. 1997). For example, in the case of UX Ori . The age of this star is about yrs (Natta et al. 1997; Rostopchina 1999) and hence the mean value of the mass accretion rate is about per year, i.e., only a few times greater then the value derived from the hydrogen lines.
#### 6.2 Origin of the redshifted absorption components in the sodium D Na I lines
Variable red-shifted absorption components in the sodium D Na I resonance lines are frequently observed in the spectra of UXORs and their origin is a matter of debate (see Grady et al. 2000 and reference therein). Their properties are very similar to those observed in the spectra of Pictoris in the Ca II resonance lines (Ferlet et al. 1987). In the case of UXORs the highest radial velocity of the infalling gas reaches 300 kms-1. The problem is connected with the fact that sodium is quickly ionized in the vicinity of HAe stars, where a sizeable amount of neutral atoms can be obtained in two cases: 1) when the clump has a strong excess of metals relative to the solar abundance, or 2) when it has normal chemical composition, but high density (Sorelli et al. 1996).
The accretion event which has been observed in the spectrum of UX Ori in the night of 28.11.96 turned out to be important for discriminating between these two possibilities. It was so strong that red-shifted absorption components were clearly seen not only in the resonance lines, such as Na D, but also in many other metallic lines and, of particular importance, in all the Balmer lines visible in the spectrum (Fig. 1a). Non-LTE modeling of the lines optical depth has shown that the infalling gas in that event could not be heavily hydrogen-depleted, as would be expected if it was produced by the evaporation of a solid body of chemical composition similar to solar system comets, but had a chemical composition approximately solar (Natta et al. 2000b). Evaporating comet-like or planetesimal bodies cannot be the main source of matter accreting onto UX Ori-type stars. Thus, these stars can be consider only as remote, rather than immediate progenitors of Pic, as indicated also by the large difference between their IR excesses: in the case of UXORs they are similar to those observed in the classical HAe stars (Grinin et al. 1991).
### 7 Summary
We have presented in this paper the high resolution spectra of five UXORs (UX Ori, CQ Tau, BF Ori, RR Tau, WW Vul), which cover the entire visual range, from 3900 to 8700 Å. There are between 4 and 7 spectra per star, obtained over a time interval of two years. Simultaneous or quasi-simultaneous photometric observations were obtained at the Crimean Astrophysical Observatory. The spectra are too sparse to form true time sequences; however, they provide an extremely useful database for studying the photospheric spectrum of the stars and determining stellar parameters such as the effective temperature, stellar gravity and rotational velocity, as well as for investigating the circumstellar environment and its variability. The complete, reduced and normalized spectra are available in electronic form. Here we have shown for each star a selection of the most interesting lines, and the full spectrum of UX Ori computed by averaging the spectra obtained when the stars were at maximum light. We show also the synthetic spectrum and provide an identification of most of the lines.
The main features of the spectra have been discussed briefly, to indicate how they can be used to study the properties of UXORs. However, we want to stress that the wealth of information they contain has by no means been entirely exploited.
Among the points we have touched, we have shown that the spectra of UXORs contain many photospheric lines that can be described in good approximation by the synthetic spectra of normal A stars with = 3.5-4. This conclusion agrees with previous estimates of the evolutionary status of UXORs as HAe stars, based on the analysis of their IR excess (Grinin et al. 1991; Natta et al. 1997).
In addition to this normal photospheric spectrum, there is a large number of absorption components of circumstellar origin (shell-components) visible in the strong metallic lines. Most of them are highly variable and red-shifted, indicating non-stationary accretion. This is specific of UXORs (also with respect to other, less variable HAEBE stars of similar spectral type) and is probably caused by the orientation of their CS disks, which in UXORs is almost edge-on for the observer. One object (WW Vul) shows simultaneous evidence of infall and outflow motions. Strong emission is seen only in the lower Balmer lines. When interpreted as forming in the inner regions of an accretion disk, they can be used to estimate the accretion rate, which turns out to be rather low (2-5 yr-1 in UX Ori; Tambovtseva et al. 2001).
Most lines vary strongly with time. The variability is in many cases not related to the brightness variation of the star and is caused by perturbations of the gas density, electron temperature and the kinematic conditions in the CS environment. The strongest variations of such a type are observed in He I 5876 and O I 7774 lines. There is, however, an additional source of spectral variability, which is related to the brightness variations that characterize UXORs. It is connected to the coronagraphic effect produced by the dust clouds that sporadically obscure the star and part of the region where the CS line is formed. Spectral variability of such a kind is seen in the present database only in one RR Tau spectrum, when the star fades by about 1.6 mag, in both H and the [O I] 6364 Å line.
We did not find any evidence of veiling of the photospheric lines by a non-stellar continuum, although we have not set an upper limit to it. This agrees with the low level of accretion activity of UXORs estimated from the non-LTE analysis of the hydrogen lines and with the absence of veiling in the spectra of non-embedded HAEBE stars in general. The recent idea of Herbst & Shevchenko (1999) that accretion plays a dominant role in UXORs seems to have no observational support.
Acknowledgements
Many thanks to the referee, Carol Grady, for her help in organizing the material shown in this paper. V. P. Grinin thanks Franco Pacini for hospitality during his stay in the Arcetri Astrophysical Observatory where part of this work was done. This work was partly supported by the RFBR grant 99-02-18520.
## References
• Bertout, C. 2000, A&A, 363, 984 NASA ADS
• Böhm, T., & Catala, C. 1993, A&ASS, 101, 629
• Böhm, T., & Catala, C. 1995, A&A, 301, 155 NASA ADS
• Corcoran, M., & Ray, T. 1994, PASPC, 62, 151
• Corcoran, M., & Ray, T. 1997, A&A, 321, 189 NASA ADS
• Edwards, S., Hartigan, P., Ghandour, L., & Andrulis, C. 1994, AJ, 108, 1056 NASA ADS
• Eiroa, C., Garzon, F., Miranda, L. F., & EXPORT 2000, in Disks, Planetesimals and Planets, ed. F. Garzon, C. Eiroa, D. de Winter, & T. J. Mahoney, PASPC, 219, 363
• Ferlet, R., Vidal-Madjar, A., & Hobbs, L. M. 1987, A&A, 185, 267 NASA ADS
• Finkenzeller, U., & Jankovics, I. 1984, A&ASS, 57, 285
• Finkenzeller, U., & Mundt, R. 1984, A&ASS, 55, 109
• Ghandour, L., Strom, S., Edwards, S., & Hillenbrand, L. 1994, PASPC, 62, 227
• Graham, J. 1992, PASP, 104, 479 NASA ADS
• Grady, C. A., Pérez, M., Talavera, A., et al. 1996, A&ASS, 120, 157
• Grady, C. A., Woodgate, B., Bruhweiler, F. C., et al. 1999, ApJ, 523, L151 NASA ADS
• Grady, C. A., Sitko, M. L., Russell, R. W., et al. 2000, in Protostars and Planets IV, ed. V. Mannings, et al. (The University of Arizona Press, Tucson), 613
• Grinin, V. P. 1988, Sov. Astron. Lett. 14, 27 NASA ADS
• Grinin, V. P., Kiselev, N. N., Minikulov, N. Kh., & Chernova, G. P. 1988, Pis'ma v Astron. Zn. 14, 514 NASA ADS
• Grinin, V. P. 1994, in The Nature and Evolutionary Status of Herbig Ae/Be stars, ed. P. S. Thé, M. R. Pérez, & E. P. J. van den Heuvel, ASP Conf. Ser., 62, 63
• Grinin, V. P., & Rostopchina, A. N. 1996, Astron. Rep., 40, 171 NASA ADS
• Grinin, V. P., & Tambovtseva, L. V. 1995, A&A, 293, 396 NASA ADS
• Grinin, V. P., Kiselev, N. N., Minikulov, N. K., Chernova, G. P., & Voshchinnikov, N. V. 1991, Ap. Sp. Sci, 186, 283 NASA ADS
• Grinin, V. P., Thé, P. S., de Winter, D., et al. 1994, A&A, 292, 165 NASA ADS
• Grinin, V. P., Kozlova, O. V., Thé, P. S., & Rostopchina, A. N. 1996, A&A, 309, 474 NASA ADS
• Hamann, F., & Persson, S. E. 1992, ApJS, 82, 247 NASA ADS
• Herbig, G. H. 1960, ApJS, 4, 337 NASA ADS
• Herbst, W., Holtzman, J. A., & Klasky, R. S. 1983, AJ, 88, 1648 NASA ADS
• Herbst, W., & Shevchenko, V. S. 1999, AJ, 118, 1043 NASA ADS
• Ilyin, I. V. 2000, Ph.D. Thesis, University of Oulu
• Ilyin, I. V., & Duemmler, R. 1999, in Precise stellar radial velocities, IAU Colloq. 170, ed. J. B. Hearnshaw, & C. D. Scarfe, PASPC, 185, 63
• Johns, C. M., & Basri, G. 1995, AJ, 449, 341
• Kolotilov, E. A. 1977, Astrofizika, 13, 33 NASA ADS
• Kovalchuk, G. U., & Pugach, A. F. 1997, A&A, 325, 1077 NASA ADS
• Kozlova, O. V., Grinin, V. P., & Rostopchina, A. N. 1995, AAT, 8, 249
• Kozlova, O. V., Grinin, V. P., & Rostopchina, A. N. 2000, Astron. Rep., 44, 36 NASA ADS
• Kupka, F., Piskunov, N. E., Ryabchikova, T. A., Stempels, T. A., & Weiss, W. W. 1998, Vienne data base VALD
• Kurucz, R. 1993, CD-ROM No. 13, Cambridge, Mass.: Smithsonian Astrophysical Observatory
• Natta, A., & Whitney, B. A. 2000, A&A, 364, 633. NASA ADS
• Natta, A., Grinin, V. P., Mannings, V., & Ungerechts, H. 1997, ApJ, 491, 885 NASA ADS
• Natta, A., Grinin, V., & Mannings, V. 2000a, in Protostars and Planets IV, ed. V. Mannings et al. (The University of Arizona Press, Tucson), 559
• Natta, A., Grinin, V., & Tambovtseva, L. 2000b, ApJ, 542, 421 NASA ADS
• Pérez, M. R., & Grady, C. A. 1997, Space Sci. Rev., 82, 407 NASA ADS
• Petrov, P. P., Gullbring, E., Ilyin, I., et al. 1996, A&A, 314, 821 NASA ADS
• Piirola, V. 1975, Ann. Acad. Sci. A, VI, 61
• Piskunov, N. E. 1992, in Stellar Magnetizm, ed. Yu.V.Glagolevsky & I. I. Romanjuk (St. Petersburg, Nauka), 92
• Pogodin, M. A. 1992, Astron. Lett., 18, 1066
• Reipurth, Bo, Pedrosa, A., & Lago, M. T. V. T. 1996, A&AS, 120, 229 NASA ADS
• Rodgers, B., Wooden, D., Grinin, V., et al. 2001, ApJ, in press
• Rostopchina, A. N. 1999, Astron. Rep. 43, 113 NASA ADS
• Rostopchina, A. N., Grinin, V. P., Okazaki, A., et al. 1997, A&A, 327, 145. NASA ADS
• Sorelli, C., Grinin, V. P., & Natta, A. 1996, A&A, 309, 155 NASA ADS
• Tambovtseva, L. V., Grinin, V. P., & Kozlova, O. V. 1999, Astrofizika, 42, 75
• Tambovtseva, L. V., Grinin, V. P., Rodgers, B., & Kozlova, O. V. 2001, Astron. Rep., 45, 445
• Thé, P. S., de Winter, D., & Pérez, M. R. 1994, A&AS, 104, 315 NASA ADS
• Tuominen, I., Ilyin, I., & Petrov, P. 1999, in Astrophysics with the NOT, ed. H. Karttunen, & V. Piirola (Univ. Turku), 47
• Wallace, L., Hinkle, K., & Livingston, W., An atlas of spectrum of the solar photosphere from 13500 to 28000 sm-1, NSO Technical Report N 98-001
• de Winter, D. 1996, Observational aspects of Herbig Ae/Be stars and of candidate young A/B stars, Thesis, Amsterdam
• de Winter, D., Grady, C., van den Ancker, M. E., et al. 1999, A&A, 343, 137 NASA ADS
### Online Material
Figure 2: a) Selected lines of CQ Tau (same as Fig. 1).
Figure 2: b) Selected lines of CQ Tau.
Figure 2: c) Selected lines of CQ Tau.
Figure 2: d) Selected lines of CQ Tau.
Figure 2: e) Selected lines of CQ Tau.
Figure 3: a) Selected lines of BF Ori (same as Fig. 1).
Figure 3: b) Selected lines of BF Ori.
Figure 3: c) Selected lines of BF Ori.
Figure 3: d) Selected lines of BF Ori.
Figure 3: e) Selected lines of BF Ori.
Figure 4: a) Selected lines of RR Tau (same as Fig. 1).
Figure 4: b) Selected lines of RR Tau.
Figure 4: c) Selected lines of RR Tau.
Figure 4: e) Selected lines of RR Tau.
Figure 5: a) Selected lines of WW Vul (same as Fig. 1).
Figure 5: b) Selected lines of WW Vul.
Figure 5: c) Selected lines of WW Vul.
Figure 5: d) Selected lines of WW Vul. | {"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": 0.9032249450683594, "perplexity": 2821.050933489041}, "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-2020-24/segments/1590347441088.63/warc/CC-MAIN-20200604125947-20200604155947-00268.warc.gz"} |
http://www.gradesaver.com/the-outsiders/q-and-a/chapter4-question-314528 | # Chapter4 question
What is Bob's definition of a greaser? From what you have read so far about the Greasers, do you think it is a fair one? | {"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": 0.8133950233459473, "perplexity": 1020.5014969395006}, "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-2017-39/segments/1505818689752.21/warc/CC-MAIN-20170923160736-20170923180736-00065.warc.gz"} |
http://math.stackexchange.com/users/39894/rex?tab=activity | # rex
less info
reputation
2
bio website location age member for 1 year, 7 months seen May 6 '13 at 12:59 profile views 3
# 12 Actions
Nov22 accepted understanding the least squares criterion Nov16 revised understanding the least squares criterion edited tags Nov16 comment understanding the least squares criterion Thanks, i will give it a try. Nov15 comment understanding the least squares criterion Well in the notes I wrote, there is something about a method for a discrete case involving partial derivatives and a matrix. Since our teacher is kind of fast, my notes are incomplete. Does that sound familiar? Nov15 asked understanding the least squares criterion Nov13 comment clairification on standard deviation Ok, thanks alot! Nov13 awarded Student Nov13 asked clairification on standard deviation Sep10 awarded Scholar Sep10 accepted finding the significant digits for relative error Sep10 comment finding the significant digits for relative error Ok, thanks for your example. I was shown in class, but the teacher went really fast and i think i messed up my notes and I had put down 3 significant digits. But your way shows cleary how it is suppose to be done. Sep9 asked finding the significant digits for relative error | {"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": 0.8114587664604187, "perplexity": 1890.580330154127}, "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-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00536-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/179628/please-prove-that-fx-0-on-a-b/179632 | Please prove that $f(x)=0$ on $[a,b]$
Suppose $f$ is a continuous function on $[a,b]$ and $$\int_a^b f(x)g(x) = 0$$ for every integrable function. Show that $f(x) = 0$ on $[a,b].$
Here is what I have so far:
Consider any $x \in [a,b].$ Consider any $y >0.$ Say $g(u) = 1$ for $x<u<x+y,$ and $g(u) = 0$ otherwise. Hence $\int_x^{x+y} f(u) du = 0.$ Hence $\frac{\int_x^{x+y} f(u) du}{y} = 0.$ tend $y = 0$ we get $f(x) = 0.$ Hence proved.
Is this correct?
• I edited your question. Please double check I did not change anything (apparently, Yuval caught one mistake already).
– user2468
Aug 6 '12 at 20:21
• Your proof sounds fine, just remember to invoke explicitly the fundamental theorem of calculus. Aug 6 '12 at 20:21
• Well "dx" is missing...
– Hawk
Aug 6 '12 at 20:27
Hint: Let $x_0$ be a point at which $f(x_0) \ne 0$; Since $f$ is continuous there is an interval about $x_0$ in which $f$ is non-zero (can you prove this?). Now, can you find a $g(x)$ for which $\int_a^b f(x) g(x) dx \ne 0$ given this information?
Note that $\int_{[a,b]}f^2=0$. Then, $f\ge 0$. If there exist $x_0$ such that $f(x_0)>0$ Then, by continuity there exist an interval such that $f^2>0$ there and $\int_{[a,b]}f^2>0$, contradiction. | {"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": 0.9576019644737244, "perplexity": 195.84661655052824}, "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-2022-05/segments/1642320305317.17/warc/CC-MAIN-20220127223432-20220128013432-00272.warc.gz"} |
http://physics.oregonstate.edu/portfolioswiki/courses:order20:vforder20:vfcalculating?rev=1555011485&do=diff | # Differences
This shows you the differences between the selected revision and the current version of the page.
courses:order20:vforder20:vfcalculating 2019/04/11 12:38 courses:order20:vforder20:vfcalculating 2019/05/30 08:08 current
Line 8: Line 8:
===== In-class Content ===== ===== In-class Content =====
+
+====Lecture: Electric Potential====
+
+Students may be familiar with the iconic equation for the electric potential (due to a point charge):
+$$\text{Iconic:} \qquad V=\frac{1}{4 \pi \epsilon_0} \frac{Q}{r}$$
+With information about the type of source distribution, one can write or select the appropriate coordinate independent equation for $V$. For example, if the source is a line of charge:
+$$\text{Coordinate Independent:} \qquad V=\frac{1}{4 \pi \epsilon_0} \int\frac{\lambda | d\vec r' |}{| \vec r - \vec r' |}$$
+Looking at symmetries of the source, one can choose a coordinate system and write the equation for the potential in terms of this coordinate system. Note that this step is often combined with the following step, though one may wish to keep them separate for the sake of careful instruction.
+$$\text{Coordinate Dependent:} \qquad V=\frac{1}{4 \pi \epsilon_0} \int\frac{\lambda |ds'\ \hat s + s'\ d\phi'\ \hat \phi + dz'\ \hat z|}{| s'^2 + s^2 +2ss' \cos(\phi-\phi') + z^2|}$$
+Using what you one about the geometry of the situation, one can possibly simplify the numerator. For example:
+$$\text{Coordinate and Geometry Dependent:} \qquad V=\frac{1}{4 \pi \epsilon_0} \int\frac{\lambda s'\ d\phi'}{| s'^2 + s^2 +2ss' \cos(\phi-\phi') + z^2|}$$
+Emphasize that "primes" (i.e., $s'$, $\phi'$, $z'$, etc.) are used to indicate the location of charge in the charge distribution.
- * To find the area under a curve, one may chop up the x-axis into small pieces (of width dx). The area under the curve is then found by calculating the area for each region of dx (which is dx times f(x)) and then summing up all of those areas. In the limit where dx is small enough, the sum becomes an integral. + * To find the area under a curve, one may chop up the x-axis into small pieces (of width $dx$). The area under the curve is then found by calculating the area for each region of $dx$ (which is $f(x) dx$) and then summing up all of those areas. In the limit where $dx$ is small enough, the sum becomes an integral.
- * One could also find the area under a curve by chopping up both the x- and y-axes (chop), calculating the area of each small area under the curve (calculate), and adding all of those together with a double sum or double integral (add)+{{courses:order20:vforder20:chop_x.png?300|}}
- * This approach can be used to find the area of a cone, where the 'horizontal' length of each area is $r d\phi$ and the 'vertical' length is dr, giving an area of $dA = r d\phi dr$. It is important to make sure that the limits of integration are appropriate so that the integrals range over the whole area of interest. + * One could also find the area under a curve by chopping up both the x- and y-axes (chop), calculating the area of each small area under the curve (calculate), and adding all of those together with a double sum or double integral.
- * If one wants to calculate something other than length, area, or volume, such as if one sprinkled charge over a thin bar, then chop, calculate, and add still works. Again, chop the bar up into small lengths of dx. Then calculate the change dQ on each length (dQ = lambda dx), and add all of the dQs together in a sum or integral. +{{courses:order20:vforder20:chop_x_y.png?300|}}
- *This also works for calculating something (such as charge) over a volume. For a thick cylindrical shell with a charge density rho(vec r), chop the shell into small volumes d tau (which will be a product of 3 small lengths), multiply this volume by the charge density at each part of the shell, and add the resulting dQs together. + * This approach can be used to find the area of a cone, where the 'horizontal' length of each area is $r d\phi$ and the 'vertical' length is $dr$, giving an area of $dA = r d\phi dr$. It is important to make sure that the limits of integration are appropriate so that the integrals range over the whole area of interest.
+{{courses:order20:vforder20:dA_for_cone.png?200|}}
+ * If one wants to calculate something other than length, area, or volume, such as if one sprinkled charge over a thin bar, then chop, calculate, and add still works. Again, chop the bar up into small lengths of $dx$. Then calculate the charge $dQ$ on each length ($dQ = \lambda dx$), and add all of the $dQ$s together in a sum or integral.
+{{courses:order20:vforder20:chop_lambda.png?300|}}
+ *This also works for calculating something (such as charge) over a volume. For a thick cylindrical shell with a charge density $\rho(\vec r)$, chop the shell into small volumes of $d \tau$ (which will be a product of 3 small lengths, e.g. $d \tau = r d\phi\ dr\ dz$), multiply this volume by the charge density at each part of the shell (defined by e.g. $r, \phi,$ and $z$), and add the resulting $dQ$s together.
+{{courses:order20:vforder20:chop_rho.png?200|}}
====Activities==== ====Activities====
Line 21: Line 38:
* [[..:..:activities:vfact:vfvring|Electrostatic potential due to a ring of charge]] (SGA - 50 min) * [[..:..:activities:vfact:vfvring|Electrostatic potential due to a ring of charge]] (SGA - 50 min)
- * [[..:..:activities:vfact:vfvring|Series expansion of potential due to a ring of charge ]] (Extension of previous SGA + 20-30 min)
* [[..:..:lecture:vflec:vflines|Lines of Charge]] (Lecture: 30 min) * [[..:..:lecture:vflec:vflines|Lines of Charge]] (Lecture: 30 min)
##### Views
New Users
Curriculum
Pedagogy
Institutional Change
Publications | {"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": 0.959373950958252, "perplexity": 534.2354997700568}, "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-2019-47/segments/1573496667945.28/warc/CC-MAIN-20191114030315-20191114054315-00236.warc.gz"} |
https://msdn.microsoft.com/en-us/library/ms164293(v=vs.85).aspx | 1500 characters remaining
.NET Framework 3.0
Determines which items in the specified item collection have paths that are in or below the specified folder.
## Parameters
The following table describes the parameters of the FindUnderPath task.
Parameter Description
Files
Specifies the files whose paths should be compared with the path specified by the Path parameter.
InPath
Contains the items that were found under the specified path.
OutOfPath
Path
Specifies the folder path to use as the reference.
## Example
The following example uses the FindUnderPath task to determine if the files contained in the MyFiles item have paths that exist under the path specified by the SearchPath property. After the task completes, the FilesNotFoundInPath item contains the File1.txt file, and the FilesFoundInPath item contains the File2.txt file.
```<Project xmlns="http://schemas.microsoft.com/developer/msbuild/2003">
<ItemGroup>
<MyFiles Include="C:\File1.txt" />
<MyFiles Include="C:\Projects\MyProject\File2.txt" />
</ItemGroup>
<PropertyGroup>
<SearchPath>C:\Projects\MyProject</SearchPath>
</PropertyGroup>
<Target Name="FindFiles">
<FindUnderPath
Files="@(MyFiles)"
Path="\$(SearchPath)">
<Output
ItemName="FilesFoundInPath" />
<Output
ItemName="FilesNotFoundInPath" />
</FindUnderPath>
</Target>
</Project>
``` | {"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": 0.9485825300216675, "perplexity": 4108.21470095569}, "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/1435375097546.28/warc/CC-MAIN-20150627031817-00146-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://www.techylib.com/el/view/tumwaterpointless/joomla_1.6_a_users_guide_building_a_successful_joomla_powered | # Joomla! 1.6 : A User's Guide Building a Successful Joomla! Powered ...
Internet και Εφαρμογές Web
4 Δεκ 2013 (πριν από 4 χρόνια και 3 μήνες)
3.862 εμφανίσεις
Praise for Previous Edition of Joomla!: A User’s Guide
“A complete guide to the powerful features of Joomla! 1.5, this book takes a holistic
approach to building a Joomla!-powered website—from the CMS itself to its many
extensions, search engine optimization, and even building your own tableless template.
The novice reader is eased into the subject and confidently guided through the basic
principles and on to the more advanced features. This guide empowers the user not
only to build a professional website but to also to make it a success.”
—Russell Walker, CEO, Netshine Software Limited
(Joomla! Development Consultancy)
“If you’ve been using or following Joomla! in the past years, you’ve most likely seen
the name Barrie North or Joomlashack. Barrie has been a member of the community
for a long time and, as such, my expectations for this book were pretty high. Besides
explaining how Joomla! works from a usability point of view, there is valuable informa-
tion for people who want to learn serious template building, and readers can stand out
of the crowd by using Barrie’s steps to make their (X)HTML and CSS optimized for
accessibility and SEO. All in all, this book is a great guide that comes at the right time
for newcomers and more experienced Joomla! users and developers alike. Well done,
Barrie!”
—Arno Zijlstra, Joomla! cofounder, custom template specialist,
www.alvaana.com
“In a time when solid, real-life Joomla! 1.5 information is rarely available, this book is
a thirst-quenching oasis of knowledge. The abundant and clear examples in the book
make Joomla! 1.5 websites within anyone’s reach. I heartily recommend Joomla! 1.5: A
User’s Guide by Barrie North.”
—Tom Canavan, author of Dodging the Bullets:
A Disaster Preparation Guide for Joomla! Based Web Sites
“Refreshing! After reading many how-to books, this one is a step beyond the rest
because of its focus on examples based on live sites. This book is well crafted for begin-
ners to advanced users with a well-organized overview that walks you through the
entire Joomla! CMS.”
—Steven Pignataro, corePHP, www.corephp.com
“As a long-time Joomla! end-user and developer, I had low expectations for anything
new I might learn from this book. However, I was pleasantly surprised to find it a great
refresher course, especially since the book is logically organized, leading beginners from
the most basic Joomla! concepts and continuing through to more complex ones, such
as tableless template design and how to write a template for Joomla! 1.5. In summary,
Barrie North has produced the gold-standard print reference for Joomla! 1.5. I highly
recommend this book for novice and intermediate users if you want to make the most
of Joomla!”
—Vicor Drover, http://dev.anything-digital.com
Prentice Hall
Upper Saddle River, NJ • Boston • Indianapolis • San Francisco
New York • Toronto • Montreal • London • Munich • Paris • Madrid
Cape Town • Sydney • Tokyo • Singapore • Mexico City
Joomla!
1.6:
A User’s Guide
Building a Successful Joomla!
Powered Website
Barrie M. North
Many of the designations used by manufacturers and sellers to distinguish their products are
claimed as trademarks. Where those designations appear in this book, and the publisher was
aware of a trademark claim, the designations have been printed with initial capital letters or in all
capitals.
The author and publisher have taken care in the preparation of this book, but make no expressed
or implied warranty of any kind and assume no responsibility for errors or omissions. No liability
is assumed for incidental or consequential damages in connection with or arising out of the use
of the information or programs contained herein.
The publisher offers excellent discounts on this book when ordered in quantity for bulk
purchases or special sales, which may include electronic versions and/or custom covers and
content particular to your business, training goals, marketing focus, and branding interests. For
U.S. Corporate and Government Sales
(800) 382-3419
[email protected]
International Sales
[email protected]
Visit us on the Web: informit.com/ph
North, Barrie M.
Joomla! 1.6 : a user’s guide : building a successful Joomla! powered website / Barrie M. North.
p. cm.
ISBN 978-0-13-248706-1 (pbk. : alk. paper)
1. Joomla! (Computer file) 2. Web sites--Authoring programs. 3. Web site development.
I. Title.
TK5105.8885.J86N67 2011
006.7’8--dc22
2010051011
by copyright, and permission must be obtained from the publisher prior to any prohibited
reproduction, storage in a retrieval system, or transmission in any form or by any means,
electronic, mechanical, photocopying, recording, or likewise. For information regarding
permissions, write to:
Pearson Education, Inc
Rights and Contracts Department
501 Boylston Street, Suite 900
Boston, MA 02116
Fax (617) 671 3447
Joomla!™ and the Joomla!® logo are registered trademarks of Open Source Matters.
Chapter 9, “Creating Pure CSS Templates,” is released under a Creative Commons Attribution-
sa/2.5/ for more details.
ISBN-13: 978-0-132-48706-1
ISBN-10: 0-132-48706-3
Text printed in the United States on recycled paper at RR Donnelley in Crawfordsville, IN.
First printing February 2011
Editor-in-Chief
Mark Taub
Executive Editor
Debra Williams Cauley
Development Editor
Songlin Qiu
Marketing Manager
Stephane Nakib
Managing Editor
Kristy Hart
Project Editor
Anne Goebel
Copy Editor
Geneil Breeze
Indexer
Heather McNeill
Kathy Ruiz
Technical Reviewers
Robert P. J. Day
Torah Bontrager
Publishing Coordinator
Kim Boedigheimer
Cover Designer
Chuti Prasertsith
Compositor
Nonie Ratcliff
For Sarah
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv
About the Author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi
Chapter 1: Content Management Systems and an Introduction to Joomla! . . . . . 1
What Is a Content Management System? . . . . . . . . . . . . . . . . . . . . . . . . . 2
Static Web Pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Web Pages with CSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Dynamic Web Pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Open Source Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
History of Joomla! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
The Joomla! Community . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Third-Party Extensions Development . . . . . . . . . . . . . . . . . . . . . . . . . 9
Joomla!’s Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Elements of a Joomla! Website . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Contents vii
Chapter 2:
Downloading and Installing Joomla! . . . . . . . . . . . . . . . . . . . . . . . . 17
How to Install Joomla! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Obtaining the Latest Joomla! File Package . . . . . . . . . . . . . . . . . . . . . . . 18
Joomla! Package Naming Conventions . . . . . . . . . . . . . . . . . . . . . . . 20
Creating a MySQL Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Unpacking the Joomla! Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Unpacking Joomla! on a Local Desktop Computer . . . . . . . . . . . . . . 21
Unpacking Joomla! on a Hosting Account . . . . . . . . . . . . . . . . . . . . 25
Running the Joomla! Installation Wizard . . . . . . . . . . . . . . . . . . . . . . . . 26
Getting to the Joomla! Installer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Step 1: Language. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Step 2: Pre-Installation Check. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Step 3: License . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Step 4: Database Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Step 5: FTP Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Step 6: Main Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Step 7: Finish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Chapter 3:
Joomla! Administration Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
What Are the Frontend and Backend of a
Joomla!-Powered Website? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
The Menu Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
The Toolbar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
The Workspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Administrator Functions in the Menu Bar . . . . . . . . . . . . . . . . . . . . . . . 40
The Site Submenu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Users Menu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
viii Contents
The Menus Menu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
The Content Menu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
The Components Submenu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
The Extensions Menu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
The Help Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
View Site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Chapter 4:
Content Is King: Organizing Your Content . . . . . . . . . . . . . . . . . . . 63
How Does Joomla! Generate Web Pages? . . . . . . . . . . . . . . . . . . . . . . . . 64
How Joomla! Organizes Content Articles . . . . . . . . . . . . . . . . . . . . . . . . 66
Uncategorized Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Categories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
A Sample Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Creating the Widget Inc. Website with Uncategorized Content . . . . . . . 70
Creating Content Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Creating Menu Items . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
The Featured Article Component . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Creating the Widget Inc. Website with Categories . . . . . . . . . . . . . . . . . 85
Creating Categories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Creating Content Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Creating Menu Items . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Linking to Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
“Read More” Links and Individual Pages . . . . . . . . . . . . . . . . . . . . . 95
Module Content. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Contents ix
Chapter 5:
Creating Menus and Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
How Menu Modules Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
What Menu Items Do . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Creating a Menu Item. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Where Does a Menu Item Link? . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
What Does a Page Look Like After a Link Is Followed?. . . . . . . . . . 108
Blog Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Blog Layout Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
List Layout for a Blog. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Category List Advanced Options . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Managing Menu Modules in the Module Manager . . . . . . . . . . . . . . . 116
Show Title. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Access . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Menu and Module Class Suffixes (Advanced Options) . . . . . . . . . . 118
Menu Assignment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Chapter 6: Extending Joomla! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Installing Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Managing Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Core Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Third-Party Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Module Display . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Core Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Third-Party Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
x Contents
Plug-ins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Core Plug-ins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Third-Party Plug-ins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Templates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Core Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Third-Party Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Chapter 7: Expanding Your Content: Articles and Editors . . . . . . . . . . . . . . . . 137
WYSIWYG Editors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Managing WYSIWYG Editors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Other Third-Party Editors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Creating and Managing Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Managing Content Through the Backend . . . . . . . . . . . . . . . . . . . . . . 144
Adding Content from the Backend . . . . . . . . . . . . . . . . . . . . . . . . . 146
Inserting Images into Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Category Descriptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Managing Content Through the Frontend . . . . . . . . . . . . . . . . . . . . . . 159
Creating a Frontend User Menu . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Limiting Access to Menus by User Level . . . . . . . . . . . . . . . . . . . . . 163
Authors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Editors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Publishers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Article Checkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Chapter 8: Getting Traffic to Your Site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .175
Start at the Beginning: Site Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
Contents xi
Organic Traffic (SEO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Introduction to Google. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
Creating Keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Keywords and Domain Name. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
Designing Your Site for Organic Traffic. . . . . . . . . . . . . . . . . . . . . . 184
Advanced SEO Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Referral Traffic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Google PageRank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Other Link-Building Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Internal Linking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Pay Per Click Traffic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
How Google AdWords Works. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Joomla! and AdWords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Email Traffic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
Third-Party Hosted Email Solutions. . . . . . . . . . . . . . . . . . . . . . . . 209
Joomla! SEF Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Quick Start SEO for Joomla! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Chapter 9:
Creating Pure CSS Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
What Is a Joomla! Template? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
The Localhost Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
Localhost Server Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
W3C and Tableless Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
Semantically Correct Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Cascading Style Sheets (CSS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Creating a Simple Template: 960TemplateTutorialStep1 . . . . . . . . . . . 220
Template File Components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
The Joomla! Page Body. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
xii Contents
Using CSS to Create a Tableless Layout: CSSTemplateTutorialStep2 . . 234
Default CSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Modules in Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
Menus in Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
Hiding Columns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Making a Real Joomla! 1.6 Template: 960TemplateTutorialStep3. . . . . 256
Slicing and Dicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
Header . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
The Banner/Message Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Column Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
Flexible Modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
Typography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Chapter 10: Creating a School Site with Joomla! . . . . . . . . . . . . . . . . . . . . . . . . 265
Why Do You Need a School Website? . . . . . . . . . . . . . . . . . . . . . . . . . 266
Students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
Teachers and Administrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Parents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Potential Students and Their Parents. . . . . . . . . . . . . . . . . . . . . . . . 267
What Features Do You Need on a School Site?. . . . . . . . . . . . . . . . . . . 268
Downloading and Installing a School Template . . . . . . . . . . . . . . . . . . 268
Fresh Template Features and Positions. . . . . . . . . . . . . . . . . . . . . . . 270
Configuring a Logo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
Configuring the Search Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
Configuring the Main Horizontal Drop-Down Menu . . . . . . . . . . 272
Organizing Content on a School Website. . . . . . . . . . . . . . . . . . . . . . . 274
Creating the Menus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Contents xiii
Building Out Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Creating Subnavigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
The Academics Submenu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Creating News Links for a Section . . . . . . . . . . . . . . . . . . . . . . . . . 287
Setting Up the Footer Area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Setting Up the Home Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
Adding Basic Functionality to a School Website . . . . . . . . . . . . . . . . . . 293
User Registration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
Events Calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Downloadable Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Staff Directory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Email Newsletter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
RSS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
Random Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
Sitemap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
Extending the School Website Beyond the Basics . . . . . . . . . . . . . . . . . 298
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
Chapter 11:
Creating a Restaurant Site with Joomla! . . . . . . . . . . . . . . . . . . . . . 301
Why Does a Restaurant Need a Website? . . . . . . . . . . . . . . . . . . . . . . . 302
What Features Does a Restaurant Website Need? . . . . . . . . . . . . . . . . . 302
Downloading and Installing a Restaurant Template . . . . . . . . . . . . . . . 304
Organizing the Content on a Restaurant Website. . . . . . . . . . . . . . . . . 306
Building Content Articles with Lorem Ipsum. . . . . . . . . . . . . . . . . . . . 309
Setting Up the Home Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
Home Page Alternative to the Featured Article Manager . . . . . . . . . 310
xiv Contents
Creating Menus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Creating Footer Content. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
Creating Module Teaser Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
Using Stock Imagery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
Extending a Restaurant Website . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
Image Gallery: JPG Flash Rotator 2 . . . . . . . . . . . . . . . . . . . . . . . . 322
Email Marketing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
Google Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
Chapter 12:
Creating a Blog with Joomla! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
What Is a Blog? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
Why Have a Blog? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
What Options Are There for Blogging?. . . . . . . . . . . . . . . . . . . . . . 327
What Features Are Needed on a Blog Site? . . . . . . . . . . . . . . . . . . . . . . 328
Downloading and Installing a Blog Template . . . . . . . . . . . . . . . . . . . . 330
Optimus Template Features and Positions. . . . . . . . . . . . . . . . . . . . 331
Configuring the Logo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
Configuring the Main Horizontal Drop-Down Menu . . . . . . . . . . 334
Organizing Content on a Blog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
Organizing a Blog Within a Larger Site. . . . . . . . . . . . . . . . . . . . . . 335
Organizing a Standalone Blog. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
About Tagging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
Creating the Menus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
Adding Dynamic Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
Adding Static Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
Contents xv
Adding Basic Functionality to a Blog . . . . . . . . . . . . . . . . . . . . . . . . . . 346
Flexible Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
Browser-Based Editing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
Automated Publishing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
Categories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
Search Engine–Friendly URLs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
Comment Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
Syndication Feeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
Email Notification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
Extending a Blog Website Beyond the Basics . . . . . . . . . . . . . . . . . . . . 351
Forums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
E-commerce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
Appendix A:
Getting Help. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
Community Forums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
Help Sites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
Getting Help from Google . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
Appendix B:
A Guide to Joomla! 1.6 ACL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
Appendix C:
A Quick Introduction to SEO . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
Keyword Use in Title Tag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
Anchor Text of Inbound Link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
Global Link Popularity of Site (PageRank) . . . . . . . . . . . . . . . . . . . 359
Age of Site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
Link Popularity Within the Site . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
xvi Contents
Topical Relevance of Inbound Links and Popularity
of Linking Site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
Link Popularity of Site in Topic Community . . . . . . . . . . . . . . . . . 361
Keyword Use in Body Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
File Size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
Clean URL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
Utilize Your Error Pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
What’s Not Here? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
Appendix D: Installing WampServer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
Preface
Joomla is an open source content management system (CMS) that anyone can down-
choice for small businesses. Don’t let the price tag fool you, though; Joomla is power-
ful and robust, and more big organizations are choosing to use open source software
solutions all the time. Its universal appeal has made Joomla hugely popular as a CMS.
As Joomla matures, it is being adopted by more and more organizations, from cor-
porations to schools and universities to government organizations to newspapers and
magazines to small businesses. Its greatest advantage is its flexibility. You can see it on
a huge variety of sites.
The Purpose of This Book
This book is about Joomla, a popular and award-winning (“Best Linux/Open Source
Project” for 2005) open source CMS. This book walks, step-by-step, through every-
thing you need to develop a successful website powered by Joomla. The book gives
a general overview of management of a CMS and teaches you key concepts regard-
ing content organization, editing, and templates. Finally, this book examines some
more general topics, such as how to maximize search engine optimization (SEO) with
Joomla and what resources are available in the Joomla web community.
This book focuses on the most current release of Joomla—version 1.6. This release
is an important update that includes some key new features such as better Access Con-
trol Levels (ACL).
This Book’s Target Audience
This book primarily targets people using Joomla to create a website, either for them-
selves or their clients. It’s easy to read and low on technical jargon. It doesn’t assume
that you know PHP or CSS.
xviii Preface
All the concepts in this book are explained with step-by-step contextual examples.
If you follow all the steps in all the chapters, you will build seven separate Joomla
websites!
How to Use This Book
You can use this book in several ways. You can start at the beginning and go chapter-
by-chapter, as you develop your own site. The book is carefully laid out so that intro-
ductory ideas in the earlier chapters are developed and built on to help you understand
more advanced concepts later. You can also use the book as a reference. If you need
some quick ideas of what newsletter extensions are available, for example, head to
Chapter 6, “Extending Joomla!” Finally, the appendixes contain valuable information
Chapter 1: Content Management Systems and an Introduction to Joomla!
In today’s fast moving web, if you have a website that doesn’t have rich functionality
or fresh content, you will find yourself at a disadvantage to those that do. The idea of
powering websites with a CMS has been around for some time, but only recently with
the advent of high-quality open source CMS scripts like Joomla have we seen these
powerful CMS tools coming into the hands of you and me.
In this chapter, I explain in detail the difference between a “traditional” website and
one using a CMS. We also look at the history of Joomla and an overview of some of
its features.
Joomla is one of the most popular open source CMSs on the planet. The first step in
becoming part of the “Joomlasphere,” the vibrant community that exists around the
This chapter shows you how to get up and running with a Joomla site. The two
steps are to find and download the latest files and to install them on a web server. This
chapter describes both a local installation—your home computer to use as you read
this book (if you don’t have a hosting account or have a slow Internet connection)—
and a real web server installation.
How to Use This Book xix
managing users, and making sure installed components and modules are running cor-
rectly. With a properly configured Joomla site, the administration burden is relatively
low. Most of the effort can be dedicated to generating that all-important content.
In this chapter, we go on a whirlwind tour of the core administrative functions you
need. I won’t be going step-by-step explaining every last button in the admin backend,
but rather picking out key functions, tips, and tricks that you need to know to keep
Chapter 4: Content Is King: Organizing Your Content
As a CMS, Joomla’s primary function is to organize and present all the content in your
site. It does this through content articles. These discrete pieces of content must be
organized into a hierarchy of categories.
This chapter provides an in-depth tutorial that explains how Joomla displays its
content articles and how you can organize the hierarchical structure of them. It details
how to plan and organize the content and user experience for the site. It also explains
how to best structure content into them for small and large sites.
Menus are perhaps the core of a Joomla site. In a static HTML site, they merely serve
as navigation. In a Joomla site, they serve that purpose, but also determine the layout
of what a dynamic page looks like and what content appears on that page when you
navigate to it. The relationship between menus, menu items, pages, and modules is
perhaps one of the most confusing in Joomla. This chapter explains this relationship so
that you can create a navigation scheme that works for your site.
site and how the different aspects interact to produce a coherent navigation structure.
Chapter 6: Extending Joomla!
It’s hard to find a Joomla powered website that has not added functionality beyond the
basics with some sort of extension. The word “extension” collectively describes com-
ponents, modules, plug-ins, and languages. Many hundreds of extensions are available
both free and commercially from third-party providers.
xx Preface
In this chapter, we look at some examples of core and third-party Joomla exten-
sions. We also examine how they are installed and managed in Joomla.
Chapter 7: Expanding Your Content: Articles and Editors
There are two main ways to add and manage content in a Joomla site: through the
frontend or backend. Part of the attraction of Joomla is the ability to easily add and
edit content through a What You See Is What You Get (WYSIWYG) editor.
In this chapter, we look at WYSIWYG and how it functions in the backend with
editors, and publishers manage content through the frontend.
Chapter 8: Getting Traffic to Your Site
Search Engine Optimization (SEO) might be one of the most maligned subjects on the
Web. From black hat SEO—people who use unethical methods to gain rank in search
engines—to their counterparts white hat SEO—the good guys—how best to get traffic
Trying to learn about SEO is difficult, to say the least. In this chapter, I emphasize
Search Engine Marketing
(SEM). I point out some obvious SEO tips and how they apply
to Joomla, but I also discuss a more holistic marketing plan including such strategies
as Pay Per Click and blogging.
Chapter 9: Creating Pure CSS Templates
In this chapter, we go through the steps of creating a Joomla template. Specifically, we
create a template that uses Cascading Style Sheets (CSS) to produce a layout without
use of tables. This is a desirable goal as it means that the template code is easier to vali-
date to World Wide Web Consortium (W3C) standards. It also tends to load faster,
be easier to maintain, and perform better in search engines. We discuss these issues in
detail later in the chapter.
Chapter 10: Creating a School Site with Joomla!
School websites tend to be medium to large in size. Two of Joomla’s defining character-
istics are its power and flexibility, but it can be time intensive to set up. This leads us
to this chapter—an extensive guide to creating and setting up a school website using
the Joomla CMS.
How to Use This Book xxi
Chapter 11: Creating a Restaurant Site with Joomla!
This chapter looks at the entire process of creating a small business website, in this
case a restaurant website, from scratch. Starting from an analysis of needs, this chapter
shows you how to organize possible content all the way through to adding photos and
considering further extensions.
Chapter 12: Creating a Blog with Joomla!
It seems like everyone has a blog these days. Many people still think of blogs as per-
sonal diaries, but more and more organizations and companies are using blogs as a
way to shape perception of who they are and what they do. Chances are, if you go to a
company’s website today, you will find a link to its blog somewhere on the site. What
is becoming more common on websites now, is a section of the site that is dedicated
to the blog.
This chapter talks about blogs in a more general sense: a dynamic communication
medium for a person or organization to interact with stakeholders. We look at creating
a blog from scratch using Joomla.
Appendix A: Getting Help
Stuck with Joomla? A tremendous amount of information is available on the Web, as
well as many active communities to ask for help.
Appendix B: A Guide to Joomla! 1.6 ACL
Access Control Levels dictate what users can perform what tasks in your Joomla web-
site. This brief guide helps you understand how ACL has been changed and improved
in Joomla 1.6.
Appendix C: A Quick Introduction to SEO
appendix.
Appendix D: Installing WampServer
This appendix provides a quick guide to installing WampServer on your home com-
puter. This package is important, so you can follow along with all the site examples in
the book.
xxii Preface
What Is a Content Management System?
A CMS is a collection of scripts that separate content from its presentation. Its main
features are the ease of creation and editing of content and dynamic web pages. CMSs
are usually sophisticated and can have newsfeeds, forums, and online stores. They are
also easily edited. More and more websites are moving toward being powered by CMSs.
Most CMSs are expensive—in the range of \$50,000 to \$300,000—but an increas-
ing number of open source alternatives are becoming available. Open source CMSs
have become increasingly more reliable and are now being used for important projects
in many companies, nonprofits, and other organizations.
A CMS separates the responsibilities involved in developing a website. A web
designer can be concerned with the design, and nontechnical people can be responsible
for the content.
A modern CMS is usually defined by its capability to manage and publish content.
Most CMSs do far more, taking advantage of a wide range of extensions and add-ons
What Is Open Source Software?
Joomla is an example of open source software; its nonprofit copyright holder is Open
Source Matters (see www.opensourcematters.org). An open source project is developed
by a community of developers around the world, all volunteering their time. Some
examples of open source software you might have heard of are Firefox, Apache, Wiki,
Linux, and OpenOffice. All these projects have challenged and even surpassed their
commercial equivalents. If you are curious about how and why people should create
• en.wikipedia.org/wiki/Open_source
• www.opensource.org
How to Use This Book xxiii
Joomla!
The full and proper name of the Joomla CMS includes an exclamation point, as shown
here. For the sake of readability, and a tree or two, I’ve kept the exclamation point in
heads but dropped it in the text.
www.joomlabook.com
downloadable versions of all the sites created in the chapters, at www.joomlabook.com.
TIP
The tip boxes give more advanced ideas about an aspect of Joomla. You usually can
find more details about the tip at compassdesigns.net.
Things to Look For
The following are specific elements to look for when reading:
NOTE
The note boxes denote cautions about an aspect of the topic. They are not appli-
cable to all situations, but you should check whether a note applies to your site.
THE LEAST YOU NEED TO KNOW
Explanations of key critical concepts can be found in the Least You Need to Know
boxes. These are worth circling in a big red pen or writing out for yourself on a cheat
sheet.
À
CAUTION
Cautions provide critical information.
xxiv Preface
As with many open source products, Joomla changes on a very short release cycle. New
maintenance releases with slight changes can often be released in as little as six weeks,
and usually the changes are difficult to find out about. This makes writing for open
source challenging. If you find minor inconsistencies in this book, chances are it is
because of these minor updates. To stay informed of recent changes to Joomla, consult
the forum at www.joomlabook.com where you can find discussions of Joomla versions.
Acknowledgments
Without the continuing support of my wife, Sarah, this book would not have been
possible. Sarah let me frequently slip off to work on the manuscript. Part of my thanks
also goes to the three boys who (mostly) managed not to bug me while I was writing.
I’d also like to thank the third-party developers I frequently annoyed on Skype with
Finally, many thanks to the guys who live on the trunk—the many developers who
selflessly contribute code to the Joomla project on a daily basis.
Barrie M. North
has more than 20 years of experience with the Internet as a user,
designer, and teacher. He has spent more than 8 years in the education field, becoming
steadily more involved in web technology, teaching web design classes to students and
technology integration to teachers. Most recently, he worked as an IT consultant for
two new schools pioneering the use of technology. As well as web design, he has pro-
vided web marketing/SEO, usability, and standards compliance expertise to his clients.
He is a founder of Joomlashack.com, one of the oldest and most popular Joomla
template providers, and SimplWeb.com, a service that provides easy-to-use, turnkey
Joomla hosting for those new to Joomla. He also maintains a blog about all things
Joomla at CompassDesigns.net. When not working, he can frequently be found on the
Joomla community boards, and he has written many free tutorials for using Joomla.
His combination of Joomla expertise, educational skills, and engaging writing has pro-
duced a book accessible to everyone. Barrie lives in South Strafford, Vermont.
1
1
Content Management
Systems and an Introduction
to Joomla!
In This Chapter
O
n today’s Internet, if your website doesn’t have rich functionality or fresh con-
tent, you will find yourself at a disadvantage. The idea of powering websites
with content management systems (CMSs) has been around for some time, but it is
only recently—thanks to high-quality open source CMS scripts such as Joomla—that
you and I can now use these powerful CMS tools.
In this chapter, I explain in detail the difference between a traditional website and
one that uses a CMS. I also provide a look at the history of Joomla and give an over-
view of some of its features. This chapter answers the following questions:
Q
What is a CMS, and how is it different from a traditional website?
Q
What is Joomla, and where did it come from?
Q
What can Joomla do?
Q
What are the basic elements of a Joomla web page?
2
What Is a Content Management System?
What exactly is a content management system (CMS)? To better understand the
power of a CMS, you need to understand a few things about traditional web pages.
Conceptually, there are two aspects to a web page: its content and the presentation
of that content. Over the past decade, there has been an evolution in how these two
pieces interact:
Static web pages—The content and presentation are in the same file.
Web pages with Cascading Style Sheets (CSS)—The content and presen-
tation are separated.
Dynamic web pages—Both content and presentation are separated from
the web page itself.
Static Web Pages
A web page is made up of a set of instructions written in Hypertext Markup Lan-
guage (
HTML)
that tells your browser how to present the content of a web page. For
example, the code might say, “Take this title ‘This is a web page,’ make it large, and
make it bold.” The results will look something like the page shown in Figure 1.1.
FIGURE 1.1 Results of code on a web page.
What Is a Content Management System? 3
This way of creating a web page is outdated, but an astonishing number of design-
ers still create sites using this method. Pages created using this method have two main
drawbacks:
Difficult to edit and maintain—All the content shown on the page (“This is
a web page”) and the presentation (big and bold) are tied together. If you want
to change the color of all your titles, you have to make changes to all the pages
in your site to do so.
Large file sizes—Because each bit of content is individually styled, the pages
are big, which means they take a long time to load. Most experts agree that
large file sizes hurt your search engine optimization efforts because most search
engines tend not to completely index large pages.
Web Pages with CSS
In an effort to overcome the drawbacks of static web pages, over the past four or five
years, more comprehensive web standards have been developed. Web standards are
industrywide “rules” that web browsers such as Internet Explorer and Mozilla Firefox
follow (to different degrees, some better than others) to consistently output web pages
onto your screen. One of these standards involves using Cascading Style Sheets (CSS)
to control the visual presentation of a web page. CSS is a simple mechanism for add-
ing style (for example, fonts, colors, spacing) to web documents. All this presentation
information is usually contained in files that are separate from the content and reusable
across many pages of a site.
Using CSS, the generated web page from Figure 1.1 might look as shown in
Figure 1.2.
Now the file containing the content is much smaller because it does not contain
presentation or style information. All the styling has been placed in a separate file that
the browser reads and applies to the content to produce the final result.
Using CSS to control the presentation of the content has big advantages:
Maintaining and revising the page is much easier. If you need to change all the
title colors, you can just change one line in the CSS file.
Both files are much smaller, which allows the data to load much more quickly
than when you create web pages using HTML.
4 Chapter 1 Content Management Systems and an Introduction to Joomla!
FIGURE 1.2 A modern web page using CSS.
The CSS file will be cached (saved) on a viewer’s local computer so that it
won’t need to be downloaded from the Web each time the viewer visits a differ-
ent page that uses the same styling rules.
NOTE
Take a look at www.csszengarden.com. Every page on this classic CSS site has
identical content but has a different CSS applied. You can browse through the
designs and see the same content styled in hundreds of different ways.
THE LEAST YOU NEED TO KNOW
Modern websites separate content from presentation by using CSS. CSS files con-
tain presentation rules that determine how content should look when it’s displayed.
The same CSS file can be used with many different pages of content to maintain a
consistent appearance and style across a site.
Dynamic Web Pages
A CMS further simplifies web pages by creating dynamic web pages. Whereas CSS
separates presentation from content, a CMS separates the content from the page.
Therefore, a CMS does for content what CSS does for presentation. It seems that between
What Is a Content Management System? 5
CSS and a CMS, there’s nothing left of a web page, but in reality what is left can be
thought of as insertion points, or placeholders, in a structural template or layout. For
example, see Figure 1.3.
FIGURE 1.3 The structure of a CMS web page.
The “put some content here” instruction tells the CMS to take some content from
a database, the “pure content,” and place it in a designated place on the page. So what’s
so useful about that trick? It’s actually very powerful: It separates out the responsibili-
ties for developing a website. A web designer can be concerned with the presentation
or style and the placement of content within the design layout—the placeholders. This
means that nontechnical people can be responsible for the content—the words and
pictures of a website—without having to know any code languages, such as HTML
and CSS, or worry about the aesthetics of how the content will be displayed. Most
CMSs have built-in tools to manage the publication of content.
It’s possible to imagine a workflow for content management that involves both
designers and content authors (see Figure 1.4).
A CMS makes the pages dynamic. A page doesn’t really exist until you follow a
link to view it, and the content might be different each time you view it. This means a
page’s content can be updated and customized based on the viewer’s interactions with
the page. For example, if you place an item in a shopping cart, that item shows up
on the shopping cart page. It was stored in a database and now gets inserted into the
“shopping cart placeholder.” Many complex web applications—for example, forums,
6 Chapter 1 Content Management Systems and an Introduction to Joomla!
shopping carts, and guest books, to name a few—are in fact mini CMSs (by this
definition).
FIGURE 1.4 A CMS manages content publication.
Other good examples of CMSs are blogs. A blog uses a template that presents all the
content (or posts, in this case), and it is easy to edit and publish. The growth in the use
of CMSs for powering websites is probably due in part to the huge rise in popularity
of CMS-based blogging tools such as Blogger and WordPress.
THE LEAST YOU NEED TO KNOW
A CMS totally separates the content of the pages from their graphical design and
layout. This makes it easy to keep the sitewide design coherent and easy to change.
It also makes adding content easy for nontechnical people.
The range of available CMSs is extensive—from enterprise-scale versions that cost
\$300,000 to open source versions, such as Joomla, that are free. Modern CMSs are
usually defined by their capability to manage and publish content. They typically have
workflow processes that start at content creation and move through editing or approval
stages to publishing. Most do far more: They have the capability to use a wide range of
extensions or add-ons to give the site more functionality. Joomla has more than 4,000
extensions available through various forums and newsletters; many of them are free
and are created by volunteer developers around the world. The official repository is at
extensions.joomla.org. Since 2009, this now lists only GPL licensed extensions. For
non-GPL extensions, you will have to resort to Google searching.
Open Source Software 7
There is one large drawback to using a CMS: From a technical viewpoint, a CMS
can become extremely complex, containing thousands of files and scripts that work
together in concert with databases to present a comprehensive and feature-rich web-
site. Normally, this means that a CMS site will be designed and created by technical
staff and managed and run by nontechnical users. Joomla is probably the easiest to set
up among currently available CMSs, allowing users with modest technical skill to har-
ness its power. The purpose of this book is to guide nontechnical users, step-by-step,
Table 1.1 sums up this concept of “difficult to set up but easy to grow.”
TABLE 1.1 Comparison of Static Websites and CMSs
Static Website
Content Management System
Creating initial web pages is easy.
Creating initial pages is time-consuming because a large
infrastructure must be installed, databases must be set up,
and templates must be created before the first page can be
created.
Content is static; changing it requires technical
Content is dynamic; it can be changed with no technical
expertise, and multiple instances of content have
knowledge, and a single change can appear or take effect
to be edited individually on each page.
sitewide.
Adding new functions is difficult and often
Most CMSs have many extensions that “plug in” easily.
requires custom code.
Open Source Software
One factor that has contributed to the rise in the popularity and ease of use of CMSs
is the growth of the open source software movement.
NOTE
extensions.joomla.org has a rating and review systems. Be careful about relying
too heavily on the ratings. The highest-rated extensions are shown at the top level
of the site, so they continue to get more traffic and then tend to get rated even
more. There are often great extensions, especially newer ones, hidden away in the catego-
ries. It’s worth taking an hour or two to browse all of them to find extensions that might
be of use to you.
The quality of the available extensions varies widely. If you are using an extension on an
important site, do the due diligence to check out the developer and visit his or her site, as
well as test the extension thoroughly before using it on a production site.
8 Chapter 1 Content Management Systems and an Introduction to Joomla!
In 1998, Netscape bucked the universal wisdom of how to develop software by
making the source code for its browser, Netscape, freely available to anyone and every-
one. This milestone was key in creating a philosophical movement among code devel-
opers in which software is created by large communities of developers and released
openly to the world. (Hence, the term open source.)
As the Web has grown explosively, we have seen open source software grow and
mature to power the Web. The most significant open source software is collectively
referred to as LAMP:
Linux—An operating system
Apache—Software to run a web server
MySQL—Powerful database software
PHP—A programming language used to write both simple and complex
scripts that create interactive functionality with databases
LAMP has allowed developers to create powerful applications using the PHP program-
ming language. One specific area of growth has been the development of CMSs that
are written in PHP, such as XOOPS, PostNuke, WordPress, Drupal, and Joomla.
THE LEAST YOU NEED TO KNOW
Joomla is an example of open source software. It’s created and maintained by a
worldwide community of developers and distributed at no charge.
History of Joomla!
Joomla is a powerful open source CMS that has grown in popularity since its rebrand-
ing from Mambo in 2006. Its two key features—ease of administration and flexibil-
ity of using templates—have made it useful for powering everything from corporate
intranets to school district sites.
Late 2007 saw the release of Joomla 1.5, which signified a major rewrite of the soft-
ware. The changes included a simplification and streamlining of the processes for users
to contribute content, add extensions, and manage sites. It was a significant enough
change that extensions had to be rewritten to operate efficiently in the new version.
That is why you see extensions listed at extensions.joomla.org (for example, 1.0 Native,
1.5 Legacy, 1.5 Native).
Joomla!’s Features 9
This third edition of this book covers the latest release of Joomla—1.6. The change
for extensions is even more than that from 1.0 to 1.5. At the time of writing, we can
anticipate that the extensions directory will show a new flag for 1.6 rated extensions.
The Joomla! Community
A large and active community is an important factor in the success of an open source
project. The Joomla community is both big and active. The official forum at forum.
joomla.org is perhaps one of the biggest forum communities on the Web. In addition,
there are many forums on Joomla’s international sites and the respective sites of its
third-party extension developers.
Third-Party Extensions Development
Joomla is unique among open source CMSs in the number and nature of the nonof-
ficial developers who create extensions for it. It’s hard to find a Joomla site that doesn’t
use at least one extension. The true power of Joomla lies in the astonishing range of
extensions that are available.
The nature of Joomla developers is interesting. There are an unusually high pro-
portion of commercial developers and companies creating professional extensions for
Joomla. Although open source and commercial development might seem unlikely bed-
fellows, many commentators have pointed to this characteristic of the Joomla project
as a significant contributor to its growth.
Joomla!’s Features
Joomla has a number of “out of the box” features. When you download Joomla from
www.joomlacode.org, you get a zip file about 5MB that needs to be installed on a web
server. Running an installation extracts all the files and enters some “filler” content into
the database. In no particular order, the following are some of the features of the base
installation:
Simple creation and revision of content using a text editor from the main
front-end website or through a nonpublic, back-end administration site
User registration and the ability to restrict viewing of pages based on user level
Control of editing and publishing of content based on various admin user
levels
10 Chapter 1 Content Management Systems and an Introduction to Joomla!
Simple contact forms
Public site statistics
Private detailed site traffic statistics
Built-in sitewide content search functionality
Email, PDF, and print capability
Simple content rating system
Display of newsfeeds from other sites
As you can see, Joomla has some tremendous features. To have a web designer create all
these features for a static site would cost tens of thousands of dollars, but it doesn’t stop
there. Joomla has a massive community of developers worldwide (more than 30,000)
who have contributed more than 5,000 extensions for Joomla, most of which are free.
The following are some of the most popular extension types:
Forums
Shopping carts
Calendars
Photo galleries
Forms
User directories and profiles
Each extension can be installed into Joomla to extend its functionality in some man-
ner. Joomla has been very popular partly because of the availability of the huge and
diverse range of extensions.
To customize your site further, you can easily find highly specialized extensions,
such as the following:
Recipe managers
Help/support desk management
Fishing tournament tracking
Elements of a Joomla! Website 11
Multiple site management
Real estate listings
Hotel room bookings
You get the idea!
THE LEAST YOU NEED TO KNVOW
Joomla provides rich functionality in its default package. It can be further extended
to almost any niche application through the availability of both free and low-cost
commercial extensions.
Elements of a Joomla! Website
A Joomla website has several elements that work together to produce a web page.
The three main elements are content, templates, and modules. The content is the core
aspect of the website; the template controls how the website’s content is presented; and
the modules add dynamic functionality around the edges of the main page content.
Think of these three elements as three legs holding up a stool. Without any one of
them, the page (the stool) would topple.
Figure 1.5 shows a page of www.compassdesigns.net, my own Joomla-powered
blog.
Figure 1.6 highlights two of the three elements of a Joomla page—the content and
the modules. The third, the template, is evident in the color, graphics, layout, and font
(which are all part of the template).
On this Joomla web page, the main page content is a large column on the left with a
blog post. Various modules are shown in the right-side column and at the top and bot-
tom. The layout and positioning of the content is managed by the template, together
with any CSS content styling files it references.
Content
The most important part of a website is the content—the meat and potatoes of your
web page, the important stuff in the middle of the page that the viewer is looking at;
you have probably heard the phrase “Content is king.” Joomla, as a CMS, helps you
efficiently create, publish, and manage your content. Content is organized into man-
ageable chunks called articles.
12 Chapter 1 Content Management Systems and an Introduction to Joomla!
FIGURE 1.5 A Joomla website, www.compassdesigns.net.
Joomla actually has a specific name for the core of the page: the mainbody of the
page. This is usually the biggest column and is placed in the middle.
The content in the mainbody is generated by what Joomla calls a component. The
biggest and most important component in Joomla is the one that handles all your
articles, the individual content items in the site. In fact, it’s so important that often you
find these referred to as content articles. In the default Joomla installation, there are also
a few other components that generate the content that appears in the main body, such
You can take advantage of the many available third-party components that can gen-
erate content in the main body. Examples include forums and shopping carts.
THE LEAST YOU NEED TO KNOW
The mainbody of a Joomla web page displays content produced by a component.
The most important component within Joomla is the one that manages and dis-
plays one or more pieces of content, stored in the database as articles.
Elements of a Joomla! Website 13
FIGURE 1.6 The elements of a Joomla web page.
Templates
A template is a set of rules about the presentation of components and modules within
a page and their placement on the screen. A template determines the layout or posi-
tioning of a web page. A template, along with its CSS files, also determines how many
columns to use and what color to make titles, for example. A template acts as a filter (or
lens) that controls the presentation aspects of a web page. It does not have any content,
but it can include logos.
NOTE
You will find that templates are also often referred to as a type of extension, along
with components and modules. The template concept is shown in Figure 1.7. Here
you can see the raw content from the database that is presented through the template to
the final viewed web page.
14 Chapter 1 Content Management Systems and an Introduction to Joomla!
FIGURE 1.7 How a web page is built from a CMS database.
Modules
Modules are small functional blocks that are usually shown around the main part of the
page, such as a poll/survey, a login form, or a newsflash. Modules may display other
content from the database that may or may not be related to the mainbody content
(such as related stories), implement features of the site such as manage your login sta-
tus, provide navigation to other pages, or provide a search capability.
The example shown previously in Figure 1.6 has modules at the top: a search and a
Components and modules are usually both referred to as extensions because they
extend the functionality of a site.
THE LEAST YOU NEED TO KNOW
A Joomla site is made up of content (articles and other content displayed by com-
ponents), a template, and modules. The template selects and positions the content
that appears on the page, and it controls all the presentation and aesthetic aspects of the
web page. It does not have any content (except that it can include logos).
Summary
Joomla is a great CMS that is capable of powering sophisticated dynamic websites—for
little or no cost. This chapter looked at the general nature of a CMS, Joomla’s history,
what Joomla can do, and what makes up a Joomla web page.
Summary 15
Here are the key points covered in this chapter:
A CMS separates the content of web pages from the graphical design. This
makes it easy to keep sitewide design coherent and makes it easy to change. It
also makes adding content easy for nontechnical people.
Joomla is an example of open source software. It is created by a worldwide
community of developers and is available free of charge.
Joomla provides rich functionality in its default package. It can be further
extended with additional features to meet the needs of almost any niche
application through the availability of free (GPL) and low-cost (commercial)
mercial extension sites.
The main body of a Joomla web page displays content generated by a com-
ponent. The most important component is the one that manages, selects, and
displays articles stored in the database.
A Joomla site is made up of content (generated by components), a template,
and modules. The template determines what content is displayed on a page
and its placement within the layout. It also controls all the presentation aspects
of the web pages. It does not have any content, but it can include logos.
Modern websites separate content from presentation by using a technology
known as Cascading Style Sheets (CSS). Templates utilize CSS style sheets to
control the aesthetic appearance of the content displayed by that template.
17
2
Installing Joomla!
In This Chapter
J
oomla is one of the most popular open source CMSs on the planet. The first step
in becoming part of the “Joomlaverse,” the vibrant community that exists around
This chapter describes the first two stages in getting a Joomla-powered site up and
This chapter describes both local installation (which you’ll want to use if you don’t
have a hosting account or if you have a slow Internet connection) to use as you read
this book and a real web server installation. This chapter covers the following topics:
Q
How do you install Joomla?
Q
Where can you find the most current Joomla files?
Q
How do you unpack the Joomla files on your desktop computer or on a
hosting account?
Q
How do you use the Joomla Installation Wizard?
Q
How can you support the Joomla project?
18
How to Install Joomla!
The process of installing Joomla involves several steps:
1.
Obtain the latest Joomla file package.
2.
Create a SQL database.
3.
Unpack the package on a server.
4.
Use a browser-based wizard to complete the installation.
We look at each of these steps in turn. There are two alternative paths you can take
for step 2: You can either unpack Joomla on a remote hosting account or create a
web server that actually runs on your desktop/laptop computer. The second tech-
nique is useful for trying out Joomla by creating a site and then transferring it to a
hosting account.
Obtaining the Latest Joomla! File Package
the Joomla project is www.joomla.org. The Joomla site is actually a
The home of
collection of separate sections and sites for different aspects of the project. With one
exception, all the sections and sites are powered by Joomla. As of this writing, the
following sections are available:
www.joomla.org—This is the main Joomla site, where you can find the
latest information and news. This site is home to the official news blog for
Joomla, which talks primarily about the project’s development plans and
progress. You can subscribe to news via RSS by clicking the link in the left-
hand column.
community.joomla.org—A portal for all the community activities of the
Joomla project, this site includes blog posts from both the Leadership Team
and community members, as well as information about events, user groups,
and the “Joomla Magazine” and JoomlaConnect—an aggregated RSS feed
from the third-party developer community.
forum.joomla.org—With more than 394,850 members at the time of this
writing, the official Joomla forum is one of the biggest forums on the Web.
Obtaining the Latest Joomla! File Package 19
You can get help from the active Joomla community, whether for templates,
translations, components, using extensions, or just help in general. When you
are asking for help, remember that the forum is all-volunteer, so provide as
much detail on your problem as you can in a concise note and be respectful.
extensions.joomla.org—The Joomla Extensions Directory is packed with
more than 5,000 third-party GPL extensions, including components, mod-
ules, and plug-ins. This is the place to look when you’re ready to extend the
functionality of your Joomla website. It even includes useful reviews and rating
tools so you can see what other people think of various extensions.
resources.joomla.org—This is a listing of third-party individuals and compa-
nies that provide products and service for Joomla, including education, exten-
sions, and support.
docs.joomla.org—This site provides documentation and help for Joomla.
This section is a community-generated Wiki that provides a lot of useful infor-
developer.joomla.org—This is where developers can find documentation on
the Joomla API.
people.joomla.org—This is a new social portal where you can join and create
a user profile, make friends, and join discussions on Joomla. You can find me
at http://people.joomla.org/my-page/compass.html.
www.joomlacode.org—This site is the only one that doesn’t run on Joomla,
but it’s the one we are interested in right now. Known as the forge, it serves as
the code repository both for the main Joomla files and many of the thousands
of GPL third-party extensions.
NOTE
Let’s take a look at the forge so you can get an idea of how to find the files you need.
have changed over various redesigns of www.joomla.org. When you click the Down-
load Joomla button, you are sent to a page where you can grab the latest zip files that
make up a base Joomla installation.
Joomla! Package Naming Conventions
Before we move on, let’s take a quick look at how the Joomla packages and releases are
named.
The naming convention for Joomla versioning is A.B.C, which represent the fol-
lowing elements:
A—This is the major release number. Currently all versions of Joomla begin
with 1 (that is, 1.B.C).
B—This is the minor release number. The current minor release number is
1.6, and this book is based on Joomla 1.6.
C—This is the maintenance release number; for example, 1.5.20 was a recent
security release of Joomla 1.5.
NOTE
At the time of writing, Joomla 1.6 was in beta, and the page http://www.joomla.
to get beta versions, you can get them at http://developer.joomla.org/code.html or
http://joomlacode.org/gf/project/joomla/frs/.
THE LEAST YOU NEED TO KNOW
The core Joomla files are available for free at www.joomlacode.org. The Joomla
make sure you are getting the correct version, either the full package or an update.
IMPORTANT NOTE
You cannot upgrade from Joomla 1.5 to Joomla 1.6. There are significant enough
changes in the code that simply overwriting files would break your site. Migrating a
site from 1.5 to 1.6 is a complex process that is beyond the scope of this book. For many
sites, it is easier to construct the site over again in 1.6 and copy and paste the content.
After you’ve located and downloaded a compressed Joomla file package of several
megabytes, what do you do with it?
Unpacking the Joomla! Package 21
Creating a MySQL Database
Whether set up at home or on a hosted server, Joomla needs a MySQL database to
serve as a repository for site content. SQL (pronounced “sequel”) stands for Structured
Query Language and has become a shorthand reference to any database structure that
responds to requests written in the SQL language. One particular brand of SQL data-
base software is the very popular MySQL, which can be set up on almost any hosted
web server, including your home computer.
If you are installing Joomla locally (on your home or office computer) with
WampServer 2 or XAMPP, as described in the following section, the wizard will have
the permissions necessary to automatically create a database.
If you are installing Joomla on a web host, you will need to pre-create a SQL data-
base. When you do, make sure to note the username, password, and database name.
The most common way to set up a database is through some sort of button/link in
Unpacking the Joomla! Package
You need to choose whether you will be setting up your Joomla site on your home
computer, on a hosted server, or both. The following section, “Unpacking Joomla! on
a Local Desktop Computer,” walks through setting up a home computer to serve as
a host and installing Joomla on it. This approach is ideal for designing a new site and
testing the extensions you might use with it. The section, “Unpacking Joomla! on a
Hosting Account,” walks through setting up a Joomla site—which will serve as your
production website—on a hosted server.
IMPORTANT NOTE
Before you begin installing Joomla, you need to have a MySQL database ready for
Joomla to use.
Unpacking Joomla! on a Local Desktop Computer
If you unzip the Joomla file package and try to run/open the main index.php file, it will
not work and will instead open in an editor where you can see all the code.
Joomla is not a self-contained program like Microsoft Word or Mozilla Firefox.
With those sorts of programs, you simply install them onto your computer by running
an installation file. Joomla is very different. Joomla is client/server software and needs
an installation of PHP in order to execute.
Joomla is a complex series of Hypertext Preprocessor (PHP) scripts that run on a
web server. When you browse a Joomla site, these scripts are generated on-the-fly and
create what you see on the pages of the site. The key term here is web server. This is
an example of client/server scripting: The software is actually running on a different
NOTE
No, the acronym PHP doesn’t quite match the words Hypertext Preprocessor. It used
to stand for Personal Home Page. PHP is principally a programming language for
web pages/servers.
Thus, you cannot download Joomla and try to run it on your computer as if it’s an
EXE file. It has to have a server, which means you need to have a hosting account at a
hosted server or a set of programs on your local computer that emulate a hosted server.
Before you shell out your hard-earned money for a hosting account, there is some-
thing else you can do first: You can run a web server on your local computer—that is,
your desktop or laptop. This is known as having a localhost. It may sound like I just
contradicted myself from the previous paragraph, but I really didn’t. You can’t “run”
Joomla itself on your own computer, but you can install a localhost web server for it
to “run on.” In this scenario, your computer is acting as both the server and the client.
from your own computer. One disadvantage is that you will have to move, or “port,”
the site to a real web host later on. Setting up a localhost is a great way to learn about
Joomla before you start to develop your site.
THE LEAST YOU NEED TO KNOW
Joomla needs a web server to run on. A good way to learn about Joomla is to run
a web server on your own computer, known as using a localhost. This makes your
“practice” site blazingly fast (if not available to the world).
To set up your localhost, you need some software that runs Apache, PHP, and
MySQL on your computer. These are the component scripts of a remote web server
on a hosting account. Two popular packages include all these scripts, and both are free:
Unpacking the Joomla! Package 23
WampServer 2—This package, available at www.wampserver.com/en, is for
Windows.
XAMPP—This package, available at www.apachefriends.org/en/xampp.html,
is for Windows, Mac OS X, and Linux.
Let’s quickly run through Joomla setup using WampServer 2 for the localhost:
NOTE
WampServer 2 is Windows-specific.
1.
the end of the installation, you will have a folder called c:\wamp\www, which
serves as the root folder of a local website. If you use XAMPPLite, the folder
will be called c:\xampplite\htdocs.
2.
\www\ or \htdocs\. It doesn’t matter what the new folder is called (for example,
c:\wamp\www\Joomla would work). Make sure that you don’t unpack it in
such a way that you end up with two folders, one inside the other (for exam-
ple, c:\wamp\www\Joomla\Joomla_1.6-Full_Package.zip).
3.
Run WampServer. You should get a handy icon in your system tray (the icons
at the bottom right of the Windows desktop). Figure 2.1 shows three possible
versions of the icon. (For XAMPP, you run start-apache.bat and then start-
mysql.bat.)
FIGURE 2.1 WampServer icons in the Windows system tray.
4.
Make sure that the dial is white.
5.
Now open a browser and go to http://localhost (no “www”) or left-click the
icon and select localhost. You should see a page that looks like Figure 2.2.
FIGURE 2.2 Browser view of WampServer http://localhost.
NOTE
If you are not seeing the page shown in Figure 2.2, you should stop and figure
out why. You have to get this page before you can proceed. The WampServer site
has some helpful troubleshooting FAQs and a forum. You should see your folder called
“Joomla” (or whatever you called it) in the Your Projects list. Click that folder, and you are
taken to that website, running locally on your computer.
Unpacking the Joomla! Package 25
At this point, all you have done is set up the localhost web server and unpacked the
Joomla files so they are ready to install. Before we look at how to install Joomla, let’s
take a quick look at what the upload process involves if you have a hosting account and
want to install Joomla on a hosted web server.
NOTE
If you get stuck with this part of the process, you can refer to Appendix D, “Install-
THE LEAST YOU NEED TO KNOW
Several free packages include all the files and scripts needed to run a web server on
a desktop computer. WampServer is one for Windows.
Unpacking Joomla! on a Hosting Account
This section assumes that you either have a hosting account or are going to get one.
Joomla has some minimum requirements to run, and Joomla 1.6.X has slightly higher
requirements than Joomla 1.5.X.
Here are the minimum requirements for Joomla 1.6:
• PHP 4.2.x or above
• MySQL 5.0.4 or above—See www.mysql.com
You must ensure that you have MySQL, XML, and Zlib support built into your PHP.
For assistance in making sure you have the proper support, refer to the Joomla Help
Forums (see help.joomla.org).
When you have a host that meets the requirements, you need to upload the main
Joomla files. There are two ways to do this:
You can upload the zip file and then extract the contents on the server by using
a shell command or Cpanel file manager.
You can extract the contents of the zip file onto your desktop and then upload
the contents individually via FTP.
If you have Cpanel with your hosting company (almost all hosting companies provide
it), the first method is usually the fastest and easiest way to do this. You can use this file
manager to upload the zip file to the public_html folder (or whatever folder you have
Running the Joomla! Installation Wizard
If you are this far along, you have unzipped the Joomla package to either a remote web
host or a localhost root folder on your local computer. Now for the fun stuff—actually
installing Joomla.
You install Joomla via a browser-based wizard that walks you through several steps.
Getting to the Joomla! Installer
Using your browser of choice (mine is Firefox), navigate to the location of your Joomla
files. (In my case on a localhost, it is http://localhost/Joomla.) You will see the first
installation screen (see Figure 2.3). If you don’t see this screen, be sure the Apache/
MySQL/PHP host software is all running, you have the Joomla files unpacked into the
root folder, and the path (folder names) to the index.php file within the Joomla folder
is typed correctly.
Before you start, make sure you have pre-created a SQL database for the site to use.
Step 1: Language
Figure 2.3 gives you a first look at some of the internationalization features of Joomla
1.6. You can select among many languages for the installation instructions.
NOTE
Many web hosts offer a tool called Fantastico , which enables you to instantly cre-
ate a Joomla website, along with all the databases needed. I actually don’t recom-
mend using Fantastico. Although it makes the process easier, many hosts don’t have the
most current releases of Joomla in the available Fantastico installation scripts.
THE LEAST YOU NEED TO KNOW
To install Joomla on a web host, your account needs to meet some minimum
requirements. Make sure your host does, or you will have problems later.
as the root folder on your host; www and htdocs are sometimes used). You can then
use it to extract the files.
Running the Joomla! Installation Wizard 27
FIGURE 2.3 The Choose Language screen.
Step 2: Pre-Installation Check
After you have selected your language, the next screen you are presented with is the
Pre-installation Check screen (see Figure 2.4). A critical part in the installation process,
this screen checks to see if all the minimum system requirements are met.
FIGURE 2.4 The Pre-installation Check screen.
The first set of checks is for the required minimums for installation. If they are red
(not met), then you need to find a new environment (change hosts) or talk your host-
ing provider into changing its environment (upgrading PHP, for example). Note that
the last item, whether configuration.php is writable, is a permissions issue that is easy
to rectify. You can usually change permissions through the Cpanel tool provided by
your host. (Cpanel is a tool that is standard with almost all hosting companies.)
The second set of checks is recommended settings. If you don’t meet them, you can
still install Joomla, but you might experience problems with functionality and security.
When all items in the Pre-installation Check screen are green, you are ready to
proceed. Click Next.
The next step of the wizard is the License screen (see Figure 2.5).
Joomla is released under a GNU/GPL license. One of the most common questions
Running the Joomla! Installation Wizard 29
It’s actually perfectly okay to do this; you just have to keep the copyright statement in
the source code.
Step 4: Database Configuration
The next screen is Database Configuration (see Figure 2.6).
FIGURE 2.6 The Database Configuration screen.
The Database Configuration screen is one of the main pages of the installation
process; it’s where you enter important information about how Joomla can access the
MySQL database that your Joomla site will use. You will see a drop-down for the data-
base type.
The hostname will almost always be localhost, and the username and password are
provided by your hosting company, usually in an email you receive when you create
the account.
If you are installing on a localhost using WampServer or XAMPP, the username is
usually root, and the password is nothing or blank.
At this point, you need to choose a name for the SQL database that Joomla will use. | {"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": 0.867433488368988, "perplexity": 165.9969026854475}, "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-2018-09/segments/1518891812873.22/warc/CC-MAIN-20180220030745-20180220050745-00440.warc.gz"} |
https://www.cs.utexas.edu/users/flame/laff/pfhp/week4-parallelizing-the-first-loop.html | Skip to main content
## Unit4.3.2Parallelizing the first loop around the micro-kernel
Let us start by considering how to parallelize the first loop around the micro-kernel:
The situation here is very similar to that considered in Unit 4.1.1 when parallelizing the IJP loop ordering. There, we observed that if $C$ and $A$ are partitioned by rows, the matrix-matrix multiplication can be described by
\begin{equation*} \left( \begin{array}{c} \widetilde c_0^T \\ \hline \vdots \\ \hline \widetilde c_{m-1}^T \end{array} \right) := \left( \begin{array}{c} \widetilde a_0^T \\ \hline \vdots \\ \hline \widetilde a_{m-1}^T \end{array} \right) B + \left( \begin{array}{c} \widetilde c_0^T \\ \hline \vdots \\ \hline \widetilde c_{m-1}^T \end{array} \right) = \left( \begin{array}{c} \widetilde a_0^T B + \widetilde c_0^T \\ \hline \vdots \\ \hline \widetilde a_{m-1}^T B + \widetilde c_{m-1}^T \end{array} \right) . \end{equation*}
We then observed that the update of each row of $C$ could proceed in parallel.
The matrix-matrix multiplication with the block of $A$ and micro-panels of $C$ and $B$ performed by the first loop around the micro-kernel instead partitions the block of $A$ into row (micro-)panels and the micro-panel of $C$ into micro-tiles.
\begin{equation*} \left( \begin{array}{c} C_0 \\ \hline \vdots \\ \hline C_{M-1} \end{array} \right) := \left( \begin{array}{c} A_0 \\ \hline \vdots \\ \hline A_{M-1} \end{array} \right) B + \left( \begin{array}{c} C_0 \\ \hline \vdots \\ \hline C_{M-1} \end{array} \right) = \left( \begin{array}{c} A_0 B + C_0 \\ \hline \vdots \\ \hline A_{M-1} B + C_{M-1} \end{array} \right) . \end{equation*}
The bottom line: the updates of these micro-tiles can happen in parallel.
###### Homework4.3.2.1.
In directory Assignments/Week4/C,
• Copy Gemm_Five_Loops_Packed_MRxNRKernel.c into Gemm_MT_Loop1_MRxNRKernel.c.
• Modify it so that the first loop around the micro-kernel is parallelized.
• Execute it with
export OMP_NUM_THREADS=4
make MT_Loop1_8x6Kernel
• Be sure to check if you got the right answer!
• View the resulting performance with data/Plot_MT_performance_8x6.mlx, uncommenting the appropriate sections.
Parallelizing the first loop around the micro-kernel yields poor performance. A number of issues get in the way:
• Each task (the execution of a micro-kernel) that is performed by a thread is relatively small and as a result the overhead of starting a parallel section (spawning threads and synchronizing upon their completion) is nontrivial.
• In our experiments, we chose MC=72 and MR=8 and hence there are at most $72/8=9$ iterations that can be performed by the threads. This leads to load imbalance unless the number of threads being used divides $9$ evenly. It also means that no more than $9$ threads can be usefully employed. The bottom line: there is limited parallelism to be found in the loop.
This has a secondary effect: Each micro-panel of $B$ is reused relatively few times by a thread, and hence the cost of bringing it into a core's L1 cache is not amortized well.
• Each core only uses a few of the micro-panels of $A$ and hence only a fraction of the core's L2 cache is used (if each core has its own L2 cache).
The bottom line: the first loop around the micro-kernel is not a good candidate for parallelization. | {"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": 1.0000070333480835, "perplexity": 1091.5369043593867}, "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-2020-24/segments/1590347396089.30/warc/CC-MAIN-20200528104652-20200528134652-00566.warc.gz"} |
https://www.physicsforums.com/threads/methods-parameters-v-undetermined-coefficients.232335/ | # (Methods)Parameters v Undetermined Coefficients
1. Apr 30, 2008
### hils0005
[SOLVED] (Methods)Parameters v Undetermined Coefficients
Can anyone tell me why I would use one technique over the other? It seems as though undetermined Coef. is much easier to do but I suppose that comes with limitations???
2. Apr 30, 2008
### rock.freak667
Well if you can guess the particular integral of a Diff eq'n then method of undetermined coefficients will work and the pi's you can usually guess are usually $e^{ax},sinax,cosax,sinax+cosax,etc.$. But if you have tanx or $\frac{1}{1+x}$ then you'll need to use variation of parameters to solve.
3. Apr 30, 2008
### hils0005
Thanks for the explanation
Similar Discussions: (Methods)Parameters v Undetermined Coefficients | {"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": 0.971015989780426, "perplexity": 2331.037763968755}, "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-2017-39/segments/1505818687938.15/warc/CC-MAIN-20170921224617-20170922004617-00655.warc.gz"} |
https://www.physicsforums.com/threads/chemistry-quick-check-please.158354/ | # Chemistry - Quick Check, Please
1. Feb 26, 2007
### Fusilli_Jerry89
Just double checking to see if the product is correct:
H2SO4 + LiOH ---> H2O + LiHSO4
or is it LiSO4?
2. Feb 27, 2007
### Gokul43201
Staff Emeritus
What the valency of lithium? What's the valency (or the net charge) on the sulfate radical (you can deduce this from H2SO4)? Use these to determine the incorrect formula among the two you've written.
3. Feb 27, 2007
### Fusilli_Jerry89
uhh by valency do U mean valnce electrons, and therefore combining capacity? I don't know what you are talking about for the 2nd half of your explanation because I am in grade 12 and we have not done anything like that.
4. Feb 27, 2007
### Fusilli_Jerry89
Im actually guessing now that I can just look at the chart and see that H2SO4 is monoprotic, and therefore the correct answer is LiHSO4
5. Feb 27, 2007
### pmahesh107
yes it is LiHSO4 because Li is 1+ and HSO4 is 1-
6. Feb 28, 2007
### Gokul43201
Staff Emeritus
Ka2 for sulfuric acid is not small. I'm most people would call it diprotic.
But saying LiHSO4 is the dominant product is not a wrong answer. On the other hand, LiSO4 is an incorrect formula. The product of the neutralization of LiHSO4 would be Li2SO4.
Similar Discussions: Chemistry - Quick Check, 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": 0.875885009765625, "perplexity": 3530.8788558932606}, "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-2017-17/segments/1492917121528.59/warc/CC-MAIN-20170423031201-00177-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/164380/does-small-forcing-preserve-ch | Does small forcing preserve CH?
Suppose CH holds and $\mathbb{P}$ is a poset of size $\omega_1$, such that forcing with $\mathbb{P}$ preserves $\omega_1$. Does forcing with $\mathbb{P}$ preserve CH? If $\mathbb{P}$ is proper then the answer is yes, see this question. Is this true for improper $\omega_1$-preserving posets of size $\omega_1$?
-
Meanwhile, if $\omega_1$ is not preserved, then the answer is no, not necessarily, since when you collapse $\omega_1$ to become countable, the size of $2^\omega$ becomes the ground model $2^{\omega_1}$, which can be very large. – Joel David Hamkins Apr 25 '14 at 19:22
Yes. Let $X$ be a name for a subset of $\omega$. It can be describe the following way: for every $i<\omega$ $\{p^i_\alpha:\alpha<\omega_1\}$ a maximal antichain, $\{\varepsilon^i_\alpha:\alpha<\omega_1\}$ where $\varepsilon^i_\alpha\in\{0,1\}$ and $p^i_\alpha$ forces $i\in X$ iff $\varepsilon^i_\alpha=1$. In $V[G]$ the function $i\mapsto h(i)$ is an $\omega\to\omega_1$ map where $p^i_{h(i)}\in G$. As $P$ preserves $\omega_1$, the range of $h$ is bounded, i.e., some $p$ forces that $h(i)<\gamma$ for all $i$, for some $\gamma<\omega_1$. Now the structure $\langle p,\langle p^i_\alpha,\varepsilon^i_\alpha:\alpha<\gamma\rangle\rangle$ fully describes $X$. | {"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": 0.9915499091148376, "perplexity": 100.44372058719121}, "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-2015-48/segments/1448398456289.53/warc/CC-MAIN-20151124205416-00086-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/difference-quotients.132264/ | # Difference Quotients
1. Sep 16, 2006
### Apost8
This isn't a homework question, but I'm wondering if the following two equations can always be used interchangeably (thanks in advance):
f(x)-f(c)/(x-c)
and
f(a+h)-f(a)/h
2. Sep 16, 2006
### Data
yes, it's just substituting x for a+h and c for a, so then h = x-c.
3. Sep 16, 2006
### arildno
Do not double post!
4. Sep 16, 2006
### Apost8
No need to get upset. I accidentally posted in the wrong section, I apologize. Thanks for your help as well.
5. Sep 17, 2006
### HallsofIvy
Staff Emeritus
The problem I have with your post is that neither of the "equations" you give is an equation! (The lack of and "= " is a sure sign!) | {"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": 0.8979576826095581, "perplexity": 2993.457649814967}, "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-2017-09/segments/1487501171620.77/warc/CC-MAIN-20170219104611-00005-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/105192/what-is-the-homotopy-type-of-the-space-of-simple-closed-curves-isotopic-to-a-give | ## What is the homotopy type of the space of simple closed curves isotopic to a given one?
For surfaces there are many statements along the lines of: if two simple closed curves are homotopic, they are isotopic. I'm interested in such questions for families of curves.
More precisely, let $\Sigma$ be a hyperbolic surface, possibly with boundary. We fix an essential simple closed curve $\gamma$ on $\Sigma$. It is true that the subspace of $Emb(S^1,\Sigma)$ consisting of those curves that are isotopic to $\gamma$ is homotopy equivalent to a circle? Here the circle would come from reparametrisation of the curves.
This statement is true if we instead look at the space of all continuous (or smooth) maps of $S^1$ into $\Sigma$ that are homotopic to $\gamma$. Also note that this seems to be false for the torus, as for any essential simple closed curve we get at least $S^1 \times S^1$.
-
Earlier than Grayson, the determination of the homotopy-types of these spaces was done by Gramain.
There are a few special cases, like the torus and sphere and the non-orientable analogue, the case of null curves. But if they're not null homotopic the components of the embedding space have the homotopy type of $S^1$ -- the reparametrizations of the given curve.
Andre Gramain, Le type d'homotopie du groupe des diffeomorphisms d'une surface compacte. Ann. Sci. l'ENS $4^e$ serie tome 6 $n^o$ 1 (1973) 53--66
-
I was just looking at this Gramain reference and it seems only to prove something which is equivalent, though perhaps not obviously so. Namely, Theorem 4 of the paper says the space of embeddings of a circle with a fixed 1-jet at a point has contractible components. This is the fiber of a map from the full embedding space which takes the 1-jet at a point. The base space of this fibration is essentially the unit tangent bundle of the surface. There's a little checking left to get the result from this, but it's not hard. Does anyone know a reference for the exact statement? – Allen Hatcher Aug 21 at 23:13
If you forget about the parametrization, the "curve shortening flow" isotopes an essential simple closed curve to THE geodesic isotopic to it (this is a celebrated result of Matt Grayson), which I believe is gives a deformation retraction of the unparametrized space to a point. When you throw the parametrization back in, you get your conjectured result.
The Grayson result is this: Shortening embedded curves MA Grayson - The Annals of Mathematics, 1989
- | {"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": 0.8900722861289978, "perplexity": 172.36496736276035}, "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/1368705749262/warc/CC-MAIN-20130516120229-00009-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://mathhelpforum.com/pre-calculus/208341-factor-completely-2-print.html | factor completely
Show 40 post(s) from this thread on one page
Page 2 of 2 First 12
• Nov 24th 2012, 07:47 PM
mathisfun26
Re: factor completely
$\frac{1}{4}x^{-1/2}(3x+1)^{-1/2}[3x+2]$ if the question said no negative exponents in the final answer how would it be?
• Nov 24th 2012, 07:52 PM
mathisfun26
Re: factor completely
$\frac{3x+2}{4^{x^1/2} (3x+1)^{1/2}}$ would it be this?
• Nov 24th 2012, 08:09 PM
MarkFL
Re: factor completely
You would have:
$\frac{1}{4}x^{-\frac{1}{2}}(3x+1)^{-\frac{1}{2}}\left((3x+1)-3x \right)$
Now simplify...what do you get?
• Nov 24th 2012, 08:10 PM
mathisfun26
Re: factor completely
$\frac{1}{4}x^{-1/2}(3x+1)^{-1/2}[1]$ ??
• Nov 24th 2012, 08:53 PM
MarkFL
Re: factor completely
Yes, and you may omit the factor of 1. You may choose to write this as:
$\frac{1}{4\sqrt{x(3x+1)}}$
• Nov 24th 2012, 09:05 PM
mathisfun26
Re: factor completely
Quote:
Originally Posted by MarkFL2
A neat feature here is that if you hover your mouse cursor over an expression written in LaTeX code, you will see the code used to generate the expression.
Quote:
Originally Posted by jakncoke
When ever you want to type latex start with [tex ] latex stuff [/ tex]. For example [tex ] \frac{1}{2} [/ tex].
[QUOTE=MarkFL2;754391]The image is a bit hard to read, but I presume you have:
$\frac{1}{2}x^{-\frac{1}{2}}(3x+1)^{\frac{1}{2}}-\frac{3}{4}x^{ \frac{1}{2}}(3x+1)^{-\frac{1}{2}}$
First, look at the constant factors...they have $\frac{1}{4}$ as factors. Then look at the variable factors and pull out the factors with the smallest exponents.
quick question how did u get 1/4 from the factors of 1/2 and 3/4
• Nov 24th 2012, 09:29 PM
MarkFL
Re: factor completely
$\frac{1}{2}=\frac{1}{4}\cdot2$ and $\frac{3}{4}=\frac{1}{4}\cdot3$
Show 40 post(s) from this thread on one page
Page 2 of 2 First 12 | {"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": 9, "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": 0.8568816781044006, "perplexity": 3735.7577279562606}, "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-2017-17/segments/1492917120206.98/warc/CC-MAIN-20170423031200-00140-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://arxiver.wordpress.com/2017/03/09/reconstruction-of-extended-inflationary-potentials-for-attractors-cl/ | # Reconstruction of extended inflationary potentials for attractors [CL]
We give the procedure to reconstruct the extended inflationary potentials for general scalar-tensor theory of gravity and use the $\alpha$ attractor and the constant slow-roll model as examples to show how to reconstruct the class of extended inflationary potentials in the strong coupling limit. The class of extended inflationary potentials have the same attractor in the strong coupling limit, and the reconstructed extended inflationary potentials are consistent with the observational constraints. We also derive the condition on the coupling constant $\xi$ for satisfying the strong coupling.
Q. Gao and Y. Gong
Thu, 9 Mar 17
41/54 | {"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": 0.9654139280319214, "perplexity": 931.2533890507377}, "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-2017-22/segments/1495463607702.58/warc/CC-MAIN-20170523202350-20170523222350-00559.warc.gz"} |
https://www.physicsforums.com/threads/help-with-problem.133239/ | # Help with problem
1. Sep 23, 2006
### Alanf718
I was doing this problem, and now I wonder where I went wrong
The problem states
A ball starts from rest and rolls down a hill iwth uniform acceleration, traveling 150m during the second 5.0s of its motion. How far did it roll during the first 5.0s of motion.
This is what I did
x = 150m
t = 5.0s
with that data I got the velocity for stage 2
v = 30m/s
with that I tried to determine the acceleration so
V = Vo + at
30m/s = 0m/s + a(10s);
I got a = 3m/s
then I tried getting the x with
X = Xo + Vot + 1/2(a*t^2);
X = 0 + 0 + 1/2(3m/s*(5^2));
and I got X to be 37.5m but the book says the answer is 50m where did I go wrong?
2. Sep 23, 2006
This would be correct if there was no acceleration, i.e. if the velocity had been constant. Write down the equation of displacement for the second interval of 5 seconds first. Then see which parts of the equation you can (and need) to express with information you know about the first 5 seconds. You'll end up having only one unknown, which will be the acceleration. Then you can easily calculate the displacement in the first 5 seconds.
3. Sep 23, 2006
### Alanf718
Iam ending up with 2 unknowns
writing it as
X- Xo = Vot + 1/2(a*t^2);
That leaves me with Vo unknown as well as the a, I cant solve it with 2 unkowns :-( and I dont know How I would get the initial velocity at stage two to solve this.
[EDIT]
Forget it i got the answer I replace Vo with a*t in other words a*5 thanks for the help!
Last edited: Sep 23, 2006
4. Sep 23, 2006 | {"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": 0.9179670214653015, "perplexity": 1220.4757484878644}, "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/1521257650993.91/warc/CC-MAIN-20180324190917-20180324210917-00718.warc.gz"} |
http://mathoverflow.net/questions/25498/examples-of-loss-of-regularity-by-creation-of-topology?sort=votes | # Examples of loss of regularity by “creation of topology”
I would like to have a list as general as possible of examples of situations where the density of smooth objects into some "natural class" (the meaning of "natural" depending on the problem considered) fails, giving rise to interesting topological classifications. What comes to my mind are the following famous facts:
1) The condition for $C^\infty(\mathbb R^k, M^n)$ to be dense in the manifold-valued Sobolev spaces $W^{1,p}(R^k, M^n)$, is that the homotopy group $\pi_{[p]}(M)$ should be trivial.(Hang-Lin)
[this was kind of general, but gives the idea of what I'm looking for, maybe!]
2) A map $u$ in $W^{1,2}(B^3,S^2)$ is in the closure of $C^\infty(B^3, S^2)$ if and only if for any $2$-form $\omega$ on $S^2$ such that $\int_{S^2}\omega\neq 0$ one has $d(u^*\omega)=0$.(Bethuel-Coron-Demengel-Helein)
3) In $4$ dimensions, if the Yang-Mills functional is finite on a connection $A\in L^2(M^4)$, then the curvature $F_A$ of $A$ realizes an integral Chern class (i.e. the number $c_2(A):=1/(8\pi^2)\int_{M^4}Tr(F_A\wedge F_A)$ is an integer).(Uhlenbeck)
(Maybe I could also formulate the question differently, asking for mathematical situations having the "loss of differentiability" via "creation of new topology" analogous to the list above.)
-
I believe the original case of this is bubbling of 2-d harmonic maps and minimal surfaces due to Sacks-Uhlenbeck. This was generalized to higher dimensional minimal hypersurfaces by Schoen-Simon-Yau. The bubbling phenomena for Einstein manifolds was studied first by L. Z. Gao and then by Anderson and Anderson-Cheeger. – Deane Yang Jun 6 '11 at 13:51
I imagine my answer will look a bit strange, but I think it fits the title of your question (may be not really what you had in mind), although it is more a description of a situation where the failure of density is used to create new objects rather than a new topological classification.
I try with some motivation. In (cyclotomic) Iwasawa theory one naturally finds algebraic objects of arithmetic interest (typically, some cohomology group of some kind of arithmetic/geometric creature) to which one would like to attach an analytic object, called $p$-adic $L$ function which should remember very well, or know a lot, about the algebraic object. The way this is achieved is
1. Realize that the algebraic object $X$ you have in your hands is module over $\mathbb{Z}_p$
2. There is a topological group $\Gamma$ which is procyclic (projective limit of finite cyclic groups) acting continuously on $X$: by some easy commutative algebra, this says $X$ is a module over $\mathbb{Z}_p[[T]]$
3. Observe that power series in $\mathbb{Z}_p[[T]]$ are naturally analytic (i.e. admitting a power series expansion...) functions on some $p$-adic space.
4. In one way or another, try to attach one function $L_p(X,s)$ in $\mathbb{Z}_p[[T]]$ to your object $X$ by using that the latter object has an action by things which are analytic functions.
Now the application. One would like that the construction above characterizes $L_p(X,s)$ uniquely: for this, the fact that locally constant functions are dense in the algebra of all continuous functions $$f:\Gamma\to\mathbb{C}_p$$ when endowed with the sup-norm, is a crucial tool (via some kind of $p$-adic Fourier transform). This uniqueness is sometimes true (for instance, if $X$ comes from a modular form which is "ordinary at $p$") but sometimes one needs to allow more general analytic functions than those whose power-series expansion have all coefficients in $\mathbb{Z}_p$ (the case of "supersingular reduction"). In order to produce these "more general" analytic functions, one endows the algebra of continuous functions with a different norm $|\cdot |_r$ (for some $r\in\mathbb{N}$) in which the locally constant ones are not dense anymore – those who becomes dense are the locally polynomial ones with degrees in $[0,r]$. Then, re-transforming back via Fourier, the topological dual of the continuous functions with the norm $|\cdot |_r$ becomes big enough to contain the $p$-adic $L$ function of modular forms with supersingular reduction. A paper where all this is beutifully desxribed is Colmez' text on functions of one $p$-adic variable (in French).
Final warning: unfortunately, the connection between the $p$-adic $L$-function one constructs this way and the algebraic object is (often) still only conjectural.
-
Not responding directly to your question, but perhaps suggesting something: to prove that a linear action of a group G on a topological vectorspace V a genuine "representation, in the sense that G x V -> V is continuous, when V is a space of functions on a space X on which G acts (continuously), usually one wants/shows that continuous compactly-supported functions (or maybe test functions...) are dense in V.
This fails in simple situations, such as L^\infty on the real line, with the action of \R on itself, for straightforward reasons (not very topological, tho').
So/but, the point is that seemingly natural density claims can fail for simple reasons.
- | {"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": 0.8660641312599182, "perplexity": 429.5087902699117}, "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-2014-49/segments/1416400381177.56/warc/CC-MAIN-20141119123301-00191-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/improper-integral-concept-question.485660/ | # Improper integral concept question
1. Mar 30, 2011
### wetwilly92
1. The problem statement, all variables and given/known data
For what values of K is the following integral improper?
$$\int\stackrel{K}{0}x^2 / (x^2-19x+90) dx$$
I'm stuck on this question. I understand mechanically, that the integration require partial fraction decomp, which results in -9ln(x-9) (from 0 to K) + 10ln(x-10) (from 0 to K). What I don't understand is what makes this integral improper. I understand that LN is undefined for all evaluations < 1. So does this mean that any K < 10 will create an improper integral?
EDIT: How does one properly display the upper and lower limits on the integration symbol?
Last edited: Mar 30, 2011
2. Mar 30, 2011
### Tangent87
To get the limits right use \int_0^K instead of stackrel.
As for the question itself, you might want to draw a sketch of the function.
3. Mar 30, 2011
### HallsofIvy
Staff Emeritus
An integral may be "improper" for one of several reasons-
1) The lower limit is $-\infty$.
2) The upper limit is $\infty$.
3) The integrand goes to $-\infty$ at some point in the interval of integration.
4) The integrand goes to $\infty$ at some point in the interval of integration.
Which of those can happen here?
What values of x make the denominator of the integrand 0? | {"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": 0.9930732846260071, "perplexity": 750.6206336833438}, "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/1454701152982.47/warc/CC-MAIN-20160205193912-00189-ip-10-236-182-209.ec2.internal.warc.gz"} |
http://antonism.arbitrary.eu/research-interest/ | # Research
X-Ray binaries (XRBs) are accreting systems hosting a star and a compact object. Giacconi et al. (1962) serendipitously discovered the first of these sources, i.e. Sco X-1, during an X-ray observation of the Moon. There are three types of compact objects that can take part in a binary system: a white dwarf (WD), a neutron star (NS), or a black hole (BH). If the compact object is either a NS or a WD, it can feature a strong surface magnetic field B~1012-15 Gauss, or B~106 Gauss, respectively. The magnetic field influences the accretion flow, by driving matter to the accreting column. Strong magnetic field in NS, also, imprints the X-ray spectrum emitted by the compact object, in the form of absorption lines, referred to as cyclotron lines. A further distinction is based on the mass (and therefore the spectral type) of the companion star. If the mass of the companion is more than 10 Msun the system is referred to as a High Mass X-ray Binary (HMXB). If the mass of the companion star is less than 1 Msun then it is referred to as a Low Mass X-ray binary (LMXB). Finally, if the mass of the companion falls in between 1 Msun and 10 Msun thenit is referred to as an Intermediate Mass X-ray Binary (IMXB). | {"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": 0.9652966260910034, "perplexity": 1031.3402717925849}, "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-2020-05/segments/1579250601040.47/warc/CC-MAIN-20200120224950-20200121013950-00518.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/180850-orthogonal-vectors.html | Math Help - Orthogonal vectors
1. Orthogonal vectors
Find a vector $(w_1,w_2,w_3,...)$ that is orthogonal to $v=(1,\frac{1}{2},\frac{1}{4},...)$. Compute its length $||w||$.
2. Originally Posted by alexmahone
Find a vector $(w_1,w_2,w_3,...)$ that is orthogonal to $v=(1,\frac{1}{2},\frac{1}{4},...)$. Compute its length $||w||$.
Orthogonal...in what vector space and with respect to what inner product??
Tonio
3. Originally Posted by tonio
Orthogonal...in what vector space and with respect to what inner product??
Tonio
Hilbert space with respect to dot product.
4. Originally Posted by alexmahone
Hilbert space with respect to dot product.
There are infinite Hilbert spaces ( to add "with inner product" is futile: ANY Hilbert space is a vector space with an inner product). I can think of at least
two different Hilbert spaces with their inner products to which your vector belongs, so again: what vector space with what iiner product?
Tonio
5. Originally Posted by tonio
There are infinite Hilbert spaces ( to add "with inner product" is futile: ANY Hilbert space is a vector space with an inner product). I can think of at least
two different Hilbert spaces with their inner products to which your vector belongs, so again: what vector space with what iiner product?
Tonio
I think the question merely requires us to find $w_1,w_2,w_3,...$ such that $w_1+\frac{w_2}{2}+\frac{w_3}{4}+...=0$.
6. Originally Posted by alexmahone
I think the question merely requires us to find $w_1,w_2,w_3,...$ such that $w_1+\frac{w_2}{2}+\frac{w_3}{4}+...=0$.
That's what I also thought, but then you already know what to look for......
Idea: $\sum\limits^\infty_{n=0}\frac{1}{4^n}=\frac{4}{3} \, , \, \, \,\sum\limits^\infty_{n=1}\frac{1}{2^{2n-1}}=2\sum\limits^\infty_{n=1}\frac{1}{4^n}=2\cdot \frac{1}{3}=\frac{2}{3}$
Tonio
7. Originally Posted by tonio
That's what I also thought, but then you already know what to look for......
Idea: $\sum\limits^\infty_{n=0}\frac{1}{4^n}=\frac{4}{3} \, , \, \, \,\sum\limits^\infty_{n=1}\frac{1}{2^{2n-1}}=2\sum\limits^\infty_{n=1}\frac{1}{4^n}=2\cdot \frac{1}{3}=\frac{2}{3}$
Tonio
I think the question merely requires us to find $w_1,w_2,w_3,...$ such that $w_1+\frac{w_2}{2}+\frac{w_3}{4}+...=0$.
You also need to ensure that $\sum|w_n|^2<\infty$, and in fact you need to be able to compute that sum in order to do the second part of the question.
You also need to ensure that $\sum|w_n|^2<\infty$, and in fact you need to be able to compute that sum in order to do the second part of the question.
Thanks. $w=(1,-2,0,...)$ and its length is $\sqrt{5}$. | {"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": 18, "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": 0.9601525664329529, "perplexity": 346.6123732900971}, "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-2015-11/segments/1424937406179.50/warc/CC-MAIN-20150226075646-00001-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://me.gateoverflow.in/969/gate2018-2-47 | A steel wire is drawn from an initial diameter $(d_i$) of $10 \: mm$ to a final diameter ($d_f$) of the $7.5 \: mm$. The half cone angle ($\alpha$) of the die is $5^{\circ}$ and the coefficient of friction ($\mu$) between the die and the wire is $0.1$. The average of the initial and final yield stress $[(\sigma_Y)_{avg}]$ is $350 \: MPa$. The equation for drawing stress $\sigma_f$, in (in MPa) is given as: $$\sigma_f=(\sigma_Y)_{avg} \bigg\{ 1+ \frac{1}{\mu \cot \alpha} \bigg\} \bigg[1- \bigg( \frac{d_f}{d_i} \bigg) ^{2 \mu \: \cot \alpha} \bigg]$$ The drawing stress (in MPa) required to carry out this operation is _____ (correct to two decimal places) | {"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": 0.9791046977043152, "perplexity": 271.16612281996515}, "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-20221129031912-20221129061912-00329.warc.gz"} |
http://emathematics.matematicas.us/ecsegundogrado.php?a=2&tipo=numero | User:
# The discriminant
Consider the general quadratic equation ax2+bx+c=0. The quantity b2-4ac is called the discriminant of the quadratic equation and determine the type of root which arises from a quadratic equation. When
• b2-4ac > 0 The equation has two real roots.
• b2-4ac < 0 The equation has no real roots.
• b2-4ac = 0 The equation has one repeated root.
(If the discriminant is negative, the roots are complex, and this page can not display the complex number. The output will be "None").
Find the number of solutions of 3x2-2x=7
First write the equation in standard form: 3x2-2x-7=0
The discriminant is b2-4ac=(-2)2-4·3·(-7)=4+84=88>0
The equation has two real solutions.
How many roots does the following equation have? $\fs12x^2-7x+2\;=\;0$
None One Two | {"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": 1, "/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": 0.9246799945831299, "perplexity": 1025.3416935094624}, "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-2017-17/segments/1492917123048.37/warc/CC-MAIN-20170423031203-00153-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/174533-i-think-i-made-mistake-somewhere.html | # Thread: I think I made a mistake somewhere
1. ## I think I made a mistake somewhere
The question states to prove beta= sqrt(pi ln 5) whereas I get beta= pi ln 5. I attached my steps below
2. You have mishandled the limits of integration when making the substitution in the integral. If $t=\theta^2$ then $t=0$ when $\theta=0,$ and $t=\beta^2$ when $\theta=\beta$. So the integral should become $\displaystyle\int_0^{\beta^2}\!\!\!\tfrac12\sqrt t e^{2t/x}\tfrac1{2\sqrt t}}\,dt.$
You have worked out the integral correctly (apart from having $\sqrt t$ instead of $\beta^2$ as the upper limit of integration), and it leads to the result $5 = e^{\beta^2/\pi}$, which is what you want. | {"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": 9, "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": 0.996126651763916, "perplexity": 248.1020064015493}, "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-2018-05/segments/1516084889681.68/warc/CC-MAIN-20180120182041-20180120202041-00596.warc.gz"} |
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=16&t=65552 | ## Chapter Readings for the 2nd Midterm
Melissa Solis 3D
Posts: 47
Joined: Wed Sep 30, 2020 10:08 pm
### Chapter Readings for the 2nd Midterm
Can someone specify which chapters we'll be focusing on for the next midterm?
Thanks :)
kateraelDis2F
Posts: 48
Joined: Wed Sep 30, 2020 9:54 pm
### Re: Chapter Readings for the 2nd Midterm
I'm wondering this too ^ Also just to be sure, Midterm 2 will only cover the stuff we learned after midterm 1 right? Will there be any content we have to remember from midterm 1?
Audrey Han 2F
Posts: 40
Joined: Wed Sep 30, 2020 9:38 pm
### Re: Chapter Readings for the 2nd Midterm
I think it'll include the next two topics (chemical bonds + molecular shape and structure) as well as the end of the quantum world because there are 6 topics total. But this is just a guess!
Liam Bertrand 3
Posts: 42
Joined: Wed Sep 30, 2020 9:53 pm
### Re: Chapter Readings for the 2nd Midterm
I would assume that it's the next two outlines as long as the remainder of outline 2
JaesalSoma1E
Posts: 40
Joined: Wed Sep 30, 2020 10:02 pm
Been upvoted: 3 times
### Re: Chapter Readings for the 2nd Midterm
I think midterm 2 will contain material from the end of outline 2, outline 3 and, outline 4.
Posts: 48
Joined: Wed Sep 30, 2020 9:51 pm
### Re: Chapter Readings for the 2nd Midterm
I think the second midterm will cover material we learn after midterm 1, starting with quantum numbers and electron configurations up until wherever we stop.
Brian Nguyen 2F
Posts: 40
Joined: Wed Sep 30, 2020 9:49 pm
### Re: Chapter Readings for the 2nd Midterm
In my opinion, it will probably cover material starting from where we left off after we took Midterm 1 up until Focus 9 if I had to guess. I don't think it'll be a cumulative test, but there's probably bound to be mention of old topics at some point since most topics that we learn probably overlap with the old ones that we learned in Midterm 1. | {"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": 0.8447008728981018, "perplexity": 3945.4766004749968}, "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-2020-50/segments/1606141205147.57/warc/CC-MAIN-20201130035203-20201130065203-00233.warc.gz"} |
https://secure.msri.org/seminars/22796 | # Mathematical Sciences Research Institute
Home » GFA Main Seminar: Duality of floating bodies and illumination bodies
# Seminar
GFA Main Seminar: Duality of floating bodies and illumination bodies November 30, 2017 (02:00 PM PST - 03:00 PM PST)
Parent Program: Geometric Functional Analysis and Applications Space Science Lab, Room 105
Speaker(s) Elisabeth Werner (Case Western Reserve University)
Description No Description
Video
Abstract/Media
Based on joint work with Olaf Mordhorst, we discuss a duality relation between floating and illumination bodies.
The definitions of these two bodies suggest that the polar of the floating body should be similar to the illumination body of the polar.
We consider this question for the class of centrally symmetric convex bodies.
We provide precise estimates for l^n_p-balls and for centrally symmetric convex bodies with everywhere positive Gauss curvature.
We also investigate the problem for the class of centrally symmetric polytopes.
Our estimates show that equality of the polar of the floating body and the illumination body of the polar can only be achieved in the case of ellipsoids. | {"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": 0.8447510004043579, "perplexity": 1638.2038584535617}, "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/1585370499280.44/warc/CC-MAIN-20200331003537-20200331033537-00460.warc.gz"} |
https://physics.stackexchange.com/questions/394210/difference-between-phonons-and-heat | # Difference between Phonons and heat?
1, if these two were the different then how we differentiate with one another?. 2, if these two were the same, then what is really vibrating?, atom in the lattice or electron in the atom or the bond between two atoms or an entire medium(sound)? The main problem is to find frequency of the phonon in a solid. many people used dispersion relation. but the omega symbol is mainly used for angular frequency. and the unit of dispersion relation is one over centimetre. which means it is might be wave number k. there are so many things confusing me to find the frequency of phonons in metal.
sorry for my english it's my second language.
I will try to clear up your confusion step by step.
First of all: Your main question.
Phonons describe a collective excitation of atoms/molecules in solids. Think of vibrations in a material. They can be treated like particles which is why phonons are often called quasiparticles.
Heat is a form of energy which is an entirely different quantity. You are comparing apples and oranges when you ask whether phonons and heat are the same thing. Usually, phonons strongly influence the heat capacity in a solid but they are not identical.
Now, let's get to the sub-questions.
Heat is energy stored in the (random) motion of particles. In a solid, the important particles are free electrons and the ions they leave behind. You could consider the motion of each ion separately, but it is usually more convenient to consider a collective motion of the ions at the same frequency. That's a phonon. That's also why there's an $\omega$ in the dispersion relation. That $\omega$ is an angular frequency --- the frequency the ions are oscillating at.
Phonons can be thought of as waves, and the dispersion relation tells you the relation between the wavevector (which can have units of $\left[cm\right]^{-1}$) of the wave and its frequency. In general, these relations can be quite complicated (especially compared to many waves in undergraduate physics, such as light in a vacuum, which has a linear dispersion relation $\omega = c k$). A good starting point is to understand the dispersion relation for a one-dimensional lattice. (For example, see the wikipedia page for phonons.) It shows an example of a non-linear dispersion relation $\omega \propto \sqrt{1 - \cos{k a}}$, where $a$ is the spacing between the atoms in the lattice. | {"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": 0.9589360356330872, "perplexity": 250.53533264823258}, "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/1610704835901.90/warc/CC-MAIN-20210128040619-20210128070619-00457.warc.gz"} |
https://www.semanticscholar.org/paper/Experimental-observation-of-the-Anderson-transition-Chab%C3%A9-Lemari%C3%A9/b8d88b0dff4c13f8dc0537b8cede3b5478183995 | # Experimental observation of the Anderson metal-insulator transition with atomic matter waves.
@article{Chab2008ExperimentalOO,
title={Experimental observation of the Anderson metal-insulator transition with atomic matter waves.},
author={Julien Chab{\'e} and Gabriel Lemari{\'e} and B. Gr{\'e}maud and Dominique Delande and Pascal Szriftgiser and Jean Claude Garreau},
journal={Physical review letters},
year={2008},
volume={101 25},
pages={
255702
}
}
We realize experimentally an atom-optics quantum-chaotic system, the quasiperiodic kicked rotor, which is equivalent to a 3D disordered system that allows us to demonstrate the Anderson metal-insulator transition. Sensitive measurements of the atomic wave function and the use of finite-size scaling techniques make it possible to extract both the critical parameters and the critical exponent of the transition, the latter being in good agreement with the value obtained in numerical simulations of…
## Figures from this paper
### Observation of the Anderson metal-insulator transition with atomic matter waves: Theory and experiment
• Physics
• 2009
Using a cold atomic gas exposed to laser pulses - a realization of the chaotic quasiperiodic kicked rotor with three incommensurate frequencies - we study experimentally and theoretically the
### Phase diagram of the Anderson transition with atomic matter waves
• Physics
• 2013
We realize experimentally a cold atom system equivalent to the 3D Anderson model of disordered solids where the anisotropy can be controlled by adjusting an experimentally accessible parameter. This
### Critical state of the Anderson transition: between a metal and an insulator.
• Physics
Physical review letters
• 2010
Using a three-frequency one-dimensional kicked rotor experimentally realized with a cold atomic gas, we study the transport properties at the critical point of the metal-insulator Anderson
### Phase diagram of the anisotropic Anderson transition with the atomic kicked rotor: theory and experiment
• Physics
• 2013
We realize experimentally a cold-atom system, the quasiperiodic kicked rotor, equivalent to the three-dimensional Anderson model of disordered solids where the anisotropy between the x direction and
### Experimental Observation of Two-Dimensional Anderson Localization with the Atomic Kicked Rotor.
This work experimentally study Anderson localization in dimension 2 and shows that the localization length depends exponentially on the disorder strength and anisotropy and is in quantitative agreement with the predictions of the self-consistent theory for the 2D Anderson localization.
### Universality of the Anderson transition with the quasiperiodic kicked rotor
• Physics
• 2009
We report a numerical analysis of the Anderson transition in a quantum-chaotic system, the quasiperiodic kicked rotor with three incommensurate frequencies. It is shown that this dynamical system
### The Anderson metal-insulator transition
The Anderson Localization is the phase transition from metal to insulator under a change in the amount of disorder in a material. In a first part, we introduce the scaling theory of the Anderson
### Multifractality of the kicked rotor at the critical point of the Anderson transition
• Physics
Physical Review A
• 2019
We show that quantum wavepackets exhibit a sharp macroscopic peak as they spread in the vicinity of the critical point of the Anderson transition. The peak gives a direct access to the mutifractal
### Field theory of Anderson transition of the kicked rotor
• Physics
• 2012
We present a microscopic theory of Anderson transition in the quantum kicked rotor. The behavior of the system is shown to depend sensitively on the value of the effective Planck constant, . For the
## References
SHOWING 1-10 OF 22 REFERENCES
### Stochastic behavior of a quantum pendulum under a periodic perturbation
• Physics
• 1979
This paper discusses a numerical technique for computing the quantum solutions of a driver pendulum governed by the Hamiltonian H = (p_\theta ^2 /2m\ell ^2 ) - [m\ell ^2 \omega _o ^2 \cos \theta
### Introduction To Percolation Theory
• Physics
• 1985
Preface to the Second Edition Preface to the First Edition Introduction: Forest Fires, Fractal Oil Fields, and Diffusion What is percolation? Forest fires Oil fields and fractals Diffusion in
### An introduction to percolation
• Physics
• 1994
An integrated approach (i.e. lecture, computer simulation and experimental measurements) to percolation theory is discussed. The computational and experimental techniques are simple enough to make
### Phys
• Rev. Lett. 82, 382
• 1999
### Condens
• Matter 6, 2511
• 1994
### Phys
• Rev. Lett. 62, 345
• 1989
### Phys
• Rev. A 29, 1639
• 1984
### Commun
• Nonlin. Sci. Num. Simul. 8, 301
• 2003
### Phys
• Rev. Lett. 75, 4598
• 1995
### Phys
• Rev. E 60, 3942
• 1999 | {"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": 0.8717592358589172, "perplexity": 3655.8990897021554}, "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-2022-49/segments/1669446710978.15/warc/CC-MAIN-20221204172438-20221204202438-00032.warc.gz"} |
https://people.maths.bris.ac.uk/~matyd/GroupNames/320/C5s(C2%5E3sC8).html | Copied to
clipboard
G = C5⋊(C23⋊C8) order 320 = 26·5
The semidirect product of C5 and C23⋊C8 acting via C23⋊C8/C22×C4=C4
Series: Derived Chief Lower central Upper central
Derived series C1 — C2×C10 — C5⋊(C23⋊C8)
Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23.2F5 — C5⋊(C23⋊C8)
Lower central C5 — C10 — C2×C10 — C5⋊(C23⋊C8)
Upper central C1 — C22 — C23 — C22×C4
Generators and relations for C5⋊(C23⋊C8)
G = < a,b,c,d,e | a5=b2=c2=d2=e8=1, bab=a-1, ac=ca, ad=da, eae-1=a3, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >
Subgroups: 546 in 98 conjugacy classes, 28 normal (22 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C23, C23, D5, C10, C10, C22⋊C4, C2×C8, C22×C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2×C10, C22⋊C8, C2×C22⋊C4, C5⋊C8, C2×Dic5, C2×Dic5, C2×C20, C22×D5, C22×D5, C22×C10, C23⋊C8, D10⋊C4, C2×C5⋊C8, C22×Dic5, C22×C20, C23×D5, C23.2F5, C2×D10⋊C4, C5⋊(C23⋊C8)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), F5, C22⋊C8, C23⋊C4, C4.D4, C2×F5, C23⋊C8, D5⋊C8, C4.F5, C22⋊F5, D10.D4, D10⋊C8, C23.F5, C5⋊(C23⋊C8)
Smallest permutation representation of C5⋊(C23⋊C8)
On 80 points
Generators in S80
(1 75 39 44 68)(2 45 76 69 40)(3 70 46 33 77)(4 34 71 78 47)(5 79 35 48 72)(6 41 80 65 36)(7 66 42 37 73)(8 38 67 74 43)(9 51 26 62 22)(10 63 52 23 27)(11 24 64 28 53)(12 29 17 54 57)(13 55 30 58 18)(14 59 56 19 31)(15 20 60 32 49)(16 25 21 50 61)
(1 20)(2 6)(4 19)(5 24)(8 23)(9 62)(10 74)(11 79)(12 61)(13 58)(14 78)(15 75)(16 57)(17 21)(25 54)(26 51)(27 43)(28 48)(29 50)(30 55)(31 47)(32 44)(33 46)(34 56)(35 53)(36 45)(37 42)(38 52)(39 49)(40 41)(59 71)(60 68)(63 67)(64 72)(65 76)(66 73)(69 80)(70 77)
(1 5)(2 17)(3 7)(4 19)(6 21)(8 23)(9 13)(10 67)(11 15)(12 69)(14 71)(16 65)(18 22)(20 24)(25 36)(26 30)(27 38)(28 32)(29 40)(31 34)(33 37)(35 39)(41 50)(42 46)(43 52)(44 48)(45 54)(47 56)(49 53)(51 55)(57 76)(58 62)(59 78)(60 64)(61 80)(63 74)(66 70)(68 72)(73 77)(75 79)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 70)(10 71)(11 72)(12 65)(13 66)(14 67)(15 68)(16 69)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(41 54)(42 55)(43 56)(44 49)(45 50)(46 51)(47 52)(48 53)(57 80)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 79)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)
G:=sub<Sym(80)| (1,75,39,44,68)(2,45,76,69,40)(3,70,46,33,77)(4,34,71,78,47)(5,79,35,48,72)(6,41,80,65,36)(7,66,42,37,73)(8,38,67,74,43)(9,51,26,62,22)(10,63,52,23,27)(11,24,64,28,53)(12,29,17,54,57)(13,55,30,58,18)(14,59,56,19,31)(15,20,60,32,49)(16,25,21,50,61), (1,20)(2,6)(4,19)(5,24)(8,23)(9,62)(10,74)(11,79)(12,61)(13,58)(14,78)(15,75)(16,57)(17,21)(25,54)(26,51)(27,43)(28,48)(29,50)(30,55)(31,47)(32,44)(33,46)(34,56)(35,53)(36,45)(37,42)(38,52)(39,49)(40,41)(59,71)(60,68)(63,67)(64,72)(65,76)(66,73)(69,80)(70,77), (1,5)(2,17)(3,7)(4,19)(6,21)(8,23)(9,13)(10,67)(11,15)(12,69)(14,71)(16,65)(18,22)(20,24)(25,36)(26,30)(27,38)(28,32)(29,40)(31,34)(33,37)(35,39)(41,50)(42,46)(43,52)(44,48)(45,54)(47,56)(49,53)(51,55)(57,76)(58,62)(59,78)(60,64)(61,80)(63,74)(66,70)(68,72)(73,77)(75,79), (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,70)(10,71)(11,72)(12,65)(13,66)(14,67)(15,68)(16,69)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(41,54)(42,55)(43,56)(44,49)(45,50)(46,51)(47,52)(48,53)(57,80)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)>;
G:=Group( (1,75,39,44,68)(2,45,76,69,40)(3,70,46,33,77)(4,34,71,78,47)(5,79,35,48,72)(6,41,80,65,36)(7,66,42,37,73)(8,38,67,74,43)(9,51,26,62,22)(10,63,52,23,27)(11,24,64,28,53)(12,29,17,54,57)(13,55,30,58,18)(14,59,56,19,31)(15,20,60,32,49)(16,25,21,50,61), (1,20)(2,6)(4,19)(5,24)(8,23)(9,62)(10,74)(11,79)(12,61)(13,58)(14,78)(15,75)(16,57)(17,21)(25,54)(26,51)(27,43)(28,48)(29,50)(30,55)(31,47)(32,44)(33,46)(34,56)(35,53)(36,45)(37,42)(38,52)(39,49)(40,41)(59,71)(60,68)(63,67)(64,72)(65,76)(66,73)(69,80)(70,77), (1,5)(2,17)(3,7)(4,19)(6,21)(8,23)(9,13)(10,67)(11,15)(12,69)(14,71)(16,65)(18,22)(20,24)(25,36)(26,30)(27,38)(28,32)(29,40)(31,34)(33,37)(35,39)(41,50)(42,46)(43,52)(44,48)(45,54)(47,56)(49,53)(51,55)(57,76)(58,62)(59,78)(60,64)(61,80)(63,74)(66,70)(68,72)(73,77)(75,79), (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,70)(10,71)(11,72)(12,65)(13,66)(14,67)(15,68)(16,69)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(41,54)(42,55)(43,56)(44,49)(45,50)(46,51)(47,52)(48,53)(57,80)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80) );
G=PermutationGroup([[(1,75,39,44,68),(2,45,76,69,40),(3,70,46,33,77),(4,34,71,78,47),(5,79,35,48,72),(6,41,80,65,36),(7,66,42,37,73),(8,38,67,74,43),(9,51,26,62,22),(10,63,52,23,27),(11,24,64,28,53),(12,29,17,54,57),(13,55,30,58,18),(14,59,56,19,31),(15,20,60,32,49),(16,25,21,50,61)], [(1,20),(2,6),(4,19),(5,24),(8,23),(9,62),(10,74),(11,79),(12,61),(13,58),(14,78),(15,75),(16,57),(17,21),(25,54),(26,51),(27,43),(28,48),(29,50),(30,55),(31,47),(32,44),(33,46),(34,56),(35,53),(36,45),(37,42),(38,52),(39,49),(40,41),(59,71),(60,68),(63,67),(64,72),(65,76),(66,73),(69,80),(70,77)], [(1,5),(2,17),(3,7),(4,19),(6,21),(8,23),(9,13),(10,67),(11,15),(12,69),(14,71),(16,65),(18,22),(20,24),(25,36),(26,30),(27,38),(28,32),(29,40),(31,34),(33,37),(35,39),(41,50),(42,46),(43,52),(44,48),(45,54),(47,56),(49,53),(51,55),(57,76),(58,62),(59,78),(60,64),(61,80),(63,74),(66,70),(68,72),(73,77),(75,79)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,17),(7,18),(8,19),(9,70),(10,71),(11,72),(12,65),(13,66),(14,67),(15,68),(16,69),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39),(41,54),(42,55),(43,56),(44,49),(45,50),(46,51),(47,52),(48,53),(57,80),(58,73),(59,74),(60,75),(61,76),(62,77),(63,78),(64,79)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80)]])
38 conjugacy classes
class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 5 8A ··· 8H 10A ··· 10G 20A ··· 20H order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 20 20 4 4 10 10 10 10 4 20 ··· 20 4 ··· 4 4 ··· 4
38 irreducible representations
dim 1 1 1 1 1 1 2 2 4 4 4 4 4 4 4 4 4 type + + + + + + + + + + image C1 C2 C2 C4 C4 C8 D4 M4(2) F5 C23⋊C4 C4.D4 C2×F5 D5⋊C8 C4.F5 C22⋊F5 D10.D4 C23.F5 kernel C5⋊(C23⋊C8) C23.2F5 C2×D10⋊C4 C22×C20 C23×D5 C22×D5 C2×Dic5 C2×C10 C22×C4 C10 C10 C23 C22 C22 C22 C2 C2 # reps 1 2 1 2 2 8 2 2 1 1 1 1 2 2 2 4 4
Matrix representation of C5⋊(C23⋊C8) in GL6(𝔽41)
1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 40 6 0 0 0 0 0 0 40 6 0 0 0 0 35 35
,
1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 35 1 0 0 0 0 0 0 1 0 0 0 0 0 6 40
,
40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
0 1 0 0 0 0 9 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 18 35 0 0 0 0 20 23 0 0
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,40,35,0,0,0,0,6,35],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,35,0,0,0,0,0,1,0,0,0,0,0,0,1,6,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,9,0,0,0,0,1,0,0,0,0,0,0,0,0,0,18,20,0,0,0,0,35,23,0,0,1,0,0,0,0,0,0,1,0,0] >;
C5⋊(C23⋊C8) in GAP, Magma, Sage, TeX
C_5\rtimes (C_2^3\rtimes C_8)
% in TeX
G:=Group("C5:(C2^3:C8)");
// GroupNames label
G:=SmallGroup(320,253);
// by ID
G=gap.SmallGroup(320,253);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,387,268,1123,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^2=e^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^3,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations
×
𝔽 | {"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": 0.9996183514595032, "perplexity": 2721.7068241330576}, "config": {"markdown_headings": false, "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-2021-21/segments/1620243989914.60/warc/CC-MAIN-20210516201947-20210516231947-00435.warc.gz"} |
http://stats.stackexchange.com/questions/107685/discrete-analog-of-cdf-cumulative-mass-function | # Discrete analog of CDF: “cumulative mass function”?
We call the integral of a probability density function (PDF) a cumulative distribution function (CDF).
But what's the cumulative sum of a probability mass function (PMF) called? I've never heard the term "cumulative mass function" before, and the Wikipedia page for it redirects to the CDF page, so I'm confused what the proper terminology is.
-
The proper terminology is Cumulative Distribution Function, (CDF). The CDF is defined as $$F_X(x) = \mathrm{P}\{X \leq x\}.$$ With this definition, the nature of the random variable $X$ is irrelevant: continuous, discrete, or hybrids all have the same definition. As you note, for a discrete random variable the CDF has a very different appearance than for a continuous random variable. In the first case, it is a step function; in the second it is a continuous function. | {"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": 0.8892319202423096, "perplexity": 265.49379053405914}, "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/1408500829421.59/warc/CC-MAIN-20140820021349-00457-ip-10-180-136-8.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/799477/left-and-right-eigenvectors-perpendicular-to-each-other | # Left and right eigenvectors perpendicular to each other
I just read in a textbook on numerical methods that you can always have that the right eigenvectors of a matrix can be taken as orthonormal to the left eigenvectors for a diagonalisable matrix. This seems suprising to me. I would have assumed that you need to have that the matrix is normal or hermitian, but is this also possible for all matrices?
-
Can you be more explicit? What exactly does the book say? – Brian Fitzpatrick May 17 '14 at 20:21
It says: If you have a matrix $A$ that is diagonalisable with right eigenvectors $\phi_1,...,\phi_n$, you can always find left eigenvectors $\psi_1,...,\psi_n$ such that $\phi_i \psi_j = \delta_{i,j}$ ( this is of course the real case) – user66906 May 17 '14 at 20:23
Is there any hypothesis of distinct eigenvalues? Ah, I guess with geometric multiplicity $>1$ we can choose the eigenvectors to make it true. – Ted Shifrin May 17 '14 at 20:34
no, there is nothing about distinct eigenvalues – user66906 May 17 '14 at 20:37
What is a standard exercise is that when $\lambda\ne\mu$, with $x$ a $\lambda$-eigenvector for $A$ and $y$ a $\mu$-eigenvector for $y$, then $x\cdot y = 0$ (compute $Ax\cdot y$ two ways). But it wasn't obvious to me what happened with the $\lambda$ eigenvectors for both.
Consider the diagonalization $P^{-1}AP=\Lambda$. Then, transposing this equation, we obtain $Q^{-1}A^\top Q=\Lambda$, where $Q=(P^{-1})^\top$. The columns of $P$ and $Q$ are eigenvectors of $A$ and $A^\top$, respectively. But let's look at $Q^\top P$: the $ij$-entry will compute the dot product of the $i^{\text{th}}$ eigenvector of $Q$ and the $j^{\text{th}}$ eigenvector of $P$. But $Q^\top P = P^{-1}P = I$, which is the desired result. | {"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": 0.9478118419647217, "perplexity": 173.3225325362852}, "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/1438042988311.72/warc/CC-MAIN-20150728002308-00064-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://arxiv.org/abs/1805.09318 | # Title:Schwinger Effect by an $SU(2)$ Gauge Field during Inflation
Abstract: Non-Abelian gauge fields may exist during inflation. We study the Schwinger effect by an $SU(2)$ gauge field coupled to a charged scalar doublet in a (quasi) de Sitter background and the possible backreaction of the generated charged particles on the homogeneous dynamics. Contrary to the Abelian $U(1)$ case, we find that both the Schwinger pair production and the induced current decrease as the interaction strength increases. The reason for this suppression is the isotropic vacuum expectation value of the $SU(2)$ field which generates a (three times) greater effective mass for the scalar field than the $U(1)$. In the weak interaction limit, the above effect is negligible and both the $SU(2)$ and $U(1)$ cases exhibit a linear increase of the current and a constant conductivity with the interaction strength. We conclude that the Schwinger effect does not pose a threat to the dynamics of inflationary models involving an $SU(2)$ gauge field.
Comments: Minor changes Subjects: High Energy Physics - Theory (hep-th); Cosmology and Nongalactic Astrophysics (astro-ph.CO) DOI: 10.1007/JHEP02(2019)041 Cite as: arXiv:1805.09318 [hep-th] (or arXiv:1805.09318v3 [hep-th] for this version)
## Submission history
From: Kaloian Lozanov [view email]
[v1] Wed, 23 May 2018 17:59:29 UTC (120 KB)
[v2] Sun, 24 Jun 2018 21:14:59 UTC (120 KB)
[v3] Tue, 20 Nov 2018 17:54:19 UTC (128 KB) | {"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": 0.826138436794281, "perplexity": 1153.3857764080885}, "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/1579251778168.77/warc/CC-MAIN-20200128091916-20200128121916-00449.warc.gz"} |
https://moodle.frankfurt-university.de/mod/glossary/showentry.php?eid=30492&displayformat=dictionary | #### Forward Error Correction
The message is encoded so that the receiver can detect and correct (some) errors without retransmission | {"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": 0.9443413019180298, "perplexity": 3959.5790901496903}, "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/1659882572220.19/warc/CC-MAIN-20220816030218-20220816060218-00555.warc.gz"} |
https://chem.libretexts.org/Bookshelves/General_Chemistry/Book%3A_ChemPRIME_(Moore_et_al.)/10Solids%2C_Liquids_and_Solutions/10.12%3A_Critical_Temperature_and_Pressure | # 10.12: Critical Temperature and Pressure
Suppose we seal a pure liquid and its vapor in a strong glass tube and heat it to a very high temperature. As we increase the temperature, the vapor pressure will rise. (It is not a good idea to heat a liquid this way unless you are sure the container can withstand the increased pressure.) The rising vapor pressure corresponds to a greater number of molecules in the limited volume of the vapor phase. In other words, the vapor becomes considerably denser. Eventually we reach a temperature at which the density of the vapor becomes the same as that of the liquid. Since liquids are usually distinguished from gases on the basis of density, at this point both have become identical. The temperature at which this occurs is called the critical temperature, and the pressure is called the critical pressure.
The accompanying videos illustrate what happens experimentally in the case of Chlorine. In the first video, as the temperature nears the critical temperature, liquid and vapor become very similar in appearance and the meniscus between them becomes difficult to distinguish. Finally, at the critical temperature the meniscus disappears completely. Above the critical temperature the sample is quite uniform and it is difficult to know whether to call it a liquid or a gas.
In the second half of the video, the flask is brought back below the critical temperature. The speed of gas molecules decreases to a point where intermolecular forces can cause a liquid phase to condense out. The meniscus reappears, and the Chlorine separates back again into a liquid and vapor phase.
Once a gas is above its critical temperature, it is impossible to get it to separate into a liquid layer below and a vapor layer above no matter how great a pressure is applied, as can be seen in the graph below. On the graph, once the temperature is higher than 300 K, it is not possible to revert to liquid form. Increasing the pressure only leads to the transition from gas to supercritical fluid.
Oxygen, for instance, is well above its critical temperature at room temperature. If we increase the pressure on it to a few thousand atmospheres, its density becomes so high that we are forced to classify it as a liquid. Nevertheless as we increase the pressure, there is no point at which drops of liquid suddenly appear in the gas. Instead the oxygen gradually changes from something which is obviously a gas to something which is obviously a liquid (a supercritical fluid). Conversely, if we gradually relax the pressure, there is no point at which the oxygen will start to boil.
The following table lists the critical temperatures and critical pressures for some well-known gases and liquids. Such data are often quite useful. Many gases are sold commercially in strong steel cylinders at high pressures. The behavior of the gas in such a cylinder depends on whether it is above or below the critical temperature. The critical temperature of propane, for instance, is 97°C, well above room temperature. Thus propane in a high-pressure cylinder consists of a mixture of liquid and vapor, and you can sometimes hear the liquid sloshing about inside.
Table $$\PageIndex{1}$$ Critical Temperatures and Pressures of Some Simple Substances.
Name Critical Temperature (K) Critical Pressure (MPa) Critical Pressure (atm)
Hydrogen (H) 33.2 1.30 12.8
Neon (Ne) 44.5 2.7 26.9
Nitrogen (N) 126.0 3.39 33.5
Carbon dioxide (CO2) 304.2 7.39 73.0
Propane (C3H8) 370 4.23 41.8
Ammonia (NH3) 405.5 11.29 111.5
Water (H2O) 647.1 22.03 217.5
The pressure of the gas in such a cylinder will be the vapor pressure of propane at namely, 9.53 atm (965.4 kPa).As long as there is some liquid left in the cylinder, the pressure will remain at 9.53 atm. Only when all the liquid has evaporated the pressure begin to drop. At that point the cylinder will be virtually empty. A very different behavior is found in the case of a cylinder of oxygen. Since oxygen is above its critical temperature at 20°C, the cylinder will contain a uniform fluid rather than a liquid-vapor mixture. As we use up the oxygen, the pressure will gradually decrease to 1 atm, at which point no more O2 will escape from the cylinder.
The principles discussed in the preceding paragraph apply to the aerosol sprays most of us encounter every day. Such spray cans contain a small quantity of the active ingredient—hair conditioner, deodorant, shaving cream, and the like—and a large quantity of propellant. The propellant is a substance, such as propane, whose critical temperature is well above room temperature. Therefore it can be liquefied at the high pressure in the spray can.
When the valve is opened, the vapor pressure of the liquid propellant causes the active ingredient and the propellant to spray out of the can. As long as liquid propellant remains, the pressure inside the can will be constant (it will be the vapor pressure), and the spray will be reproducible. It should be obvious why such cans always bear a warning against throwing them in a fire--vapor pressure increases more rapidly at higher temperatures, and so heating an enclosed liquid is far more likely to produce an explosion than heating a gas alone. (The latter case was described in an earlier example.) | {"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": 0.8634942770004272, "perplexity": 569.9094231315474}, "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/1547583700734.43/warc/CC-MAIN-20190120062400-20190120084400-00515.warc.gz"} |
https://notes.reasoning.page/html/matrices | A calculus of the absurd
17.2 Matrices
17.2.1 Matrix multiplication
Example: given that
$$A = \begin{pmatrix} 2 & 3 \\ 0 & 2 \end {pmatrix}$$
consider $$A$$, $$A^2$$, $$A^3$$ and $$A^4$$ to make a conjecture about $$A^n$$, and then prove this conjecture by induction.
Solution
Although it’s tempting to reach for one’s calculator to compute $$A$$, $$A^2$$, $$A^3$$ and $$A^4$$, it makes things harder.
When we multiply $$A$$ by $$A$$, we get that
$$A^2 = \begin{pmatrix} 2^2 & 2\cdot 3 + 3\cdot 2 \\ 0 & 2^2 \end {pmatrix}$$
If we then multiply this by $$A$$ again, we get that
$$A^3 = \begin{pmatrix} 2^3 & 2(2\cdot 3 + 3\cdot 2) + 3\cdot 2^2 \\ 0 & 2^3 \end {pmatrix}$$
We can multiply out the bracket and get that
\begin{align} 2(2\cdot 3 + 3\cdot 2) + 3\cdot 2^2 &= 2^2\cdot 3 + 2^2\cdot 3 + 3\cdot 2^2 \\ &= 2^2\cdot 3 + 2^2\cdot 3 + 2^2\cdot 3 \\ &= 3\cdot 2^2\cdot 3 \end{align}
Simplifying here is a bad idea, as we’re looking for patterns, and these are easier to spot when we have more information to work with (which we do when we think about the factors of our numbers.)
Multiplying to find $$A^4$$ we get that
$$A^4 = \begin{pmatrix} 2^4 & 2(3\cdot 2^2\cdot 3) + 3\cdot 2^3 \\ 0 & 2^4 \end {pmatrix}$$
We can simplify this a bit, because
\begin{align} 2(3\cdot 2^2\cdot 3) + 3\cdot 2^3 &= 3\cdot 2^3\cdot 3 + 3\cdot 2^3 &= 4\cdot 2^3\cdot 3 \end{align}
This implies that
$$A^4 = \begin{pmatrix} 2^4 & 4\cdot 2^3\cdot 3 \\ 0 & 2^4 \end {pmatrix}$$
Looking at these four matrices, the general shape is something like
$$A^n = \begin{pmatrix} 2^n & n\cdot 2^{n-1}\cdot 3 \\ 0 & 2^n \end {pmatrix}$$
There’s another way of doing this, but the maths behind it isn’t on the A Level specification. It’s the way I originally did it.
From the initial multiplication, we can see that the general shape of $$A^n$$ is
$$\begin{pmatrix} 2^n & ? \\ 0 & 2^n \end {pmatrix}$$
Here the $$?$$ denotes the confusion about what $$A_{1,2}$$ is. In terms of notation, it’s slightly more handy to call $$?$$ something more like $$f(n)$$.
We can also write the matrix $$A^n$$ as $$A A^{n-1}$$.
$$A^n = \begin{pmatrix} 2 & 3 \\ 0 & 2 \end {pmatrix} \begin{pmatrix} 2^{n-1} & f(n-1) \\ 0 & 2^{n-1} \end {pmatrix}$$
We can apply the standard rules of matrix multiplication to the above.
$$A^n = \begin{pmatrix} 2^{n} & 2f(n-1) + 3\cdot 2^{n-1} \\ 0 & 2^{n} \end {pmatrix}$$
We can compare the value of $$A^n$$ we have computed here (in terms of $$f(n-1)$$) to the one we’d like to know (that of $$f(n)$$). This gives
$$f(n) = 2f(n-1) + 3\cdot 2^{n-1}$$
We also know that $$f(1) = 3$$. To find a closed-form expression for $$f(n)$$, we can just keep substituting:
\begin{align} f(n) &= 2(2f(n-2)+3\cdot 2^{n-2}) + 3\cdot 2^{n-1} \\ &= 2^2f(n-2) + 3\cdot 2^{n-1} + 3\cdot 2^{n-1} \\ &= 2^2(2f(n-3) + 3\cdot 2^{n-3}) + 3\cdot 2^{n-1} + 3\cdot 2^{n-1} \\ &= 2^3f(n-3) + 3\cdot 2^{n-1} + 3\cdot 2^{n-1} + 3\cdot 2^{n-1} \\ &= n \cdot 3 \cdot 2^{n-1} \end{align}
The last step isn’t necessarily immediately obvious (doing the expansion by hand might make it easier to understand). The other way to think about this is by using one of the helpful heuristics looked at in the heuristics section, in this case “wherever possible, draw a diagram” 108108 Unfortunately, it’s not a great diagram (but I tried)..
The sum of each node $$f(n)$$ in the tree is $$2f(n-1) + 3\cdot 2^{n-1}$$ - the $$2f(n-1)$$s are drawn as children of the parent node. We want to find the sum of all the nodes in the tree. To do this, note that we have $$n$$ levels of the tree and at each level the nodes sum to $$3\cdot 2^{n-1}$$, and thus overall we have that $$f(n)$$ is equal to the sum of all the nodes, which is equal to $$n\cdot 3\cdot 2^{n-1}$$.
It’s all downhill (in difficulty) from here!
For the proof by induction, if we set
$$P(n) = \begin{pmatrix} 2^n & n\cdot 2^{n-1}\cdot 3 \\ 0 & 2^n \end {pmatrix}$$
then we want to show that $$A^n = P(n)$$ for every natural number.
We start with the basis case: as $$A^1 = A$$, it is clear that $$A^1 = P(1)$$.
For the inductive step, we assume that $$P(k) = A^k$$, and then we consider $$P(k+1)$$, which is equal to $$A^{1}A^{k}$$. To this expression, we can now apply the inductive hypothesis (always be looking at $$P(k+1)$$ to see where $$P(k)$$ has been hidden!) and thus this expression is equal to $$A P(k)$$. We can then carry out the multiplication
$$\begin{pmatrix} 2 & 3 \\ 0 & 2 \end {pmatrix} \begin{pmatrix} 2^k & k\cdot 2^{k-1}\cdot 3 \\ 0 & 2^k \end {pmatrix} = \begin{pmatrix} 2\cdot 2^k & 2(k\cdot 2^{k-1}\cdot 3) + 3\cdot 2^{k} \\ 0 & 2\cdot 2^k \end {pmatrix}$$
This can then be rewritten as
$$\begin{pmatrix} 2^{k+1} & k\cdot 2^{k}\cdot 3 + 3\cdot 2^{k} \\ 0 & 2^{k+1} \end {pmatrix}$$
and as $$k\cdot 2^{k}\cdot 3 + 3\cdot 2^{k}$$ is equal to $$2^{k}\cdot 3\cdot (k+1)$$ we have that
$$\begin{pmatrix} 2^{k+1} & 2^{k}\cdot 3\cdot (k+1) \\ 0 & 2^{k+1} \end {pmatrix}$$
The final step is to rewrite the $$2^{k}$$ as $$2^{(k+1)-1}$$, and then we have
$$\begin{pmatrix} 2^{k+1} & (k+1) \cdot 2^{(k+1)-1}\cdot 3 \\ 0 & 2^{k+1} \end {pmatrix}$$
Thus as $$P(n)$$ is true for $$n=1$$ and $$P(n)$$ implies that $$P(n+1)$$ is true, $$P(n)$$ must hold for all $$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": 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": 3, "equation": 15, "x-ck12": 0, "texerror": 0, "math_score": 0.9997833371162415, "perplexity": 183.62249151910763}, "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-2022-40/segments/1664030335396.92/warc/CC-MAIN-20220929225326-20220930015326-00395.warc.gz"} |
https://mathoverflow.net/questions/74976/acceleration-via-smoothing/75148 | # Acceleration via smoothing
Is the following approach to accelerating the rate of convergence of $(1+1/2+\dots+1/n)- \ln n$ (with $n=1,2,3,\dots$), and other sequences like it, in the literature?
Let $f(t)=(\sum_{1 \leq n \leq t} 1/n) - \ln t$ for $t \geq 0$, so that $f(t)$ tends to Euler's constant $\gamma$ as $t \rightarrow \infty$. It follows that the continuous piecewise-differentiable function $F(t) := \int_0^t f(s) \: ds$ grows like $\gamma t$ so that $F(t)/t \rightarrow \gamma$. Note that $F(t)= (\sum_{1 \leq n \leq t} (t-n)/n) - t \ln t + t$ so that $F(n)/n$ is about as easy to compute as $f(n)$.
(1) Can someone provide asymptotic estimates for $f(n)-\gamma$ and $F(n)/n-\gamma$? The former is undoubtedly classical (and, I suspect, easy), but I don't know if anyone's looked at the latter (though it's probably not too hard). It appears empirically that $F(n)/n$ converges to $\gamma$ faster than $f(n)$.
(2) Is this "smoothing" approach to acceleration of sequence-convergence discussed in the literature? (I don't know any numerical analysis; maybe this trick is "well-known to those who know it".)
(3) The smoothing method can be iterated, but there often comes a point of diminishing returns. Is anything known about the optimal amount of smoothing to apply? (Is it related to the rate of decay of the coefficients of some Fourier series or something like that?)
Of course there are much better ways to estimate Euler's constant than smoothing, and for sequences in general there are much better ways to accelerate convergence. What's distinctive about the above method is that it's linear.
The method is related to "smoothed summation" (as described in http://www.tricki.org/article/Smoothing_sums for instance) via integration by parts. Specifically, $(n+1)$-fold iteration of the operation $f \mapsto F$ gives the function $\int_0^t [(t-s)^n/n!] f(s) \: ds$, which can be viewed as the convolution of $f(t)$ with the kernel $t^n/n!$. In some contexts this kind of convolution is called "filtering". Perhaps there's more to be said about these connections.
I also have a feeling that the method is related to Cesaro summation. Can anyone provide details of the relation?
-
The function $F(t)= (\sum_{1 \leq n \leq t} (t-n)/n) - t \ln t + t\;$ for the integer values of the argument simplifies to $F(n)=n \sum_{1 \leq k \leq n} 1/k -n\ln n=nf(n)\;$. Is there a misprint in $F$? – Andrew Sep 9 '11 at 13:06
It's more than just a misprint; I mis-remembered which kernels accelerate convergence! I'll need to check back through the Mathematica notebook in which I did the calculations and revise this posting accordingly. Or should I just retract the question and re-post it with the needed corrections? Is there even a way to retract a question? Can experienced MathOverflow members suggest the right course of action in a case like this? You can either post a comment on this thread or, if you think your suggestion might be of interest only to me, send me an email; my email address is easy to find. – James Propp Sep 9 '11 at 21:44
Andrew is 100% correct; $f(n)$ and $F(n)$ are equal! There are examples where the kind of filtering I described can improve convergence of a sequence, but this isn't one of them. I'm going to close this thread and open a new one once I've figured out what I should say. Apologies to those who spent time reading the thread; I should've spend more time reading it myself before I posted.
In case you care, I'm pretty sure $f(n)-\gamma$ is $1/2n$ plus change, based on a piecewise linear approximation to $1/n$. – Will Sawin Oct 9 '11 at 21:10 | {"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": 0.8300030827522278, "perplexity": 192.15589910613824}, "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-2015-48/segments/1448398446286.32/warc/CC-MAIN-20151124205406-00210-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://chem.libretexts.org/Bookshelves/General_Chemistry/Book%3A_ChemPRIME_(Moore_et_al.)/03%3A_Using_Chemical_Equations_in_Calculations/3.02%3A_Equations_and_Mass_Relationships/3.2.01%3A_Cultural_Connections-_Berthollides-_A_Challenge_to_Chemical_Stoichiometry | # 3.2.1: Cultural Connections- Berthollides- A Challenge to Chemical Stoichiometry
## Stoichiometry
Chemists are committed to the idea that a balanced chemical equation such as
$2 Fe + O_2 \rightarrow 2 FeO \label{1}$
not only tells how many atoms or molecules of each kind are involved in a reaction, it also indicates the amount of each substance that is involved. Equation (1) says that 1 Fe atom can react with 2 O2 molecules to give 2 formula units of FeO. Here we're using the term "formula unit" to indicate that the substance may not be a molecule, but rather an ionic compound or ["network crystal"]. A "formula unit" gives the composition of the substance without specifying the type of bonding.
The equation also says that 1 mol of Fe atoms would react with 2 mol O2 yielding 2 mol FeO.
We have become so accustomed to the idea of atoms, that it seems logical that equations should represent whole numbers of atoms. We now talk about the stoichiometric ratios of atoms when we want to indicate that they must combine in small whole number ratios, like 1:1 in FeO. The word stoichiometric comes from the Greek words stoicheion, “element,“ and metron, “measure.“ Hence the stoichiometric ratio measures one element (or compound) against another.
## A Challenge to Stoichiometric Reasoning
But the idea of whole number stoichiometric ratios was strongly opposed in the early nineteenth century. This is understandable, because the Law of Definite Proportions is quite anti-intuitive. After all, it seems that you can generally mix things in virtually any ratio to get desired results. Since Dalton's atomic theory implied that atoms should combine in definite ratios, many of his contemporaries opposed the atomic theory. Dalton's theory explained Proust's earlier Law of constant composition (Law of Definite Proportions or "Proust's Law"(1797)) [1],
There are in fact many important compounds that are "non-stoichiometric", including the product of Equation (1)! They are often called "Berthollides" after French inorganic chemist Claude Louis Berthollet (1748–1822), who attacked Dalton's atomic theory (1803-5) and Proust's Law [2]. N. S. Kurnakov introduced the terms "Daltonides" and "Berthollides" in 1912–14 to designate chemical compounds of constant composition (Daltonides) and variable composition (Berthollides)[3].
Claude Louis Berhollet. Opponent of the Law of Constant Composition Joseph Proust (1754-1826). Discoverer of Law of Constant Composition
This episode in the history of chemistry is a good example of the claim by philosophers of science that counterexamples to an accepted theory may not be recognized. The concept of stoichiometry was used to cover up the most glaring exceptions, and, for more than a century, substances that could not be made to fit these new rules were ignored until Kurnakov called attention to Berthollides[4][5].
## Non-stoichiometric Compounds
Berthollet opposed Dalton's atomic theory, pointing to many reactions, like the reaction of many transition metals with oxygen, that are not stoichiometric. Some of his writings are available online. The reaction in Equation (1) often does not give FeO. In nature, FeO is the mineral wüstite, which has the actual stoichiometry closer to Fe0.95O. For each "missing" Fe2+ ion, the crystal contains two Fe3+ ions to supply the missing 2+ charge for charge balance[6]. In the diagram below, one vacancy (Fe2+) compensated by two substitutions (Fe3+) at lateice points. The "Frenkel Pair" occurs when a lattice ion is replaced by an interstitial ion in some nonstoichiometric compounds. The composition of a non-stoichiometric compound may vary only slightly, as in wüstite where the formula may be written as Fe1-xO, where x is a small number (~0.05). In some cases, like copper sulphides, the variation can be much larger[7]. For practical purposes, the term describes materials where the non-stoichiometry is at least 1% of the ideal composition.
Types of defects in crystals leading to non-stoichiometry
## Details of Stoichiometric Reasoning
To contrast stoichiometric with nonstoichiometric compounds, refer to Equation (2),
$3 Fe + 2 O_2 \rightarrow Fe_3O_4 \label{2}$
In addition to atom ratios, it also tells us that 2 × 3 mol = 6 mol Fe will react with 2 × 2 mol = 4 mol O2, and that ½ × 3 mol = 1.5 mol Fe requires only ½ × 2 mol = 1 mol O2. In other words, the equation indicates that exactly 2 mol O2 must react for every 3 mol Fe consumed. For the purpose of calculating how much O2 is required to react with a certain amount of Fe therefore, the significant information contained in Eq. (2) is the ratio
$\dfrac{\text{2 mol O}_{\text{2}}}{\text{3 mol Fe}}$
We shall call such a ratio derived from a balanced chemical equation a stoichiometric ratio and give it the symbol S. Thus, for Eq. (2),
## Stoichiometric Reasoning and Non-Stoichiometric Compounds
Under normal circumstances, the stoichiometric ratio holds closely (Fe3O4 is a combination of Fe2O3 and FeO, so it may incorporate the nonstoichiometry of FeO described above). But unusual geological or synthetic conditions lead to other stoichiometries. For example, a technique called "sputter deposition" or "Sputtering" involves heating one reactant in a vacuum with a small amount of the second reactant present as a vapor, as shown in the Figure at right. The product is collected on a cold substrate.
This leads to berthollides when an iron sample is sputtered in the presence of water vapor. Oxide compositions ranging from Fe3O4 to Fe2O3, were obtained, depending on the temperature and pressure [8]. They would have stoichiometric ratios ranging from for Equation (2).
to for the reaction 2 Fe + 1.5 O2 → Fe2O3, and most of the ratios would not be easily reduced to whole number ratios.
It is remarkable that stoichiometric ratios are used to understand and guide the synthesis of even non-stoichiometric compounds, so this is truly an important area of chemistry.
## Example 1
Derive all possible stoichiometric ratios from Eq. (2)
Solution Any ratio of amounts of substance given by coefficients in the equation may be used:
When any stoichiometric chemical reaction occurs, the amounts of substances consumed or produced are related by the appropriate stoichiometric ratios. Using Eq. (2) as an example, this means that the ratio of the amount of O2 consumed to the amount of Fe consumed must be the stoichiometric ratio S(O2/Fe): Similarly, the ratio of the amount of Fe3O4 produced to the amount of Fe consumed must be
S(Fe3O4/Fe):
In general we can say that or, in symbols,
Note that in the word Eq. (4a) and the symbolic Eq. (4b), X and Y may represent any reactant or any product in the balanced chemical equation from which the stoichiometric ratio was derived. No matter how much of each reactant we have, the amounts of reactants consumed and the amounts of products produced will be in appropriate stoichiometric ratios.
## Example 2
Find the amount of Fe3O4 produced when 3.68 mol Fe is consumed according to Eq. (2).
Solution The amount of Fe3O4 produced must be in the stoichiometric ratio S(Fe3O4/Fe) to the amount of Fe consumed:
Multiplying both sides nFe consumed, by we have
= 1.23 mol Fe3O4
This is a typical illustration of the use of a stoichiometric ratio as a conversion factor. Example 2 is analogous to Examples 1 and 2 from Conversion Factors and Functions, where density was employed as a conversion factor between mass and volume. Example 2 is also analogous to Examples 2.4 and 2.6, in which the Avogadro constant and molar mass were used as conversion factors. As in these previous cases, there is no need to memorize or do algebraic manipulations with Eq. (4) when using the stoichiometric ratio. Simply remember that the coefficients in a balanced chemical equation give stoichiometric ratios, and that the proper choice results in cancellation of units. In road-map form or symbolically.
When using stoichiometric ratios, be sure you always indicate moles of what. You can only cancel moles of the same substance. In other words, 1 mol Fe cancels 1 mol Fe but does not cancel 1 mol Fe3O4.
The next example shows that stoichiometric ratios are also useful in problems involving the mass of a reactant or product.
## Example 3
Calculate the mass of Oxygen (O2) consumed when 3.68 mol Fe reacts according to Equation (2).
Solution The problem asks that we calculate the mass of O2 consumed. As we learned in Example 2 of The Molar Mass, the molar mass can be used to convert from the amount of O2 to the mass of O2. Therefore this problem in effect is asking that we calculate the amount of O2 consumed from the amount of Fe consumed. This is the same problem as in Example 2. It requires the stoichiometric ratio
The amount of O2 consumed is then
The mass of O2 is With practice this kind of problem can be solved in one step by concentrating on the units. The appropriate stoichiometric ratio will convert moles of Fe to moles of O2 and the molar mass will convert moles of O2 to grams of O2. A schematic road map for the one-step calculation can be written as Thus
These calculations can be organized as a table, with entries below the respective reactants and products in the chemical equation. You may verify the additional calculations of the mass of product.
3 Fe + 2 O2 1 Fe3O4
m (g) 205.5 78.5 284.0
M (g/mol) 55.845 32.0 231.54
n (mol) 3.68 2.45 1.23
## Example 4
We noted above that when iron is sputtered in the presence of water vapor, a range of products from Fe3O4 to Fe2O3 (common rust) may be produced. Prepare a table similar to the one above for the reaction in which 0.200 g of Fe is converted to Fe2O3.
Solution First, write a balanced equation
2 Fe + 3 H2O → Fe2O3 + 3 H2 The problem gives the mass of Fe. Thinking the problem through before trying to solve it, we realize that the molar mass of Fe could be used to calculate the amount of Fe consumed. Then we need a stoichiometric ratio to get the amount of O2 consumed and the amount of Fe2O3 and H2 produced. Finally, the molar masses of O2, Fe2O3, and H2 permit calculation of the mass of O2, Fe2O3, and H2.
We might start the table by entering the given mass, and the molar masses which we calculate from atomic weight tables:
2 Fe + 3 H2O → 2 Fe2O3 + 3 H2
m (g) 0.200
M (g/mol) 55.845 32.0 159.69 2.016
n (mol)
Now we can calculate the amount of Fe present:
Then the stoichiometric ratios are used to calculate the amounts of water and products:
So the amount of water required is
0.00358 mol Fe x (3 mol H2O / 2 mol Fe) = 0.00537 mol H2O
Similarly, we use the stoichiometric ratio to calculate the amount of product:
So the amount of Fe2O3 produced is
0.00358 mol Fe x (1 mol Fe2O3 / 2 mol Fe) = 0.00179 mol Fe2O3.
The amount of H2 produced must be the same as the amount of water consumed, since they are in the ratio 3:3 from the equation.
We can add these to the table:
2 Fe + 3 H2O → 2 Fe2O3 + 3 H2
m (g) 0.200
M (g/mol) 55.845 32.0 159.69 2.016
n (mol) 0.00358 0.00537 0.00179 0.00537
Finally, we can use the molar masses to convert from amounts (in mol) to masses (in g):
2 Fe + 3 H2O → 2 Fe2O3 + 3 H2
m (g) 0.200 0.097 0.286 0.0108
M (g/mol) 55.845 18.015 159.69 2.016
n (mol) 0.00358 0.00537 0.00179 0.00537
Note that the sum of the masses of reactants equals the sum of the masses of the products.
## References
1. en.Wikipedia.org/wiki/Joseph_Proust
2. en.Wikipedia.org/wiki/Claude_Louis_Berthollet
3. http://encyclopedia2.thefreedictionary.com/Daltonides+and+Berthollides
4. J.P. Suchet, Daltonides, berthollides and inorganic glasses"; Journal of Non-Crystalline Solids, V.6, #4, (1971) 370-392
5. www.sciencedirect.com/science...5&searchtype=a
6. en.Wikipedia.org/wiki/Non-stoichiometric_compound
7. en.Wikipedia.org/wiki/Copper_sulfide
8. jap.aip.org/resource/1/japiau...sAuthorized=no | {"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": 0.8715851902961731, "perplexity": 2130.4401149756077}, "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/1634323585280.84/warc/CC-MAIN-20211019171139-20211019201139-00051.warc.gz"} |
http://math.stackexchange.com/questions/134468/how-to-identify-naturally-two-subspaces | # How to identify “naturally” two subspaces?
I am self-studying Hoffman and Kunze's book Linear Algebra. This is exercise $1$ from page $111$.
If $W$ is a subset of a vector space $V$, we define $W^{0}=\{f\in V^{\star}|f(w)=0 \operatorname{for all}w\in W\}.$
Let $n$ be a positive integer and $\mathbb{F}$ be a field. Let $W$ be a set of all vectors $(x_{1},\ldots,x_{n})$ in $\mathbb{F}^{n}$ such that $x_{1}+\cdots+x_{n}=0$.
a) Prove that $W^{0}$ consists of all linear functionals $f$ of the form $$f(x_{1},\ldots,x_{n})=c(\sum_{j=1}^{n}x_{j}).$$
b) Show that the dual space $W^{\star}$ of $W$ can be "naturally" identified with the linear functionals $$f(x_{1},\ldots,x_{n})=c_{1}x_{1}+\cdots+c_{n}x_{n}$$ on $\mathbb{F}^{n}$ which satisfy $c_{1}+\cdots+c_{n}=0$.
My approach for part a) First, we note that $e_{i}-e_{j}\in W$ for all $i,j\in\{1,2,\ldots,n\}$. If $f\in W^{0},$ then we have $f(e_{i})-f(e_{j})=f(e_{i}-e_{j})=0.$ Therefore $f(e_{i})=f(e_{j})$ and we have $f(e_{i})=c$, for all $i\in \{1,2,\ldots,n\}$, where $c$ is a constant. Therefore, if $w=(x_{1},\ldots,x_{n})\in W$, then $f(x_{1},\ldots,x_{n})=c(\sum_{j=1}^{n}x_{j}).$ The converse is easy.
I was not able to solve the part b).
-
Let $H = \{f \in (\mathbb F^n)^* \mid f(x_1, \ldots, x_n) = \sum c_ix_i \text{ and } \sum c_i = 0 \}.$ There's a map $(\mathbb F^n)^* \to W^*$ given by restriction—if you like, this is the dual $i^*$ of the inclusion $i\colon W \to \mathbb F^n$. Restrict this to a map $H \to W^*$ and show that you obtain an isomorphism. For this, you could use the fact that the kernel of $i^*$ is precisely $W^0$. What is $W^0 \cap H$?
I take the word "natural" here to mean, "Don't just identify $W$ and $H$ because they have the same dimension." The word often means "basis-free" (or functorial), but here it seems to me that we've chosen a basis by merely writing $\mathbb F^n$. In general, if I have a vector space $V$ and a subspace $W \subset V$ then the dual of $W$ is most naturally identified with the quotient $V^*/W^0$. (In the same spirit, can you find something naturally isomorphic to $(V/W)^*$?)
-
I'm slightly worried about the characteristic of the field. Will update when I figure out the confusion. – Dylan Moreland Apr 22 '12 at 18:47
Not sure if the OP knows about the first isomorphism theorem... – Benja Apr 22 '12 at 22:54
@BenjaminLim: I know those Theorems. – spohreis Apr 22 '12 at 23:26
Well, there are other ways to verify that this. This just seemed to be the quickest one. And I flipped through Hoffman-Kunze and it appears to me that fields are assumed to be subfields of $\mathbf C$ unless otherwise noted, so we're safe. – Dylan Moreland Apr 22 '12 at 23:34
@DylanMoreland $\dim W^{\circ} \leq \dim V$. Therefore we have a surjective map $f : V \longrightarrow W^{\circ}$ with kernel $W$. Therefore by the FIT we have that $V/W \cong W^{\circ}$. How do we conclude from here that the isomorphism $(V/W)^{\ast} \cong W^{\circ}$ is natural? – Benja Apr 23 '12 at 0:00
show 1 more comment
This does not address OP's concern, but this does something irrelevant and trivial.
The part $(b)$ is largely about writing the details down:
$$W=\{\underline w =(w_1,\dots,w_n) \mid \sum_{i=1}^nw_i=0\} \tag{1}$$ $$W^{\ast}=\{f \mid f\;\; \mbox{linear function from } W \;\;\mbox{to}\;\;\Bbb F\} \tag{2 }$$
Let's also define $f_{\underline x}:W \to \Bbb F$ given by $f_{\underline x}(\underline w)=\underline x \cdot \underline w$. Let's use this to define the following map:
\begin{align}W &\to W^\ast \\ \underline x&\mapsto f_{\underline x}\end{align}
Proving that the above map is an injective homomorphism identifies $W$ with $W^\ast$. Since, $\dim W=\dim W^\ast$, we also have that this map is an isomorphism.
But OP wants an altogether different isomorphism, so, I GOOFED IT UP.
-
Why are you defining a map from $W$ into $W^{\star}?$ – spohreis Apr 20 '12 at 18:39
I am trying to identify $W$ as a guy sitting in $W^\ast$. So, an injectivehomomorphism does that, it renames every element in $W$ by an equivalent name in $W^\ast$. So, $W$ sits inside $W^\ast$... – user21436 Apr 20 '12 at 18:46
But, why do you want to define an injective homomorphism from $W$ into $W^{\star}?$ If I understood the question correctly, we want to define an isomorphism from $W^{\star}$ onto $\{f:V\rightarrow \mathbb{F}|f(x_{1},\ldots,x_{n})=\sum_{i=1}^{n}c_{i}x_{i} \operatorname{and} \sum_{i=1}^{n}c_{i}=0\}.$ I will be back soon. – spohreis Apr 20 '12 at 18:53
You definitely won't have an isomorphism. (They asked you to identify $W$ as a subspace of $W^\ast$...) You have linear functionals of the form $\underline x \mapsto \sum_{i=1}^n c_i x_i$ where $\sum c_i \neq 0$.. – user21436 Apr 20 '12 at 18:57
@KannappanSampath Actually I am trying to understand why $W^{\ast}$ is not isomorphic to $W$ under your map. – Benja Apr 22 '12 at 14:33 | {"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": 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": 0.9786127209663391, "perplexity": 198.2689232889147}, "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-2013-48/segments/1387345774929/warc/CC-MAIN-20131218054934-00078-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/simple-efficiency-question.308281/ | Simple efficiency question
1. Apr 18, 2009
diegzumillo
Hello again, all! I have another basic thermodynamics question :P This one came from a 2008 admission test. Seems simple enough, but as usual, my answer doesnt match =P
1. The problem statement, all variables and given/known data
A monoatomic ideal gas performs, reversibly, the cycle shown in the diagram attached in this post. The values of $$P_0$$ and $$V_0$$ are, respectively, $$1\times 10^5 Pa$$ and $$100cm^3$$. The area of the interior of the cycle is 15J. What is the efficiency of this cycle?
2. Relevant equations
$$e=\frac{W}{Q_h}$$
e is the efficiency, W is work done by the system and $$Q_h$$ is the energy absorbed by the system (heat).
This is probably the only equation relevant here..
3. The attempt at a solution
The problem seemed straigh forward: The area enclosed by the cycle is the work done. The area under the curve $$ab$$ is the $$Q_h$$, right? If so, calculating this is trivial and results in an efficiency of, approximately, $$0,43$$.
However, this is not right. The correct answer should be $$0,136$$.
Attached Files:
• problem_efficiency_engine.jpg
File size:
4.4 KB
Views:
67
2. Apr 19, 2009
diegzumillo
No ideas? :P
3. Apr 19, 2009
Redbelly98
Staff Emeritus
Ah, no. That area would be the work done from a to b.
Q is T dS, and W is P dV.
The easiest way to get Q, for any single path of this cycle, is to use
ΔU = Q - W
since ΔU and W are fairly straightforward to calculate.
Last edited: Apr 19, 2009
4. Apr 19, 2009
abhikesbhat
Redbelly you are smart!!
5. Apr 19, 2009
diegzumillo
Thanks Redbelly :) I got carried away by the relation $$W=Q_h-Q_c$$ and assumed something wrong :P
Can I classify this system as cyclic? If so, isn't the internal energy variation supposed to be zero?
6. Apr 20, 2009
Redbelly98
Staff Emeritus
Gosh, thank you.
Yes, the entire process is cyclic, and ΔU is zero for the entire cycle.
But ... to calculate Qh, you'll need to consider each individual subpath, and whether heat is flowing into or out of the system for that path. Heat in contributes to Qh, while heat out does not.
Since ΔU is not necessarily zero for each subpath, it needs to be considered.
7. Apr 24, 2009
diegzumillo
Hello again! I let go of this problems for a few days. But today I took another look at it, and I think I have a solution :)
We'll use this equation for heat
$$Q=C\Delta T$$
where this C depends on the path.
We can start by calculating the heat for the path BC and CA, we'll call them $$Q_{bc}$$ and $$Q_{ca}$$, respectively. We cannot calculate directly $$Q_{ab}$$ because we only know $$c_p=3/2 R$$, for constant pressure, and $$c_v=5/2 R$$, for constant volume. So we have
$$Q_{bc}=\frac{c_v}{R}(P_c V_c - P_b V_b)=-45J$$
$$Q_{ca}=\frac{c_p}{R}(P_a V_a - P_c V_c)=-50J$$
(I'm calling $$P_i$$ and $$V_i$$ for better understanding)
We can see that both of those values represent heat leaving the system. Now we'll use the internal energy relations to find $$Q_{ab}$$. The internal energy relations for each path are:
$$\Delta U_{bc}=Q_{bc}-W_{bc}$$ (Yes, $$W_{bc}$$ is zero)
$$\Delta U_{ca}=Q_{ca}-W_{ca}$$
Since we know that
$$\Delta U=\Delta U_{ab}+\Delta U_{bc}+\Delta U_{ca}=0$$
and
$$\Delta U_{ab}=A_{ab}-W_{ab}$$
We have
$$Q_{ab}=W_{ab}-(Q_{bc}-W_{bc}+Q_{ca}-W_{ca})$$
$$Q_{ab}=W-(Q_{bc}+Q_{ca})=110J$$
Wich is our heat transfered into the system! ($$Q_h$$)
In possession of these values, we can now calculate the efficiency of this cycle:
$$e=\frac{W}{Q_h}=13,6%$$
=D
(Btw, for some strange reason, I can't visualize correctly the formulas. So there may be some errors. I'll correct them as soon as I'm able to read what I wrote :P)
8. Apr 24, 2009
Redbelly98
Staff Emeritus
Latex equations have been having problems for several days now.
Much of what you wrote can be done without Latex. You can get the Greek letter Δ here:
https://www.physicsforums.com/blog.php?b=347 [Broken]
Also:
[noparse]a[/noparse] for subscript a
[noparse]2[/noparse] for superscript 2
Last edited by a moderator: May 4, 2017
9. Apr 24, 2009
diegzumillo
Yep.. just saw the anouncement. What a bummer!
Since there is no estimate for the return, I'll post the solution again! (But I'll keep that one there... it took me a while to do! :P)
10. Apr 24, 2009
diegzumillo
The solution again. Simplified notation version :D
We'll use this equation for heat
Q=CΔT
where this C depends on the path.
We can start by calculating the heat for the path BC and CA, we'll call them Qbc and Qca, respectively. We cannot calculate directly Qab because we only know cp=3/2 R, for constant pressure, and cv=5/2 R constant volume. So we have
Qbc=cv/R (PcVc-PbVb)=-45J
Qca=cp/R (PaVa-PcVc)=-50J
(I'm calling Pi and Vi for better understanding before using the values given by P0 and V0)
We can see that both of those values represent heat leaving the system. Now we'll use the internal energy relations to find Qab. The internal energy relations for each path are:
ΔUbc=Qbc-Wbc (Yes, Wbc is zero)
ΔUca=Qca-Wca
Since we know that
ΔU=ΔUab+ΔUbc+ΔUca=0
and
ΔUab=Qab-Wab
We have
Qab=Wab-(Qbc-Wbc+Qca-Wca)
Qab=W - (Qbc+Qca) = 110J
Wich is our heat transfered into the system! (Qh)
In possession of these values, we can now calculate the efficiency of this cycle:
e=W/Qh=13,6%
=D
Thanks again Redbelly! And btw, that's a really helpful list of symbols! :) Especially for this dark times without latex.. lol
11. Apr 24, 2009
Redbelly98
Staff Emeritus
It looks like the values of cv and cp have been swapped here, but you did use the correct values in the calculation. Good job, you nailed this one!
You're welcome. "Dark times", LOL | {"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": 0.9484282732009888, "perplexity": 1312.1753428170323}, "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-2017-43/segments/1508187823731.36/warc/CC-MAIN-20171020044747-20171020064747-00816.warc.gz"} |
https://www.mathgenealogy.org/id.php?id=12829 | ## Norman Christian Meyer
Ph.D. University of Oregon 1970
Dissertation: The Ring of Holomorphic Functions on an Open Riemann Surface | {"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": 0.8323232531547546, "perplexity": 1568.1068589242323}, "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/1659882573242.55/warc/CC-MAIN-20220818154820-20220818184820-00431.warc.gz"} |
http://mathhelpforum.com/differential-equations/280413-diagonalize-matrix.html | 1. ## Diagonalize the matrix
$\textsf{ Diagonalize the matrix:}$
$$A=\left[\begin{array}{rrrr} 2& 2& -1\\1& 3& -1\\-1& -2& 2 \end{array}\right]$$
$\textit{ a. Charateristic equation:}$
$$A-\lambda I= \left[\begin{array}{rrrr} 2& 2& -1\\1& 3& -1\\-1& -2& 2 \end{array}\right]-\left[\begin{array}{rrrr} \lambda& 0& 0\\0& \lambda& 0\\0& 0& \lambda \end{array}\right]$$
$$-\lambda^3+7\lambda^2-\lambda+5$$
$\textit{ b.$D=$,$P=$}$
$$v_1=\begin{bmatrix}-1\\-1\\1\end{bmatrix} v_2=\begin{bmatrix}1\\0\\1\end{bmatrix} v_3=\begin{bmatrix}-2\\1\\0\end{bmatrix}$$
$$\therefore P=(v_1,v_2,v_3)\begin{bmatrix} \,\, 1&1&0\\-1&0&1\\ \,\,1&1&0 \end{bmatrix}$$
$\textit{ c. Verify}\\$
ok I got the Characteristic equation and $v_1,v_2,v_3$ from W|A
but didn't understand the example?
And didn't know how to verify it
2. ## Re: Diagonalize the matrix
you haven't done any of it.
come up with matrices, $P, ~D$ such that
$PDP^{-1} = A$
$P$ is the matrix of normalized eigenvectors as columns
$D$ is a diagonal matrix of associated eigenvalues, i.e. element $d_{1,1}$ is the eigenvalue corresponding to column 1 of P, etc.
3. ## Re: Diagonalize the matrix
Originally Posted by bigwave
$\textsf{ Diagonalize the matrix:}$
$$A=\left[\begin{array}{rrrr} 2& 2& -1\\1& 3& -1\\-1& -2& 2 \end{array}\right]$$
$\textit{ a. Charateristic equation:}$
$$A-\lambda I= \left[\begin{array}{rrrr} 2& 2& -1\\1& 3& -1\\-1& -2& 2 \end{array}\right]-\left[\begin{array}{rrrr} \lambda& 0& 0\\0& \lambda& 0\\0& 0& \lambda \end{array}\right]$$
$$-\lambda^3+7\lambda^2-\lambda+5$$
$\textit{ b.$D=$,$P=$}$
$$v_1=\begin{bmatrix}-1\\-1\\1\end{bmatrix} v_2=\begin{bmatrix}1\\0\\1\end{bmatrix} v_3=\begin{bmatrix}-2\\1\\0\end{bmatrix}$$
$$\therefore P=(v_1,v_2,v_3)\begin{bmatrix} \,\, 1&1&0\\-1&0&1\\ \,\,1&1&0 \end{bmatrix}$$
$\textit{ c. Verify}\\$
ok I got the Characteristic equation and $v_1,v_2,v_3$ from W|A
but didn't understand the example?
And didn't know how to verify it
Look At This.
You may get some very useful verification.
4. ## Re: Diagonalize the matrix
The point is that a 3 by 3 matrix, A, is diagonalizable if and only all 3 eigenvectors are independent so form a basis for $R^3$. In that case, A is similar to the diagonal matrix having the eigenvalues of A on its diagonal. When you were finding the eigenvectors of A didn't you find the corresponding eigenvalues? Saying that A is similar to a diagonal matrix, D, means that there exist an invertible matrix, P, Such that $PAP^{-1}= D$. And P can be constructed by taking the eigenvectors of A as columns.
5. ## Re: Diagonalize the matrix
The point is that a 3 by 3 matrix, A, is diagonalizable if and only all 3 eigenvectors are independent so form a basis for $R^3$. In that case, A is similar to the diagonal matrix having the eigenvalues of A on its diagonal. When you were finding the eigenvectors of A didn't you find the corresponding eigenvalues? Saying that A is similar to a diagonal matrix, D, means that there exist an invertible matrix, P, Such that PAP^{-1}= D. And P can be constructed by taking the eigenvectors of A as columns.
To "verify" this, actually do that multiplication.
6. ## Re: Diagonalize the matrix
I'm going to interject a quick question here.
In my work in Physics I usually don't bother with finding P, I simply find the eigenvalues and immediately write the diagonal matrix. Am I losing any generality by doing it this way?
-Dan
7. ## Re: Diagonalize the matrix
The characteristic polynomial you listed is wrong. $\displaystyle -\lambda^3+7\lambda^2-11\lambda+5 = -(\lambda-5)(\lambda-1)^2$ is the characteristic polynomial.
So, how did you find the correct eigenvectors? Setting your characteristic polynomial equal to zero would give only one real root, not three.
8. ## Re: Diagonalize the matrix
Originally Posted by topsquark
I'm going to interject a quick question here.
In my work in Physics I usually don't bother with finding P, I simply find the eigenvalues and immediately write the diagonal matrix. Am I losing any generality by doing it this way?
-Dan
If you find n distinct eigenvalues for a nxn matrix then you can immediately write down the diagonal matrix.
What do you do when you have fewer than n eigenvalues?
9. ## Re: Diagonalize the matrix
Actually the number of distinct eigenvalues is irrelevant. It is the number of independent eigenvectors that is important. Yes, if you have n distinct eigenvalues, for an n by n matrix, the matrix is diagonalizable because you necessarily have n independent eigenvectors- eigenvectors corresponding to distinct eigenvalues are independent. But if some or even all of the n eigenvalues are the same, there still might be n independent eigenvectors.
10. ## Re: Diagonalize the matrix
Originally Posted by Idea
If you find n distinct eigenvalues for a nxn matrix then you can immediately write down the diagonal matrix.
What do you do when you have fewer than n eigenvalues?
The usual. I construct the space with the degenerate eigenvalues and take an appropriate linear combinations.
-Dan
11. ## Re: Diagonalize the matrix
Once again, if an n by n matrix has fewer than n eigenvalues it might still have n independent eigenvectors and so be diagonalizable just as I said before. If there are fewer than n independent eigenvectors, then the matrix is not diagonalizable. One can, in that case, find the "Jordan normal form" where we have "block submatrices" with copies of the single eigenvalue on the diagonal and "1"s just above the diagonal. For example, if a matrix has
'1' as a double eigenvalue, 2 as a triple eigenvalue, and 3 as a single eigenvalue (so 6 eigenvalues and the matrix is 6 by 6) [b]and there exist only 1 eigenvector for each eigenvalue (not counting dependent vectors) then the Jordan Normal form is
$\displaystyle \begin{bmatrix}1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 3\end{bmatrix}$.
However if there exist two independent eigenvectors with eigenvalue 1 and 3 with eigenvalue 2, then the matrix is diagonalizable, to matrix
$\displaystyle \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 3 \end{bmatrix}$.
If there are other "combinations" of number of eigenvector, if, say, the was only one eigenvector corresponding to eigenvalue 1 and 2 corresponding too eigenvalue 2, then the matrix would be
$\displaystyle \begin{bmatrix}1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 3 \end{bmatrix}$. | {"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": 0.9695229530334473, "perplexity": 417.3196181522847}, "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/1550247479967.16/warc/CC-MAIN-20190216065107-20190216091107-00494.warc.gz"} |
http://mathhelpforum.com/advanced-math-topics/3931-help-print.html | # Help!
• June 30th 2006, 12:19 PM
Refujoi
Help!
A wooden block slide directly down an inclined plane velocity of 6.0 m/s How large is the coefficient of kinetic friction, if the plane makes an angle of 25 deegre with the horizontal?
• July 1st 2006, 01:29 AM
CaptainBlack
Quote:
Originally Posted by Refujoi
A wooden block slide directly down an inclined plane velocity of 6.0 m/s How large is the coefficient of kinetic friction, if the plane makes an angle of 25 deegre with the horizontal?
As the block slides with constant velocity down the plane the Normal Reaction
$N$, the Frictional force $F$, and the Gravitational force $mg$ on the block are in equilibrium.
Resolve the gravitational force into components down the plane and normal to
the plane and set these equal to the frictional force and the normal reaction
(allowing for the appropriate signs):
$F=mg \sin(\theta)$,
$N=mg \cos(\theta)$
where $\theta$ is the inclination of the plane.
Now $F=\mu N$, where $\mu$ is the coefficient of
kinetic/sliding ... friction. So:
$\mu\ mg \cos(\theta)=mg \sin(\theta)$
rearranging:
$\mu=\tan(\theta)$.
In this problem $\theat=25^{\circ}$, so:
$\mu=\tan(25^{\circ}) \approx 0.466$.
RonL | {"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": 0.9012454152107239, "perplexity": 637.9822352595279}, "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-2016-26/segments/1466783404382.73/warc/CC-MAIN-20160624155004-00147-ip-10-164-35-72.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/questions/149404/bold-text-in-newenvironment/149406 | # Bold text in newenvironment
I want to define my own environment, like
\newenvironment{question}
{\begin{tiny}}
{\end{tiny}}
That may not make much sense, but it works. How can I do the same with bold font?
-
There is no environment tiny. There is a command \tiny. The same can be achieved with \bfseries: \newenvironment{foo}{\bfseries}{} – Marco Daniel Dec 10 '13 at 20:55
You need to use a "switch", achieved by issuing \bfseries at the start of the environment (say boldenv):
\newenvironment{boldenv}
{\bfseries}% \begin{boldenv}
{}% \end{boldenv}
The scope of this switch is limited to the environment, since \begin{boldenv} ... \end{boldenv} naturally forms a group. Also, italicized text inside this environment will inherit the bold switch (for example, if you use \textit{<stuff>}). To avoid this, you'd have to restore the font using something like {\normalfont\itshape <stuff>}.
-
There is no environment tiny. There is a command \tiny. The same can be achieved with \bfseries: \newenvironment{foo}{\bfseries}{}
I think it's important to mention a fact which is mostly unknown: Don't use font commands as environments.
Explanation:
\documentclass{article}
\usepackage[ngerman]{babel}
\usepackage{blindtext}
\begin{document}
\blindtext
\begin{quote}\tiny\blindtext\end{quote}
\blindtext
\begin{quote}\begin{tiny}\blindtext\end{tiny}\end{quote}
\end{document}
What happened
First you can see the different line spacing. In the first example (the correct one) \begin{quote}\tiny\blindtext\end{quote} the command \par is executed with the font size \tiny. So the space is correct. In the second example \begin{quote}\begin{tiny}\blindtext\end{tiny}\end{quote} the inner environment makes a local switch to \tiny and the closing \par by quote uses the default font size. So the complete paragraph is getting the wrong spacing.
- | {"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": 0.9964133501052856, "perplexity": 1777.9931645485615}, "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/1454701154682.35/warc/CC-MAIN-20160205193914-00240-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/center-of-mass-using-machs-restatement-of-newtonian-mechanics.795433/ | # Center of mass using Mach's restatement of Newtonian Mechanics
Tags:
1. Feb 1, 2015
### AllRelative
1. The problem statement, all variables and given/known data
Use Mach’s restatement of Newtonian mechanics to show that if we define the centre of mass of two particles according to,
⃗ r = (m1 ⃗ r1 + m2 ⃗ r2) / (m1 + m2)
then the center of mass moves according to the equation,
⃗r = [(m1 ⃗u1 + m2 ⃗u2) / (m1 + m2)]t + ⃗r0
where ⃗r0 is a constant vector.
2. Relevant equations
This is a condensed version of the notes that were given in class but everything is there :
Mach's approach was to use Newton’s third law, which is the only law of the three to actually state a law or prescription of nature, as a starting point.From Newton's third law we can get the expression
a1/a2 = k12 and k12 is a constant.
So then Mach’s reformulation of Newton’s mechanics states that for any two of two or more interacting particles, the ratio of their acceleration will be constant. If we apply a Galilean transformation we obtain,
a1'/a2' = k12 = a1/a2
showing the equations are form invariant.
We relate the constant with the inertial mass .
Mach’s version of Newtonian mechanics is free from definitions and has therefore simplified the theory by making reference only to acceleration a geometric quantity. All other standard notion of Newtonian mechanics can be derived from Mach’s restatement.
3. The attempt at a solution
I fully understand Newton's three laws and I get what Mach's restatement. I just can't figure for the life of me how I can use it to get the desired equation. If anyone has an idea of where to start that would be great!
Cheers!
2. Feb 1, 2015
### TSny
Look at the 2nd time derivative of
⃗ r = (m1 ⃗ r1 + m2 ⃗ r2) / (m1 + m2)
3. Feb 1, 2015
### elsayed
derive R relative to t and then at t=0 R=R0 and m1u1+m2u2=constant
4. Feb 2, 2015
### AllRelative
I did what you said but I get (constant/m1+m2)t + R0 = R
How can a scalar + a vector = a vector ??
5. Feb 2, 2015
### TSny
Since Mach's statement deals with the accelerations of the two objects, it might be good to somehow relate the formula for the position of the center of mass to the accelerations.
Note $\frac{d\vec{r_1}}{dt} = \vec{v_1}=$ the velocity vector.
What if you take one more derivative? What does $\frac{d^2\vec{r_1}}{dt^2}$ give you?
6. Feb 2, 2015
### elsayed
the first derivative to big R relative to t (dR/dt) in left hand side will give u already m1u1+m2u2 in right hand side then then multiply two side by dt and then make integrations two side u will get what u need
7. Feb 2, 2015
### lazytofindname
hey, im pretty sure i am in your class cause you quoted the exact pdf the teacher gave us, i really don't understand this question either and don't get any of those answers too..
8. Feb 2, 2015
### lazytofindname
@elsayed , it doesn't use mach restatement though
9. Feb 2, 2015
### TSny
It might help if we knew the mathematical level of this class. Is this a class where you can use calculus?
Draft saved Draft deleted
Similar Discussions: Center of mass using Mach's restatement of Newtonian Mechanics | {"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": 0.9381587505340576, "perplexity": 1062.4303650357601}, "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/1512948588072.75/warc/CC-MAIN-20171216123525-20171216145525-00263.warc.gz"} |
https://koreauniv.pure.elsevier.com/en/publications/magnetic-and-magnetocaloric-properties-of-lasub07subcasub03submns | # Magnetic and magnetocaloric properties of La0.7Ca0.3Mn1−xZnxO3
T. A. Ho, S. H. Lim, P. T. Tho, T. L. Phan, S. C. Yu
Research output: Contribution to journalArticlepeer-review
10 Citations (Scopus)
## Abstract
The magnetic Mn3+ ions in La0.7Ca0.3MnO3 are partially replaced by nonmagnetic Zn2+ ions to form La0.7Ca0.3Mn1−xZnxO3 compounds (x=0.0, 0.06, 0.08, and 0.1), and their magnetic and magnetocaloric properties are investigated. The Curie temperature decreases drastically from 245 to 70 K as x increases from 0 to 0.1. An analysis using the Banerjee's criterion of the experimental results for magnetization as a function of temperature and magnetic field indicates that the first-to-second order magnetic phase transformation occurs at a threshold composition of x=0.06, which is further supported by the universal curves of the normalized entropy change versus reduced temperature. The maximum magnetic entropy change measured at a magnetic field span of 50 kOe, which occurs near the Curie temperature, decreases from 10.30 to 2.15 J/kg K with the increase of x from 0.0 to 0.1. However, the relative cooling power, an important parameter for practical applications, shows a maximum value of 404 J/kg at x=0.08, which is 1.5 times greater than that observed for the undoped sample.
Original language English 18-24 7 Journal of Magnetism and Magnetic Materials 426 https://doi.org/10.1016/j.jmmm.2016.11.050 Published - 2017 Mar 15
## Keywords
• Magnetic phase transformation
• Magnetic properties
• Magnetocaloric effect
• Perovskite manganites
• Spin glass
## ASJC Scopus subject areas
• Electronic, Optical and Magnetic Materials
• Condensed Matter Physics
## Fingerprint
Dive into the research topics of 'Magnetic and magnetocaloric properties of La0.7Ca0.3Mn1−xZnxO3'. Together they form a unique fingerprint. | {"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": 0.8588994145393372, "perplexity": 3721.053157094399}, "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/1674764501555.34/warc/CC-MAIN-20230209081052-20230209111052-00408.warc.gz"} |
http://mathoverflow.net/questions/72536/asymptotics-for-primality-of-sum-of-three-consecutive-primes/72545 | # Asymptotics for primality of sum of three consecutive primes
We consider the sequence $R_n=p_n+p_{n+1}+p_{n+2}$, where $\{p_i\}$ is the prime number sequence, with $p_0=2$, $p_1=3$, $p_2=5$, etc..
The first few values of $R_n$ are: 10, 15, 23, 31, 41, 49, 59, 71, 83, 97, 109, 121, 131, 143, 159, 173, 187, 199, 211, 223, 235, 251, 269, 287, 301, 311, 319, 329, 349, 371, 395, 407, 425, 439, 457, ...
Now, we define $R(n)$ to be the number of prime numbers in the set $\{R_0, R_1 , \dots , R_n\}$.
What I have found (without justification) is that $R(n) \approx \frac{2n}{\ln (n)}$.
My lack of programming skills, however, prevents me from checking further numerical examples. I was wondering if anyone here had any ideas as to how to prove this assertion.
As a parting statement, I bring up a quote from Gauss, which I feel describes many conjectures regarding prime numbers: "I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of."
-
Is it called "recursion" when the sequence you're defining and the one you're using to define it are two different sequences? – Michael Hardy Aug 10 '11 at 1:52
Im sorry, I dont understand what you are asking, Michael. First I have a recursion that defines an infinite set of numbers (R_n=sum of three consecutive primes). Then I consider the number of prime numbers in the set {R_0,...,R_n}. That is R(n). I hope that clears up your question – Raj Aug 10 '11 at 2:13
@Raj: Your $R_n$ is not recursively defined, but defined directly in terms of the primes. You have no recursion in your message. – GH from MO Aug 10 '11 at 2:15
ohhhh wow my bad, that was terrible wording... ughh well I think I'm still getting the main point of the question across – Raj Aug 10 '11 at 2:17
I've taken the liberty of changing the title. – Gerry Myerson Aug 10 '11 at 5:32
I believe proving (or disproving) such a statement is beyond current technology. On the other hand, by a crude heuristics the conjecture must be right: $R_1,\dots,R_n$ are $n$ odd numbers up to $\sim 3n\ln(n)$ which are rather evenly distributed in size and in residue classes. In this range the density of primes among odd numbers is $\sim 2/\ln(n)$, so $R(n)$ should be $\sim 2n/\ln(n)$.
EDIT: As Noam Elkies points out in a comment below, there should be a fine tuning similar to the twin prime constant.
-
I dont understand how this is not a proof? Assuming everything you stated is known... I understand its not fully rigorous, but my question only asked that it is APPROXIMATELY 2n/ln(n), and you showed just that...? – Raj Aug 10 '11 at 2:20
@Raj: My response contained vague statements like "rather evenly distributed". It is very far from a rigorous proof, and I would be surprised if anyone could give a rigorous proof. – GH from MO Aug 10 '11 at 2:24
@Raj: Even the statement that your $R(n)$ tends to infinity seems hopeless to verify rigorously. – GH from MO Aug 10 '11 at 2:28
The factor of $2$ isn't quite right, because even for prime $l>2$ the sum of three consecutive primes will diverge from equidistribution $\bmod l$ even if we believe that the residues of $p_n,p_{n+1},p_{n+2}\bmod l$ are independent. If I did this right, the probability that the sum of $3$ independent elements of $({\bf Z}/l{\bf Z})^*$ is nonzero $\mod l$ is $(l^2-3l+3)/(l-1)^2$, which exceeds $(l-1)/l$ by a factor $1 + 1/(l-1)^3$. So the constant should be $\prod_l \bigl(1+1/(l-1)^3\bigr) = 2.30096+$. This $15\%$ discrepancy should be experimentally detectable though it will take some care. – Noam D. Elkies Aug 10 '11 at 2:40
@Will: That seems like an overvaluation of proofs to me. It can't be proved, but it seems highly likely that it does, and something interesting might be learned from what the limit seems to be quite independent of whether its existence can be proved. – joriki Aug 10 '11 at 6:43
For what it's worth, I calculated a bunch of values of $R(n)$ and the claimed densities. I also had into consideration Noam Elkies' possible correction to the factor of $2$ (i.e., use $\lambda n/\log(n)$ instead of $2n/\log(n)$, where $\lambda = \prod_l (1+1/(l-1)^3)\cong 2.30096\ldots$; see the comments in GH's answer). $$n \quad|\quad R(n) \quad | \quad 2n/\log(n) \quad | \quad (2.30096)n/\log(n)$$ $$10 \quad | \quad 7 \quad | \quad 8.685\ldots \quad | \quad 9.992\ldots$$ $$100 \quad | \quad 44 \quad | \quad 43.429\ldots \quad | \quad 49.964\ldots$$ $$1000 \quad | \quad 339 \quad | \quad 289.529\ldots \quad | \quad 333.098\ldots$$ $$10000 \quad | \quad 2437 \quad | \quad 2171.472\ldots\quad | \quad 2498.235\ldots$$ $$100000 \quad | \quad 18892 \quad | \quad 17371.779\ldots \quad | \quad 19985.884\ldots$$
-
It may seem from this data that $2.30...$ is too large, but there are several small-number effects here. $n/\log n$ should be $\sum_{m=1}^n 1/\log(3 p_n)$, which is asymptotically the same but still smaller by $24\%$ at $n=10^5$ (6589 vs. 8686). This makes $R(10^5) = 18892$ seem too large by a substantial margin. It turns out that for small $n$ and the smallest odd primes $l$ it's much rarer than expected to have $l\mid p_n+p_{n+1}+p_{n+2}$; e.g. up to $10^5$ we get only $16401$ for $l=3$ and $15856$ for $l=5$, not the expected $25000$ and $18750$. Presumably this decays for large $n$... – Noam D. Elkies Aug 10 '11 at 13:58
[correction: naturally what I meant is the sum over $m$ of $1/\log(3p_m)$, not $1/\log(3p_n)$.] – Noam D. Elkies Aug 10 '11 at 14:05
I verified Noam's calculation of the factor $\lambda=2.30096\ldots$ and Álvaro's computations, extended the latter and calculated the corresponding ratios:
$$\begin{array}{|c|c|c|c|c|} n & R(n) & 2n/\log n & \lambda n / \log n & R(n)\log n/n\\\\ \hline 10 & 7 & 9 & 10 & 1.61181\\\\ 100 & 44 & 43 & 50 & 2.02627\\\\ 1000 & 339 & 290 & 333 & 2.34173\\\\ 10000 & 2437 & 2171 & 2498 & 2.24456\\\\ 100000 & 18892 & 17372 & 19986 & 2.17502\\\\ 1000000 & 157183 & 144765 & 166549 & 2.17156\\\\ 10000000 & 1346797 & 1240841 & 1427564 & 2.17078\\\\ 30000000 & 3784831 & 3484987 & 4009410 & 2.17208\\\\ \end{array}$$
(The values in the third and fourth columns are rounded to the nearest integer, the values in the last column are rounded to 5 digits after the decimal point.)
I don't think we can deduce anything from the ratio in this form, however, since it shows convergence in the "random" fluctuations but not with respect to the asymptotic approximations made, e.g. dropping a term $\log\log n$, which at this stage is still comparable to $\log n$; a more detailed analysis will be required to test the independence hypothesis in this case.
[Update:] With reference to Noam's comments below, here are some data for the relative frequencies of the sum of three consecutive primes being divisible by the first four odd primes. These are averaged over samples of $400,000$ primes beginning at powers of ten, which are given in the first column; note that these refer to the numbers $x$ themselves, not the indices $n$ of the primes.
$$\begin{array}{|c|c|c|c|c|} \log_{10}x&3&5&7&11\\\\ \hline 8 &0.183&0.165&0.130&0.087\\\\ 9 &0.189&0.169&0.131&0.087\\\\ 10&0.195&0.170&0.133&0.087\\\\ 11&0.198&0.172&0.133&0.088\\\\ 12&0.203&0.173&0.133&0.088\\\\ 13&0.208&0.175&0.134&0.087\\\\ \hline \text{limit?}&0.250&0.188&0.139&0.090 \end{array}$$
I also looked at the joint distribution of the residues modulo $3$ for the three primes. There's a significant preference for avoiding repeated residues; for instance, at $x=10^9$, the repeating patterns $1,1,1$ and $2,2,2$ have relative frequencies around $0.095$, the alternating patterns $1,2,1$ and $2,1,2$ have relative frequencies around $0.150$, and the remaining mixed patterns have relative frequencies around $0.128$, which is almost completely explained by $1,1$ and $2,2$ having relative frequencies $0.445$ and $1,2$ and $2,1$ having relative frequencies $0.555$. I'm trying to work out a probabilistic model for these effects.
-
See my comment on Álvaro Lozano-Robledo's computation. What happens when $n / \log n$ is replaced by the better estimate $\sum_{m=1}^n 1/\log(3p_m)$? Up to $n=10^5$ this made the observed constant much larger than $\lambda$ (about $2.8$). It seems that for the smallest odd primes $l$ it takes a while for the probability of $l \mid p_n + p_{n+1} + p_{n+2}$ to approach the expected value $(l^2-3l+3)/(l-1)^2$. – Noam D. Elkies Aug 10 '11 at 14:07
@Noam: Yes, I made the same observations and was just doing some experiments on that -- the approach is quite slow; even at the end of the table the probability for $l=3$ is only about $0.2$ instead of the asymptotic $3/4$; this is due to repeated residues being significantly less likely than alternating residues, and this effect only decays slowly as the gaps between the primes become wider. – joriki Aug 10 '11 at 15:22
@joriki: I guess you mean $0.2$ instead of $1/4$, not $3/4$. Also -- how close are $l=5$ and $l=7$, and does there seem to be significant dependence between $l=3$ and $l=5$? Presumably all these effects eventually disappear but it looks like it would require lots of computation to see the ratio converge to 2.30. – Noam D. Elkies Aug 10 '11 at 15:42
@Noam: Yes, sorry, $1/4$. Regarding you other questions, see above. Yes, seeing the ratio converge to $2.30$ isn't feasible, I think; what might be feasible, though, is to model these effects well enough to support the conclusion that they eventually disappear. – joriki Aug 10 '11 at 17:45 | {"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": 0.9380452632904053, "perplexity": 254.8196655348485}, "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-2013-48/segments/1387345766127/warc/CC-MAIN-20131218054926-00012-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://link.springer.com/article/10.1007%2Fs11538-008-9367-5 | Bulletin of Mathematical Biology
, 71:399
Can a Species Keep Pace with a Shifting Climate?
• H. Berestycki
• O. Diekmann
• C. J. Nagelkerke
• P. A. Zegeling
Open AccessOriginal Article
DOI: 10.1007/s11538-008-9367-5
Berestycki, H., Diekmann, O., Nagelkerke, C.J. et al. Bull. Math. Biol. (2009) 71: 399. doi:10.1007/s11538-008-9367-5
Abstract
Consider a patch of favorable habitat surrounded by unfavorable habitat and assume that due to a shifting climate, the patch moves with a fixed speed in a one-dimensional universe. Let the patch be inhabited by a population of individuals that reproduce, disperse, and die. Will the population persist? How does the answer depend on the length of the patch, the speed of movement of the patch, the net population growth rate under constant conditions, and the mobility of the individuals? We will answer these questions in the context of a simple dynamic profile model that incorporates climate shift, population dynamics, and migration. The model takes the form of a growth-diffusion equation. We first consider a special case and derive an explicit condition by glueing phase portraits. Then we establish a strict qualitative dichotomy for a large class of models by way of rigorous PDE methods, in particular the maximum principle. The results show that mobility can both reduce and enhance the ability to track climate change that a narrow range can severely reduce this ability and that population range and total population size can both increase and decrease under a moving climate. It is also shown that range shift may be easier to detect at the expanding front, simply because it is considerably steeper than the retreating back.
Keywords
Climate changeReaction–diffusion equationTraveling waveMoving favorable patchCo-moving population profilePersistenceExtinctionPrincipal eigenvalue
1 1. Introduction
The area occupied by a species is to a large extent determined by the climatic circumstances with temperature playing a major role. The global warming phenomenon, there-fore, has a great impact on survival and location of such species. See Walther et al. (2002) for a review of ecological responses to recent climate change.
We idealize the world by putting the North Pole at +∞ and the equator at −∞. This ignores the finiteness of the Earth, but it offers a good framework for a theoretical analysis. Warming and its effect can be seen as a shift in the profile of local climatic suitability, which the population density profile of a species tries to track. If a species keeps pace, its area expands as much in the north as it loses in the south. However, if it lags behind too much, it will go extinct.
Which of these two scenarios applies? How does the answer depend on the mobility of the species, on the extensiveness of the area, the local population dynamics, and on the speed of climate shift and the way climate actually acts on a species? If the species survives, what happens to the size and form of its population profile?
The recent research of one of us (Nagelkerke, 2004) tackles these issues in the context of a relatively realistic metapopulation model, using simulations as the main tool. The aim of the present paper is to address the same issues for a continuous population using an analytical approach. We study a simplified model, taking the form of a reaction-diffusion equation. Within this framework, our findings confirm the ones that had been observed in simulations. Here, we establish these results for a large class of equations, with rigorous mathematical proofs, thus proving their robustness and shedding light on the mathematical properties behind them.
From an ecological point of view, our main results are the following:
• An explicit formula (23) and in different forms in (24) and (25), for the critical size of the favorable patch for persistence, as a function of the Malthusian parameters, the diffusion constant and the climate speed. The formula pertains to the juxtaposition of two types of homogeneous habitat, the favorable patch being a bounded interval outside of which the environment is unfavorable.
• Revelation of a striking asymmetry in the comoving population profile: the north front is much steeper than the south tail and the population maximum occurs near to the northern border of the population profile.
• The observation that if the climate does not move too fast, the size of the total population as well as the range of the population may actually grow, relative to the situation in a static climate. But when the climate speed is further increased, an abrupt collapse may follow (see Figs. 7, 8 and 9).
The model we study here takes the form of the following reaction-diffusion equation:
(1)
. Here, −∞<x<+∞ and c is a given positive number. Here, u is the population density of the species of interest and we have assumed that dispersal is adequately described by diffusion with constant D. The function f describes the net effect of reproduction and mortality and how this depends on population density and on the local climatic conditions. Hence, it expresses the suitability profile. Note that D is assumed to be independent of climate. The situation before the climate shift sets in corresponds to c = 0. We assume that
(2)
, where the per capita growth rate g is negative for large values of x, both negative and positive. More precisely, we shall incorporate only negative density dependence in the model (i.e., we do not incorporate an Allee effect, as discussed in Shi and Shivaji, 2006 for instance) and so the suitable area is
(3)
which we assume to be an interval of length L.
The key questions concerning (1) are: does a positive solution of the form
(4)
exist and is it a stable solution? What is the form of the solution? If no such solution exists, does it follow that u converges to zero for t→∞? How do the answers depend on c, D, L, and other parameters characterizing f?
Such questions are a bit reminiscent of the “critical patch length” problem (cf. Okubo and Levin, 2001 and Ludwig et al., 1979), the “traveling wave invasion” problem (cf. Kolmogorov et al., 1937; Fisher, 1937; Aronson and Weinberger, 1978; Berestycki and Hamel, 2009; Thieme and Zhao, 2003; Rass and Radcliffe, 2003) and the “heterogeneous environment” problem (cf. Berestycki et al., 2005a, 2005b; Roques and Stoica, 2007; Shigesada and Kawasaki, 1997; Shigesada et al., 1986; Weinberger, 2002). Yet, the mix of ingredients (in particular, the fact that c is prescribed, and hence amounts to an external forcing) makes it different from each of these and apart from Pease et al. (1989), where a quantitative genetics approach is adopted, we could not find any references. After most of the work described here was finished, however, we came across the preprint version of Potapov and Lewis (2004), which addresses exactly the same question, but with emphasis on the effect of a moving climate on the outcome of competitive interaction between two species. In fact, the special case that we treat in Sections 2 and 3 is also treated by Potapov and Lewis. Yet, we decided to include our analysis of this case in this paper as (1) the method is more geometrical (essentially phase plane analysis), (2) we deliver an analytical solution for the population profile and (3) we present additional results, leading to further biological insights. Other related work can be found in Dahmen et al. (2000), Deasi and Nelson (2005), and Pachepsky et al. (2005). It is known that diffusion enhances invasion speed but is counter productive for population growth on a finite stationary patch. Consequently, for a moving patch, there is a conflict between gain due to colonization of newly favorable habitat and loss due to migration into unfavorable habitat. A key result, formula (23) below, provides a quantitative algorithm for deciding which of these two effects is the stronger one.
We employ two different methods. If we assume that g, as a function of x, is piecewise constant we can glue phase portraits corresponding to the second order traveling wave ODE:
(5)
.
Making use of the linearization at w = 0 we thus derive rather explicit answers to the key questions.
A more qualitative analogue of these answers for quite general g is obtained by a PDE approach. The information provided by the linearization at zero is again crucial. Using various methods, notably the comparison principle, we derive, in Section 4, a dichotomy from this information:
• Either no positive traveling wave exists and zero is the global attractor,
• Or such a wave does exist and it attracts all orbits starting from nonnegative (≢ 0) initial data.
The biological insights derived from our analysis are explained in detail in Sections 2 and 3 while taking for granted that the results of Section 4 demonstrate the correctness and the robustness of the conclusions. More ecological consequences are discussed in Section 5. Readers who are looking for theorems and proofs will find the rigorous theory for a general class of equations presented in Section 4.
2 2. Glueing phase portraits
Throughout this section, we assume that for given positive parameters , r, K, and L,
(6)
, while requiring that solutions are C1 (indeed, to guarantee that diffusion conserves mass, the flux should be continuous). In this model, the underlying assumption is that spatial heterogeneity is fully described by two abrupt changes taking places at the positions x = 0 and x = L. By scaling u, t, and x, we can reduce the number of parameters from six to three. We choose to do this in such a way that the new values of K, , and D are all one. To facilitate the interpretation of our final results, we list how the new r, c, and L relate to the six original parameters:
(7)
.
Already at this stage we can conclude that K only sets the scale for u, but that it is irrelevant for the answers to our questions.
In the outer regions ζ <0 and ζ > L, the function w that we seek to construct should satisfy the linear equation
(8)
. Define
(9)
then any solution to (8) is a linear combination of exp(μ+ζ) and exp(μζ). Since μ+ > 0 and μ < 0 and we want w to be bounded, the solution for ζ <0 should be a multiple of exp(μ+ζ) while the solution for ζ > L should be a multiple of exp(μζ).
If we rewrite (8) as the first order system:
(10)
, and think in terms of orbits in (w, v)-space, the solution for ζ<0 corresponds to motion away from the origin along the half-line v = μ+w,w > 0, while the solution for ζ >L corresponds to motion toward the origin along the half-line v = μw,w > 0 (see Fig. 1).
The analogue of (10) for 0 ≤ ζ ≤ L is
(11)
.
This system has equilibria (w, v) = (0, 0) and (w, v) = (1, 0). (Incidentally, orbits connecting these two equilibria yield the classical KPP-Fisher traveling waves Fisher, 1937; Kolmogorov et al., 1937. These exist if and only if c ≥ 2√r. The lowest possible wave speed, 2√r, is the invasion speed Aronson and Weinberger, 1978; Berestycki and Hamel, 2009.)
The linearization at (1, 0) has eigenvalues . So one is positive and the other negative or, in other words, (1, 0) is a saddle point. The linearization at (0, 0) has eigenvalues:
(12)
.
Provided these form a complex conjugate pair and then since c > 0, (0, 0) is a stable spiral point (see Fig. 2).
If on the other hand, , then (0, 0) is a stable node. Since μ < σ±, orbits of (10) that approach the origin from the positive half plane w>0 do so “above” the line υ = μw. It is known (see Hadeler and Rothe, 1975; Aronson and Weinberger, 1978; Volpert et al., 1994, or Diekmann and Temme, 1982) that the unstable manifold of (1, 0) that lies in the region w ≤ 1 does, in fact, approach the origin in this manner, and that it lies entirely above the line v = μw (in fact above v = σw).
Our task is to make a connection between the line v = μ+w and the line v = μw by way of a piece of orbit of (11) that is completed in a ζ-interval of exactly length L. The preceding paragraph established that this is impossible for , since then the connecting orbit between (1, 0) and (0, 0) forms an obstruction. (Note that this implies the nonsurprising fact that a species can never track a climate that moves faster than the invasion speed of that species into the favorable habitat.) So, we focus our attention on the situation obtained by combining Figs. 1 and 2 (see Fig. 3).
In view of the results of Schaaf on two-point boundary value problems (Schaaf, 1990), it is to be expected that the length of the ζ-interval of an orbit piece connecting the two lines increases with increasing distance (along either line) from the origin (with limit +∞ if we approach the connection via pieces of the stable—and unstable manifold of (1, 0)).
Indeed, let us now prove this monotonicity property.
Lemma 2.1.Let1, w1) and2, w2) be two solutions of (11) defined on (a1, b1) and (a2, b2), respectively, and satisfying
as well as
for i = 1, 2. Suppose that v2(a2) ≥ v1(a1). Then b2-b2 > b1-a1
Proof: By shifts of the solutions w1 and w2, (taking wi(x + ai)) there is no loss in generality in assuming that a1 = a2 = 0. Then we want to show that b2 > b1.
Recalling that wx = υ, Eq. (11) reads
(13)
, where g(w) = r(1 − w). From (13) and integration by parts, we see that for any α, 0 < α<Min⨑b1, b2⫂, the following relation holds:
(14)
. Suppose that w1 < w2 in (0,α) (which is certainly true for small α>0). Then formula (14) shows that
(15)
.
Indeed, g(w1 > g(w2) in (0,α). Now, if α < min⨑ub;b1, b2⫂ub; is such that w1 <w2 in (0,α) while w1(α) = w2(α), then (15) yields w2x(α) α w1x1 and w2 do not cross each other in (0, min⨑ub;b1, b2}⫂).
Assume now by way of contradiction that b2b1. Then choosing α = b2 in (15), we get
again a contradiction. Therefore, b2 >b1 and the lemma is proved.
So, the shortest feasible L is obtained in the limit where the points on the line υ = μ±w approach the origin. In that limit, we can replace the nonlinear term −rw(1 − w) in the second equation of (11) by its linearization −rw.
So, we want to connect the half-lines υ = μ±w, w> 0, by a piece of orbit corresponding to
(16)
that is traversed in an interval of length L or less. The general real solution of (16), for , is given by
(17)
, where k is an arbitrary complex number. If we require
(18)
, to determine the unknown k and l, it seems that we have one real unknown too much, as k counts for two. Note, however, that the system (18) is real homogeneous of degree one: If k satisfies the equation, so does every real multiple of k. This reflects the fact that for the linear system (16), the “time” (i.e., the length of the independent variable interval) needed to cross the area between the lines υ = μ±w is independent of the starting point on the line ν = μ+w. So, we may add to (18) a condition that serves to normalize k. As such we choose
(19)
. The first equation of (18) then implies that
(20)
, and now that k is known, we can consider the second equation of (18) as determining l. After some manipulation, it can be rewritten as
(21)
. Provided we adopt the convention that the function arctan takes its values in (0,π] we can now formulate the conditions for the existence of a traveling wave solution as c< 2√r and
(22)
. These conditions are summarized in Fig. 4. Thus, for the right-hand side takes the value , while for the right-hand side goes to infinity like . Using the scaling relations (7), we can rewrite (22) in terms of the original parameters as
where
(23)
which should hold for .
We shall rewrite the expression for Lcrit in a somewhat more informative form which, moreover, facilitates the comparison with the formula at the end of Section 4 in Potapov and Lewis (2004). To do so, we introduce
and recall that this so-called Fisher-speed is the asymptotic speed of propagation of disturbances (also called spreading or invasion speed) if all of the real line corresponds to favorable habitat (see Aronson and Weinberger, 1978). At the same time c0 is the lowest speed for which for such a homogeneous favorable habitat, traveling wave solutions exist. The expression
(24)
has a factor with the dimension of length, but is otherwise in terms of dimensionless quantities. The function arctan takes here values in [0,π). Using the doubling formula
we can derive the alternative expression
(25)
, where now arctan takes values in . The corresponding expression in Potapov and Lewis (end of Section 4) has an extra factor √D in the denominator of the argument of arctan, but is otherwise identical. We claim that this factor should not be there.
In the limit of an extremely hostile environment outside of the favorable patch, i.e., in the limit , we obtain the much simpler expression
(26)
. Formula (26) is related to the classical critical domain size problem when c = 0 see Okubo and Levin (2001), Sections 9.1 and 10.2.2, and the references given therein. Indeed, the results there are recovered from ours by putting c = 0.
Note that here for the right-hand side of (26) decreases as a function of D, meaning that a species can survive on a smaller patch provided it increases its dispersal propensity, whereas for small c it has to decrease D. (See also Fig. 5.) The reader should not be misled by the notation when verifying this statement: c0 actually also depends on D. We can also rewrite the inequality L >Lcrit in the form
.
From this it easily follows that in order to see persistence of the species, the diffusion constant should be neither too small nor too large, depending on c. This is illustrated in Fig. 5 which shows L(D) achieving a minimum at a positive value of D, depending on c, for any c > 0. Figure 6 displays the curve for various values of r with the asymptotic value as given by formula (26) above. It further shows that for given L, r has to have a minimum size for persistence.
Additional information can be obtained by computing the shape of the moving profile. Under our assumptions, the profile is symmetric when the climate does not move. In particular, there is no shape difference between the north and the south tail which are both maintained by migration from the favorable into the unfavorable area. The movement of the climate introduces asymmetry. As far as the tails are concerned, this is reflected in (9), which in terms of the original parameters and variables, implies that the spatio-temporal features of the tails are described by the expressions
(27)
. By analogy with c0, the Fisher speed of invasion into the favorable habitat, we introduce a speed defined by the formula
(28)
. This speed can be thought of as representing the speed of retreat from the unfavorable region. Then we measure c in terms of by putting
(29)
and write (27) in the form
(30)
and conclude that the decay for positive x is faster than the decay for negative x by a factor
(31)
.
Numerical results (see Fig. 7) show that when c is increased, the point at which the population achieves its maximum density shifts toward the south boundary of the patch. However, due to a tracking lag, it becomes closer to the north boundary of the population profile. Clearly this “body” effect strongly reinforces the asymmetry exhibited by the tails. Note that the steepness of the north front will make it relatively easy to determine from population census data that a shift took place and that in contrast, it may be much harder to do so on the basis of a time series of observations in the south tail. This asymmetry is particularly visible on the two panels corresponding to the values c = 5 and c = 6.05 of Fig. 7. Parmesan et al. (1999) find exactly such a north-south asymmetry in a sample of 35 non-migratory European butterflies. They offer some speculations on possible causes. As explained above, our results provide a simple explanation on the basis of just the way in which the climate shift manifests itself in the (moving) population distribution. (See also the concluding remarks below.)
Another consequence of the asymmetry is that the range of the species may increase when the climate starts moving, when one defines the range as the spatial domain in which the population density exceeds a certain, somewhat arbitrarily chosen, lower bound (see Fig. 7). This phenomenon too derives from the relatively slow decay in the south tail. From Fig. 8, it is clear that the range keeps increasing until close to the critical speed, after which the range collapses fast. A distressing consequence of this threshold behavior is that a relatively small increase in climate speed can cause extinction with little advance warning.
The shape of the moving population profile is one aspect, total population size is another. A numerical “shooting” method to compute the total size of the “traveling” popu lation as a function of the parameters is the following. Solve, for positive values of the parameter Q, the initial value problem
(32)
up to , then Q is too high. If , then Q is too low. By using a bisection-type technique, one can find an approximate solution for Q to the equation . In the last step of this iterative procedure, one adds to (32) the equation
(33)
. The total population size then is given by
(34)
, where the three contributions correspond to, respectively, the left tail, the middle part, and the right tail (see also Fig. 9).
The (counter intuitive) conclusion is that an increase in c may lead to an increase of the total population size whenever the unsuitable area outside the favorable core area is not too harsh. This is due to a lag effect in the left tail: the decay of the population in the region that was favorable until recently may be slow while meanwhile, the rise of the population in the right region that just became favorable is relatively fast. This possibility of increases in both range and population size was not shown in the related work of Potapov and Lewis (2004).
In conclusion of this section, we formulate an insight deriving from (23): for small c, an increase of D entails an increase of the minimal interval length, since diffusion creates a net loss over the boundary of the favorable region. For larger c, however, the influence of D on the minimal interval length may be opposite, since increased mobility helps to track the moving climate.
3 3. The linearization at zero
In this section, we investigate formally the stability of the extinct state. We find that the principal eigenvalue switches sign exactly at the codimension one manifold in parameter space that separates the domain of existence of a nontrivial solution from the domain of nonexistence. In the next section, we shall see that the principal eigenvalue for a general equation characterizes the existence and nonexistence of nontrivial solutions and determines as well the large time dynamics of this model. Note that within the domain of non-existence this eigenvalue further yields information about the rate of decay to zero, i.e., the rate at which the population declines on its way to extinction.
Returning to the general problem (1) with f of the form (2), we note that the linearization at the trivial steady state u ≡ 0 is given by
(35)
. To investigate the stability of u ≡ 0, we focus on solutions of (35) of the particular form
(36)
. By substitution we deduce that such a solution exists if and only if ϕ is an eigenfunction corresponding to eigenvalue λ for the linear differential operator L defined by
(37)
.
In the next section, we will see that the sign of the principal (or dominant) eigenvalue of this operator, when properly defined, yields the long term dynamics in Eq. (1). Its sign gives a criterion for either extinction or persistence.
Therefore, methods to determine the sign of the dominant eigenvalue are of great interest, as are methods to give more quantitative estimates in case it is negative (at which time scale will the extinction happen?).
For the caricatural case of Section 2, we can derive an explicit equation for the dominant eigenvalue. The derivation follows the same pattern as the analysis leading to (22). In particular, we adopt the same scaling, which allows us to take D = 1 in (37) and
(38)
. Hence, the bounded solution of
(39)
, which is normalized by the condition
(40)
, is given by
(41)
, for ξ < 0 whenever the expression under the square root is positive. For 0 ≤ ξ ≤ L, on the other hand, the solution is represented by
(42)
, where k is a complex number and, by assumption, the expression under the square root is now negative. Finally, for ξ > L, we should have
(43)
. It remains to determine k and C from linking conditions that should guarantee that ϕ is continuously differentiable at both ξ = 0 and ξ = L. From the smoothness condition at ξ = 0, we deduce
(44)
. Eliminating C from the smoothness condition at ξ = L, we end up with one equation for the unknown λ. This equation is the analogue of (21). It reads
(45)
. As a consequence of the more general results in the next section, it can be shown, that the condition λ = 0 in (45) is equivalent to the critical length condition of the previous section.
Note that λ and c only occur in the combination . In terms of the unscaled time and parameters, this means that
(46)
In other words, the dominant eigenvalue is a quadratically decreasing function of c, with a coefficient of the quadratic term which is inversely proportional to D but independent of all other parameters. The relation (46) can be derived by a Liouville transformation which eliminates the first order derivative from the eigenvalue problem Lϕ = λϕ. So, it holds for general functions g(0, ξ), not just for (38).
4 4. Analysis of a general class of equations
So far, we have considered a particular type of heterogeneity, that which is obtained by juxtaposing two homogeneous media—the favorable and unfavorable ones—with an abrupt transition at the two end points of the favorable interval. Are the results which we have derived previously robust? And is the co-moving nontrivial solution stable if it exists? Here we give very strong affirmative answers to both these questions in a rather general setting. The motivation for considering general types of nonlinearities is twofold. First, the assumptions made in Section 2 are rather contrived from a modeling point of view and one would like to consider more complex transitions e.g. gradual transitions between recognizable but not necessarily strictly homogeneous zones. Second, a general mathematical theory sheds much more light on the underlying mechanisms, since the proofs reveal the role that various assumptions play in yielding the conclusions. Here, for instance, the linearization at the trivial steady state, in particular the sign of the associated principal eigenvalue, will be seen to fully account for the ability to keep pace with a shifting climate.
In this section, we consider Eq. (1). As was already mentioned, there is no loss in generality in assuming that D = 1, which we do henceforth.
The functions f and g (related through (2)) will be assumed to satisfy the following set of conditions.
1. (a)
Negative density dependence: ug(u, x) is decreasing for all x ∈ ℝ and strictly decreasing for xI0, where I0 is a nonempty open interval.
2. (b)
Allow for multiple discontinuities, e.g., several patches: there is a finite set of points F = {a1, … , ap} in ℝ such that g is continuous on ℝ+ × (ℝ\F) and both g(u, x) and g(u, x) exist, uniformly for u in compact subsets of ℝ+.
3. (c)
Existence of a linearization: There existsδ > 0 such that ug(u, x) is C1 on [0,δ] for all x ∈ ℝ, gu is continuous on [0,δ] × (ℝ∖F) and both gu(u, x) and gu(u, x) exist, uniformly for u ∈ [0,δ].
4. (d)
Unfavorable outer regions: g(0, x) → −1 as x → ±∞.
5. (e)
Saturation: There existsM > 0 such that g(u, x) ≤ 0 for all x ∈ ℝ whenever uM.
The properties formulated in (a)–(e) above are the standing hypotheses on the function g throughout this section. The last one means that everywhere the population declines when it exceeds some level M, i.e., negative density dependence guarantees that the population stays bounded. The limits at ±∞ in (d) are taken to be the same in order to simplify the formulation, but the statements and proofs can readily be adapted to the case of different limits. Note that the values of the limit can be changed by a scaling of the time variable t. Accordingly, the value −1 is representative for general negative values. Similarly it is no restriction that we take D = 1, as this can always be achieved by a scaling of the spatial variable x (after the scaling of time). Since g may have discontinuities with respect to x, we consider generalized solutions. These are functions u which, as a function of x, are globally of class C1 and piecewise of class C2 and satisfy the equation at each point with xai, i = 1, … , p.
For studies of solutions of (1) on bounded domains, without an imposed translation speed (i.e., c = 0), we refer to Murray and Sperb (1983), Cantrell and Cosner (1991, 1998, 2003), Cano-Casanova and López-Gómez (2003) and the references given there. Recently, the effect of a heterogeneous but spatially periodic environment has been studied by Berestycki et al. (2005a, 2005b). Lastly, periodic stochastic environments are considered by Roques and Stoica (2007).
The problem we study here involves a lack of compactness (the problem is set on the whole real line) as well as the difficulty deriving from the fact that c is imposed.
Our first aim is to give necessary and sufficient conditions for the existence of a traveling wave solution, that is, of a bounded solution w > 0 of (5). We shall find that such a solution exists if and only if the zero steady state of the equation
(47)
(which is just (1) rewritten in terms of a moving coordinate system) is linearly unstable in the sense that an associated dominant eigenvalue is positive. Next, we settle the uniqueness issue by showing that there is at most one traveling wave solution. Concerning the large time asymptotic behavior of solutions of the initial value problem for (1), we then formulate a dichotomy:
• If no traveling wave solution exists, every positive solution of (1) converges to zero for t → ∞, uniformly in x.
• If a traveling wave solution w exists, every nontrivial positive solution u(t, x) of (1) converges for t → ∞ to w(xct), uniformly in x.
4.1 4.1. A priori estimates for the “far out” asymptotic behavior of traveling waves
We will replace the symbol ξ by the symbol x in order to facilitate the reference to the literature. Thus, we write (5) with D = 1 as
(48)
. We start by analyzing the limiting behavior as x → ±∞ that any bounded positive solution necessarily has. Indeed, note that no conditions at infinity, other than being bounded are imposed here on solutions.
Recall that the quantities μ± are defined in (9) and that they are the roots of λ2 + − 1 = 0. It is tempting to conjecture that for some positive constants a, b, w(x) ∼ aeμ+x for x→ −∞ and w(x) ∼ beμ−x for x → ∞. This is indeed true if, for instance, g(0, x) = −1 for large |x| as was the case in Sections 2 and 3 above. (More general results in this direction can be found in Berestycki and Nirenberg, 1991.) But, in general, it is not so. Indeed, if g(0, x) converges slowly to −1 we do not, in general, obtain exact exponential behavior. For instance,
is a solution on ℝ+ of the equation
with
This observation motivates us to formulate that w(x) behaves like eμ+x for x→ − ∞ and like eμ−x for x→ +∞in a weaker sense that we now make precise.
Proposition 4.1.Let w be a bounded positive solution of (48). Then w(x)→ 0 for x → ±∞. In fact, for any ɛ > 0
(49)
and
(50)
Proof: We start by proving that w(x)→ 0 for x→ ∞. There exists R > 0 such that for all xR the inequality
holds for, say, ν = ½. Hence, (a) implies that for x > R,
and, by the maximum principle, it follows that for all a > 0, for x ∈ (R,R + a) the inequality
holds where ψa is defined by the boundary value problem
with
It should be noted here, also for future use, that even though f may be discontinuous, the maximum principle still applies to the C1 solutions that we consider. In the present one-dimensional setting, this can be verified rather directly. More general statements in Gilbarg and Trudinger (1998) also cover the multi-dimensional situation.
A direct computation shows that
where
(i.e., ρ± are the roots of r2 + cr −ν = 0 with ρ < 0 < ρ+). Since
we obtain, by taking the limit a→ ∞, the inequality
(51)
which shows that w(x)→ 0 for x→ ∞.
Similarly, one derives for some R the inequality
(52)
and concludes from this that w(x)→ 0 for x→ −∞.
Next, in order to derive the more precise description of the limiting behavior of w given by (49) and (50), we first observe that in the previous argument ν should be less than 1, but can otherwise be chosen as close to 1 as we wish: For all ν with 0 < ν < 1, there exists R = R(ν) such that g(0, x) ≤ −ν for xR. Since ρ± → ν± as ν ↑ 1, it follows from (50) and (51) that for allδ > 0 there exists R = R(δ) > 0 such that
Next, we want to derive lower bounds. We first formulate an auxiliary result that will also be used later.
Proposition 4.2.Let ν : ℝ → ℝ be positive, piecewise C2and such that for all but at most finitely many, x in an interval of the form (R,∞) the inequality
holds, where c and ν are such that} . Then there exists a positive constant K such that
.
If, similarly, the differential inequality holds for x in an interval of the form (−∞, −R) then there exists a positive constant K such that
.
We omit the (easy) proof, since it follows exactly the same line of argumentation that we used to prove the preceding proposition. Now let w be a bounded positive solution of (48). Then since w(x) → 0 for x → ±∞, for everyδ > 0, there exists a R = R(δ) such that
for xR and for x ≤ −R. Thus, as a corollary of Proposition 4.2, we obtain the following proposition.
Proposition 4.3.Let w be a bounded positive solution of (48). Then for any ɛ > 0, we have that
and
To conclude this subsection, we formulate an estimate for the derivative of w.
Proposition 4.4.Let w be a bounded positive solution of (48). For every ɛ > 0, there exists R = R(ɛ) > 0 such that
.
Proof: We restrict our attention to large positive x, the case of negative x being the same. From (48), we get
. This shows that the limit of wx(z) exists when z → ∞, hence that wx (∞) = 0. Therefore, letting y → ∞ in this relation yields the identity
. The result now follows from the properties of g and the estimates for w obtained in Proposition 4.1.
4.2 4.2. The eigenvalue problem
As another preparatory step, we shall make precise how, in the present case, the principal (or dominant) eigenvalue of the linearized problem at w ≡ 0 is defined. Since we consider solutions defined on the whole real line, some special care is needed.
Let LR denote the differential operator:
(53)
. Let λR denote the principal eigenvalue of LR associated with zero Dirichlet boundary conditions
(54)
. Then the corresponding eigenfunction ϕR is strictly positive on (−R, R). It is well known that R ↦ λR is increasing (see, for instance Berestycki et al., 1994 for a monotonicity proof in a more general framework). So, it is meaningful to formulate the following definition.
Definition 4.5.
(55)
.
There are alternative ways to define λ, see Berestycki et al. (1994), Pinsky (1995), or Berestycki et al. (2007) for a formula that applies to general operators in unbounded domains. As a special case of the results established in Berestycki et al. (1994), we obtain the characterization
Note that the ϕ that we consider here are allowed to grow beyond any bound for |x| → ∞. By restricting the “test” functions ϕ to those that are bounded on ℝ, we obtain a different generalized eigenvalue that may be smaller. For instance, if g(0, x)=−1 for all x, then while the additional requirement that the functions be bounded yields a generalized eigenvalue equal to 1; so when c ≠ = 0, these are not equal. We refer to Berestycki et al. (2007) for a general study of these themes.
In the current context, the motivation to call λ the principal eigenvalue derives from the results presented in the next subsection and in Section 4.5. In the proof, we shall need that a positive ϕWloc2,∞ exists such that
(56)
on ℝ\F. Such a ϕ is obtained as the limit, uniformly on compact subsets, of a sequence for some sequence Rj such that Rj → ∞ as j → ∞. The idea is to first normalize ϕR by requiring that for instance, ϕR(0) = 1. Next, we invoke the Harnack inequality (see Ladyženskaja et al., 1968, Section III.10, p. 209 or Krylov, 1987, Section 4.2, p. 130), stating that on a given bounded interval (−A, A) and for R sufficiently large, the maximum of ϕR is bounded by a constant (depending on A, but not on R) times the minimum of ϕR. Since the minimum of ϕR is bounded by 1, we obtain an R independent bound on the maximum of ϕR. The regularity theory of elliptic equations next guarantees that any sequence has a converging subsequence and that we can pass, for such a subsequence, to the limit in the differential equation.
We conclude this subsection by describing, in a crude manner, the “far out” asymptotic behavior of ϕ when λ is either negative or zero. We only state the properties that we shall use in the next subsection.
Proposition 4.6. Let ϕbe positive and satisfy (56) on ℝ\F with λ < 0. For every δ with max {0,−1−λ} < δ, there exist R(δ) and K(δ) such that
(57)
,
(58)
Proof: Again, we restrict our attention to large positive x. For anyδ > 0, the function ϕ satisfies for large enough x the differential inequality
So the inequality (57) follows from Proposition 4.2, provided the argument of the squareroot is positive. This requires thatδ > − 1 − λ and since we already required thatδ >0, we should restrict toδ > max {0, −1 − λ}.
Proposition 4.7.Let ϕbe positive and satisfy (56) on ℝ\F with λ = 0. Then ϕhas the properties formulated in terms of w as (49), (50), and in Proposition 4.4.
Sketch of the proof: The proof is essentially identical to the proofs of Propositions 4.1 and 4.4. Here, however, we do not a priori know that ϕ is bounded. The idea is to replace the ψa from the proof of Proposition 4.1 by functions zR which satisfy
where p is such that g(0, x) ≤ −ν for x > p and α := supR ϕR(p). One then combines the inequality ϕR(x) ≤ zR(x), for x ∈ (p, R), with the fact that for R→∞
4.3 4.3. The solvability condition
Theorem 4.8.Equation (48) has a bounded positive solution if and only if λ > 0.
Proof: Assume that λ > 0. We shall prove that a solution exists by constructing both a sub- and a supersolution. Recall that ϕR denotes the principal eigenvalue of ϕR defined by (53)–(54). Again, we denote by ϕR the associated positive eigenfunction, but this time we normalize by requiring that the maximum of ϕR equals one. Now define
Then for −R < x < R,
We claim that for R large enough and for ɛ > 0 small enough, υ is a subsolution, i.e., the right-hand side is positive. To substantiate the claim, we first note that λR > 0 for large R since λR → λ for R → ∞ and λ > 0. Next, observe that g(υ(x), x)−g(0, x) → 0 for ɛ ↓ 0. Lastly, it is known, that since ϕRR) = 0, extending ɛϕR by 0 outside (−R, R) yields a subsolution υ.
Assumption (e) guarantees that the constant function taking the value M is a supersolution. Clearly, υ(x) < M for small ɛ. We conclude that a solution exists.
It remains to verify the necessity of the condition λ 〉 0. We assume that a bounded positive solution w of (48) exists and that λ ≤ 0 and then try to reach a contradiction. We start by making the stronger assumption λ < 0. Let ϕ be positive and satisfy (56) on ℝ\F. We claim that
Indeed, this follows by combining (57) with (48) and (58) with (49), if we chooseδ in Proposition 4.6 such that not onlyδ > max{0, −1 −λ} but alsoδ < −λ. The point is that in this case so that by choosing next ɛ in Proposition 4.1 sufficiently small, the quotient ϕ(x)/w(x) has a positive exponent for large positive x and a negative exponent for large negative x.
Since ϕ∞ > 0, w > 0, and for large |x|, the function w is “dominated” by ϕ, the set
is nonempty. Let α0 be the infimum of this set. Then α0 > 0 and by continuity, α0ϕw on ℝ, i.e., α0 belongs to the set. Since ϕ(x)/w(x) → ∞ for |x| → ∞, there exists R > 0 such that αϕw for |x| ≥ R and ½α0 < α < α0. If min{α0ϕ(x)−w(x) : −RxR} would be positive, we arrive at a contradiction with the definition of α0. So, this minimum must be zero, i.e., the positive function υ := α0ϕw assumes its minimum value zero. Since
and the right-hand side of this identity is nonpositive, the strong maximum principle states that this is only possible if υ ≡ 0, which is clearly impossible. So, λ∞ < 0 precludes the existence of w.
Now assume that λ = 0. We rewrite the equation for ϕ in self-adjoint form as
The analogue form of the equation for w reads
If we multiply the equation for w by ϕ, the equation for ϕ by w, integrate by parts over [−A, A] and then subtract, we obtain the identity
Now g(0, x) − g(w(x), x) ≥ 0 for all x, but with strict inequality for xI0 (compare condition (4.1)). Hence, the right-hand side is strictly positive as soon as (−A, A) ∩ {I0 ≠ ≠ ∅. The estimates presented in Propositions 4.1, 4.4, and 4.7 guarantee that the left-hand side tends to zero for A. But the right-hand side is an increasing function of A which takes positive values, so is bounded away from zero for large A. We thus reached a contradiction and conclude that the existence of a positive bounded solution of (48) implies that λ > 0.
4.4 4.4. Uniqueness of traveling waves
Theorem 4.9. Equation (48) has at most one bounded positive solution.
Proof: The argument follows some ideas in Berestycki (1981). The new difficulty is that here we have to deal with an unbounded domain.
Assume there are two distinct solutions wi , i = 1, 2. Writing the differential equation in the form
and manipulating as in the end of the proof of Theorem 4.8 we obtain for any α, β with −∞ < α < ∞ < +∞, the identity
Now assume that {x : w2(x) > w1(x) is nonempty and let (a, b) be a connected component of this set, then w1(a) = w2(a) and w1(b) = w2(b), where it is understood that −∞ ≤ a < b ≤ +∞ and wi (±∞) = 0 (recall Proposition 4.1). Suppose first that a and b are finite and take α = a and β = b. Then since ug(u, x) is decreasing, the right-hand side is positive. In fact, it is strictly positive (because the only way in which it could be zero is that g(w1(x), x) = g(w2(x), x)) for almost all x ∈ (a, b), but then the wi’s satisfy one and the same linear equation, as well as the same boundary conditions, so w1w2 on (a, b)). On the other hand, we must have that wx2(a) > wx1(a) and wx2(b) < wx1(b), so the left-hand side is strictly negative which is a contradiction. If either β =+∞ or α = −∞ or both, we use the estimates of Propositions 4.1 and 4.4 to establish that the integral converges and that the corresponding terms at the left-hand side vanish in the limit β→ ∞ and/or α → −∞. So, then too we arrive at the contradiction that the right-hand side is strictly positive while the left-hand side is at most zero.
Corollary 4.10. Equation (48) has exactly one bounded positive solution if λ > 0 and no such solution if λ ≤ 0.
4.5 4.5. Large time behavior
We now return to the evolution Eq. (1) and investigate the asymptotic behavior (for large time) of solutions of the initial value problem obtained by supplementing (1) by the initial condition
(59)
where u0 is a given bounded nonnegative function defined on ℝ. Our assumptions on g guarantee that the initial value problem has a unique, globally defined, solution u = u(t, x).
Theorem 4.11. Let u be the solution of the Cauchy problem (1)–(59).
1. (i)
If λ ≤ 0, then u(t, x) → 0 for t → ∞, uniformly for x ∈ ℝ. That is, any population is bound to go extinct, no matter what the initial distribution is.
2. (ii)
If λ > 0 andu0is nontrivial, then u(t, x)w(xct) → 0 fort → ∞, uniformly for x ∈ ℝ. Here, w is the unique bounded positive solution of (48). So, any population is bound to persist by traveling along with the shifting climate.
Proof: Define υ(t, x) := u(t, x +ct) then (1) may be reformulated as
(60)
Let M′ > max{M, supxu0(x)} and let z = z(t, x) be the solution of
Since M′ is a supersolution of the elliptic operator at the right-hand side of the differential equation, we know that zt < 0 (see, for instance Sattinger, 1973, p. 33). Since z is bounded from below by zero, z(t, ·) must converge for t → ∞ to a nonnegative solution of (48). If λ ≤ 0, the only such solution is zero. So, in that case, z(t, x) converges to zero for t → ∞. Additional arguments (explained in detail below) lead to the conclusion that the convergence is uniform for x ∈ ℝ. Since 0 ≤ u(t, x) = v(t, x - ctz(t, x - ct), we conclude that if λ ∞ ≤ 0, u(t, x)→0 for t → ∞, uniformly for x ∈ ℝ.
It remains to prove (ii). If u0 is nontrivial, u(δ, x) is strictly positive forδ > 0, and hence so is υ(δ, x). So, for any given R > 0 and ɛ sufficiently small we have υ(δ, x) ≥ ɛϕR(x) for −RxR. Now assume that λ > 0. Recall from the proof of Theorem 4.8 that for R large enough, we obtain a subsolution if we extend ɛϕR(x) by zero outside the interval [−R, R]. Accordingly, z(t, ·) cannot converge to zero for t → ∞ when λ > 0 and, therefore, the limit must be the unique bounded positive solution w of (48), (compare Corollary 4.10). Likewise, the subsolution converges to w and since υ is sandwiched in between; it too must converge to w.
We now show that the convergence is uniform for x ∈ ℝ. Again, we concentrate on the supersolution z. Suppose z does not converge uniformly to w for t → ∞. This means thatδ >0 exists as well as sequences tj → ∞ and xj ∈ ℝ such that
By possibly restricting to a subsequence, we may assume that
where x is either finite, +∞ or −∞. Define
and note that for each j, zj is a decreasing function of t. Since zj is uniformly bounded, standard parabolic estimates guarantee that we can extract once more a subsequence, still denoted by zj , such that zj converges uniformly on compact subsets to a function z(t, x). Clearly z is a nonincreasing function of t and z(tj, 0)−w(x) ≥δ.
If x is finite, then z is a solution of
and so its limit for t → ∞ is a nontrivial solution of
By uniqueness, we must have
which, however, would imply that whereas taking the limit j → ∞ z(tj, 0) - w(x) ≥ δ in we deduce that . So, we ruled out the possibility that x is finite.
Next, assume that x =∞. Since g(z, x) ≤ g(0, x) and g(0, x) → −1 for x → ∞, we now deduce that z satisfies the inequality
and once more taking the limit t → ∞ that
But a function satisfying this inequality cannot have a positive maximum, and hence no nontrivial bounded positive solution can exist. Since we must have (note that w(xj) → 0 for j → ∞ when x = ∞), we conclude that x = ∞ is impossible as well. The possibility that x =−∞ is ruled out in exactly the same manner.
So, the assumption that z does not converge uniformly to w for t → ∞ leads to a contradiction and we conclude that the convergence is, in fact, uniform. The proof that the subsolution converges uniformly to w follows exactly the same pattern. Hence, the true solution υ, which lies in between, must converge uniformly to w and the proof of (ii) is completed.
Finally, we note that the proof that z(t, ·) converges uniformly to zero when λ ≤ 0 is based on precisely the same arguments as used above.
5 5. Concluding remarks
Mathematical studies of simplified models can yield ecological insights, and at the same time, shed light on basic mechanisms. In that spirit, we have analyzed the effect that a shifting climate may have on the persistence of a species. A patch of favorable habitat, surrounded by unfavorable habitat, is able to sustain a population provided the gain by reproduction can balance the losses due to mortality inside the patch and dispersal away from the patch. If the patch itself moves in space, an additional loss term is created, since individuals may be left behind. Dispersing individuals, on the other hand, may be fortunate enough to land where conditions are changing for the better. As a result, the critical size that a patch should have in order to sustain a population, does not only depend on reproduction, mortality and dispersal rates, but also on the speed with which the patch moves through space. In Section 2, we have derived an explicit expression in formulas (23), (24), and (25) for the dependence which produces valuable insights. In Section 4 we have rigorously established several mathematical properties for a large class of models.
Persistence in a moving patch is facilitated when the rate of climate change is low, the rate of population growth within the patch is high, and the climate outside the patch not too hostile (Fig. 5). Migration, however, is a double-edged sword. Both too much and too little dispersal can lead to extinction and the optimal dispersal rate increases with patch speed (Fig. 5). The results imply that a small latitudinal range diminishes the maximal rate of climate change a species should be able to track. This means that the conventional approach (see Skellam, 1951) of using the invasion (Fisher) speed as an estimate of this maximal rate can lead to a severe overestimation when ranges are small or D is large.
A moving climate can have dramatic effects on the size and form of the population profile. When the favorable region moves to the north, the population becomes more concentrated toward the north end of the population profile. Interestingly, if the habitat outside the favorable patch is not too hostile, the south tail becomes considerably thicker and longer as a result of the movement, since it takes a while before the marooned local population disappears. As a consequence, movement may result in increases in both the total population size and the population range (Figs. 7, 8 and 9).
In unpublished simulations of a metapopulation model, Nagelkerke (2004) obtained results similar to those reported here on our continuous population model. This demonstrates the structural robustness of our sometimes counterintuitive findings. For example, he modeled jump dispersal of propagules. This leads us to believe that our results are not restricted to movement by simple diffusion. Distance dispersal is relevant for many organisms.
Here, we have concentrated on the long time dynamics. Nagelkerke (2004) also studied the transient dynamics shortly after the climate starts to move. He found that generally the northern border initially moves faster than the southern border, both for surviving populations and for those that were doomed to go extinct. In the case of ultimate extinction, the southern border catches up after a while and then moves even faster than the climate, until it collides with the northern border. Note that another reason for not being too confident about an increasing range is the threshold behavior shown in Fig. 8. A small additional increase in climate speed can cause total collapse. The initial asymmetry between the velocities of both borders is in agreement with the outcome of an extensive analysis of butterfly data by Parmesan et al. (1999) that found more evidence for moving northern borders than for southern borders, suggesting that this is a transient phenomenon (see also Collingham et al., 1996). In addition, it could be easier to observe the move of the steep north front than that of the far less steep south back.
Our analysis was relatively simple, since we considered a one-dimensional spatial domain. Two-dimensional models give rise to new subtleties. Some mathematical issues involved in higher-dimensional versions of this problem will be discussed in Berestycki and Rossi (2008). Of particular interest is to understand the effect of the geometry on the ability to persist despite a climate change. For instance, a bottle-neck may occur when the extension of the patch in the lateral direction has a local minimum-giving rise to a narrow strait. Actually, one could mimic this effect in the one-dimensional setting by allowing the diffusion coefficient to depend on the spatial variable x; there would then be both an x and an x-ct dependence, making the problem inhomogeneous even modulo time translation. We plan to analyze such problems in further works.
Acknowledgement
Part of the research presented in this paper was carried out while Henri Berestycki was visiting the Department of Mathematics at the University of Chicago, which he thanks for its hospitality. Odo Diekmann thanks Frithjof Lutscher for bringing the work of Potapov and Lewis to his attention and A.B. Potapov for stimulating discussions during a Spatial Ecology meeting in Miami organized by Cantrell, Cosner, and DeAngelis. Lastly, the authors are indebted to Lionel Roques (INRA, Avignon, France), for providing the computations shown in Figs. 5, 6, 7 and 8.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
© The Author(s) 2008
Authors and Affiliations
• H. Berestycki
• 1
• O. Diekmann
• 2
• C. J. Nagelkerke
• 3
• P. A. Zegeling
• 2
1. 1.École des hautes études en sciences socialesCAMSParisFrance
2. 2.Department of MathematicsUtrecht UniversityUtrechtThe Netherlands
3. 3.Institute for Biodiversity and Ecosystem Dynamics (IBED)University of AmsterdamAmsterdamThe Netherlands | {"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": 0.9665971994400024, "perplexity": 667.052355673342}, "config": {"markdown_headings": false, "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/1474738661327.59/warc/CC-MAIN-20160924173741-00214-ip-10-143-35-109.ec2.internal.warc.gz"} |
http://slideplayer.com/slide/4308433/ | # Joint University Accelerator School
## Presentation on theme: "Joint University Accelerator School"— Presentation transcript:
Joint University Accelerator School
Non-linear effects Yannis PAPAPHILIPPOU Accelerator and Beam Physics group Beams Department CERN Joint University Accelerator School Archamps, FRANCE 22-24 January 2014
Bibliography Books on non-linear dynamical systems
M. Tabor, Chaos and Integrability in Nonlinear Dynamics, An Introduction, Willey, 1989. A.J Lichtenberg and M.A. Lieberman, Regular and Chaotic Dynamics, 2nd edition, Springer 1992. Books on beam dynamics E. Forest, Beam Dynamics - A New Attitude and Framework, Harwood Academic Publishers, 1998. H. Wiedemann, Particle accelerator physics, 3rd edition, Springer 2007. Lectures on non-linear beam dynamics A. Chao, Advanced topics in Accelerator Physics, USPAS, 2000. A. Wolski, Lectures on Non-linear dynamics in accelerators, Cockroft Institute 2008. W. Herr, Lectures on Mathematical and Numerical Methods for Non-linear Beam Dynamics in Rings, CAS 2013. L. Nadolski, Lectures on Non-linear beam dynamics, Master NPAC, LAL, Orsay 2013.
Contents of the 1st lecture
Accelerator performance parameters and non-linear effects Linear and non-linear oscillators Integral and frequency of motion The pendulum Damped harmonic oscillator Phase space dynamics Fixed point analysis Non-autonomous systems Driven (damped) harmonic oscillator, resonance conditions Linear equations with periodic coefficients – Hill’s equations Floquet solutions and normalized coordinates Perturbation theory Non-linear oscillator Perturbation by periodic function – single dipole perturbation Application to single multipole – resonance conditions Examples: single quadrupole, sextupole, octupole perturbation General multi-pole perturbation– example: linear coupling Resonance conditions and working point choice Summary Appendix I: Multipole expansion
Contents of the 1st lecture
Accelerator performance parameters and non-linear effects Linear and non-linear oscillators Integral and frequency of motion The pendulum Damped harmonic oscillator Phase space dynamics Fixed point analysis Non-autonomous systems Driven (damped) harmonic oscillator, resonance conditions Linear equations with periodic coefficients – Hill’s equations Floquet solutions and normalized coordinates Perturbation theory Non-linear oscillator Perturbation by periodic function – single dipole perturbation Application to single multipole – resonance conditions Examples: single quadrupole, sextupole, octupole perturbation General multi-pole perturbation– example: linear coupling Resonance conditions and working point choice Summary Appendix I: Multipole expansion
Accelerator performance parameters
Colliders Luminosity (i.e. rate of particle production) Νb bunch population kb number of bunches γ relativistic reduced energy εn normalized emittance β* “betatron” amplitude function at collision point High intensity accelerators Average beam power mean current intensity Ε energy fN repetition rate Ν number of particles/pulse Synchrotron light sources (low emittance rings) Brightness (photon density in phase space) Νp number of photons εx,,y transverse emittances I Non-linear effects limit performance of particle accelerators but impact also design cost
Non-linear effects in colliders
Limitations affecting (integrated) luminosity Particle losses causing Reduced lifetime Radio-activation (super-conducting magnet quench) Reduced machine availability Emittance blow-up Reduced number of bunches (either due to electron cloud or long-range beam-beam) Increased crossing angle Reduced intensity Cost issues Number of magnet correctors and families (power convertors) Magnetic field and alignment tolerances At injection Non-linear magnets (sextupoles, octupoles) Magnet imperfections and misalignments Power supply ripple Ground motion (for e+/e-) Electron (Ion) cloud At collision Insertion Quadrupoles Magnets in experimental areas (solenoids, dipoles) Beam-beam effect (head on and long range)
Non-linear effects in high-intensity accelerators
Limitations affecting beam power Particle losses causing Reduced intensity Radio-activation (hands-on maintenance) Reduced machine availability Emittance blow-up which can lead to particle loss Cost issues Number of magnet correctors and families (power convertors) Magnetic field and alignment tolerances Design of the collimation system Non-linear magnets (sextupoles, octupoles) Magnet imperfections and misalignments Injection chicane Magnet fringe fields Space-charge effect
Non-linear effects in low emittance rings
Limitations affecting beam brightness Reduced injection efficiency Particle losses causing Reduced lifetime Reduced machine availability Emittance blow-up which can lead to particle loss Cost issues Number of magnet correctors and families (power convertors) Magnetic field and alignment tolerances Chromaticity sextupoles Magnet imperfections and misalignments Insertion devices (wigglers, undulators) Injection elements Ground motion Magnet fringe fields Space-charge effect (in the vertical plane for damping rings) Electron cloud (Ion) effects
Contents of the 1st lecture
Accelerator performance parameters and non-linear effects Linear and non-linear oscillators Integral and frequency of motion The pendulum Damped harmonic oscillator Phase space dynamics Fixed point analysis Non-autonomous systems Driven (damped) harmonic oscillator, resonance conditions Linear equations with periodic coefficients – Hill’s equations Floquet solutions and normalized coordinates Perturbation theory Non-linear oscillator Perturbation by periodic function – single dipole perturbation Application to single multipole – resonance conditions Examples: single quadrupole, sextupole, octupole perturbation General multi-pole perturbation– example: linear coupling Resonance conditions and working point choice Summary Appendix I: Multipole expansion
Reminder: Harmonic oscillator
Described by the differential equation: The solution obtained by the substitution and the solutions of the characteristic polynomial are which yields the general solution The amplitude and phase depend on the initial conditions Note that a negative sign in the differential equation provides a solution described by an hyperbolic sine function Note also that for no restoring force , the motion is unbounded
Integral of motion Rewrite the differential equation of the harmonic oscillator as a pair of coupled 1st order equations which can be combined to provide or with an integral of motion identified as the mechanical energy of the system Solving the previous equation for , the system can be reduced to a first order equation
The last equation can be be solved as an explicit integral or “quadrature” , yielding or the well-known solution Note: Although the previous route may seem complicated, it becomes more natural when non-linear terms appear, where a substitution of the type is not applicable The ability to integrate a differential equation is not just a nice mathematical feature, but deeply characterizes the dynamical behavior of the system described by the equation
Frequency of motion The period of the harmonic oscillator is calculated through the previous integral after integration between two extrema (when the velocity vanishes), i.e : The frequency (or the period) of linear systems is independent of the integral of motion (energy) Note that this is not true for non-linear systems, e.g. for an oscillator with a non-linear restoring force The integral of motion is and the integration yields This means that the period (frequency) depends on the integral of motion (energy) i.e. the maximum “amplitude”
The pendulum An important non-linear equation which can be integrated is the one of the pendulum, for a string of length L and gravitational constant g For small displacements it reduces to an harmonic oscillator with frequency The integral of motion (scaled energy) is and the quadrature is written as assuming that for
Solution for the pendulum
Using the substitutions with , the integral is and can be solved using Jacobi elliptic functions: For recovering the period, the integration is performed between the two extrema, i.e and , corresponding to and , for which i.e. the complete elliptic integral multiplied by four times the period of the harmonic oscillator
Damped harmonic oscillator I
is the ratio between the stored and lost energy per cycle with the damping ratio is the eigen-frequency of the harmonic oscillator General solution can be found by the same ansatz leading to an auxiliary 2nd order equation with solutions
Damped harmonic oscillator II
Three cases can be distinguished Overdamping ( real, i.e or ): The system exponentially decays to equilibrium (slower for larger damping ratio values) Critical damping (ζ = 1): The system returns to equilibrium as quickly as possible without oscillating. Underdamping ( complex, i.e or ): The system oscillates with the amplitude gradually decreasing to zero, with a slightly different frequency than the harmonic one: Note that there is no integral of motion, in that case, as the energy is not conserved (dissipative system)
Contents of the 1st lecture
Accelerator performance parameters and non-linear effects Linear and non-linear oscillators Integral and frequency of motion The pendulum Damped harmonic oscillator Phase space dynamics Fixed point analysis Non-autonomous systems Driven (damped) harmonic oscillator, resonance conditions Linear equations with periodic coefficients – Hill’s equations Floquet solutions and normalized coordinates Perturbation theory Non-linear oscillator Perturbation by periodic function – single dipole perturbation Application to single multipole – resonance conditions Examples: single quadrupole, sextupole, octupole perturbation General multi-pole perturbation– example: linear coupling Resonance conditions and working point choice Summary Appendix I: Multipole expansion
Phase space dynamics Valuable description when examining trajectories in phase space Existence of integral of motion imposes geometrical constraints on phase flow For the harmonic oscillator (left), phase space curves are ellipses around the equilibrium point parameterized by the integral of motion (energy) By simply changing the sign of the potential in the harmonic oscillator (right), the phase trajectories become hyperbolas, symmetric around the equilibrium point where two straight lines cross, moving towards and away from it For the damped harmonic oscillator (above), the phase space trajectories are spiraling towards equilibrium with a rate depending on the damping coefficient
Non-linear oscillators
Conservative non-linear oscillators have quadrature with potential being a general (polynomial) function of positions The equations of motion are Equilibrium points are associated with extrema of the potential Considering three non-linear oscillators Quartic potential (left): two minima and one maximum Cubic potential (center): one minimum and one maximum Pendulum (right): periodic minima and maxima
Fixed point analysis Consider a general second order system
Equilibrium or “fixed” points are determinant for topology of trajectories at their vicinity The linearized equations of motion at their vicinity are Fixed point nature is revealed by eigenvalues of , i.e. solutions of the characteristic polynomial Jacobian matrix
Fixed point for conservative systems
For conservative systems of two dimensions the second order characteristic polynomial has too solutions: Two complex eigenvalues with opposite sign, corresponding to elliptic fixed points. Phase space flow is described by ellipses, with particles evolving clockwise or anti-clockwise Two real eigenvalues with opposite sign, corresponding to hyperbolic (or saddle) fixed points. Flow described by two lines (or manifolds), incoming (stable) and outcoming (unstable) hyperbolic elliptic
Pendulum fixed point analysis
The “fixed” points for a pendulum can be found at The Jacobian matrix is The eigenvalues are Two cases can be distinguished: , for which corresponding to elliptic fixed points , for which corresponding to hyperbolic fixed points The separatrix are the stable and unstable manifolds passing through the hyperbolic points, separating bounded librations and unbounded rotations elliptic hyperbolic
Contents of the 1st lecture
Accelerator performance parameters and non-linear effects Linear and non-linear oscillators Integral and frequency of motion The pendulum Damped harmonic oscillator Phase space dynamics Fixed point analysis Non-autonomous systems Driven (damped) harmonic oscillator, resonance conditions Linear equations with periodic coefficients – Hill’s equations Floquet solutions and normalized coordinates Perturbation theory Non-linear oscillator Perturbation by periodic function – single dipole perturbation Application to single multipole – resonance conditions Examples: single quadrupole, sextupole, octupole perturbation General multi-pole perturbation– example: linear coupling Resonance conditions and working point choice Summary Appendix I: Multipole expansion
Non-autonomous systems
Consider a linear system with explicit dependence in time Time now is not an independent variable but can be considered as an extra dimension leading to a completely new type of behavior Consider two independent solutions of the homogeneous equation and The general solution is a sum of the homogeneous solutions and a particular solution, , where the coefficients are computed as with the Wronskian of the system Tabor 33-34
Driven harmonic oscillator
Consider periodic force pumping energy into the system General solution is a combination of the homogeneous and a particular solution found as Obviously a resonance condition appears when driving frequency hits the oscillator eigen-frequency. In the limit of the solution becomes The 2nd secular term implies unbounded growth of amplitude at resonance Tabor 35
Damped oscillator with periodic driving
Consider periodic force pumping energy into the system The solution of the homogeneous system is The particular solution is The homogeneous solution vanishes for , leaving only the particular one, for which there is an amplitude maximum for but no divergence In that case the energy pumped into the system compensate the friction, a steady state representing a limit cycle Tabor 36
Phase space for non-autonomous systems
Plotting the evolution of the driven oscillator in provides trajectories that intersect each other (top) The phase space has time as an extra 3rd dimension By rescaling the time to become and considering every integer interval of the new time variable, the phase space looks like the one of the harmonic oscillator (middle) This is the simplest version of a (Poincaré) surface of section, which is useful for studying geometrically phase space of multi-dimensional systems The fixed point in the surface of section is now a periodic orbit (bottom) defined by In that case, one can show the existence of two integrals of motion, but when a non-linearity is introduced, the system becomes non-integrable
Contents of the 1st lecture
Accelerator performance parameters and non-linear effects Linear and non-linear oscillators Integral and frequency of motion The pendulum Damped harmonic oscillator Phase space dynamics Fixed point analysis Non-autonomous systems Driven (damped) harmonic oscillator, resonance conditions Linear equations with periodic coefficients – Hill’s equations Floquet solutions and normalized coordinates Perturbation theory Non-linear oscillator Perturbation by periodic function – single dipole perturbation Application to single multipole – resonance conditions Examples: single quadrupole, sextupole, octupole perturbation General multi-pole perturbation– example: linear coupling Resonance conditions and working point choice Summary Appendix I: Multipole expansion
Linear equation with periodic coefficients
George Hill A very important class of equations especially for beam dynamics (but also solid state physics) are linear equations with periodic coefficients with a periodic function of time These are called Hill’s equations and can be thought as equations of harmonic oscillator with time dependent (periodic) frequency There are two solutions that can be written as with periodic but also with a constant which implies that is periodic The solutions are derived based on Floquet theory
Amplitude, phase and invariant
Differentiating the solutions twice and substituting to Hill’s equation, the following two equations are obtained The 2nd one can be integrated to give , i.e. the relation between the “phase” and the amplitude Substituting this to the 1st equation, the amplitude equation is derived (or the beta function in accelerator jargon) By evaluating the quadratic sum of the solution and its derivative an invariant can be constructed, with the form
Normalized coordinates
Recall the Floquet solutions for betatron motion Introduce new variables In matrix form Hill’s equation becomes System becomes harmonic oscillator with frequency or Floquet transformation transforms phase space in circles
Perturbation of Hill’s equations
Hill’s equations in normalized coordinates with harmonic perturbation, using and where the F is the Lorentz force from perturbing fields Linear magnet imperfections: deviation from the design dipole and quadrupole fields due to powering and alignment errors Time varying fields: feedback systems (damper) and wake fields due to collective effects (wall currents) Non-linear magnets: sextupole magnets for chromaticity correction and octupole magnets for Landau damping Beam-beam interactions: strongly non-linear field Space charge effects: very important for high intensity beams non-linear magnetic field imperfections: particularly difficult to control for super conducting magnets where the field quality is entirely determined by the coil winding accuracy
Contents of the 1st lecture
Accelerator performance parameters and non-linear effects Linear and non-linear oscillators Integral and frequency of motion The pendulum Damped harmonic oscillator Phase space dynamics Fixed point analysis Non-autonomous systems Driven (damped) harmonic oscillator, resonance conditions Linear equations with periodic coefficients – Hill’s equations Floquet solutions and normalized coordinates Perturbation theory Non-linear oscillator Perturbation by periodic function – single dipole perturbation Application to single multipole – resonance conditions Examples: single quadrupole, sextupole, octupole perturbation General multi-pole perturbation– example: linear coupling Resonance conditions and working point choice Summary Appendix I: Multipole expansion
Perturbation theory Completely integrable systems are exceptional
For understanding dynamics of general non-linear systems composed of a part whose solution is known and a part parameterized by a small constant , perturbation theory is employed The general idea is to expand the solution in a power series and compute recursively the corrections hoping that a few terms will be sufficient to find an accurate representation of the general solution This may not be true for all times, i.e. facing series convergence problems In addition, any series expansion breaks in the vicinity of a resonance Tabor 90-96
Perturbation of non-linear oscillator
Consider a non-linear harmonic oscillator, This is just the pendulum expanded to 3rd order in Note that is a dimensionless measure of smallness, which may represent a scaling factor of (e.g without loss of generality) Expanding and separating the equations with equal power in : Order 0: Order 1: The 2nd equation has a particular solution with two terms. A well behaved one and the first part of which grows linearly with time (secular term) But this cannot be true, the pendulum does not present such behavior. What did it go wrong? Tabor 90-96
Perturbation of non-linear oscillator
It was already shown that the pendulum has an amplitude dependent frequency, so the frequency has to be developed as well (Poincaré-Linstead method): Assume that the solution is a periodic function of which becomes the new independent variable. The equation at zero order gives the solution and at leading perturbation order becomes The last term has to be zero, if not it gives secular terms, thus which reveals the decrease of the frequency with the oscillation amplitude Finally, the solution is the leading order correction due to the non-linear term
Perturbation by periodic function
In beam dynamics, perturbing fields are periodic functions The problem to solve is a generalization of the driven harmonic oscillator, with a general periodic function , with frequency The right side can be Fourier analyzed: The homogeneous solution is The particular solution can be found by considering that has the same form as : By substituting we find the following relation for the Fourier coefficients of the particular solution There is a resonance condition for infinite number of frequencies satisfying Tabor 35
Perturbation by single dipole
Hill’s equations in normalized coordinates with single dipole perturbation: The dipole perturbation is periodic, so it can be expanded in a Fourier series Note, as before that a periodic kick introduces infinite number of integer driving frequencies The resonance condition occurs when i.e. integer tunes should be avoided (remember orbit distortion due to single dipole kick)
Perturbation by single multi-pole
For a generalized multi-pole perturbation, Hill’s equation is: As before, the multipole coefficient can be expanded in Fourier series Following the perturbation steps, the zero-order solution is given by the homogeneous equation Then the position can be expressed as with The first order solution is written as
Resonances for single multi-pole
Following the discussion on the periodic perturbation, the solution can be found by setting the leading order solution to be periodic with the same frequency as the right hand side Equating terms of equal exponential powers, the Fourier amplitudes are found to satisfy the relationship This provides the resonance condition or which means that there are resonant frequencies for and “infinite” number of rationals
Tune-shift for single multi-pole
Note that for even multi-poles and or , there is a Fourier coefficient , which is independent of and represents the average value of the periodic perturbation The perturbing term in the r.h.s. is which can be obtained for (it is indeed an integer only for even multi-poles) Following the approach of the perturbed non-linear harmonic oscillator, this term will be secular unless a perturbation in the frequency is considered, thereby resulting to a tune-shift equal to with This tune-shift is amplitude dependent for
Consider single quadrupole kick in one normalized plane: The quadrupole perturbation can be expanded in a Fourier series Following the perturbation approach, the 1st order equation becomes with For , the resonance conditions are i.e. integer and half-integer tunes should be avoided For , the condition corresponds to a non-vanishing average value , which can be absorbed in the left-hand side providing a tune-shift: or
Single Sextupole Perturbation
Consider a localized sextupole perturbation in the horizontal plane After replacing the perturbation by its Fourier transform and inserting the unperturbed solution to the right hand side with Resonance conditions: Note that there is not a tune-spread associated. This is only true for small perturbations (first order perturbation treatment) Although perturbation treatment can provide approximations for evolution of motion, there is no exact solution 3rd integer integer Show also octupole…
General multi-pole perturbation
Equations of motion including any multi-pole error term, in both planes Expanding perturbation coefficient in Fourier series and inserting the solution of the unperturbed system on the rhs gives the following series: The equation of motion becomes In principle, same perturbation steps can be followed for getting an approximate solution in both planes
Example: Linear Coupling
For a localized skew quadrupole we have Expanding perturbation coefficient in Fourier series and inserting the solution of the unperturbed system gives the following equation: with The coupling resonance are found for Linear sum resonance Linear difference resonance
General resonance conditions
The general resonance conditions is or , with order The same condition can be obtained in the vertical plane For all the polynomial field terms of a pole, the main excited resonances satisfy the condition but there are also sub-resonances for which For normal (erect) multi-poles, the main resonances are whereas for skew multi-poles If perturbation is large, all resonances can be potentially excited The resonance conditions form lines in the frequency space and fill it up as the order grows (the rational numbers form a dense set inside the real numbers)
Systematic and random resonances
If lattice is made out of identical cells, and the perturbation follows the same periodicity, resulting in a reduction of the resonance conditions to the ones satisfying These are called systematic resonances Practically, any (linear) lattice perturbation breaks super-periodicity and any random resonance can be excited Careful choice of the working point is necessary
Contents of the 1st lecture
Accelerator performance parameters and non-linear effects Linear and non-linear oscillators Integral and frequency of motion The pendulum Damped harmonic oscillator Phase space dynamics Fixed point analysis Non-autonomous systems Driven (damped) harmonic oscillator, resonance conditions Linear equations with periodic coefficients – Hill’s equations Floquet solutions and normalized coordinates Perturbation theory Non-linear oscillator Perturbation by periodic function – single dipole perturbation Application to single multipole – resonance conditions Examples: single quadrupole, sextupole, octupole perturbation General multi-pole perturbation– example: linear coupling Resonance conditions and working point choice Summary Appendix I: Multipole expansion
Summary Accelerator performance depends heavily on the understanding and control of non-linear effects The ability to integrate differential equations has a deep impact to the dynamics of the system Phase space is the natural space to study this dynamics Perturbation theory helps integrate iteratively differential equations and reveals appearance of resonances Periodic perturbations drive infinite number of resonances There is an amplitude dependent tune-shift at 1st order for even multi-poles Periodicity of the lattice very important for reducing number of lines excited at first order
Magnetic multipole expansion
From Gauss law of magnetostatics, a vector potential exist Assuming transverse 2D field, vector potential has only one component As. The Ampere’s law in vacuum (inside the beam pipe) Using the previous equations, the relations between field components and potentials are x y iron rc i.e. Riemann conditions of an analytic function Exists complex potential of with power series expansion convergent in a circle with radius (distance from iron yoke)
Multipole expansion II
From the complex potential we can derive the fields Setting Define normalized coefficients on a reference radius r0, 10-4 of the main field to get Note: is the US convention | {"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": 0.9468271136283875, "perplexity": 1438.9341559172524}, "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/1521257649627.3/warc/CC-MAIN-20180324015136-20180324035136-00699.warc.gz"} |
http://internetdo.com/2023/01/solving-lesson-6-page-27-math-workbook-10-kite/ | Solving Lesson 6 Page 27 Math Workbook 10 – Kite>
Topic
a) Round the number 865 549 to hundreds. How accurate is the approximate number received?
b) Round the number -0.526 to the nearest hundredth. How accurate is the approximate number received?
Solution method – See details
Round numbers and calculate d
We say $$a$$ is an approximation of $$\overline a$$ with precision $$d$$ if $${\Delta _a} = \left| {a – \overline a } \right| \le d$$
Detailed explanation
a) The rounded number of 865 549 to hundreds is 865 500
We have: $${\Delta _a} = \left| {865{\rm{ }}500 – 865{\rm{ }}549} \right| = 49 \le 50 = d$$
b) The rounded number of -0.526 to the hundredths is -0.53
We have: $${\Delta _a} = \left| { – 0.53 – \left( { – 0.526} \right)} \right| = 0.004 \le 0.005 = 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": 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": 0.8828188180923462, "perplexity": 2440.633191665246}, "config": {"markdown_headings": false, "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-2023-06/segments/1674764499524.28/warc/CC-MAIN-20230128054815-20230128084815-00520.warc.gz"} |
https://www.physicsforums.com/threads/two-concentric-conducting-spherical-shells-find-outer-q.826101/ | # Homework Help: Two concentric conducting spherical shells, find outer Q
1. Aug 3, 2015
### mattz205
1. The problem statement, all variables and given/known data
Two concentric conducting spherical shells produce a radially outward electric field of magnitude 49,000 N/C at a point 4.10 m from the center of the shells. The outer surface of the larger shell has a radius of 3.75 m. If the inner shell contains an excess charge of -5.30 μC, find the amount of charge on the outer surface of the larger shell. (k
2. Relevant equations
closed surface int(E dot dA) = Q encl/epsilon0 (gauss's law)
3. The attempt at a solution
I plugged in the given electric field into gauss's law in the form E=Q encl/ 4pir^2 epsilon0, with Q encl= q(inner)+ q(outer) and solved for q(outer) and arrived at ~ 96.9 μC but that is wrong according to MP (my homework site)
any help would be greatly appreciated
Last edited: Aug 3, 2015
2. Aug 3, 2015
### SammyS
Staff Emeritus
First of all it's not a magnetic field (I suppose that was a typo).
How much charge is there on the inner surface of the outer larger shell?
3. Aug 3, 2015
### mattz205
Yes, sorry about that i meant electric field, and wouldn't the charge on the inner surface of the outer shell just be the negative of the outer? If that's not it then I'm not entirely sure.
4. Aug 4, 2015
### Qwertywerty
No , not necessarily . This would be true if the outer shell itself had no excess charge .
I couldn't understand your solution - have you equated electric field to kq/r2 (where q is charge on the outer surface of the outer sphere ) ?
5. Aug 4, 2015
### SammyS
Staff Emeritus
The charge on the inner surface of the outer shell would just be the negative of the outer shell only if the outer shell has a net charge of zero. That's not stated anywhere. Besides if that was true, then the total charge inside the Gaussian surface (the closed surface you mention) would have to be -5.30 μC .
Consider a Gaussian surface totally embedded in the conducting material of the large sphere. What's the total charge enclosed by this surface ?
6. Aug 4, 2015
### mattz205
the electric field inside a conductor is always zero according to my lecture... and the charge enclosed there would be the -5.3 + whatever the inner surface of the conductor is, which i dont know, nor do i know how to find it...
7. Aug 4, 2015
### SammyS
Staff Emeritus
If the electric field is zero, then the flux through that Gaussian surface is zero. That tells you total charge enclosed is ___ .
8. Aug 4, 2015
### mattz205
0? would that make the charge on the inside of the conductor +5.3?
9. Aug 4, 2015
### Pyrus
I tried this by using E=V/r. I found potential at point at distance 4.1 m.
V=Er => V=20900 V
Then applying V= 1/(4πε) * {q/4.1 + Q/4.1}
V=(9*10^9 )[-5.3/4.1 + Q/4.1]*10^-6
Solving it I also got answer 96.6μC
10. Aug 4, 2015
### mattz205
i tried that method too and got the same answer as i did using the other method, the answer is wrong regardless though
11. Aug 4, 2015
### mattz205
ok, ok, i got it, thank you very much for your help!
12. Aug 4, 2015
### Pyrus
So simple and I got it too complex....
13. Aug 4, 2015
### SammyS
Staff Emeritus
Yes.
Notice that your final answer is independent of the amount of charge on the inner sphere. The conductor effectively shields the region outside of the sphere from the region inside the sphere.
14. Aug 4, 2015
### mattz205
That makes sense, in lecture my instructor wasn't very clear about how conductors worked, but it makes sense now. Thnk you again for your help, i greatly appreciate 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": 0.9744697213172913, "perplexity": 1154.7124332880353}, "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/1526794867094.5/warc/CC-MAIN-20180525141236-20180525161236-00251.warc.gz"} |
http://mathoverflow.net/questions/106612/concavity-of-spectral-mean?sort=votes | # Concavity of Spectral mean
The geometric mean of two positive definite matrices $A, B$ is defined by $A\sharp B=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}$. The following inequality holds true $$\left(\sum_{i=1}^n A_i\right)\sharp \left(\sum_{i=1}^n B_i\right)\ge\sum_{i=1}^n A_i\sharp B_i$$ for positive definite matrices $A_i, B_i$, $i=1\ldots, n$.
The spectral mean is defined by Fiedler and Ptak as $A\natural B=(A^{-1}\sharp B)^{1/2}A(A^{-1}\sharp B)^{1/2}$. Is the spectral mean also concave? That is, whether
$$\left(\sum_{i=1}^n A_i\right)\natural \left(\sum_{i=1}^n B_i\right)\ge\sum_{i=1}^n A_i\natural B_i$$ for positive definite matrices $A_i, B_i$, $i=1\ldots, n$?
The inequality here is Loewner order.
-
I'd consider changing the notation. The two symbols are too easily confused. – Felix Goldberg Sep 7 '12 at 16:02
These notation is standard, see e.g. Volume 1, Number 3 (2007), 443–447 of Journal of Mathematical Inequalities. – Betrand Sep 7 '12 at 16:08
I know it's standard; I just think it's no good... :( – Felix Goldberg Sep 7 '12 at 18:44
@Suvrit, I deleted that confusing words. – Betrand Sep 7 '12 at 19:09
Some experiments reveal that the said inequality does not hold for the spectral mean.
Here is a random (i.e., mindless) counterexample:
\begin{equation*} A_1=\begin{bmatrix} 29 & 15\\\\ 15 & 26 \end{bmatrix},\quad A_2=\begin{bmatrix} 5 & 0\\\\ 0 & 5 \end{bmatrix},\quad B_1=\begin{bmatrix} 4 & -16\\\\ -16 & 113 \end{bmatrix},\quad B_2=\begin{bmatrix} 18 & -12\\\\ -12 & 16 \end{bmatrix}. \end{equation*}
With this choice, we have
\begin{equation*} (A_1+A_2)\natural (B_1+B_2) = \begin{bmatrix} 22.3606 & 0.4475\\\\ 0.4475 & 58.3661 \end{bmatrix},\quad (A_1\natural B_1)+(A_2\natural B_2) =\begin{bmatrix} 15.2404 & -2.2975\\\\ -2.2975 & 58.5164 \end{bmatrix}. \end{equation*}
A quick calculation shows that the difference between the two matrices is an indefinite matrix, so the alleged inequality does not hold.
-
@Betrand: For the above example, I used trivial matlab code using eigendecomposition. But for more accurate evaluation, or larger matrices, I tend to use Bruno's matrix means toolbox: bezout.dm.unipi.it/software/mmtoolbox – Suvrit Sep 8 '12 at 10:14
Many thanks for the link. – Betrand Sep 8 '12 at 16:22 | {"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": 0.9672226309776306, "perplexity": 2737.2680938575}, "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-2015-14/segments/1427131298692.48/warc/CC-MAIN-20150323172138-00172-ip-10-168-14-71.ec2.internal.warc.gz"} |
http://artent.net/2013/03/page/2/ | # March 2013
You are currently browsing the monthly archive for March 2013.
## “Machine Learning Cheat Sheet (for scikit-learn)”
I love this diagram created by Peekaboo: Andy’s Computer Vision and Machine Learning Blog.
## “An Empirical Evaluation of Thompson Sampling”
In “An Empirical Evaluation of Thompson Sampling”, Chapelle and Lihong Li (NIPS 2011) compare Upper Confidence Bound (UCB), Gittins, and Thompson methods for the multi-armed bandit problem. More theoretical analysis has been done for UCB and Gittins, but Thompson sampling is simple (as opposed to Gittins) and seems to work well, sometimes performing significantly better that the other common algorithms. Both synthetic and real world (advertising) results are presented.
## “Exact Sparse Recovery with L0 Projections”
Thanks to Nuit Blanche for pointing me towards the paper “Exact Sparse Recovery with L0 Projections” by Ping Li and Cun-Hui Zhang (2013). The paper describes an algorithm (Min+Gap(3)), which perfectly recovers $x\in R^N$ from $M_0$ random measurements $y_i\in R$. The vector $y$ is generated by
$y = S x$
where $S\in R^{N\times M},\ ||x||_0 = K,\ K\ll N,\ s_{ij} \sim S(\alpha, 1)$, and $S(\alpha, 1)$ denotes an $\alpha$-stable distribution with unit scale. Typically the number of measurements required for exact recovery is “about $5 K$”. More specifically,
“$M_0 = K \log ((N − K)/\delta)$ measurements are sufficient for detecting all zeros with at least probability $1 − \delta$.”
Li and Zhang illustrate the effectiveness of their algorithm by applying it to a simple video stream exactly recovering the video from measurements constructed from the differences between frames. They also discuss an application to finding “heavy hitters” (“elephant detection”) in marketing data (see e.g “Compressive sensing medium access control for wireless lans” Globecom, 2012 and “A data streaming algorithm for estimating entropies of od flows” IMC, San Diego, CA, 2007.).
The idea for the algorithm seems to have stemmed from Piotr Indyk’s paper “Stable distributions, pseudorandom generators, embeddings, and data stream computation.” Li and Zang state that Indyk estimated “the $\alpha$-th frequency moment $\sum_{i=1}^N |x_i|^\alpha$” using random measurements generated from an $\alpha$-stable distribution instead of the harder problem of recovering $x$ exactly.
In numerical simulations, they compare their algorithm to orthogonal matching pursuit and linear programming. Additionally, detailed proofs of the their algorithm’s effectiveness are provided.
## “Linear Bandits in High Dimension and Recommendation Systems”
Thanks to Nuit Blanche for pointing me towards the presentation by Andrea MontanariCollaborative Filtering: Models and Algorithms” and the associated Deshpande and Montanari paper “Linear Bandits in High Dimension and Recommendation Systems” (2012). In the presentation, Montanari reviews Spectral, Gradient Descent, Stochasitc Gradient Descent, Convex Relaxation, and Linear Bandit methods for approximating the standard linear model for recommendation systems and some accuracy guarantees.
Assuming the $j$th movie has features $v_{j1}, v_{j2}, \ldots, v_{jr}$, then the $i$th viewer gives the rating
$R_{ij} = \langle u_i, v_j \rangle +\epsilon_{ij}$
where $u_i$ is an $r$ dimensional vector representing the preferences of $i$th viewer and $\epsilon_{ij}$ is Gaussian noise.
The paper introduces a new Linear Bandit method, Smooth Explore, better suited for recommendation systems. Their method is motivated by the three objectives:
• Constant-optimal cumulative reward,
• Constant-optimal regret, and
• Approximate monotonicity (rewards approximately increase with time).
Smooth Explore estimates the user preferences vectors with a regularized least squares regression. Proofs of optimality and numerical results are provided.
## Lunokhod 2 Moon Rover
This Russian rover traveled over 20 miles on the moon in 1973! | {"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": 0.8268720507621765, "perplexity": 3318.762648527086}, "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-2017-43/segments/1508187825399.73/warc/CC-MAIN-20171022165927-20171022185927-00841.warc.gz"} |
https://www.spp2026.de/members-guests/28-member-pages/jprof-dr-maria-beatrice-pozzetti | # Members & Guests
## JProf. Dr. Maria Beatrice Pozzetti
Ruprecht-Karls-Universität Heidelberg
E-mail: pozzetti(at)mathi.uni-heidelberg.de
Telephone: +49-6221-54-14208
Homepage: https://www.mathi.uni-heidelberg.de/~poz…
## Project
28Rigidity, deformations and limits of maximal representations
71Rigidity, deformations and limits of maximal representations II
## Publications within SPP2026
We study the Thurston–Parreau boundary both of the Hitchinand of the maximal character varieties and determine therein an open setof discontinuity for the action of the mapping class group. This result isobtained as consequence of a canonical decomposition of a geodesic cur-rent on a surface of finite type arising from a topological decompositionof the surface along special geodesics. We show that each componenteither is associated to a measured lamination or has positive systole. Fora current with positive systole, we show that the intersection function onthe set of closed curves is bi-Lipschitz equivalent to the length functionwith respect to a hyperbolic metric.
Related project(s):
28Rigidity, deformations and limits of maximal representations
In this paper we investigate the Hausdorff dimension of limitsetsof Anosov representations. In this context we revisit and extend the frameworkof hyperconvex representations and establish a convergence property for them,analogue to a differentiability property. As an applicationof this convergence,we prove that the Hausdorff dimension of the limit set of a hyperconvex rep-resentation is equal to a suitably chosen critical exponent. In the appendix, incollaboration with M. Bridgeman, we extend a classical result on the Hessianof the Hausdorff dimension on purely imaginary directions.
Related project(s):
28Rigidity, deformations and limits of maximal representations
We study Anosov representation for which the image of the bound-ary map is the graph of a Lipschitz function, and show that theorbit growthrate with respect to an explicit linear function, the unstable Jacobian, is inte-gral. Several applications to the orbit growth rate in the symmetric space areprovided.
Related project(s):
28Rigidity, deformations and limits of maximal representations
We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite dimensional totally geodesic subspace on which the action is maximal. In the opposite direction we construct examples of geometrically dense maximal representation in the infinite dimensional Hermitian symmetric space of tube type and finite rank. Our approach is based on the study of boundary maps, that we are able to construct in low ranks or under some suitable Zariski-density assumption, circumventing the lack of local compactness in the infinite dimensional setting. | {"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": 0.9700185656547546, "perplexity": 741.6387265639618}, "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-2021-49/segments/1637964363332.1/warc/CC-MAIN-20211207014802-20211207044802-00030.warc.gz"} |
http://math.stackexchange.com/questions/309894/how-can-i-simplify-frac-sqrtx-1x-sqrtx-x-sqrtx-frac1x2 | How can I simplify $\frac{\sqrt{x} + 1}{x\sqrt{x} + x + \sqrt{x}} : \frac{1}{x^2-\sqrt{x}}$?
$$\frac{\sqrt{x} + 1}{x\sqrt{x} + x + \sqrt{x}} : \frac{1}{x^2-\sqrt{x}}$$
As I'm trying to study calculus so I will be thankfull to just a hint, not full solution.
Thanks.
-
Hint: Let $\sqrt{x}=y$. Chances are good you will find things so much more familiar that everything will become clear.
Thanks. $\sqrt{x}=y$ and multiplying the first fraction on $\frac{a-1}{a-1}$ helped me. – demas Feb 21 '13 at 6:18
I think of it this way. After cancelling the obvious $y$ we get $\frac{(y+1)(y^3-1)}{y^2+y+1}$. But $y^3-1$ factors as $(y-1)(y^2+y+1)$. Another cancellation. – André Nicolas Feb 21 '13 at 6:34
Here is my solution: $\frac{a+1}{a(a^2+a+1)} \frac{a(a^3-1)}{1} = \frac{(a+1)(a-1)}{a(a^2+a+1)(a-1)} \frac{a(a^3-1)}{1}=\frac{(a^2-1)}{a(a^3-1)} \frac{a(a^3-1)}{1} = a^2 - 1$ Thanks :) – demas Feb 21 '13 at 6:44 | {"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": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9251778721809387, "perplexity": 619.1781489454627}, "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-2014-35/segments/1408500833115.44/warc/CC-MAIN-20140820021353-00392-ip-10-180-136-8.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/83061-solved-manipulating-power-series-ln-1-x.html | # Thread: [SOLVED] manipulating power series ln(1+x)
1. ## [SOLVED] manipulating power series ln(1+x)
Find a power series representation for the following function:
$f(x) = ln (1+x)$
Here is how I started this:
$f'(x) = \frac{1}{1+x}$
$\sum^{\infty}_{n=0} (-x)^n$
Then I started to integrate, but I didn't know if I should do a definite or indefinite integral. If definite, what limits? Or, is this just not a good way to approach this problem? lol
CAn someone get me straight here?
Thanks!!
2. Originally Posted by mollymcf2009
Find a power series representation for the following function:
$f(x) = ln (1+x)$
Here is how I started this:
$f'(x) = \frac{1}{1+x}$
$\sum^{\infty}_{n=0} (-x)^n$
Then I started to integrate, but I didn't know if I should do a definite or indefinite integral. If definite, what limits? Or, is this just not a good way to approach this problem? lol
CAn someone get me straight here?
Thanks!!
find the indefinite integral, without the + C though. note also, that you have $\sum (-1)^n x^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": 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": 0.9232304692268372, "perplexity": 446.66313219423057}, "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/1476988720062.52/warc/CC-MAIN-20161020183840-00170-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://physicslens.com/course/motion-in-a-circle/lessons/linear-velocity-for-circular-motion/ | # Motion in a Circle
Seng Kwang
H2 Physics
Free
• 6 lessons
• 1 quizzes
• 10 week duration
• ##### Angular displacement and velocity
Learn how to describe circular motion quantitatively. The kinematics of circular motion.
• ##### Centripetal force
Learn what causes circular motion. The dynamics of circular motion.
## Centripetal Acceleration
In the following interactive, you can adjust the radii and angular velocities of two different objects and observe their changes in centripetal acceleration according to the relationship $a = r\omega^2$.
See if you could obtain the same magnitude of acceleration with different values of $r$ and $\omega$. | {"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": 0.85096275806427, "perplexity": 2111.4872761525703}, "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/1600401614309.85/warc/CC-MAIN-20200928202758-20200928232758-00480.warc.gz"} |
https://electronics.stackexchange.com/questions/304909/calculate-the-thevenin-resistance-with-diagonal-resistors | # Calculate the thevenin resistance with diagonal resistors
The question is:
Replace the circuit with a Thevenin Equivalent Circuit and find the Thevenin Resistance (Rt) in ohms and find the current in R3.
I've used Thevenin before and solved quite a few equivalent circuits as well.
I've tried to redrawing the circuit, but I getting very confused on you would exactly go about combining these resistors. I found this exercise in a book, however, at the end of the book, it only shows the answer and not the approach. So the thevenin resistance should be 51.61 ohms.
You need to do the Thevenin’s theorem in the orange box.
So my first try was to consider R4 and R6 in series, then in parallel with R5, then in series with R1 to finish in parallel with R2. But obviously, that wasn't the right way to do it.
• try searching harder electronics.stackexchange.com/questions/131604/… – Tony Stewart EE75 May 12 '17 at 6:50
• I would use source transformations to find Rth – Mike May 12 '17 at 6:52
• $R_2$ can be ignored. It sits across the voltage supply and cannot impact anything $R_3$ sees. Start there and redraw the schematic. – jonk May 12 '17 at 7:06
• The key to solving this is as per what @jonk says. – Andy aka May 12 '17 at 7:43
• "my first try was to consider R4 and R6 in series, then..." You are doing it backwards. You should be looking for the resistance as seen from the load R3, not as seen from the 80V generator. Short the generator and then redraw your circuit without R3. You want to find the resistance seen from what were R3's terminals. – Sredni Vashtar May 12 '17 at 8:28
Try the following sequence of steps. If you don't follow any of them, let me know:
simulate this circuit – Schematic created using CircuitLab
simulate this circuit
simulate this circuit
simulate this circuit
simulate this circuit
simulate this circuit
Done.
• @Liquidox Hopefully, you see it. $V_{TH}\approx 17.2\:\textrm{V}$ and $R_{TH}\approx 51.6\:\Omega$. – jonk May 12 '17 at 17:18
The method here is fairly straightforward: ask what would be the resistance looking into the circuit from the load point of view, if the voltage source was replaced by a short ?
Imagine placing an ohm-meter at the terminals of R4. That turns out to look like R4 || ( R6 + ( R5|| R1 ) ). Question then is, where did R2 go ? Well, if you remember, we shorted out the voltage source, thus shorting out R2 as well.
If you go through calculations, that's:
• R5 || R1 = (150*120)/(150+120) = 66.67 Ohm
• R6 + ( R5|| R1 ) = 40 + 66.67 = 106.67 Ohm
• R4 || ( R6 + ( R5|| R1 ) ) = (106.67*100)/(106.67+100) = 51.61 Ohm
I'm not doing the work for you but just list two different ways to solve this:
1. You can apply Y→$\Delta$ transformation (or $\Delta$→Y transformation) and transform the circuit step by step into its Thevenin Equivalent or
2. you can use the standard approach for finding the Thevenin (or Norton) Equivalent (together with one of the general circuit analysis methods, i.e. Nodal or Mesh Analysis):
• find short circuit current $i_{sc}$ (i.e. replace R3 by short and find current through that connection)
• find open circuit volatge $v_{oc}$ (i.e. remove R3 and find voltage across the two nodes R3 was connected to)
• $v_{th} = v_{oc}$, $R_{th}=\frac{v_{oc}}{i_{sc}}$ | {"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": 0.8575741052627563, "perplexity": 1398.5568798109289}, "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-2021-17/segments/1618039398307.76/warc/CC-MAIN-20210420122023-20210420152023-00270.warc.gz"} |
http://nextbigfuture.com/2011/09/dr-richard-nebel-now-at-tibbar.html | ## September 05, 2011
### Dr Richard Nebel now at Tibbar Technologies
Dr. Richard Nebel used to be the lead researcher at EMC2 Fusion. EMC2 Fusion is funded by the Navy to develop inertial electrostatic fusion. The EMC2 fusion processed were developed by Dr Bussard (Bussard Ramjet and a lot of other work). Dr Nebel is now at Tibbar Technologies. (H/T Talk polywell
Electrostatic Mode Locking and Mode Suppression in RFPs and Tokamaks by Richard Nebel (x lead researcher at EMC2 Fusion)
In this paper we show that it is possible to lock and amplify m=1 modes from the boundary in an RFP by using electrostatic fields. Furthermore, it is possible to do this without any magnetic field lines penetrating the boundary (i.e. the normal component of the magnetic field vanishes at the boundary).
These can result in single-helicity states which have good flux surfaces everywhere. The key to forming these states is to drive one of the unstable RFP modes. For the unstable modes, perturbations from the boundary amplify into the interior (Resonant Field Amplification). This is consistent with the theory developed 20+ years ago that boundary perturbations can be described by the marginal ideal MHD equations.
We believe that these same ideas can be applied to suppressing modes as well, such as edge modes in Tokamaks. We derive the required phasings to implement this scheme. We also present a conceptual feedback control scheme for suppressing instabilities. This system consists of an array of electrostatic plates and radial magnetic field probes. This system forms an electrostatic smart shell which essentially uses the plasma to heal itself. This system eliminates the inductive losses and resistive timescale restrictions present in magnetic field systems
Name
Email *
Message * | {"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": 0.8720439672470093, "perplexity": 2216.9473443169636}, "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/1438042988048.90/warc/CC-MAIN-20150728002308-00201-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/fourier-series.214340/ | # Fourier series
1. Feb 10, 2008
### buzzmath
1. The problem statement, all variables and given/known data
1. f is a function defined on the interval -a<x<a and has Fourier coefficients an=0 bn=1/n^(1/2) what can you say about the integral from -a to a of f^2(x)dx?
2. Show that as n goes to infinity the fourier sine coefficients of the function f(x)=1/x -pi<x<pi tend to a nonzero constant. Use the fact that the integral from 0 to infinity of sin(t)/t dt=pi/2
2. Relevant equations
3. The attempt at a solution
1. Persevals equality says 1/a * integral from -a to a of f^2(x)dx = 2a0^2 + susm n=1 to infinity of an^2 + bn^2 so I was thinking that since to a's are zero that we could just say that f^2(x) converges to the sum of the b's but the book says the answer is f^2(x) doesn't converge since sum n=1 to infinity of (|an|+|bn|) is infinity but the a's are zero and bn = 1/n^(1/2) I thought these bn's would go to zero not infinity. What am I doing wrong?
2. f I was just going to compute the bn's from the integral directly and show that as n goes to infinity they go to a nonzero constant but i'm not sure how I would do this problem with the integral given. How does an integral from 0 to infinity come into this problem?
thanks
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
2. Feb 10, 2008
### quasar987
1. The sequence 1/n goes to 0 as n goes to infinity, but the series $\sum 1/n$ does not!
2. Consider doing a change of coordinate.
3. Feb 10, 2008
### buzzmath
I know that the series isn't zero but they say that the series goes to infinity which I don't understand since the sequence goes to zero. I mean how would they come up with this? is there something i'm not seeing?
What do you mean by doing a change of coordinates? how would I get integral from 0 to infinity?
4. Feb 10, 2008
### quasar987
1. That the general term of a series goes to 0 is not a sufficient condition to assure that the series converge. The standard example used to illustrate this fact is precisely the series of 1/n, called the harmonic series: http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#Divergence_of_the_harmonic_series
2. Maybe you don't see where this is going yet, but you gotta try something. A logical first step seems to me to try to get the integrand in the same form as the integral in the hint. In the integralof the Fourier coefficient, you have sin(nx)/x... you'd like to get sin(t)/t. What change of variables do you suggest? And what does that leave you with?
Similar Discussions: Fourier series | {"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": 0.9928929805755615, "perplexity": 394.0044656031163}, "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/1502886109803.8/warc/CC-MAIN-20170822011838-20170822031838-00235.warc.gz"} |
http://tex.stackexchange.com/questions/62122/how-can-i-change-the-width-of-the-subsection-headings-to-4-in | # How can I change the width of the subsection headings to 4 in?
I am using `\@startsection` in `pdflatex` to define the formatting of the subsection headings. I would like their width to be limited to `4in`, and centered. I can change the third item `{\z@}` to indent the left hand side, but it doesn't center. Is there another way of doing this?
``````\renewcommand{\subsection}
{\@startsection
{subsection}
{2}%
{\z@}%
{\li}%
{1pt}%
{\reset@font\centering\normalsize}}
``````
-
have a look at format-title-section-number-flush-left-and-title-centered/49320#49320; it's not exactly the same, but it's pretty close – cmhughes Jul 4 '12 at 1:30
What about the subsection number? – egreg Jul 4 '12 at 6:31
Using the titlesec package you can do this as follows, by first calculating the necessary margins:
``````\usepackage{titlesec}
\newlength{\mymargin}
\setlength{\mymargin}{0.5\textwidth}
This uses the default `hang` style with the label hanging to the left of the title. Use `\titleformat{\subsection}[block]{...}` to move the label in, so the first line as a whole is centered, or `[display]` to have the label on the line above. | {"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": 0.8405169248580933, "perplexity": 767.0890520446868}, "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-2015-06/segments/1422122030742.53/warc/CC-MAIN-20150124175350-00050-ip-10-180-212-252.ec2.internal.warc.gz"} |
https://zenodo.org/record/4567660 | Poster Open Access
Preliminary mass loss results from a survey of stellar winds for young, Sun-like stars
Evensberget, Dag; Carter, Brad; Marsden, Stephen; Brookshaw, Leigh
Editor(s)
Wolk, Scott
We present current results from a project aiming to clarify how stellar winds evolve during the first billion years of the main sequence evolution of solar-type stars, and the implications for early solar evolution. This project is based on the extensive TOUPIES survey of stellar magnetic fields, with surface radial field maps used to model stellar winds via Space Weather Modelling Framework codes. We also explore numerical methods for producing wind models verifiable through direct comparisons with observations of the modern-day Sun. We find that a modelling approach used with solar magnetograms can be efficiently extended to survey a range of young solar-type stars with magnetic field strengths up to two orders of magnitude greater, but with some caveats regarding the assumptions needed and the interpretations gained from solar as well as stellar models. In our preliminary models wind mass loss does not exceed 1010 kg/s.
Files (7.6 MB)
Name Size
cool-stars-205-poster.pdf
md5:71152d2891bc230f338ca1212fa438a8
7.3 MB
thumbnail.jpg
md5:70f4d6a28ff047598faf26091de14aff
251.6 kB
• Folsom C. P., et al., 2016, MNRAS, 457, 580
• Folsom C. P., et al., 2018, MNRAS, 474, 4956
61
68
views | {"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": 0.8218652606010437, "perplexity": 3269.7863279916182}, "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-2022-33/segments/1659882572021.17/warc/CC-MAIN-20220814083156-20220814113156-00066.warc.gz"} |
https://elo.mastermath.nl/course/info.php?id=252 | ### Statistical Theory for High- and Infinite-dimensional Models - 8EC - M2
Prerequisites
• Measure theoretic probability
• Asymptotic Statistics
• Stochastic Processes
• Functional Analysis
Aim of the course
In most introductory courses on statistics the focus is on classical statistical models in which the number of unknown parameters is small relative to the sample size. If such models depend smoothly on the parameter of interest, then under regularity conditions we have that as the sample size n tends to infinity, maximum likelihood or Bayesian estimators converge at the rate n^(1/2)
to the parameter corresponding to the true distribution that generated the data. Moreover, asymptotically such estimators are normally distributed and efficient, in the sense that they have minimal asymptotic variance.
There are many situations in which it is natural to consider models with an unknown parameter that is very high-dimensional compared to sample size, or even infinite-dimensional. In this course we will see that in such models we usually have completely different behaviour of statistical procedures. Convergence rates are typically slower than n^(1/2), asymptotic normality is not guaranteed, and optimality of procedures can not be assessed in terms of minimal asymptotic variance.
This course provides a rigorous introduction to the mathematics of high-dimensional and nonparametric statistical models. Specific possible topics include high-dimensional linear models, nonparametric regression and classification, penalisation, regularisation, model selection, optimal convergence rates, minimax lower bounds for testing and estimation, adaptation.
At the end of this course the student is familiar with:
• the fundamental statistical issues related to high-dimensional statistical models
• the role of regularisation, penalisation etc.
• methods to achieve regularisation
• approaches to assess the performance and optimality of procedures in high-dimensional statistics
Lecturers
Harry van Zanten (UvA) | {"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": 0.8738633990287781, "perplexity": 657.7213610256525}, "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-2020-05/segments/1579251779833.86/warc/CC-MAIN-20200128153713-20200128183713-00259.warc.gz"} |
https://amathew.wordpress.com/2011/03/04/gila-3-probabilistic-algorithms-for-primality-testing/ | Last time, we showed that prime numbers admit succinct certificates. Given a number containing ${k}$ bits, one could produce a “proof” that the number is prime in polynomial in ${k}$ space. In other words, the language ${PRIMES}$ containing all primes (represented in the binary expansion, say) lies in the complexity class ${\mathbf{NP}}$.
In practice, though, being in ${\mathbf{NP}}$ does not give a good way of solving the problem by itself; for that, we want the problem to be in ${\mathbf{P}}$. Perhaps, though, that is too strict. One relaxation is to consider classes of problems that can be solved by randomized algorithms that run in polynomial time—randomized meaning that the right output comes out with high probability, that is. This is the class ${\mathbf{BPP}}$: by definition, something belongs to the class ${\mathbf{BPP}}$ if there is a polynomial-time nondeterministic Turing machine such that on any computation, at least ${2/3}$ of the computation tree leads to the right answer. So the idea of ${\mathbf{BPP}}$ is that, to decide whether a string ${x}$ belongs to our language ${\mathcal{L}}$, we generate a bunch of random bits and then run a deterministic algorithm on the string ${x}$ together with the random bits. Most likely, if the random bits were not too bad, we will get the right answer.
In practice, people tend to believe that ${\mathbf{BPP} = \mathbf{P}}$ while ${\mathbf{P} \neq \mathbf{NP}}$. But, in any case, probabilistic algorithms may be more efficient than deterministic ones, so even if the former equality holds they are interesting.
Today, I want to describe a probabilistic algorithm for testing whether a number ${N}$ is prime. As before, the algorithm will rely on a little elementary number theory.
Namely, we can assume first of all that ${N}$ is odd. Then it makes sense to talk about the quadratic symbol
$\displaystyle \left( \frac{M}{N} \right)$
for any ${M \in \mathbb{Z}}$. Recall that if ${N = \prod q_i}$ is a product of primes, then this is just the product of the usual quadratic symbols ${\prod \left( \frac{M}{q_i} \right)}$. When ${N}$ is a prime, then the fact that the group ${(\mathbb{Z}/N \mathbb{Z})^*}$ is cyclic implies that the quadratic symbol is given by
$\displaystyle \left( \frac{x}{N} \right) = x^{(N-1)/2},$
but this is not true in general. In fact:
Lemma 1 The equality $$\left( \frac{x}{N} \right) = x^{(N-1)/2}$$ holds for all ${x}$ prime to ${N}$ if and only if ${N}$ is prime.
We know that this holds if ${N}$ is prime. Conversely, let us suppose ${N}$ is composite. Then from the equality and the fact that the Jacobi symbol is always ${\pm 1}$, we find that ${x^{N-1} \equiv 1}$, implying ${N-1}$ is a period for the multiplicative group.
Suppose first ${N}$ is divisible by the square of a prime ${p}$. But this multiplicative group has order ${\phi(N)}$, which is divisible by ${p}$; thus ${(\mathbb{Z}/N\mathbb{Z})^*}$ has a subgroup of order ${p}$. But then the period cannot be ${N-1}$, which is prime to ${p}$.
Suppose next that ${N =\prod q_i}$ for the ${q_i}$ distinct primes. We can choose ${M}$ coprime to ${N}$ such that ${M}$ is congruent to a nonresidue ${r_1}$ mod ${q_1}$ and to ${1}$ mod ${q_i, i > 1}$ by the Chinese remainder theorem. Then the expression ${\left( \frac{M}{N} \right) = \prod \left( \frac{M}{q_i} \right)}$ shows that ${\left( \frac{M}{N} \right) = -1}$. If we reduce mod ${q_2}$, we get
$\displaystyle 1 \equiv -1 \mod q_2.$
This is a contradiction. In particular, we find:
Corollary 2 If ${N}$ is not prime, then for at least half of ${x \in (\mathbb{Z}/N\mathbb{Z})^*}$, we have
$\displaystyle \left( \frac{x}{N} \right) \neq x^{(N-1)/2}.$
Indeed, we have two different group-homomorphisms, so the subgroup on which they agree is index at least two.
In this way, we do have a probabilistic algorithm for testing whether a number ${N}$ is prime. Generate a random ${x \in [1, N-1]}$. Apply the euclidean algorithm to check whether ${x, N}$ are relatively prime; if not, return that ${N}$ is composite. Otherwise, compute the symbol ${\left( \frac{x}{N} \right)}$ and the power ${x^{(N-1)/2}}$; if they are equal, output yes, and if not, output no (composite). If ${N}$ is prime, the above algorithm will always output true. If ${N}$ is composite, the above algorithm will output false with probability ${\frac{1}{2}}$ at least. If we choose multiple ${x}$‘s randomly, we can make this probability better.
If we can show that this algorithm is efficient, then we will have seen that ${PRIMES \in \mathbf{BPP}}$. Now ${x^{(N-1)/2} \mod N}$ can be computed in polynomial time by the method of repeated squaring. The quadratic symbol can be computed by using the quadratic reciprocity law ${\left( \frac{M}{N} \right) \left( \frac{N}{M} \right)}$ (which is still true even if we don’t assume primeness) and the expression for ${\left( \frac{2}{N} \right) = (-1)^{(N^2 -1)/8}}$. Namely, to compute ${\left( \frac{M}{N} \right)}$ where ${M < N}$, we reduce this to computing ${\left( \frac{N}{M} \right)}$. But we can take the remainder of ${N}$ mod ${M}$ to compute this, and so are reduced to a smaller problem. So our algorithm runs pretty fast. | {"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": 75, "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": 0.9485653638839722, "perplexity": 102.34751968954308}, "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/1627046154085.58/warc/CC-MAIN-20210731074335-20210731104335-00404.warc.gz"} |
https://unapologetic.wordpress.com/2008/11/ | # The Unapologetic Mathematician
## The Category of Representations of a Group
Sorry for missing yesterday. I had this written up but completely forgot to post it while getting prepared for next week’s trip back to a city. Speaking of which, I’ll be heading off for the week, and I’ll just give things here a rest until the beginning of December. Except for the Samples, and maybe an I Made It or so…
Okay, let’s say we have a group $G$. This gives us a cocommutative Hopf algebra. Thus the category of representations of $G$ is monoidal — symmetric, even — and has duals. Let’s consider these structures a bit more closely.
We start with two representations $\rho:G\rightarrow\mathrm{GL}(V)$ and $\sigma:G\rightarrow\mathrm{GL}(W)$. We use the comultiplication on $\mathbb{F}[G]$ to give us an action on the tensor product $V\otimes W$. Specifically, we find
\begin{aligned}\left[\left[\rho\otimes\sigma\right](g)\right](v\otimes w)=\left[\rho(g)\otimes\sigma(g)\right](v\otimes w)\\=\left[\rho(g)\right](v)\otimes\left[\sigma(g)\right](w)\end{aligned}
That is, we make two copies of the group element $g$, use $\rho$ to act on the first tensorand, and use $\sigma$ to act on the second tensorand. If $\rho$ and $\sigma$ came from actions of $G$ on sets, then this is just what you’d expect from linearizing the product of the $G$-actions.
Symmetry is straightforward. We just use the twist on the underlying vector spaces, and it’s automatically an intertwiner of the actions, so it defines a morphism between the representations.
Duals, though, take a bit of work. Remember that the antipode of $\mathbb{F}[G]$ sends group elements to their inverses. So if we start with a representation $\rho:G\rightarrow\mathrm{GL}(V)$ we calculate its dual representation on $V^*$:
\begin{aligned}\left[\rho^*(g)\right](\lambda)=\left[\rho(g^{-1})^*\right](\lambda)\\=\lambda\circ\rho(g^{-1})\end{aligned}
Composing linear maps from the right reverses the order of multiplication from that in the group, but taking the inverse of $g$ reverses it again, and so we have a proper action again.
November 21, 2008
## Cocommutativity
One things I don’t think I’ve mentioned is that the category of vector spaces over a field $\mathbb{F}$ is symmetric. Indeed, given vector spaces $V$ and $W$ we can define the “twist” map $\tau_{V,W}:V\otimes W\rightarrow W\otimes V$ by setting $\tau_{V,W}(v\otimes w)=w\otimes v$ and extending linearly.
Now we know that an algebra $A$ is commutative if we can swap the inputs to the multiplication and get the same answer. That is, if $m(a,b)=m(b,a)=m\left(\tau_{A,A}(a,b)\right)$. Or, more succinctly: $m=m\circ\tau_{A,A}$.
Reflecting this concept, we say that a coalgebra $C$ is cocommutative if we can swap the outputs from the comultiplication. That is, if $\tau_{C,C}\circ\Delta=\Delta$. Similarly, bialgebras and Hopf algebras can be cocommutative.
The group algebra $\mathbb{F}[G]$ of a group $G$ is a cocommutative Hopf algebra. Indeed, since $\Delta(e_g)=e_g\otimes e_g$, we can twist this either way and get the same answer.
So what does cocommutativity buy us? It turns out that the category of representations of a cocommutative bialgebra $B$ is not only monoidal, but it’s also symmetric! Indeed, given representations $\rho:B\rightarrow\hom_\mathbb{F}(V,V)$ and $\sigma:B\rightarrow\hom_\mathbb{F}(W,W)$, we have the tensor product representations $\rho\otimes\sigma$ on $V\otimes W$, and $\sigma\otimes\rho$ on $W\otimes V$. To twist them we define the natural transformation $\tau_{\rho,\sigma}$ to be the twist of the vector spaces: $\tau_{V,W}$.
We just need to verify that this actually intertwines the two representations. If we act first and then twist we find
\begin{aligned}\tau_{V,W}\left(\left[\left[\rho\otimes\sigma\right](a)\right](v\otimes w)\right)=\tau_{V,W}\left(\left[\rho\left(a_{(1)}\right)\otimes\sigma\left(a_{(2)}\right)\right](v\otimes w)\right)\\=\tau_{V,W}\left(\left[\rho\left(a_{(1)}\right)\right](v)\otimes\left[\sigma\left(a_{(2)}\right)\right](w)\right)\\=\left[\sigma\left(a_{(2)}\right)\right](w)\otimes\left[\rho\left(a_{(1)}\right)\right](v)\end{aligned}
On the other hand, if we twist first and then act we find
\begin{aligned}\left[\left[\sigma\otimes\rho\right](a)\right]\left(\tau_{V,W}(v\otimes w)\right)=\left[\sigma\left(a_{(1)}\right)\otimes\rho\left(a_{(2)}\right)\right]\left(w\otimes v\right)\\=\left[\sigma\left(a_{(1)}\right)\right](w)\otimes\left[\rho\left(a_{(2)}\right)\right](v)\end{aligned}
It seems there’s a problem. In general this doesn’t work. Ah! but we haven’t used cocommutativity yet! Now we write
$a_{(1)}\otimes a_{(2)}=\Delta(a)=\tau_{B,B}\left(\Delta(a)\right)=\tau_{B,B}\left(a_{(1)}\otimes a_{(2)}\right)=a_{(2)}\otimes a_{(1)}$
Again, remember that this doesn’t mean that the two tensorands are always equal, but only that the results after (implicitly) summing up are equal. Anyhow, that’s enough for us. It shows that the twist on the underlying vector spaces actually does intertwine the two representations, as we wanted. Thus the category of representations is symmetric.
November 19, 2008
## The Category of Representations of a Hopf Algebra
It took us two posts, but we showed that the category of representations of a Hopf algebra $H$ has duals. This is on top of our earlier result that the category of representations of any bialgebra $B$ is monoidal. Let’s look at this a little more conceptually.
Earlier, we said that a bialgebra is a comonoid object in the category of algebras over $\mathbb{F}$. But let’s consider this category itself. We also said that an algebra is a category enriched over $\mathbb{F}$, but with only one object. So we should really be thinking about the category of algebras as a full sub-2-category of the 2-category of categories enriched over $\mathbb{F}$.
So what’s a comonoid object in this 2-category? When we defined comonoid objects we used a model category $\mathrm{Th}(\mathbf{CoMon})$. Now let’s augment it to a 2-category in the easiest way possible: just add an identity 2-morphism to every morphism!
But the 2-category language gives us a bit more flexibility. Instead of demanding that the morphism $\Delta:C\rightarrow C\otimes C$ satisfy the associative law on the nose, we can add a “coassociator” 2-morphism $\gamma:(\Delta\otimes1)\circ\Delta\rightarrow(1\otimes\Delta)\circ\Delta$ to our model 2-category. Similarly, we dispense with the left and right counit laws and add left and right counit 2-morphisms. Then we insist that these 2-morphisms satisfy pentagon and triangle identities dual to those we defined when we talked about monoidal categories.
What we’ve built up here is a model 2-category for weak comonoid objects in a 2-category. Then any weak comonoid object is given by a 2-functor from this 2-category to the appropriate target 2-category. Similarly we can define a weak monoid object as a 2-functor from the opposite model 2-category to an appropriate target 2-category.
So, getting a little closer to Earth, we have in hand a comonoid object in the 2-category of categories enriched over $\mathbb{F}$ — our algebra $B$. But remember that a 2-category is just a category enriched over categories. That is, between $H$ (considered as a category) and $\mathbf{Vect}(\mathbb{F})$ we have a hom-category $\hom(H,\mathbf{Vect}(\mathbb{F}))$. The entry in the first slot $H$ is described by a 2-functor from the model category of weak comonoid objects to the 2-category of categories enriched over $\mathbb{F}$. This hom-functor is contravariant in the first slot (like all hom-functors), and so the result is described by a 2-functor from the opposite of our model 2-category. That is, it’s a weak monoid object in the 2-category of all categories. And this is just a monoidal category!
This is yet another example of the way that hom objects inherit structure from their second variable, and inherit opposite structure from their first variable. I’ll leave it to you to verify that a monoidal category with duals is similarly a weak group object in the 2-category of categories, and that this is why a Hopf algebra — a (weak) cogroup object in the 2-category of categories enriched over $\mathbb{F}$ has dual representations.
November 18, 2008
## Representations of Hopf Algebras II
Now that we have a coevaluation for vector spaces, let’s make sure that it intertwines the actions of a Hopf algebra. Then we can finish showing that the category of representations of a Hopf algebra has duals.
Take a representation $\rho:H\rightarrow\hom_\mathbb{F}(V,V)$, and pick a basis $\left\{e_i\right\}$ of $V$ and the dual basis $\left\{\epsilon^i\right\}$ of $V^*$. We define the map $\eta_\rho:\mathbf{1}\rightarrow V^*\otimes V$ by $\eta_\rho(1)=\epsilon^i\otimes e_i$. Now $\left[\rho(a)\right](1)=\epsilon(a)$, so if we use the action of $H$ on $\mathbf{1}$ before transferring to $V^*\otimes V$, we get $\epsilon(a)\epsilon^i\otimes e_i$. Be careful not to confuse the counit $\epsilon$ with the basis elements $\epsilon^i$.
On the other hand, if we transfer first, we must calculate
\begin{aligned}\left[\left[\rho^*\otimes\rho\right](a)\right](\epsilon^i\otimes e_i)=\left[\rho^*\left(a_{(1)}\right)\otimes\rho\left(a_{(2)}\right)\right](\epsilon^i\otimes e_i)\\=\left[\rho\left(S\left(a_{(1)}\right)\right)^*\otimes\rho\left(a_{(2)}\right)\right](\epsilon^i\otimes e_i)\\=\left[\rho\left(S\left(a_{(1)}\right)\right)^*\right](\epsilon^i)\otimes\left[\rho\left(a_{(2)}\right)\right](e_i)\end{aligned}
Now let’s use the fact that we’ve got this basis sitting around to expand out both $\rho\left(S\left(a_{(1)}\right)\right)$ and $\rho\left(a_{(2)}\right)$ as matrices. We’ll just take on matrix indices on the right for our notation. Then we continue the calculation above:
\begin{aligned}\left[\rho\left(S\left(a_{(1)}\right)\right)^*\right](\epsilon^i)\otimes\left[\rho\left(a_{(2)}\right)\right](e_i)=\epsilon^j\rho\left(S\left(a_{(1)}\right)\right)_j^i\otimes\rho\left(a_{(2)}\right)_i^ke_k\\=\epsilon^j\otimes\rho\left(S\left(a_{(1)}\right)\right)_j^i\rho\left(a_{(2)}\right)_i^ke_k\\=\epsilon^j\otimes\left[\rho\left(\mu\left(\left[S\otimes1_H\right]\left(\Delta(a)\right)\right)\right)\right](e_j)\\=\epsilon^j\otimes\left[\rho\left(\iota\left(\epsilon(a)\right)\right)\right](e_j)=\epsilon^j\otimes\epsilon(a)e_j\end{aligned}
And so the coevaluation map does indeed intertwine the two actions of $H$. Together with the evaluation map, it provides the duality on the category of representations of a Hopf algebra $H$ that we were looking for.
November 14, 2008
## The Coevaluation on Vector Spaces
Okay, I noticed that I never really gave the definition of the coevaluation when I introduced categories with duals, because you need some linear algebra. Well, now we have some linear algebra, so let’s do it.
Let $V$ be a finite-dimensional vector space with dual space $V^*$. Then if we have a basis $\left\{e_i\right\}$ of $V$ we immediately get a dual basis $\left\{\epsilon^i\right\}$ for $V^*$ (yet another $\epsilon$ to keep straight), defined by $\epsilon^j(e_i)=\delta_i^j$. We now define a map $\eta:\mathbf{1}\rightarrow V^*\otimes V$ by setting $\eta(1)=\epsilon^i\otimes e_i$. That is, we take the tensor product of each dual basis element with its corresponding basis element, and add them all up (summation convention).
But this seems to depend on which basis $\left\{e_i\right\}$ we started with. What if we used a different basis $\left\{f_i\right\}$ and dual basis $\left\{\phi^i\right\}$? We know that there is a change of basis matrix $f_i=t_i^je_j$, so let’s see how this works on the dual basis.
The dual basis is defined by the fact that $\phi^j(f_i)=\delta_i^j$. So we use this new expression for $f_i$ to write $\phi^j(t_i^ke_k)=t_i^k\phi^j(e_k)=\delta_i^j$. That is, $\phi^j(e_k)$ must be the inverse matrix to $t_i^k$, which we’ll write as $\left(t^{-1}\right)_k^j$. But now we can check
$\left[\left(t^{-1}\right)_i^j\epsilon^i\right](e_k)=\left(t^{-1}\right)_i^j\delta_k^i=\left(t^{-1}\right)_k^j$
And so we find that $\phi^i=\left(t^{-1}\right)_j^i\epsilon^j$ when we change bases.
Now we can use the same definition for $\eta$ above with our new basis. We set $\eta(1)=\phi^i\otimes f_i$, and then substitute our expressions in terms of the old bases:
$\eta(1)=\left(t^{-1}\right)_j^i\epsilon^j\otimes t_i^ke_k=\left(t^{-1}\right)_j^it_i^k\left(\epsilon^j\otimes e_k\right)=\delta_j^k\left(\epsilon^j\otimes e_k\right)=\epsilon^k\otimes e_k$
which is what we got before. That is, this map actually doesn’t depend on the basis we chose!
Okay, now does this coevaluation — along with the evaluation map from before — actually satisfy the conditions for a duality? First, let’s start with a vector written out in terms of a basis: $v=v^ie_i$. Now we use the coevaluation to send it to $v^ie_i\otimes\epsilon^j\otimes e_j$. Next we evaluate on the first two tensorands to find $v^i\delta_i^je_j=v^je_j$. So we do indeed have the identity here. Verifying the other condition is almost the same, starting from an arbitrary covector $\lambda=\lambda_i\epsilon^i$.
So now we know that the category $\mathbf{FinVect}(\mathbb{F})$ has duals. Tomorrow we can promote this to a duality on the category of finite-dimensional representations of a Hopf algebra.
November 13, 2008 Posted by | Algebra, Linear Algebra | 2 Comments
## Representations of Hopf Algebras I
We’ve seen that the category of representations of a bialgebra is monoidal. What do we get for Hopf algebras? What does an antipode buy us? Duals! At least when we restrict to finite-dimensional representations.
Again, we base things on the underlying category of vector spaces. Given a representation $\rho:H\rightarrow\hom_\mathbb{F}(V,V)$, we want to find a representation $\rho^*:H\rightarrow\hom_\mathbb{F}(V^*,V^*)$. And it should commute with the natural transformations which make up the dual structure.
Easy enough! We just take the dual of each map to find $\rho(h)^*:V^*\rightarrow V^*$. But no, this can’t work. Duality reverses the order of composition. We need an antiautomorphism $S$ to reverse the multiplication on $H$. Then we can define $\rho^*(h)=\rho(S(h))^*$.
The antiautomorphism we’ll use will be the antipode. Now to make these representations actual duals, we’ll need natural transformations $\eta_\rho:\mathbf{1}\rightarrow\rho^*\otimes\rho$ and $\epsilon_\rho:\rho\otimes\rho^*\rightarrow\mathbf{1}$. This natural transformation $\epsilon$ is not to be confused with the counit of the Hopf algebra. Given a representation $\rho$ on the finite-dimensional vector space $V$, we’ll just use the $\eta_V$ and $\epsilon_V$ that come from the duality on the category of finite-dimensional vector spaces.
Thus we find that $\epsilon_\rho$ is the pairing $v\otimes\lambda\mapsto\lambda(v)$. Does this commute with the actions of $H$? On the one side, we calculate
\begin{aligned}\left[\left[\rho\otimes\rho^*\right](h)\right](v\otimes\lambda)=\left[\rho\left(h_{(1)}\right)\otimes\rho^*\left(h_{(2)}\right)\right](v\otimes\lambda)\\=\left[\rho\left(h_{(1)}\right)\otimes\rho\left(S\left((h_{(2)}\right)\right)^*\right](v\otimes\lambda)\\=\left[\rho\left(h_{(1)}\right)\right](v)\otimes\left[\rho\left(S\left(h_{(2)}\right)\right)^*\right](\lambda)\end{aligned}
Then we apply the evaluation to find
\begin{aligned}\left[\left[\rho\left(S\left(h_{(2)}\right)\right)^*\right](\lambda)\right]\left(\left[\rho\left(h_{(1)}\right)\right](v)\right)=\lambda\left(\left[\rho\left(S\left(h_{(2)}\right)\right)\right]\left(\left[\rho\left(h_{(1)}\right)\right](v)\right)\right)\\=\lambda\left(\left[\rho\left(h_{(1)}S\left(h_{(2)}\right)\right)\right](v)\right)=\lambda\left(\left[\rho\left(\mu\left(h_{(1)}\otimes S\left(h_{(2)}\right)\right)\right)\right](v)\right)\\=\lambda\left(\left[\rho\left(\iota\left(\epsilon(h)\right)\right)\right](v)\right)=\epsilon(h)\lambda(v)\end{aligned}
Which is the same as the result we’d get by applying the “unit” action after evaluating. Notice how we used the definition of the dual map, the fact that $\rho$ is a representation, and the antipodal property in obtaining this result.
This much works whether or not $V$ is a finite-dimensional vector space. The other direction, though, needs more work, especially since I waved my hands at it when I used $\mathbf{FinVect}$ as the motivating example of a category with duals. Tomorrow I’ll define this map.
November 12, 2008
## Representations of Bialgebras
What’s so great about bialgebras? Their categories of representations are monoidal!
Let’s say we have two algebra representations $\rho:A\rightarrow\hom_\mathbb{F}(V,V)$ and $\sigma:A\rightarrow\hom_\mathbb{F}(W,W)$. These are morphisms in the category of $\mathbb{F}$-algebras, and so of course we can take their tensor product $\rho\otimes\sigma$. But this is not a representation of the same algebra. It’s a representation of the tensor square of the algebra:
$\rho\otimes\sigma:A\otimes A\rightarrow\hom_\mathbb{F}(V,V)\otimes\hom_\mathbb{F}(W,W)\cong\hom_\mathbb{F}(V\otimes W,V\otimes W)$
Ah, but if we have a way to send $A$ to $A\otimes A$ (an algebra homomorphism, that is), then we can compose it with this tensor product to get a representation of $A$. And that’s exactly what the comultiplication $\Delta$ does for us. We abuse notation slightly and write:
$\rho\otimes\sigma:A\rightarrow\hom_\mathbb{F}(V\otimes W,V\otimes W)$
where the homomorphism of this representation is the comultiplication $\Delta$ followed by the tensor product of the two homomorphisms, followed by the equivalence of $\hom$ algebras.
Notice here that the underlying vector space of the tensor product of two representations $\rho\otimes\sigma$ is the tensor product of their underlying vector spaces $V\otimes W$. That is, if we think (as many approaches to representation theory do) of the vector space as fundamental and the homomorphism as extra structure, then this is saying we can put the structure of a representation on the tensor product of the vector spaces.
Which leads us to the next consideration. For the tensor product to be a monoidal structure we need an associator. And the underlying linear map on vector spaces must clearly be the old associator for $\mathbf{Vect}(\mathbb{F})$. We just need to verify that it commutes with the action of $A$.
So let’s consider three representations $\rho:A\rightarrow\hom_\mathbb{F}(U,U)$, $\sigma\rightarrow\hom_\mathbb{F}(V,V)$, and $\tau:A\rightarrow\hom_\mathbb{F}(W,W)$. Given an algebra element $a$ and vectors $u$, $v$, and $w$, we have the action
\begin{aligned}\left[\left[(\rho\otimes\sigma)\otimes\tau\right](a)\right]((u\otimes v)\otimes w)=\\\left(\left[\rho\left(\left(a_{(1)}\right)_{(1)}\right)\right](u)\otimes\left[\sigma\left(\left(a_{(1)}\right)_{(2)}\right)\right](v)\right)\otimes\left[\tau\left(a_{(2)}\right)\right](v)\end{aligned}
On the other hand, if we associate the other way we have the action
\begin{aligned}\left[\left[\rho\otimes(\sigma\otimes\tau)\right](a)\right](u\otimes(v\otimes w))=\\\left[\rho\left(a_{(1)}\right)\right](u)\otimes\left(\left[\sigma\left(\left(a_{(2)}\right)_{(1)}\right)\right](v)\otimes\left[\tau\left(\left(a_{(2)}\right)_{(2)}\right)\right](v)\right)\end{aligned}
Where we have used the Sweedler notation to write out the comultiplications of $a$. But now we can use the coassociativity of the comultiplication — along with the fact that, as algebra homomorphisms, the representations are linear maps — to show that the associator on $\mathbf{Vect}(\mathbb{F})$ intertwines these actions, and thus acts as an associator for the category of representations of $A$ as well.
We also need a unit object, and similar considerations to those above tell us it should be based on the vector space unit object. That is, we need a homomorphism $A\rightarrow\hom_\mathbb{F}(\mathbb{F},\mathbb{F})$. But linear maps from the base field to itself (considered as a one-dimensional vector space) are just multiplications by field elements! That is, the $\hom$ algebra is just the field $\mathbb{F}$ itself, and we need a homomorphism $A\rightarrow\mathbb{F}$. This is precisely what the counit $\epsilon$ provides! I’ll leave it to you to verify that the left and right unit maps from vector spaces intertwine the relevant representations.
November 11, 2008
## Sweedler notation
As we work with coalgebras, we’ll need a nice way to write out the comultiplication of an element. In the group algebra we’ve been using as an example, we just have $\Delta(e_g)=e_g\otimes e_g$, but not all elements are so cleanly sent to two copies of themselves. And other comltiplications in other coalgebras aren’t even defined so nicely on any basis. So we introduce the so-called “Sweedler notation”. If you didn’t like the summation convention, you’re going to hate this.
Okay, first of all, we know that the comultiplication of an element $c\in C$ is an element of the tensor square $C\otimes C$. Thus it can be written as a finite sum
$\displaystyle\Delta(c)=\sum\limits_{i=1}^n(c)a_i\otimes b_i$
Now, this uses two whole new letters, $a$ and $b$, which might be really awkward to come up with in practice. Instead, let’s call them $c_{(1)}$ and $c_{(2)}$, to denote the first and second factors of the comultiplication. We’ll also move the indices to superscripts, just to get them out of the way.
$\displaystyle\Delta(c)=\sum\limits_{i=1}^n(c)c_{(1)}^i\otimes c_{(2)}^i$
The whole index-summing thing is a bit awkward, especially because the number of summands is different for each coalgebra element $c$. Let’s just say we’re adding up all the terms we need to for a given $c$:
$\displaystyle\Delta(c)=\sum\limits_{(c)}c_{(1)}\otimes c_{(2)}$
Then if we’re really pressed for space we can just write $\Delta(c)=c_{(1)}\otimes c_{(2)}$. Since we don’t use a subscript in parentheses for anything else, we remember that this is implicitly a summation.
Let’s check out the counit laws $(1_M\otimes\epsilon)\circ\Delta=1_M=(\epsilon\otimes1_M)\circ\Delta$ in this notation. Now they read $c_{(1)}\epsilon(c_{(2)}=c=\epsilon(c_{(1)})c_{(2)}$. Or, more expansively:
$\displaystyle\sum\limits_{(c)}c_{(1)}\epsilon\left(c_{(2)}\right)=c=\sum\limits_{(c)}\epsilon\left(c_{(1)}\right)c_{(2)}$
Similarly, the coassociativity condition now reads
$\displaystyle\sum\limits_{(c)}\left(\sum\limits_{\left(c_{(1)}\right)}\left(c_{(1)}\right)_{(1)}\otimes\left(c_{(1)}\right)_{(2)}\right)\otimes c_{(2)}=\sum\limits_{(c)}c_{(1)}\otimes\left(\sum\limits_{\left(c_{(2)}\right)}\left(c_{(2)}\right)_{(1)}\otimes\left(c_{(2)}\right)_{(1)}\right)$
In the Sweedler notation we’ll write both of these equal sums as
$\displaystyle\sum\limits_{(c)}c_{(1)}\otimes c_{(2)}\otimes c_{(3)}$
Or more simply as $c_{(1)}\otimes c_{(2)}\otimes c_{(3)}$.
As a bit more practice, let’s write out the condition that a linear map $f:C\rightarrow D$ between coalgebras is a coalgebra morphism. The answer is that $f$ must satisfy
$f\left(c_{(1)}\right)\otimes f\left(c_{(2)}\right)=f(c)_{(1)}\otimes f(c)_{(2)}$
Notice here that there are implied summations here. We are not asserting that all the summands are equal, and definitely not that $f\left(c_{(1)}\right)=f(c)_{(1)}$ (for instance). Sweedler notation hides a lot more than the summation convention ever did, but it’s still possible to expand it back out to a proper summation-heavy format when we need to.
November 10, 2008 Posted by | Algebra | 7 Comments
## Hopf Algebras
One more piece of structure we need. We take a bialgebra $H$, and we add an “antipode”, which behaves sort of like an inverse operation. Then what we have is a Hopf algebra.
An antipode will be a linear map $S:H\rightarrow H$ on the underlying vector space. Here’s what we mean by saying that an antipode “behaves like an inverse”. In formulas, we write that:
$\mu\circ(S\otimes1_H)\circ\Delta=\iota\circ\epsilon=\mu\circ(1_H\otimes S)\circ\Delta$
On either side, first we comultiply an algebra element to split it into two parts. Then we use $S$ on one or the other part before multiplying them back together. In the center, this is the same as first taking the counit to get a field element, and then multiplying that by the unit of the algebra.
By now it shouldn’t be a surprise that the group algebra $\mathbb{F}[G]$ is also a Hopf algebra. Specifically, we set $S(e_g)=e_{g^{-1}}$. Then we can check the “left inverse” law:
\begin{aligned}\mu\left(\left[S\otimes1_H\right]\left(\Delta(e_g)\right)\right)=\mu\left(\left[S\otimes1_H\right](e_g\otimes e_g)\right)=\\\mu(e_{g^{-1}}\otimes e_g)=e_{g^{-1}g}=e_1=\iota(1)=\iota\left(\epsilon(e_g)\right)\end{aligned}
One thing that we should point out: this is not a group object in the category of vector spaces over $\mathbb{F}$. A group object needs the diagonal we get from the finite products on the target category. But in the category of vector spaces we pointedly do not use the categorical product as our monoidal structure. There is no “diagonal” for the tensor product.
Instead, we move to the category of coalgebras over $\mathbb{F}$. Now each coalgebra $C$ comes with its own comultiplication $\Delta:C\rightarrow C\otimes C$, which stands in for the diagonal. In the case of $\mathbb{F}[G]$ we’ve been considering, this comultiplication is clearly related to the diagonal on the underlying set of the group $G$. In fact, it’s not going too far to say that “linearizing” a set naturally brings along a coalgebra structure on top of the vector space structure we usually consider. But many coalgebras, bialgebras, and Hopf algebras are not such linearized sets.
In the category of coalgebras over $\mathbb{F}$, a Hopf algebra is a group object, so long as we use the comultiplications and counits that come with the coalgebras instead of the ones that come from the categorical product structure. Dually, we can characterize a Hopf algebra as a cogroup object in the category of algebras over $\mathbb{F}$, subject to a similar caveat. It is this cogroup structure that will be important moving forwards.
November 7, 2008 Posted by | Algebra | 9 Comments
## Bialgebras
In yesterday’s post I used the group algebra $\mathbb{F}[G]$ of a group $G$ as an example of a coalgebra. In fact, more is true.
A bialgebra is a vector space $A$ equipped with both the structure of an algebra and the structure of a coalgebra, and that these two structures are “compatible” in a certain sense. The traditional definitions usually consist in laying out the algebra maps and relations, then the coalgebra maps and relations. Then they state that the algebra structure preserves the coalgebra structure, and that the coalgebra structure preserves the algebra structure, and they note that really you only need to require one of these last two conditions because they turn out to be equivalent.
In fact, our perspective allows this equivalence to come to the fore. The algebra structure makes the bialgebra a monoid object in the category of vector space over $\mathbb{F}$. Then a compatible coalgebra structure makes it a comonoid object in the category of algebras over $\mathbb{F}$. Or in the other order, we have a monoid object in the category of comonoid objects in the category of vector spaces over $\mathbb{F}$. And these describe essentially the same things because internalizations commute!
Okay, let’s be explicit about what we mean by “compatibility”. This just means that — on the one side — the coalgebra maps are not just linear maps between the underlying vector spaces, but actually are algebra homomorphisms. On the other side, it means that the algebra maps are actually coalgebra homomorphisms.
Multiplication and comultiplication being compatible actually mean the same thing. Take two algebra elements and multiply them, then comultiply the result. Alternatively, comultiply each of them, and the multiply corresponding factors of the result. We should get the same answer whether we multiply or comultiply first. That is: $\Delta\circ\mu=(\mu\otimes\mu)\circ(1_A\otimes\tau_{A,A}\otimes1_A)\circ(\Delta\otimes\Delta)$, where $\tau$ is the twist map, exchanging two factors.
Let’s check this condition for the group algebra $\mathbb{F}[G]$:
\begin{aligned}\left[\mu\otimes\mu\right]\left(\left[1_A\otimes\tau_{A,A}\otimes1_A\right]\left(\left[\Delta\otimes\Delta\right](e_g\otimes e_h)\right)\right)=\\\left[\mu\otimes\mu\right]\left(\left[1_A\otimes\tau_{A,A}\otimes1_A\right](e_g\otimes e_g\otimes e_h\otimes e_h)\right)=\\\left[\mu\otimes\mu\right](e_g\otimes e_h\otimes e_g\otimes e_h)=e_{gh}\otimes e_{gh}=\\\Delta(e_{gh})=\Delta\left(\mu(e_g\otimes e_h)\right)\end{aligned}
Similarly, if we multiply two algebra elements and then take the counit, it should be the same as the product (in $\mathbb{F}$) of the counits of the elements. Dually, the product of two copies of the algebra unit should be the algebra unit again, and the counit of the algebra unit should be the unit in $\mathbb{F}$. It’s straightforward to verify that these hold for $\mathbb{F}[G]$.
November 6, 2008 Posted by | Algebra, Category theory | 5 Comments | {"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": 215, "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": 0.9676056504249573, "perplexity": 225.25384730153775}, "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/1469257828322.57/warc/CC-MAIN-20160723071028-00015-ip-10-185-27-174.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/706778/an-extension-of-a-game-with-two-dice | # An extension of a game with two dice
This question is an extension of a previous question already correctly answered: A game with two dice. In that question, we had two dice: the first one, when rolled, determined the number of times that the second die would be rolled. The goal was to quantify, before any dice was rolled, the probabilities of the sum of the second die rolls (the sum is always between 1 and 36). Now we have two imaginary dice, both with infinite faces. Again, we roll the first die that will determine how many times we’ll roll the second die. The probability functions for each die are power functions:
For the first die $P(x) = ax^{-k}$ and for the second die $P(x) = bx^{-q}$, where $a, b, k$ and $q$ are constants, $x\in{N^+}$, $k, q>1$, and $0<a, b<1$
We want to calculate the probabilities of the sum of the second die, before any dice is rolled.
-
What have you tried? Specifically, have you tried to understand and generalize the generating function approach to the previous question? – Greg Martin Mar 10 '14 at 17:20
Also note that you must have $k,q>1$, not $k,q>0$, for convergence, and that necessarily $a=1/\zeta(k)$ and $b=1/\zeta(q)$. (I'm assuming the infinitely many faces are labeled with $x$ in the positive integers.) – Greg Martin Mar 10 '14 at 17:21
@GregMartin. IMHO, the function also converge for k,q>0. I can do the PGF but the problem is to group all of them by the same sum – Luis Gonilho Mar 10 '14 at 17:33
I'd say the PGF will be $$ab\sum_{i=1}^{n}{i^{-(k+iq)}}\$$ The next question is how to group all the combinations – Luis Gonilho Mar 10 '14 at 17:57
For a single die, if the probability of the positive integer $x$ being chosen is $ax^{-k}$, then the total probability is $\sum_{x=1}^\infty ax^{-k}$. This sum does not converge if $x\le1$. – Greg Martin Mar 10 '14 at 21:34 | {"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": 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": 0.9355428814888, "perplexity": 340.73893064134194}, "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-2015-06/segments/1422118059355.87/warc/CC-MAIN-20150124164739-00017-ip-10-180-212-252.ec2.internal.warc.gz"} |
https://pure.mpg.de/pubman/faces/ViewItemFullPage.jsp?itemId=item_1300726_1&view=EXPORT | English
# Item
ITEM ACTIONSEXPORT
On the stability of the massive scalar field in Kerr space-time
Beyer, H. R. (2011). On the stability of the massive scalar field in Kerr space-time. Journal of Mathematical Physics, 52(10): 102502. doi:10.1063/1.3653840.
Item is
### Basic
show hide
Genre: Journal Article
### Files
show Files
hide Files
:
1105.4956 (Preprint), 246KB
Name:
1105.4956
Description:
File downloaded from arXiv at 2012-01-10 13:00
Visibility:
Public
MIME-Type / Checksum:
application/pdf / [MD5]
-
-
:
JMP52_102502.pdf (Any fulltext), 833KB
Name:
JMP52_102502.pdf
Description:
-
Visibility:
Public
MIME-Type / Checksum:
application/pdf / [MD5]
-
-
-
show
### Creators
show
hide
Creators:
Beyer, Horst Reinhard1, Author
Affiliations:
1Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, Golm, DE, ou_24012
### Content
show
hide
Free keywords: Mathematical Physics, math-ph,General Relativity and Quantum Cosmology, gr-qc,Mathematics, Analysis of PDEs, math.AP,Mathematics, Mathematical Physics, math.MP,
Abstract: The current early stage in the investigation of the stability of the Kerr metric is characterized by the study of appropriate model problems. Particularly interesting is the problem of the stability of the solutions of the Klein-Gordon equation, describing the propagation of a scalar field in the background of a rotating (Kerr-) black hole. Results suggest that the stability of the field depends crucially on its mass $\mu$. Among others, the paper provides an improved bound for $\mu$ above which the solutions of the reduced, by separation in the azimuth angle in Boyer-Lindquist coordinates, Klein-Gordon equation are stable. Finally, it gives new formulations of the reduced equation, in particular, in form of a time-dependent wave equation that is governed by a family of unitarily equivalent positive self-adjoint operators. The latter formulation might turn out useful for further investigation. On the other hand, it is proved that from the abstract properties of this family alone it cannot be concluded that the corresponding solutions are stable.
### Details
show
hide
Language(s):
Dates: 2011-05-252011
Publication Status: Published in print
Pages: 30 pages, 2 figures
Publishing info: -
Rev. Method: -
Identifiers: arXiv: 1105.4956
DOI: 10.1063/1.3653840
Degree: -
show
show
show
### Source 1
show
hide
Title: Journal of Mathematical Physics
Source Genre: Journal
Creator(s):
Affiliations:
Publ. Info: Woodbury, N.Y. [etc.] : American Institute of Physics
Pages: - Volume / Issue: 52 (10) Sequence Number: 102502 Start / End Page: - Identifier: ISSN: 0022-2488
CoNE: https://pure.mpg.de/cone/journals/resource/954922836227 | {"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": 0.8773407340049744, "perplexity": 2897.402514884324}, "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-2020-29/segments/1593655883439.15/warc/CC-MAIN-20200703215640-20200704005640-00025.warc.gz"} |
https://tex.stackexchange.com/questions/65626/misalignment-in-custom-big-symbol-using-mathop | # Misalignment in custom big symbol using mathop
Following the advice of this question, I created the \itsep symbol:
\newcommand{\itsep}{\mathop{\scalebox{1.5}{\raisebox{-0.2ex}{$\circledast$}}}}
When used with single character sub/super-scripts, it displays properly, like this:
a \itsep^f_{g} b
If I place multiple characters in the sub-script the symbol behaves incorrectly:
a \itsep^f_{g \, a \, b} b
Is there a way to make this symbol truly behave like \sum or \bigwedge where the "big" symbol is centered over the subscript and centered under the superscript?
if, instead of using the basic \mathop you use \DeclareMathOperator* from amsmath, this "insulates" the result from its context:
\DeclareMathOperator*{\itsep}{\scalebox{1.5}{\raisebox{-0.2ex}{$\circledast$}}}
thus
$a \itsep^f_{g} b \qquad a \itsep^f_{g \, a \, b} b$
results in
For reference, mathtools provides \mathclap which allows for zero-width centred overlap of math content, based on the size its used in:
\documentclass{article}
\usepackage{mathtools}% http://ctan.org/pkg/mathtools
\usepackage{amssymb}% http://ctan.org/pkg/amssymb
\usepackage{graphicx}% http://ctan.org/pkg/graphicx
\begin{document}
\newcommand{\itsep}{\mathop{\scalebox{1.5}{\raisebox{-0.2ex}{$\circledast$}}}}
$a \itsep^f_{g} b \qquad a \itsep^f_{g \, a \, b} b \qquad a \itsep^f_{\mathclap{g \, a \, b}} b$
\end{document}
It depends on your use case whether this would be a problem when having excessively long subscripts, which may extend into the horizontal domain of the "operands" a and b.
• I've used \mathclap before and it certainly is an interesting solution to this problem (that I hadn't thought of), but it is my preference to simply make the symbol behave like other "big" symbols and only have to present a fix at the definition rather than fixing it at each instance as this mathclap solution does. – Arlen Cox Aug 2 '12 at 22:01 | {"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": 0.8485478162765503, "perplexity": 2658.6347097804055}, "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-2020-10/segments/1581875145774.75/warc/CC-MAIN-20200223123852-20200223153852-00451.warc.gz"} |
https://www.physicsforums.com/threads/how-to-solve-a-2-ab-b-2.868333/ | # How to solve: a^2 - ab + b^2
• #1
85
3
## Main Question or Discussion Point
How do you solve 4a2 - 8ab + 3b2=0?
I know there are general formulae, but I'm not sure how to use them:
a2-2ab+b^2 = (a-b)2
a2+2ab+b^2= (a+b)2
a2-b2 = (a+b)(a-b)
• #2
berkeman
Mentor
56,814
6,782
How do you solve 4a2 - 8ab + 3b2=0?
I know there are general formulae, but I'm not sure how to use them:
a2-2ab+b^2 = (a-b)2
a2+2ab+b^2= (a+b)2
a2-b2 = (a+b)(a-b)
Is this for schoolwork?
• #3
Math_QED
Homework Helper
2019 Award
1,391
514
Consider it as a second grade equation in a in the form: da^2 + ea + f = 0. Solve for a.
• #4
Homework Helper
Gold Member
4,522
1,926
I worked something similar a few years back=perhaps you will find this relevant. a^2 +2ab +b^2 is a perfect square. It can always be written as (a+b)^2. The question I had (probably since algebra class), but never explored, was could a^2+ab+b^2 ever be a perfect square? i.e. a^2+ab+b^2=c^2 where a, b, and c are integers. I did find some solutions. I expect you could do the same for a^2-ab+b^2. In general, there are only a couple expressions like a^2+2ab+b^2 and a^2-b^2 that factor in such a simple manner. You can even do the same thing for a^2+b^2=c^2 and there are a number of solutions like the 3,4, 5 and 5,12, 13 right triangles. (You should recognize the Pythagoren theorem equation a^2+b^2=c^2 for this latter case.) And of course, any integral multiples of these solutions will also be solutions.
• #5
85
3
Is this for schoolwork?
Partially. I'm just studying for a physics exam. While I understand the physics behind this particular question, I'm just stuck on how to solve for the final answer from the above expression.
Perhaps I should have posted it in the homework questions folder?
• #6
Math_QED
Homework Helper
2019 Award
1,391
514
Let us know when you find the answer :)
• #7
85
3
Consider it as a second grade equation in a in the form: da^2 + ea + f = 0. Solve for a.
I'm not quite sure I understand what you mean.
• #8
berkeman
Mentor
56,814
6,782
I'm not quite sure I understand what you mean.
Yeah, I didn't understand his reply either.
You need to factor the equation to find the roots. Have you learned much about how to factor polynomials yet?
Perhaps I should have posted it in the homework questions folder?
It sounds more like a general question, so it's probably okay here for now. Definitely all homework-like questions should be posted in the Homework Help forums, though.
• #9
Math_QED
Homework Helper
2019 Award
1,391
514
I'm not quite sure I understand what you mean.
Do you know how to solve the equation: ax^2 + bx+ c = 0 for x?
• #10
Homework Helper
Gold Member
4,522
1,926
One additional item: The form a^2-ab+b^2 appears in the factoring of a^3+b^3=(a+b)(a^2-ab+b^2). I think many people ask themselves upon seeing the expression a^2-ab+b^2 here, does it factor further, and the answer is no. And your other question, how do you factor (4a^2-8ab +3b^2)? It is a polynomial that happens to factor. That one you should be able to get an answer for.
Last edited:
• #11
129
41
I'm not quite sure I understand what you mean.
Ok I'll bite, but not sure I get any closer :-)
Sticking it through the old x = -b +/- root(b squared..... etc
a = b + or - 2b squared
(note to self.. must learn how to post powers!!)
• #12
Homework Helper
Gold Member
4,522
1,926
• #13
Math_QED
Homework Helper
2019 Award
1,391
514
One additional item: The form a^2-ab+b^2 appears in the factoring of a^3+b^3=(a+b)(a^2-ab+b^2). I think many people ask themselves upon seeing the expression a^2-ab+b^2 here, does it factor further, and the answer is no. And your other question, how do you factor (4a^2-6ab +3b^2)? It is a polynomial that happens to factor. That one you should be able to get an answer for.
• #14
Homework Helper
Gold Member
4,522
1,926
I miscopied the -6ab". It is a "-8ab". See my correction (edited). 4a^2-8ab+3b^2 factors quite easily.
• #15
berkeman
Mentor
56,814
6,782
(note to self.. must learn how to post powers!!)
See the LaTeX turorial in the Help/How-To list under INFO at the top of the page.
• #16
berkeman
Mentor
56,814
6,782
How do you solve 4a2 - 8ab + 3b2=0?
It is a polynomial that happens to factor.
It is indeed. Now let's let the OP try to figure it out please...
• #17
85
3
3x^2 -8rx +4r^2
=3x^2 -2rx -6rx +4r^2
=x(3x-2r) -2r(3x-2r)
=(3x-2r)(x-2r)
:)
• #18
SammyS
Staff Emeritus
Homework Helper
Gold Member
11,248
968
Consider it as a second grade equation in a in the form: da^2 + ea + f = 0. Solve for a.
Do you possibly mean second degree?
I don't know any second graders who can do this. I'm sure that they are rare if not non-existent .
• #19
Math_QED
Homework Helper
2019 Award
1,391
514
Do you possibly mean second degree?
I don't know any second graders who can do this. I'm sure that they are rare if not non-existent .
Oh yes second degree. I'm sorry English is not my native language.
• #20
129
41
So how does this factor? I got as far as I could with the traditional quadratic formula but I'm pretty sure it was a fail!
Q1 Did anyone solve it?
Q2 If so... Was it by intuition/brute force; or is there a reusable technique?
Thanks
Matt
• #21
SammyS
Staff Emeritus
Homework Helper
Gold Member
11,248
968
So how does this factor? I got as far as I could with the traditional quadratic formula but I'm pretty sure it was a fail!
Q1 Did anyone solve it?
Q2 If so... Was it by intuition/brute force; or is there a reusable technique?
Thanks
Matt
Well, there's the thread title. Seems to asks about factoring a^2 - ab + b^2, or solving a^2 - ab + b^2 = 0. (It doesn't fully state either.)
But the opening line in the OP asks about solving 4a2 - 8ab + 3b2 = 0 .
The left hand side of the latter equation can be factored to get the solution.
The expression in the title cannot be factored.
• #22
OmCheeto
Gold Member
2,114
2,499
I cheated and used Wolfram Alpha to get the answers.
But that was only so I could make sure I did my maths right.
It's been decades since I've studied maths, so I wasn't sure if solving the problem with the quadratic method would work.
To my surprise, it did work.
• #23
129
41
I'm still missing something here, please bear with me ....
I also used the quadratic formula, but it gives solutions for a that includes factors of b. How did you move to simple factors?
Thanks
• #24
OmCheeto
Gold Member
2,114
2,499
I'm still missing something here, please bear with me ....
I also used the quadratic formula, but it gives solutions for a that includes factors of b. How did you move to simple factors?
Thanks
One of the reasons you will seldom see me in the maths forum, is because I don't speak maths.
I don't know what "move to simple factors" means.
Sorry.
ps. My answers, a=f(b), were not the same as Wolfram Alpha's, b=f(a). But it only took me about 3 seconds to reconcile.
• #25
SammyS
Staff Emeritus
Homework Helper
Gold Member
11,248
968
I'm still missing something here, please bear with me ....
I also used the quadratic formula, but it gives solutions for a that includes factors of b. How did you move to simple factors?
Thanks
If you treat a as the variable, then yes, it's solution will include b .
Alternatively, divide your equation by b2 giving:
$\displaystyle 4\left(\frac ab \right)^2-8\left(\frac ab \right)+3=0$
Solve that for $\displaystyle \ \frac ab \$ either by factoring, or by the quadratic formula. (Yes, this is factorable.)
• Last Post
Replies
26
Views
18K
• Last Post
Replies
9
Views
824
• Last Post
Replies
4
Views
9K
• Last Post
Replies
5
Views
971
• Last Post
Replies
8
Views
17K
• Last Post
Replies
21
Views
3K
• Last Post
Replies
18
Views
8K
• Last Post
Replies
9
Views
9K
• Last Post
Replies
9
Views
28K
• Last Post
Replies
8
Views
6K | {"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": 0.8028037548065186, "perplexity": 1631.3880892397767}, "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/1585370496227.25/warc/CC-MAIN-20200329201741-20200329231741-00333.warc.gz"} |
https://socratic.org/questions/a-triangle-has-sides-a-b-and-c-sides-a-and-b-are-of-lengths-5-and-2-respectively-3 | Trigonometry
Topics
# A triangle has sides A, B, and C. Sides A and B are of lengths 5 and 2, respectively, and the angle between A and B is (pi)/8 . What is the length of side C?
Feb 10, 2016
≈ 3.24
#### Explanation:
In this triangle , 2 sides and the angle between them are known , hence use the 'cosine rule '.
for this triangle the cosine rule is :
${C}^{2} = {A}^{2} + {B}^{2} - \left(2 A B \cos \left(\frac{\pi}{8}\right)\right)$
${C}^{2} = {5}^{2} + {2}^{2} - \left(2 \times 5 \times 2 \times \cos \left(\frac{\pi}{8}\right)\right)$
$= 25 + 4 - \left(20 \cos \left(\frac{\pi}{8}\right)\right) = 29 - \left(18.478\right) = 10.522$
C^2 = 10.522 rArr C =sqrt10.522 ≈ 3.24
##### Impact of this question
77 views around the world | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 4, "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": 0.8470085263252258, "perplexity": 1138.8938748348621}, "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-2019-13/segments/1552912204300.90/warc/CC-MAIN-20190325194225-20190325220225-00300.warc.gz"} |
https://istopdeath.com/evaluate-integral-of-1-x-1x3-with-respect-to-x/ | Evaluate integral of 1/((x-1)(x+3)) with respect to x
Reorder the factors of .
Write the fraction using partial fraction decomposition.
Simplify.
Split the single integral into multiple integrals.
Since is constant with respect to , move out of the integral.
Let . Then . Rewrite using and .
Let . Find .
Differentiate .
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
Rewrite the problem using and .
The integral of with respect to is .
Since is constant with respect to , move out of the integral.
Since is constant with respect to , move out of the integral.
Let . Then . Rewrite using and .
Let . Find .
Differentiate .
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
Rewrite the problem using and .
The integral of with respect to is .
Simplify.
Substitute back in for each integration substitution variable.
Replace all occurrences of with .
Replace all occurrences of with .
Evaluate integral of 1/((x-1)(x+3)) with respect to x
Scroll to top | {"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": 0.996511697769165, "perplexity": 1036.9807320818989}, "config": {"markdown_headings": false, "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-2023-06/segments/1674764500392.45/warc/CC-MAIN-20230207071302-20230207101302-00643.warc.gz"} |
http://mathhelpforum.com/calculus/141435-determine-whether-series-absolutely-convergent-conditionally-convergent.html | # Thread: Determine whether the series is absolutely convergent, conditionally convergent ....
1. ## Determine whether the series is absolutely convergent, conditionally convergent ....
Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
2. Originally Posted by alin1916
Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
Ratio test:
$\lim_{k \to \infty}\left|\frac{a_{k + 1}}{a_k}\right|$
$= \lim_{k \to \infty}\left|\frac{\frac{(-1)^{k + 2}\cdot 1 \cdot 4 \cdot 7 \cdot \dots \cdot (3k - 2) \cdot (3k + 1)}{(k + 1)!2^{k + 1}}}{\frac{(-1)^{k + 1}\cdot 1 \cdot 4 \cdot 7 \cdot \dots \cdot (3k - 2)}{k!2^k}}\right|$
$= \lim_{k \to \infty}\frac{\frac{1 \cdot 4 \cdot 7 \cdot \dots \cdot (3k - 2) \cdot (3k + 1)}{(k + 1)!2^{k + 1}}}{\frac{1 \cdot 4 \cdot 7 \cdot \dots \cdot (3k - 2)}{k!2^k}}$
$= \lim_{k \to \infty}\frac{3k + 1}{2(k + 1)}$
$= \lim_{k \to \infty}\frac{3k + 1}{2k + 2}$
$= \lim_{k \to \infty}\frac{3}{2}$ by L'Hospital's Rule
$= \frac{3}{2}$
$> 1$.
So the series is divergent. | {"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": 0.9186670184135437, "perplexity": 438.9510623105318}, "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/1368707773051/warc/CC-MAIN-20130516123613-00060-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://physicscatalyst.com/graduation/newtons-laws-motion-applications/ | # Newton’s laws of motion and applications
This article is about Newton’s laws of motion and applications for B.Sc first year students and is also applicable if you are preparing for exams like JAM , GATE, JEST , CSIR-NET etc. It comes under the subject Mechanics. If you like the content do help us by sharing it among your fellow students.
## Introduction
Sir Isaac Newton expressed his ideas regarding the motion of bodies in the form of three laws which are considered as the basic laws of mechanics. In fact Mechanics is the study of certain general relations that describe the interactions of material bodies and the Laws governing them. One general property of a material body is its inertial mass. The concept of describing interactions between material bodies if ‘force’. These two concepts of inertial mass and force were first defined in a quantitative manner by Isaac Newton.
Newton’s Laws of motion are of utmost importance in the field of physics.
It is important to understand which part of Newton’s laws are based on experimet and which parts are matter of definitions. While studying these laws we must learn to apply them as their application is essential for real understanding of underlying concepts. (An Introduction to Mechanics )
Newton’s Laws gives us the direct introduction of classical mechanics but it is not the only theory to do so. There are the formulations of Lagrange and Hamilton which take energy instead of force as the fundamental concept. As for the results are concerned these methods are equivalent to Newtonian approach and understanding of Newtonian mechanics is helpful in understanding other treatments of mechanics.
As such Newton’s laws of motion forms the core of classical dynamics. These laws are explained as follows.
## Newton’s First Law of motion
Everybody continues to be in its state of rest or of uniform motion in a straight line unless acted upon by an external unbalanced force
Newton’s first law of motion talk about two things:-
(a) force (or its absence): which is an expression of the interaction between different objects.
(b) measurement of rest or uniform motion of the body or object
This law of states that body would either remain at rest or continue to move with constant velocity. But the question arises “with respect to whom”? For this, we need a frame of reference.
Now consider a situation, where the observer is sitting in an accelerated frame ( velocity of the frame changing with time). Then, in this case, the object can not always remain at rest or in uniform motion w.r.t. the observer. A reference frame w.r.t. which first law holds is an inertial frame of reference. So, not all coordinate systems or frame of reference are inertial. It is always possible to find a coordinate system with respect to which isolated bodies move uniformly. This is the essence of Newton’s first law of motion. So this law of motion is the assertion that the inertial system exists. An inertial system must be a frame moving with a constant velocity (non accelerating). The question whether a given frame of reference is inertial or not is a matter of observation and experiment.
Note:- For most of the observations made on the surface of the earth, the frame of reference attached to the surface of Earth behaves as an inertial frame to a very good approximation.
## Newton’s Second Law of motion
The rate of change of linear momentum is proportional to the impressed force and takes place in the direction of the applied force.
We can also say that, the time rate of change of linear momentum of an object is proportional to the external force acting on it and occurs in the direction of the resultant force. That is
$\vec F = k\frac{{d\vec p}}{{dt}} = km\vec a$
Here $\vec{F}$ is the resultant external force, m is the inertial mass of the object, and $\vec p = m\vec v$ is the linear momentum of the object. K is the constant of proportionality which could be taken as 1 by defining appropriate units.
Here let us be careful about a point that Newton’s second law is not a definition of force. The equation $\vec F = m\vec a$ is an equality which says that the force acting on a body determines the rate of change of velocity. What is the nature and value of $\vec{F}$ is unanswered in in second law.
Newton’s second law of motion is valid for an object having constant inertial mass and moving with velocity small compared to that of light. If the velocity of the object is significantly high (i.e., a significant fraction of the velocity of light), we enter into the domain of special theory of relativity and Newton’s laws were revised in relativistic dynamics.
## Newton’s Third Law of motion
When two objects interact, the mutual force of interaction on each object is equal in magnitude and opposite in direction. That is ‘to every action there is an equal and opposite reaction’.
In the third law of motion, the action means the force due to the body (1) on the body (2) and may be expressed by ${\vec F_{12}}$. The reaction here means the force due to the body (2) on the body (1) and may be expressed as ${\vec F_{21}}$. The third law can be expressed as
$\vec{F}_{12}=-\vec{F}_{21}$
There is no such thing as lone force without a partner. Newton’s third law is the statement about the nature of the force. Third law of motion leads directly to the law of conservation of momentum.
Newton’s third law of motion implies that the forces exerted by the two interacting bodies over each other are equal and opposite, provided both the forces are measured simultaneously. A simultaneous measurement of the two forces is possible for ordinary practical purposes if the time taken by the interaction is sufficiently large compared with the time taken by light signal to travel one to the other. This means that the law ceases to hold good for particles of atomic dimensions, for which simultaneous measurement of two forces is almost impossible.
Solved Example 1. Show that Newton’s first law of motion is simply a special case of Newton’s second law.
Solution. Newton’s second law of motion gives a measure of a force as the rate of change of linear momentum. If $\vec{p}$ is the momentum of the particle, given by, where m is mass and $\vec{p}$ its velocity, then
$\vec{F}=\frac{d\vec{p}}{dt}=\frac{d}{dt}\left( m\vec{v} \right)$
$\vec{F}=m\frac{d\vec{v}}{dt}=m\vec{a}$ (m being constant)
so that, if $\vec{F}=0$, then, $\vec{a}=0$ which is the first law of motion, i.e. if no force acts, there will be no change in velocity and hence no acceleration. Thus, Newton’s first law of motion is simply a special case of the second law.
## Conservation of linear momentum
According to Newton’s first law of motion, ‘a body remains at rest or continues to move at a uniform velocity, as long as no external force is acting on it.’ According to Newton’s second law of motion, ‘the rate of change of linear momentum of a body is proportional to the force acting on it.’
Using Newton’s first law of motion, if a particle of mass m is moving with velocity $\vec{v}$, its linear momentum is $p=m\vrc{v}$. When a force $\vec{F}$ is applied to it, its momentum changes at the rate $\frac{d\vec{p}}{dt}$.
$\vec{F}=\frac{d\vec{v}}{dt}=\frac{d(m\vec{v})}{dt}$
Clearly when
$\vec{F}=0,\frac{d\vec{p}}{dt}=0$
Or.
$\vec{p}=m\vec{v}$= a constant
i.e., if no force acts, the linear momentum is conserved. Thus Newton’s first law of motion is only a special case of second law.
Using Newton’s second law of motion, now consider a system of two particles which are acted only by forces of interaction like Coulomb’s forces or gravitational forces and no external forces act on the system.
Let $\vec{F}_{12}$ be the force exerted by the first particle on the second, known as action and $\vec{F}_{21}$ be the force exerted by the second on the first, known as reaction, then according to Newton’s third law of motion, action and reaction being equal and opposite. That is,
$\vec{F}_{12}=-\vec{F}_{21}$
According to Newton’s second law of motion
${{\vec{F}}_{12}}=\frac{d{{{\vec{p}}}_{2}}}{dt}=\frac{d({{m}_{2}}{{{\vec{v}}}_{2}})}{dt}={{m}_{2}}\frac{d\vec{v}_2}{dt}$
where $m_2$, $\vec{v}_2$ and $\vec{p}_2$ are the mass, velocity and momentum of the second particle.
Similarly ,
${{\vec{F}}_{21}}=\frac{d{{{\vec{p}}}_{1}}}{dt}=\frac{d({{m}_{1}}{{{\vec{v}}}_{1}})}{dt}={{m}_{1}}\frac{d\vec{v}_1}{dt}$
where $m_1$, $\vec{v}_1$ and $\vec{p}_1$ are the mass, velocity and
$\vec{F}_{12}=-\vec{F}_{21}$
Or
${{\vec{F}}_{12}}+{{\vec{F}}_{21}}=0$.
So, we have
$\frac{d{{{\vec{p}}}_{1}}}{dt}+\frac{d{{{\vec{p}}}_{2}}}{dt}=0$
or,
$\frac{d}{dt}\left( {{{\vec{p}}}_{1}}+{{{\vec{p}}}_{2}} \right)=0$
Integrating, we have
$\vec{p}_1+\vec{p}_2=$ a constant or $m_1\vec{v}_1+m_2\vec{v}_2 =$ a constant
The quantity
$\vec{p}_1+\vec{p}_2= m_1\vec{v}_1+m_2\vec{v}_2$
represents the total linear momentum of the system. Hence we conclude that ‘If Newton’s second and third law of motion holds good, the total linear momentum of the system of two particles remains constant’. The above law can be extended to a system of three or more interacting particles. The law of conservation of linear momentum is, therefore, a basic law and is stated as under. “The total linear momentum of a system of particles free from the action of external forces and subjected only to their mutual interaction remains constant, no matter how complicated the forces are.” | {"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": 0.9624282717704773, "perplexity": 181.6011309762217}, "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/1550249490870.89/warc/CC-MAIN-20190223061816-20190223083816-00399.warc.gz"} |
http://math.stackexchange.com/questions/117932/proof-that-x24xyy2-1-has-infinitely-many-integer-solutions/117938 | # Proof that $x^2+4xy+y^2=1$ has infinitely many integer solutions
The question would, naturally, be very straight forward if there was a $2xy$ instead of a $4xy$. Then it would simply be a matter of doing:
$$x^2+2xy+y^2=1\\ (x+y)^2=1\\ \sqrt{(x+y)^2}=\sqrt{1}\\ |x+y|=1$$
Clearly, there are infinitely many pairs of integers that differ by one so there is an infinite number of integer solutions to $x^2+2xy+y^2=1$. Unfortunately, the same principle does not apply to $x^2+4xy+y^2=1$ where if I attempt to construct a similar proof all I can do is:
$$x^2+4xy+y^2=1\\ (x+y)^2+2xy=1\\ 2xy=1-(x+y)^2\\ 2xy=(1+x+y)(1-x-y)$$
And, from there, I have no idea how to proceed to complete the proof that there are an infinite number of integer solutions. I'm wondering whether I'm approaching the question entirely in the wrong way or if I am simply missing something. A nudge in the right direction would be much appreciated!
-
oeis.org/A001353 Scroll down to the 2nd reference to Michael Somos – Byron Schmuland Mar 8 '12 at 16:41
Hint $\$ Consider $\,f(x) = x^2 + 4y\ x + y^2\!-\!1\,$ as a quadratic in $\,x,$ where $y$ is constant. By Vieta its roots $\,x,x'$ satisfy $\ x+x' = -4y.\,$ Thus if $\,x\,$ is a root then so too is $\,x' = -4y-x.$
This yields a reflection symmetry $\ \, (x,y) \mapsto (-4y-x,\,y)\,$ on the solution space. Composing
this with the reflection symmetry $\,(x,y)\mapsto (y,x)\,$ yields the map $\,(x,y)\mapsto (-4x-y,x)$
which, iterated starting at solution $(1,0),\,$ yields infinitely many solutions
$$(1,0),\ (-4,1),\ (15,-4),\ (-56,15),\ (209,-56),\ (-780,209),\ (2911,-78),\ \ldots$$
Sequence $\, 0,1,4,15,56,209,\ldots$ satsifies the recursion $\,f_{n+2} = 4 f_{n+1} - f_{n}\,$ as is easily derived.
This can be transformed to the Pell equation $\ X^2\! - 3 Y^2 = 1\$ and studied using standard results on Pell equations. See also the comments on this sequence at OEIS sequence A001353.
-
Pardon the silly question... what does "$x$ is a root of $x^2 + bx + c$" mean? Isn't that just saying that $x^2 + bx + c = 0$?? – The Chaz 2.0 Apr 21 '12 at 21:53
To the anonymous user suggesting edits: they are being rejected by other users before I have a chance to see them. If you have questions please post them in comments and/or a new question. – Bill Dubuque Feb 24 at 5:48
Hint:
Have you learnt about Pell's equation?
Try adding a multiple of $y^2$ (on both sides) to complete the square on the left.
-
Using Pell's equation you would get $(x + 2y)^2 -3y^2 = 1$. How would you then prove it after this? – Vishwa Iyer Jan 22 at 17:03
@Vishwa: The equation $a^2 - 3b^2 = 1$ has infinitely many solutions. Now $y=b$, $x = a -2b$ and you are done, aren't you? – Aryabhata Jan 22 at 21:16
Following Aryabhata, write it as $x^2+4xy+4y^2-3y^2=1=(x+2y)^2-3y^2$ Let us define $z=x+2y$, so this becomes $z^2-3y^2=1$. Clearly $z=1,y=0$ is a solution. Now if we have a solution $(z,y)$ we observe that $(z',y')=(2z+3y,z+2y)$ is also a solution, because \begin {align}z'^2-3y'^2&=(2z+3y)^2-3(z+2y)^2\\&=4z^2+12zy+9y^2-3z^2-12yz-12y^2\\&=z^2-3y^2\\&=1 \end{align} Given any solution we can find a larger one, so there are infinitely many.
How do you know that $x$ will be an integer (or even real) solution, for a fixed $y$? This seems to be a vast oversimplification of the problem. – user61527 Feb 6 at 3:16 | {"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": 0.9689657092094421, "perplexity": 229.65969791407483}, "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-2014-41/segments/1412037663167.8/warc/CC-MAIN-20140930004103-00083-ip-10-234-18-248.ec2.internal.warc.gz"} |
https://chemistry.stackexchange.com/questions/23609/whats-the-difference-between-entropy-and-the-disorder-of-a-system | # Whats the difference between entropy and the (dis)order of a system?
Entropy is often verbally described as the order/disorder of the thermodynamic system. However, I've been told that this description is a vague "hand-waving" attempt at describing what entropy is. For example, a messy bedroom doesn't have greater entropy than a tidy room
My question is why is this the case? Also, what would better describe entropy verbally?
Briefly, spontaneous processes tend to proceed from states of low probability to states of higher probability. The higher-probability states tend to be those that can be realized in many different ways.
Entropy is a measure of the number of different ways a state with a particular energy can be realized. Specifically, $$S=k\ln W$$ where $k$ is Boltzmann's constant and $W$ is the number of equivalent ways to distribute energy in the system. If there are many ways to realize a state with a given energy, we say it has high entropy. Often the many ways to realize a high entropy state might be described as "disorder", but the lack of order is beside the point; the state has high entropy because it can be realized in many different ways, not because it's "messy".
Here's an analogy: if energy were money, entropy would be related to the number of different ways of counting it out. For example, there, there are only two ways of counting out two dollars with American paper money (2 1-dollar bills, or 1 two-dollar bill). But there are five ways of counting out two dollars using 50-cent or 25-cent coins (4 50-cent pieces, 3 50-cent pieces and 2 quarters, and so on). You could say that the "entropy" of a system that dealt in coins was higher than that of a system that dealt only in paper money.
Let's look the change in entropy for a reaction $\rm A\rightarrow B$, where A molecules can take on energies that are multiples of 10 energy units, and B molecules can take on energies that are multiples of 5 units.
Suppose that the total energy of the reacting mixture is 20 units. If we have 3 molecules of A, there are 2 ways to distribute our 20 units among energy levels with 0, 10, and 20 units:
If we have 3 molecules of B, there are 4 ways to distribute 20 units among energy levels with 0, 5, 10, 15, and 20 units:
The entropy of B is higher than the entropy of A because there are more ways to distribute the same amount of energy in B than in A. Therefore, $\Delta S$ for the reaction $\rm A\rightarrow B$ will be positive.
what would better describe entropy verbally?
We can also use "measurement of randomness" or "amount of chaos" or " energy dispersion"
Initially in 1862, Rudolf Clausius asserted that thermodynamic process always "admits to being reduced to the alteration in some way or another of the arrangement of the constituent parts of the working body" and that internal work associated with these alterations is quantified energetically by a measure of entropy change.
But later after few years Ludwig Boltzmann translated word alteration from Rudolf Clausius' assertion to order and disorder in gas phase molecular systems.
But if you see latest books they use concept of energy dispersion instead of order or disorder to explain entropy.
Source: Wikipedia
Also have look at physics S.E.
• measurement of randomness?? – user7484 Jan 11 '15 at 15:20
• @RutvikSutaria I have added link have a look there. :) – Freddy Jan 11 '15 at 15:26
I will take a crack at this although I admit that this topic sometimes confuses me as well. Here is how I like to think about entropy. Consider a box containing equal amounts of gases A and B. So, we have a fixed volume and fixed number of molecules. Let us also isolate the box from its surroundings so that it has a fixed energy as well. We have just created a microcanonical ensemble (constant NVE). The ensemble consists of every possible configuration of molecules having the same NVE. Now, if we were able to step back and take a broad view of the ensemble, we would observe that the vast majority of boxes contain a rather bland homogeneous mixture of gases A and B. In fact, they would be indistinguishable for all practical purposes. Let us count the number of boxes in this state and call the number $$W_1$$. Continuing with the box counting, we find that there are only a handful of distinguishable boxes left, but they are quite interesting! One box, for instance, might have all of the A molecules crowded together in one corner and all of the B molecules in another corner! This one box can be labelled $$W_2$$. We start to understand that some configurations of molecules will be extremely improbable because they represent such a small fraction of the total number of possible configurations. Boltzmann quantified this relationship as $$S=k \log W$$. If we plug $$W_1$$ into this equation we get a high value for the entropy, $$S$$, because $$W_1$$ is so large. On the other hand, if we plug in $$W_2$$ the entropy we get is very low.
So, to sum up, entropy is really the measure of how likely a given system configuration is when compared to all of the possible configurations. In this sense, I would argue that you could say a messy room has a higher entropy than a clean room because there are so many more ways a room could be considered "messy" than "clean".
• Can you make it bit more readable. This seems too boring! :'( – user7484 Jan 11 '15 at 15:21
• That's probably about the best I can do. I am a rather boring person. – Qubit1028 Jan 11 '15 at 16:14
• Breaking down the wall of text might be helpful. – Peter Mortensen Jan 13 '15 at 1:12 | {"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": 7, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8567327857017517, "perplexity": 369.9866101151827}, "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/1623487635724.52/warc/CC-MAIN-20210618043356-20210618073356-00591.warc.gz"} |
http://mathoverflow.net/questions/67630/convergence-of-the-series-sum-p-p-s-p-prime-and-s1 | # Convergence of the series $\sum_p p^{-s}$ ($p$ prime and $s>1$)
I know that $\sum_p p^{-s}$, $s>1$, converges. Now, I define $J(s) = \sum_p p^{-s}$. Are there any "well known" values for $J(2)$, $J(3)$, $J(4)$, etc? We all know that $\zeta(2)= \frac{\pi^2}{6}$, $\zeta(4)=\frac{\pi^4}{90}$, etc.
-
@Yemon: True, I misunderstood the question. I thought, he was asking formula known for $\zeta(s)$. – S.C. Jun 13 '11 at 6:26
As an aside, one way of showing that the sum over primes diverges is via the identity $\sum_{p}{p^{-s}} = \log \zeta(s) - \sum_{p}\sum^{\infty}_{n=2}{n^{-1} p^{-ns}}$, which is valid for all $\Re(s) > 1$. This shows the connection between special values of $\sum_{p}{p^{-s}}$ and of $\zeta(s)$, but I don't think it's possible to obtain nice closed-form values for $\sum_{p}\sum^{\infty}_{n=2}{n^{-1} p^{-ns}}$ (though it is of course easy to show that it is uniformly bounded as $s \to 1$). – Peter Humphries Jun 13 '11 at 6:31
This page: mathworld.wolfram.com/RiemannZetaFunction.html answers some for values for $\zeta(s)$ – S.C. Jun 13 '11 at 6:40
I believe $J(s)$ is sometimes called "the prime zeta function" and information about it can be found by using that search term. – Gerry Myerson Jun 13 '11 at 12:12
From the identity Peter mentioned and Mobius inversion one has $J(s) = \sum_{n=1}^\infty \frac{\mu(n)}{n} \log \zeta(ns)$. So for even $s$ one can get a series formula for $J$ this way, but it is unlikely to lead to any particularly compact closed form. – Terry Tao Feb 22 '15 at 19:58
No, in the sense that there are (for all I know) no identies along the lines of those for $\zeta(s)$, that you recalled, known.
Your function $J$ is sometimes called the prime zeta function.
You can find some information, some approciamte numerical values, plots, and pointers to the literature, e.g., at
http://mathworld.wolfram.com/PrimeZetaFunction.html
and
http://en.wikipedia.org/wiki/Prime_zeta_function
Two related hand-waving/heuristic arguments for the difficulty (not sure how good/convincing they are):
1. The values would 'encode' quite precise information on the set of primes.
2. The arithmetic function you are summing, that is, $f(n) = n^{-s}$ if $n$ is prime, and $f(n)=0$ if $n$ is not prime, is not a 'nice' arithmetic function; for example it is not multiplicative.
A related note that might interest you, in case you are not aware of it:
As you say $\sum_p p^{-1}$ diverges. However, the rate of divergence is fairly precisely known. Namely, by Mertens's Second Theorem $$\lim_{n \to \infty} \left ( \sum_{p\le n} p^{-1}\right ) - \log \log n$$ exists, and is equal to (or perhaps, rather defines) the Meissel--Mertens constant, which is approxiamtely $0.2614972$.
-
There is one explicit special value concerning $J$ I can think of (as long as one allows analytic continuation and regularization): $J'(0)=-2\log (2\pi)$. See the 'Product over all primes' reference given in the Mathworld link. – dke Jun 13 '11 at 13:06 | {"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": 0.973820149898529, "perplexity": 425.6142040646374}, "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/1454701147492.21/warc/CC-MAIN-20160205193907-00274-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://large-numbers.fandom.com/wiki/Octadecation?oldid=5230 | ## FANDOM
1,080 Pages
Octadecation refers to the 18th hyperoperation starting from addition. It is equal to $$a \uparrow^{16} b$$ in Knuth's up-arrow notation.
Octadecation can be written in array notation as $$\{a,b,16\}$$ and in chained arrow notation as $$a \rightarrow b \rightarrow 16$$.
Octadecational growth rate is equivalent to $$f_17(n)$$ in the fast-growing hierarchy.
Community content is available under CC-BY-SA unless otherwise noted. | {"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": 0.9950832724571228, "perplexity": 1129.7879577266067}, "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/1579250599718.13/warc/CC-MAIN-20200120165335-20200120194335-00504.warc.gz"} |
http://link.springer.com/chapter/10.1007%2F978-3-642-25510-6_13 | Lecture Notes in Computer Science Volume 7090, 2011, pp 146-157
# Social Learning in a Changing World
* Final gross prices may vary according to local VAT.
## Abstract
We study a model of learning on social networks in dynamic environments, describing a group of agents who are each trying to estimate an underlying state that varies over time, given access to weak signals and the estimates of their social network neighbors.
We study three models of agent behavior. In the fixed response model, agents use a fixed linear combination to incorporate information from their peers into their own estimate. This can be thought of as an extension of the DeGroot model to a dynamic setting. In the best response model, players calculate minimum variance linear estimators of the underlying state.
We show that regardless of the initial configuration, fixed response dynamics converge to a steady state, and that the same holds for best response on the complete graph. We show that best response dynamics can, in the long term, lead to estimators with higher variance than is achievable using well chosen fixed responses.
The penultimate prediction model is an elaboration of the best response model. While this model only slightly complicates the computations required of the agents, we show that in some cases it greatly increases the efficiency of learning, and on complete graphs is in fact optimal, in a strong sense. | {"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": 0.8103930354118347, "perplexity": 394.1171609744751}, "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/1405997892648.47/warc/CC-MAIN-20140722025812-00046-ip-10-33-131-23.ec2.internal.warc.gz"} |
https://www.mathplanet.com/education/algebra-1/radical-expressions/radical-equations | When you want to solve an equation with containing a radical expression you have to isolate the radical on one side from all other terms and then square both sides of the equation.
Example
$3\sqrt{x}=9$
$\sqrt{x}=\frac{9}{3}=3$
$\begin{pmatrix} \sqrt{x} \end{pmatrix}^{2}=\begin{pmatrix} 3 \end{pmatrix}^{2}$
$x=9$
When you square a radical equation you sometimes get a solution to the squared equation that is not a solution to the original equation. Such an equation is called an extraneous solution. Remember to always check your solutions in the original equation to discard the extraneous solutions.
Example
$\sqrt{2-x}=x$
$\left ( \sqrt{2-x} \right )^{2}=x^{2}$
$2-x=x^{2}$
$x^{2}+x-2=0$
$x=\frac{-1\pm \sqrt{1^{2}-4\cdot \left ( -2 \right )}}{2}$
$x=\frac{-1\pm \sqrt{1+8}}{2}$
$x_{1}=\frac{-1+\sqrt{9}}{2}=\frac{-1+3}{2}=\frac{2}{2}=1$
$x_{2}=\frac{-1-\sqrt{9}}{2}=\frac{-1-3}{2}=\frac{-4}{2}=-2$
Here we've got two solutions x = 1 or x = (-2). We check both solutions in the original equation to test whether they are true solutions or extraneous solutions.
$\begin{matrix} \sqrt{2-1}\overset{?}{=}1& \: or\: & \sqrt{2-\left ( -2 \right )}\overset{?}{=}-2\\ 1=1& &2=-2 \\ & & {\color{red} {Wrong!}} \end{matrix}$
As we could see when we checked our numbers in the original equation x =1 is the only true solution for this equation and that x = -2 is an extraneous solution.
## Video lesson
$\sqrt{10-x}=x+2$ | {"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": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8576408624649048, "perplexity": 205.41790850453714}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2018-26/segments/1529267859817.15/warc/CC-MAIN-20180617213237-20180617233237-00398.warc.gz"} |
http://fourier.eng.hmc.edu/e84/lectures/ch4/node13.html | # Metal-Oxide-Semiconductor Field-Effect Transistors
A metal-oxide-semiconductor field-effect transistor (MOSFET) has three terminals, source, gate, and drain. In an n-MOSFET (or p-MOSFET), both the source S and drain D are N-type (or P-type) and the substrate between them is P-type (or N-type). The gate and the P-type substrate is insulated by a thin layer of silicon dioxide (). Due to this insulation, there is no gate current to either the source or drain.
Typically the polarities of the voltages applied to the MOS transistor are such that
and (150)
The n-MOSFET can be considered as a voltage-controlled current channel. When sufficient voltage is applied between gate and source, the positive potential at the gate will induce enough electrons from the P-type substrate (minority carriers) to form an electronic channel called an inversion layer between source and drain, and a current between source and drain is formed.
The MOS transistor can be used in either analog circuits or as a switch in binary logic circuit:
(151)
More acturately, the drain current and the gate voltage can be modeled by
(152)
The current is affected by voltage as well as . It can therefore considered as a function of both and plotted below (similar to a bipolar transistor ):
This function can be divided into three different regions:
• Cutoff region:
(153)
where is a threshold voltage, , i.e., no current flows through S and D (due to the two back-to-back PN-junctions).
• Triode region:
and (154)
Voltages and at both the S and D ends of the inversion layer exceeds , some electrons in the P-type substrate (minority carriers) are pulled toward the gate to form an inversion layer close to the gate to form an N-type channel with certain resistance between S and D. increases linearly as increases, with a coefficient (Ohm's law), and nonlinearly as increases (to pull more electrons toward the gate to enhance the conductivity of the n-channel). Note that as , the inversion layer is narrower at the D end than the S end.
• Saturation region:
but (155)
Voltage at the S end exceeds , but at the D end is lower than . On the one hand, the increased voltage tends to increase , on the other hand, the reduced makes the inversion layer at the D end narrow to the extend that it is nearly closed (pinch-off). As the result, higher voltage does not cause more current (saturated), and it is only affected by .
In the plot of vs , the triode region and the saturation region are separated by the curve of .
Example: Assume .
• when , the MOSFET is in cutoff region with independent of .
• when and , the MOSFET is in linear or triode region with affected by both and .
• when , the MOSFET is in saturation region with determined only by .
Example: Assume and , and both MOSFETs in the following circuit are in the saturation region. Find output voltage .
Since both MOSFETs are in saturation region with the same which is determined only by but independent of , their must be the same. The upper MOSFET must have the same as the lower one , i.e., the output voltage has to be .
Comparison between BJT and FET
• BJT has a low input resistance . But as MOSFET's gate is insulated from the channel ( ), it draws virtually no input current and therefore its input resistance is infinity in theory.
• BJT is current ( or ) controlled, but MOSFET is voltage () controlled. Consequently, the power consumption of MOSFETs is lower than BJTs.
• MOSFETs are easy to fabricate in large scale and have higher element density than BJTs.
• MOSFETs have thin insulation layer which is more prone to statics and requires special protection.
• BJTs have higher cutoff frequency and higher maximum current than MOSFETs.
• MOSFETs are much more widely used (especially in computers and digital systems) than BJTs.
• Both majority and minority carriers are used in BJTs, but only majority carriers are used in FETs. Consequently FETs have better temperature stability (minority carriers are more sensitive to temperature).
The BJT and FET can be compared with the old technology of vacuum tube. Although the specific physics of each of these devices is quite different from others, the working principles of these devices are essentially the same. In all three devices, a small AC input voltage (signal) is applied to the input terminal of the device (base, gate, or grid) to control the current that flows through the device (from collector, drain, or plate to emitter, source, or cathode, respectively), causing a much amplified voltage to appear at the output terminal (collector, drain, or plate) of the device. | {"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": 0.913063108921051, "perplexity": 1773.43124691438}, "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-2020-10/segments/1581875141653.66/warc/CC-MAIN-20200217030027-20200217060027-00193.warc.gz"} |
https://www.physicsforums.com/threads/integral-what-has-gone-wrong-here.112428/ | # Integral what has gone wrong here?
1. Feb 27, 2006
### mmh37
I am still struggling my way through line integrals, and this here is one where I do not understand what has gone wrong - does anyone see what it is ( I really want to understand all of this)?
$$\int_{0}^{1}{(y*e^{xy} + 2x + y)dx + (x*e^{xy} + x )dy }$$
the curve joins (0,0) to (1,1)
where x= t and y=t (0<= t <= 1)
so, I said that
$$\frac {dx} {dt} = \frac {dy} {dt} = 1$$
therefore:
$$\int_{0}^{1} {( t*e^{t^2} + 2t + t + t*e^t^2 + t)dt} = \int_{0}^{1}{ (2t*e^{t^2} + 4t) dt} = e^1 + 1$$
However, the solution is supposed to be -2pi
Last edited: Feb 28, 2006
2. Feb 27, 2006
### assyrian_77
What do you mean by $$e^{t2}$$? Is it $$e^{2t}$$ or $$e^{t^2}$$?
I guess it comes from the first expression, viz. $$ye^xy$$. Is it $$y^2e^x$$ or $$ye^{xy}$$?
Never mind, going through your calculation, I can see what you meant.
Hmmm... I obtain the same thing as you. Are you sure about that parametrization?
Last edited: Feb 27, 2006
3. Feb 28, 2006
### mmh37
I am so sorry about that.....I fixed it.
That's the parametrization the question stated:
(....) the curve joins (0,0) to (1,1)
where x= t and y=t (0<= t <= 1)
( I just double checked this)
Last edited: Feb 28, 2006
4. Feb 28, 2006
### mmh37
does it mabe help that the field is conservative?
Last edited: Feb 28, 2006
5. Feb 28, 2006
### HallsofIvy
Staff Emeritus
A bit. That of course is the reason the path did not need to be given!
And it verifies that the correct answer is e+ 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": 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": 0.9747453331947327, "perplexity": 2565.9092230040173}, "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-2016-50/segments/1480698541864.44/warc/CC-MAIN-20161202170901-00268-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://eprints.soton.ac.uk/38409/ | The University of Southampton
University of Southampton Institutional Repository
# On the notion of canonical dimension for algebraic groups (in special volume in honor of Michael Artin: part I)
Berhuy, Grégory and Reichstein, Zinovy (2005) On the notion of canonical dimension for algebraic groups (in special volume in honor of Michael Artin: part I). Advances in Mathematics, 198 (1), 128-171.
Record type: Article
## Abstract
We define and study a numerical invariant of an algebraic group action which we call the canonical dimension. We then apply the resulting theory to the problem of computing the minimal number of parameters required to define a generic hypersurface of degree d in pn-1.
Published date: 1 December 2005
Keywords: algebraic group, g-variety, generic splitting field, essential dimension, canonical dimension, homogeneous forms
## Identifiers
Local EPrints ID: 38409
URI: http://eprints.soton.ac.uk/id/eprint/38409
ISSN: 0001-8708
PURE UUID: 40ae26ec-05f2-407d-892d-341c1465456d
## Catalogue record
Date deposited: 08 Jun 2006
## Contributors
Author: Grégory Berhuy
Author: Zinovy Reichstein | {"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": 0.8496799468994141, "perplexity": 4565.174885599896}, "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/1652662587158.57/warc/CC-MAIN-20220525120449-20220525150449-00078.warc.gz"} |
https://collaborate.princeton.edu/en/publications/pairing-of-zeros-and-critical-points-for-random-polynomials | # Pairing of zeros and critical points for random polynomials
Research output: Contribution to journalArticlepeer-review
8 Scopus citations
## Abstract
Let pN be a random degree N polynomial in one complex variable whose zeros are chosen independently from a fixed probability measure μ on the Riemann sphere S2. This article proves that if we condition pN to have a zero at some fixed point ζ ϵ S2, then, with high probability, there will be a critical point ωζ at a distance N-1 away from ζ. This N-1 distance is much smaller than the N-1/2 typical spacing between nearest neighbors for N i.i.d. points on S2. Moreover, with the same high probability, the argument of ωζ relative to ζ is a deterministic function of μ plus fluctuations on the order of N-1.
Original language English (US) 1498-1511 14 Annales de l'institut Henri Poincare (B) Probability and Statistics 53 3 https://doi.org/10.1214/16-AIHP767 Published - Aug 2017 Yes
## All Science Journal Classification (ASJC) codes
• Statistics and Probability
• Statistics, Probability and Uncertainty
## Keywords
• Critical points
• Gauss-Lucas
• Random polynomials
• Zeros | {"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": 0.9164420366287231, "perplexity": 2096.646713629906}, "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/1623487586239.2/warc/CC-MAIN-20210612162957-20210612192957-00099.warc.gz"} |
http://mathoverflow.net/questions/147299/higher-commutators-in-e-n-algebras-and-the-maurer-cartan-equation | # Higher commutators in E_n algebras and the Maurer--Cartan equation
Let $A$ be an associative algebra in $dgVect_k$. Then the commutator $[\cdot,\cdot]:A\otimes A\to A$ defined by $[x,y]=xy-(-1)^{|x||y|}yx$ gives $A$ the structure of a (dg-)Lie algebra. The Maurer--Cartan equation: $$d\alpha+\frac 12[\alpha,\alpha]=0$$ for $\alpha\in A$ (necessarily of degree $1$) plays an important role in deformation theory.
Does anyone recognize the following generalization of the above construction from associative algebras (that is, $E_1$-algebras) to $E_n$-algebras?
Let $A$ be an $E_n$-algebra in $dgVect_k$. The $E_n$-algebra structure gives a map $A\otimes A\otimes C_\bullet(Conf_2(D^n))\to A$ where $Conf_2(D^n)\simeq S^{n-1}$ is the configuration space of two distinct points in $D^n$. By picking a cycle in $C_\bullet(Conf_2(D^n))$ representing $[S^{n-1}]$, we get a pairing $[\cdot,\cdot]:A\otimes A\to A$ which I will think of as a sort of "higher commutator" (of course, this recovers the usual notion of commutator when $n=1$).
Now for some questions:
1. Does this "higher commutator" endow $A$ with the structure of a dg-Lie algebra (or, more likely, an $L_\infty$-algebra), or is it something more exotic?
2. What is the significance of the solutions of the Maurer--Cartan equation (using the higher commutator) to the given $E_n$-algebra?
-
I've always assumed the answer to 1 is "yes". I don't know a reference, however. One place these MC equations arise in deformation theory is the following. Any $E_{n-1}$ algebra $B$ has an "endomorphism ring" $A = \mathrm{End}(B)$ which is an $E_n$ algebra: it is a certain Hochschild complex, and by design MC elements in $A$ are "the same" as deformations of $B$ as an $E_{n-1}$ algebra. I've left this as a comment because someone with more expertise should be the one to leave an answer. – Theo Johnson-Freyd Nov 8 '13 at 3:38
This operation appears prominently in the theory of $n$-fold loop spaces, where it is called a Browder operation and is related by suspension to the Samelson product. In characteristic $p$, this operation accompanies Dyer-Lashof operations, and these operations together give enough structure to compute $H_*(\Omega^n \Sigma^n X;\mathbf{F}_p)$ as an explicit functor of $H_*(X;\mathbf{F}_p)$ for any space $X$. This was part of Fred Cohen's 1972 PhD thesis and appears in "The homology of iterated loop spaces" http://www.math.uchicago.edu/~may/BOOKS/homo_iter.pdf. This does not directly answer the questions, but I thought the history of these operations might be of some interest. The answer to question 1 in characteristic zero is well-known and is summarized in Section 5 of "Operads, algebras, and modules", http://www.math.uchicago.edu/~may/PAPERS/mayi.pdf. The algebras over the homology of an $E_{n+1}$-operad $\mathcal{C}_{n+1}$ are $n$-braid algebras, which are commutative algebras and $n$-Lie algebras that satisfy the Poisson formula. When $n=1$, we see Batalin-Vilkovisky algebras. The free $n$-braid algebras are described explicitly in Theorem 5.6 op cit, where the description is deduced from topology.
If the bracket is of the wrong degree, then we can fix this by shifting $A$. It seems like the MC equation then does make sense. Or are you referring to other "degree reasons"? – John Pardon Nov 11 '13 at 1:40
Sorry, you're right, the only problem is that MC elements have degree other than $0$: $|d(a)|=|a|+1$ and $|1/2[a,a]|=2|a|+n-1$ so the MC equation has sense for $|a|=2-n$. I'd say it's wiser to include higher brackets in the MC equation. The MC equation for L-infinity algebras has been much studied (I remember a paper by Getzler in Annals). However I don't know how to interpret solutions in this case. Maybe also as deformations by general Koszul duality principles? It looks like sonething that somebody must have looked at! :-) – Fernando Muro Nov 11 '13 at 7:24 | {"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": 0.9400787949562073, "perplexity": 217.80402991423915}, "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-2016-26/segments/1466783395679.18/warc/CC-MAIN-20160624154955-00135-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://arxiv.org/abs/1404.4240 | math.AG
(what is this?)
# Title: The matrix model for dessins d'enfants
Abstract: We present the matrix models that are the generating functions for branched covers of the complex projective line ramified over $0$, $1$, and $\infty$ (Grotendieck's dessins d'enfants) of fixed genus, degree, and the ramification profile at infinity. For general ramifications at other points, the model is the two-logarithm matrix model with the external field studied previously by one of the authors (L.Ch.) and K.Palamarchuk. It lies in the class of the generalised Kontsevich models (GKM) thus being the Kadomtsev--Petviashvili (KP) hierarchy $\tau$-function and, upon the shift of times, this model is equivalent to a Hermitian one-matrix model with a general potential whose coefficients are related to the KP times by a Miwa-type transformation. The original model therefore enjoys a topological recursion and can be solved in terms of shifted moments of the standard Hermitian one-matrix model at all genera of the topological expansion. We also derive the matrix model for clean Belyi morphisms, which turns out to be the Kontsevich--Penner model introduced by the authors and Yu. Makeenko. Its partition function is also a KP hierarchy tau function, and this model is in turn equivalent to a Hermitian one-matrix model with a general potential. Finally we prove that the generating function for general two-profile Belyi morphisms is a GKM thus proving that it is also a KP hierarchy tau function in proper times.
Comments: 18 pages, 3 figures; reference and introduction section expanded, remark 3.5 edited Subjects: Algebraic Geometry (math.AG); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Combinatorics (math.CO) MSC classes: 05A15, 14N10, 60B20 Cite as: arXiv:1404.4240 [math.AG] (or arXiv:1404.4240v2 [math.AG] for this version)
## Submission history
From: Leonid Chekhov O [view email]
[v1] Wed, 16 Apr 2014 13:39:30 GMT (21kb)
[v2] Wed, 18 Jun 2014 15:02:09 GMT (22kb) | {"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": 0.8460873365402222, "perplexity": 1272.9457608404546}, "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-2018-43/segments/1539583509845.17/warc/CC-MAIN-20181015205152-20181015230652-00287.warc.gz"} |
https://www.transtutors.com/questions/three-ideal-polarizers-are-oriented-as-follows-the-axis-of-the-second-polarizer-is-a-1482009.htm | # Three ideal polarizers are oriented as follows: The axis of the second polarizer is at an angle of 5
Three ideal polarizers are oriented as follows: The axis of the second polarizer is at an angle of 59.0A?° relative to the first one. The axis of the third polarizer is at an angle of 31.0A?° relative to the second one, so the axis of the axis of the third polarizer is perpendicular to the axis of the first one. Unpolarized light of intensity is incident on the first polarizer.
(a) What is the intensity of the light after it passes through all three polarizers?
(b) What is the intensity of the transmitted light if the second polarizer is removed? | {"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": 0.8021284937858582, "perplexity": 217.54216930130957}, "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-2019-47/segments/1573496668534.60/warc/CC-MAIN-20191114182304-20191114210304-00019.warc.gz"} |
https://socratic.org/questions/how-do-you-find-the-horizontal-asymptote-for-4x-x-3 | Precalculus
Topics
# How do you find the horizontal asymptote for (4x)/(x-3) ?
Dec 1, 2015
I found $y = 4$
#### Explanation:
The horizontal asymptote is a horizontal line of equation: $y = \text{constant}$
towards which the curve described by your function TENDS to get closer and closer maybe not immediately but as $x$ becomes sufficently big.
To find this line there is a trick!
Take your function and try to "see" its behavior very far from the origin...i.e. when $x$ becomes VEEEEERY big!
In your case consider a $x$ value very big, say, $x = 1 , 000 , 000$:
you get:
$y = 4 \cdot \frac{1 , 000 , 000}{1 , 000 , 000 - 3} \approx 4 \cdot \frac{1 , 000 , 000}{1 , 000 , 000} =$ the $3$ is negligible;
$y = 4 \cdot \frac{\cancel{1 , 000 , 000}}{\cancel{1 , 000 , 000}}$
So, you get $y = 4$ that is the equation of a horizontal line that your function tends to become for $x$ VEEEERY large!!
The two branches of your function will get as near as possible to the horizontal line $y = 4$! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 12, "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": 0.9394293427467346, "perplexity": 959.6872898497447}, "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/1593655891654.18/warc/CC-MAIN-20200707044954-20200707074954-00065.warc.gz"} |
http://mathoverflow.net/questions/56990/probabilities-independent-of-zfc/56992 | # Probabilities independent of ZFC?
Hi guys,
is it possible to change the probability of an event via forcing? More precisely, is there an innocent looking question on the probability of "something" whose answer is independent of ZFC?
All the best, Sebastian
-
There are several issues.
On the one hand, any set can be made countable by forcing, and this process will certainly affect the measure of the set, if it did not have measure zero in the ground model.
But in the context of the Lebesgue measure on the reals, say, it is natural to consider not the set itself, but the Borel description of the set, interpreted first in the ground model and then reinterpreted in the forcing extension. (For exampe, the "unit interval" of $V$ is not necessarily the same as the unit interval of a forcing extension $V[G]$, but we have a borel code that correctly picks out the unit interval when interpreted in any model of ZFC.) In this case, one gets a positive solution for preservation of measure. The reason is that the assertion that the measure of the set with Borel code $b$ is $x$ has complexity at most $\Sigma^1_2(b,x)$ and hence is absolute to all forcing extensions by the Shoenfield absoluteness theorem. In this sense, the measure of a measurable set cannot be affected by forcing.
Meanwhile, the use of other non-absolute descriptions can lead again to a negative answer, where the measure can be affected by forcing. For example, consider the set $X$ of all binary sequences $x$ whose sequence of digits is realized somewhere in the GCH pattern of cardinals, in the sense that there is an ordinal $\beta$ such that $x(n)=1$ iff $2^{\aleph_{\beta+n}}=\aleph_{\beta+n+1}$. If the Generalized Continuum Hypothesis holds, then $X$ has measure zero, since only one pattern is realized. But one can force the GCH pattern to realize all patterns, and so there are forcing extensions in which $X$ has full measure.
Here is another comparatively concrete example. Consider the set of reals that are constructible, in the sense of Gödel's constructible universe. This set has complexity $\Sigma^1_2$ in the descriptive set-theoretic hierarchy, which is just a step up from Borel. The set has full measure in the constructible universe, of course, but it is easily made to have measure zero in a forcing extension. Thus, the probability that a randomly chosen real number is constructible has an answer that is independent of ZFC, because in some models of set theory this probability is 1 and in others it is 0.
-
thanks a lot. that is already in the direction I am looking for. Do you know of any non-absolute description that one could use that is still "simple"? More precisely, the example above is a non-absolute description by basically providing a direct link from the sequences to the cardinal patterns. is there something more "basic" or "elementary"? I understand that it has to be somewhat more complex than the Borel code. thx a lot again! – sebastian Mar 1 '11 at 13:28
I added a more concrete example. – Joel David Hamkins Mar 1 '11 at 14:06
@Joel: I have the following naive question: is it true that the the set you defined is at least measurable in all models of set theory? Or in other words, is it possible to re-state your conclusion as "because in some models of set theory this probability is 1 and in others it is 0, and in others is not defined at all". – Matteo Mio Mar 1 '11 at 15:22
Matteo, that is an interesting question. More generally, if $V\subset V[G]$ is any forcing extension, then must $\mathbb{R}^V$ be measurable in $V[G]$? Must it always have either full measure or measure $0$? Hmmmmm... – Joel David Hamkins Mar 1 '11 at 20:42
There are models of set theory in which the set of constructible reals is not Lebesgue measurable. If I remember correctly, examples include the models obtained from $L$ by adjoining a random real, a Laver real, a Sacks real, or a Miller real. These models satisfy CH, but countable support iterations of Laver, Miller, or Sacks forcing give models where the continuum has cardinality $\aleph_2$ yet the constructible reals are again non-measurable. And you can get the same result with even larger continuum by forcing with a big measure algebra to add a lot of random reals. – Andreas Blass Mar 3 '11 at 1:55 | {"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": 0.9609659314155579, "perplexity": 230.31616173846126}, "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-2014-15/segments/1397609533957.14/warc/CC-MAIN-20140416005213-00543-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/165780/is-a-deligne-mumford-curve-defined-over-qbar-if-and-only-if-its-coarse-moduli-sp | # Is a Deligne-Mumford curve defined over Qbar if and only if its coarse moduli space is
Let $\mathcal X$ be a smooth proper finite type Deligne-Mumford stack over $\mathbb C$ that is generically a scheme. Let $X$ be its coarse moduli space.
If $\mathcal X$ can be defined over $\overline{\mathbb Q}$, then $X$ can be defined over $\overline{\mathbb Q}$. This is because "the base-change of the coarse moduli space is the coarse moduli space of the base-change".
Does the converse hold?
What if we drop the properness condition?
My motivation is mere curiosity.
-
No. Even if $X$ is "defined over $\overline{\mathbb{Q}}$" in the sense that $X$ is isomorphic to $X_0\otimes_{\overline{\mathbb{Q}}} \mathbb{C}$ for some variety $X_0$ over $\overline{\mathbb{Q}}$, nonetheless the stack $\mathcal{X}$ may not be defined over $\overline{\mathbb{Q}}$.
For instance, let $X_0$ be $\mathbb{P}^1_{\overline{\mathbb{Q}}}$ so that $X$ is $\mathbb{P}^1_{\mathbb{C}}$. Now let $t_0,t_1,t_2\in \mathbb{P}^1_{\mathbb{C}}(\mathbb{C})\setminus\{0,1,\infty\}$ be three distinct points such that at least one is transcendental, e.g., $2$, $3$, $\pi$. Denote by $$f:Y\to X,$$ the genus $2$, hyperelliptic curve branched over $$\{0,1,\infty,t_0,t_1,t_2\}.$$ There is an action of $\mathbb{Z}/2\mathbb{Z}$ on $Y$ via the hyperelliptic involution. Let $\mathcal{X}$ be the quotient Deligne-Mumford stack, $$\mathcal{X} = [Y /(\mathbb{Z}/2\mathbb{Z})].$$ The coarse moduli space is $X$, via the unique morphism $\pi: \mathcal{X}\to X$ that factors $f$. Of course $X$ is defined over $\overline{\mathbb{Q}}$. However, the branch divisor of the morphism $\pi$ is not defined over $\overline{\mathbb{Q}}$ as a subscheme of $X$. Thus, also $\mathcal{X}$ is not defined over $\overline{\mathbb{Q}}$.
It might be useful to (slightly) generalize the above construction.
Let $X=\mathbb P^1_{\mathbb C}$ and let $f:Y\to X$ be a hyperelliptic curve with branch locus $D$ (and $Y$ of positive genus). Write $U=X\backslash D$. Let $\mathcal X$ be the stack $[Y/G]$. Then the unique morphism $\pi:\mathcal X \to X$ that factors $f:Y\to X$ is the coarse moduli space of $\mathcal X$. Suppose that $U$ can't be defined over Qbar. Then $Y\to X$ (and $Y$) can't be defined over $\overline{\mathbb Q}$ (as the hyperelliptic involution is unique). Therefore, $\mathcal X\to X$ and $\mathcal X$ can't be defined over Qbar.
-
Thank you very much. – Ariyan Javanpeykar May 11 '14 at 17:00
LateX wasn't rendering so good anymore (on my pc), so I changed some of the "$\overline{\mathbb{Q}}$" into "Qbar". – Ariyan Javanpeykar May 11 '14 at 23:32
It seems to me that hyperelliptic is overkill and you could do exactly the same thing in the elliptic case, ramified at $0,1,\infty,\pi$. – Ben Wieland May 12 '14 at 0:07
@Ari: What you added contains a mistake. "Bonus: we can find $\mathcal{X}$ such that $Y$ and $X$ are defined over $\mathbb{Q}$, but $\mathcal{X}$ isn't. (Just take $Y = X = \mathbb{P}^1$)". That is impossible. This is the reason that I chose the example I chose, where the morphism $\pi$ is "characteristic". – Jason Starr May 12 '14 at 12:58
@JasonStarr I see. Thank you. – Ariyan Javanpeykar May 12 '14 at 15:28
Let $\mathcal{X}$ be an algebraic stack over $k$. A coarse moduli space for $\mathcal{X}$ over $k$ is a morphism $\pi:\mathcal{X}\rightarrow X$, where $X$ is a scheme such that:
• the morphism $\pi$ is universal for morphisms to schemes;
• $\pi$ induces a bijection between $|\mathcal{X}|$ and the closed points of $X$, where $|\mathcal{X}|$ denotes the set of isomorphism classes in $\mathcal{X}$.
If $\mathcal{X}$ admits a coarse moduli space $\pi:\mathcal{X}\rightarrow X$ then this is unique up to unique isomorphism. A separated algebraic stack has a coarse moduli space which is a separated algebraic space.
Let $\mathcal{X}$ be a separated stack admitting a scheme $X$ as coarse moduli space $\pi:\mathcal{X}\rightarrow X$. The map $\pi$ is universal for morphisms in schemes, that is for any morphism $f:\mathcal{X}\rightarrow Y$, with $Y$ scheme, there exists a unique morphisms of schemes $g:X\rightarrow Y$ such that $f = g\circ\pi$.
Now, if $\mathcal{X}$ is defined over $k$ we have a morphism $f:\mathcal{X}\rightarrow Spec(k)$. By universality there is a morphism of schemes $g:X\rightarrow Spec(k)$ such that $g\circ\pi = f$. In particular $X$ is defined over $k$.
Conversely, if $X$ is defined over $k$ then there is a morphism $g:X\rightarrow Spec(k)$. By composition we get a morphism $f:=g\circ\pi:\mathcal{X}\rightarrow X$. Therefore $\chi$ is defined over $k$.
For instance both the stack $\overline{\mathcal{M}}_{g,n}$ and its coarse moduli space $\overline{M}_{g,n}$ parametrizing Deligne-Mumford stable curves are defined over $\mathbb{Z}$. In particular over any commutative ring and hence over any field.
-
I think I'm missing the argument. Assume that the coarse moduli space $X$ has a model $X_0$ over $\overline{\mathbb {Q}}$. Where exactly do you show that $\mathcal X$ has a model over $\overline{\mathbb {Q}}$? – Ariyan Javanpeykar May 10 '14 at 23:08
@CamSar your definition of "$X$ is defined over $\overline{\mathbb Q}$" is not correct. – John Pardon May 10 '14 at 23:48
Indeed, every $X$ over $\mathbb C$ has a morphism to $\operatorname{Spec}\overline{\mathbb Q}$ since there is a morphism $\operatorname{Spec}\mathbb C\to\operatorname{Spec}\overline{\mathbb Q}$. – John Pardon May 10 '14 at 23:49
If $X$ is a scheme over $\mathbb C$, then "$X$ is defined over $\overline{\mathbb Q}$" means there exists a scheme $Y$ over $\overline{\mathbb Q}$ and an isomorphism $X\cong Y\times_{\overline{\mathbb Q}}\mathbb C$ (and similarly for stacks, etc.). – John Pardon May 10 '14 at 23:51 | {"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": 0.9916025400161743, "perplexity": 147.2702457612298}, "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-2015-14/segments/1427131300441.51/warc/CC-MAIN-20150323172140-00158-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://eprint.iacr.org/2018/605 | ## Cryptology ePrint Archive: Report 2018/605
N-term Karatsuba Algorithm and its Application to Multiplier designs for Special Trinomials
Yin Li and Yu Zhang and Xiaoli Guo and Chuanda Qi
Abstract: In this paper, we propose a new type of non-recursive Mastrovito multiplier for $GF(2^m)$ using a $n$-term Karatsuba algorithm (KA), where $GF(2^m)$ is defined by an irreducible trinomial, $x^m+x^k+1, m=nk$. We show that such a type of trinomial combined with the $n$-term KA can fully exploit the spatial correlation of entries in related Mastrovito product matrices and lead to a low complexity architecture. The optimal parameter $n$ is further studied. As the main contribution of this study, the lower bound of the space complexity of our proposal is about $O(\frac{m^2}{2}+m^{3/2})$. Meanwhile, the time complexity matches the best Karatsuba multiplier known to date. To the best of our knowledge, it is the first time that Karatsuba-based multiplier has reached such a space complexity bound while maintaining relatively low time delay.
Category / Keywords: foundations / N-term Karatsuba Algorithm, Specific trinomials, Bit-parallel Multiplier | {"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": 0.9108729958534241, "perplexity": 1109.0701351628481}, "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-2021-49/segments/1637964358078.2/warc/CC-MAIN-20211127013935-20211127043935-00173.warc.gz"} |
https://examplemath.com/integration-problems | Full of Math Examples
FULL OF MATH EXAMPLES
# Integration Problems
Here is the integration problem set to test your integration skills.
## Integration by Substitution problems
These Problems can be easily solve using integration by substitution .
Explore these integration by substitution examples to learn.
1. $\frac{x}{e^{x^{2}}}$
2. $\frac{2 x}{1+x^{2}}$
3. $\frac{x}{9-4 x^{2}}$
4. $x \sqrt{1+2 x^{2}}$
5. $\frac{1}{x-\sqrt{x}}$
6. $\cot x \log \sin x$
7. $\frac{\sin x}{1+\cos x}$
8. $\frac{1}{1-\tan x}$
9. $\int\left(x^{2}+10\right)^{50} 2 x d x$
10. $\int 8 x^{2}\left(3 x^{3}-1\right)^{16} d x$
11. $\frac{e^{2 x}-1}{e^{2 x}+1}$
12. $\frac{e^{t a n^{-1} x}}{1+x^{2}}$
13. $\sec ^{2}(7-4 x)$
14. $\frac{1}{x+x \log x}$
## Integration by Parts problems
These Problems can be easily solve using integration by parts .
Explore all integration by parts examples to learn.
1. $\int \frac{x \sin ^{-1} x}{\sqrt{1-x^{2}}} d x$
2. $\int x e^{-x} d x$
3. $\int\left(1+x^{2}\right) e^{-x} d x$
4. $\int e^{x} \sin x d x$
5. $x \sec ^{2} x$
6. $\int 3 x e^{4 x} d x$
7. $\int x \ln x d x$
8. $\tan ^{-1} x$
9. $x^{2} \log x$
10. $e^{x}(\sin x+\cos x)$
11. $e^{2 x} \sin x$
12. $\sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)$
13. $\frac{(x-3) e^{x}}{(x-1)^{3}}$
14. $x \log 2 x$
Solve these math problems by yourself.
If you could not solve a problem ask for help using post your 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": 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": 0.8769674301147461, "perplexity": 781.7722529759553}, "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-2023-06/segments/1674764499758.83/warc/CC-MAIN-20230129180008-20230129210008-00775.warc.gz"} |
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=75&t=40761 | ## Phase Changes and Temperature
Nicholas Le 4H
Posts: 74
Joined: Fri Sep 28, 2018 12:24 am
### Phase Changes and Temperature
Why does the temperature not change during a phase change even though more heat energy is being added?
Ethan Yi 1K
Posts: 62
Joined: Fri Sep 28, 2018 12:28 am
### Re: Phase Changes and Temperature
that energy is being used to break bonds
Rami_Z_AbuQubo_2K
Posts: 89
Joined: Thu Jun 07, 2018 3:00 am
### Re: Phase Changes and Temperature
The reason that temperature stays the same is that, energy required to break bonds is greater than the energy needed to raise the temperature. Since the temperature is already high enough in the process of changing phases, the rest of the energy is put into breaking the bonds of the substance to change phases.
Ray Guo 4C
Posts: 90
Joined: Fri Sep 28, 2018 12:15 am
### Re: Phase Changes and Temperature
Heat energy is used to break intermolecular bonds, so kinetic energy stays constant. | {"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": 0.9245581030845642, "perplexity": 2479.2049873396277}, "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/1566027316194.18/warc/CC-MAIN-20190821194752-20190821220752-00181.warc.gz"} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.